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Spin Dependent Transport Through a Carbon Nanotube Quantum dot in the Kondo Regime 23
Blanter, Ya. M. & Büttiker, M. (2000). Shot noise in mesoscopic conductors. Phys. Rep. 336, pp.
1-166
Binasch G., Grünberg, P., Saurenbach, F. & Zinn W. (1989). Enhanced magnetoresistance in
layered magnetic structures with antiferromagnetic interlayer exchange. Phys. Rev.
B, 39, pp. 4828-4830
Brataas, A., Tserkovnyak, Y., Bauer, G. E. W. & Halperin, B. I. (2002). Spin battery operated by
ferromagnetic resonance. Phys. Rev. B, 66, pp. 060404-1-4
Buitelaar, M. R., Bachtold, A., Nussbaumer T., Iqbal, M. & Schönenberger, C. (2002). Multiwall
carbon nanotubes as quantum dots. Phys. Rev. Lett., 88, pp. 156801-1-4
Bułka, B. R. & Lipi´nski, S. (2003). Coherent electronic transport and Kondo resonance in
magnetic nanostructures. Phys.Rev.B, 67, pp. 024404-1-8
Büsser, C. A. & Martins, G. B. (2007). Numerical results indicate a half-filling SU(4) Kondo
state in carbon nanotubes. Phys. Rev. B, 75, pp. 045406-1-7
Choi, M. S., Sanchez, D. & Lopez, R. (2004). Kondo effect in a quantum dot coupled to
ferromagnetic leads: A numerical renormalization group analysis. Phys.Rev.Lett,
92, pp. 056601-1-4
Choi, M., López, R., Aguado, R. (2005). SU(4) Kondo effect in carbon nanotubes. Phys. Rev.
Lett. 95, pp. 067204-1-4
Cobden, D. H. & Nygård, J. (2004). Shell filling in closed single-wall carbon nanotube quantum
dots. Phys.Rev.Lett., 89, pp.046803-1-4
Coleman, P. (1987). Mixed valence as an almost broken symmetry. Phys. Rev. B 35, 5072-5116
Cottet, A. & Choi, M. S. (2006). Magnetoresistance of a quantum dot with spin-active
interfaces. Phys.Rev.B74, pp. 235316-1-9
Cottet, A., Kontos, T., Sahoo, S., Man, H. T., Choi, M. S., Belzig, W., Bruder, C., Morpurgo, A. F.
& Schönenberger, C. (2006). Nanospintronics with carbon nanotubes. Semicond. Sci.
Technol., 21, pp. 78-95
Cronewett, S. M., Oesterkamp, T. H. & Kouwenhoven, L. P. (1998). A tunable Kondo effect in
quantum dots. Science 291, pp. 540-544
Das Sarma, S. (2001). Spintronics. American Scientist, 89, pp. 516-523
De Franceschi, S., Hanson, R., van der Wiel, W. G., Elzerman, J. M., Wijpkema, J. J.,
Fujisawa, T., Tarucha, S. & Kouwenhoven (2002). Out-of-equilibrium Kondo effect
in mesoscopic devices. Phys. Rev. Lett. 89, pp. 156801-1-4
de Haas, W. J, de Boer, J. & van den Berg G. J. (1933). The electrical resistance of gold, copper
and lead at low temperatures. Commun. Kamerlingh Onnes Lab., No. 233b; Physica,1,
pp. 1115-1124
Delattre, T., Feuillet-Palma, C., Herman, L. G., Morfin, P., Berroir, J. M., Fève, G., Plaçais, B.,
Glattli, D. C., Choi, M. S., Mora, C. & Kontos, T. (2009). Noisy Kondo impurities.
Nature Physics, 5, pp. 208-212
Dresselhaus, M. S., Dresselhaus, G. & Avouris, Ph. (2001). Carbon Nanotubes. Synthesis,
Structure, Properties, and Applications, Springer-Verlag, Berlin
Foros, J., Brataas, A., Tsevkovnyak, Y. & Bauer, G. E. W. (2005). Magnetization noise in
magnetoelectronic nanostructures. Phys.Rev.Lett., 95, pp. 016601-1-4
Frolov, S. M., Venkatesan, A., Yu, W., Folk, J. A. & Wegscheider, W. (2009). Electrical Generation
of Pure Spin Currents in a Two-Dimensional Electron Gas. Phys. Rev. Lett. 102, pp.
116802-1-4
Galpin, M. R., Logan, D. E., Krishnamurthy, H. R. (2006). Dynamics of capacitively coupled
double quantum dots. J. Phys.: Condens. Matter, 18, pp. 6571-6583
305
Spin Dependent Transport Through a Carbon Nanotube Quantum Dot in the Kondo Regime
24 Will-be-set-by-IN-TECH
Galpin, M. R., Jayatilaka, F. W., Logan, D. E. & Anders F. B. (2010). Interplay between Kondo
physics and spin-orbit coupling in carbon nanotube quantum dots. Phys.Rev.B, 81,
pp. 075437-1-14
Goldhaber-Gordon, D., Shtrikman, H., Mahlul, D., Abuschmagder, D., Meiraev, U. & Kastner,
M. A. (1998). Kondo effect in a single-electron transistor. Nature 391, pp. 156-159
Grobis, M., Rau, I. G., Potok, R. M., Shtrikman, H. & Goldhaber-Gordon, D. (2008). Universal
scaling in nonequilibrium transport through a single channel Kondo dot. Phys. Rev.
Lett., 100, pp. 246601-1-4
Grove-Rasmussen, K., Jørgensen, H. J. & Lindelof, P. E. (2007). Fabry-Perot interference,
Kondo effect and Coulomb blokade in carbon nanotubes. Physica E, 40, pp. 92-98
Hai, P. N., Ohya, S., Tanaka, M., Barnes, S. E. & Maekawa, S. (2009). Electromotive force and
huge magnetoresistance in magnetic tunnel junctions. Nature, 458, pp. 489-492
Haldane, F. D. M. (1978). Scaling theory of the asymmetric Anderson Model. Phys.Rev.Lett.,
40, 416-419
Haug, H. & Jauho, A. P. (1998).Quantum Kinetics in Transport and Optics of Semiconductors,
Springer, Berlin
Hauptmann, J. R., Paaske, J. & Lindelof, P. E. (2008). Electric-field-controlled spin reversal in a
quantum dot with ferromagnetic contacts. Nature Physics, 4, pp. 373-376
Hewson, A. C. (1993). The Kondo problem to heavy fermions, pp.1-472, Cambridge University
Press, ISBN 978-0521599474, Cambridge
Holleitner, A., Chudnovskiy, A., Pfannkuche, D., Eberl, K. & Blick R. H. (2004). Pseudospin
Kondo correlations versus hybridized molecular states in double quantum dots.
Phys.Rev.B, 70, pp. 075204-1-5
Iijima, S. (1991). Helical microtubules of graphitic carbon. Nature, 354, pp. 56-58
Iijima, S. & Ichihashi, T. (1993). Single-shell carbon nanotube of 1-nm diameter. Nature, 363,
pp. 603-605
Jarillo-Herrero, P., Sapmaz, S., Dekker, C., Kouwenhoven, L. P. & van der Zant, H. S. J. (2004).
Electron-hole symmetry in a semiconducting carbon nanotube quantum dot. Nature,
429, 389-392
Jarillo-Herrero, P., Kong, J., van der Zant, H. S. J., Dekker, C., Kouvenhoven, L. P. & De
Franceschi, S. (2005). Orbital Kondo effect in carbon nanotubes. Nature, 434, pp.
484-488
Julliére, M. (1975). Tunneling between ferromagnetic films. Phys. Rev. Lett., 54A, pp. 225-226
Kondo, J. (1964). Resistance minimum in dilute magnetic alloys. Prog. Theor. Phys., 32, pp.37-49
Krychowski, D., Lipi´nski, S. & Krompiewski., S. (2007). Spin dependent transport through a
carbon nanotube quantum dot in magnetic field. J. Alloys and Compd., 442, pp. 379-381
Krychowski, D. & Lipi´nski, S. (2008). Tunnel magnetoresistance in carbon nanotube quantum
dots. Acta Phys. Pol. A, 113, 545-548
Krychowski, D. & Lipi´nski, S. (2009). Kondo effect in carbon nanotube quantum dot in a
magnetic field. Acta Phys. Pol. A, 115, pp. 293-29
Lacroix, C. (1998). Density of states for the Anderson model. J. Phys. F: Metal Phys., 11, pp.
2389-2397
Langreth, D. C. (1966). Friedel sum rule for Anderson’s model of localized impurity states.
Phys. Rev., 150, pp. 516-518
Liang, W., Bockrath, M., Bozovic, D., Hafner, J. H., Tinkham, M. & Park, H. (2001). Fabry-Perot
interference in a nanotube electron waveguide. Nature, 411, pp. 665-669
306
Electronic Properties of Carbon Nanotubes
Spin Dependent Transport Through a Carbon Nanotube Quantum dot in the Kondo Regime 25
Liang, W., Bockrath, M. & Park, H. (2002). Shell filling and exchange coupling in metallic
single-walled carbon nanotubes. Phys. Rev. Lett., 88, pp. 126801-1-4
Liang, W. J., Shores, M. P., Bockrath, M., Long, J. R. & Park (2002). Kondo resonance in a
single-molecule transistor. Nature 417, pp. 725-729
Lim, J. S., Choi, M., Choi, M. Y., López, R. & Aguado, R. (2006). Kondo effect in carbon
nanotubes: From SU(4) to SU(2) symmetry. Phys.Rev.B74, pp. 205119-1-16
Lipi´nski, S. & Krychowski, D. (2005). Orbital Kondo effect and spin polarized transport
through quantum dots. Phys. S tatus. Solidi B 242, pp. 206-209
Lipi´nski, S. & Krychowski, D. (2010). Spin-polarized current and shot noise in a carbon
nanotube quantum dot in the Kondo regime. Phys.Rev.B, 81, pp. 115327-1-19
Loss, D. & DiVincenzo, D. P. (1998). Quantum computation with quantum dots. Phys. Rev. A,
57, 120-126
Makarowski, A., Zhukov, A., Liu, J. & Finkelstein, G. (2007). SU(2) amd SU(4) Kondo effect in
carbon nanotube quantum dots. Phys. Rev. B, 75, 241407(R)-1-4
Martinek, J., Utsumi, Y., Imamura, H., Barna´s, J., Maekawa, S., König, J. & Schön, G. (2003).
Kondo Effect in Quantum Dots Coupled to Ferromagnetic Leads. Phys. Rev. Lett. 91,
pp. 127203-1-4
Martinek, J., Sindel, M., Borda, L., Barna´s, Bulla, R., König, J., Schön, G., Maekawa, S. & von
Delft, J. (2005). Gate-controlled spin splitting in quantum dots with ferromagnetic
leads in the Kondo regime. Phys. Rev. B., 72, pp. 121302-1-4
Mizumo, M., Kim, E. H. & Martins G. B. (2009). Transport and strong-correlation phenomena
in carbon nanotube quantum dots in a magnetic field. J. Phys.: Condens. Matter 21, pp.
292208-1-5
Ng, T. K. (1996). AC response in the nonequilibrium Anderson impurity model. Phys. Rev.
Lett., 76, pp. 487-490
Novoselov, K. S., Geim, A. K., Morozov, S. V., Jiang, D., Katsnelson, M. I., Grigorieva, I. V.,
Dubonos, S. V. & Frisov, A. A.(2005). Two-dimensional gas of massless Dirac fermions
in graphene. Nature, 438, pp. 197-200
Nygård, J., Cobden, D. H. & Lindelof, P. E. (2000). Kondo physics in carbon nanotube. Nature,
408, pp. 342-346
Paaske, J., Rosch, A., Wölfe, P., Mason, N., Marcus, C. M. & Nygård, J. (2006). Non-equilibrium
singlet-triplet KOndo effect in carbon nanotubes. Nature Physics 2, pp. 460-464
Park, J., Pasupathy, A. N., Goldsmith, J. I., Chang, C., Yaish, Y., Petta, J. R., Rinkoski, M.,
Sethna, J. P., Abruna, H. D., McEuen, P. L. & Ralph, D. C. (2002). Coulomb blokade
and the Kondo effect in single-atom transistor. Nature 417, pp. 722-725
Sasaki, S., Amaha, S., Asakawa, N., Eto, M. & Tarucha S. (2004). Enhanced Kondo effect
via tuned orbital degeneracy in a spin 1/2 artificial atom. Phys.Rev.Lett.93, pp.
017205-1-4
Sergueev, N., Sun, Q. F., Guo, H., Wang, B. G. & Wang, J. (2002). Spin-polarized transport
through a quantum dot: Anderson model with on-site Coulomb repulsion. Phys. Rev.
B, 65, 165303-1-9
Shen, S. Q. (2008). Spintronics and spin current. AAPPS Bulletin, Vol. 18, No. 5, pp. 29-36
´
Swirkowicz, R., Wilczy´nski, M. & Barna´s, J. (2006). Spin-polarized transport through a
single-level quantum dot in the Kondo regime. J. Phys.: Condens. Matter., 18, pp.
2291-1-9
Tanaka, M. & Higo Y. (2001). Large tunneling magnetoresistance in GaMnAs/AlAs/GaMnAs
ferromagnetic semicontuctor tunnel junction. Phys. Rev. Lett., 87, pp. 026602-1-4
307
Spin Dependent Transport Through a Carbon Nanotube Quantum Dot in the Kondo Regime
26 Will-be-set-by-IN-TECH
Wallace, P. R. (1947). The band theory of graphite. Phys. Rev., 71, pp. 622-634
Wei, Ch., Cho K. & Srivastava D. (2003). Tensile strength of carbon nanotube under realistic
temperature and strain rate. Phys.Rev.B, 67, pp. 115407-1-6
Wilhelm, U., Schmid, J., Weis, J. & v. Klitzing, K. (2002). Experimental evidance for spinless
Kondo effect in two electrostically coupled quantum dot system. Phisics E, 14, pp.
385-390
Wu, F., Danneau, R., Queipo, P., Kauppinen, E., Tsuneta, T. & Hakonen, P. J. (2009).
Single-walled carbon nanotube weak links in Kondo regime with zero-field splitting.
Phys.Rev.B, 79, pp. 073404-1-4
Xiong, Z. H., Wu, D., Valy Vardeny, Z. & Shi, J. (2004). Giant magnetoresistance in organic
spin-valves. Nature, 427, pp. 821-824
Žuti´c, I., Fabian, J. & Das Sarma, S. (2004). Spintronics: Fundamentals and applications. Rev.
Mod. Phys., 76, pp. 323-410
308
Electronic Properties of Carbon Nanotubes
14
A Density Functional Theory Study of Chemical
Functionalization of Carbon Nanotubes -
Toward Site Selective Functionalization
Takashi Yumura
Department of Chemistry and Materials Technology, Kyoto Institute of Technology,
Matsugasaki, Sakyo-ku, Kyoto
Japan
1. Introduction
Single walled carbon nanotubes exhibit metallic or semiconducting characters, depending
on how to roll-up graphene sheet (Iijima, 1991; Iijima & Ichihashi, 1993; Dresselhaus et al.,
1996; Saito et al., 1998). In addition of the unique electronic properties, functionalization of
nanotubes can expand the versatility because of enhancing their solubility as well as feature
expansion by attached functional groups (Hirsch, 2002; Haddon, 2002; Niyogi et al., 2002;
C.A. Dyke & Tour, 2004; Tasis et al., 2006; Prato et al, 2008; Vzquez & Prato, 2009; Strano et
al., 2009; Karousis et a;., 2010). The functionalization is roughly categorized into two types;
noncovalent- and covalent-functionalization. In covalent functionalization, only highly
reactive reagents are available. In a pioneering paper discussing the covalent bond
formation, Haddon et al. reported that nanotubes were successfully functionalized by a
divalent carbene-derivative CCl
2
(Chen et al., 1998; Chen et al., 1998; Kamarás et al., 2003). In
the experiments, its divalent atom binds into two carbon atoms of a nanotube surface. After
the publication, different types of cyclopropanized nanotube were generated by a single
Bingel reaction (Coleman et al., 2003; Worsley et al., 2004; Umeyama et al., 2007). Note that
Bingel reaction has been also utilized for selective functionalization of fullerenes (Diederich
& Thilgen, 1996). Another type of covalent functionalization of nanotubes is radical
additions. In this case, radical species, whose precursors are alkyhalides, diazonium salts,
alky-lithium and so on, can make one covalent bond with a nanotube surface (Holzinger et
al., 2001; Bahr & Tour , 2001; Bahr et al., 2001; Bahr et al., 2001; Ying et al., 2003; Saini et al.,
2003; Peng et al., 2003; Stevens et al., 2003; Peng et al., 2003).
From a viewpoint of the electronic properties of nanotubes, covalent functionalization can
modulate their band structures. If their bands in the vicinity of the Fermi level are
significantly perturbed upon the functionalization, one can detect a sizable change of
physically observable values, such as the conductivity. Recent theoretical studies evaluated
the conductance of monovalent or divalent functionalized nanotubes (Park et al., 2006; Lee
& Marzari, 2006; Lpez-Bezanilla et al., 2009). The theoretical findings suggested that divalent
functionalization can preserve the conductance of its pristine nanotube, whereas
monovalent case destroys it (Park et al., 2006; Lpez-Bezanilla et al., 2009). The theoretical
Electronic Properties of Carbon Nanotubes
310
results inspired experimentalists to measure the conductance of functionalized nanotubes.
In fact, changes of the conductance of nanotubes upon the functionalization were observed
experimentally (Goldsmith et al., 2007; Kanungo et al., 2009).
Although our knowledge on functionalization of nanotubes has been expanded, “site-
selective” functionalization in either divalent or monovalent manners has been still a
challenging target in nanotube chemistry. The difficulty in achieving site selective
functionalization comes from high reactivity of functional groups. Of course, higher reactive
species are necessary to attach to a rigid tube-bond. In an opposed action they would attack
randomly at almost equivalent tube-bonds. If one can control binding sites for carbene-
derivatives on a nanotube surface to pattern the functional groups, such functionalization
can open the door toward designing novel tube-based nanomachines. For example, Globus
et al. proposed that “paddle wheels” can be constructed conceptually by addition of
benzynes onto a nanotube circumferentially, whereas the addition in the axial direction
results in “gear teeth” (Han & Globus, 1997; Globus et al., 1998). Accordingly, selective
functionalization of nanotubes is now required to design the unique nanomachines.
For the purpose of achieving site-selective functionalization of nanotubes, it is indispensable
to elucidate how functional groups interact with an outer or inner surface of a nanotube, as
well as how the resultant interactions perturb its sp
2
bonding frameworks. In this direction,
recent quantum chemistry calculations would help to obtain an insight of changes of
nanotube surfaces upon the functionalization. In particular, recent advances in ab initio
density functional theory (DFT) calculations, where Khon-Sham equations including
electron correlation are solved iteratively (Hohenberg & Kohn, 1964; Kohn & Sham, 1965;
Parr & Yang, 1996), allow us to accurately calculate large-scale systems like nanotubes.
Enhancing the reliability of DFT calculations is due to improvement in the quality of
exchange-correlation functionals (Becke, 1988; Lee et al., 1988; Becke, 1992; Becke, 1992;
Becke, 1993; Stephens et al., 1994; Vosko et al., 1980). Actually, recent DFT methods can
provide reliable data, such as energies and force in materials. Thus one can utilize the DFT
calculations to look at CC bondings in nanotubes “on an atomic level”, and to devise a
plausible strategy toward site-selective functionalization of nanotubes.
Because of retaining the unique electronic properties of nanotubes after the divalent
functionalization, we discuss in this chapter the properties of nanotubes functionalized by
carbene-derivatives from a viewpoint of computational chemistry based on DFT
calculations. The aim in this study is finding out an approach to how site-selective
functionalization of nanotubes can be achieved. This chapter is organized as follows. In
section 2.1, we briefly present how computational methods influence a description of its CC
bonding network by investigating the addition of carbene into naphthalene as a simple test
case. After the discussion, results of mono- or double functionalization of nanotubes are
given. First, we discuss chemical reactivity of inner or outer tube surface toward carbene
with the focus on surface modification by the addition in section 2.2. In sections 2.3, we
propose two strategies for site-selective double addition into a nanotube, based on the
findings in section 2.2. In section 2.3.1, we investigate whether surface modification induced
by the inner carbene addition can direct the next carbene into a specific site. On the other
hand, another strategy is introduced in section 2.3.2; the use of geometrical constraints of a
bismalonate for site-selective addition into a nanotube. Finally we conclude this chapter in
Section 3.
A Density Functional Theory Study of Chemical Functionalization
of Carbon Nanotubes - Toward Site Selective Functionalization
311
2.1 Choice of computational methods
Prior to discussing functionalization of nanotubes by carbene-derivatives, we briefly explain
computational methods suitable to accurately describe the interactions. As a simple and
relevant test case, we focused on the addition of carbene into naphthalene with ten
electrons (Choi & Kertesz, 1998). In the test case, we employed three different calculations;
Hartree-Fock (HF), the second-order Møller-Plesset perturbation theory (MP2) (Szabo &
Ostlund, 1996), and density functional theory (DFT) methods. In the DFT calculations, GGA-
based PW91 functional and hybrid B3LYP functional are considered as a standard
functional. When the divalent atom of carbene (labeled by C11) attacks at the bridgehead
atoms of naphthalene (labeled by C1 and C6), the valence tautomerization of 1,6-
methano[10]annulene occurs, as shown in the upper of Figure 1.
Fig. 1. Total energies of 1,6-methano[10]annulene as a function of the d
1,6
value, obtained
from Hartree-Fock (black circles), second order Møller-Plesset perturbation theory (MP2)
(blue triangles), and density functional theory calculations. PW91 and B3LYP functionals
(purple open and red closed squares, respectively) are considered in the DFT calculations.
The basis set used is the 6-31G* basis set.
In the tautomerization, there are two types of structure; one is bisnorcaradiene form, and the
other is aromatic form. As shown in Figure 1, the two isomers can be distinguished by the
separation between the C1 and C6 atoms (the d
1,6
bond); the bisnorcaradiene form has a
shorter d
1,6
bond, whereas the aromatic form has a longer d
1,6
bond. Reflecting the d
1,6
values, the two tautomers have different number of electrons in naphthalene-moiety. In
the aromatic form, ten electrons remain in its naphthalene-moiety, while two electrons
transfer from naphthalene to form two CC bonds in the bisnorcaradiene form.
The d
1,6
value is not only a geometrical parameter, but it is key in relative stability of the two
tautomers of 1,6-methano[10]annulene. Figure 1 displays total energies of 1,6-
methano[10]annulene against the d
1,6
value, depending on computational methods (HF,
Electronic Properties of Carbon Nanotubes
312
MP2, and DFT (B3LYP and PW91 (Perdew & Wang, 1992) methods). As shown in Figure 1,
there are two local minima that are energetically equivalent in the HF calculations. The
minima correspond to the bisnorcaradiene and aromatic forms. However, we obtained from
DFT and MP2 calculations only an aromatic form as a stable conformation. The calculated
d
1,6
values are around 2.3 Å, which can reproduce that obtained experimentally (2.235 Å)
(Bianchi et al., 1980). The DFT and MP2 methods include electron correlation, but the HF
method does not include. Thus, the differences indicate that electron correlation should be
important in relative stability of the two tautomers, and describing the geometrical change
of naphthalene induced by the interactions with carbene.
In general, MP2 calculations are more time-consuming than DFT calculations. Since
nanomaterials, such as nanotubes, contain few hundred of atoms in the unit cell, MP2
methods are not doable to estimate the properties of nanomaterials. Actually our systems
introduced in this chapter contain up to 240 carbon atoms of a nanotube plus a few
functional groups in the unit cell in periodic boundary condition (PBC) calculations.
Instead, DFT methods, which can describe their properties in a relatively accurate
manner, are less time-consuming. Accordingly, we will investigate interactions between
carbene-derivatives and a nanotube surface by using DFT methods with B3LYP or PW91
functionals.
2.2 Addition of carbene into a nanotube, and concomitant surface modification
Let us first discuss the addition of carbene into the outer (Chen et al., 2004; Bettinger, 2004;
Zhao et al., 2005; Bettinger, 2006; Zheng et al., 2006; Yumura & Kertesz, 2007; Yumura et al.,
2007; Lee & Marzari, 2008) or inner (Yumura & Kertesz, 2007; Yumura et al., 2007) surface of
the (5,5) or (10,10) nanotube, whose optimized geometries are seen in Figures 2 and 3.
As mentioned above, the divalent atom of carbene binds into a CC bond of sp
2
systems.
The (5,5) and (10,10) nanotubes have two types of CC bond (a CC bond orthogonal or
slanted relative to the tube axis). Accordingly, there are two types of optimized structure
for an inner (outer) carbene binding into an armchair nanotube. Figures 2 and 3 show the
preferences of an orthogonal site over a slanted site as the single carbene addition,
irrespective of an inner or outer addend. In terms of the geometrical features, a slanted
bond retains after the addition of an inner or outer carbene, as shown in Figures 2 and 3
(Yumura & Kertesz, 2007; Yumura et al., 2007). When carbene binds into an orthogonal
site, different behaviors were found between the inner and outer surfaces. The inner
addition retains the orthogonal bond, whereas the outer addition breaks it. The different
behaviors come from more rigorous restriction of surface modification toward the tube
center. This is understandable from a strain analysis based on -orbital axial vector
(POAV) (Haddon, 1998), where the strain energy is proportional to the square of a POAV
value. In fact, a POAV value would increase by the surface modification toward the
center. Then the strain energy would be more pronounced, if carbene freely binds into the
inner surface rather than into the outer surface. As a result, the concave nanotube surface
prohibits inner carbene from freely binding into the inner surface to keep the CC bond at
the binding site.
Breaking or retaining the CC bond at the binding site of a nanotube is reminiscent of the
valence tautomerization of 1,6-methano[10]annulene. From an analogy to 1,6-
methano[10]annulene whose tautomers have different number of electrons, we can
understand the modification of a nanotube upon the carbene addition obtained from DFT
A Density Functional Theory Study of Chemical Functionalization
of Carbon Nanotubes - Toward Site Selective Functionalization
313
Fig. 2. Optimized structures for the addition of carbene into an inner or outer surface of the
(5,5) nanotube. Their relative energies are also given in kcal/mol, lengths of the CC bond at
the binding site (the d
1,6
values) in Å. These values were obtained from PW91 PBC
calculations (Yumura & Kertesz, 2007).
Fig. 3. Optimized structures for the addition of carbene into an inner or outer surface of the
(10,10) nanotube. Their relative energies are also given in kcal/mol, lengths of the CC bond
at the binding site (the d
1,6
values) in Å. These values were obtained from PW91 PBC
calculations (Yumura et al., 2007).
calculations. To interpret the surface modification, we applied a basic concept in organic
chemistry, the Clar concept (Clar, 1972). Based on Clar valence bond (VB) representation, all
electrons in the pristine armchair nanotubes can be given by aromatic sextet (Matsuo et al.,
Electronic Properties of Carbon Nanotubes
314
2003; Ormsby & King, 2004; Ormsby & King, 2007). In the inner addition into a nanotube
where the d
1,6
CC bond retains, two electrons migrate from the nanotube. Due to the
electron migration, the CC bondings of the nanotube attached by inner carbene are
significantly perturbed. Based on the Clar representation, the perturbed surface consists of
two butadiene (B) patterns near the binding site plus quinonoid (Q) pattern in the
circumferential direction (Figure 4(a)).
Fig. 4. Clar representation of modified nanotube surface upon the functionalization by inner
(a) or outer (b) carbene. Aromatic sextets are given by circles. Note that pristine armchair
nanotubes have all electrons represented by aromatic sextets.
Basically, the DFT-optimized geometries for the nanotubes attached by inner CH
2
are
explainable from the Clar representation, as shown in Table 1 and Figure 5. Actually, bond-
length alternations in the circumferential direction follow the Q pattern in Figure 5, but the
alternations are significant only near the binding site. Similar perturbation patterns can be
seen in a defected nano-peapod (Figure 6) where a defected C
60
(C
1
-C
59
) makes two covalent
bonds with the inner surface of the (10,10) nanotube (Yumura et al., 2007). Intrinsic reaction
coordination (IRC) analyses based on B3LYP calculations suggest that C
60
directly converts
into a C
1
-C
59
defect as the initial intermediate with a barrier of 8.4 eV (Yumura et al., 2007).
Experimentally, the formation of defective fullerenes has been actually observed under
electron irradiation in transmission electron microscopy measurements (TEM) (Urita et al.,
2004; Sato et al., 2006). Thus, one can utilize the activation of guest molecules (e.g. by
heating-treatment or electron irradiation) to locally perturb a host nanotube surface from
inside out.