- Email: [email protected]
Third-order optical nonlinearities of chiral graphene tubules
5 November 1999
Chemical Physics Letters 313 Ž1999. 211–216 www.elsevier.nlrlocatercplett
Third-order optical nonlinearities of chiral graphene tubules Rui Hua Xie a
a,)
, Qin Rao
b
Max-Planck-Institut fur Bunsenstr. 10, D-37073 Gottingen, Germany ¨ Stromungsforschung, ¨ ¨ Nanchang Telecommunications Bureau, Nanchang 330003, People’s Republic of China
b
Received 7 June 1999; in final form 1 September 1999
Abstract The third-order optical nonlinearities, characterized by the second-order hyperpolarizabilities g , of chiral graphene tubules are studied. The average contribution h 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 h ; the metallic chiral graphene tubule favors larger g values; chiral graphene tubules can compete with the conducting polymer achieving a large g value which is needed for photonic applications. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Molecules with large third-order optical nonlinearities, characterized by large second-order hyperpolarizabilities g , are required for photonic applications including all-optical switching, data processing, and eye and sensor protection. However, the g magnitudes of most third-order materials are usually smaller than those needed for photonic devices. Hence, finding third-order materials with large g magnitudes has been a hot topic in physics and chemistry. Theoretical and experimental studies w1–6x have shown that the conjugated p-electron organic systems and quantum dots are potentially important in photonics owing to their large nonlinear optical ŽNLO. response. Since fullerenes C n and carbon nanotubes w7–10x possess a large number of delocalized p electrons and do not have any residual infrared absorption, the intense research effort devoted to fullerenes and carbon nanotubes has recently been )
Corresponding author. E-mail: [email protected]
extended to a consideration of fullerenes and carbon nanotubes as potential nonlinear optical materials. The third-order optical nonlinearities of C 60 have been extensively studied both theoretically and experimentally w9,11x, but its g magnitude is smaller than those needed for photonic devices. Then, it has been shown, both theoretically w12–15x and experimentally w16–18x, that higher fullerenes possess larger nonlinear optical response. Recently, our theoretical studies w19–24x have predicted that two kinds of C 60 -based tubules, named armchair and zigzag tubules w9x, are also potentially important in photonics owing to their larger NLO response. Based on their experimental observations, Iijima and co-workers w25x claim that most single-wall carbon nanotubes show chirality. Using a similar technique to Iijima’s, Dravid et al. w26x have found that most of their carbon nanotubes have a chiral structure. Therefore, it is interesting to study the thirdorder optical nonlinearities of chiral graphene tubules in view of their practical application. This is the purpose of the present Letter.
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 0 4 4 - 1
212
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
2. Theoretical model Chiral graphene tubules w27–29x can be formed with a screw axis along the axis of the tubule and with a variety of ‘hemispherical’-like caps. These tubules can be specified mathematically in terms of the tubule diameter d t and chiral angle u , which is shown in Fig. 1, where the chiral vector C h s pa1 q qa 2 . The vector C h connects two crystallographically equivalent sites O and A on a two-dimensional graphene structure. The construction in Fig. 1 shows the chiral angle u of the tubule with respect to the zigzag direction Ž u s 0. and two units, a1 and a 2 , of the hexagonal honeycomb lattice. An ensemble of possible chiral vectors can be specified by C h in terms of pairs of integers Ž p,q . w27x. Each pair of integers Ž p,q . defines a different way of rolling the graphene sheet to form a carbon tubule. In detail, the cylinder connecting the two hemispherical caps is formed by superimposing the two ends OA of the vector C h , and the cylinder joint is made by joining the line ABX to the parallel line OB, where lines OB and ABX are perpendicular to the vector C h 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 u and in the tubule diameter d t give rise to differences in the properties of the various chiral graphene tubules. In the Ž p,q . notation for specify-
ing the chiral vector C h , the vectors Ž p,0. denote zigzag tubules, the vectors Ž q,q . denote armchair tubules, and all other vectors Ž p,q . correspond to chiral tubules. In terms of the integers Ž p,q ., the tubule diameter d t is given by w9,27–29x dt s
(3 Ž p q pq q q . a s 2
Ch
p
p
c–c
,
Ž 1.
where a c – c is the nearest-neighbor c–c distance ˚ in graphite., C h is the length of the Žs 1.421 A chiral vector C h , and the chiral angle u is given by w9,27–29x
u s tany1
ž
'3 q qq2 p
/
.
Ž 2.
Let m be the largest common divisor in p and q. Then, the atom number n per unit cell is equal to w30x ns
4 Ž p 2 q pq q q 2 .
Ž 3.
m
if Ž p y q . is not a multiple of 3m, or w30x ns
4 Ž p 2 q pq q q 2 .
Ž 4.
3m
if Ž p y q . is a multiple of 3m. Recently, an extended version of the Su– Schrieffer–Heeger model has been applied to describe C 60 , C 70 , and C 60-based tubules and conducting polymer w15,19–24,31–34x. In this Letter, we use this model to study the third-order optical nonlinearities of chiral graphene tubules. The model can be written as follows Hs
Ý Ý Žyt y a yi j . Ž c†i , s c j, s q h.c. . s
² ij :
K q 2
Ý yi2j q U Ý c†i ,≠ c i ,≠ c†i ,x c i ,x i
² ij :
qV Ý
Ý c†i , s c i , s c†j, s c j, s ,
² ij : s, s
Fig. 1. The chiral vector OA or C h s pa1 q qa 2 is defined on the honeycomb lattice of carbon atoms by unit vectors a1 and a 2 and the chiral angle u with respect to the zigzag axis Ž u s 0.. Also shown is the lattice vector T s OB of the 1D tubule unit cell.
2
X
X
Ž 5.
X
where the sum ² ij : is taken over the nearest neighbors for the c–c bond; t represent the hopping integrals for the c–c bond; a is the electron-phonon coupling constants related to the c–c bond; K is the spring constants corresponding to the c–c bond; yi j
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
is the change of the bond length between the ith and jth atoms; the operator c i, s Ž c†i, s . annihilates Žcreates. a p electron at the ith atom with spin s Žs ≠,x.; U is the usual on-site Coulomb repulsion strength; V is the Coulomb interaction between the nearest and next-nearest atoms. Using the Hartree–Fock approximation, we are able to transform the above Eq. Ž5. into Hs
s
K q 2
Ý yi2j q U Ý Ý ri , s c†i , s c i , s y ri ,≠ ri ,x
qV Ý ² ij :
ž
i
² ij :
s
Ý Ý r j, s c†i , s c i , s y ri , s Ý r j, s
ž
s
X
X
/
/
Ž 6.
where r i, s s ² c†i, s c i, s : is the electron density. This equation is solved by the adiabatic approximation for phonons. The Schrodinger equation for the p elec¨ tron is
Ý Ž yt y a yi j y V² c†i , s c j, s : . Zk , s Ž j . ² ij :
q Ur i , s q V Ý X
s
Ý r j, s
Zk , s Ž i . ,
X
Ž 7.
j
where e k is the k th eigenvalue. The self-consistent equation for the lattice is yi j s y2 a Ky1 Ý k,s
yNy1
Ý ² ml :
½
)
Zk , s Ž i . Zk , s Ž j .
5
Zk , s Ž m . Zk , s Ž l . ,
Ž 8.
where the mark ‘ U ’ denotes the sum over all occupied states, the second term originates from the constraint condition
Ý yi j s 0 ,
5
g Ž y3 v ; v , v , v . s
Ý g i Ž y3 v ; v , v , v . , Ž 10.
where g i is given in detail in the literatures w20,34x. Since the ratios between different components of g are not known, a spatial average of g is given by w13,20,34x:
s
y² c†i , s c j, s : c†i , s c j, s q ² c†i , s c j, s :2 ,
e k Zk , s s
Within the independent electron approximation and sum-over-state approach, the second-order hyperpolarizability g of molecules is given by
g s 15 gx x x x q gy y y y q gz z z z
X
X
s
3. Numerical results
is1
Ý Ý Ž yt y a yi j . Ž c†i , s c j, s q h.c. . ² ij :
213
Ž 9.
² ij :
and N is the number of p bonds. Then, the electron eigenstates z k, s Ž i ., eigenenergies e k , and bond variables yi j can be obtained self-consistently through Eqs. Ž7. and Ž8..
q2 Ž gx x y y q gy y z z q gz z x x . .
Ž 11 .
In this Letter we study chiral graphene tubules with finite atom numbers Nc . Obviously, the tubule edge effects cannot be neglected in such cases. A finite chiral graphene tubule 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, here we apply periodic boundary condition for the tubule axis, and each carbon atom at the end of the finite chiral graphene tubule can still find its three neighbors by imaging that the two ends of the tubule are connected. In Fig. 1, we have defined a unit cell along the tubule axis, and C h and T s 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 h passes. The length of T is '3 Chrm if Ž p,q . is not a multiple of 3m, and C hrm if Ž p y q . is a multiple of 3m. In addition, the parameters are chosen to be the same as those of our previous work w19–24x, i.e., ˚ Žbond length., U s 2V s t, t s 2.5 eV, a s 1.4225 A ˚ and K s 49.7 eVrA˚ 2 . a s 6.31 eVrA, Based on the electronic structures obtained, we calculate the static g magnitudes of 17 chiral graphene tubules, which have different diameter d t and chiral angle u . Then, we calculate the average contribution h Žs grNc . of one carbon atom to the
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
214
third-order optical nonlinearity of a chiral graphene tubule, where Nc is the total atom number in the studied tubule. We note that for carbon nanotubes with smaller diameters than that of C 60 , there are no caps containing only pentagons and hexagons which can be fit continuously to such a small carbon nanotube Ž p,q .. For this reason, it is expected that the ˚ . carbon observation of very small diameter Ž- 7 A nanotubes is very unlikely w9,10x. For example, the Ž4,2. 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 Ž6,5. tubule which is the smallest diameter chiral tubule. In detail, our numerical results are listed in Table 1. Meanwhile, we have also given the corresponding diameter d t , the chiral angle u , the atom number n per unit cell, and the total atom number Nc 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 1, s and m in Ž p,q . s and Ž p,q . m denote the semiconducting and metallic tubules, respectively. Table 1 The static g and corresponding h values of 17 chiral graphene tubules, where s and m in Ž p,q . s and Ž p,q . m denote semiconducting and metallic tubules, respectively. n, Nc , d t , and u are the atom number per unit cell, the total atom number calculated, the diameter, and the chiral angle of a chiral graphene tubule, respectively Ž p,q .
n
Nc
˚. u d t ŽA
g Ž10y3 3 esu. h Ž10y35 esu.
Ž6,5. s Ž9,1. s Ž7,4. m Ž8,3. s Ž9,2. s Ž7,5. s Ž10,1. m Ž8,4. s Ž9,3. m Ž10,2. s Ž7,6. s Ž8,5. m Ž9,4. s Ž10,3. s Ž8,6. s Ž9,5. s Ž10,4. m
364 364 124 388 412 436 148 112 156 248 508 172 532 556 296 604 104
364 364 372 388 412 436 444 448 468 496 508 516 532 556 592 604 624
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
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
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
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
It has been shown that the optical nonlinearities of fullerenes and C 60 -based tubules will decrease with the increasing of space dimension w34,15,23,24x. For example, 3D C 60 molecule possesses smaller g values than 1D conducting polymer with the same atom number w9,11x. The substitute doping effect reduces the effective space dimension of C 60 , C 70 and C 60based tubules and thus their optical nonlinearities are greatly enhanced w15,23,24,34x. Here we see that the 1D chiral graphene tubule will greatly become a 2D graphite sheet with the increase of its diameter. So it is expect that the smaller the diameter of a chiral graphene tubule, the larger the g value of the chiral graphene tubule is available. Indeed, Table 1 tells us that the h value for semiconducting graphene tubules Ž p,q . s increases with decreasing their diameters, i.e., 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 Ž p,q . m . From Table 1, we know that both Ž6,5. s and ˚. Ž9,1. s tubules have the same diameter Ž d t s 7.47 A and the same atom number per unit cell Ž n s 364. as well as the semiconducting properties. The only difference between both tubules is the chiral angle u Ž u s 0.4711 for Ž6,5. s tubule and u s 0.0909 for Ž9,1. s tubule.. In this case, if the total atom number Nc is the same, both tubules will have the same height but the length of the base helix of Ž6,5. s tubule is shorter than that of Ž9,1. s tubule. Thus, the atoms in Ž6,5. s tubule are situated along a more straight line than those in Ž9,1. s tubule and thus Ž6,5. s tubule has lower space dimension than Ž9,1. s tubule. Therefore, we expect Ž6,5. s tubule possesses a larger g value than Ž9,1. s tubule. From Table 1, we find that the value h of Ž6,5. s tubule is bigger than that of Ž9,1. s tubule, i.e., the g value of Ž6,5. s tubule is larger than that of Ž9,1. s tubule if their total atom number Nc is the same. The periodic boundary conditions for the 1D graphene tubules permit only a few wave vectors to exist in the circumferential direction w9x. 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 w9x. Recently, it has been shown that when the total atom number Nc of carbon nanotube is in-
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216 Table 2 Seven different molecular weights ŽMW. with the experimentally derived g values and the average contribution h of a well characterized polyenic polymer. The number Nc 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. Adapted from Ref. w14x MW
Nc
g Ž10y3 3 esu.
h Ž10y35 esu.
2800 4100 5400 7500 10000 17600 27900
230 340 450 620 830 1460 2320
3.5 5.4 7.8 15.5 26.7 62.9 85.4
1.5 1.6 1.7 2.5 3.2 4.3 3.7
215
our theoretical results remain for the moment purely theoretical predictions and are only of academic interest. Recently, many research groups have successfully yielded significant quantities of the singlewall carbon nanotube w9,10x, which are now available at prices starting from US$ 100 per g Žroughly ten times the price of gold.. Surely, they are too costly for most applications, but it is low enough to allow laboratory testing. We expect the experimental studies of the nonlinear optical properties of carbon nanotubes.
4. Summary
creased greatly, a large energy gap for a semiconducting graphene tubule is still available but the energy gap for a metallic graphene tubule approaches zero w9x. In this case, it is expect that if the total number Nc of carbon nanotubes is the same, a metallic tubule possesses a larger g value than a semiconducting one. Indeed, it is seen from Table 1 that the h value of a metallic graphene tubule Ž p,q . m is larger than that of a semiconducting graphene tubule Ž p,q . s. Well characterized conjugated p-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 w1–5x. As an example, in Table 2, we list the experimentally derived g values and the average contribution h of a well characterized polyenic polymer for seven different molecular weights w5x. Comparing their h 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 which 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. To the best of our knowledge, there is no experimental data available regarding the nonlinear optical response of carbon nanotubes. In light of this point,
We have studied the third-order optical nonlinearities of chiral graphene tubules. The average contribution of one carbon atom to the third-order optical nonlinearities of a chiral graphene tubule is calculated and compared to 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 h ; the metallic chiral graphene tubule favors larger g values; the chiral graphene tubules can compete with polyenes reaching a large g value which is needed for photonic devices.
Acknowledgements The authors would like to thank Q. Huang for helpful discussions. This work is supported by Alexander von Humboldt-Stiftung.
References w1x J. Messier, F. Kajzar, P.N. Prasad, D. Ulrich, Nonlinear Oprical Effects in Organic Polymers, Kluwer Academic, Dordrecht, 1991. w2x J. Messier, F. Kajzar, P.N. Prasad, Organic Molecules for Nonlinear Optics and Photonics, Kluwer Academic, Dordrecht, 1991. w3x J.I. Sakai, Phase Conjugate Optics, McGraw-Hill, Singapore, 1992. w4x J. Zyss, Molecular Nonlinear Optics, Academic Press, New York, 1994. w5x I.D.W. Samuel, I. Ledoux, C. Dhenaut, J. Zyss, H.H. Fox, R.R. Schrock, R.J. Silbey, Science 265 Ž1994. 1070. w6x U. Woggon, Optical Properties of Semiconductor Quantum Dots, Springer-verlag, Berlin, 1997.
216
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
w7x H.W. Kroto, J.R. Heath, S.C. Obrien, R.E. Curl, R.E. Smalley, Nature ŽLondon. 318 Ž1985. 162. w8x S. Iijima, Nature ŽLondon. 354 Ž1991. 56. w9x M. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, 1996. w10x T.W. Ebbesen, Carbon Nanotubes, CRC Press, Boca Raton, FL, 1997. w11x P. Norman, Y. Luo, D. Jonsson, H. Agren, J. Chem. Phys. 106 Ž1997. 8788, and related references therein. w12x M. Fanti, G. Orlandi, F. Zerbetto, J. Am. Chem. Soc. 117 Ž1995. 6101. w13x C.E. Moore, B.H. Cardelino, X.Q. Wang, J. Phys. Chem. 100 Ž1996. 4685. w14x M. Fanti, P.W. Fowler, G. Orlandi, F. Zerbetto, J. Chem. Phys. 107 Ž1997. 5072. w15x R.H. Xie, Phys. Lett. A 258 Ž1999. 51. w16x H. Huang, G. Gu, S. Yang, J. Fu, P. Yu, G.K.L. Wong, Y. Du, Chem. Phys. Lett. 272 Ž1997. 427. w17x H. Huang, G. Gu, S. Yang, J. Fu, P. Yu, G.K.L. Wong, Y. Du, J. Phys. Chem. 102 Ž1998. 61. w18x T. Zhang, F. Wang, H. Yang, Q. Gong, X. An, H. Chen, D. Qiang, Chem. Phys. Lett. 301 Ž1999. 343.
w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x
R.H. Xie, J. Jiang, Appl. Phys. Lett. 71 Ž1997. 1029. R.H. Xie, J. Jiang, Chem. Phys. Lett. 280 Ž1997. 66. R.H. Xie, Q. Rao, Appl. Phys. Lett. 72 Ž1998. 2358. R.H. Xie, J. Chem. Phys. 108 Ž1998. 3626. R.H. Xie, Z. Naturforsch. A 54 Ž1999. 348. R.H. Xie, Chem. Phys. Lett. 310 Ž1999. 379. S. Iijima, T. Ichihashi, Nature ŽLondon. 363 Ž1993. 603. V.P. Dravid, X. Lin, Y. Wang, X.K. Wang, A. Yee, J.B. Ketterson, R.P.H. Chang, Science 259 Ž1993. 1601. R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Appl. Phys. Lett. 60 Ž1992. 2204. M.S. Dresselhaus, G. Dresselhaus, R. Saito, Phys. Rev. B 45 Ž1992. 6234. R.A. Jishi, L. Venkataraman, M.S. Dresselhaus, G. Dresselhaus, Phys. Rev. B 51 Ž1995. 11176. R.A. Jishi, M.S. Dresselhaus, G. Dresselhaus, Phys. Rev. B 47 Ž1993. 16671. K. Harigaya, J. Phys.: Condens. matter 3 Ž1991. 8855. K. Harigaya, M. Fujita, Phys. Rev. B 47 Ž1992. 16563. K. Harigaya, S. Abe, Phys. Rev. B 49 Ž1994. 16746. J. Dong, J. Jiang, J. Yu, Z.D. Wang, D.Y. Xiong, Phys. Rev. B 52 Ž1995. 9066.
Chemical Physics Letters 313 Ž1999. 211–216 www.elsevier.nlrlocatercplett
Third-order optical nonlinearities of chiral graphene tubules Rui Hua Xie a
a,)
, Qin Rao
b
Max-Planck-Institut fur Bunsenstr. 10, D-37073 Gottingen, Germany ¨ Stromungsforschung, ¨ ¨ Nanchang Telecommunications Bureau, Nanchang 330003, People’s Republic of China
b
Received 7 June 1999; in final form 1 September 1999
Abstract The third-order optical nonlinearities, characterized by the second-order hyperpolarizabilities g , of chiral graphene tubules are studied. The average contribution h 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 h ; the metallic chiral graphene tubule favors larger g values; chiral graphene tubules can compete with the conducting polymer achieving a large g value which is needed for photonic applications. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Molecules with large third-order optical nonlinearities, characterized by large second-order hyperpolarizabilities g , are required for photonic applications including all-optical switching, data processing, and eye and sensor protection. However, the g magnitudes of most third-order materials are usually smaller than those needed for photonic devices. Hence, finding third-order materials with large g magnitudes has been a hot topic in physics and chemistry. Theoretical and experimental studies w1–6x have shown that the conjugated p-electron organic systems and quantum dots are potentially important in photonics owing to their large nonlinear optical ŽNLO. response. Since fullerenes C n and carbon nanotubes w7–10x possess a large number of delocalized p electrons and do not have any residual infrared absorption, the intense research effort devoted to fullerenes and carbon nanotubes has recently been )
Corresponding author. E-mail: [email protected]
extended to a consideration of fullerenes and carbon nanotubes as potential nonlinear optical materials. The third-order optical nonlinearities of C 60 have been extensively studied both theoretically and experimentally w9,11x, but its g magnitude is smaller than those needed for photonic devices. Then, it has been shown, both theoretically w12–15x and experimentally w16–18x, that higher fullerenes possess larger nonlinear optical response. Recently, our theoretical studies w19–24x have predicted that two kinds of C 60 -based tubules, named armchair and zigzag tubules w9x, are also potentially important in photonics owing to their larger NLO response. Based on their experimental observations, Iijima and co-workers w25x claim that most single-wall carbon nanotubes show chirality. Using a similar technique to Iijima’s, Dravid et al. w26x have found that most of their carbon nanotubes have a chiral structure. Therefore, it is interesting to study the thirdorder optical nonlinearities of chiral graphene tubules in view of their practical application. This is the purpose of the present Letter.
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 1 0 4 4 - 1
212
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
2. Theoretical model Chiral graphene tubules w27–29x can be formed with a screw axis along the axis of the tubule and with a variety of ‘hemispherical’-like caps. These tubules can be specified mathematically in terms of the tubule diameter d t and chiral angle u , which is shown in Fig. 1, where the chiral vector C h s pa1 q qa 2 . The vector C h connects two crystallographically equivalent sites O and A on a two-dimensional graphene structure. The construction in Fig. 1 shows the chiral angle u of the tubule with respect to the zigzag direction Ž u s 0. and two units, a1 and a 2 , of the hexagonal honeycomb lattice. An ensemble of possible chiral vectors can be specified by C h in terms of pairs of integers Ž p,q . w27x. Each pair of integers Ž p,q . defines a different way of rolling the graphene sheet to form a carbon tubule. In detail, the cylinder connecting the two hemispherical caps is formed by superimposing the two ends OA of the vector C h , and the cylinder joint is made by joining the line ABX to the parallel line OB, where lines OB and ABX are perpendicular to the vector C h 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 u and in the tubule diameter d t give rise to differences in the properties of the various chiral graphene tubules. In the Ž p,q . notation for specify-
ing the chiral vector C h , the vectors Ž p,0. denote zigzag tubules, the vectors Ž q,q . denote armchair tubules, and all other vectors Ž p,q . correspond to chiral tubules. In terms of the integers Ž p,q ., the tubule diameter d t is given by w9,27–29x dt s
(3 Ž p q pq q q . a s 2
Ch
p
p
c–c
,
Ž 1.
where a c – c is the nearest-neighbor c–c distance ˚ in graphite., C h is the length of the Žs 1.421 A chiral vector C h , and the chiral angle u is given by w9,27–29x
u s tany1
ž
'3 q qq2 p
/
.
Ž 2.
Let m be the largest common divisor in p and q. Then, the atom number n per unit cell is equal to w30x ns
4 Ž p 2 q pq q q 2 .
Ž 3.
m
if Ž p y q . is not a multiple of 3m, or w30x ns
4 Ž p 2 q pq q q 2 .
Ž 4.
3m
if Ž p y q . is a multiple of 3m. Recently, an extended version of the Su– Schrieffer–Heeger model has been applied to describe C 60 , C 70 , and C 60-based tubules and conducting polymer w15,19–24,31–34x. In this Letter, we use this model to study the third-order optical nonlinearities of chiral graphene tubules. The model can be written as follows Hs
Ý Ý Žyt y a yi j . Ž c†i , s c j, s q h.c. . s
² ij :
K q 2
Ý yi2j q U Ý c†i ,≠ c i ,≠ c†i ,x c i ,x i
² ij :
qV Ý
Ý c†i , s c i , s c†j, s c j, s ,
² ij : s, s
Fig. 1. The chiral vector OA or C h s pa1 q qa 2 is defined on the honeycomb lattice of carbon atoms by unit vectors a1 and a 2 and the chiral angle u with respect to the zigzag axis Ž u s 0.. Also shown is the lattice vector T s OB of the 1D tubule unit cell.
2
X
X
Ž 5.
X
where the sum ² ij : is taken over the nearest neighbors for the c–c bond; t represent the hopping integrals for the c–c bond; a is the electron-phonon coupling constants related to the c–c bond; K is the spring constants corresponding to the c–c bond; yi j
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
is the change of the bond length between the ith and jth atoms; the operator c i, s Ž c†i, s . annihilates Žcreates. a p electron at the ith atom with spin s Žs ≠,x.; U is the usual on-site Coulomb repulsion strength; V is the Coulomb interaction between the nearest and next-nearest atoms. Using the Hartree–Fock approximation, we are able to transform the above Eq. Ž5. into Hs
s
K q 2
Ý yi2j q U Ý Ý ri , s c†i , s c i , s y ri ,≠ ri ,x
qV Ý ² ij :
ž
i
² ij :
s
Ý Ý r j, s c†i , s c i , s y ri , s Ý r j, s
ž
s
X
X
/
/
Ž 6.
where r i, s s ² c†i, s c i, s : is the electron density. This equation is solved by the adiabatic approximation for phonons. The Schrodinger equation for the p elec¨ tron is
Ý Ž yt y a yi j y V² c†i , s c j, s : . Zk , s Ž j . ² ij :
q Ur i , s q V Ý X
s
Ý r j, s
Zk , s Ž i . ,
X
Ž 7.
j
where e k is the k th eigenvalue. The self-consistent equation for the lattice is yi j s y2 a Ky1 Ý k,s
yNy1
Ý ² ml :
½
)
Zk , s Ž i . Zk , s Ž j .
5
Zk , s Ž m . Zk , s Ž l . ,
Ž 8.
where the mark ‘ U ’ denotes the sum over all occupied states, the second term originates from the constraint condition
Ý yi j s 0 ,
5
g Ž y3 v ; v , v , v . s
Ý g i Ž y3 v ; v , v , v . , Ž 10.
where g i is given in detail in the literatures w20,34x. Since the ratios between different components of g are not known, a spatial average of g is given by w13,20,34x:
s
y² c†i , s c j, s : c†i , s c j, s q ² c†i , s c j, s :2 ,
e k Zk , s s
Within the independent electron approximation and sum-over-state approach, the second-order hyperpolarizability g of molecules is given by
g s 15 gx x x x q gy y y y q gz z z z
X
X
s
3. Numerical results
is1
Ý Ý Ž yt y a yi j . Ž c†i , s c j, s q h.c. . ² ij :
213
Ž 9.
² ij :
and N is the number of p bonds. Then, the electron eigenstates z k, s Ž i ., eigenenergies e k , and bond variables yi j can be obtained self-consistently through Eqs. Ž7. and Ž8..
q2 Ž gx x y y q gy y z z q gz z x x . .
Ž 11 .
In this Letter we study chiral graphene tubules with finite atom numbers Nc . Obviously, the tubule edge effects cannot be neglected in such cases. A finite chiral graphene tubule 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, here we apply periodic boundary condition for the tubule axis, and each carbon atom at the end of the finite chiral graphene tubule can still find its three neighbors by imaging that the two ends of the tubule are connected. In Fig. 1, we have defined a unit cell along the tubule axis, and C h and T s 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 h passes. The length of T is '3 Chrm if Ž p,q . is not a multiple of 3m, and C hrm if Ž p y q . is a multiple of 3m. In addition, the parameters are chosen to be the same as those of our previous work w19–24x, i.e., ˚ Žbond length., U s 2V s t, t s 2.5 eV, a s 1.4225 A ˚ and K s 49.7 eVrA˚ 2 . a s 6.31 eVrA, Based on the electronic structures obtained, we calculate the static g magnitudes of 17 chiral graphene tubules, which have different diameter d t and chiral angle u . Then, we calculate the average contribution h Žs grNc . of one carbon atom to the
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
214
third-order optical nonlinearity of a chiral graphene tubule, where Nc is the total atom number in the studied tubule. We note that for carbon nanotubes with smaller diameters than that of C 60 , there are no caps containing only pentagons and hexagons which can be fit continuously to such a small carbon nanotube Ž p,q .. For this reason, it is expected that the ˚ . carbon observation of very small diameter Ž- 7 A nanotubes is very unlikely w9,10x. For example, the Ž4,2. 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 Ž6,5. tubule which is the smallest diameter chiral tubule. In detail, our numerical results are listed in Table 1. Meanwhile, we have also given the corresponding diameter d t , the chiral angle u , the atom number n per unit cell, and the total atom number Nc 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 1, s and m in Ž p,q . s and Ž p,q . m denote the semiconducting and metallic tubules, respectively. Table 1 The static g and corresponding h values of 17 chiral graphene tubules, where s and m in Ž p,q . s and Ž p,q . m denote semiconducting and metallic tubules, respectively. n, Nc , d t , and u are the atom number per unit cell, the total atom number calculated, the diameter, and the chiral angle of a chiral graphene tubule, respectively Ž p,q .
n
Nc
˚. u d t ŽA
g Ž10y3 3 esu. h Ž10y35 esu.
Ž6,5. s Ž9,1. s Ž7,4. m Ž8,3. s Ž9,2. s Ž7,5. s Ž10,1. m Ž8,4. s Ž9,3. m Ž10,2. s Ž7,6. s Ž8,5. m Ž9,4. s Ž10,3. s Ž8,6. s Ž9,5. s Ž10,4. m
364 364 124 388 412 436 148 112 156 248 508 172 532 556 296 604 104
364 364 372 388 412 436 444 448 468 496 508 516 532 556 592 604 624
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
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
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
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
It has been shown that the optical nonlinearities of fullerenes and C 60 -based tubules will decrease with the increasing of space dimension w34,15,23,24x. For example, 3D C 60 molecule possesses smaller g values than 1D conducting polymer with the same atom number w9,11x. The substitute doping effect reduces the effective space dimension of C 60 , C 70 and C 60based tubules and thus their optical nonlinearities are greatly enhanced w15,23,24,34x. Here we see that the 1D chiral graphene tubule will greatly become a 2D graphite sheet with the increase of its diameter. So it is expect that the smaller the diameter of a chiral graphene tubule, the larger the g value of the chiral graphene tubule is available. Indeed, Table 1 tells us that the h value for semiconducting graphene tubules Ž p,q . s increases with decreasing their diameters, i.e., 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 Ž p,q . m . From Table 1, we know that both Ž6,5. s and ˚. Ž9,1. s tubules have the same diameter Ž d t s 7.47 A and the same atom number per unit cell Ž n s 364. as well as the semiconducting properties. The only difference between both tubules is the chiral angle u Ž u s 0.4711 for Ž6,5. s tubule and u s 0.0909 for Ž9,1. s tubule.. In this case, if the total atom number Nc is the same, both tubules will have the same height but the length of the base helix of Ž6,5. s tubule is shorter than that of Ž9,1. s tubule. Thus, the atoms in Ž6,5. s tubule are situated along a more straight line than those in Ž9,1. s tubule and thus Ž6,5. s tubule has lower space dimension than Ž9,1. s tubule. Therefore, we expect Ž6,5. s tubule possesses a larger g value than Ž9,1. s tubule. From Table 1, we find that the value h of Ž6,5. s tubule is bigger than that of Ž9,1. s tubule, i.e., the g value of Ž6,5. s tubule is larger than that of Ž9,1. s tubule if their total atom number Nc is the same. The periodic boundary conditions for the 1D graphene tubules permit only a few wave vectors to exist in the circumferential direction w9x. 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 w9x. Recently, it has been shown that when the total atom number Nc of carbon nanotube is in-
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216 Table 2 Seven different molecular weights ŽMW. with the experimentally derived g values and the average contribution h of a well characterized polyenic polymer. The number Nc 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. Adapted from Ref. w14x MW
Nc
g Ž10y3 3 esu.
h Ž10y35 esu.
2800 4100 5400 7500 10000 17600 27900
230 340 450 620 830 1460 2320
3.5 5.4 7.8 15.5 26.7 62.9 85.4
1.5 1.6 1.7 2.5 3.2 4.3 3.7
215
our theoretical results remain for the moment purely theoretical predictions and are only of academic interest. Recently, many research groups have successfully yielded significant quantities of the singlewall carbon nanotube w9,10x, which are now available at prices starting from US$ 100 per g Žroughly ten times the price of gold.. Surely, they are too costly for most applications, but it is low enough to allow laboratory testing. We expect the experimental studies of the nonlinear optical properties of carbon nanotubes.
4. Summary
creased greatly, a large energy gap for a semiconducting graphene tubule is still available but the energy gap for a metallic graphene tubule approaches zero w9x. In this case, it is expect that if the total number Nc of carbon nanotubes is the same, a metallic tubule possesses a larger g value than a semiconducting one. Indeed, it is seen from Table 1 that the h value of a metallic graphene tubule Ž p,q . m is larger than that of a semiconducting graphene tubule Ž p,q . s. Well characterized conjugated p-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 w1–5x. As an example, in Table 2, we list the experimentally derived g values and the average contribution h of a well characterized polyenic polymer for seven different molecular weights w5x. Comparing their h 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 which 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. To the best of our knowledge, there is no experimental data available regarding the nonlinear optical response of carbon nanotubes. In light of this point,
We have studied the third-order optical nonlinearities of chiral graphene tubules. The average contribution of one carbon atom to the third-order optical nonlinearities of a chiral graphene tubule is calculated and compared to 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 h ; the metallic chiral graphene tubule favors larger g values; the chiral graphene tubules can compete with polyenes reaching a large g value which is needed for photonic devices.
Acknowledgements The authors would like to thank Q. Huang for helpful discussions. This work is supported by Alexander von Humboldt-Stiftung.
References w1x J. Messier, F. Kajzar, P.N. Prasad, D. Ulrich, Nonlinear Oprical Effects in Organic Polymers, Kluwer Academic, Dordrecht, 1991. w2x J. Messier, F. Kajzar, P.N. Prasad, Organic Molecules for Nonlinear Optics and Photonics, Kluwer Academic, Dordrecht, 1991. w3x J.I. Sakai, Phase Conjugate Optics, McGraw-Hill, Singapore, 1992. w4x J. Zyss, Molecular Nonlinear Optics, Academic Press, New York, 1994. w5x I.D.W. Samuel, I. Ledoux, C. Dhenaut, J. Zyss, H.H. Fox, R.R. Schrock, R.J. Silbey, Science 265 Ž1994. 1070. w6x U. Woggon, Optical Properties of Semiconductor Quantum Dots, Springer-verlag, Berlin, 1997.
216
R.H. Xie, Q. Rao r Chemical Physics Letters 313 (1999) 211–216
w7x H.W. Kroto, J.R. Heath, S.C. Obrien, R.E. Curl, R.E. Smalley, Nature ŽLondon. 318 Ž1985. 162. w8x S. Iijima, Nature ŽLondon. 354 Ž1991. 56. w9x M. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, 1996. w10x T.W. Ebbesen, Carbon Nanotubes, CRC Press, Boca Raton, FL, 1997. w11x P. Norman, Y. Luo, D. Jonsson, H. Agren, J. Chem. Phys. 106 Ž1997. 8788, and related references therein. w12x M. Fanti, G. Orlandi, F. Zerbetto, J. Am. Chem. Soc. 117 Ž1995. 6101. w13x C.E. Moore, B.H. Cardelino, X.Q. Wang, J. Phys. Chem. 100 Ž1996. 4685. w14x M. Fanti, P.W. Fowler, G. Orlandi, F. Zerbetto, J. Chem. Phys. 107 Ž1997. 5072. w15x R.H. Xie, Phys. Lett. A 258 Ž1999. 51. w16x H. Huang, G. Gu, S. Yang, J. Fu, P. Yu, G.K.L. Wong, Y. Du, Chem. Phys. Lett. 272 Ž1997. 427. w17x H. Huang, G. Gu, S. Yang, J. Fu, P. Yu, G.K.L. Wong, Y. Du, J. Phys. Chem. 102 Ž1998. 61. w18x T. Zhang, F. Wang, H. Yang, Q. Gong, X. An, H. Chen, D. Qiang, Chem. Phys. Lett. 301 Ž1999. 343.
w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x
R.H. Xie, J. Jiang, Appl. Phys. Lett. 71 Ž1997. 1029. R.H. Xie, J. Jiang, Chem. Phys. Lett. 280 Ž1997. 66. R.H. Xie, Q. Rao, Appl. Phys. Lett. 72 Ž1998. 2358. R.H. Xie, J. Chem. Phys. 108 Ž1998. 3626. R.H. Xie, Z. Naturforsch. A 54 Ž1999. 348. R.H. Xie, Chem. Phys. Lett. 310 Ž1999. 379. S. Iijima, T. Ichihashi, Nature ŽLondon. 363 Ž1993. 603. V.P. Dravid, X. Lin, Y. Wang, X.K. Wang, A. Yee, J.B. Ketterson, R.P.H. Chang, Science 259 Ž1993. 1601. R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Appl. Phys. Lett. 60 Ž1992. 2204. M.S. Dresselhaus, G. Dresselhaus, R. Saito, Phys. Rev. B 45 Ž1992. 6234. R.A. Jishi, L. Venkataraman, M.S. Dresselhaus, G. Dresselhaus, Phys. Rev. B 51 Ž1995. 11176. R.A. Jishi, M.S. Dresselhaus, G. Dresselhaus, Phys. Rev. B 47 Ž1993. 16671. K. Harigaya, J. Phys.: Condens. matter 3 Ž1991. 8855. K. Harigaya, M. Fujita, Phys. Rev. B 47 Ž1992. 16563. K. Harigaya, S. Abe, Phys. Rev. B 49 Ž1994. 16746. J. Dong, J. Jiang, J. Yu, Z.D. Wang, D.Y. Xiong, Phys. Rev. B 52 Ž1995. 9066.