Chapter 6 ^

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES

Rui Hua Xie Department of Atomic and Molecular Physics^ Max-Planck-Institut fur Stromungsforschung, Bunsenstr 10, D-37073 Gottingen, Germany

Contents

1. Quantum Theory of Molecular Polarizability and Hyperpolarizability 267 1.1. Nonlinear Optical Interactions 267 1.2. Quantum Theory of Molecular Polarizability and Hyperpolarizability 269

2. Second-Order Optical Nonlinearities of FuUerene 272 2.1. Second-Order Optical Nonlinearity of €50 272 2.2. Second-Order Optical Nonlinearity of Cgo Derivatives 274 2.3. Second-Order Optical Nonlinearity of C70 275

3. Third-Order Optical Nonlinearity of Fullerene 275 3.1. Background 275 3.2. Third-Order Optical Nonlinearity of C60 276 3.3. Third-Order Optical Nonlinearity of Higher FuUerenes 278

4. Third-Order Optical Nonlinearities of Tubular Fullerene and Carbon Nanotubes 284 4.1. Structure of Fullerene and Carbon Nanotubes 284 4.2. Model of Fullerene and Carbon Nanotubes 286 4.3. Third-Order Optical Nonlinearity of Tubular FuUerenes 288 4.4. Third-Order Optical Nonlinearity of Chiral Carbon Nanotubes 291

5. Third-Order Optical Nonlinearity of Doped FuUerenes 293 5.1. Model of Doped FuUerenes 293 5.2. Third-Order Optical Nonlinearity of Doped Tubular FuUerenes 295

6. Coherent Control of Molecular Polarizability and HyperpolarizabUity 297 7. Remarks 300

Appendix A: Details of Interference Terms 301 Acknowledgments 304 References 304

1. QUANTUM THEORY OF MOLECULAR optical field E(0. Why does the polarization play an important role POLARIZABILITY AND HYPERPOLARIZABILITY in the description of nonlinear optical phenomena? One reason is

that a time-varying polarization can act as the source of new com-1.1. Nonlinear Optical Interactions ponents of the electromagnetic field. For example, the wave equa-

Nonlinear optics [1-15], which focuses mainly on the study of non-tion in nonlinear optical media often has the form [9,16]

linear optical phenomena that occur as a consequence of the mod- / ^2 ^2 ^2 ^2 ^2 \ 4 ^ ^2p ification of the optical properties of a material owing to the pres- ( J^ "*• T ^ + ^ ~ ~^'Jfl]^~ ~/l 'Jfi ^^^ ence of light, plays an important role in many areas of science and ^ ' technology. To show what an optical nonlinearity means, we should where n is the refractive index of the medium and c is the speed of consider how the dipole moment per unit volume, or polarization light in vacuum. This expression is usually interpreted as an inho-P(0, of the material system depends on the strength of the applied mogeneous wave equation in which the polarization P drives the

Handbook of Advanced Electronic and Photonic Materials and Devices, edited by H.S. Nalwa Volume 9: Nonlinear Optical Materials Copyright © 2001 by Academic Press

ISBN 0-12-513759-1/$35.00 All rights of reproduction in any form reserved.

267

268 XIE

electric field E. Whenever d^V/dfi is nonzero, charges are acceler­ated. Then, according to Larmor's theorem of electromagnetism, accelerated charges generate electromagnetic radiation [17].

In linear optics, it is known that the induced polarization of a material system depends linearly on the electric field strength as [9]

P(0 = X^^Mt) (2)

where x^^"^ is named as the linear optical susceptibility. In nonhn-ear optics, one can describe the nonlinear optical (NLO) response of a material system by expressing the polarization P(0 in a power series of the field strength E(0 [9]:

»(0)r >(1) (2)/ ?0)(

where

P(0 = P ^ CO + P^'^(0 + P^^\0 + P^^(0 +

p( l ) (0 = ;^(1^E(0

p(2)(^) = x^^'^^^it)

p(3) r-^^(0 = x^^^ it) 0W(

(3)

(4)

(5)

(6)

P^^^ and P^^^ are the second-order and third-order nonlinear op­tical polarizations, respectively, and the corresponding quantities, ; ^ ^ and x^^\ ^^^ known as the second-order and third-order nonlinear optical susceptibilities of the material system. Avail­able studies [1-4, 18] show that the second-order nonlinear opti­cal interaction occurs only in noncentrosymmetric crystals that do not display inversion symmetry, whereas the third-order nonlinear optical interactions occur for both centrosymmetric and noncen­trosymmetric media.

Here, as an example, a rough order-of-magnitude estimate of the size of both x^'^^ and x^'^^ quantities is made for the common case, where the nonlinearity is electronic in origin. It is assumed that the amplitude of the applied field strength E is of the order of the characteristic atomic electric field strength

^atom — ~^ (7)

where e is the charge of the electron and AQ is the Bohr radius of the hydrogen atom. Then, under conditions of nonresonant ex­citation, x^'^'^ will be of the order of A'^^V^atom- For condensed matter, x^^^ is of the order of unit, then we have [9]

X^'^'^ ^ 5 X 10~^ cm/statvolt = 5 x 10~^ esu (8)

Similarly, x^^"^ will be of the order of AT^ V^atom- ^^^ condensed matter, it will be [9]

X^'^'> ^ 3 X 10-1^ cm^/statvolt^ = 3 x 10"^^ esu (9)

This chapter follows standard conventions, where the dimensions of physical quantities are not given explicitly, but instead Gaus­sian units or electrostatic units abbreviated esu are used to quote values.

The nonlinear polarization mentioned earlier is only for a ma­terial system that is without loss and dispersion. In the general case of a material with dispersion and/or loss, the nonlinear optical sus-ceptibihty becomes a complex quantity that is related to the com­plex amplitudes of the electric field and polarization. For conve­nience, the formal definition of the nonlinear optical susceptibility is introduced briefly. It is assumed that the electric field vector of

the optical wave can be represented as the discrete sum of a num­ber of frequency components as the compact form [9]

E ( r , 0 = ^E(6;„)e-^ '^«^ n

= ^A(a)„)e'(''''-''-'-«'> (10)

where the sum is the over-all frequency, both positive and neg­ative, E{-o)n) = E{o)nT, and A(-o)n) = ^(o^n)*. Thus, the quantity E(r, 0 will be real as it represents a physical field. The /:th-order nonlinear polarization P^^^ can be expressed in a simi­lar notation as

P(^)(r,0 = i : Z Pi('^n)e -liOnt (11)

l=x,y,z

where the sum extends over all positive and negative frequency components. Then, the components of the second-order optical

(2) susceptibihty tensor x]-i^ are defined as the proportional constant

that relates the amplitude of the nonlinear polarization to the product of field amplitudes according to [9]

pPc^.) 2Z Yl ^\ik^^^ = 0)n + (Om', 0)n, 0)m) j,k=x,y,z(nm)

xEj{(0n)Ej^((0m) (12)

where the indices ijk refer to the cartesian components of the fields and the notation (nm) indicates that the sum con + o)m is held fixed in performing the summation over n and m, although (On or (Om is each allowed to vary. Similarly, the components of the third-order nonlinear optical susceptibility are defined as the coefficients relating the amplitude of the nonlinear polarization to a product of three electric field amphtudes according to the ex­pression [9]

},k,l=x,y,z{onm)

xEj(o)o)Ek(o)n)Ei((Om) (13)

where the indices ijkl refer to the cartesian components of the fields and the notation (nmo) indicates that the sum con + com + (x)o is held fixed in performing the summation over n, m, and o, although o)n, or o)m, or coo is each allowed to vary.

In the following section, several representative examples of nonlinear optical interactions are introduced briefly. For more de­tails about them and other interesting cases of nonlinear optical interactions, for example, optical Kerr effect, see literatures given in the standard textbooks [1-5, 9]. Second-Harmonic Generation. In this process, it is assumed that the electric field strength is represented as

E{t) = Ee''"^^ -h c.c. (14)

which is incident upon a material for which the second-order op­tical susceptibility ; ^ ^ is nonzero. Then, the second-order nonlin­ear polarization is given by [9]

p(^)(t) = Ix^^^EE'^ + (x^^^E^e-'^^^ + c.c.) (15)

The second term in the preceding equation leads to the generation of radiation at the second-harmonic frequency.

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 269

Sum- and Difference-Frequency Generation. In this case, the opti­cal field incident upon a nonlinear optical medium, characterized by a nonlinear optical susceptibility x^'^\ consists of two distinct frequency components

E(t) = Eie-'"^^^ 4- E2e-''^^^ + c.c. (16)

Then, the second-order polarization is [9]

+ 2X^^HE^EI+E2E^2^ (17)

Given the usual notation

p(2)(/) = ^P(a)„)e-^'^n^ (18) n

the following physical processes are arrived at: (i) second-harmon­ic generation (SHG) [9]

P(2w2 + «i) = 3A'(^*£?£2 (32)

(19)

(20)

(21)

P(2«i) = X^^^EI

P(2w2) = A- ' 2

(ii) sum-frequency generation (SFG) [9]

Pi(oi + (02) = 2x^^'^EiE2

(iii) difference-frequency generation (DFG) [9]

/>(«! - W2) = Ix^^^EiE^ (22)

and (iv) optical rectification (OR) [9]

P(0) = 2X'-^\EIEI+E2E*) (23)

Third-Harmonic Generation. As introduced earlier, the third-order contribution to the nonlinear polarization of a material system is

P^^\t) = x^^^E(t)^ (24)

If the applied field is monochromatic, for example, given by E(t) = E cos cot, then the third-order nonlinear optical polarization is [9]

p(3)(0= f MA:^3)E^COS3W^+ {^JAf^^^E^cosw^ (25)

The first term describes a response at frequency 3ct> that is due to an applied field at frequency o). This term leads to the process of third-harmonic generation (THG). Third-Order Polarization (General Case). The general case of the third-order nonlinear optical polarization of a material system can be considered as follows. It is assumed that E(t) is made up of several different frequency components [9]

E(t) = ^i^-^'^i^ + Eie-'"^^^ + £3e-^"^3^ + c.c. (26)

If E{t)^ is calculated, then 44 different frequency components are got [9]. As an example, six cases are listed below [9]:

P(3cui) = X^^^EI (27)

P0w2) = X^^^EI (28)

P(3cu3) = X^^^EI (29)

P{o)l + ^2 + ^3) = 6;tr " ^ 1^2^3 (30)

P{2oyi + C02) = 3X^^^EIE2 (31)

If we adopt the standard convention that a negative frequency is associated with a field amplitude that appears as a complex con­jugate, then the frequency argument of P in each case is equal to the sum of the frequencies associated with the field amphtudes appearing on the right-hand side of the equation. The numerical factor that appears in each term on the right-hand side of each case is equal to the number of distinct permutations of the field frequencies that contribute to that term. Degenerate Four-Wave Mixing. The basic experimental arrange­ment of degenerate four-wave mixing (DFWM) is illustrated as follows. In this experiment, a polarization Pi is induced in the sample by the interaction between three light waves (Ei, E2, and E3) and the electrons in the nonlinear medium characterized by a third-order nonlinear optical susceptibility x^^^ through the rela­tion

jkl

(33)

where two wave beams Ei and E2 with electric field amplitudes Ej and Ej^, respectively, are incident at the same region on the sample and have equal and opposite wave vectors and the same frequency co. Both Ei and £2 wave beams from, for example, a He-Ne laser, set up an interference grating that interacts with a third light beam E3, with the electric field amplitudes E* coming from the back side of a transparent sample. A fourth beam E^, resulting from this interaction with wave vector equal and opposite to that of £*, is detected, that is, if the frequency of wave E^ is o), the wave E4 will be generated with the same frequency co and its complex amplitude everywhere will be the complex conjugate of wave £3. The polarization P/ corresponds to the fourth signal light beam. For more details about the geometry of the four light waves in the DFWM experiment see literatures in the two recent books [5,19].

1.2. Quantum Theory of Molecular Polarizability and Hyperpolarizability

In this section, based on the laws of quantum mechanics, we cal­culate explicit expressions for the nonlinear optical polarizabil­ity of molecules. The motivation for obtaining these expressions is [9] that (i) these expressions show us how the nonlinear op­tical polarizability of molecules depends on some important pa­rameters such as dipole transition moments and molecular en­ergy levels; (ii) these expressions are used to obtain numerical values of the nonlinear optical polarizabilities of molecules; and (iii) it is possible to obtain very large values of the nonlinear op­tical polarizability through the techniques of resonance enhance­ment. The formulas derived in this section demonstrate that all three procedures (one-photon, two-photon, and three-photon res­onance transitions) are equally effective at increasing the value of the third-order nonlinear optical susceptibihty [9]. However, two-photon transition is usually the preferred way to generate the third-harmonic field with high efficiency [9]. For the case of a one-photon or three-photon resonance, the incident field experi­ences linear absorption and is rapidly attenuated as it propagates through the medium [9]. But for the case of a two-photon reso­nance, attenuation occurs only as a result of two-photon absorp­tion process, which occur with much lower efficiency than one-photon process [9].

270 XIE

The radiation-free molecular Hamiltonian, its discrete or con­tinuous set of energy^igenvalues, and corresponding eigenfunc-tions are denoted by HQ, e„, and |4)„(r)), respectively, which sat­isfy the time-independent Schrodinger equation

where

Hol^nir)) = €n\^n(r))

€n = hcon

(34)

(35)

and wn is the angular frequency. Here it is assumed that these solutions are chosen in such a manner that they constitute a com­plete, orthonormal set satisfying the condition

{^m(r) I <^n(r)) = 8mn (36)

In the presence of the radiation field, the time evolution of the molecule-field coupling system is governed by the Hamiltonian

H(t)=Ho-^V(t) (37)

where the interaction Hamiltonian, V{t), which describes the in­teraction of the molecule with the radiation field, is given in the dipole approximation by

K(r, 0 = -w • E(0 (38)

where u = -er(t) is the electric dipole moment operator, -e is the charge of the electron, and E(0 represents the radiation field. Assuming that all of the properties of the molecule-field coupling system are described by the wave function |^(r, 0)? we have

eft (39)

In general, the preceding time-dependent Schrodinger equation cannot be solved exactly. So, it is adequate to solve this equa­tion through the use of perturbation theory. To solve the time-dependent Schrodinger equation systematically in terms of a per­turbation expansion, we replace the Hamiltonian 7/(0 by

H(t) = i/o + AK(0 (40)

where A is a continuously varying parameter ranging from zero to unity that characterizes the strength of the interaction, and the value A = 1 corresponds to the actual physical situation. Moreover, we seek a solution to the time-dependent Schrodinger's equation in the form of a power series in A:

k(r, t)\ = \^^^\r, t)\ + AI^^^HF, t)\ + A l ^^ r, t)\

+ --- + A A | ^^^ ( r ,0 ) (41)

By introducing it into the time-dependent Schrodinger equation and requiring that all terms that are proportional to A^ satisfy the equality separately, we obtain the set of equations

(42)

AT = 1 , 2 , 3 , . (43)

The preceding equations are solved by making use of the fact that the energy eigenfunctions for the free molecule constitute a com­plete set of basic functions in terms of which any function can be expanded. Therefore, the full time-dependent wave function

|^^^(r , 0) can be expanded in the term of eigenfunction |^«(r)) of the radiation-free molecular Hamiltonian HQ

|>pW(r, o) = EC^^(t)e-^'^'^^\^i(r)) (44)

where Q gives the probability amplitude that, to the Mh or­der in the perturbation, the molecule is in the energy eigenstate |4)/(r)> at time t. Furthermore, we find that the probabiHty ampH-tudes are given by

where

'^T,f V„,{t')CJ^-'\t')e''-''n"'dt' (45)

^ml = i^m - €i)/h (46)

is the transition frequency between eigenstates |<I>m(r)> and |<l>/(r)> of HQ, and we have introduced the matrix elements of the perturbing Hamiltonian, which are defined by

Vml(t') = {^m(r) I V(t') I (l>/(r)) (47)

The form of C^ \t) given earher demonstrates the usefulness of the perturbation techniques. Once the probability amplitudes of order (A' - 1) are determined, the amplitude of the next higher order (N) can be obtained by straightforward time integration.

Eventually we are interested in determining the linear, second-order and third-order optical polarizabilities, a, j8, and y, of molecule. To do so, we require explicit expressions for the prob­abiHty amplitudes up to the third-order in the perturbation expan­sion. Now the form of these amphtudes can be determined. We assume that the molecule-field coupling system is initially in an eigenstate | <l>g(r)> of ^ Q

|^F(r,^ = 0))=|<Dg(r)) (48)

and E(r, 0 can be represented as a discrete sum of (positive and negative) frequency components

E(r,0 = ^E(o>«)e-^'^«^

Then, [9]

<^(1)/ i(o>lg-(op)t

^(2) (0 = ^-2 E L pqm

cf\t) = h-

{o)lg-(op-a)q){o)mg-(t)p)

i{ailg-(op-(Oq-(or)t

^ pqrmn i(^lg-(op-o)q-ior){<ong-(Op-(oq)ia)mg •'ip)

where Uij = {<^i{r)\mjix)

(49)

(50)

(51)

(52)

(53)

(54)

is the transition-dipole matrix element between |0/(r)) and |c|>y(r))ofHo.

The expectation value of the electric dipole moment of mole­cule is exactly given by [9]

Pit) = (^(r, mmr, O) = P^^\t) + kp^^\t) + \^p^^\t)

+AV^^(0 + - (55)

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 271

where

p(^\t) = l^¥^\r, 0|w|^^^^(r, O) (56)

H-(^(l>(r,0|w|^^^^(r,0) (57)

p(^\t) = h^^\r,t)\u\^^^\r,t)\

-\-i^¥^Hr,t)\u\¥^Hr,t)^ (58)

+ l^¥^Hr,t)\u\¥^\r,t)^ (59)

The preceding results can be used to determine the Unear optical properties of a material system. The first-order contribution to the induced dipole moment per molecule is given by [9]

P^'Ht)

p m

Ugm[Umg*^((Op)]

(Omg — 0)p

[Ugm^^{o>p)Wmg

(^mg + (^p + L-/a>p/ (60)

In the preceding equation, we have formally allowed the possibil­ity that the transition frequency comg is a complex quantity [9]. We have done this because a crude way of incorporating damping phe­nomena into the theory is to take comg to be the complex quantity

iTm/2, where co^^ is the real transition frequency o)mg = (o mg and Tm is the population decay rate of the upper level m. Next we take the linear polarization to be [9]

V^^\t) = pp^^\t) (61)

where p is the number density of molecules, then express the po­larization P^^^(0 in terms of its complex amplitude as [9]

P l=x,y,z

^^^'(op)e-'''''' (62)

and finally introduce the linear optical polarizability a defined through the relation

(63) Pi (o)p)= ^ paij((Op)Ej{cop)

We thereby find the linear optical polarizability a(o)p) of molecule

(64) aij((Op) = h ^Y^ 'gm'^mg ^gm^mg

Omg (Or 0) mg + 0)l

The first and second terms in the preceding equation can be inter­preted as the resonant and antiresonant contributions to the linear optical polarizabiHty [9]. It should be noted that if "g" denotes the ground state it is impossible for the second term to become reso­nant, which is why it is called antiresonant contribution [9].

The expression for the second-order nonlinear polarizability P of molecule is derived in a manner analogous to that used for the linear optical polarizability a of molecule. The second-order

contribution to the induced dipole moment per molecule is given by [9]

pq mn

+ •

UgnWnm • E(o)q)][Umg • Ejcop)]

{(Ong - (Op- 0)q){o)mg - 0)p)

[Ugn • E{a)q)]Unm[Umg • ^((Op)]

(o)ng -\- (Oq)(o)mg - 0)p)

[Ugn • E{(Oq)][Unm • E(o)p)]Umg ]

{a)*g + 0)q)i0)*ig + ft>p + (Oq) J

X e-'^'^P'^'^^^^ (65)

Next we take the second-order polarization to be [9]

(66)

(67)

and represent it in terms of its frequency components as [9]

p(2)(0 = ^ E ^r\^cr)e-^'^^' o- l=x,y,z

Using the formal definition of the second-order nonlinear optical polarizability (3 of molecule

Pp(ft)o-) = E E PPijk(^^ = 0)p + (Oq', (Oq, (Op) j\k=x,y,z(pq)

xEj{(0q)Ej,((0p) (68)

we find that j8(wo-) is given by

Pijk((^CT', (Oq, (Op)

TP U^ U^ ^ gn^nm^mg

{(Ong -(Op- (Oq){(Omg - (Op)

+ • U^ IP TI^ ^gn^nm^mg

((Ong + (Oq)((Omg - (Op)

+ -TTJ Tjk Tji ^gn^nm^mg

{(Olg + (Oq){(0%g -\-(Op + (Oq) (69)

where 0 denotes the intrinsic permutation operator [9]. This op­erator tells us to average the expression that follows it over both permutations of frequencies (Op and (Oq of the appUed field. The cartesian indices ; and k should be permuted simultaneously.

Finally, the third-order nonlinear optical polarizability y of molecule can be calculated. The dipole moment per molecule, cor­rect to the third order in perturbation theory, is given by [9]

Let [9]

(o)ng-(Or-(oq-(op)((i}ng-op-a)q){(i)nig-(i>p)

^ [Ugo»E{a)r)]Uon[Unm»Ei(oq)][Umg»E{a)p)]

((OQg+(Or)((Ong-(Oq-(t)p){a)mg-(Op)

^ [Ugo»E((or)][Uon»E{a)q)]Unm[Umg»Eia)p)]

{a)Qg+a)r){a)^g+(Or+(oq)i(Omg-(o p)

[Ugo»E{a)r)][Uon*E((oq)][Unm»E{(op)]Umg i(o'^g+(Or)i(o*ig+(Or+(Oq)i(o*f^g+(Or+(Oq+o)p)

y_^-ii(op+(^q+(Or)t

o- l=x,y,z

(70)

(71)

272 XIE

Using the formal definition of the third-order nonhnear optical polarizabihty 7 of a molecule

h,i,j=x,y,zipqr)

xEj((i)r)Ei(a)q)Efj((Op) (72)

we obtain the third-order nonlinear optical polarizabihty y(o)o-)of molecule

ykiih(^(T'^ (^r,o}q, (^p)

:h-^Q E Tjk TjJ Tji Tjh

'j)ng-(^r-(Oq-(op){(Ong-(Op-coq){(Omg~o)p)

h TTJ Tjk Tjl TJ '-'go'-'on'-'nm'-'mg

{o)og+^r){(Ong-ojq-oJp)io)mg-o>p)

TjJ Tji Tjk Tjh *-'ga*-'on^nm^mg

((Oog + (Or)((o^g+(Or+(Oq)((Omg-o)p)

Tji Tji Tjh Tjk

{o}'^g^-(Or){o)'^g^-(x)r^(^q){(o%ig+(x)r+(oq+(tip) (73)

Here we have again made use of the intrinsic permutation opera­tor 0 defined before. The complete expression for the third-order nonlinear optical polarizabihty y contains twenty-four terms, of which only four are displayed explicitly; the others can be obtained through permutations of the frequencies (and cartesian indices) of the applied fields.

Until now, the basic structure-property relations for the sec­ond-order optical materials are both relatively well understood and explored. However, a comparable level of understanding of the third-order nonlinear optical materials, in contrast to the second-order ones, is only just emerging. In this chapter, we briefly review recent theoretical and experimental studies on the second-order optical nonlinearities (characterized by the first-order hyper-polarizability j8) of C6o and higher fullerene C70 in Section 2, and then the third-order optical nonlinearities (characterized by the second-order hyperpolarizability y) of C60 and higher fuUerenes including C70, €75, C78, 0^4, C^^, C90, C94, and C% in Section 3. Then, we introduce the extended Su-Schrieffer-Heeger (ESSH) model, where the Coulomb interaction is included, and apply this model to describe the fullerenes and fullerene-related nanotubes. On the basis of the electronic structure obtained from the ESSH model, we have studied the third-order optical nonlinearities of tubular fullerenes and chiral carbon nanotubes in Section 4. Later, we extend the ESSH model to include the effect of the dopant ions and then study the doping effect on the third-order optical nonlin­earities of tubular fullerenes in Section 5. Finally, we develop a coherent control theory of molecular polarizabihty a and hyper­polarizability j8 and y in Section 6. On the basis of the coherent control theory, we are able to achieve a large third-order optical nonlinearity of molecule at a desired frequency, which is required for photonic application.

2. SECOND-ORDER OPTICAL NONLINEARITIES OF FULLERENE

Among interesting properties of fullerenes are their strong non­linear optical properties. In the recent years, third-order nonlinear optical responses of fullerenes have been studied by, for example, THG and DFWM on their thin films or in their solutions (see next section). Because C^Q and higher fullerene C70 possess inversion

symmetry, the second-order nonlinear optical processes are for­bidden within the dipole approximation. Recently, however, a sig­nificant second-order nonlinear optical response in C60 and C70 has been observed by several groups.

2.1. Second-Order Optical Nonlinearity of C60

Hoshi et al. [20] made SHG measurement on the C60 film. In their experiment, the method of Kratschmer et al. [21] was applied to prepare Cgo powders [22], a 600-A-thick film of C50 was deposited on a silica substrate at 30°C by the molecular-beam epitaxy system (with 1 X 10~^ torr for the base pressure) [23], and C^Q powder was sublimed at about 400° C. SHG was measured by the same system as described by them in a separate paper [24]. To study the symmetrical aspect of the NLO response, the dependences of the harmonic intensities on the polarization of light and on the angle of incidence were measured in the transmission geometry. Unfo­cused polarized NdrYAG laser fundamental light (1064 nm) was impinged on the film, which could be rotated around a vertical axis to change the angle 9 of incidence. Transmitted SHG (532 nm) was detected through a polarization analyzer and a monochromator. Hoshi et al. confirmed that there is no detectable SHG emitted from the blank substrate. If the polarization state of light is ex­pressed by X and Y of (X, y, Z) coordinates (where Z is in the di­rection of the incident laser beam and X is the rotation axis), and the symbol X > Y is used to denote the experimental condition in which the incident beam is X polarized and the analyzer is Y po­larized, they found that no SHG was observed under X > X and Y> X, irrespective of the angle 6 of incidence. On the other hand, for the polarizations oiX >Y and Y> Y, there was no detectable SHG at ^ = 0, but it was observed that SHG started to emerge as the C60 film was rotated from ^ = 0. Their results are similar to those for phthalocyanine films [24] and can be explained by assum­ing an appropriate point group symmetry of the film structure. If the normal axis of the C60 film is assumed to possess a Coo sym­metry, the point group will be one of the three possibihties: Cooi;, ^ooh^ or Ki^. If SHG is due to the electrical dipolar responses, only the C^ov symmetry is consistent with their observed results. This symmetry implies that asymmetry exists along the direction normal to the substrate. One possible cause of this asymmetry may be the presence of the substrate. On the other hand, if SHG arises from an electrical quadrupolar response, either Coov or C^i^ lead to a consistent account of the observed results. As for the origin of SHG, more quantitative analysis of the angle dependence is re­quired. However, the observable SHG was too weak to make a complete analysis.

Wang et al. [25] studied the second-order optical nonlinearity of C60 film by the SHG measurement. In their experiment, the C50 powder was prepared by contact-arc evaporation of graphite in a He environment [21]; the soluble components of the resulting soot were extracted with toluene and separated from other fullerenes by column chromatography [26]; the purity of C60 was better than 99% measured by Raman, IR absorption, high-performance liquid chromatography, and fast atom bombardment mass spectroscopy; and C o thin films were deposited on glass substrates by ther­mal sublimation at 350^00° C under an ambient chamber pres­sure of 1.5 X 10~^ torr. First, Wang et al. measured the reflection SHG using the equipment and techniques of Zhang, Zhang, and Wong [27]. It was found that the C60 film is isotropic about its nor­mal and the ratio of Xz^yy/xi):z is 0.46 [25]. Then, they carried out the transmission SHG measurements at 1064 nm provided by

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 273

a Nd:YAG laser in the ;?-polarized geometry using instrumenta­tion and calibration techniques of Dai et al. [28] and found that the ratio of x^z,yylXz,zz is 0.48 [25], which is consistent with the value obtained by the reflection SHG measurements. Using a Ten-cor Alpha-Step film profiler, they measured the C6o film thick­ness and studied the SHG on this film with different thicknesses. They found that the square root of SHG intensity increases lin­early with the film thickness [25], which imphes that C6o film shows a bulk second-order optical nonlinearity. Using the formalism of Jerphagnon and Kurtz [29] for uniaxial materials and the x^^^ component ratio obtained from their C6o films, they further deter­mined the value of x^^^ for their C6o films by comparison with the

bulk Xx^xx contribution to the transmission SHG of a quartz plate (2) measured under the same conditions, and found Xz,zz of their C^Q

films at room temperature is 2.1 x 10~^ esu [25], which is about L5 times that of quartz. Also, they found that the SHG intensity of C60 films starts to rise at a temperature of 90° C, reaches a max­imum at 140° C, and then falls with increasing temperature [25]. The largest value observed at the nominal temperature of 140° C is about 2 x 10~^ esu [25], which is about 10 times larger than that at room temperature. In their experimental work, two poten­tial mechanisms for SHG were pointed out to explain the large second-order nonlinear optical response of C6o films: (i) small amounts of impurities such as Cgo isomers, other fullerenes or ox­ides that have noncentral symmetry may be present; (ii) the SHG is caused by electric quadrupole (or magnetic dipole) contributions, that are allowed in centrosymmetric materials (the electric dipole contribution is forbidden).

Using a combination of frequency-, rotational-, angular-, and film-thickness-dependent measurement, Koopmans et al. [30] studied the optical-second-harmonic generation from thin C60 films. It is the first spectroscopic SHG study of pure Cgo films or C60 surfaces. They found the C6o films to show a high SHG effi­ciency and a strong and sharp resonance 3X2ha) = 3.60 eV, which is close to an allowed optical transition.

Kuhnke et al. [31] have shown the SHG spectrum of C6o films over a wide spectra range employing a tunable OPG/OPA appa­ratus (their thin-film samples are prepared by evaporation of C6o on amorphous quartz discs). Between the fundamental photon en­ergies of 1.0 and 2.3 eV, they have found three pronounced res­onances at 1.18, 1.82, and 2.02 eV and a weak one at 1.35 eV in their SHG spectrum. Comparison with their experimental stud­ies shows that the resonance peak observed at 1.82 eV by Kuhnke et al. confirms the position and width of the resonance first ob­served by Kooopmans et al. In the work of Kuhnke et al. [31], a significant SHG signal was also found for a fundamental energy of 1.165 eV, which was the energy of the study at fixed wavelength by Wilk et al. [32]. The experimental measurement demonstrates that this study was really done close to the maximum of the resonance. The observation of a resonance at 2.02 eV in a SFG measurement was only recently reported [33]. In that work, the resonance could, however, not be observed in SHG, and it was argued that a reso­nance at the doubled energy was missing. The reason for the dis­crepancy with this work is not yet clear.

It is known that attenuated total reflection (ATR) is a very sen­sitive method for determining the dielectric constant e of thin films [34, 35]. When an electrical field is appHed to a thin film, the vari­ation of the dielectric constant would be [36]

Ae = Ae' -\- i^8' = x^^H-(^\ oj, 0)E (74)

where E is the peak-to-peak amplitude of the electrical field, s^ and s^^ are the real and imaginary parts of the relative dielectric constant, respectively, and ; ^ ^ is the second-order nonlinear op­tical susceptibility. For details about the experimental set-up, see the literature of Wang et al. [36]. Recently, using the experimental technique of electro-optically modulated ATR spectroscopy, Wang et al. [36] have studied the second-order optical nonlinearity of Langumir-Blodgett (LB) films of C^o by electro-optical (Pockels) characterization for the first time and measured the second-order nonlinear optical susceptibility of the mixed C6o and arachidic acid (AA) LB monolayers. In their experiment, fuUerene-containing carbon soot was also synthesized by the arc discharge of graphite rods in the presence of helium [21], and C^Q^ whose purity was also better than 99% measured by high-performance liquid chro­matography, was separated from the higher fullerenes by liquid column chromatography [26]. The absorption spectra of their so­lution samples were found to match the spectra of Hare et al. [37]. In addition, they used a KSV 5000 Langmuir trough to prepare the LB film samples. They measured the conventional ATR spectra of the prism/Ag/air and prism/Ag/mixed LB monolayer/air systems at the wavelength 633 nm by the angle-scanning method. To obtain the second-order nonlinear optical susceptibility x^'^^ of C6o, they measured the angular dependence of the differential reflectivity from the mixed C60-AA LB monolayer at 633 nm. They found a linear relation between the amplitudes of the measured differen­tial reflectivity and the applied voltage, which confirmed that the electro-optical response observed in their experiment was indeed the Pockels effect. Finally, they deduced that the second-order nonlinear optical susceptibility x^^"^ of C60 film was 1.3 x 10~^ esu, which is in agreement with that of Wang et al. [25]. On the ba­sis of the work of Rashing et al. [38], they calculated the molecular hyperpolarizability jS of the C^Q molecule, which was found to be 1.1 X 10~^^ esu [36]. Because they found that the electro-optical response of a pure AA monolayer was about two orders of magni­tudes smaller than that of the mixed monolayer under the same ex­perimental conditions, the Pockels effect, which was observed for the mixed monolayer, in fact, originated from the second-order op­tical nonlinearity of the C o molecule. Once more, their observed large electro-optical response of C50 confirmed that it is a promis­ing NLO material regardless of the molecular inversion symmetry.

The second-order nonlinear optical susceptibility of Cgo films is interesting because of the inversion symmetry of 059 molecules. As shown earlier, several groups have observed the SHG in C50 films [20, 25, 30, 36]. These films are known to crystalize in fee structure at room temperature [39, 40], and are therefore centrosymmetric. Within the electric-dipole approximation, one would predict that the SHG of the C^Q film should vanish, so that only the surface contribution could be detected. However, Wang et al. [25] reported that the growth of the SHG signal is propor­tional to the square of the film thickness. On the basis of their ex­periment, we can conclude that the surface contribution to SHG is very small. In the light of this point, it seems that the origin of the SHG signal from a C^Q film can only be an electric quadrupole or a magnetic dipole contribution, which is allowed in a centrosym­metric material, or from an impurity associated with the fuUerene species. The recent work by Koopmans et al. [41] showed convinc­ingly that a fuUerene's SHG originates from a magnetic-dipole in­teraction, and the theoretical calculation by Qin et al. [42] has also demonstrated that the second-order quadrupole nonlinear contri­bution was large enough to explain the experimental results ob­tained by Wang et al. [25]. In addition, it is interesting to note that

274 XIE

in one recent experiment, Liu et al. [43] have detected not only the SHG signal from €50 films but also found a change of the SHG sig­nal with the structural phase transition of C^Q. If the SHG of a C^Q film in the fee and sc phases was induced simply by the impurity in the €50 material, they should not observe the phase transition of €50 around 245 K since the impurity cannot display the property of C60. Therefore, the SHG from C^Q films cannot be attributed solely to the impurity and the SHG response of impurity, in C^Q films cannot explain the change in SHG at 245 K.

Qin et al. [42] determined theoretically the dispersion of the quadrupole response for single C^Q molecule. They find two reso­nances at 1.09 and 1.86 eV, respectively: the first one is only about 0.1 eV lower than the lowest peak that Kuhnke et al. [31] observed; the second one is in good agreement with the experimental mea­surement of Kuhnke et al. [31]. The calculated peak is of asym­metric shape, but there is no separate feature appearing near 2 eV. This indicates that the peak observed at 2.02 eV by Kuhnke et al. [31] is not because of the electric quadrupole transition. Between the two calculated peaks there is a sharp minimum that is caused by a change of amplitude sign. In the SHG spectrum of Kuhnke et al. [31], a similarly sharp minimum at 1.57 eV was also observed. The overall agreement with the theoretical calculation is good, al­though one may suggest that the peak calculated for 1.86 eV has a different physical origin than that suggested by the assignment of the observed resonance at 1.82 eV. Finally, no SHG peak is found at the spin-forbidden excitation of the lowest triplet exciton at 1.55 eV [44].

2.2. Second-Order Optical Nonlinearity of C60 Derivatives

It has been shown that charge-transfer interaction is the most ef­fective mechanism to enhance the second-order optical nonlinear­ity of organic molecule [45, 46], and fuUerenes are excellent elec­tron acceptors [47, 184]. Therefore, by forming charge-transfer complexes with appropriate donors, the center of symmetry in €50 or C70 is broken and significant second-order optical non-linearity may be induced. Recently, Wang and Cheng [49] have used A , A/ -diethylaniline (DEA), which is known to form charge-transfer complexes with various aromatic acceptor molecules [50], as the electron donor molecule and measured the second-order optical property of €50 charge-transfer complexes with DEA us­ing the standard dc electric field-induced second harmonic gener­ation (EFISH) technique (for the details of this technique see the literature of Meredith et al. [51] and Cheng et al. [52]). The funda­mental laser wavelength is at 1.91 [xm and the second harmonic at 0.955 jiim. They found that the formation of charge-transfer com­plexes with DEA indeed breaks the center of symmetry in C60 and induces its second-order optical nonlinearity [49]. Although all 27 components of the (3 tensor can be computed, only the vec­tor component in the dipolar direction (j8^) is sampled in their experiment. (3^ is given by [53]

j8^(-2cu; 0), 0)) = Y^^iiPi/\fi\ i

where

Pi = Piii-^\^l]^(l^jii-^Piji + Piij) ii,j)e(x,y,z) (76)

It was found that the value of (3^ of the C60/DEA is determined to be (6.7 ± 2) X 10~^^ esu, if (y>////(-2w; cu, w, 0) is approxi­

mately the same in toluene and DEA. The origin of the second-order optical nonlinearity of fuUerene/DEA complexes is likely to be similar to that of the previously reported pyridine-/2 (PI) and related complexes [45, 46], where the charge-transfer interaction induces asymmetry in the electronic polarizability. In the PI case, a n-a complex is formed and its j8^ was shown to be dominated by the oscillator strength of the charge-transfer transition. However, unlike the PI case, the fuUerene/DEA complex is likely to be of the 77—77 type [49], and inductive polarization effects by the large dipole moment of the complex on the large number of hyperpolar-izable 77 electrons, as indicated by the fuUerene third-order opti­cal nonlinearity, may also contribute significantly to the higher (3^ value [49].

It is known that the monofunctionized derivative C60— (C4H8N2) has no central symmetry [54]. Gan et al. [55] have mea­sured its SHG spectrum in transmission with a Y-cut quartz plate as reference and with a Nd:YAG laser beam (A = 1064 nm) at an angle of 77/2 to the film surface [56]. The film thickness is 1.5 nm, which is slightly larger than the nearest-neighbor distance 1.0 nm. Assuming that the refractive index is similar to that (= 1.90) of C60 [57] and using the methods of Ashwell et al. [58] and Lupo et al. [59], they found that the second-order nonlinear optical susceptibility x^^^ and molecular hyperpolarizability (3 are deter­mined to be (1.8 di 0.8) X 10"' esu and (3.6 ± 1.2) x 10"^^ esu, respectively.

Kajzar et al. [60] have studied the second-order optical nonlin-earities of C6o-based composites and multilayered charge-transfer structures with 5,10,15,20-tetraphenyl-21H, 23H-porphine (TPP) and 5,6,11,12-tetraphenylnaphthacene (rubrene) (TPN) acting as electron donors. It is found that SHG is observed in thin films containing C60 molecule only, and a significant enhancement in SHG signals is observed in the multilayered structures. Their re­sults can be interpreted in terms of a ground-state permanent elec­tron transfer from the electron-donating molecules to C^Q [60].

The INDO/CI (intermediate neglect of differential overlap/ configuration interaction) technique [61] and sum-over-states (SOS) approach are of proven reliability in the description of molecular nonlinear optical properties [61-63]. On the basis of the INDO/CI-SOS method, Feng and his co-workers have suc­cessfully calculated the second-order nonlinear optical suscepti­bilities Pij/^ of many molecules [64-66]. Using this method, Feng and his co-workers [53] have also carried out the calculation of the second-order nonlinear optical susceptibility j8/y and (3^ of C6o/aniline (AN) (a theoretical model of the experimentally stud­ied C60/DEA [49]) on the basis of having predicted the electronic spectra of C60/AN correctly. The nonzero, independent 10 com­ponents of j8/y , namely, Pzzz, Pxxx, l^xxz, Pxyy, Pxzz, Pyxy, l^yyz, Pzxx, Pzxz, and Pzyy (in the units of 10"- ^ esu) at a> = 1.91 jjum are calculated to be 1.726, 0.515, 0.623, 0.332, 0.436, -0.153, 1.109, 0.205, 0.604, and 0.800, respectively. The calculated value of PiJi(-2(o; 0), co) ^t (0 = 1.91 fim is 3.217 x 10"" ^ esu, which is

(75) in good agreement with the determined value (6.7±2) x 10"^^ esu for EFISH experiment of C60/DEA solution [49] (considering the DEA is a stronger donor than AN). The calculated fi • j8^ and fi values of the C60/AN complex are 3.955 x lO"" ^ esu and 12.293 D, respectively, which are also in good agreement with the observed values (6 ± 2) X 10"'*^ esu and (9 ± 2) D, respectively.

Preceding theoretical and experimental studies have shown that by forming charge-transfer complexes between fuUerenes and amines, their center of symmetry is broken and the second-order

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 275

Table I. Calculated Dipole Moments and First-Order Hyperpolarizability of the Closed-Shell Singlet States of C60H2 Isomers

Table II. The Static and Dynamical Values of the First-Order Hyperpolarizabihty j8 (in the units of 10"- ^ esu) For C59N and C59B

Calculated by Using the SOS Method C60H2 isomer IX (10-^^ esu) j8 (10-29 ggy)

C60H2-2

C60H2-3

C60H2-6

C60H2-7

C60H2-9

C60H2-I3

C60H2-I4

C60H2-I5

C60H2-I6

C60II2-23

C60H2-24

^60^2-31

C60H2-32

C60H2-33

C60H2-34

Q0H2-35

C60H2-4I

C60H2-49

C60H2-5O

C60f^2"52

C60H2-56

Q0H2-58

C60H2-6O

21

21

26

24

24

25

30

23

20

17

23

13

22

10

17

9

10

20

10

11

0

11

80

0.12

0.43

0.40

0.12

0.11

0.29

-0.06

0.18

0.29

0.11

0.01

0.12

-0.84

0.83

0.62

0.40

0.17

-14.8

-0.43

-0.13

-0.22

-1.51

-31.5

Source: Adapted from N. Matsuzawa et al., /. Phys. Chem. 96, 7584 (1992).

Optical nonlinearity is induced. The charge separation in substi­tuted C60 leading to enhancement of the second-order nonlin­ear optical susceptibility has also been discussed in the theoretical work of Matsuzawa et al. [67], and Jiang et al. [68]. It was found that further charge transfer takes place during the excitation pro­cess from the ground state to the excited state[68]. It is this kind of strong intramolecular charge transfer that causes large second-order nonlinear optical susceptibility in the substituted fuUerene. Table I and Table II collect some of their calculated results [67,68]. It is clear that among 050, Cgo/DEA, C^Q/AN, and C59Z, 050 has the smallest j8 value and C59Z has the largest one. This means that substitute doping is a good means to achieve a large second-order nonlinear optical susceptibility of pure C^Q.

2.3. Second-Order Optical Nonlinearity of C70

Kuhnke et al. [31] have also obtained the SHG spectrum of C70 in the same way as described for C60 in their experiment. In their spectrum, a weak resonance peak is observed at the one photon energy E = 1.26 eV and especially a steep signal at the one pho­ton energy E = 1.85 eV arises and indicates a broad resonance feature where there is no resolved structure. To the best of our knowledge, no SHG intensity was yet observed for C70 until the work of Kuhnke et al. [31]. We may understand this based on the following two facts: (1) the broad structure at the one photon en­ergy E > 1.83 eV is still very weak, and (2) the peak observed

Pijki~2(o; a>, (o)[a) (eV)] C59N C59B

Pzzz [0.0]

Pxxx [0.0]

Pxxz [0.0]

Pxyy [0.0]

Pxzz [0.0]

Pyxy [0.0]

Pyy, [0.0]

Pzxx [0.0]

Pzxz [0.0]

Pzyy [0.0]

Pzzz [0.3]

Pzzz [0.6]

Pzzz [1.3]

Pzzz [2.6]

145.37

0.16

25.09

21.08

75.84

41.29

90.09

0.64

53.43

37.95

4300

1050

300

700

17.59

14.38

9.98

2.54

6.85

14.54

15.88

1.19

25.99

0.61

30

760

840

440

Source: Adapted from J. Jang et al., Z. Phys. D 37,341 (1996).

at the one photon energy E = 1.26 eV does not extend to the fundamental frequency (1.165 eV) of Nd:YAG lasers. It has been shown that the lowest single (Si) and triplet (Ti) excitons were as­signed for isolated molecules in an Ne matrix to 1.93 and 1.56 eV, respectively [69]. If there is a shift between matrix isolated C70 and the solid, of about 0.1 eV, the singlet corresponds very well to the onset of SHG intensity. The proximity of the next higher sin­glet exciton (S2) and higher excitonic states may lead to the broad structure above 1.85 eV. However, the resonance peak at 1.26 eV is too low in energy to allow an assignment to the triplet exciton. Shuai and Bredas [70] have calculated the SHG spectra for C70. In their calculations, the contribution of the electric quadrupole cal­culated for C70 has nonvanishing values in a sharp peak at 1.2 eV and in a broad structure between 2.0 and 2.4 eV [70], which is in good agreement with the SHG spectrum [31] observed by Kuhnke et al. We know that the lower symmetry of C70, with respect to C^Q, makes the low-energy part of the absorption spectrum of the solid richer in electric-dipole-allowed transitions. In contrast, the SHG spectrum shows only two features with low intensity. Numerous dipole allowed transitions are absent. This and the agreement with calculated spectra suggest that, as in the case of C60, the forbidden transitions may be observed most easily. In fact, a dipole-forbidden transition at 2.49 eV was predicted by Kajzar et al. [71, 72]. It would be in even better agreement with the 1.26 eV peak than the dipole-allowed transitions.

3. THIRD-ORDER OPTICAL NONLINEARITY OFFULLERENE

3.1. Background

Molecules with large third-order optical nonlinearities, character­ized by large second-order hyperpolarizabilities 7, are required for photonic applications including all-optical switching, data pro­cessing, and eye and sensor protection [6-8,10-15]. However, the y magnitudes of most materials are usually smaller than those

276 XIE

needed for photonic devices. Therefore, recent research effort in physics and chemistry has been devoted to finding potential third-order optical materials with large nonlinear optical response.

Theoretical and experimental studies [6-8, 11-15] have shown that conjugated 7r-electron organic systems and quantum dots are potentially important in photonics owing to their large nonlin­ear optical (NLO) response. For example, polydiacetylenes, poly-acetylenes, and polythiophenes, because of their delocalized TT electrons, are characterized by large second-order hyperpolariz-abilities [6-8, 12, 14]. Garito and co-workers [73] suggested a mechanism to increase the second-order hyperpolarizability y of linear 7r-electron-conjugated molecules. Marder and co-workers [74] have recently shown that larger third-order optical nonlinear-ities could be elicited by manipulating the electronic character of small molecules embedded in the polymers. Furthermore, the scal­ing of second-order hyperpolarizability y for the polyenes (a class of prototypical nonlinear optical organic materials) with the num­ber of double bonds A has received considerable attention, both theoretically and experimentally [75-84]. A power law dependence for small values of N was predicted at different levels of theory [75-84]:

r = kN"" (77)

where 3 < a < 6. Experiments on well-characterized polyenic derivatives seem to favor the lower value of the exponent [77]. When A' becomes sufficiently large, a saturation effect is reached and the scaling becomes linear with N. Theory sets the onset of sat­uration in the range 20 < A/ < 60 [75, 76, 78-84], whereas exper­iments on polyene-related systems indicate a substantially higher value [77]. Recently, Lu et al. [82] have presented an analysis of the difference between experiment and theory on the basis of ab initio results, in which the possible role of disorder and polymer defects in enhancing nonlinear optical response was pointed out. However, those organic materials absorb in the near-infrared due to the overtones of high-energy C—H and O—H vibrations, which limit their applications in the infrared region (telecommunication window).

On the other hand, the advent of the technology for produc­tion of bulk quantities of fullerenes Cn [19, 85-91], which has a three-dimensional cage structure, has provided us another class of completely conjugated materials. These materials have quan­tum dot nature and possess a large number of delocalized TT elec­trons, but are uniquely composed of carbon atoms. A further desir­able characteristic of this all-carbon molecule in comparison with organic or polymer nonlinear optical material is that no bonds such as C—H or O—H are present, and thus there is no such ab­sorption as observed in the usual polymers, which would other­wise limit their use in nonlinear optics. In addition, the electronics bands in fullerenes are narrower [92] than in conjugated polymers, where they are broadened because of the confirmations, polymer chain length distributions, and vibronic couplings to electronic lev­els. Consequently, the resonances in fullerenes are expected to be narrower too. Naturally, these features make fullerenes-appealing NLO materials for photonic applications and stimulates the theo­retical and experimental researchers to study the third-order opti­cal nonlinearities of fullerenes.

In the following, we briefly review the recent experimental and theoretical studies on the third-order optical nonlinearities of C^Q and higher fullerenes are made.

3.2. Third-Order Optical Nonlinearity of €50

The extensive studies of the third-order optical nonlinearity of C60 were triggered by the first experimental measurement by Blau et al. [93] in 1991, although it was later found out that an er­ror by more than three orders of magnitude for the final hyper­polarizability was incorporated in their measurements [94, 95]. This error seems to be quite indicative of the later development, with numerous measurements using different techniques obtain­ing data for the third-order nonlinear optical susceptibilities x^^'^ of C60 [20, 49, 96-116]. In Table III (also shown in Fig. 1) we col­lect some of the reported third-order nonlinear optical suscepti-

(3) bilities Xxxxx of Qo ^t a few selected wavelengths measured by DFWM, THG, and electric-field-induced second-harmonic gen­eration (EFISHG). It is important to recognize that because the third-order optical responses are very sensitive to many experi­mental factors such as the measurement techniques adopted, the incident laser power, and even the sample preparation method, the direct comparison of experimental results on nonlinear op­tical properties of Qo obtained from different groups by differ­ent techniques is rather difficult. In general, there is reasonable agreement between the values of Xxxxx of Qo measured by THG and EFISHG, although some variation in the magnitudes of Xxxxx is apparent. Moreover, as shown in Figure 1, two interesting fea­tures are observed from the wavelength (A)-dispersed THG spec­trum of C60: a sharp decrease of the third-order nonlinear opti­cal susceptibility x^'^'^ at shorter wavelength and strong resonance-ments in x^^^ with two peaks at the fundamental wavelengths of 1.06 and 1.21 / m, respectively. On the basis of three-level model, Kajzar et al. [72] have shown that the first resonance observed at A = 1.06 /im may be interpreted as a three-photon resonance with Tii^ state of C^Q. There are several states lying in the gaps, be­tween 2.3 and 2.5 eV, which are forbidden for a one-photon tran­sition. Thus, the second resonance observed at A = 1.21 [xm could be interpreted as a two-photon resonance [72]. Their theoretical calculations, as shown in Figure 1, agrees well with other experi­mental observations as well as theirs.

Furthermore, despite the discrepancy of several orders in the third-order nonhnear optical susceptibilities existed in the experi-

3 CO 0

>^

~\ I I I T" - I 1 1—

_ I I I I L _

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

^(|im) Fig. 1. Wavelength (A) dependence of the third-order nonlinear optical susceptibility x^^\-3o); o), co, (o) in C6o. Solid line is the calculated re­sults of Kajzar et al. [72] within three-level model, and "+" denotes the experimental values of €50 films or solutions listed in Table III.

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 277

Table III. Third-Order Optical Susceptibilities Xxxxx of Cgo Measured by Different Experimental Techniques

Wavelength (jitm)

0.532

0.597

0.602

0.633

0.675

0.816

0.825

0.834

0.843

0.85

0.852

0.861

0.870

0.882

0.891

0.900

0.909

1.022

1.030

1.039

1.056

1.064

1.074

1.083

1.092

1.138

1.158

1.165

1.177

1.236

1.245

1.254

1.263

1.269

1.278

1.287

1.291

1.296

; t ixL(10- l l esu)

0.01

38

22

20

8.2

1.322

0.747

1.025

0.723

1.5

1.865

1.526

1.456

1.341

1.840

1.472

2.051

7.148

7.254

7.253

7.414

0.7

6000

330

1.4

20

8.2

8.201

7.2

2.0

8.710

7.632

7.422

6.295

6.727

7.763

5.122

5.319

5.091

5.151

5.385

5.175

5.751

5.709

6.000

5.736

Technique

DFWM

DFWM

DFWM

DFWM

DFWM

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

DFWM

DFWM

DFWM

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

State of Material

solution

thin film

solution

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

solution

solution

thin film

thin film

thin film

thin film

solution

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

Reference

[138]

[106]

[101]

[110]

[106]

[72]

[72]

[72]

[72]

[96,97]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72] [100]

[93]

[99]

[102]

[20]

[96,97]

[72]

[104]

[25]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

{continues)

278 XIE

Table III. (continued)

Wavelength (/im)

1.305 1.323 1.33

1.332 1.344 1.356 1.368 1.381 1.413 1.437 1.456 1.50 1.907 1.91

2.00 2.38

X^LaO-^^ esu)

5.643 6.072 3.0 6.1 5.661 6.030 6.000 5.817 5.670 5.676 5.460 5.610 3 3.242 1.6 0.9 3.2 2.0 3.7 0.4

Technique

THG THG THG THG THG THG THG THG THG THG THG THG THG THG

EFISHG THG THG THG THG THG

State of Material

thin film thin film thin film thin film thin film thin film thin film thin film thin film thin film thin film thin film solution thin film solution thin film thin film thin film solution thin film

Reference

[72] [72] [102]

[96,97] [72] [72] [72] [72] [72] [72] [72] [72] [104] [72] [49] [102]

[96, 97] [98] [104] [102]

Source: Adapted from H. S. Nalwa, in "Nonlinear Optics of Organic Molecules and Polymers," (H. S. Nalwa and S. Miyata, Eds.). Chap. 11, pp. 611-797. CRC Press, Boca Raton, FL, 1997; F. Kajzar et al., Synth. Met. 11,257 (1996); J. R. Lindle et 2A.,Phys. Rev. B 48, 9447 (1993).

mental data shown earlier, the third-order nonlinear optical non-linearity of C60 has recently been proven to be a small value [117,118]. Upper bound of 3.7 x 10~^^ esu of the second-order hyperpolarizability for C6o was measured by nondegenerate four-wave mixing by Geng and Wright [117], and a 9.0 x 10"- ^ esu up­per limit was determined by the femtosecond optical Kerr effect (OKE) by Gong and his co-workers [118]. Moreover, a strong en­hancement on third-order optical nonlinearity was reported when C60 was chemically modified to form a charge transfer (CT) com­plex [49, 118] or was chemically reduced to anions [119, 120]. An increase of from several decades to 100 times on the y value was even observed.

3.3. Third-Order Optical Nonlinearity of Higher FuUerenes

The synthesis and isolation of higher fuUerenes is now an active research field [19], and it is expected that as the availability of large amounts of these materials becomes widespread, more extensive measurements of all kinds of properties, such as nonlinear optical properties, of these higher fuUerenes will be carried out.

Compared with the Qo molecule, higher fuUerenes have at­tracted less attention on their third-order optical nonlinearities. Theoretical calculations [108,109, 111, 121-136] and a few exper­imental results [49, 104-106, 110, 112, 115, 137, 138] have shown that higher fuUerenes possess large third-order optical nonlinear­ity than Qo- C70, the easiest purified higher fuUerene, has been experimentally proven to have a y magnitude of 1.6-3 times that of C60 [49,104-106,110,112,115]. Table IV (also shown in Fig. 2), Table V, and Table VI collect the third-order nonlinear optical

10

8

3

CD 6

O 4

co"

1

L

L + 1 1 1

1 1

+

j^ + / V +

+ +

1 — 1

' 1 ' T

+ >

+ / +

r

+

+

^ ^

1

+

+

-0.8 1.2

?i(|Lim) 1.6 1.8

Fig. 2. Wavelength (A) dependence of the third-order nonlinear optical susceptibility x^^H-^^^', (^, ^, ^) in C70. Solid line is the calculated re­sults of Kajzar et al. [71, 72] within three-level model, and "+" denotes the experimental values of C70 films or solutions listed in Table IV.

susceptibUities x^^^ oi C70 and higher fuUerenes studied by dif­ferent experimental or theoretical techniques. In the foUowing, we briefly review recent theoretical and experimental studies on the third-order optical nonlinearities of higher fuUerenes. Other re­views on the nonlinear optical properties of fuUerene (mainly C^Q and C70 molecules) are also avaUable in the recent literatures by Nalwa [13], Kajzar et al. [72], Rustagi et al. [139], and Belousov et al. [140].

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 279

Table IV.

Wavelength (/xm)

0.532

0.597

0.633

0.675

0.799

0.816

0.825

0.834

0.843

0.852

0.861

0.870

0.882

0.891

0.900

1.013

1.022

1.039

1.047

1.056

1.064

1.074

1.129

1.131

1.138

1.148

1.158

1.167

1.177

1.187

1.216

1.226

1.247

1.267

1.278

1.289

1.300

1.322

1.333

1.344

1.368

1.378

1.390

1.401

1.413

Third-Order Optical Susceptibilities

X^Li^O-'^^ esu)

0.043

210

30

64

0.825

1.132

1.252

0.571

0.636

0.697

1.024

0.568

0.729

1.165

1.122

2.974

2.928

2.881

2.940

2.558

1.2

0.56

140

2.6

2.577

3.584

3.521

2.569

3.357

3.861

4.262

3.657

3.026

4.740

4.383

4.344

1.875

2.895

5.106

6.663

6.910

7.056

6.840

8.307

6.579

7.649

8.970

7.587

Xxxxx of C70 Measured by ]

Technique

DFWM

DFWM

DFWM

DFWM

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

DFWM

DFWM

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

THG

Different Experimental Tech

State of Material

solution

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

solution

solution

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

thin film

niques

Reference

[138]

[106]

[110]

[106]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[112]

[105]

[104]

[71]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

[72]

(continues)

280 XIE

Table IV. (continued)

Wavelength (/mm)

1.42 1.425 1.437 1.449 1.50 1.907 1.91

2.00

;tSL(10-ll esu)

9.0 7.157 7.776 8.292

54 2.428 4.4 2.4 9.1

Technique

THG THG THG THG THG THG

EFISHG THG THG

State of Material

thin film thin film thin film thin film solution thin film solution thin film solution

Reference

[71] [72] [72] [72] [104] [72] [49] [71] [104]

Source: Adapted from H. S. Nalwa, in "Nonlinear Optics of Organic Molecules and Polymers," (H. S. Nalwa and S. Miyata, Eds.). Chap. 11, pp. 611-797. CRC Press, Boca Raton, FL, 1997; F. Kajzar et al, Synth. Met. 77, 257 (1996); J. R. Lindle et al^Phys. Rev. B 48, 9447 (1993).

Table V. Second-Order Hyperpolarizabilities y//// of Cyg, Cyg, €34, Cgg, C90, C94, and C96 Measured by Different Experimental Techniques

Molecule Wavelength (fim)

0.532 0.532 0.532 0.647 0.532 0.532 0.532 0.532

r////(10-30 esu)

0.8 ±0.3 1.5 ±0.3 1.2 ±0.3

0.16 1.3 ±0.5 1.8 ±0.6 1.9 ±0.6 2.1 ±0.6

Technique

DFWM DFWM DFWM

OKE DFWM DFWM DFWM DFWM

Reference

[138] [138] [138] [115] [138] [137] [138] [138]

C76

C78

C94

C96

Theoretical Calculations

On the basis of the geometries optimized by AMI semiempirical technique [141], Shuai and Bredas [127] exploited the valence-effective-Hamiltonian (VEH) method to study the electronic structures of €50 and C70. The valence-electronic density-of-states calculated is found to be in excellent agreement with the high-resolution energy-distribution curves obtained from synchrotron-photoemission experiments in terms of both positions and relative intensities of the peaks [127]. The maximum difference in peak position between theory and experiment is 0.4 eV, which shows that the VEH method provides a very reasonable description of both €50 and C70. Further they applied the VEH-Sum-Over-State (SOS) approach to study the nonlinear optical response of €50 and C70. Their numerical results showed that the off-resonance third-order nonlinear optical susceptibility x^^"^ is in the order of 10~^^ esu [127]. Their calculations are fully consistent with the EFISHG and THG measurements by Wang and Cheng [49] and the DFWM measurements by Kafafi et al. [95, 100], but about three to four orders of magnitude lower than the data reported by Blau et al. [93] and Yang et al. [105]. Also they investigated the dynamic nonlinear optical response by calculating the THG spectrum. It is found that the lowest two-photon and three-photon resonances occur at almost the same frequency for €50 because of the symmetry of the molecule [127].

Recently, available theoretical studies predicted that the sec­ond-order hyperpolarizability of a higher fuUerene scales with the mass of the all-carbon molecule [121, 125, 128-135]. Knize [121]

calculated the linear polarizability and the second-order hyper­polarizability of a free electron gas confined to a spherical shell. This free electron gas model is applied to determine the polariz-abilities of fuUerene molecules. The magnitude of the calculated polarizabilities of C^Q and C70 molecules were found to be in rea­sonable agreement with some of the experiments, and the second-order hyperpolarizability of the larger fuUerene molecules is pre­dicted to increase as the cube of the number of carbon atoms [121]. With the SOS-CNDO/S CI approach, a power dependence of the second-order hyperpolarizabilities of fuUerenes versus the number of carbon atoms was observed by Fanti et al. [128], where the observed exponent is four. Then, using a sum-over-molecular-orbitals (SOMO) approach at the Hartree-Fock level with a 6— 31G* basis set, Fanti et al. [129] further calculated the static second-order hyperpolarizabUities of the eight carbon cages now avaUable in macroscopic quantities: C60, C70, one isomer of C76, three isomers (1, 2, 3) of C78, and two isomers (22 and 23) of C84. It is found that the static y value increases as a nonlinear func­tion of the number N of carbon atoms [129]. In their work, once more, they showed the trends rather than individual values of the second-order hyperpolarizability y. They concentrated on either the molecular mass (the number N) or the number of partial dou­ble bonds ( |A/^-60) , and got two different scaling laws [129]:

(mass) _ \ 3.5

6 0 / ^ (78)

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 281

Table VI. Static Second-Order Hyperpolarizabilities 7 (in the unit of 10" ^ esu) of Higher FuUerenes Calculated by Different

Theoretical Techniques

Molecule

C70

C76

C78(^3)

C78(C2.(1))

C78(C2^(2))

C78(^3/r(l)) C78(D3;,(2))

C84

C84(^2)

C84(^2^)

C84

7 (10-^6 esu)

45.2

862.3

1300

857

54.5

640

294.5

1050

75.4

368.1

361.9

681

369.9

2034

440.7

507

480

2096

1812

63.6

426.9

460.6

82.4

Technique

FFA

SOS

SOS

SOS

FFA

SOS

SOS

SOS

RPA

SOMO

SOMO

SOS

SOMO

SOS

SOMO

SOS

SOS

SOS

SOS

FFA

SOMO

SOMO

RPA

Reference

[109]

[127]

[108]

[128]

[126]

[122]

[129]

[142]

[111]

[129]

[129]

[142]

[129]

[142]

[129]

[142]

[142]

[142]

[128]

[126]

[129]

[129]

[111]

and

(bond) jN-60

~ 3 0

1.5 w60 (79)

where y^^(= 1.546 x 10~^^ esu) is the static y value of C^Q. Later, based on the extended Su-Schrieffer-Heeger model, which will be introduced in the following section Xie and his co-workers [130,131] predicted that the static y values of armchair and zigzag tubular fullerenes of small size scale with about three power of the carbon atom N. Also, Harigaya [132-135] theoretically inves­tigated the nonlinear optical properties of 050, extracted higher fullerenes C70, C76, C7g, and Cg^ by using the exciton formal­ism and the SOS method. It is found that the off-resonant third-order nonlinear optical susceptibilities of higher fullerenes are a few times larger than those of C^Q [132-135], where the magni­tude of the optical nonlinearity increases as the optical gap de­creases for higher fullerenes and the optical nonlinearity is nearly proportional to the fourth power of the carbon number when the on-site Coulomb repulsion is 2t or 4t (t being the nearest-neighbor-hopping integral). The theoretical calculations of both Xie [130,131] and Harigaya [132-135] indicate the important roles of Coulomb interactions in higher fullerenes and agree very well with quantum chemical calculations for higher fullerenes. Luo [125] reexamined the reported data for the second-order hyper-polarizabihties of fullerenes calculated by Fanti et al. [129] and Jonsson et al. [111]. It is found that if C^Q is excluded, a perfect power law dependence for the second-order hyperpolarizability

versus the number of carbon atoms is observed [125]

y = 1.1 X IQ-' AT -

for the data of Fanti et al. [129] and

y = 3.2 X IQ-^ATO-^S

(80)

(81)

for the data of Jonsson et al. [111]. Luo [125] also indicated that C60 has the most exceptional electron localization among fullerenes.

It is well known that molecular symmetry has great effect on the nonlinear optical properties of molecules. To identify this is­sue, C78 molecule would be very interesting and a better candi­date for examing this effect than others, because C78 molecule has five topologically distinct structures: two with C2v symmetry, two with D^h symmetry, and one with D3 symmetry, each hav­ing 29 six-membered rings and 12 five-membered rings. For con­venience, the five isomers of C78 are denoted by D^hW, ^3/1 (2), Ds, C2vi2), C2vW' Using the SOS method, Wan et al. [142] cal­culated the third-order optical nonlinearities for the five C78 iso­mers. Although the Coulomb interactions among 7r-electrons have not been taken into account, their numerical results really indicate that both the symmetry and the arrangement of atoms have great influences on the third-order optical nonlinearities of C78 isomers. It is found that the static y values (in units of 10~^^ esu) of D^hW, D^hil), D^, C2u(l), and C2v{2) are 0.0480,0.2096,0.0681,0.2034, and 0.0507, respectively [142]. As we know, the D^hW structure is similar to the D^ structure except that the former has one more symmetry plane than the latter, which makes its energy levels move to each other and the degeneracy of its energy levels be higher than that of D3 isomer. But the THG spectra of both D^hW and D3 isomers are greatly different because of their different symmetry. In the THG spectrum of the D^ isomer, there is a big peak lo­cated at 3w = 6.08 eV and the y magnitude is 342.49 x 10"- ^ esu [142]. However, in the THG spectrum of the D^hil) isomer, the highest peak at 3ft> = 6.592 eV and the corresponding y mag­nitude is 17.50 X 10~^^ esu, which is 20 times smaller than that of D3 [142]. Furthermore, they have also shown that the atom ar­rangement of the C78 isomers has a great effect on the nonlinear optical properties. We know that both I>3/ (1) and D^hi^) isomers have the same symmetry, but their arrangements of atoms are dif­ferent. It is found that their THG spectra are different, too. In the THG spectrum of the D^hW isomer, the y magnitude of the highest peak is only 17.50 x 10"^^ esu [142]. In the THG spectrum of the £>3/i(2) isomer, its y magnitudes at the frequency region, 2.944 eV < 3w < 3.584 eV, are greater than 80 x 10"^^ esu, and its largest y magnitude even reaches 509.01 x 10"^^ g^u, which is 30 times larger than that of the D^hW isomer [142]. Because of the shape effect, the y magnitudes of C2y{2) and C2yil) are also greatly different. In the THG spectrum of the C2i;(l) isomer, there are three peaks with y magnitudes greater than 50 x 10~^^ esu, but the largest y magnitude in the THG spectrum of the C2v(2) isomer is only 15.995 x 10"^^ esu [142].

From a geometric point of view, IVIoore et al. [126, 136] ana­lyzed the static third-order nonlinear optical polarizabilities y of Qo» C70, five isomers of C78, and two isomers of Cg^ in terms of three properties: (i) symmetry; (ii) aromaticity; (iii) size. The polarizability values were based on the finite field approximation (FFA) using a semiempirical Hamiltonian and applied to molecu­lar structures obtained from density functional theory calculations. Symmetry was characterized by the molecular group order. The

282 XIE

selection of six-member rings as aromatic was determined from an analysis of bond lengths. Maximum interatomic distance and sur­face area were the parameters considered with respect to size. On the basis of triple linear regression analysis, it was found that the static linear polarizability a and y in these molecules respond dif­ferently to geometrical properties [126,136]: a depends almost ex­clusively on surface area, whereas y is affected by a combination of number of aromatic rings, length, and group order, in decreasing importance. In the case of a, valence electron contributions pro­vide the same information as all-electron estimates [126,136]. For the second-order hyperpolarizability y, the best correlation coeffi­cients are obtained when all-electron estimates are used and when the dependent parameter is ln{y) instead of y [126,136].

Recently, Jonsson et al. [Ill] have studied the third-order op­tical nonlinearities of €50, C70, and C^^ at the random phase approximation (RPA) level with a 6—31 + +G basis set. The calculated static y values for Cgo, C70, and €34 are 55.0, 75.4, 82.4 X 10"^^ esu, respectively.

Theoretical and experimental studies [143-154] have shown that (i) the lattice and electronic structures of fullerenes are mod­ified because of the substitute doping effect; (ii) the bandgaps be­tween the highest occupied molecular orbitals (HOMO) and low­est unoccupied molecular orbitals (LUMO) and the electronic po­larization of the substituted fullerenes vary greatly with different substitute doping; (iii) the distribution of TT electrons on the sur­face of fullerene is changed because of the substitute doping effect, and the original delocalized TT electrons in pure fullerene become more localized around the substituted atoms. The three factors have a large effect on the NLO properties of fullerenes. Recently, it has been shown, both theoretically [150-154] and experimen­tally [119, 120], that the y value of €50 and higher fullerenes can be enhanced greatly by substituting the carbon atom of fullerenes with boron or nitrogen or other atoms. This opens a new approach to achieving a large nonlinear optical response of fullerenes.

In Table VI we collect the static second-order hyperpolarizabil-ities of higher fullerenes calculated by different theoretical tech­niques. A direct comparison between their results shows signifi­cant differences. All of the SOS approaches give a second-order hyperpolarizabihty, which is about an order of magnitude larger than that predicted by the ab initio calculations. This discrepancy arises from the truncation of the expansion in excited states in the explicit summation of contributions to the second-order hyperpo­larizability in SOS calculations [111]. In contrast, the second-order hyperpolarizability obtained in the FFA methods [109, 126] are of the same order of magnitude. The described problems are absent in the analytical RPA approach [111].

In preceding theoretical studies, screening effect due to elec­tron-electron interaction has not been considered. It should be in fact considered because the polarization of the electron cloud also modifies the charge density and therefore, the self-consistent potential that is seen by each electron. Using the RPA approach and neglecting the radial spread in electron density, Bertsch et al. [155] found that the screening effect reduces the linear response a of C60 by a factor of 6. Using the empirical pseudopotential method, Nair [156] also found that the screening effect is very sensitive to the radial spread in the electron density, a factor of / = (1 -h aa/R^) was calculated for the linear response and the screening correction for y is slightly smaller than f^. From the measured refractive index of 059 films [157], it was estimated that "screened ^ 100 A^ [139] by using the Claussius-Mossotti relation. With a^are ^ 200 A^, it was estimated that the screening factor /

is about 2, which impHes a reduction of 7 by a factor of 16 from the unscreened values. Bertsch et al. [155] analysed the screening effect on the frequency dependent response using RPA approach and found that this effect is more dramatic: the absorption spectra are strongly modified because of the screening of the applied elec­tric field by the electrons themselves; all the absorption peaks were shifted to 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. In short, extension of these analyses to the nonlinear optical responses of fullerenes would be very interesting. Using the standard analysis of the local field correction to the nonlinear optical susceptibili­ties of condensed media [158], we would expect the screening cor­rection to, for example, the third-order nonlinear optical polariz-abihty to be given by a product of three linear screening factors that correspond to the four frequencies involved [139]. This im­plies many unusual features in the near-resonant NLO response of fullerenes, for example, the one- and three-photon resonances that correspond to the same pair of single-particle states may occur at slightly different frequencies. Detailed analysis of these features are expected in the near future.

Experimental Measurements

Neher et al. [104] studied the second-order hyperpolarizabihties y of the C70 at three different wavelengths measured by THG in a toluene solution, and the third-order nonlinear optical suscep-tibilities Xxxxx of C70 at 1.06, 1.50, and 2.00 /xm are determined to be 1.4 X 10~^ esu, 5.4 x lO'^^ esu, and 9.1 x lO'^^ esu, re­spectively. Their experimental results were compared with those of €50 • Based on this, they observed the strong effects with y ex­ceeding 5 X 10~^^ esu in the three-photon resonant regime, and the nonresonant measurements indicate a negative real hyperpo­larizability in C70 and a positive y for €50, both of the order of 10"^^ esu.

Wang and Cheng [49] reported the determination of the third-order optical nonlinearities of fullerenes and fullerene/DEA charge-transfer complexes by EFISHG measurements with 1.91-ixm radiation. The third-order nonlinear optical polarizabilities, 'yilll(~2ct>, w, w,0), of €50 and C70 are determined to be (7.5 ± 2) X 10" " esu and (1.3 ± 0.3) x 10~^^ esu, respectively. With due consideration to molecular size, the optical nonlineari­ties of fullerenes are comparable to those of linearly conjugated organics. Their experimental results are encouraging enough to warrant further study of the nonlinear optical properties of fullerenes and properly substituted fullerene derivatives.

Kafafi and co-workers [106, 112] studied the third-order op­tical nonlinearities of C70. Time-resolved DFWM experiments were conducted on films of pure C70 by using a picosecond tun­able dye laser. The fullerenes exhibit a two-photon resonantly en­hanced third-order optical response that is primarily laser-pulse-limited [112], their dynamics show wavelength and fluence depen­dence [106], and in detail the third-order nonlinear optical suscep­tibilities Xxxxx (ii the unit of 10"^^ esu) of C70 at 1.064,0.675, and 0.597 ^m are measured to be 1.2 [112], 64 ± 20 [106], and 210 ± 10 [106], respectively. At high laser intensities, a fifth-order compo­nent to the nonlinear optical signal is observed and is attributed to a two-photon excited-state transient grating [112]. The large two-photon absorption coefficients measured for C70 at 1.064 cm by nonlinear transmittance are consistent with this assignment.

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 283

Gong and his co-workers [105] studied the nonhnear third-order nonUnear optical susceptibility of C70 in a toluene solution, which was measured for the first time by the method of DFWM using 10 ns laser pulses at 1.06 fim. The third-order nonlinear op­tical susceptibility A'lin is measured to be 5.6 x 10~^^ esu for a C70 toluene solution at a concentration of 0.476 g/1. The corre­spondent magnitude of the second-order hyperpolarizability y i m of the C70 molecule is 1.2 x 10"^^ esu.

Thin film of pure C70 has also been studied by time-resolved DFWM using femtosecond optical pulses with a wavelength of 0.633 fim [110]. Large optical nonlinearities of C70 were measured with x?Jxx = 3 X 10-1^ esu at 0.633 /xm.

Kajzar et al. [71, 72] reported wave dispersed optical THG studied in the fundamental wavelength range of 0.8-1.9 ixm in thin films of C70. In their THG spectrum, a broad resonance en­hancement in cubic susceptibility ; ^ ^ with a maximum value of ^^^^^-3^; 0), (X), (o) = (0.9 ±0.1) X 10"!^ esu located at 1.42 fim and a dramatic decrease at higher energies were observed.

In Figure 2, we also show the wavelength dependence of the third-order nonlinear optical susceptibility of C70 listed in Ta­ble IV. Two interesting features are observed: a sharp decrease of X^ at shorter wavelength and strong resonance enhancement in ; ( ) at A = 1.41 fim, which are similar to those of C60 shown in Figure 1. Especially, it is observed that there is a shoulder around 1.06 /xm. On the basis of the three-level model, Kajzar et al. [71, 72] indicated that the observed resonance at 1.41 ixm can also be interpreted as a three-photon resonance with a one-photon allowed transition lying at 0.47 fim above the fundamental state, and the second resonance (the shoulder at 1.06 jim) is interpreted as a two-photon resonance. This assignment of their theoretical calculations agrees well with other experimental observations as well as theirs.

Sun et al. [115] studied the third-order optical nonlinearity of Cg4 by using the time-resolved optical Kerr effect (OKE). They got a large instantaneous nonlinear optical response of C84 using 150-fs laser pulses with a wavelength of 647 nm. Comparing the nonlin­ear optical response with that of the CS2 reference, they acquired large second-order hyperpolarizabihty for C84, C70, and C60 with 71111 of 5.2 X 10~^^ 4.7 X 10~^^ and 1.6 x 10"^^ esu, respec­tively [115]. In contrast to the y i m increment of C70 to C60, the small accretion of the optical Kerr response of C84 in comparison to C70 seems to be a puzzle. It is claimed that the sample purity was not satisfactory [137]. In their experiment, C84 was obtained from fullerite soot. The separation and purification were carried out by means of liquid phase chromatography combined with a recrys-tallization technique. Their nuclear magnetic resonance (NMR) spectra showed that the main impurities include C78, C82, C86, and C90, although a purity of C84 is greater than 85%.

Yang and his co-workers [137] have made DFWM measure­ments on the third-order optical nonlinearities of the higher pu­rity C90 (>97% in their mass spectrum) and C60. They reported that C90 shows a large third-order nonlinear optical response at 0.532 jxm in comparison to that of C60. The concentrations of fullerenes and the incident laser powers were found to be vitally important in the DFWM measurements and are therefore opti­mized. Figure 3 presents the UV-VIS absorption spectra of C^Q and C90 in CS2 [137]. The spectrum of C^Q is identical to that of Diederich et al. [159], confirming the high purity of their sample. The spectrum of C90 exhibits the strongest peak around 450 nm in the spectra range 400-800 nm. It is apparent, from a comparison

1 . 0

0 . 0 "

500

Wavelength (nm)

Fig. 3. UV-VIS absorption spectra of C90 and €50 in CS2, where the arrow above the horizontal axis indicates the excitation wavelength (0.532 /Am) of the DFWM measurements. The concentration of C90 and Cgo are 0.09 and 0.42 g/1, respectively. Reprinted from H. Huang et al, Chem. Phys. Lett. Ill, All (1997). Copyright 1997, with permission from Elsevier Science.

o

h •

1 * * r •

\ • •• • • •

r*

—^—1

1 •

1 4 u

„«L

• •

1 > 1

• ' ' ' 1 1 - y

- i ft--—0 Q" — t ,» 1 > L _ i _

- T — • ^ 1 — , 1 — 1 —

c„

Ceo

. * t 1 1 1 — k .

» ' 1 » ' — « — « — 1

^

-

- - - & . . ] ^

Concentration of C^ and Cg in CS^ (g/1 )

Fig. 4. The measured second-order hyperpolarizabilities y\\\\ of €50 and C90 as a function of concentration. The data points at each concen­tration were measured at different incident laser powers. Reprinted from H. Huang et al., Chem. Phys. Lett. 212, 427 (1997). Copyright 1997, with permission from Elsevier Science.

of the two spectra in Figure 3 that the long wavelength absorption edge is significantly red-shifted from €50 to C90. This may be at­tributed to the narrowing of the HOMO-LUMO gap due to the increased size of the 7r-conjugated system and the reduction in the fullerene symmetry. Since their excitation wavelength (0.532 tm) is near the absorption peaks of €50 and C90, the linear absorption should not be ignored in their DFWM measurements. From the

(3) set of A'liii values, the second-order hyperpolarizability of C90 is determined to be (1.8±0.6) x 10~^^ esu through a statistical anal­ysis in which three times the standard deviation (±3(7) was taken for the uncertainty [137]. For comparison, the second-order hyper­polarizability of €50 was measured to be (2.2 i 0.6) x 10 esu using the same procedure. The statistical results of y i m of €50 and C90 are shown in Figure 4 [137]. The most striking observation

284 XIE

from Figure 4 is the approximate eightfold difference in y im be­tween C90 and C^Q. Since there are more highly delocalized 7r-con-jugated electrons over the spherical-like surface in C90 compared to €50, the larger y value of C90 is expected. The eightfold in­crease in the third-order nonlinear optical polarizability from Q Q to C90 is consistent with the trend in the theoretical predictions [128-135]. However, the predicted increase in the second-order hyperpolarizability y from C^Q to C90 is somewhat lower than the measured eightfold increase. This suggests that other contributing factors to the increased second-order hyperpolarizability may be important as well.

Furthermore, Yang and his co-workers [138] have system­atically studied the nonlinear optical response of other higher fuUerenes including €75, Cyg, C^4, 035, C94, and C96 by perform­ing the DFWM measurement on those fullerene series dissolved in CS2. The DFWM measurements were carried out by using 70 ps laser pulses at the wavelength of 0.532 fim under the optimized experimental conditions. The second hyperpolarizabilities 71 m (in the unit of IQ-^^ esu) of C^Q, C70, C76, C78, €34, C86, C94, and C96 are 0.22 ± 0.06,1.3 ± 0.4, 0.8 ± 0.3,1.5 ± 0.3,1.2 ± 0.3, 1.3±0.5,1.9±0.6, and 2.1±0.6, respectively, that is in the order of 10-^1 to 10-^0 esu [138], and the ratios for the second-order hy­perpolarizability for C70/C60, C84/C60, and C84/C70 were found to be 3.63, 5.45, and 0.92, respectively, where overall they increase smoothly with the carbon cage size except for C70 and C78. Their experiments confirmed the high nonlinear efficiency of C70 and they found that C78 possesses a large value for y compared with the other fullerene cages. The number of vr-conjugated electrons, geometrical structure, and resonance enhancement are discussed as possible factors responsible for the observed third-order optical nonlinearities of the fullerenes [138].

Gong and his co-workers [160], stimulated by the strong en­hancement on the third-order optical nonlinearity of C^Q when its CT complexes were formed [49, 118], inferred that C70 CT complexes should exhibit a larger nonlinear optical response. The higher fullerene CT complex, C70 poly-aminonitrile, was synthe­sized and a second-order hyperpolarizability as high as 1.6 x 10~^^ esu was measured by using the femtosecond time-resolved OKE technique [160]. Similar to the case of C^Q and its poly-aminonitrile derivative, the y of the new derivative was more than 30 times larger than that of C70 [160]. And the value for the C70 CT complex was about five times larger than that of the C^Q CT complex. Since the CT interaction has a similar effect of introduc­ing electrons to fullerene, it is easily understood that the second-order hyperpolarizability of fullerene is enhanced in a CT com­plex. Also the CT interaction might induce anisotropy in the elec­tronic polarizability of fullerene, which enhanced the delocalized movement of the IT electrons [49]. This may also induce an en­hancement on the optical nonlinearity [11, 49].

Gu and his co-workers [161] have experimentally studied the third-order optical nonlinearities of Dy@C^2^ which has been prepared by an efficient solvent-extraction procedure and char­acterized by negative-ion desorption chemical ionization mass spectrometry, and UV-VIS-near-IR absorption. DFWM measure­ments on the CS2 solution ofDy@Cs2 were performed using 70 ps laser pulses at 0.532 fjum. The magnitude of the second-order hy­perpolarizability 71111 of the Dy@C^2 molecule was (3.0 ± 0.5) x 10~^^ esu at 0.532 fim. The resonant enhancement and the elec­tron transfer from the trapped dysprosium to the C82 cage are sug­gested to bring about its larger nonlinear optical response relative

to the empty fullerene. Their results are in agreement with the the­oretical predictions [150-154, 161].

4. THIRD-ORDER OPTICAL NONLINEAWTIES OF TUBULAR FULLERENE AND CARBON NANOTUBES

4.1. Structure of Fullerene and Carbon Nanotubes

The discovery of C60 molecule has a long and interesting his­tory. The structure of the regular truncated icosahedron was already known to Leonardo da Vinci in about the year 1500 [162,163]. In this century, many theoretical suggestions for icosa-hedral molecules predated their experimental discovery by several decades [164]. In 1933, Tisza [165] considered the point group symmetry for icosahedral molecules. In 1970, Osawa [85] sug­gested that an icosahedral C60 molecule be stable chemically. Later, Russian workers showed that C60 molecule should have a large electronic gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) [166, 167]. Regretfully, all these theoretical suggestions for C o were not widely appreciated. Some of this literature was rediscovered only after the experimental work established the sta­bility of the C o molecule in the gas phase in 1985 [86].

Now it is known that the 60 carbon atoms in C50 are located at the vertices of a truncated icosahedron, where all carbon sites are equivalent. This is in agreement with the experimental observation of a single sharp line in the nuclear magnetic resonance spectrum [168, 169]. If a first approximation is made for C60, this molecule may be considered as a roUed-up graphene sheet because of three reasons [19]: (i) the average nearest-neighbor carbon-carbon dis­tance flc-c in Cgo (^ 1.44 A) is almost identical to that in graphite (^ 1.42 A); (ii) each carbon atom in graphite and in C60 is trigo-nally bonded to three other carbon atoms in an 5/? -derived bond­ing configuration; (iii) most of the faces on the regular truncated icosahedron are hexagons. It is known that a regular truncated icosahedron has totally 90 edges of equal length, 60 equivalent vertices, 20 hexagonal faces, and 12 additional pentagonal faces to form a closed shell, which satisfies the well-known Ruler's the­orem. C60 molecule has two single C—C bonds with bond length ^5, which are located along a pentagonal edge at the fusion of a hexagon and a pentagon, and one double bond with a bond length B^, which is located at the fusion between two hexagons. The sin­gle bond length B^ is 1.46 A measured by nuclear magnetic reso­nance [168-171] or 1.45 A by neutron scattering [172]. The double bond length B^ is measured to be 1.40 A by nuclear magnetic reso­nance [168-171] and 1.391 A by neutron diffraction [172]. As there exists a small difference between ^5 and B^ (B^ — B^^ 0.06 A), the vertices of the Cgo molecule form a truncated icosahedron rather than a regular truncated one. However, in many Uteratures related to the molecule C^Q, the small difference between B5 and B^ is neglected and C60 is thus often called a regular truncated icosahedron.

C o has filled molecular levels because of the satisfaction of the bonding requirements of all the valence electrons. The nom­inal sp^ bonding between adjacent carbon atoms, owing to the closed-shell properties of C60 (and also other fullerenes), occurs on a curved surface, in contrast to the case of graphite where the sp^ trigonal bonds are truly planar. This curvature of the trigo­nal bonds in C60 leads to some admixture of sp^ bonding, which

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 285

is characteristic of tetrahedrally bonded diamond but absent in graphite. Further inspection of the 059 molecular structure tells us that every pentagon of 059 is surrounded by five hexagons, and the pentagon, together with its five neighboring hexagons, has the form of the corannulene molecule [19]. The double bonds in corannulene are in different positions relative to those in C^Q be­cause the edge carbons in corannulene are bonded to hydrogen atoms. Furthermore, the pyracylene [19], which consists of two pentagons and two hexagons, is another molecular subunit on C^Q. Also, the double bonds in pyraclene are in different positions rel­ative to those of C60 because of the edge hydrogens in pyracylene.

A fuUerene [19, 85-91], by definition, is a closed cage molecule, which contains only hexagonal and pentagonal faces. This defini­tion intentionally leaves out possible heptagons, which are respon­sible for the concave parts and are treated as defects. Like any simple polyhedron, a fuUerene cage satisfies Euler's theorem re­lating the number of vertices (carbon atoms: v), edges (covalent bonds: e), and faces (/) [19]:

v-e-\-f = 2 (82)

If the number of pentagons is p and the other (/ - p) faces are all hexagonal, then the doubled number of edges (each edge be­longs to two faces) is5p-\-6(f - p), which also equals the tripled number of vertices (each trivalent carbon is shared by three adja­cent faces). A simple accounting then yields p = 12. Therefore, we conclude that a nice, defectless fuUerene must have exactly 12 pentagons, the same dozen as in the buckyball, and the number of hexagonal faces is arbitrary. The addition of each hexagonal face adds two carbon atoms to the total number N of carbon atoms in a fuUerene, so the number of hexagonal faces in a C v fuUerene can readily be determined and must be even, which is consistent with the observable mass spectra [173]. However, it is energeti­cally unfavorable for two pentagons to be adjacent to each other, since this would lead to higher local curvature on the fuUerene baU, and hence more strain. The resulting tendency for pentagons not to be adjacent to one another is called the isolated pentagon rule [174,175]. The smaUest fuUerene C^ to satisfy the isolated pen­tagon rule is C60. For this reason, fuUerenes with many fewer than 60 carbon atoms are less likely. In fact, no fuUerenes with fewer than 60 carbon atoms are found in the soot commonly used to ex­tract fuUerenes. Although the smaUest fuUerene to satisfy the iso­lated pentagon rule is 050, the next high fuUerene to do so is C70, which is in agreement with the absence of fuUerenes between C60 and C70 in fuUerene-containing soot. With increasing number of carbon atoms in the fuUerenes, the number of distinct carbon sites and bond lengths increases, as does the complexity of the geomet­ric structure. In fact, it is easy to specify all possible icosahedral fuUerenes using the projection method [19], where an icosahedron consists of 20 eqmlateral triangles, each specified by a pair of inte­gers (;, k) such that A in C ^ is given by [19]

and the diameter d of the icosahedron is given by [19]

d = ITT -cic-c

(83)

(84)

where AC-C is the nearest-neighbor carbon-carbon distance on the fuUerene. In comparing the diameters of various fuUerenes, it is useful to use an average value of ac-c, which can be estimated

from detaUed measurements on C60- Then, the diameters of sev­eral icosahedral fuUerenes can be found. The synthesis and isola­tion of higher fuUerenes is now an active research field [19], and it is expected that as the avaUabUity of large amounts of these materials becomes widespread, more extensive properties mea­surements of these higher fuUerenes wiU be carried out. Whereas smaU-mass C« fuUerenes {n < 100) are likely to be single-walled, it is not known presently whether high-mass fuUerenes (n > 200) are single-waUed or perhaps may be multiwalled and described as a fuUerene onion.

In addition to fuUerenes, it is possible to synthesize tubular fuUerenes and nested concentric fuUerenes. The field of carbon nanotube research was greatly stimulated by the discovery of the existence of carbon tubules or nanotubes [176] and the subsequent report of conditions for the synthesis of large quantities of nan­otubes [19, 177-197]. Theoretically, there are three major clas­sifications of C6o-based tubules or nanotubes [19, 198-202], de­pending on whether they are related to fivefold, a threefold, or a twofold axis relative to the C50 molecule. Generally speaking, aU of them consist of a graphene sheet roUed up in one dimension to form a cylinder with top and bottom edges that fit perfectly on to a cap at either end, the caps being formed by appropriately cutting the C60 molecule into half.

To be explicit, first we consider the tube formed along a fivefold axis, which is the easiest to visualize and can be represented by the formula C6o+/xlO' where / is a positive integer. We can think of this tubule as foUows: cutting a C60 molecule into two parts along its equatorial line and then inserting one row of five arm­chair hexagons, one obtains C70. More generally, adding / rows of armchair hexagons, one then obtains a C6o+/xlO molecule (arm­chair tubule [19, 198-202]). Figure 5a [199] shows an example of an armchair tubule. Closely related to armchair tube based on a fivefold axis is C6o+/xl8» which is based on a threefold axis. In de-taU, it is formed as follows: cutting C50 into two parts along the zigzag edges and inserting / rows of nine zigzag hexagons, one gets a C60+/XI8 molecule (zigzag tubule [19,198-202]). Figure 5b [199] shows an example of a zigzag tubule. These tubules are of scientific interest as carbon fibers, which are today commercially important for their extraordinary high modulus and strength.

In addition to the armchair and zigzag tubules, a large num­ber of chiral carbon nanotubes as shown in Figure 5c [199] can be formed with a screw axis along the axis of the tubule and with a variety of "hemispherical"-like caps. These general carbon nan­otubes can be specified mathematically in terms of the tubule di­ameter dt and chiral angle 6, as shown in Figure 6, where the chiral vector Ch [19]

^h = P^l + ci^l (85)

is shown as weU as the basic translation vector T for the tubule. The vector C/j connects two crystallographically equivalent sites O and ^ on a two-dimensional graphene structure. The construction in Figure 6 shows the chiral angle d of the tubule with respect to the zigzag direction {0 = 0) and two units, ai and a2, of the hexagonal honeycomb lattice. An ensemble of possible chiral vectors can be specified by C/ in terms of pairs of integers {p, q) [19, 198-202]. Each pair of integers {p, q) defines a different way of roUing the graphene sheet to form a carbon nanotube. In detaU, the cylinder connecting the two hemispherical caps of Figure 5a, b, c is formed by superimposing the two ends OA of the vector C/ . The cylin­der joint shown in Figure 6 is made by joining the line AB' to the paraUel line OB, where lines OB and AB' are perpendicular to the

286 XIE

chiral angle 0 is given by [19]

Fig. 5. A C60-based tubule is formed by rolling a graphene sheet into a cylinder and capping each end of the cylinder with half of a fuUerene molecule. Here is a schematic theoretical model for a single-wall carbon tubule with the tubule axis normal to: (a) the 6 = 7r/6 direction (an arm­chair tubule), (b) the 0 = 0 direction (a zigzag tubule), and (c) a general direction 0 < ^ < 7r/6 (a chiral tubule). The actual tubules shown in the figure correspond to {n, m) values of: (a) (5, 5), (b) (9, 0) and (c) (10, 5). Reprinted from M. S. Dresselhaus, G. Dresselhaus, and R. Saito, Carbon 33, 883 (1995). Copyright 1995, with permission from Elsevier Science.

Fig. 6. The chiral vector OA or C/ = pai + q2i2 is defined on the honey­comb lattice of carbon atoms by unit vectors aj and a2 and the chiral angle B with respect to the zigzag axis {6 = 0). Also shown is the lattice vector T = OBoi the ID tubule unit cell.

vector C/ at each end. The chiral tubule thus generated has no dis­tortion of bond angles other than distortions caused by the cylin­drical curvature of the tubule. Differences in chiral angle d and in the tubule diameter dt give rise to differences in the properties of the various carbon nanotubes. In the {p, q) notation for specifying the chiral vector C/ , the vectors (/?, 0) denote zigzag tubules, and the vectors {q, q) denote armchair tubules, and all other vectors {p, q) correspond to chiral tubules [19,198-202]. Since both right and left handed chirality is possible for chiral tubules, it is expected that chiral tubules are optically active to either right or left circu­larly polarized light propagating along the tubule axis. In terms of the integers {p, q), the tubule diameter dt is given by [19]

dt = Q / T T = ,l?>{p'^^-pq-Vqh

«c-c (86)

where UQ-C is the nearest-neighbor c-c distance (= 1.421 A in graphite [19]), Q is the length of the chiral vector C/ , and the

e = tan" - l / V3q

q + 2p (87)

For example, the armchair tubule specified by (5,5) in Figure 5a has dt = 6.88 A and 6 = 7r/6, the zigzag tubule specified by (9, 0) in Figure 5b has a theoretical tubule diameter of dt = 7.15 A and 6 = 0, and the chiral tubule specified by (10, 5) in Figure 5c has dt = 10.36 A and 6 = 0.7137, all derived from hemispherical caps for the C^Q molecule and assuming an average AC-C = 1.44 A appropriate for Q Q - Let m be the largest common divisor in p and q. Then the atom number n per unit cell is equal to [198-202]

4(p^ + pq-\-q^)

m (88)

(89)

if (p - q) is not a multiple of 3m or [198-202]

4(p2 + pq^ q^)

3m

if ip - q) is a multiple of 3m. Because of the small diameter of a carbon nanotube (^ 10 A) and the large length-to-diameter ratio (>10'^), carbon nanotubes provide an important system for studying one-dimensional physics, both theoretically and experi­mentally. Especially, they have been recognized as a fascinating material about to trigger a revolution in nanodevices, optical com­puting, optical communication, carbon chemistry, and new func­tional structural materials, generating intense research activities in recent years [19,181-184].

4.2. Model of Fullerene and Carbon Nanotubes

The Su-Schrieffer-Heeger (SSH) model [203] has been success­fully applied to describe conducting polymer and Q Q [150, 151, 204-206]. Recently, it has been shown that the Coulomb interac­tion effect plays an important role in a physical understanding of the optical absorption spectra and third-order optical nonlineari-ties of C^Q, C70, C76, C78, C84, and carbon nanotubes [132-135, 207-210]. Therefore, with the Coulomb interaction included, we have extended the SSH model to describe higher fuUerenes and C60-based nanotubes [122-124, 130,131,152-154, 211]. The total Hamiltonian can be written as follows [122,123]:

^ = ED-^o-«oy.i)(ic,, + h.c.) + X:> -(ij) ' iij)

+ uoJ2 ^U^iA'^li'^iA + ^0 Z ! Z ) is^hscl,^''j,s' (90) i (ij) s,s'

where the sum {/)) is taken over the nearest neighbors for the c—c bond; tQ represent the hopping integrals for the c—c bond; ag is the electron-phonon coupling constants related to the c—c bond; ko is the spring constants corresponding to the c—c bond; y/y is the change of the bond length between the ith and ;th atoms; the op­erator c/5 (cj ^) annihilates (creates) a TT electron at the ith atom with spin s (s = t , i ) ; WQ is the usual on-site Coulomb repulsion strength; VQ is the Coulomb interaction between the nearest and next-nearest atoms. Using the Hartree-Fock approximation, we are able to transform the preceding equation into [123]

H = E E ( - ^ o - ao>'//)(4,cy,.+h.c.) + J2yl m (ij)

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 287

i ^ s

{ij) ' ' s

- Pi.s E Pj.s' - ^ijAs'hs + rls) (91) ("« - + ^)("«/ + ^^^("-^ + ^^^

" The other terms are given in detail as follows [127]:

(99)

where Pi,s = {clci,s) (92) y3i-3<o;co,co,co)

is the electron density and a,r,seunocc teocc

Tij^s = [c]^Cj^s) (9^) where ^3(6;) is analogous to 52(a>) with the substitution of e y by ej-i and e^^ by e^/;

is the bond order parameter. This equation is solved by the adi-abatic approximation for phonons. The Schrodinger equation for 7i(-3ft>; w, co, w)

the 77 electron is [123] = - 2 ^ ^ M,aMa./^;7Mr>5i(a,) (101)

^^Z/t,. = E ( - ' - "Oyy - V^Tij,,)Zk,s{}) a,reunoccij^occ

(^y) where 5*1 (cu) is analogous to 52(w) with the substitution of e^j by

+ U P , > + « 'OEEP;> ' ]2M(0 (94) ^ri^-de^kbyerj; s' j y^(—3(o; 0), o), (o)

where ej^ is the /:th eigenvalue. The self-consistent equation for = - 2 ^ E P'iaP'jif^arlJi>rj^4(<^) (102) the lattice is [123] ajeunocc ijeocc

. . ' 'y7(—3ft>; ct), w, w)

^^,5(^)^^,5(0 [ (95) (m/)

where 54(w) is analogous to 52(tt>) with the substitution of e^^ by

1 r7(-36>; ct), o), (o) - n - 1 E ^ M ( ^ ) ^ M ( ^ ) (95) 9 Y - Y - w ^ nn^^

'^ a,reunoccij^occ where the prime denotes the sum over all occupied states, the sec­ond term originates from the constraint condition [123] ^here Sjioj) is analogous to ^2(0)) with the substitution of 6«y by

€ai + €rj and €ak by e y

E>'(/ = 0 (96) , , ^ r8(-3w; 0), 0), (o)

and n is the number of ir bonds. Then, by performing the self- = " ^ E E P'ial^rjt^ajP^riS^M (104) , . , , . a,reunocc / /gocc

consistent mteraction, we are able to obtam the electron eigen- ' states zyt,5(0, eigenenergies e , and bond variables ytj. where 53(w) is analogous to 52(w) with the substitution of e y by

The third-order nonlinear optical polarizabiHty y can be ex- e^/+e;.y and e^j, by e .,. The double excitation channels 77 + 73 can pressed as follows. Within the independent electron approxima- be cast into a single term after simple algebra by noting e«/ + e^j = tion and the SOS approach discussed by Orr and Ward [212], the ^ . _ ^ . [127]: second-order hyperpolarizability 7 for the THG process can be rewritten as [see Fig. 2 of the literature [213], taking into account ^^ + ^g = _2 V" V" fXiafJirj/HajHr six diagrams in total] [127, 213-215] a,reunocc ij^occ

7(-3w; 0), 0), cu) = 72 + 73 + 71 + 74 + 77 + 78 (97) f 1

where 7/ is given in detail as [127] I ( « ' " ^^)^^aj - o))(^ri - ^ )

72(—3co; cu, CO, (o) +

= 2 E E P^aiNjP'jkP'kaSlM (98) aeunocc. ij^keocc. +

(e^/ + w)(e^y + ft;)(e^y + 3w)

1

(e^/ + w)(e^y + o))ieri - (o)

i, 7,and k denote occupied ones. The ^2(6^) function is defined -\ I (105) Hereafter, a, r, and s denote unoccupied molecular levels, and /, 7,and k denote occupied ones. The ^2(6^) function is defined as [127] (eai-ho)){€aj-(o)ieri-o))

^ 1 In the preceding formulae, €p is the one-electron energy, €pn(= ~ {€ai - ?>o)){eaj - 2o)){eak - o)) ^P~ ^n) is the transition energy, and (Xmn is the transition matrix

288 XIE

element between one-electron states [127]:

l^mn = Y^Z*^s(j)(-er)Znis(j) (106) js

In the numerical calculations, a lifetime broadening factor rj (= 1.6 X 10~^ eV) is included. Since the ratios between different com­ponents of y are not known, a spatial average of y is given by [126]:

yxxxx + yyyyy + Jzzzz + ^Jxxyy + "^Jyyzz + ^yzzxx 5 y = (107)

4.3. Third-Order Optical Nonlinearity of Tabular FuUerenes

On the basis of the electronic structure obtained in the earUer mentioned ESSH model, one can calculate the second-order hy-perpolarizabihties of armchair and zigzag tubular fullerenes. In the actual calculation, the z axis in armchair tubular fullerenes is taken along the direction from the bottom pentagon to the top one, and the z axis in zigzag tubular fullerenes is taken along the direction from the bottom hexagon to the top one. If the length of every bond is given, the coordinates of every atom will be ob­tained. However, we find that the y magnitude is not sensitive to small changes of atomic coordinates. So, for simplicity, we take the same bond length (= 1.4225 A, which is the average bond length of C70 given by the experiment [216]) for all tubular fullerenes. The Coulomb interaction is assumed to be UQ = 2i;o = t, which is not strong, so the effective hopping integral is the same as that in the free-electron case. Since three parameters, tQ, aQ, and k^, in the ESSH model do not sensitively depend on the shape and size of the molecule [142, 217], we let tQ = 2.5 eV, ao = 6.31 eV/A, and kQ = 49.7 eV/A^, which are the same as those in €50 and C70.

The theoretical static y magnitudes of several armchair and zigzag tubular fullerenes and €50 are listed in Table VII [211]. For C60J the difference between z and x (or y) components of y is very small because C50 is almost a sphere. But this difference is pronounced for large tubular fullerenes, which contain a large number of carbon atoms. Wang and Cheng [49] reported that the nonresonant y magnitudes of C60 and C70 are 0.75 x 10" - esu and 1.3 X 10~^^ esu, respectively. Our theoretical calculations for C60 and C70 are in agreement with their experimental measure­ments. In Table VII, we have listed the theoretical results of C70 obtained by Shuai and Bredas [127]. Our numerical results are also consistent with theirs.

In our numerical calculation, we found that when the carbon number of tubular fullerenes increases, the interval between the neighboring level decreases and the dipole matrix, at the same time, increases. So the static y magnitude of tubular fullerenes may be enhanced. Table VII shows that y of C^Q+QXIO is about 14 times larger than that of C70.

Because ir electrons in the fullerenes C^ with higher symmetry are more delocalized, their static y magnitude will be larger than that containing an identical carbon number but with lower symme­try. Now we define a quantity [130,131, 211]

^Q = N

1 (108)

to measure the degree of 7r-electron distribution of fullerenes de­viating from its homogeneous distribution, where [130,131, 211]

7>

(109)

Table VII. SSH-SOS Theoretical Static y (in the unit of 10"^^ esu) Tensor Components for C70, C6o+9xio» Cgo+SxlS? ^ id C144 (uncapped zigzag tubule). The Mark # Denotes the VEH-SOS Theoretical Results

of C70 by Shuai and Bredas

C60 -70 --70 ^60+9x10 C6o+5xl8 C144

yxxxx 0.048 0.051 0.05163 0.133 0.148 1.413 yyyyy 0.048 0.051 0.05163 0.133 0.148 1.413 yzzzz 0.053 0.063 0.08165 3.12 2.783 25.668 yxxyy 0.013 0.016 0.01411 0.046 0.05 0.447 yyyzz 0.027 0.031 0.05435 0.194 0.062 1.323 yzzxx 0.027 0.031 0.05435 0.194 0.062 1.323 y 0.056 0.064 0.08623 0.85 0.686 6.976

Source: Reprinted with permission from R. H. Xie and J. Jiang, /. Appl. Phys. 83, 3001 (1998). Copyright 1998, American Institute of Physics.

is the TT-electron density at the /th site. As an example, our calcu­lated results show AQ = 0.12, 0.18 for 050+9x 10 ^^^ Qo+5xl8» which have 2)5/ and D2,h symmetries, respectively. This means that 77 electrons in 059+9 x 10 ^^^ more delocalized than those in C5o+5xl8- ^^^ numerical results show that the static y of ^60+9x10 is 1.2 times bigger than that of C60+5xl8-

We have also studied the trends of the static second-order hy-perpolarizabilities in armchair and zigzag tubular fullerenes. In Figure 7 and Figure 8 [131], we show the relationship between the carbon number U and the static y magnitudes of 18 armchair tubular fullerenes (C^Q+ZXIO' ^ = 1, 2 , . . . , 18) and 10 zigzag tubu­lar fullerenes (C6o+/xl8» ' = 1,2,..., 10), respectively. It is seen that the static y magnitude of tubular fullerenes is enhanced when the carbon number is increased. Moreover, fitting these data, we found that the static y magnitudes of armchair and zigzag tubular fullerenes obey their own exponent laws given by [131]

R = v60

and

^ = 70 =

/ i X 10 \

/ i X 18 \ 2.98

(110)

(111)

respectively, where y^^ is the static y magnitude of C6o-Various studies have shown that the nonresonant y magnitude

of C60 or C70 is about 10" - esu, which is several orders of magni­tude smaller than those needed in photonic applications. Accord­ing to the preceding theoretical calculation, we see that the static y magnitude of C60+i8xl0 is about 80 times larger than that of C60 or C70. Then, using the above exponent laws, we predict that the static y magnitudes of armchair and zigzag tubular fullerenes with length 40 A are about 10" ^ esu, which is an appropriate value needed in photonic devices. Moreover, comparing their y magni­tudes with those of polyenic polymers, we find that armchair and zigzag tubular fullerenes can compete with polymers for NLO ap­plications because of their large third-order optical nonlinearities.

Next, the dynamical NLO responses of armchair and zigzag tubular fullerenes are studied in detail. The calculated results are shown in Figure 9a, b, c [211], and some details of the first peaks are listed in Table VIII [211].

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 289

uu

80

60

40

20

n

"T 1 1 —I r 1 1 r

^^'<r-^"^ 1 1 1 I 1 J_

/

•I 1"

oJ

/ j J

-J U 60 80 100 120 140 160 180 200 220 240

U

Fig. 7. The relationship between the carbon number U and the ratio R for 18 armchair nanotubes Cu (f/ = 60 + / x 10, / = 1,2,..., 18), where the dashed Une is the fit result. Reprinted with permission from R. H. Xie, /. Chem. Phys. 108, 3626 (1998). Copyright 1998, American Institute of Physics.

uu

80

60

40

20

(\

-\ 1 1 1 T 1 1 r 1 1

A y 1

1 X

," ^^.'<> H

^..''''O

^ -•"• ' --'-^ o--• 4f -J 1 1 L 1 . 1 1 U

60 80 100 120 140 160 180 200 220 240 U

Fig. 8. The relationship between the carbon number U and the ratio R for 18 armchair nanotubes Cy (C/ = 60 + / x 18, / = 1,2,..., 10), where the dashed line is the fit result. Reprinted with permission from R. H. Xie, /. Chem. Phys. 108, 3626 (1998). Copyright 1998, American Institute of Physics.

Table VIII. The Optical Frequency Positions 3w, y Magnitudes, and Resonant Transition Energies e^b (= e - e^) of the First Peaks Shown in Figures 9a, b, and c, where 3w, y, and e^b are in Units of eV, 10~^^ esu, and eV, Respectively, and Eg (in units of eV) is the Energy Gap of

the Corresponding Tubule

Cn

C70

C60+9>

C60+5>

10

18

Eg

2.08

1.16

1.18

3co

2.27

1.36

1.28

y

3.41

15.6

11.4

^ab

^36,34 =

^77,75 =

^76,74 =

2.27

1.35

1.25

3 " 30 CO

^ 20 0

^--^ 10

^-

0

15

(0 10 CD

0

0

E ' >~

:

\ 1

: AI 3 1 2 3

\ ^^^

1 11 J 1 J 4 5 6 7

3(D(eV)

':

J

1 ''M

l i v , 0 1 2 3 4 5 6 7

3C0(eV)

3 0)

o

5-

Source: Reprinted with permission from R. H. Xie and J. Jiang, /. Appl. Phys. 83, 3001 (1998). Copyright 1998, American Institute of Physics.

3C0(eV)

Fig. 9. The second-order hyperpolarizability 7 spectra for armchair and zigzag nanotubes. (a) C70; (b) C6o+9xlO, and (c) Ceo+SxlS- Reprinted with permission from R. H. Xie and J. Jiang, /. Appl. Phys. 83,3301 (1998). Copyright 1998, American Institute of Physics.

It is seen from Table VIII that the first peaks appearing in Figure 9a, b, c are located at 3w = 2.27,1.36,1.28 eV, respec­tively, and their corresponding y magnitudes are 3.41,15.6, and 11.4 X 10~^^ esu. Then, our further studies have shown that the first peak in the y spectrum of tubular fuUerenes is caused by a three-photon resonance between two energy levels (e^ and e/,) near Fermi levels with one in the conduction band and the other in the valence band, namely, e^^ ^ 3w.

Moreover, Figure 9a, b, c show that the highest peaks in the y spectra of C70, C6O+9X10J and C6o+5xl8 ^^^ located at co ^ 1.21,1.12, and 1.38 eV, respectively, and their corresponding ymax magnitudes are about 34, 1344, and 660 x 10 esu. Actually, these peaks are caused by one-, two- and three-photon resonance enhancement (OTTPRE) processes. As an example. Figure 10 shows the Feynman diagram of the OTTPRE process of ymax in Qo+5xl8- The physical meaning of the Feynman diagram is obvi-

290 XIE

Table IX. The Resonant (the first three cases) and Nonresonant (the last two cases) y Magnitudes of C70 at Different Wavelength A (optical

frequency (o). The Words "The" and "Exp" Mean Theoretical and Experimental Results, Respectively. A, (w, and y are in Units of fim, eV,

and 10~^^ esu, Respectively

Fig. 10. The Feynman diagrams of the OTTPRE processes of ymax of Qo+5xl8 shown in Figure 9c. The number n in the electron loop repre­sents the nth energy level of single electron. Reprinted with permission from R. H. Xie and J. Jiang, /. Appl. Phys. 83, 3301 (1998). Copyright 1998, American Institute of Physics.

ous: initially, an electron at the 75th level absorbs a photon with energy (o and transfers to the 77th level, then to a virtual level and to the 94th level after absorbing a photon with energy (o, consec­utively, and finally, the electron at the 94th level transfers to the 75th level and emits a photon with energy 3w. The y magnitude coming from this Feynman diagram is 404 x 10~^^ esu, which is two-thirds of the total ymax magnitudes.

When the carbon number in tubular fuUerenes is large enough, the intervals between neighboring energy levels decrease, and thus a lot of OTTPRE processes are observed. But, only those transi­tion processes that occur between the energy levels near Fermi levels contribute large y magnitudes because of their large dipole matrix and small transition energies. Figure 9b, c shows that for Qo+9xlO ^^^ Qo+5xl8» the major response peaks with larger y magnitudes concentrate on a narrow region, where the optical fre­quency is near the energy gap, namely, ?>o) ^ 3Eg.

According to the above numerical results, it is obvious that large resonant y magnitudes of armchair or zigzag tubular ful-lerenes could be obtained in the infrared region by increasing their carbon number, for example, for those containing 150 car­bons, ymax ^ lO"-^^ esu at the fundamental wavelength A = 1.1 - 1.3 fjLxn, which is two orders of magnitude larger than that of C70. Naturally, these properties make these tubular fullerenes become important NLO materials for photonic applications.

In Table IX [211], we list recent experimental and our the­oretical results of C70. It is seen that our resonant y magni­tudes at w = 1.97,1.08, and 0.76 eV, and nonresonant ones, 0.0064 X 10~^^ esu < y < 0.47 x 10~^^ esu at the range of 0 < (0 <0.1 eV, are in agreement with the resonant and nonreso­nant experimental measurements by Wang and Cheng [49], Flom et al. [106], and Neher et al. [104]. So the model parameters cho­sen earlier and our theoretical predictions are reasonable.

Aexp(^exp)

0.59 (2.06)^

1.06 (1.16)^

1.50 (0.82)^

1.91 (0.64)^

2.0 (0.62)^

^the

1.97

1.08

0.76

0.70

7exp

%20

5.75 lb 0.4

2.2 ±1 .5

0.13 d= 0.03

0.38 ±0.11

rthe

19.57

3.47

3.41

0.47

^Four-wave mixing measurement by Flom et al.

^Third-harmonic generation measurement by Nether et al.

^Electric-field-induced second-harmonic generation measurement by Wang and Cheng.

Source: Reprinted with permission from R. H. Xie and J. Jiang, /. Appl. Phys. 83, 3001 (1998). Copyright 1998, American Institute of Physics.

Experimental studies have shown that in the synthesis of car­bon nanotubes, their caps may be destroyed partially or com­pletely [19, 181, 182]. Such effects will greatly influence the geo­metric and electronic structures of carbon nanotubes. The change of these structures will have a large effect on the NLO proper­ties of carbon nanotubes. Therefore, it would be very interesting to discuss the cap effect on the third-order optical nonlinearities of tubular fullerenes. As an example, we study the static y mag­nitude of uncapped zigzag tubular fullerene C144. The results are shown in Table VII [211]. We find that its static y magnitude is about ten times larger than that of capped zigzag tubular fullerene ^60+5x18- Compared with the symmetry effect discussed above, the cap effect is more obvious. The reason is as follows. It is known that the NLO response for the C^Q system is mainly produced by delocalized IT electrons as in conjugated polymer chains. How­ever, the three-dimensional character of C60 causes severe limita­tions on its NLO property and thus make its y magnitude become about two orders of magnitudes smaller than those of linear poly­mers containing a similar number of carbon atoms. For a capped tubular fullerene, a TT electron on a site can transfer to the site's three neighbors. If both caps are cut, a ir electron on the site at the edge of a cylinder can only transfer to the site's two neighbors. This kind of edge effect will reduce the effective space dimension of 77 electrons and thus enhance the static y magnitude of tubular fullerenes.

In conclusion, we have studied the third-order optical nonlin­earities (characterized by the second-order hyperpolarizabilities y) of tubular fullerenes. It is found that the static y magnitudes of armchair and zigzag tubular fullerenes obey their own expo­nent laws given by y/y^^ = (1 + / x 10/60)^-1^ and y/y^^ = (1 + / X 18/60)^-^^, respectively, where y^^ is the static y magni­tude of C60 and / is a positive integer. Also, the dynamical nonlin­ear optical responses of armchair and zigzag tubular fullerenes are studied in detail. It is found that large third-order optical nonlin­earities in those fullerenes could be obtained in the infrared, and the symmetry and caps of these fullerenes have a large effect on their third-order optical nonlinearities. By these detailed studies, we find that armchair and zigzag tubular fullerenes can compete

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 291

with polymeric materials for photonic applications because of their large nonresonant and resonant third-order optical nonlinearities.

4.4. Third-Order Optical Nonlinearity of Chiral Carbon Nanotubes

On the basis of their experimental observations, lijima and co­workers [178] claimed that most of the single-walled carbon nan­otubes show chirality. Using a similar technique to lijima's, Dravid et al. [179] also found that most of their carbon nanotubes have a chiral structure. Therefore, it is very interesting to examine the chiral effect on the third-order optical nonlinearities of chiral graphene tubules in the view of their practical application.

Here one can study chiral carbon nanotubes with finite atom numbers N. In these cases, the tubule edge effects cannot be ne­glected. A finite chiral carbon nanotube with one or several unit cells is open with a row of dangling bonds at each end. So an atom at an edge site may have fewer than three neighbors. However, for the real carbon nanotubes, the tubule length is long enough to ne­glect the edge effect. Taking these into account, we have applied periodic boundary condition for the tubule axis, and each carbon atom at the end of the finite chiral carbon nanotube can still find its three neighbors by imagining that the two ends of the tubule are connected. In Figure 6, a unit cell is defined along the tubule axis, and C/j and T = OB construct the basis vectors of the unit cell, where B is the first lattice point of the 2D graphitic sheet through which a line through O and perpendicular to C^ passes. The length ofT is [198-202]

|T| = —-^ (112)

Table X. The Static y Value and Average Contribution T of 17 Chiral Graphene Tubules

m if (/?, q) is not a multiple of 3m and

|T| = Ch (113)

if (;? - ^) is a multiple of 3m. On the basis of the electronic structures obtained in the ESSH

model mentioned earUer, we calculate the static y magnitudes of 17 chiral carbon nanotubes, which have different diameter dt and chiral angle 0. Then, we calculate the average contribution F of one carbon atom to the third-order optical nonlinearity of a chiral graphene tubule

r = N (114)

where A is the total atom number in the studied tubule. To com­pare our results with that of C6o, we also list the static y and T value of C60 and the ratio r

£1 f60

(115)

where T^ and T^^ denote the F values of chiral tubules and C6o, respectively. We note that for carbon nanotubes with smaller di­ameters than that of C6o, there are no caps containing only pen­tagons and hexagons that can be fit continuously to such a small carbon nanotube (p, q). For this reason, it is expected that the ob­servation of very small diameter (< 7 A) carbon nanotubes is very unlikely [19, 181, 182]. For example, the (4,2) chiral vector does not have a proper cap and therefore is not expected to correspond to a physical carbon nanotube. Therefore, in the view of practi­cal application, we pay our attention to physical tubules including (6,5) tubule that is the smallest diameter chiral tubule. In detail, our numerical results are listed in Table X. Meanwhile, we have

ip,q)

C60 (6,5)

(9,1)

(7, 4)'"

(8,3)^

(9,2)^

(7,5)^

(10,1)'"

(8,4)^

(9,3)^"

(10,2)^

(7,6)^

(8,5r (9,4)^

(10,3)^

(8,6)^

(9,5)^

(10, 4)^"

n

—

364

364

124

388

412

436

148

112

156

248

508

172

532

556

296

604

104

A

60 364

364

372

388

412

436

444

448

468

496

508

516

532

556

592

604

624

dt(A)

7.10

7.47

7.47

7.56

7.72

7.95

8.18

8.25

8.29

8.47

8.72

8.83

8.90

9.03

9.24

9.53

9.68

9.79

d

—

0.4711

0.0909

0.3674

0.2669

0.1715

0.4276

0.0822

0.3334

0.2425

0.1561

0.4792

0.3911

0.3050

0.2221

0.4413

0.3601

0.2810

7(10"^^ esi]

0.5612

6.3556

5.1997

22.9996

5.4013

5.7198

6.0172

25.0398

6.1766

22.1879

6.7049

6.8504

18.1596

6.9155

6.8811

6.8157

6.7642

14.8830

i) FdO-^^esu)

0.9353

1.7461

1.4285

6.1827

1.3921

1.3883

1.3801

5.6396

1.3787

4.7419

1.3518

1.3485

3.5193

1.2999

1.2376

1.1513

1.1199

2.3851

r

1.0

1.8669

1.5273

6.6104

1.4884

1.4843

1.4756

6.0297

1.4741

5.0699

1.4453

1.4418

3.7627

1.3898

1.3232

1.2309

1.1974

2.5501

^s and m in {p, qY and {p, q)^ denote semiconducting and metallic tubules, respectively, n, N, dt, and 6 are the atom number per unit cell, the total atom number calculated, the diameter, and the chiral angle of a chiral graphene tubule, respectively.

also given the corresponding diameter dt, the chiral angle 0, the atom number n per unit cell, and the total atom number N in the calculated tubule from which we may examine the effect of size, chiral angle, and diameter on the third-order optical nonlineari­ties of chiral graphene tubules. In Table X, s and m in (p, qY and (p, q)^ denote the semiconducting and metallic tubules, respec­tively.

It has been shown that the optical nonlinearities of fuUerenes will decrease with the increasing of space dimension [19,122-124, 130,131,150-154, 211]. For example, 3D C6o molecule possesses smaller y values than ID conducting polymer with the same atom number [19]. The substitute doping effect reduces the effective space dimension of fuUerenes and thus their optical nonlineari­ties are greatly enhanced [150-154]. Here we see that the ID chi­ral graphene tubule will greatly become a 2D graphite sheet with the increase of its diameter. So it is expected that the smaller the diameter of a chiral graphene tubule, the larger the y value of the chiral graphene tubule is available. Table X tells us that the F value for semiconducting graphene tubules (p, qY increases with decreasing diameters, that is, the average contribution of one car­bon atom to the third-order optical nonlinearity of a chiral semi­conducting graphene tubule is gradually enhanced with the de­crease of its diameter. Similar conclusion is got for metallic chiral tubules (p, q)^.

From Table X, we know that both (6,5)^ and (9,1)^ tubules have the same diameter (dt = 7.47 A) and the same atom number per unit cell (n = 364) as well as the semiconducting properties. The only difference between both tubules is the chiral angle 6[6 = 0.4711 for (6,5)^ tubule and 6 = 0.0909 for (9,1)^ tubule]. In this case, if the total atom number N is the same, both tubules will

292 XIE

have the same height but the length of the base helix of (6,5)"^ tubule is shorter than that of (9, l)- tubule. Thus, the atoms in (6, 5)^ tubule are situated along a more straight line than those in (9, l)" tubule and thus (6,5)"^ tubule has lower space dimension than (9,1)^ tubule. Therefore, we expect (6,5)"^ tubule possesses a larger y value than (9, l)" tubule. From Table X, we find that the r value of (6,5)"^ tubule is bigger than that of (9,1)^ tubule, that is, the y value of (6,5)*^ tubule is larger than that of (9,1)^ tubule if their total atom number N is the same.

The periodic boundary conditions for the ID graphene tubules permit only a few wave vectors to exist in the circumferential di­rection [19]. If one of these passes through the zone corner in the Brillouin zone, then metallic conduction results; otherwise, the graphene tubule is semiconducting and has a band gap. Recently, it has been shown that when the total atom number N of carbon nanotube is increased greatly, a large energy gap for a semicon­ducting graphene tubule is still available, but the energy gap for a metaUic graphene tubule approaches zero [19]. In this case, it is expected that if the total number N of carbon nanotubes is the same, a metallic tubule possesses a larger y value than a semicon­ducting one. It is seen from Table X that the F value of a metallic graphene tubule (/?, qf^ is larger than that of a semiconducting graphene tubule (p^qY. Finally, comparing the F value of C^Q with that of carbon nanotubes, we see that carbon nanotube has a larger F value than Q Q . This means that carbon nanotubes pos­sess a larger NLO response than C^Q.

Well-characterized conjugated 7r-electron organic systems are important materials for nonlinear optics because their typically large third-order optical nonlinearities makes them likely candi­dates for components of technological devices. As an example, in Table XI, we list the experimentally derived y values and the average contribution F of a well-characterized polyenic polymer for seven different molecular weights [77]. Comparing their F val­ues with our calculated ones for 17 chiral graphene tubules, one may find that chiral graphene tubules also predict much higher NLO responses and can compete with polyenes for nonlinear op­tical applications. Since chiral graphene tubules are uniquely com­posed of carbon atoms and therefore do not have any residual in­frared absorption that the polymeric materials possess due to over­tones of C—H stretching vibrations, they will be ideal candidates among all third-order materials for photonic applications. Dur­ing the recent years, large quantities of single-walled [188-197] and multiwalled [19, 181-184] carbon nanotubes have been pro­duced by experimental researchers. Despite this progress, a num­ber of physical properties of single- or multiwalled carbon nan­otubes have not been examined carefully so far. In particular, this applied to the nonlinear optical properties of carbon nanotubes, which determine the nonlinear dependence of the polarizability of carbon nanotubes on the intensities of incident electromagnetic waves. In the view of photonic applications, experimental studies on the NLO properties of carbon nanotubes are expected to be performed.

Ye and his co-workers [218] have studied the third-order op­tical nonlinearity of carbon nanotubes by using the technique of backward DFWM. The light source was a Nd: YAG laser with a 30-ps-wide single pulse output or a Nd:YAG laser with an 8-ns-wide single pulse output. The wavelengths used in the experiments for each laser were 1064 and 532 nm. The magnitudes for the tensor component A^nii of their solutions and solvent were measured via a comparison with that of reference sample CS2. First, the typi­cal absorption spectrum of carbon nanotubes, after they removed

Table XL Seven Different Molecular Weights (MW) with the Experimentally Derived y Values of a Well-Characterized

Polyenic Polymer

M W

2800

4100

5400

7500

10000

17600

27900

A

230

340

450

620

830

1460

2320

7(10-3^ esu)

3.5

5.4

7.8

15.5

26.7

62.9

85.4

r(10-3^ esu)

1.5

1.6

1.7

2.5

3.2

4.3

3.7

^r is the average contribution of a carbon atom to the third-order optical nonlinearity of polymer. The number N of carbon atoms is obtained from the molecular weight by dividing by 12 and rounding up the result to the closest integer multiple of 10.

the contribution of the solvent, was measured with one of their solutions. Ye and his co-workers found that there exists some ab­sorption at both 1064 nm and 532 nm [218]. Then, with the 30-ps laser, A im was obtained for their solutions and solvent at both

1064 nm and 532 nm. They observed that Afim of all their so­lutions was larger than that of the solvent, especially that a lin­ear concentration dependence of Afim exists for their solutions at both 1064 nm and 532 nm. By subtracting Afiin of the solvent from

that of their solutions, Af ^^ of carbon nanotubes in different so­lutions can be obtained. Typical results for one of their solutions at 1064 nm and 532 nm are 6.460 x IQ- " esu and 6.303 x lO- " esu, respectively [218]. The results of similar experiments by using an 8-ns-wide laser at 1064 nm and 532 nm are 1.179 x 10~^^ esu and 0.309 X 10~^^ esu, respectively [218]. Because it is very difficult to know the exact number of carbon nanotubes solved in their solu­tion because of the difficulty in obtaining carbon nanotubes with the same size, they could not calculate the third-order optical non-linearity of a single nanotube. But instead, from the mass of carbon solved in the solution, they evaluated the average contribution of one carbon atom to the third-order optical nonlinearity of carbon

(3) nanotubes. As shown before, one result of A'lin of carbon nan­otubes at 1064 nm is about 6.460 x 10" " esu, and considering the local field correction, it was found that the average contribution T per carbon atom to the third-order optical nonlinearity of carbon nanotube is about 0.6 x 10"^^ esu [218]. For €50 whose y m i is 3.0 X 10" " esu at 1064 nm [100], the average contribution F per atom to the third-order optical nonlinearity of €50 is 0.5 x 10~^^ esu. The average contribution of one carbon atom in carbon nan­otubes is more than that in C^Q. Enhancement of the third-order optical nonlinearity occurs in carbon nanotubes. This conclusion is in agreement with the theoretical prediction mentioned earlier.

In summary, the third-order optical nonlinearities of chiral graphene tubules are also studied. The average contribution F of one carbon atom to the third-order optical nonlinearity of each chiral graphene tubule is calculated and compared with that of a well-characterized polyenic polymer. It is found that the smaller the diameter of a chiral graphene tubule, the larger the average contribution F; the metallic chiral graphene tubule favors larger y values; chiral graphene tubules can compete with the conduct-

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 293

ing polymer achieving a large y value that is needed for photonic applications.

5. THIRD-ORDER OPTICAL NONLINEARITY OF DOPED FULLERENES

5.1. Model of Doped FuUerenes

Doped fullerenes [19] and carbon nanotubes [219] have stimu­lated a great interest of researchers in physics and chemistry to in­vestigate their structural, electronic, optical, and other properties. Besides the alkah metal doping, there is another type of doping, named substitute doping (SD), that is, substituting one or more car­bon atoms of fullerenes and carbon nanotubes by other atoms. For example, boron and nitrogen atoms have been successfully used to replace carbon atoms of fullerenes [19] and carbon nanotubes [219]. Available studies [143-154] have shown that the lattice and electronic structures of fullerenes change with substituted doping; the bandgaps between the HOMO LUMO and the electronic po­larization of the substituted fullerenes vary greatly with different SD; the distribution of TT electrons on the surface of fuUerene is changed due to the SD effect; the original delocalized TT electrons in pure fuUerene become more localized around the substituted atoms. These factors have also a large effect on the NLO proper­ties of carbon tubules as well as fullerenes. Therefore, it would be interesting and useful to investigate theoretically the SD effect on the NLO properties of tubular fullerenes and chiral carbon nan­otubes from the view point of practical apphcation. In this section, we pay our attention to the doping effect on the third-order optical nonlinearity of tubular fullerenes.

The ESSH model has been used to describe fullerenes and car­bon nanotubes. But it should be modified to include the effect of the dopant ions in order to describe the substituted tubular fullerenes and carbon nanotubes. Here we named this model as doped ESSH (DESSH) model. In detail, the total Hamiltonian for the single substituted tubules can be written as [152]

H = Hi%+H^^l^ (116)

^0 V^ „2 H^c = EE(-'0-«0)'(/)(ic,;, + h.C.) + |X:^S

(y) (y>

(117) (y) s,s'

4^^c = ED-'i-«i3'y)(icy, + h.c.) + ^ i : 4 (y) <y)

(118) {ij) s,s'

where X denotes the substituted atom, and the sum {/;) is taken over the nearest neighbors for both the c—c and X—c bonds. tQ (or ti), aQ (or a^), and k^ (or ki) represent the hopping integrals, the electron-phonon coupling constants, and the spring constants corresponding to the c-c (or X - c ) bonds, respectively, MQ ^^ "i is the usual on-site Coulomb repulsion strength, and VQ or vi is the Coulomb interaction between the nearest and next-nearest atoms.

Because there is only one substituted impurity atom (i.e., X) in tubules, H^L^ plays a perturbational role, and as an approxima­tion the original empirical parameters (^Q, ao» ^0» "0» ^O) ^^ c—c are assumed not to change because of the substitute doping (taken to be the same in this numerical calculation as those in the pure cases studied in the preceding section), wj ^ UQ, and vi ^ VQ-

As before, we use the Hartree-Fock approximation to trans­form preceding equations into [152]

H - 7:/(0) . 1/(1) (119)

Hj) ' Hj)

+"0 E ( E Pi,s^ls^i,s - Pi^PUi ) i ^ s ' /

+^oEE(E^y>^45^^^

-Pi,s E Ph^' - ^ijA^s'^hs + ^l,s) (120) s'

4 ^ = EE(-^i-i^^y)(i^;>+h.c.) + ^ i : 4 m ' iij)

+"1 E ( EPi,sclsCi,s - PiAPiA )

+^IEE(E^7>^4^'^^> {ij) ' ^ s'

where

(121)

(122)

(123)

Pi,s = (<^ls^i,s)

is the electron density, and

is the bond order parameter. This equation is also solved by the standard adiabatic approximation method, which leads to the Schrodinger equation for the TT electron [152]

(ij)

+ s' j

iij)

+ WlP/,5 + ^1 (0 (124)

where y^^j ^ and y^^. ^ denote the yij for c-c and X-c bonds, re­spectively; e^ is the eigenvalue of the kth eigenstate; Z^ ^ is the electronic wave function. The total energy of the system is a func­tional of the set of yij [152]:

(y) (y>

294 XIE

Table XII. The Energy Gaps Eg between LUMO and HOMO and the Maximum Atomic Distortions 8 from the C50 Calculated with the

Theoretical Method of Kurita et al. (denoted by *) and the DESSH Model

C59N C59N* C59B C59B*

Eg 8

0.2977 0.0583

0.30 0.06

1.0594 0.0681

1.06 0.07

Table XIII. The Total Energies E (eV) for C59X (X = B, N) and the J^) Excess Electron Densities p^ at and around the Impurity Atoms

C59N C59B

E

P^''

pf pf>

-219.54

0.2745

-0.016

0.117

-219.17 -0.5128

-0.039

-0.066

^The number n represents the site number in the €50 molecule, for exam­ple, p^ ^ means the excess electron density at the impurity atom site.

where the first sum runs over only the occupied states. Minimizing the total energy E over yij and using the constraint condition [152]

we obtain the self-consistent equation for yij [152]

.,(0)

(126)

k,s^

- n - i ; ^ Z ^ , , ( m ) Z ^ , , ( / ) ] (127)

{ml) ^

- 2 a i A : - l ^ { z ^ , , ( 0 Z ^ , , ( ; ) k,s^

- n - 1 ^ Z^,,(m)Z^,,(/) (128) {ml) ^

where the first sum also runs over only the occupied states, and 11 is the number of TT bonds. The preceding coupled equations can be solved iteratively, and the final result should be independent of choosing the different initial values of the set y/y.

The choice of the three parameters (^i, ^ i , ki) for the X—c bonds is important. The best way to do it is to determine them by comparison between theoretical calculations and experimen­tal measurements. But to the best of our knowledge, there has not been an experimental measurement on the nonlinear optical properties of doped carbon tubules. Recently, by using a molecu­lar orbital method with Harris functional and spin-restricted ap­proximations [220], where the total electron density of the system can be approximated by a superposition of electron densities of the isolated atoms with a first-order energy correction of the den­sity error and quadratic errors in the electron Coulomb repulsion and exchange-correlation energies are partially canceled, Kurita et al. [145] have optimized the structures of C59N and C59B and at

20

Q

-0.2 -0.1

^y(A) 0.1 0.2

Fig. 11. The distribution Diyij) of the bond variables yij for a single sub­stituted €50 by nitrogen: C59N.

the same time investigated their electronic properties. They found that the optimized structures and binding energies for C59N and C59B were almost the same as those for C60, and the energy levels near the Fermi level were remarkably changed by doping. Here, by using the DESSH model shown earlier, we also investigated the structural and electronic properties of the same substituted doped fuUerenes by carefully adjusting the values of the three parame­ters {ti,ai, and ki) and found that our numerical calculations can accurately reproduce the results obtained by Kurita et al. if ti = 111 eV, a i = 6.04 eV/A, and ki = 51.1 eV/A^ for C59B, and ti = 1.05 eV, a i = 6.13 eV/A, and ki = 49.6 eV/A^ for C59N. In detail. Table XII presents our numerical results for the bandgaps between HOMO and LUMO, and the maximum atomic distor­tions from the C^Q structure. Numerical results are compared with those obtained by Kurita et al. It is seen from Table XII that the band gaps of C59X will be able to vary greatly with different sub­stituted impurity atoms (here, the gaps for X = B, N are equal to 1.0594 eV and 0.2977 eV, respectively), and the present model with appropriately chosen parameters for the substitute impurity atoms is able to accurately reproduce the results of Kurita et al. [145]. The total energies of the molecules and the excess electron densities at and around the positions of substitute impurity atom are listed in Table XIII. It is found that both C59N and C59B have nearly the same total energy, which means that they are equally stable. The excess electron density of the boron atom is -0.5128 but that of the nitrogen atom is 0.2745, which implies that electron deficiency is produced at the doped boron atom site and the boron atom gives up its electronic charge to its neighbors and exists as a donor. It is obvious that electronic charge accumulates on the doped nitrogen site and the nitrogen atom exists as an acceptor. The signs of excess electron density for the boron and nitrogen atoms are different, which makes C59N and C59B have opposite electronic polarization. Then, the distributions Diyij) of the bond variables in C59N and C59B are shown in Figure 11 and Figure 12, respectively, where a two-peak structure indicates the presence of the dimerization. The narrow peaks in the negative y/y region cor­respond to the distortion part around the impurity ions. The area of the extended portion for C59B is greater than that of C59N, which means that the doped boron atom will produce stronger dis­tortions around it than that in the nitrogen atom. Finally, the site-occupying probabilities of the HOMO states for C59B and C59N

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 295

30

20

Q 10

^ ^ I u 1 . . . „ .

f

-0.1

y//(A) 0.2

KJA 1

0.3 [

~

° -0 .1

0 [

— . 1 1 , — ^ _ __, _..,,,

J 1 1 1 1 1 1 1

J

. 1 , 1 1 , 1 1 1 , . . 1 . , I . , > J 1 1 1 1 1

10 20 30

site 40 50

T- 1-. ^u J-.-u .• / / X r.u u J • ui r • , u Fig. 14. Thc sitc-occupying probability of thc HOMO statc for a Single Fig. 12. The distribution D(y//) of the bond variables v,/for a single sub- ? .v . ^ ^ u u . ir* Ji . Ai^ uu /- o ^ ^ substituted C60 by boron: C59B stituted CgQ by boron: C59B. ° - ^

Fig. 13. The site-occupying probability of the HOMO state for a single substituted €50 by nitrogen: C59N.

are plotted in Figure 13 and Figure 14, respectively. There are two large peaks at point 1 (the position of the impurity atom) and point 9 for both atoms. The fact that the peak at point 1 in C59N is higher than that in C59B means that the localization effect coming from the nitrogen impurity atom is stronger than that from the boron atom. These conclusions are also well consistent with the exper­imental observations [146, 147]. Therefore, in numerical calcula­tions, we adopt the same values of all the parameters for C59B and C59N. After more experimental observations are made for doped carbon nanotubes, it will be possible to determine more accurately all the parameter values in our DESSH model.

5.2. Third-Order Optical Nonlinearity of Doped Ihbular FuUerenes

On the basis of the electronic structure obtained in the DESSH model mentioned earlier, we calculate the static second-order hyperpolarizabilities (y^^) of the single substituted armchair (C59+/XI0X) and zigzag (C59^_^xi8X) tubular fuUerenes, where X = B or N is at the site 1 [19]. The results are shown in Ta­ble XIV and Table XV, respectively. Here the x axis is directed to the impurity ion X. To see the substitute doping effect on the NLO

Table XIV. The Ratio q = y^^/yP of Several Doped Armchair Tubular FuUerenes^

X / = 0 i=l i = 2 i = 9 i = lS

N B

30.5 3.9

30.7

4.3

32.4 4.9

36.8 7.6

41.2 8.5

ayim jg j g calculated static y value of the doped tubule C59^_/xioX (X = B or N), yP is the static y value of the corresponding pure case and given by the empirical formula y^ = (1 + / x 10/60) - ^ y^ , and y^^ = 5.6 X 10" " esu is the static y value of C50.

Table XV. The Ratio q = f'^jyP of Several Doped Zigzag Ibbular FuUerenes^

X

N B

/ = 0

30.5 3.9

i = \

31.1 4.7

i = 2

33.2 5.1

i = 5

36.0 6.9

/ = 10

40.4 in

^y^^ is the calculated static y value of the doped tubule C59_|./xl8X (X = B or N), yP is the static y value of the corresponding pure case and given by the empirical formula 7^ = (1 + / x 18/60)^-^^7^^, and y^ = 5.6 X 10~^^ esu is the static y value of C50.

response of tubular fuUerenes, we list the ratio q between y^^ and yP in both tables:

<1 = yP (129)

where yP denotes the static y value of the corresponding pure case, which is given by the empirical formula yP = (1 + / x 10/60)^1^ 7^0 for armchair tubular fuUerene [131] and y P = (1 + / X 18/60)2-98760 fQj. zigzag tubular fuUerenes [131] and 76O = 5.6 X 10~^^ esu is the static 7 value of C60. It is seen that the static 7 value of 059.^/^ 10^ ^^ C59_^/xl8^ is several times larger than that of the corresponding pure one; and the static 7 value of C59+/XI0N or C59+/XI8N is more than 30 times larger than that of the corresponding pure one. This means that the substitute dop­ing, especially for the case of X = N, greatly increases the nonlin-

296 XIE

ear optical polarizability of carbon tubules. Here it should be men­tioned that owing to the distortion of the IT electron distribution in the substituted tubules, especially around the substituted dopant B or A atoms, the difference between the z and x (or y) components of y for those substituted tubules is much more pronounced than that for pure cases.

The dynamical nonlinear optical response of doped armchair tubule €594.9 XIQX has also been investigated by calculating the THG spectrum, and the results are shown in Figure 15 and Fig­ure 16 for X = N, B, respectively, and some details of the first peaks are listed in Table XVI.

It is seen from Table XVI that the first peaks appearing in Fig­ure 15 and Figure 16 are located at 3w = 0.129, and 0.653 eV, respectively, and their corresponding y magnitudes are 47.9 and 25.3 X 10"" ^ esu, which are three and 1.6 times larger than that (= 15.6 X 10"- ^ esu [122, 211]) of pure armchair tubules. As in the pure case, we find that the first peak in the y spectrum of the substituted armchair tubule is also caused by a three-photon reso­nance between two energy levels {a and b) near Fermi levels with one in the conduction band and the other in the valence band, namely, e^^ ^ 3a>.

0) 0

CO

250

150

100

50

0

< 1 1 1 1

-

-

^K^ i i i

J f ,...,1

! ' '

• \

1 \ -NVt i, _. 1 ._

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

3o)(eV) Fig. 15. The second-order hyperpolarizability y spectra of a single substi­tuted armchair tubular fuUerene by nitrogen: C59_|.9xioN.

100

30) (eV)

Moreover, Figure 15 and Figure 16 show that the highest peaks in the y spectra are located at 3o) = 0.437,1.783 eV, respectively, and their corresponding ymax magnitudes are about 792.45 x 10-^^ esu and 172.18 x 10"^^ esu, which are about 60 and 13 times larger than that (= 13.44 x 10"" ^ esu [122,211]) of the high­est peak of the pure armchair tubule. These peaks are caused by OTTPRE. The other peaks with larger y magnitudes are produced by two- or three-photon resonance. For example, in Figure 15, the three-photon peaks with larger y magnitudes are located at 3ft; = 0.387, 0.423, 0.465 eV, and the two-photon peaks with large y magnitudes are located at 3a> = 0.409, 0.451, 0.487 eV Also, we notice from Figure 15 and Figure 16 that the major response peaks with large y magnitudes concentrate on a narrow region, where the optical frequency is near the energy gap, that is, 3w ^ 3Eg. The reason is the same as that for the pure case, that is, a lot of OTTPRE processes can be observed in the doped tubule, but only those transition processes, which occur between the energy levels near Fermi levels, are able to contribute large y magnitudes.

We have shown that the rather larger NLO response for arm­chair and zigzag tubules is mainly produced by its delocalized TT electronics as in conjugated polymer chains. However, their three-dimensional character, which distinguishs them from linear poly­mers, causes severe limitations on their nonlinear optical proper­ties and makes their y values smaller than those of linear polymers containing a similar number of carbon atoms. On the basis of the preceding calculation, we know that the substituted dopant ions (X = B or N) attract or repel electrons and cause a distortion of TT electron distribution on the tubule's surface, which mainly hap­pens around the dopant ions (this effect can be called an inductive effect). On the other hand, the dopant ions cause greater local­ization of the original delocalized TT electrons around them, and therefore may reduce the effective space dimensions of fuUerene tubules (this effect can be called reduction effect). Our numerical results have shown that both effects make the NLO properties of the substituted tubule different from the corresponding pure one and greatly enlarge its y magnitudes. In addition, as shown before, the localization effect of the N impurities atom is stronger than that of the B atom. So the inductive and reduction effects of the effective space dimension in nitrogen-doped tubule are stronger than those in boron-doped tubule, which explains why nitrogen-doped tubule has larger y values than boron-doped one. It would be very interesting to see what would happen if heavier substitute doping is done. On the basis of the calculated results, this process will raise the y magnitude further.

In summary, using the DESSH model, we have studied the sub­stitute doping effect on the static and dynamical second-order hy-perpolarizabilities of tubular fullerenes. It is found that the sub­stitute doping effect greatly increases the y magnitudes of tubu-

Table XVI. The Optical Frequency Position (3a)), y Magnitudes, and Resonant Transition Energies (e^b = ^a - ^b) of The First Peaks Shown in Figure 15 and Figure 16 for Doped Tubule C59_|_9xioX. 3a>, y, and e^b are in Units of eV, 10~^^ esu, and eV, Respectively. Eg (in units of eV) is

the Energy Gap of the Corresponding Tubule

Fig. 16. The second-order hyperpolarizability y spectra of a single substi­tuted armchair tubular fuUerene by boron: C59_^9xioB.

N

B

0.136

0.647

0.129

0.653

47.9

25.3

677,75=0.133

^76,74 = 0-651

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 297

lar fuUerenes. The generation of boron- or nitrogen-doped carbon nanotubes [219] has been reported. In Ught of the experimental progress, theoretical studies mentioned earlier are significant for photonic applications of those nanotubes.

6. COHERENT CONTROL OF MOLECULAR POLARIZABILITY AND HYPERPOLARIZABILITY

Young's double slit experiment, as shown in Figure 17a, is not only is a key experiment to demonstrate wave behavior but also provides the basis for the rapidly developing field of physical sci­ence, coherent control of quantum dynamics of atoms and molecules [221-224]. The essence of coherent control lies in the interference principle of waves when there are two or more competing path­ways, which can quantum mechanically interfere, to arrive at the same final state.

It is known that the frequency and intensity properties of lasers have long been used to probe and even alter properties of matter. However, the most characteristic property of laser light, namely its well-defined phase, is rarely regarded as a control parameter. The possibility of so employing laser phase was first proposed by Manykin and Alfanas'ev [225], who considered the interference of quantum mechanical transition amplitudes for two pathways cou­pling the same initial and final state of an atom. In particular, they suggested that simultaneous single- and three-photon absorption could control state populations and hence the transmission of the medium. This is analogous to the interference of two beams in a Young's double slit experiment except that now one has an effec­tive "matter interferometer," because it is electrons that are being

light source

Fig. 17. (a). Interference effects in a Young's double slit experiment, (b). Illustration of introducing controllable quantum interference effects into the studied system. Two cw lasers at wi and w? interact with a molecule in a superposition of |$fl(r)) and |<I>fo(r)) of HQ. The superpo­sition state could be created by the pump laser at cjp, where 2Aw^ is the detuning of the two photon absorption.

controlled. Recently, the coherent control approach has been suc­cessfully demonstrated, both theoretically and experimentally, in unimolecular breakdown reactions [226-228], reactive scattering [229], electron distribution excited in a metal [230], the energy and angular distribution of autoionized electrons [231, 232], photocur-rent generation in bulk semiconductors [233], spontaneous emis­sion near a photonic band edge [234], light absorption and tera­hertz radiation in semiconductor nanostructures [235,236], optical phonon emission rates and optical gain from electronic intersub-band transitions in semiconductors [237, 238], optical dynamics in semiconductor microcavities [239], the polarization of an optical field [240], the total ionization yield in a two-color ionization pro­cess [241], and off-resonance refractive index of nitrogen molecule [251]. Among the coherent control scenarios , a successful one is to employ two laser fields to induce a transition in an atom, a molecule, or a solid, and control is achieved by varying the "ex­ternal parameters" (such as the relative phase and amplitude of the two fields, which can be adjusted experimentally) so that the induced transition amplitudes interfer constructively or destruc­tively.

In this section, we propose a coherent control approach to study the molecular polarizability a and hyperpolarizability /3 and y at a desired frequency. In our coherent control scenario, an ini­tial state, which comprises a superposition of two eigenstates of the radiation-free molecular Hamiltonian, is prepared, and both states are excited to the same final state by using two cw laser fields. As an example, it is demonstrated that the y magnitude of nitrogen molecule at a desired frequency can be coherently controlled, ei­ther constructively enhanced or strongly decreased, by varying the relative amplitude and phases between two laser fields and those in the initially prepared superposition state. This approach opens a door to achieving large y magnitude of molecules at a desired frequency, which is required for photonic applications. On the ba­sis of this control approach, one can get large y value of fuUerenes at a desired frequency.

The coherent control scenario is shown in Figure 17b. The molecule-field coupling system is initially in a superposition of two eigenstates, | ^ ( r ) ) and |<E> (r)), of the radiation-free molecular Hamiltonian H^ [242-252]

where

mr. ^ = 0)) = Ca\^a{r)) + C^|(D^(r)>

|Cal + |Qp = l

and the relative phase between both states is defined by

4> = arg

(130)

(131)

(132)

Both states are excited to the same final state (denoted by the en­ergy e) by using two cw laser fields that are in the form [222, 252]

where

E(0 = E(cui)e-^'^i^ -f- E(a)2)^"'"'^2^ (133)

ho)i = e- ea (134)

ha)2 = e- e^ (135)

and the relative amplitude and phase of both cw laser fields are defined by

|E(cu2)| ^ =

E(a)i) (136)

298 XIE

and

ilj = argl -—^ I E{o)i)J

(137)

respectively. Then, based on the perturbation theory and the quantum the­

ory of molecular polarizabiHty and hyperpolarizability introduced before, we got the probability amplitudes:

(138)

C, {t)-Lan l^ w,^-(On p=l (oia-o)p

where / is summed over all the dipole allowed transitions. To de­scribe the dephasing processes, which are not accompanied by the transfer of population, it would be appropriate to pursue this work with the density matrix formalism [9]. In this equation, the last sum term is a "satellite term," which does not affect the observ­able dipole moment at the desired angular frequency a>i, or ct>2-Separating the average value of the induced dipole moment into its desired angular frequency (a>i and ^2) components [9, 252] and based on the quantum theory of molecular polarizabihty intro­duced earlier, the linear polarizabihty a((o) of molecules at the de­sired angular frequencies w and (02, respectively was finally got:

p=l ""Ib-^^p (139) aij{o)i) ^-1

c\^\t) = Cah-^

2 2

E \(Ola-0)1 +

A ^ ^ [Ui^.E{(oq)][Uma*n(Op)]e' }{(^la-^p-<^q)t

p=\q=im (<^la~^P~^q'>(^ma~^P^ •+Cbh-

(140)

-^Kl,h

•^Kab

^(3) 2 2

(Oil, - ^ 1

^arib

+

+ ^Ib + ^ 1 ^la na-<^i) (144)

cY\t) = Cah-' E E p=lq=l

-imh i(oia-^p-^q-^r)i<ona-(^p-o)q)io)ma-(^p)

aij(a)2) = h ^J2

+Ct,h-^ E E E p=lq=lr=l

y [Uln»E{a>r)][Unm»E{(oq)][Umh*^{(Op)]e'^'^ib-^p-<^q ^r)t

where

Uij = (<D,-(r)|w|(Dy(r))

-^'<bb

+ '<ba

HK +

Kl^lb 0)lb - ^ 2 ^//, + ^ 2

^IK +

lb

(oib - (02 (oj^ + (02 (145)

(142)

is the transition-dipole matrix element between |<l>/(r)) and \<l>j(r))ofHo.

First, using above results, we got the linear dipole moment p^^Ht) of molecule:

where the five parameters in these equations are defined as fol­lows:

f<aa — \Ca\

^bb = 1 ~ ^aa

pa)(t) = h-^j:

I

+ \^b\ y ojit^-co^ + ^*^+^^ J

([Ulf,.E{aj2Whl I [^fo/*E(^2)]^/fo\

Kab = ^KaaKbb^e' id

Kba = ^/KaaKbbi ^

6 = i / f - 0

- U - / 5

(146)

(147)

(148)

(149)

(150)

MCb\' ^/6+^2 /

+/i-^E I

/ [^;^>E(a>2)]^/,/ , [^/,/«E(a>2)]^/^

^^-/(2a,2-wi)^

^^-i(2a;i-W2)^

For convenience, the preceding equations can be rewritten in a succinct form:

a{(Oi) = KaaOt^a^(Oi) + Kbb0^^b^(Oi) + Katot[i,{o)\) (151)

a{(02) = KaaOt^a(<^2)-^'<bbOi^b^(02)-\-KbaOil^(o)2) (152)

Similarly, we can get the second- and third-order nonlinear optical polarizabilities of molecules at their desired angular frequencies in succinct forms. In detail, the second-order optical polarizabihty P of molecules at the desired frequency (Os = 2(oi, 2(02, (oi + (02 are given by

I3{(0s=2(0i) = KaaPaa(^s) -\- f<bbP^b^(Os)

-^^abPib(^s)

(143)

(153)

KbbPbb(^s)

^^baPba(^s) (154)

^(ftJ

-^KabPib((^s)-^KbaPia((^s) (155)

P((Os = 2(02) = KaaPaa(^s) + KbbPbb^oJs)

p((Os = wi + (02) = KaaP^a(^s) + KbbP^b^O)s)

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 299

and the third-order nonlinear optical polarizabilities y of molecule at the desired angular frequencies w = 3wi, 3^2, 2wi+W2, w + 2^2 are given by

y{(Os = 3o)2) = Kaay^a(^s)-^ KtbJhb^^s)

-^^bayia('^s)

(156)

(157)

y(o)s = 2a)i + W2) = Kaay^a(^s) + Kbby^b^0)s)

•^'<abyib(^s)-^ Kbayia(<^s) (158)

y(o)s = cui + 2a>2) = Kaay^a(^s) + ^bby^b^^^^^

+^abyib(^s) + Kbayba(^s) (159)

In these succinct equations, af., pf., and yf^ denote the normal polarizabihty with all kinds of transition process both starting and ending in the state |4>/(r)). a|., p]- and y]- denote the new types of interference terms with all kinds of transition process starting at the state |4>/(r)) and ending at the state |^y(r)> (details about these terms are given in Appendix). The structure of the preced­ing equations is of the type desired: each has one Kaai- • •) term associated with the excitation of the |<l>«(r)> state, one K^^(. ..) term associated with the excitation of the |<l> (r)) state, one K^b term and/or one /<^Q(. ..) term corresponding to the interference between the two excitation routes. The interference term, which can be either constructive or destructive, is in general different for each case. What makes these equations so important in practice is that the interference term has coefficients whose magnitude and sign depend upon experimentally controllable parameters. Thus, by varying the magnitude and the relative phase ( of the coef­ficients Ca and Cb, the relative phase i/f and and the amplitude ratio ^ of two cw laser fields, we can alter the interference term di­rectly and hence control the a, j8, and y of molecules at a desired frequency. The desired superposition state could be carried out by a number of different schemes [221-224, 242-250], for example, the scheme involving a two photon absorption from a pump laser as shown in Figure 17b. Because of collisions, spontaneous emis­sion, or other dephasing effects, the loss of the coherence between |<|)fl(r)) and |^^(r)) would cause the reduction in magnitude or the disappearance of the interference term.

For simplicity, we apply this approach to nitrogen molecule [253], and as an example report the results of the coherent con­trol over the second-order hyperpolarizability y at the desired fre­quency 30 1 and 3^2. Here we consider the nonresonant case and thus do not include the population decay rate in our numerical cal­culation. Since we expect substantial enhancements of the y mag­nitude of molecules at a desired frequency, we focus on the calcu­lation of the ratio R{o)s) between the magnitudes of our controlled y{o)s) and normal y^{o)s)

R((Os) = y(o)s)

Y^((Os) (160)

In Figure 18, we show the results for the ratio R{3coi) as a func­tion of the phase difference 8 and the amplitude parameter

rj = Kbb = \Cbr (161)

The system is initially in a superposition of the state \v = 0,J = 0,M = 0) 2ind\v = 0,J = 2,M = 0) of the ground electronic

-360 -270 -180 270 360

'(deg.)

(a)

'(deg.)

(b)

Fig. 18. Contour plot of the ratio ROcoi) as a function of the phase dif­ference 8 and the parameter rj defined in the text, where coi = 3.0 x 10 ^ Hz and W2 = 2.99775 x 10 ^ Hz. The system is initially in a superpo­sition of\v = 0,J = 0,M = 0) and |j = 0, / = 2, M = 0) of the ground electronic state ZXg of the nitrogen molecule, (a) = 1; (b) ^ = 10; (c) C = 100.

300 XIE

-360 -270 -180 -90 0 90 180 270 360

5(deg.)

(c)

Fig. 19. Contour plot of the ratio R{3(02) as a function of the phase dif­ference 8 and the parameter 17 defined in the text, where cui = 3.0 x 10 ^ Hz and (02 = 2.99115 x 10 ^ Hz. The system is initially in a superpo­sition of \v = 0,J = 0,M = 0) and |i/ = 0, / = 2, M = 0) of the ground electronic state XXg of the nitrogen molecule, (a) ^ = 1; (b) ^ = 10; (c) c = 100.

state XXt of the nitrogen molecule, wi = 3.0 x 10 ^ Hz, and 0)2 = 2.99775 X 10 ^ Hz. Varying 17 from zero to one corresponds to changing the initial superposition from the pure state \^aW) to |<I> (r)). For ^ = 1, Figure 18a shows that the control over the ratio i?(3wi), due to quantum interference, is observed, but it is not very large compared with the uncontrolled ratio of one. That is, varying r) and 8 allows a change in the ratio R{3o)i) from 0.9 to 1.07. Moreover, with increasing f, as shown in Figure 18b for ^ = 10, a considerable control over i?(3wi) is achieved, that is, the

ratio R{?>o)i) changes from 0.2 to 1.8, and it can be increased for a factor of 9 just by changing the phase difference 8 from 0 to ±77 at t) = 0.5. Furthermore, if f is increased greatly, it is seen from Fig­ure 18c that varying 17 and 8 allows a substantial control over the ratio R{3(oi), which changes from 0.05 to 9.5, and in particular, the y(3wi) magnitude at the desired frequency 3a>i can be en­hanced in most of the parameter space of 8 and 77. Therefore, for powerful laser field E{o)2) and weak laser field E{(x)i), quantum interference will lead to constructive enhancements of the y(3wi) magnitude of nitrogen molecule at the desired frequency.

In Figure 19, the numerical results for the ratio R{?>a)2) with the same parameters as in Figure 18 is also shown. For f = 1, the control over R{?>o)2), as shown in Figure 19a, is similar to that over R{?>o)i), that is, varying r] and 8 allows a change in R{?>0)2) from 0.89 to 1.08. It is noted from the equations mentioned ear­lier that the parameter ^ in y(3w2) is inverted to that in y(3ft>i). Hence, when ^ is increased gradually, the interference term in yOo)2) would vanish gradually and R{3o)2) does not experience a substantial control as i?(3a>i) does, which can be seen clearly from Figure 19b and c. This means that the control over R{3o)i), with increasing f, would occur to the detriment of the control over R{?>o)2). Vice versa, the more powerful laser field E{a)i) allows a substantial control over the ratio R(3o)2) experienced by the weaker laser field E{o)2)-

In summary, a means of achieving coherent control of the molecular polarizability a and hyperpolarizability j8 and y at the desired frequency has been introduced. Because of quantum inter­ference, as an example, the nonresonant y magnitude of molecule at a desired frequency can be coherently controlled, either con­structively enhanced or strongly decreased, by varying the relative amplitude and phase between two laser fields and those in the ini­tially prepared superposition state. This approach has opened a door to getting large y magnitude of molecules at a desired fre­quency, which is required for photonic applications. On the basis of this theory, we can also get large y value of fuUerenes at a de­sired frequency.

7. REMARKS

In summary, the nonlinear optical interactions, formal definition of nonlinear optical susceptibility, and quantum theory of molec­ular polarizability a and hyperpolarizability /3 and y have been introduced. Theoretical studies and experimental measurements on the second-order optical nonlinearities of 059, C60 derivatives and C70, and the third-order optical nonlinearities of C^^Q and higher fullerenes including C70, C76, Cyg, 034, Cge, C90, C94, and C96 are briefly reviewed. In this chapter, the ESSH model, where the Coulomb interaction has been included, has been introduced in detail. We have apphed the ESSH model to describe higher fullerenes and fuUerene-related nanotubes, and investigated the dynamical and static third-order optical nonlinearities of armchair and zigzag tubular fullerenes and chiral carbon nanotubes, where the cap, symmetry, size and chiral effects are discussed and two scaling laws of the static second-order hyperpolarizability of arm­chair and zigzag tubular fullerenes are empirically arrived at, re­spectively. By including the effect of the dopant ions into the ESSH model, the doping effect on the third-order optical nonlinearities of tubular fullerenes is investigated. Finally, coherent control the­ory of molecular hyperpolarizability has been introduced. Our nu­merical experiment has demonstrated that the second-order hy-

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 301

perpolarizability of molecule at a desired frequency can be co­herently controlled, either greatly enhanced or heavily decreased, which opens a door to achieving a large nonlinear optical response of molecule needed for photonic application.

As discussed in Section 2, C6o can act as electron donors or acceptors. This feature plus the intrinsic SHG activity has led researchers to explore the possibility of fabrication of charge-transfer complexes with creation of a permanent dipole moment at ground state and an enhanced quadratic NLO response with a good temporal stability. As addressed by Kajzar et al. [60], the fab­rication of stable and efficient noncentrosymmetric molecules for quadratic NLO is an interesting challenge because of a large class of possible applications of these materials in tunable light sources, frequency converters, electro-optic modulation, ultrashort electric pulse generation, and so on. Because of the problems and espe­cially the cost that were encountered with fabrication of noncen­trosymmetric structures, artificial structures such as poled poly­mers have attracted increased interest. Controlled charge-transfer multilayered structures may offer an alternative solution, well ar-gumented by the controlled fabrication and expected enhanced stability. In this aspect, Kajzar et al. [60] have made the prelim­inary studies. Further studies on such structures are expected in the near future.

In our previous studies, we limited our attention to carbon nan-otubes of small size, which is described by the ESSH model. In fact, our tubes studied are not typical of tubes generically seen in exper­iment, where the single-wall carbon nanotubes that experimental researchers have claimed to make plenty of are a bit bigger, for example, (10,10) carbon nanotube. So it would be very interest­ing and significant to develop the present ESSH model because of computational complexity that arises because of the big size of carbon nanotubes.

It is known that bond-length alternation has been a good struc­tural parameter to describe the electrical properties of molecules. When the electron-lattice interaction is introduced in fuUerenes and carbon nanotubes, bond-length alternation is expected. So, what are the calculated values? Is it essential to take account of the electron-lattice interaction to obtain a large second-order hy-perpolarizability y? On the other hand, the bond-length alterna­tion should depend on odd or even numbers of carbon rings to be added to the C6o molecule. Then, what is the effect of odd or even number of rings on the second-order hyperpolarizability y of fuUerenes and carbon nanotubes? This is an interesting question since the y magnitude of carbon nanotubes may be considered as the limit.

In the case of graphite, the valence orbital is a irilpz) orbital and there is no interaction between the TT and cr(2s amd 2px,y) or-bitals because of their different symmetries. In the case of a tubule, the curvature of the tubule gives rise to some mixing of a and TT bands [254], that is, the bottom of antibonding o-* bands exits in the energy band of antibonding TT bands. Saito et al. [255] inves­tigated the electronic structure of graphene tubules based on C6o and examined the mixing effect of a and TT bands. It is found that this effect is small at the Fermi energy within the tight-binding ap­proximation for a and TT bands, based on which, the mixing of a and 77 orbitals has been neglected and the transition from TT to cr* in our present work has not been considered. It would be inter­esting to study further what is the effect on the second-order hy­perpolarizability y of carbon nanotubes because of the transition from TT to cr*?

Electrical conductivity measurements on single- and multi-walled carbon nanotubes have revealed that they may behave as metallic, insulating, or semiconducting nanowires, which depend on the method of production. In our previous work, we have con­sidered the chiral effect on the second-order hyperpolarizabilities of single-walled carbon nanotubes. Then what's the chiral effect on the third-order optical nonlinearities of the multi-walled car­bon nanotubes?

Experimental studies have shown that the caps of carbon nan­otubes may be destroyed partially or completely in the process of synthesis. Such cap effects may greatly influence their geomet­ric and electronic structures because their wave functions may be considered to consist of cap and tube states that have a large am­plitude around the cap and the cylindrical region. On the other hand, a TT electron on a site can transfer its effect to three neigh­boring sites. If caps are cut, a TT electron on the site at the edge of the cylinder can only transfer to two neighboring sites. The edge effect will reduce the effective space dimensions of TT elec­tron and thus enhance the second-order hyperpolarizabilities of carbon nanotubes [211]. However, it is noted that the bonds in the open tubes observed in experiments are passivated by hydrogen or something else. What's this effect on the second-order hyperpolar­izabilities of those carbon nanotubes?

On the basis of the theoretical studies and calculations, a scal­ing law of the static y values of armchair and zigzag tubular fuUerenes of smaU size was got [131]. What is the relationship be­tween carbon nanotube size and its y magnitude, that is, the trends rather than individual y magnitudes of carbon nanotubes?

Also, it would be interesting to explore structure property rela­tionships for optical nonlinearities of these unusual TT electron sys­tems, fuUerenes, and fuUerene-related nanotubes, from two com­plementary angles [139]: (i) to explore the possibility of obtaining structural information from nonlinear optical experiments and (ii) to assess the potential of fuUerenes, fuUerites, and their derivatives for the practical nonlinear optical devices. Rustagi et al. [139] have indicated that the screening of the optical fields due to electron-electron interaction may reduce the optical susceptibilities of these molecules by a large factor. Experimental work is expected to clar­ify this issue.

In this chapter, our coherent control theory to study the molec­ular hyperpolarizability at a desired frequency has been intro­duced. On the basis of the preceding studies, it has been demon­strated that the nonresonant y value of molecules at a desired frequency can be coherently controlled, either enhanced greatly or decreased heavily. Of course, it is possible for us to get larger y value of fuUerenes at the desired frequency based on this the­ory. How is the superposition state in fuUerenes established? how to control and enhance their second-order hyperpolarizability by changing the magnitude or relative phase of the coefficient of the superposition state and those of two incident coherent lasers? In addition to these questions, a number of other factors should be considered in our coherent control theory, for example, degrading processes due to Doppler broadening and collisions, and the deco-herence due to collisions. This work is stiU in progress and wUl be reported in the near future.

APPENDIX A: DETAILS OF INTERFERENCE TERMS

The detaUs of the cartesian components of interference terms, ""ib^ «L ' Pib^ ^ba^ yib^ ^^^ yia ^^ ^^^ ^ ^ ^ ^ frequency ws,

302 XIE

which are presented in the chapter, are summarized below, where ijkh denote the Cartesian components.

I. Interference terms of molecular polarizability a:

oL[j,{oys = (oi)\ij = h-^ ^ { - ~ ^ r ^ + KA ^Ib + ^ 1 ^la - ^ 1

ot[^{o)s = o)2)\ij = h ^Y.

I +

Hi^L (oih - (02 o)J^ + W2

II. Interference terms of molecular hyperpolarizabihty j8:

= h-Im

KKA {o)ia-2a)i)(o)^lj-a)i)

+ Tjj Tjk Tji

(w* + 2 w i ) ( w * j « + w i )

+ -TI^ TJ^ TT^

+

((Ola -2a)i)(o)ma - Ct>i)

TTJ Tjk Tji

+

mb

TJ^ TP TT^

{o)lia + 0)l){o)ia-o)i)

l^ia(^s=2o)2)\ijk

Im

uJi^uL (o)ib - 2o)2){o)mb - ^ 2 )

+

+

+

+

+

Km^l^la (o)*^-h2o)2)((0%ia-^o)2)

TJ bm ml la

u^ TP U^

((Oih -2a)2)(o)ma - ^ 2 )

(0)* +2(02X0)*. +(02) mb

Km<i< (o)%ia + o)2)(o)ia - ^ 2 )

Pia((^s = 0)1-^ 0)2)\ijk

Im

u'biULuL {(Oiij - ft)i - 0)2)((Oma - 2ft>i)

+ ^bm<l<

(o)*^ + wi + a)2)(o)%ia + 2a>2)

+ KmKi<

(o)ma+2o)2)((Oia -2(0i)

(Al) Pibi^s = wi + 0)2)\ijk

= h -2

(A2) E tm

Ki^LuL (o)la - wi - 0)2){0)ma - 2ft>i)

+

+

Tji Tjk Tji

(O)* + wi + (02)(o)*na + 2 ^ 2 )

TjJ Tji Tjk

(o)%ia-\-2(02)(o)ia -2a)i)

III. Interference terms of molecular hyperpolarizabihty y:

yit(^s = o)i)\ijkh

= h- 'E Imn

Tji Tji Tjk Tjh ^arin^nm^rnb

{(Ola - ^0)l)(0)nb - 2wi)(ft>m^ - (Oi)

(A3)

(A4)

+

+

+

+

+

+

+

+

+

+

+

Tjh Tjk TjJ Tji ^ am'-'mn'^ nl lb

{(o* + 3(0i)((0*a + 2(Oi)((o*ia + ^ l )

^^anKi«b ((oih - 2(Oi)((o^h - ^0(^nb + ^ 1 ^

TTk TTJ Tji Tjh ^am^mrin^nb

(ct),* + 2o)i)){(0%a + o)l){o)na - ^ l )

Ki<uLuL ((Ola - 3o)i)((Ona - 2(0i)((0^ij - (Oi)

Tjh Tjk TjJ Tji ^am^mn^nl lb

((o*^ + 3(Oi)((o''^^ + 2(oi)((o%ia + ^ l )

UanKl<^L ((Ola - '^0)l)(o)mb - O)l)((ola + ^ l )

Tjk Tji Tji Tjh ^ arn^ mr In^ nb

(ft)]' + 2(0i)((0%ia + (Oi)((^nb + ^ l )

TJ^ TT^ TJ^ Tjh ^arin^nm^rnb

(o)la - 3(Oi)((Ona - 2(Oi)((Oma - ^ l )

Tjh Tjk TjJ Tji ^ am'-'mn^ nl lb

(ft)* + 3(oi)((o* + 2a>i)(co* + (oi) ""nb

UanKMm<b ((Ola - 2(0i)((0ma - ^ i X ^ L + ^ l )

Tjk TjJ Tji Tjh ^am^rnrin^nb

((O* + 2(0i)((0* + (Oi)((Onb + w i )

yia((^s = 3(02)\ijkh

= h-Imn

""mb

IP U^ U^ U^ ^bl In n^ ^^

(A5)

(A6)

(A7)

{(Oih - 3(02X(0„b - 2(02X(Omb - <^2)

NONLINEAR OPTICAL PROPERTIES OF FULLERENES AND CARBON NANOTUBES 303

+

+

+

+

+

+

-h

+

+

+

+

^L^LKiuL (o)"^^ + 3(O2)(0)%a + 2o)2)(o)%ia + (02)

Tjh Tji TjJ Tjk

{0)iiy - 2o)2)(o)mb - ^2)(^lb + ^ 2 )

TI^ TJ^ TP U^

(wj^^ + 2o)2)(o)%ia + (02)((0na + ^ 2 )

TP U^ II^ TI^ ^ bl^ In^ nm^ ma

(w/^ - ?>0)2){o)nh - 2a)2){o)ma - ^ 2 ) Tlh Tjk TTJ Tji

(0)1 + 3o)2){(ola + 2 c 0 2 ) ( < ^ + C02)

TTH Tji TTJ Tjk

(co/^ - 2o)2)(o)ma - <^2)((^lb + ^ 2 )

(0)'^^-^2(O2){0)*^f^-\-(O2)((Ona-^(O2)

TP TJ^ tl^ TJ^ ^bl In nm^ma

((Oily - ^o)2){o)na - 2o)2)(o)ma - ^ 2 )

Tjh Tjk Tji Tji

(^L + ^2)(<^ + 2^2)(<ft + ^2) Tjh Tji Tji Tjk

(0)ia - 2o)2)((Oma - ft>2)(^L + ^ 2 )

(0)*b + 2a)2)(w;;^ + 0)2)((0nb - 0)2)

+

+

+

Tjk TjJ Tjh Tji

(w*^ + 2ft>2 + 0}i)(o)la + 2o)2)(o)%a + ^ 2 )

U^nKiUL<b (coiij - 2o)2)(o)rnb " ^ 2 ) ( ^ n a + ^ l )

Tjk Tji Tji Tjh ^am ^rnl ^ In nb

(w*^ + 2o)2)((o%a + 0)2)io)nb + ^ l ) (A9)

yba^o)s = 2o)i^-o)2)\ijkh

= h-

Imn

TP TJ^ JJ^ Jjh

((Ola - 3(0i)((0na - 2(Oi)((Oma - 0)\)

+

+

+

Tjh Tjk TjJ Tji ^bm^rnn^nria

((oj^ + 3(0i)((0*^^ + 2(Oi)((ol^^ + (Oi)

Tjh Tji Tji Tjk

((Ola - 2(Oi)((Oma - ^ l ) (^^fo + ^ l )

UL<i<Una (w*^ + 2(0i)((0*^j^ + (Oi)((Ona + ^ l )

(AlO)

yba^0)s = (Oi-\-2(O2)\ijkh

Imn

TP TJ^ TJ^ Tjh ^bl In nm^ma

((Ola - 2(0i - (02)((0na - 2(Oi)((Oma - ^ l )

(A8)

yah(o)s = 2(01-\-(02)\ijkh

= h-TP U^ TJ^ U^ ^arin^nm^rnb

Imn ' ((Oily -(Oi- 2(02)((Ona - 2(Oi)((Oynb - ^ l )

+

+

+

+

+

+

+

+

Tjh Tjk TjJ Tji ^am^mn^nl^lb

(ft)*, + ft>i + 2a>2)(w*^ + 2(Oi)((o%ia + ^ l )

^anKMm<b ((Ola - ^0)\)io)mb - ^ i X ^ L + 0)2)

Tjh Tji Tji Tjk ^am^mrin^nb

(cu*^ + 2(0i)((0%a + o)l)io)nb + ^ 2 )

U^ TT^ Tjh Tjk ^arin^nm^rnb

{o)lb - 2 ^ 2 - o)\){o)na - 2(0i)((0^i, - (02)

+

+

+

+

Tjh Tjk Tji Tji ^bm^rnn^nria

(a>* H-2(oi + o)2)io)lh + 2 ^ l ) ( ^ m ; , + ^ l ) nb Tjh Tji TjJ Tjk ^bn^nrim^rna

""mb

((Ola - 2(0i)((0ma - 0)l)(o)lu + 0)2)

^bn^L^i^na ((0*j^ + 2(Oi)((o''^^ + o)i)(o)na + 6^2)

TJ^ TJ^ TJ^ Tjh ^bl In nm^ma

((Ola - 2 ^ 1 - (02)((0nb - 2(02)((Oma - ^ i )

+ Tjh Tjk TjJ Tji

(w*^ + 2(Oi + (02)((ola + 2(02)((ol^ij + ^ i )

Tjh Tji Tji Tjk ^bn^nrim^rna

((Oil, - 2(02)((Oma - ^ l ) ( ^ ! / , + ^ l )

Tjk Tjh Tji Tji ^am^mn^nl lb

(w*^ + 2^2 + 0)\){0)%i, + 2(Oi)((o%,a + 0)2)

Tjk Tji Tjh Tji ^an^nrim^b

((Ola - ^0)i)((O^ij - (02)((0%a + ^ 2 )

Tji Tjh Tji Tjk ^a^^mrin^nb

(w* + 2(0i)((0%a + 0)2)(o)nb + ^ 2 )

KPln^Luib ((Oiij - 2(02 - o)i)((Onb - ^o)2)((o^ij - (02)

+

+

+

+

+

ULKlVna ((O*^ + 2(02)((o''^^ + 0)i)((Ona + ^ l )

Tji TjJ Tjk Tjh ^bl In nm^ma

(^/fl - ^ 2 - 2cui)(ce)„^ - 2(02)((Oma - 0)2)

Tjh Tjk Tji Tji ^bm^mn^nl^la

(o)*^ + W2 + 2cui)(w*« + 2(02)((ol^i, + (02)

Tjh Tji Tji Tjk ^bn^nrim^ma

((Oil) - 2(02)((Oma - ^ 2 ) ( ^ * / , + ^ l )

Tjk TjJ Tji Tjh ^bm^mrin^na

( ^ L + ^ ^ 2 ) ( ^ ; ^ ^ + 0)2)((Ona + ^ i ) ( A l l )

304 XIE

= r Imn

Tji TfJ Tjk Tjh

i^lb - ^(^2)(<^nb - 2w2)(^m^ " ^l)

+ rrh Tjk TTJ jji ^am^mn^ nl lb

(o)J^ + 3o)2){o)%a + 2a>2)(^mfl + ^ 2 )

(w/ -2o)2)i<^mb

+ ^am^rnrin^nb

(o)J^ + 2a)2)(wmfl + (^2)(^nb + ^ 2 ) (A12)

Acknowledgments

The author is grateful to the kind hospitaUty of Dr. Rein-hard Schinke at Max Planck Institute, to Dr. H. S. Nalwa and Q. Rao for their good suggestions about the present topic, to Dr. A. C. Perlman and Dr. B. Dey for helpful communications, and to Dr. K. Suenaga and Professor Dr. Q. H. Gong for their reprints on fullerenes and carbon nanotubes. This work was finan­cially supported by Alexander von Humboldt Foundation.

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