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Totally symmetric vibrations of armchair carbon nanotubes
Computational Materials Science 49 (2010) S231–S234
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Totally symmetric vibrations of armchair carbon nanotubes N.A. Poklonski a,*, E.F. Kislyakov a, Nguyen Ngoc Hieu b, S.A. Vyrko a, O.N. Bubel’ a, Nguyen Ai Viet b a b
Belarusian State University, Minsk 220030, Belarus Institute of Physics and Electronics, Hanoi, Viet Nam
a r t i c l e
i n f o
Article history: Received 1 October 2009 Accepted 30 December 2009 Available online 8 February 2010 Keywords: Carbon nanotubes Raman scattering Radial breathing mode Peierls distortions Kekule structure
a b s t r a c t The influence of bond length dimerization due to Peierls distortions on totally symmetric vibrations of armchair carbon nanotubes is considered theoretically. It is shown that the radial and tangential motions for quinoid type dimerization are mixed and the radial breathing mode for Kekule type dimerization is splitted. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNT) have unique mechanical and electrical properties which are important for applications [1,2]. They can be either metallic or semiconducting depending on their chirality and diameter. One of the fundamental features of metallic CNTs is the possibility of Peierls distortions [3] of the lattice structure at low temperature. Until now this phenomenon has not been properly investigated and experimentally identified, in spite of the fact that it has important consequences for the thermal and electrical conductivity of CNTs [4]. Usually it is considered that all bond lengths in CNTs are equal [1,2], and possible bond length dimerization due to Peierls distortions at low temperatures is ignored. In [5,6] it was shown within the framework of semiempirical quantum chemical PM3 [7] calculations that the ground state of the (5,5) armchair CNT is Kekule type distorted with 0.003 nm bond length difference and that the transition state between the equivalent Kekule structures is a quinoid type structure having an excess energy of 3 meV per carbon atom. A question arises, how this feature can be verified experimentally. The Raman spectroscopy is widely used for experimental characterization of CNTs [8]. The most intense Raman lines in the CNTs’ spectrum correspond to totally symmetric vibrations in the Brillouin zone center (C point). Among these vibrations the radial breathing mode is of special interest because of its uniqueness for CNTs and diameter dependence of its frequency. Today Raman
* Corresponding author. Tel.: +375 17 2095110. E-mail addresses: [email protected], [email protected] (N.A. Poklonski). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.12.033
scattering experiments for individual CNTs are possible [9]. However, in the liquid helium temperature range, where manifestation of the bond length dimerization is possible, they are lacking. The aim of this article is to qualitatively elucidate the consequences of bond length dimerization of armchair CNTs on their totally symmetric vibrations in the Brillouin zone center, visible in Raman scattering experiments. For this purpose we employ symmetry considerations and a simple analytical model taking into account only nearest neighbor harmonic interactions of the CNT’s carbon atoms. We consider the (5,5) CNT as a characteristic example. For quantitative estimations we use PM3 calculations within the MOPAC 6.0 package [10].
2. Quinoid structure The (5,5) CNT with quinoid structure of chemical bonds is shown in Fig. 1. Its geometry is completely determined by two equilibrium bond lengths a0 and b0 and by the translational period along its axis L ¼ 2b0 cosðh0 =2Þ or, equivalently, by the angle h0 between bonds. The elementary translational cell of the (5,5) CNT contains 20 carbon atoms and has a D5d point symmetry group, which gives three totally symmetric vibrations in the Brillouin zone center. The infinitely long (5,5) CNT also has mirror symmetry planes rv and rh , glide symmetry plane r0v and rotoreflection plane r0h (shown in Fig. 1), which exclude displacements along nanotube z axis from totally symmetric vibrations and leave only two totally symmetric vibration modes [11,12]. Let us consider totally symmetric motions of (5,5) CNT in a cylindrical coordinate system ðq; u; zÞ (Fig. 2), with the origin of z
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σv
a θ
Geometrical considerations show that a ¼ 2q sinð18 uÞ; b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðq sin uÞ2 þ z2 ,
z b
θ
h ¼ 2 arctan½z=ðq sin uÞ; w ¼ 2 arctan½z=ðq sin u sin 18 Þ,
L
0
h ¼ 90 þ 2 arcsinð2q sin u cos 18 =bÞ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 w0 ¼ arcsinð2z= b 4q2 sin u cos2 18 Þ þ arcsinð2 cos 18 Þ.
σh σh
Da ¼
a0
q0
Dq
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4q20 a20 Du;
Db ¼
pffiffiffi 3Dz þ Dðq sin uÞ;
2d 8d sin 18 ; Dw ¼ ; b0 b0 f pffiffiffi 3Dz cos 18 Dw pffiffiffi ; Dw0 ¼ ; Dh 0 ¼ 2 b0 f
Dh ¼ σv CNT axis Fig. 1. (5,5) CNT with C–C-bond dimerization of quinoid type; L is the translational period.
(ρ, ϕ, z)
z
a
ρ
U¼ ϕ 36°
σv
D5d (5,5) CNT
σv
Fig. 2. Cylindrical coordinates ðq; u; zÞ of (5,5) CNT atoms; C–C-bonds a and b are shown.
and u in the middle of the sloped bond, corresponding to the intersection of the bond with glide plane. During totally symmetric vibrations z and u coordinates of the atoms of this bond differ only in sign and their distances to the nanotube axis are equal, therefore the motion of the nanotube is completely determined by the coordinates of a single carbon atom in the chosen coordinate system. The Hamiltonian H of totally symmetric vibrations of the (5,5) CNT translational unit cell, taking into account only nearest neighbor carbon atom interactions, is written as
" _ Þ2 þ mz_ 2 þ H ¼ 10 mq_ 2 þ mðq0 u 2
! 2 a20 sin 18 2
b0
f2
;
ð3Þ
and the potential energy U of totally symmetric vibrations of the (5,5) CNT takes a simple form (depending only on two parameters, bond stretching stiffnesses K a and K b ):
σh
0
ð2Þ
pffiffiffi 2 where the notations d ¼ Dz 3Dðq sin uÞ; f ¼ 3 þ sin 18 are introduced, and the equilibrium value h0 ¼ 120 is chosen. Substituting Eq. (2) into Eq. (1), we can see that the coupling between motions along the nanotube axis and perpendicular to it vanishes when
K b ¼ 4K h þ 32 2K w þ K w0
b θ
For
the small deviations of geometrical parameters from their equilibrium values in the linear approximation this gives:
ða a0 Þ2 K a þ ðb b0 Þ2 K b 2 2
þ ðh h0 Þ2 b0 K h þ 2ðh0 h00 Þ2 a20 K h0 þ ðw w0 Þ2 b0 K w i þ2ðw0 w00 Þ2 a20 K w0 ¼ T þ U;
ð1Þ
where T and U are kinetic and potential energy of the atom vibrations, K a and K b are stiffnesses of a and b bonds, K h and K h0 are angle deformational stiffnesses, K w and K w0 are dihedral angle deformational stiffnesses, m is the mass of a carbon atom, q0 ; a0 ; b0 ; h0 ; h00 ; w0 ; w00 are the equilibrium values of the CNT radius, bonds and angles, 10 is the number of horizontal bonds in the (5,5) CNT unit cell.
! 2 K a a20 K b b0 þ ðDqÞ2 2q20 4q20 " !# 2 a20 b0 þ K a 2 2 þ K b 4 2 ðDðq0 uÞÞ2 2q0 4q0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 a2 b0 b0 a0 @ 1 Ka þ Kb 1 02 ADqDðq0 uÞ: q0 q0 16q20 4q0
ð4Þ
From Eq. (4) one can see that q and u are not normal coordinates for totally symmetric vibrations of armchair CNTs. Even in the limit of equal bonds (absence of dimerization) there exists a small coupling between the radial and tangential motions of carbon atoms. Quinoid type dimerization makes an additional contribution to this coupling proportional to K b b0 K a a0 . This is essential [13] for the electron–phonon matrix element, which determines the intensity p offfiffiffi Raman scattering. The frequencies (x ¼ k for ½x ¼ cm1 Þ of the totally symmetric vibrations of CNT can be obtained from the characteristic equation, for Hamiltonian (4),
k2 2kðK a þ 2K b Þ þ K a K b a2 ¼ 0;
ð5Þ
where
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u ffi 2 a0 u b b 1 a2 1 02 a ¼ t2 1 0 2 þ 0 q0 q0 2 16q0 4q0 is determined by the CNT geometry. When dimerization is absent ða0 ¼ b0 ; K a ¼ K b ¼ KÞ, the roots of Eq. (5) are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k1;2 ¼ 3K 1 1 a20 =9 :
ð6Þ
Here k1 corresponds to the high energy transverse optical (TO) mode (component of Raman G line) with frequency xTO ¼ 1590 cm1 [8], and k2 corresponds to the radial breathing mode (RBM), a0 corresponds to the non-dimerized case. If we choose
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z
σh
LK
σh
σv
σv
Ch
Fig. 3. Plane evolvement of the translational unit cell of the (5,5) CNT with Kekule structure. Ch is the chiral vector. Dotted lines show the primitive two-dimensional unit cell; LK is the translational period along CNT axis z; rv , rh are the mirror, r0v is the glide and r0h is the rotoreflection planes.
k1 ¼ x2TO , then K ¼ 430692 cm2 ¼ 30:45 104 dyn=cm which is comparable with the value K ¼ 36:50 104 dyn=cm from [14]. For this K value the RBM frequency xRBM ¼ 236:754 cm1 , which is somewhat lower than the commonly [15] accepted value 338:9 cm1 . Two parameters K a and K b of our model for the quinoid type structure can be adjusted to the frequencies of two totally symmetric modes of the (5,5) CNT. From Eq. (5) their ratio is
Ka a 1 ¼ bþ b Kb 4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 1 2; bþ b 16
where b ¼ xTO =xRBM . For example, when xRBM ¼ 236:754 cm1 ; xTO ¼ 1590 cm1 and the geometrical parameters are taken from [5], then K a =K b 1:01, with their mean value as in the non-dimerized case.
in Fig. 2 and coordinates q1 ; u1 of the adjacent atom. Coordinate z1 of this atom is fixed by the mirror plane rh . The (5,5) CNT has five totally symmetric vibrations according to the number of degrees of freedom. They are shown in Fig. 4. In addition to the two totally symmetric modes of non-dimerized and quinoid structures, the Kekule structure has a longitudinal totally symmetric mode, and RBM and transverse modes become splitted. In order to estimate quantitatively the magnitude of this splitting, we have carried out PM3 calculations of the (5,5) CNT totally symmetric modes in the Brillouin zone center for the Kekule structure obtained in [5] using MOPAC 6.0 package [10]. Both RBM and TO modes are splitted according to this calculations. The frequencies obtained are xRBM1 ¼ 376 cm1 and xRBM2 ¼ 388 cm1 for RBM, and xTO1 ¼ 1666 cm1 and xTO2 ¼ 1672 cm1 for TO modes. 4. Conclusions
3. Kekule structure The plane evolvement of the translational unit cell of the (5,5) CNT with Kekule structure is shown in Fig. 3. Kekule structure has the same D5d symmetry as quinoid structure but the translational period in this case is tripled. The translational unit cell of Kekule structure contains 60 carbon atoms instead of 20 in the case of quinoid type dimerization. The mirror ðrv ; rh Þ, glide ðr0v Þ and rotoreflection ðr0h Þ symmetry planes are also shown in Fig. 3. Due to these planes only five independent variables are needed for complete description of the totally symmetric vibrations of an armchair CNT with Kekule structure. We can choose for these variables cylindrical coordinates ðq; u; zÞ of one of the carbon atoms as
In this paper we propose to explore the ground state properties of armchair carbon nanotubes in low temperature Raman scattering experiments. In order to qualitatively elucidate totally symmetric vibrations of armchair CNTs in the Brillouin zone center, visible in Raman scattering experiments, we have developed a simple analytical model employing symmetry considerations and nearest neighbor harmonic interactions of CNT’s carbon atoms. Parameters of this model (bond stiffnesses) can be adjusted to experiment. We show that quinoid type dimerization (without Peierls gap opening) does not lead to the breathing mode splitting, but to the mixing of radial and tangential carbon atom displacements in the totally symmetric vibrations of armchair CNTs. Kekule type distortions (with Peierls gap) lead to the breathing mode splitting.
σh σh
σv
(a)
σv
(b)
(c)
(d)
(e)
Fig. 4. Totally symmetric vibrations of the armchair CNT with Kekule structure: radial (a, b); transverse (c, d); longitudinal (e).
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According to our PM3 calculations using MOPAC 6.0 package, a quantitative estimate of this splitting for the case of geometrical structure of (5,5) CNT calculated in [5] is found to be 12 cm1 . Thus, the problem of the ground state structure of armchair CNTs can be solved experimentally. Acknowledgment This work has been supported by the BFBR (Grant Nos. F08Vn003 and F08R-061). References [1] A. Jorio, G. Dresselhaus, M.S. Dresselhaus (Eds.), Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties, and Applications, Springer, Berlin, 2008. [2] M.J. O’Connell (Ed.), Carbon Nanotubes: Properties and Applications, Taylor & Francis, New York, 2006.
[3] R.E. Peierls, Quantum Theory of Solids, Oxford University Press, London, 2001. chap. 5. [4] S. Piscanec, M. Lazzeri, J. Robertson, A.C. Ferrari, F. Mauri, Phys. Rev. B 75 (2007) 035427. [5] N.A. Poklonski, E.F. Kislyakov, Nguyen Ngoc Hieu, O.N. Bubel’, S.A. Vyrko, A.M. Popov, Yu.E. Lozovik, Chem. Phys. Lett. 464 (2008) 187. [6] N.A. Poklonski, E.F. Kislyakov, Nguyen Ngoc Hieu, O.N. Bubel’, S.A. Vyrko, Tran Cong Phong, Mol. Simulat. 35 (2009) 681. [7] J.J.P. Stewart, J. Comput. Chem. 10 (1989) 209. [8] M.S. Dresselhaus, G. Dresselhaus, R. Saito, A. Jorio, Phys. Rep. 409 (2005) 47. [9] M. Oron-Carl, F. Hennrich, M.M. Kappes, H.v. Lohneysen, R. Krupke, Nano Lett. 5 (2005) 1761. [10] MOPAC 2006 Version 1.0, J.J.P. Stewart, Fujitsu Limited, Tokyo, Japan. [11] M. Damnjanovic´, I. Miloševic´, T. Vukovic´, R. Sredanovic´, Phys. Rev. B 60 (1999) 2728. [12] E.B. Barros, A. Jorio, G.G. Samsonidze, R.B. Capaz, A.G.S. Filho, J.M. Filho, G. Dresselhaus, M.S. Dresselhaus, Phys. Rep. 431 (2006) 261. [13] M. Mohr, M. Machón, C. Thomsen, I. Miloševic´, M. Damnjanovic´, Phys. Rev. B 75 (2007) 195401. [14] R.A. Jishi, L. Venkataraman, M.S. Dresselhaus, G. Dresselhaus, Chem. Phys. Lett. 209 (1993) 77. [15] J. Kürti, V. Zólyomi, M. Kertesz, G. Sun, New J. Phys. 5 (2003) 125.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Totally symmetric vibrations of armchair carbon nanotubes N.A. Poklonski a,*, E.F. Kislyakov a, Nguyen Ngoc Hieu b, S.A. Vyrko a, O.N. Bubel’ a, Nguyen Ai Viet b a b
Belarusian State University, Minsk 220030, Belarus Institute of Physics and Electronics, Hanoi, Viet Nam
a r t i c l e
i n f o
Article history: Received 1 October 2009 Accepted 30 December 2009 Available online 8 February 2010 Keywords: Carbon nanotubes Raman scattering Radial breathing mode Peierls distortions Kekule structure
a b s t r a c t The influence of bond length dimerization due to Peierls distortions on totally symmetric vibrations of armchair carbon nanotubes is considered theoretically. It is shown that the radial and tangential motions for quinoid type dimerization are mixed and the radial breathing mode for Kekule type dimerization is splitted. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNT) have unique mechanical and electrical properties which are important for applications [1,2]. They can be either metallic or semiconducting depending on their chirality and diameter. One of the fundamental features of metallic CNTs is the possibility of Peierls distortions [3] of the lattice structure at low temperature. Until now this phenomenon has not been properly investigated and experimentally identified, in spite of the fact that it has important consequences for the thermal and electrical conductivity of CNTs [4]. Usually it is considered that all bond lengths in CNTs are equal [1,2], and possible bond length dimerization due to Peierls distortions at low temperatures is ignored. In [5,6] it was shown within the framework of semiempirical quantum chemical PM3 [7] calculations that the ground state of the (5,5) armchair CNT is Kekule type distorted with 0.003 nm bond length difference and that the transition state between the equivalent Kekule structures is a quinoid type structure having an excess energy of 3 meV per carbon atom. A question arises, how this feature can be verified experimentally. The Raman spectroscopy is widely used for experimental characterization of CNTs [8]. The most intense Raman lines in the CNTs’ spectrum correspond to totally symmetric vibrations in the Brillouin zone center (C point). Among these vibrations the radial breathing mode is of special interest because of its uniqueness for CNTs and diameter dependence of its frequency. Today Raman
* Corresponding author. Tel.: +375 17 2095110. E-mail addresses: [email protected], [email protected] (N.A. Poklonski). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.12.033
scattering experiments for individual CNTs are possible [9]. However, in the liquid helium temperature range, where manifestation of the bond length dimerization is possible, they are lacking. The aim of this article is to qualitatively elucidate the consequences of bond length dimerization of armchair CNTs on their totally symmetric vibrations in the Brillouin zone center, visible in Raman scattering experiments. For this purpose we employ symmetry considerations and a simple analytical model taking into account only nearest neighbor harmonic interactions of the CNT’s carbon atoms. We consider the (5,5) CNT as a characteristic example. For quantitative estimations we use PM3 calculations within the MOPAC 6.0 package [10].
2. Quinoid structure The (5,5) CNT with quinoid structure of chemical bonds is shown in Fig. 1. Its geometry is completely determined by two equilibrium bond lengths a0 and b0 and by the translational period along its axis L ¼ 2b0 cosðh0 =2Þ or, equivalently, by the angle h0 between bonds. The elementary translational cell of the (5,5) CNT contains 20 carbon atoms and has a D5d point symmetry group, which gives three totally symmetric vibrations in the Brillouin zone center. The infinitely long (5,5) CNT also has mirror symmetry planes rv and rh , glide symmetry plane r0v and rotoreflection plane r0h (shown in Fig. 1), which exclude displacements along nanotube z axis from totally symmetric vibrations and leave only two totally symmetric vibration modes [11,12]. Let us consider totally symmetric motions of (5,5) CNT in a cylindrical coordinate system ðq; u; zÞ (Fig. 2), with the origin of z
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σv
a θ
Geometrical considerations show that a ¼ 2q sinð18 uÞ; b ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðq sin uÞ2 þ z2 ,
z b
θ
h ¼ 2 arctan½z=ðq sin uÞ; w ¼ 2 arctan½z=ðq sin u sin 18 Þ,
L
0
h ¼ 90 þ 2 arcsinð2q sin u cos 18 =bÞ, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 w0 ¼ arcsinð2z= b 4q2 sin u cos2 18 Þ þ arcsinð2 cos 18 Þ.
σh σh
Da ¼
a0
q0
Dq
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4q20 a20 Du;
Db ¼
pffiffiffi 3Dz þ Dðq sin uÞ;
2d 8d sin 18 ; Dw ¼ ; b0 b0 f pffiffiffi 3Dz cos 18 Dw pffiffiffi ; Dw0 ¼ ; Dh 0 ¼ 2 b0 f
Dh ¼ σv CNT axis Fig. 1. (5,5) CNT with C–C-bond dimerization of quinoid type; L is the translational period.
(ρ, ϕ, z)
z
a
ρ
U¼ ϕ 36°
σv
D5d (5,5) CNT
σv
Fig. 2. Cylindrical coordinates ðq; u; zÞ of (5,5) CNT atoms; C–C-bonds a and b are shown.
and u in the middle of the sloped bond, corresponding to the intersection of the bond with glide plane. During totally symmetric vibrations z and u coordinates of the atoms of this bond differ only in sign and their distances to the nanotube axis are equal, therefore the motion of the nanotube is completely determined by the coordinates of a single carbon atom in the chosen coordinate system. The Hamiltonian H of totally symmetric vibrations of the (5,5) CNT translational unit cell, taking into account only nearest neighbor carbon atom interactions, is written as
" _ Þ2 þ mz_ 2 þ H ¼ 10 mq_ 2 þ mðq0 u 2
! 2 a20 sin 18 2
b0
f2
;
ð3Þ
and the potential energy U of totally symmetric vibrations of the (5,5) CNT takes a simple form (depending only on two parameters, bond stretching stiffnesses K a and K b ):
σh
0
ð2Þ
pffiffiffi 2 where the notations d ¼ Dz 3Dðq sin uÞ; f ¼ 3 þ sin 18 are introduced, and the equilibrium value h0 ¼ 120 is chosen. Substituting Eq. (2) into Eq. (1), we can see that the coupling between motions along the nanotube axis and perpendicular to it vanishes when
K b ¼ 4K h þ 32 2K w þ K w0
b θ
For
the small deviations of geometrical parameters from their equilibrium values in the linear approximation this gives:
ða a0 Þ2 K a þ ðb b0 Þ2 K b 2 2
þ ðh h0 Þ2 b0 K h þ 2ðh0 h00 Þ2 a20 K h0 þ ðw w0 Þ2 b0 K w i þ2ðw0 w00 Þ2 a20 K w0 ¼ T þ U;
ð1Þ
where T and U are kinetic and potential energy of the atom vibrations, K a and K b are stiffnesses of a and b bonds, K h and K h0 are angle deformational stiffnesses, K w and K w0 are dihedral angle deformational stiffnesses, m is the mass of a carbon atom, q0 ; a0 ; b0 ; h0 ; h00 ; w0 ; w00 are the equilibrium values of the CNT radius, bonds and angles, 10 is the number of horizontal bonds in the (5,5) CNT unit cell.
! 2 K a a20 K b b0 þ ðDqÞ2 2q20 4q20 " !# 2 a20 b0 þ K a 2 2 þ K b 4 2 ðDðq0 uÞÞ2 2q0 4q0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 a2 b0 b0 a0 @ 1 Ka þ Kb 1 02 ADqDðq0 uÞ: q0 q0 16q20 4q0
ð4Þ
From Eq. (4) one can see that q and u are not normal coordinates for totally symmetric vibrations of armchair CNTs. Even in the limit of equal bonds (absence of dimerization) there exists a small coupling between the radial and tangential motions of carbon atoms. Quinoid type dimerization makes an additional contribution to this coupling proportional to K b b0 K a a0 . This is essential [13] for the electron–phonon matrix element, which determines the intensity p offfiffiffi Raman scattering. The frequencies (x ¼ k for ½x ¼ cm1 Þ of the totally symmetric vibrations of CNT can be obtained from the characteristic equation, for Hamiltonian (4),
k2 2kðK a þ 2K b Þ þ K a K b a2 ¼ 0;
ð5Þ
where
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u ffi 2 a0 u b b 1 a2 1 02 a ¼ t2 1 0 2 þ 0 q0 q0 2 16q0 4q0 is determined by the CNT geometry. When dimerization is absent ða0 ¼ b0 ; K a ¼ K b ¼ KÞ, the roots of Eq. (5) are
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k1;2 ¼ 3K 1 1 a20 =9 :
ð6Þ
Here k1 corresponds to the high energy transverse optical (TO) mode (component of Raman G line) with frequency xTO ¼ 1590 cm1 [8], and k2 corresponds to the radial breathing mode (RBM), a0 corresponds to the non-dimerized case. If we choose
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z
σh
LK
σh
σv
σv
Ch
Fig. 3. Plane evolvement of the translational unit cell of the (5,5) CNT with Kekule structure. Ch is the chiral vector. Dotted lines show the primitive two-dimensional unit cell; LK is the translational period along CNT axis z; rv , rh are the mirror, r0v is the glide and r0h is the rotoreflection planes.
k1 ¼ x2TO , then K ¼ 430692 cm2 ¼ 30:45 104 dyn=cm which is comparable with the value K ¼ 36:50 104 dyn=cm from [14]. For this K value the RBM frequency xRBM ¼ 236:754 cm1 , which is somewhat lower than the commonly [15] accepted value 338:9 cm1 . Two parameters K a and K b of our model for the quinoid type structure can be adjusted to the frequencies of two totally symmetric modes of the (5,5) CNT. From Eq. (5) their ratio is
Ka a 1 ¼ bþ b Kb 4
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 a2 1 2; bþ b 16
where b ¼ xTO =xRBM . For example, when xRBM ¼ 236:754 cm1 ; xTO ¼ 1590 cm1 and the geometrical parameters are taken from [5], then K a =K b 1:01, with their mean value as in the non-dimerized case.
in Fig. 2 and coordinates q1 ; u1 of the adjacent atom. Coordinate z1 of this atom is fixed by the mirror plane rh . The (5,5) CNT has five totally symmetric vibrations according to the number of degrees of freedom. They are shown in Fig. 4. In addition to the two totally symmetric modes of non-dimerized and quinoid structures, the Kekule structure has a longitudinal totally symmetric mode, and RBM and transverse modes become splitted. In order to estimate quantitatively the magnitude of this splitting, we have carried out PM3 calculations of the (5,5) CNT totally symmetric modes in the Brillouin zone center for the Kekule structure obtained in [5] using MOPAC 6.0 package [10]. Both RBM and TO modes are splitted according to this calculations. The frequencies obtained are xRBM1 ¼ 376 cm1 and xRBM2 ¼ 388 cm1 for RBM, and xTO1 ¼ 1666 cm1 and xTO2 ¼ 1672 cm1 for TO modes. 4. Conclusions
3. Kekule structure The plane evolvement of the translational unit cell of the (5,5) CNT with Kekule structure is shown in Fig. 3. Kekule structure has the same D5d symmetry as quinoid structure but the translational period in this case is tripled. The translational unit cell of Kekule structure contains 60 carbon atoms instead of 20 in the case of quinoid type dimerization. The mirror ðrv ; rh Þ, glide ðr0v Þ and rotoreflection ðr0h Þ symmetry planes are also shown in Fig. 3. Due to these planes only five independent variables are needed for complete description of the totally symmetric vibrations of an armchair CNT with Kekule structure. We can choose for these variables cylindrical coordinates ðq; u; zÞ of one of the carbon atoms as
In this paper we propose to explore the ground state properties of armchair carbon nanotubes in low temperature Raman scattering experiments. In order to qualitatively elucidate totally symmetric vibrations of armchair CNTs in the Brillouin zone center, visible in Raman scattering experiments, we have developed a simple analytical model employing symmetry considerations and nearest neighbor harmonic interactions of CNT’s carbon atoms. Parameters of this model (bond stiffnesses) can be adjusted to experiment. We show that quinoid type dimerization (without Peierls gap opening) does not lead to the breathing mode splitting, but to the mixing of radial and tangential carbon atom displacements in the totally symmetric vibrations of armchair CNTs. Kekule type distortions (with Peierls gap) lead to the breathing mode splitting.
σh σh
σv
(a)
σv
(b)
(c)
(d)
(e)
Fig. 4. Totally symmetric vibrations of the armchair CNT with Kekule structure: radial (a, b); transverse (c, d); longitudinal (e).
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According to our PM3 calculations using MOPAC 6.0 package, a quantitative estimate of this splitting for the case of geometrical structure of (5,5) CNT calculated in [5] is found to be 12 cm1 . Thus, the problem of the ground state structure of armchair CNTs can be solved experimentally. Acknowledgment This work has been supported by the BFBR (Grant Nos. F08Vn003 and F08R-061). References [1] A. Jorio, G. Dresselhaus, M.S. Dresselhaus (Eds.), Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties, and Applications, Springer, Berlin, 2008. [2] M.J. O’Connell (Ed.), Carbon Nanotubes: Properties and Applications, Taylor & Francis, New York, 2006.
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