Chapter 9 Modeling and simulations of carbon nanotubes

Nanomaterials: Design and Simulation P. B. Balbuena & J. M. Seminario (Editors) © 2007 Elsevier B.V. All rights reserved.

Chapter 9

Modeling and Simulations of Carbon Nanotubes Alper Buldum Department of Physics, The University of Akron, Akron, OH 44325

1. Introduction Carbon is a special element as it provides a wide variety of molecules and materials in different shapes and forms. Nanotube, an interesting form of carbon, was discovered by Iijima [1] in 1991. It was in the shape of multiple coaxial carbon fullerene shells (multi-wall nanotubes, or MWNTs). Later, in 1993, single fullerene shells (singlewall nanotubes, or SWNTs) were synthesized [1, 2]. Theoretical studies followed the discoveries quickly, predicting novel electronic [3–5] and mechanical properties of nanotubes [6, 7]. Many experiments confirmed these predictions including electronic structure, remarkable strength and toughness of nanotubes [8–10]. A nanotube can be simply described as a sheet of graphite (or graphene), coaxially rolled to create a cylindrical surface (as shown in Figure 1(a)). In this way the 2D hexagonal lattice of graphene is mapped onto a cylinder of radius R. The mapping can be realized with different helicities resulting in different nanotubes. Each nanotube is characterized by a set of two integers (n m) related to the components of the chiral vector C = na1 + ma2 . The 2D hexagonal Bravais lattice vectors of graphene, a1 and a2 and C vectors are illustrated in Figure 1(b). The chiral vector is a circumferential vector and the tube is obtained by folding the graphene such that the two ends of C are coincident. The radius of the tube is given in terms of (n m) through the relation R = √ 2 2 ao n + m + nm/2 where a1  = a2  = a0 . When C involves only a1 (corresponding to (n 0)) the tube is called ‘zigzag’, and if C involves both a1 and a2 with n = m (corresponding to (n n)) the tube is called ‘armchair’ [11]. The chiral (n n) vector is rotated by 30 relative to that of the zigzag (n, 0) tube. SWNTs are found in the form of nanoropes, each rope consisting of up to a few hundred nanotubes arranged in a hexagonal lattice structure [12]. 227

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(5, 5)

C a2 a1

(a)

(8, 0)

(b)

Figure 1 (a) A single-wall nanotube is graphene wrapped on a cylinder surface. (b) Nanotubes are described by a pair of integers (n m) which indicate the graphite lattice vector components. A chiral vector can be defined as C = na1 + ma2 . Tubes are called ‘zigzag’ if either one of the integers is zero, ((n 0) or (0 m)), or called ‘armchair’ if both integers are equal (n n)

2. Electronic Properties 2.1. Electronic Structure As a nanotube is in the form of a wrapped sheet of graphite, its electronic structure is analogous to the electronic structure of a graphene [3–5]. Graphene has the lowest  ∗ -conduction band and the highest -valence band, which are separated by a gap in the entire hexagonal Brillouin zone (BZ) except at its K corners where they cross. In this respect, graphene lies between a semiconductor and a metal with Fermi points at the corners of the BZ. One can imagine an unrolled, open form of nanotube, which is graphene subject to periodic boundary conditions on the chiral vector. This in turn imposes quantization on the wave vector. This is known as ‘zone folding’, whereby the BZ is sliced with parallel lines of wave vectors, leading to subband structure. A nanotube’s electronic structure can thus be viewed as a zone-folded version of the electronic band structure of the graphene. When these parallel lines of nanotube wave vectors pass through the corners, the nanotube is metallic. Otherwise, the nanotube is a semiconductor with a gap of about 1 eV, which reduces as the diameter of the tube increases. Within this simple approach, (n m) nanotubes are metallic if n − m = 3× an integer. Consequently, all armchair tubes are metallic. Thus, the electronic structures of nanotubes are determined by their chirality and diameter, i.e. simply by their chiral vectors C. The first theoretical calculations were performed and the above simple understanding was provided much earlier than the first conclusive experiments [3–5]. In these early calculations, a simple one-band -orbital tight-binding model was used. Ab initio methods have also been used to investigate the electronic structure of SWNTs. However, different calculations have been at variance on the values of the band gap. For example, while the  ∗ – ∗ hybridization due to the curvature can be treated well by ab initio calculations [13], simple tight-binding methods are useful but may have limitations for small-radius nanotubes. In Figures 2(a) and 2(b), the band structure and density of states (DOS) of a (10, 10) tube are given, based on a tight-binding calculation. Samples prepared by laser vaporization consist predominantly of (10, 10) metallic armchair SWNTs.

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229

5 4 3 2

E (eV)

1 0 –1 –2 –3 –4 –5

Γ

kz

X0

5

10

15 0

(a)

(b)

5

10 15 20

G (2e2/h)

DOS (c)

Figure 2 (a) The band structure, (b) density of states and (c) conductance of a (10, 10) nanotube. The tight-binding model is used to derive the electronic structure. The conductance is calculated using the Green’s function approach with the Landauer formalism (from [14])

As it can be seen in Figure 2(a), band crossing is allowed, and the bonding - and antibonding  ∗ -states cross the Fermi level at kz = 2/3. In Figure 2(b) the DOS is plotted for the (10, 10) tube. The E −1/2 –singularities, which are typical for 1D energy bands, appear at the band edges [5]. STM spectra of nanotubes show densities of states with similar singularities to those obtained by band calculations [15, 16]. One-dimensional metallic wires are generally unstable and have Peierls distortion. The energy gain after Peierls distortion is found to be very small for nanotubes and the Peierls energy gap can be neglected. The curvature of nanotubes introduces hybridization also between sp2 and sp3 orbitals, but these effects are small when the radius of an SWNT is large [17]. However, the  ∗ - and  ∗ -state mixing was enhanced for smallradius zigzag SWNTs [13]. Recently, it has been shown that the electronic properties of an SWNT can undergo dramatic changes owing to the elastic deformations [18–21]. For example, the band gap of a semiconducting SWNT can be reduced or even closed by the elastic radial deformation. The gap modification and the eventual strain-induced metallization seem to offer new alternatives for reversible and tunable quantum structures and nanodevices [22].

2.2. Quantum Transport Properties A nanotube can be an ideal quantum wire for electronic transport; two subbands crossing at the Fermi level should nominally give rise to two conducting channels. Under ideal conditions each channel can carry current with unit quantum conductance 2e2 /h; the total

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resistance of an individual SWNT would be h/4e2 or ∼6 k. The contribution of each subband to the total conductance is clearly seen in Figure 2(c), illustrating the calculated conductance of the (10, 10) nanotube. The first electronic transport measurements of nanotubes were carried out using MWNTs [23–25]. These measurements found MWNTs to be highly resistive due to defect scattering and weak localization. The first electronic transport measurements of individual SWNTs and nanoropes were performed by Tans et al. [26] and Bockrath et al. [27]. In these measurements nanotubes or nanoropes were placed on an insulating (oxidized silicon) substrate containing metallic electrodes. The measurements were performed at 5 mK and step-like features are observed in current– voltage curves. The steps were not due to the quantized increase of conductance with subbands but due to resonant tunnelling of electrons to the states of a finite nanotube [26, 27]. Another important behavior observed is the monotonic decrease of conductance with decrease of temperature, which is a signature of correlated electron liquids. It is well known that the electrons in 1D systems may form not Fermi liquids but the so-called Luttinger liquids with electron correlation effects [28]. An important feature of Luttinger liquids is the separation of spin and charge by the formation of quasiparticles. Theoretical investigations of correlation effects in nanotubes were performed by Kane et al. [29] and Egger and Gogolin [30]. They argued that electrons in armchair SWNTs form a Luttinger liquid. Bockrath et al. [31] confirmed these arguments by observing power-law dependences of the conductance on voltage G ∝ V  ) and also on temperature (G ∝ T  ), which are typical for Luttinger liquids. An interesting transport experiment was performed by Frank et al. [32]. MWNTs were dipped into liquid metal with the help of a scanning probe microscope tip and the conductance was measured simultaneously. The nanotubes were straight with lengths of 1–10 m. As the nanotubes were dipped into the liquid metal one by one, the conductance increased in steps of (2e2 /h). Each step corresponds to an additional nanotube coming into contact with the liquid metal. The electronic transport is found to be ballistic, since the step heights do not depend on the different lengths of nanotubes coming into contact with the metal. On the theory side, different techniques are used to calculate quantum effects on the conductance of nanotubes [33–40]. These techniques are based on linear response theory and use the Landauer formalism. Tian and Datta [37] studied the tip–nanotube– substrate system in STM and investigated the Aharonov–Bohm effect in nanotubes using the Landauer formula. They treated the transmission coefficient using a semiclassical approach. Saito et al. [38] studied tunnelling conductance of nanotube junctions which are joined by a connecting region with a pentagon–heptagon pair. They used a method for directly calculating the current density. Tamura and Tsukada [39] used effective-mass theory with envelope functions for similar junctions. Choi and Ihm [33] presented an ab initio pseudopotential method with a transfer-matrix approach and studied nanotubes with pentagon–heptagon pair defects. Among these methods, Green’s function methods [34–36] are found to be the most effective, and hence are widely used for nanotubes with local basis sets. Chico et al. [35] studied nanotubes having vacancies and other defects by using a Green’s function technique with a surface Green’s function matching method. Carbon nanotubes with disorder were studied by Anantram and Govindan [36] using a similar Green’s function technique with an efficient numerical procedure. Recently, Nardelli [34] presented an approach using a surface Green’s function matching method. Iterative calculations of transfer matrices are combined with the Landauer formula to

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calculate the conductance. In these techniques, electronic structures of nanotubes are derived from the -orbital tight-binding model. In the Green’s function method within the tight-binding representation using empirical energy parameters, a conducting nanowire or a nanotube is divided into three parts [34, 41]. These are the left (L), right (R) and central (C) regions; the left and right regions are coupled to two semi-infinite leads. The central region is the computationally important part, which may include tube junctions, defects or vacancies. Partitioning the Green’s function into submatrices due to the left, right and central regions, one can obtain the Green’s function for the central region as GrC =  − HC − L − R −1

(1)

The self-energy terms, L and R , describe the effect of semi-infinite leads on the central region. The functions for the coupling can be obtained as LR = i rLR − aLR in terms of the retarded (r) and advanced (a) self-energies. An important part of the problem is calculating the self-energy terms. The surface Green’s function matching method [34, 35] or computational algorithms [36] are used to calculate the Green’s functions of semi-infinite leads and the self-energy terms. These terms can be obtained by using wave functions of ideal leads also [41]. The transmission function T that represents the probability of transmitting an electron from one end of the conductor to the other end can be calculated using Green’s functions of the central region and couplings to the leads: T = Tr  L GrC R GaC 

(2)

Here Gra C are the retarded and advanced Green’s functions of the center, and LR are functions for couplings to the leads. This approach is used efficiently to treat nanotube junctions and nanodevices as discussed in the next subsection.

2.3. Nanotube Junctions and Electronic Devices Current trends in microelectronics are to produce smaller and faster devices. Owing to their novel and unusual mechanical and electronic properties, carbon nanotubes appear to be potential candidates for meeting the demands of nanotechnologies. There are already nanodevices which use nanotubes. Single-electron transistors were produced [26, 27] by using metallic tubes; the devices formed from there have operated at low temperatures. Tans et al. [42] demonstrated a field-effect transistor that consists of a semiconductor nanotube and operates at room temperature. One approach used in nanodevice design was based on the fact that the defects in a nanotube can form an intermolecular junction from an individual tube [38, 43, 44]. It was shown that topological defects like pentagon–heptagon pair defects can change the helicity and hence the electronic structure of the nanotube. This raises the possibility of fabricating metal–metal (M–M), metal–semiconductor (M–S) and semiconductor– semiconductor (S–S) junctions on a single nanotube. In Figure 3(a), the atomic structure of an M–S junction is shown [43]. An (8, 0) tube (semiconductor) is joined to a (7, 1) tube (metal) with a pentagon–hexagon defect (highlighted in grey). The local

232

A. Buldum cell 1

DOS

0.3 0.2 0.1 0 –10

–5

DOS

5

10

5

10

5

10

cell 2

0.3 0.2 0.1 0 –10

–5

0

cell 3

0.3

DOS

0

0.2 0.1 0 –10

–5

0

Energy (eV)

(a)

(b)

Figure 3 (a) Atomic structure of a semiconductor–metal junction ((8, 0)–(7, 1)) with a pentagon– hexagon pair defect. Grey balls denote the atoms forming the defect. (b) The local density of states (LDOS) at cells 1, 2 and 3. Cell 3 is on the (8, 0) side of the tube and cell 1 is at the interface (Reproduced from [43])

density of states (LDOS) based on a -orbital tight-binding model is presented in Figure 3(b). The LDOS is distorted close to the interface, but recovers the DOS of a perfect tube away from the interface on both sides. There is a conductance gap in M–S and S–S junctions [45] and diode-like behavior can be achieved. However, in M–M junctions, the conductance was found to depend on the arrangement of the defects [35]. The junction is insulating for symmetric arrangement, but conducting for asymmetric arrangement. Kink structures, which consist of such defects, have been observed experimentally [45–47]. Even electronic transport measurements were reported, by Yao et al. [48], on nanotubes with M–M and M–S junctions. They found that an M–S junction behaves like a rectifying diode with non-linear transport characteristics and the conductance of the M–M junction was suppressed. Other intermolecular junctions include nanotube–silicon nanowire junctions [49] and Y-junctions [50] which exhibit rectification. So far we have discussed nanotube junctions and devices that used individual nanotubes. Two or more nanotubes can form nanoscale junctions with unique properties [51]. Another simple intermolecular nanotube junction can be formed by bringing two ends of the tube together. Figure 4(a) illustrates such a junction with two semi-infinite (10, 10) tubes in parallel and pointing in opposite directions. Buldum and Lu [51] showed that these junctions have high conductance values and exhibit negative differential resistance behavior. Interference of electron waves reflected and transmitted at the tube ends gives rise to the resonances in conductance shown in Figure 10(b). The current–voltage characteristics of this junction presented in Figure 10(c) show a negative differential

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I (μA)

G (2e2/h)

Modeling and Simulations of Carbon Nanotubes

0.4

–5

0.2 l

0

0.0 –0.5

0.0

–10 –0.8

0.5

–0.4

E (eV) (a)

0.4

0

0.8

V (V)

(b)

(c)

Figure 4 (a) An intermolecular nanotube junction formed by bringing two semi-infinite (10, 10) tubes together; l is the contact length. (b) The conductance, G, of a (10, 10)–(10, 10) junction as a function of energy, E, for l = 64 Å. Interference of electron waves yields resonances in conductance. (c) Current–voltage characteristics of an (10, 10)–(10, 10) junction at l = 46 Å (Reproduced from [51])

resistance effect, which may have applications in high-speed switching, memory and amplification devices. A four-terminal junction can be constructed by placing one nanotube perpendicular to another and forming a cross-junction (Figure 5(a)). Fuhrer et al. [52] reported electronic transport measurements of cross-junctions and presented M–M, M–S and S–S fourterminal devices. The M–M and S–S junctions had high conductance values (∼0 1e2 /h) and the M–S junction formed a rectifying Schottky barrier. Buldum and Lu [51] found that the conductance of such intermolecular junctions strongly depends on the atomic structure in the contact region. Conductance between the tubes was found to be high when the junction region structure was commensurate and conductance was low when the junction was incommensurate. Figure 5(b) shows the variation of the resistance with the rotation of one of the tubes. The junction structure is commensurate at angles

600

R (h/2e2)

θ

y

2

1

400

3 x

4

200

0 0

(a)

30

60

90

120

θ (deg.)

150 180

(b)

Figure 5 (a) A four-terminal cross-junction with two nanotubes perpendicular to each other. (b) The resistance of an (18, 0)–(10, 10) junction as a function of tube rotation. The rotation angle, , and terminal indices are shown in the inset. The tube which is labeled by 2 and 4 is rotated by . The contact region structure is commensurate at  = 30, 90, 150 (Reproduced from [51])

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30, 90 and 150, and hence the resistance values are lower. Similar variation of the resistance with the atomic structure in the contact region is observed in nanotube–surface systems [53]. This significant variation of transport properties with atomic scale registry was found also in mechanical/frictional properties of nanotubes [54, 55]. Recently, Rueckes et al. [56] introduced a random-access memory device for molecular computing which is based on cross-junctions. They showed that cross-junctions can have bistable, electrostatically switchable on/off states and an array of such junctions can be used as an integrated memory device.

2.4. Electron Field Emission from Nanotubes One of the most promising applications of carbon nanotubes is employing them as electron emitters in field emission devices. Recent experiments have shown that nanotubes have excellent field emission properties with high current density at low electric fields [57–61]. Their emission characteristics and durability are found to be far better than other electron emitters [62]. Nanotubes offer promising device applications such as flat panel displays and microwave power amplifiers. ‘Field emission’ can be described as the emission of electrons from surfaces by high electric fields and/or at high temperature. It has been extensively studied since the late 1920s. Fowler and Nordheim developed a general model for electron emission from planar surfaces and their model has been widely used for electron emission from large objects. According to the Fowler–Nordheim model, emission from planar or large surfaces produces straight lines in so-called Fowler–Nordheim (FN) plots [i.e., a log(I/V 2 ) vs 1/V ]. However, almost all experimental FN plots for nanotube field emitters deviate from straight lines [62]. Localized electronic states at the nanotube cap and at the apex of nanotips are found to be very important for field emission [63–66]. Buldum and Lu employed a model which incorporates the geometry and electronic structure of nanotubes [67, 68]. By solving Laplace’s equation numerically and calculating the effective electronic potential using self-consistent field (SCF)-pseudopotential electronic structure calculation method, the variation of the local potential energy is obtained. The electronic structure of a nanotube is derived using a -orbital tight-binding Hamiltonian. Furthermore, current–voltage characteristics for different tube sizes and lengths were calculated using the WKB approximation. [19]. In Figures 6(a) and 6(b) calculated electric field lines and spatial variation of electric field magnitude near the nanotube’s cap are presented. In these calculations carbon nanotubes are considered to be metallic and Laplace’s equation is solved numerically to obtain the variation in electrostatic potential and electric field. Similar numerical calculations were performed for nanoscale emitting. Our theoretical investigations produced non-linear FN plots similar to experiments as shown in Figure 7(c). In this figure, FN plots are presented for different nanotube lengths. As seen from the figure, longer nanotubes emit first and they have low turnon (for an emission current of 1 nA) and threshold (for an emission current of 1 A) field values. The same behavior with very similar FN plots was observed in recent experiments [69]. In our investigations, the deviation from linear FN behavior is not due to the localized states but due to changes in the tunneling barriers as a result of the spatial variation of the electric field.

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Figure 6 Electric field lines and spatial distribution of field intensity near the nanotube’s cap. (a) Calculated electric field lines near a closed (5, 5) tube’s cap. (b) Spatial distribution of field intensity near a (10, 10) tube’s cap. The field is dramatically enhanced near the cap (from [67])

–4

–2 28 Å 55 Å 72 Å 90 Å

–6

–5 –6

log(I )

log(I/Eapp2)

–4

–8 –10

–8

–12

–9

(5, 5) 72 Å (10, 10) 72 Å

–10

–14 0

4

8

12

0

16

1/Eapp (A/V)

(c)

–7

0.2

0.4

0.6

0.8

Eapp (V/A)

(d)

Figure 7 (c) Fowler–Nordheim plots of electron emission from closed (5, 5) tubes of different lengths. Hollow circles, filled squares, hollow triangles, and filled circles are for tubes with 38, 55, 72, and 90 Å lengths, respectively. (d) Current vs applied electric field characteristics of a closed (5, 5) and a (10, 10) tube of similar length. Hollow circles are for a (5, 5) tube with 72 Å length and dark circles are for a (10, 10) tube with 75 Å length (from [67])

The model employed to investigate field mission enables us to study field emission locally and determine the emission current from individual atomic sites. Thus, the net current from the nanotube is the superposition of the currentfrom each site. Using this approach a microscopic picture of field emission is obtained and contributions from atomic sites at the apex of the cap and pentagonal rings to the emission current are found. In Figure 8, the atoms on the nanotube’s cab are represented and labeled by colors with respect to their contributions to the emission current. The panels are ordered with respect to increased applied field. At low applied field only (panel (a)) the atoms at the apex of the cap are emitting. There is a pentagonal ring at the apex and the atoms of the pentagonal ring are the first to start emitting electrons. By increasing the applied field, more atoms begin to emit, as indicated by the color change representing the emission from the other atoms. There are five other pentagonal rings in the cap, shown in Figure 8(d), and they have a high LDOS for field emission. However, not only do these atoms emit a significant amount of current, the other cap atoms bonded to the pentagons also make significant contributions to the total current.

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Figure 8 Top views of a closed (10, 10) tube’s cap during field emission. Different colors of atoms denote their electron emission intensity (violet = low, red = high). The panels are ordered with respect to increases in the applied field with values of 0.20, 0.30, 0.40, and 0.45 V/A with total current values of 0.9 nA, 65 nA, 0 6 A, and 1 2 A, respectively. At low voltage values, only the atoms at the apex of the cap start to emit; then more atoms begin to emit as the applied field increases (from [67])

3. Mechanical Properties Carbon nanotubes have attracted great interest as they are light, flexible, stiff and they have very large aspect ratios. The tubular shape and the network of carbon atoms in nanotubes with very strong covalent bonds make them very special materials. Some calculations and measurements showed that the tensile strength of carbon nanotubes can be as high as 300 times that of steel and a nanotube can be as stiff as diamond.

3.1. Elastic Properties First experimental measurements showed very high values of Young’s modulus for MWNTs. Using TEM images and investigating thermal vibrations, Treacy et al. [70] determined the Young’s modulus values in between 0.4 and 4 TPa. A more direct study employing an atomic force microscope (AFM) tip measured Y ∼ 1.2 TPa [71] which is close to graphite’s Young’s modulus (1 TPa). Dai et al. [72] and Falvo et al. [73] obtained similar values of Young’s modulus. First theoretical calculations on mechanical properties were performed before the experiments and some focused on the shape and energetics of nanotubes [74–77]. The mechanical properties of small single-wall nanotubes including Young’s modulus have been studied by several groups using molecular dynamics simulations [74, 77]. Young’s modulus values were predicted to be several times greater than that of the diamond.

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Using the empirical force-constant model, Lu performed extensive investigations of elastic properties of nanotubes [78]. These investigations included calculations of elastic constants of single-wall and multiwall nanotubes of various size and geometry, and that of crystalline nanoropes composed of single-wall tubes. Empirical force-constant model had been used to calculate the phonon spectrum and elastic properties of the graphite [79]. In an empirical force-constant model, the atomic interactions near the equilibrium structure are approximated by the sum of pairwise harmonic potentials between atoms. The force constants are empirically determined by fitting to measured elastic constants and phonon frequencies [79]. In the model, the elastic constants are calculated from the second derivatives of the energy density with respect to various strains. The tensile stiffness as measured by the Young’s modulus is defined as the stress/strain ratio when a material is axially strained. An important quantity in determining the values of elastic constants is the wall thickness h of nanotubes and it was taken as 3.4 Å. Table 1 presents some of the calculated bulk, Young’s, and shear moduli of SWNTs and MWNTs from [78]. In the study, it was found that elastic moduli are insensitive to the size and chiral angle of the nanotubes. The Young’s and shear moduli of nanotubes are found to be comparable to that of the diamond and that of the graphite. A crystalline rope of nanotubes is appeared to be very anisotropic in its elastic properties as it is soft on basal plane but stiff along the axial direction.

3.2. Morphology of Multi-walled Nanotubes In general, nanotubes are considered to be perfect cylinders. However, MWNTs with large radii can be facetted that planar regions can occur in the outer shells of the nanotubes [80–83]. A recent experiment by Gogotsi et al. [83] clearly showed facetted Table 1 Elastic coefficients and moduli of SWNTs and MWNTs from [78]. R – radius in nm. B Y M are bulk, Young’s, and shear moduli in units of TPa (1013 dyn/cm2 ). n is the Poisson ratio. MWNTs are constructed from (5n 5n) with n = 1 2 3    series of single-wall tubes. Experimental values for the graphite and the diamond are listed for comparison (from [78]) (n1  n2 ) (5, 5) (10, 0) (10, 10) (50, 50) (100, 100) MWNTs (10, 10) (20, 20) (30, 30) (40, 40) Graphite a Graphite b Diamond c

R

C11

C33

B

Y

M

n

0.34 0.39 0.68 3.39 6.78

0 397 0 396 0 398 0 399 0 399

1 054 1 058 1 054 1 054 1 054

0 191 0 190 0 191 0 192 0 192

0 971 0 975 0 972 0 972 0 972

0 436 0 451 0 457 0 458 0 462

0 280 0 280 0 278 0 277 0 277

0.68 1.36 2.03 2.71

0 412 0 412 0 411 0 410 1 06  1 07

1 13 1 17 1 18 1 19 ··· 0 036 1 07

0 194 0 194 0 194 0 194 0 008 3 0 008 3 0 442

1 05 1 09 1 10 1 11 1 02 0 036 5 1 063

0 455 0 472 0 491 0 514 0 44 0 004 0 575 8

0 270 0 269 0 269 0 269 0 16 0 012 0 104 1

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structures of MWNTs and graphite polyhedral crystals. In order to understand the faceting of MWNTs, we carried out computer simulations for MWNTs with different radii and thickness [84]. Using empirical force fields [85], first energy minimization using conjugate-gradient algorithm and then simulated annealing calculations with MD were performed. In Figure 9, a diagram that indicates facetted and cylindrical nanotubes with respect to their radii and thickness is presented. The reason for faceting is found to be due to competition between stretching and bending energies of nanotube layers and the interlayer van der Waals interaction. A sufficiently large nanotube is facetted in order to have more benefit from interlayer Van der Waals interaction, which is optimum in the case of planar graphite structure. By faceting, an MWNT decreases its total potential energy.

3.3. Bending Deformation of Nanotubes Carbon nanotubes behavior under very large deformations appeared to be very remarkable. MD simulations showed that they can have up to 15% tensile strain, which is extremely large comparing to other materials. Yakobson et al. performed MD simulations of nanotubes [77] under generic modes of mechanical load: axial compression, bending and torsion. The atomic interaction was modeled by the Tersoff–Brenner potential [86], which reproduces the lattice constants, binding energies, and the elastic constants of graphite and diamond. The Tersoff–Brenner potential is widely used to explore fullerenes, carbon nanotubes and their properties. The investigators noted that when the nanotubes of greater lengths axially compressed, the tubes buckle sideways as a whole. After that, the compression results with bending and a local buckling inward. They simulated bending, applying torque to the ends of a nanotube.

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thickness (nm)

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LI

SO

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4 R (nm)

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8

Figure 9 MWNTs morphology from MD simulations for different radii and thickness. Filled circles and open diamonds represent MWNTs in cylindrical and facetted shapes, respectively. The straight line with ‘SOLID’ represents the MWNTs that have their radii equal to their wall thickness. Two cross-sectional views of cylindrical and facetted MWNTs are also presented in the figure (From [84])

Modeling and Simulations of Carbon Nanotubes 1.2

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Figure 10 Bending of a nanotube of (13, 0) helicity, 8 nm long and 1 nm wide. (a) The strain energy curve (normalized to its second derivative) switches from harmonic to linear at the buckling point, while the force (dashed line) drops and remains almost constant afterwards. (b) Beyond the buckling point, a distinct kink shape develops (from [77])

In Figure 10, the strain energy curve and the atomic structure of the tube with a formed kink due to bending are presented. A notch in the energy plot (Figure 10(a)) occurred due to bending and its derivative dE/d shows an increase in tube compliance, which is a signature of a buckling event. Their simulations show that carbon nanotubes are remarkably resilient, sustaining extreme strain.

3.4. Plastic Deformation of Nanotubes: The Stone–Wales Defects Plastic deformation of nanotubes and mechanical failure under a tensile load was investigated by Nardelli et al. [87]. Their study was based on the use of classical, tight-binding and ab initio molecular dynamics simulations. They found that the ductile or brittle behaviors can be observed in nanotubes of different symmetry under the same external conditions. Furthermore, it appeared that the behavior of nanotubes under large tensile strain strongly depends on their symmetry and diameter. For example, they found that beyond a critical value of the tension, an armchair nanotube in ‘transverse’ tension released its excess strain via spontaneous formation of topological defects. A transverse tension found a natural release in the rotation of the C–C bond perpendicular to it (the Stone–Wales transformation [88]) and produced two pentagons and two heptagons coupled in pairs (5–7–7–5) [89] as it can be seen in Figure 11(a). Once nucleated, the (5–7–7–5) dislocation loop can ease further relaxation by separating the two dislocation

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7 5

5 7

(a) 7 5

(b)

5

7

Figure 11 The snapshots of molecular dynamics simulations of a (10, 10) nanotube under axial tension. (a) Formation of a Stone–Wales defect at T = 2000 K and 10% strain; (b) plastic flow behavior after 2.5 ns at T = 3000 K and 3% strain. The shaded area indicates the migration path of the (5–7) edge dislocation (from [87])

cores, which glide through successive Stone–Wales bond rotations [87]. This corresponded to a plastic flow of dislocations as it can be seen in Figure 11(b) and gave rise to ductile behavior. On the other side, in the case of a zigzag nanotube (longitudinal tension), the same C–C bond would be parallel to the applied tension and was already the minimum energy configuration for the strained bond. The formation of the Stone–Wales defect is then limited to rotation of the bonds oriented 120 with respect to the tube axis. It is well known that the introduction of topological defects can change the overall structure of nanotubes, e.g., the tube index or even the chirality. The 5–7 defect is especially important because it is a defect that can change the tube index without drastically altering the local curvature of the nanotube [43].

3.5. Nanotubes on Surfaces In various experimental systems containing nanotubes and in many studies targeting nanodevices, carbon nanotubes are grown or placed on surfaces. Thus, the interaction between nanotubes and underlying surfaces is important, and understanding this interaction and its effects is crucial. The nature of the interaction between a nanotube and an underlying surface depends on which material the surface is made of. The interaction energy may consist of shortrange, attractive interaction energy due to chemical bonding or overlapping electronic states; short-range repulsive energy due to ion–ion repulsion and long-range, attractive van der Waals energy. The interaction between a carbon nanotube and a graphite surface is similar to that between two graphite planes, which is weak and van der Waals in origin. In a recent experiment, Falvo et al. showed that it is possible to manipulate carbon nanotubes on graphite surfaces [90]. Using AFM, they were able to slide, rotate and

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roll MWNTs on a highly oriented pyrolytic graphite (HOPG) substrate. They found certain discrete orientations of nanotubes in which nanotubes ‘lock’ on the surface. After locking, further push with an AFM tip resulted with rolling motion of the nanotubes. Performing molecular statics and dynamics calculations, we investigated the interaction between nanotubes and graphite surfaces and studied modes of motion of nanotubes [84, 91]. The interaction between the tube and the graphite surface atoms were represented by an empirical potential of Lennard–Jones type [92], which was used extensively to study solid C60 and nanotubes [93]. We found that the effective contact area and the interaction energy scale with the square root of the radius of the nanotube. It appeared that a nanotube has sharp potential energy minima leading to orientational locking. Figure 12 shows the interaction energy as a function of the rotation angle between the tube axis and the graphite lattice (all the data given in this study is for per Å length of the nanotubes). Each nanotube has unique equilibrium orientations repeating in every 60, reflecting the lattice symmetry of the graphite. The variation of energy near the minimum is very sharp, which causes atomic scale locking of the nanotube. Locking angles are different for different nanotubes, and they are the direct measure of the chiral angle. This provides a novel method for measuring the chiralities of carbon nanotubes. Once the nanotube is locked on the surface, a constant lateral force can be applied on the nanotube, which resulted with rolling motion. Interestingly, the rolling motion in the atomic scale appeared to be different than the rolling motion of macroscopic objects. Consecutive sliding and spinning motions of the nanotubes were observed and overall motion appeared to be rolling. In Figure 13, snapshots during this slide–spin motion are presented. It appeared that when the atoms in the contact region are in atomic scale registry it is easier for the nanotube to slide. Then, the nanotube atoms move from in-registry to out-of-registry positions they are more close to each other and they will be repelling each other. At this atomic configuration, it is easier for the nanotube to spin or rotate. –0.160 (10, 10) –0.161

–0.207

EP (eV)

(30, 0) –0.208 –0.209

(20, 10) –0.196 –0.197 –60

–30

0

30

60

90

Figure 12 The interaction energy as a function of rotation angle between the nanotube axis and the graphite lattice for (10, 10), (30, 0) and (20, 10) nanotubes. Each nanotube has unique minimum energy orientations repeating in every 60 (from [91])

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Slide

Spin

Slide

Figure 13 Snapshots from MD simulations: a rolling nanotube on graphite surface. Nanoscale rolling of SWNT is a combination of atomic scale sliding and spinning motion

By this spinning the nanotube decreases its potential energy and the atoms recover the in-registry positions. Ideal macroscopic rolling in the case of smooth surfaces would be a perfect overlap of sliding and spinning; however, the presence of the atoms and sliding–spinning motion brought us the nanoscale mechanism of rolling.

4. Conclusions Modeling and simulation efforts for the investigations of physical properties of carbon nanotubes and their potential applications were briefly reviewed. It is clear that modeling and simulation have played important roles in the advancement of nanotube science. There are many unsolved problems and there are many challenges ahead. Novel modeling approaches have been developed to face these challenges and solve many computationally intensive problems related with nanotubes.

Acknowledgements We would like to acknowledge support from NSF grants DMS – 0407361 and CCF – 0403130.

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