A Tight-Binding Molecular Dynamics Simulation

Copyright © 2006 American Scientific Publishers All rights reserved Printed in the United States of America

Journal of Computational and Theoretical Nanoscience Vol. 3, 1–6, 2006

A Tight-Binding Molecular Dynamics Simulation Analysis of Carbon Nanotube Growth Process Parameters Shin-Pon Ju,1 Cheng-I Weng,2 ∗ Kuan-Chuan Fang,2 and Chuan-Sheng Lee2 1

Department of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, P. R. China 2 Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, P. R. China

Keywords: Carbon Nanotube, Tight-Binding Molecular Dynamics, Process Parameter. 1. INTRODUCTION The unique mechanical and electric properties of carbon nanotubes render them suitable for applications in nanodevices.1 For example, carbon nanotubes with high aspect ratios and small radii of curvature can access narrow, sharp recesses, where they can then be used to probe the internal features with significantly improved lateral resolution.2 Furthermore, the high flexibility of carbon nanotubes not only prevents the tip from crashing, but also minimizes the damage caused to soft samples during the probing process.2 Regarding electrical devices, carbon nanotubes can serve as tiny electronic devices and temperature transistors at room temperature, etc.3 Consequently, potential applications and fabrication techniques for carbon nanotubes have attracted considerable research interest over the past decade. Two commonly adopted fabrication techniques for carbon nanotubes are the arc-discharge and laser ablation processes,1 which deposit carbon atoms on functionalized templates or pre-designed patterns.3 4 Since this deposition process involves a wide variety of atomic ∗

Author to whom correspondence should be addressed.

J. Comput. Theor. Nanosci. 2006, Vol. 3, No. 2

behaviors, including the formation and destruction of covalent bonds between carbon atoms, the rearrangement of covalent bonds, and the migration of adatoms, etc., it is difficult to investigate these behaviors directly. However, the quality of a carbon nanotube is primarily dependent on the behaviors of its carbon atoms, and hence an investigation of these behaviors is required. Numerical simulation methods can yield valuable insights into the growth mechanisms of carbon nanotubes. Molecular dynamics simulation is a powerful technique for developing an understanding of nano-scale phenomena. Consequently, a large number of researchers have adopted this method to explain experimental results from an atomistic perspective in a number of different fields, including the fabrication of carbon nanotubes. On the basis of ab initio and classical molecular dynamics calculations, Maiti et al. stated that a large electric field at the tube tip does not contribute significantly to the growth of the carbon nanotube. However, differences in the growth mechanisms were noted when carbon atoms were deposited on the edges of carbon nanotubes with different radii.5 6 For narrower nanotubes, pentagonal

1546-198X/2006/3/001/006

doi:10.1166/jctn.2006.004

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This paper employs tight-binding molecular dynamics simulations to investigate the effects upon the final nanotube morphology of three principal carbon nanotube growth process parameters, namely the radii of the initial carbon nanotube patterns, the temperatures of the carbon nanotube patterns, and the energies of the incident carbon atoms. The simulation results provide an understanding of the roles played by these parameters in the fabrication of carbon nanotubes from an atomic perspective. It is shown that carbon nanotube patterns with small radii tend to form close-capped ends, while patterns with larger radii tend to promote lifted-up growth. Furthermore, the effect of pattern temperature is more obvious in the case of larger radii carbon nanotube patterns, in which disordered structures are evident at low pattern temperatures. Finally, it is shown that incident atoms with higher energies not only form unstable bonds with other carbon atoms, but may also destroy the covalent bonds of previously deposited carbon atoms. The present results provide a valuable insight into the promotion of close-capped nanotubes or lifted-up growth during the fabrication of carbon nanotubes.

RESEARCH ARTICLE

A Tight-Binding Molecular Dynamics Simulation Analysis of Carbon Nanotube Growth Process Parameters

structures at the edges result in the closure of the carbon nanotube, whereas in the case of wide carbon nanotubes, the deposited carbon atoms tend to bond with the other carbon atoms to form hexagonal rings, thereby resulting in lifted-up growth and an open-ended carbon nanotube. Brabec et al. applied the Tersoff-Brenner potential model to study the bond switching and ring migration processes as the incident atoms interacted with the other carbon atoms.7 The simulation results revealed that pentagonal structures tend to form on the step edges of narrower nanotubes, while heptagonal structures tend to grow on the step edges of nanotubes with wider radii. In related studies, Oh and Lee investigated the stability and cap formation mechanisms of single-walled armchair and zigzag carbon nanotubes by means of tight-binding molecular dynamics simulations.8 Their results indicated that a close-capped end would form whenever a pentagonal ring developed at the armchair or zigzag carbon nanotube edge. Che et al. studied the energetics and thermodynamic stability of the dangling-end atoms of small zigzag nanotubes using tightbinding molecular dynamics simulations.9 The instability of the end atoms in the high temperature region of 2500 K resulted in reconstruction by the release of C2 dimmer. Several researchers have also employed molecular dynamics simulations to investigate lip–lip interactions and atomic migration during the growth of double-walled carbon nanotubes.10 11 In the studies presented above, the use of molecular dynamics simulations has yielded a detailed understanding of the mechanisms of adatom migration and bond switching. However, most of these studies focused on the rearrangement of the carbon nanotube configuration caused by intermolecular interactions rather than upon the relationships between the macro-process parameters and the growth mechanisms. Consequently, the present study adopts a tight-binding molecular dynamics simulation approach to investigate the influence of three principal carbon nanotube growth parameters upon the final morphologies of the tubes. The present results provide a valuable reference for promoting close-capped ends or lifted-up growth during carbon nanotube fabrication.

2. SIMULATION MODEL Figure 1 provides a schematic representation of the current simulation model, in which incident carbon atoms are deposited upon a substrate in the form of a pre-designed Incident carbon atom Thermal layers

Table I.

Hopping integral parameters for carbon atom.

Hopping integral parameters

Fig. 1.

2

Schematic diagram of simulation model.

Vss (eV)

Vsp (eV)

Vpp (eV)

Vpp (eV)

−5.0

4.7

5.5

−1.55

open-ended carbon nanotube pattern. As shown, the lowest layer is fixed to prevent the system from shifting during the simulation. Meanwhile, the layers above the fixed layer are thermal control layers, which are controlled by means of the scaling method to maintain the environmental temperature required for the growth of the carbon nanotubes. In the simulation, the velocities of the incident atoms are calculated from their incident energies, i.e., 2 · Eatom (1) Vatom = M where Eatom represents the incident energy and M is the atomic mass of carbon. Individual incident atoms are generated sequentially in the x-y plane and deposited randomly at a specified deposition rate. Furthermore, all incident atoms are introduced into the simulation system at a distance from the substrate which far exceeds the truncation distance of the atoms in order to prevent initial interaction between the incident atoms and the substrate atoms. As the precision of the simulation is largely dependent upon the accuracy of the adopted interatomic model, this study employs the tight-binding potential model for carbon since it is known to provide accurate results.12 In this model, the total energy of the atoms within the simulation system is comprised of two elements, i.e., Etot = Ebs + Erep

(2)

where Ebs represents the summation of the ground electronic states occupied by the outer shell electrons. These ground states correspond to the eigenvalues of the empirical tight-binding Hamiltonian model, which is constructed from a set of hopping integral parameters for carbon atoms, i.e., Vss , Vsp , Vpp , and Vpp 12 using the twocenter approximation method. 13 The corresponding values of these parameters are listed in Table I. Meanwhile, Erep denotes the repulsive energy, and is expressed as: rij (3) Erep = f i

j

where rij is the interatomic potential induced between two atoms, i and j, which are separated by a distance of Table II.

Parameters for Sr and r.

Sr Fixed layer

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r

0 (eV) 8.18555

n 2.0

nc 6.5

rc (Å) 2.18

r0 (Å) 1.536329

r1 (Å) 2.45

m 3.30304

mc 8.6655

dc (Å) 2.1052

d0 (Å) 1.64

d1 (Å) 2.57

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A Tight-Binding Molecular Dynamics Simulation Analysis of Carbon Nanotube Growth Process Parameters Table III. Parameters for f x. −2.5909765118191 0.5721151498619 −1.7896349903996 × 10−3 2.3539221516757 × 10−5 −1.24251169551587 × 10−7

c0 c1 c2 c3 c4

(a)

less than the truncation distance, i.e., 2.6 Å. The complete form of rij is given in Eq. (4). Furthermore, as shown in Eq. (5), f x can be expressed in the form of a 4th-order polynomial.12 m

mc

(b)

mc

r = d /r expm−r/dc + d /dc (4) f x =

4 n=0

cn x n

(5)

The function sr used for scaling Etot is given as follows:12 sr = r /rn expn−r/rc nc + r /rc nc

(6)

(c)

The parameters for functions r and sr are presented in Table II, while those of f x are listed in Table III. It is noted that the present simulation uses the Verlet velocity method to calculate the trajectories of the carbon atoms.14

(d)

This section discusses the influences of the initial pattern radius, the pattern temperature and the incident energy of the deposited carbon atoms upon the final morphologies of the grown carbon nanotubes. 3.1. Influence of Initial Pattern Radius Figures 2(a)–2(d) present top and side views of the final carbon nanotube morphologies grown from (5,5), (7,7), (10,10), and (15,15) armchair carbon nanotube patterns with radii of 7.2, 9.6, 13.6, and 20.3 Å, respectively. The elapsed times to form these final morphologies are 5.4 ps, 20 ps, 45.4 ps, and 55 ps, respectively. In each case, the incident energy is 1-eV, and the substrate temperature is 3000 K. From Figure 2, it is clear that an open end is maintained for the (15,15) carbon nanotube pattern, but that close-capped ends are formed for the smaller patterns. Accordingly, close-capped end nanotube structures appear to form more readily on patterns with smaller radii. Table IV summarizes the arrangements of the deposited carbon atoms when form covalent bonds with the atoms on the step edge. The results indicate that pentagonal structures tend to form on the step edge of the (5,5) carbon nanotube pattern, whereas hexagonal structures form on pattern edges with larger radii in general, and upon the edge of the largest radius (15,15) carbon nanotube pattern J. Comput. Theor. Nanosci. 3, 1–6, 2006

Fig. 2. Side and top views of final morphologies for different initial pattern radii with incident energy of 1-eV and pattern temperature of 3000 K: (a) (5,5); (b) (7,7); (c) (10,10); and (d) (15,15) armchair structure patterns.

in particular. It is reasonable to propose that the highly curved structure of the pentagonal ring results in the closure of the nanotube as deposition continues. Conversely, the structure of the heptagonal ring tends to grow in the longitudinal direction and hence results in an open-ended morphology.5–8 Table IV. Number of pentagonal, hexagonal, and heptagonal carbon rings formed from the incident atoms for nanotube patterns of different radii. The diameter of the pattern Å 7.2 9.6 13.6 20.3

Number of carbon ring Pentagon:5 Pentagon:3 Pentagon:2 Pentagon:2

Hexagon:0 Heptagon:0 Hexagon:3 Heptagon:1 Hexagon:8 Heptagon:0 Hexagon:11 Heptagon:2

End structure Close Close Close Open

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3. RESULTS AND DISCUSSION


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(a)

(a)

(b)

(b)

Fig. 3. Side and top views of final morphologies of (5,5) carbon nanotube patterns with incident energy of 1-eV and pattern temperatures of: (a) 2000 K and (b) 2500 K.

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3.2. Influence of Pattern Temperature (5,5), (7,7), and (10,10) carbon nanotubes are chosen as patterns in an investigation of the influence of pattern temperature on the final carbon nanotube morphology. Simulation is performed with an incident energy of 1-eV and pattern temperatures of 2000 K, 2500 K, and 3000 K. Figures 3–5 present the simulation results for (5,5), (7,7), and (10,10) carbon nanotube patterns, respectively, with pattern temperatures of 2000 K and 2500 K in each case. It is noted that the corresponding morphologies for a pattern temperature of 3000 K have been presented previously

(a)


in Figure 2. In Figure 2(a) and Figure 3, the morphologies of the three (5,5) carbon nanotubes are similar, with pentagonal rings forming on the pattern edges to construct close-capped ends. However, for the (7,7), and (10,10) patterns, the pattern temperature has a clear influence on the final morphologies, as shown in Figure 2(b) and Figure 4 for the (7,7) carbon nanotube pattern and in Figure 2(c) and Figure 5 for the (10,10) carbon nanotube pattern. Figure 4(a) indicates that a disordered cap structure is evident for the (7,7) carbon nanotube pattern at the lowest temperature of 2000 K. Meanwhile, Figures 2(c) and 4(b) suggest that higher pattern temperatures of 3000 K and 2500 K yield a more ordered arrangement. In the case of the largest radius carbon nanotube (i.e., the (10,10) tube), the results of Figure 5(b) indicate that a disordered structure still remains even when the pattern temperature is increased to 2500 K. Table V summarizes the elapsed time required to form a closed end for nanotube patterns of Table V. Elapsed time to form close-capped end structures for different pattern temperatures. Pattern temperature (K)

Elapsed time for forming closed-end (ps)

(5,5)

2000 2500 3000

7 5.5 5.4

(7,7)

2000 2500 3000

31 (disorder) 27.1 20

(10,10)

2000 2500 3000

45.4 ps (disorder) 45.4 ps (disorder) 45.4

(b) Pattern type


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(a) (a)

(b)

(b)

(c)

(c)

Fig. 6. Side and top views of final morphologies of (5,5) carbon nanotube patterns with pattern temperature of 2000 K and incident energies of: (a) 3-eV; (b) 5-eV; and (c) 7-eV.

3.3. Influence of Incident Energy To explore the influence of incident energy on the final nanotube morphology, incident energies of 1-eV, 3-eV, 5-eV, and 7-eV are simulated for a pattern temperature of 2000 K. Figures 6 and 7 present the final morphologies of the (5,5) and (7,7) carbon nanotube patterns, respectively, for incident energies of 3-eV, 5-eV, and 7-eV. It is noted that the corresponding morphologies for an incident energy of 1-eV have been presented previously in Figures 3(a) and 4(a). Figure 3(a) and Figure 6, which together show the morphologies of the (5,5) carbon nanotube pattern at incident energies of 1-eV, 3-eV, 5-eV, and 7-eV, indicate that a disordered structure is formed at the highest incident energy of 7-eV, while ordered close-capped end structures are formed at incident energies of 1-eV, 3-eV, and 5-eV. Regarding the (7,7) carbon nanotube pattern, a closecapped end structure is not formed at incident energies exceeding 5-eV. Cornwell15 stated that the energy required to remove carbon atoms from the tube edge is larger in the case of a smaller carbon nanotube. This explains why an ordered close-capped end structure can be formed on J. Comput. Theor. Nanosci. 3, 1–6, 2006

Fig. 7. Side and top views of final morphologies of (7,7) carbon nanotube patterns with pattern temperature of 2000 K and incident energies of: (a) 3-eV; (b) 5-eV; and (c) 7-eV.

the (5,5) carbon nanotube at an incident energy of 5-eV, but not on the (7,7) carbon nanotube pattern. In the latter case, carbon atoms with higher incident energies scatter the atoms bonded with neighboring carbon atoms or fail to form stable bonds with other atoms on the pattern. Table VI summarizes the final morphologies grown from (5,5), and (7,7) carbon nanotube patterns using different incident energies and a pattern temperature of 2000 K. The results show that close-capped end structures are no longer formed when the incident energies exceed 7-eV and 5-eV for the (5,5) and (7,7) carbon nanotube patterns, respectively. Table VI. Elapsed time to form close-capped end structures for different incident energies. Pattern type

Incident energy (eV)

Elapsed time for forming closed-end (ps)

(5,5)

1 3 5 7

7 5.5 5 7 (Open end with disorder structure)

(7,7)

1 3 5 7

31 (disorder) 31.7 (disorder) 42.5 (Open end with disorder structure) 42.5 (Open end with disorder structure)

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different types and temperatures. The results suggest that a higher pattern temperature promotes the migration ability of the incident atoms such that they attain their equilibrium positions, hence prompting the formation of ordered structures.


4. CONCLUSIONS

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This study has employed tight-binding molecular dynamics simulations to investigate the effects of the initial pattern radius, the pattern temperature, and the incident energy intensity upon the final morphologies of carbon nanotubes from an atomistic perspective. The conclusions of this investigation may be summarized as follows: (1) In the case of carbon nanotube patterns with smaller radii, the deposited atoms tend to form pentagonal rings with the atoms on the edge of the carbon nanotube. This structure becomes curved as it bonds with subsequently deposited atoms, thus forming a close-capped end structure. For carbon nanotube patterns of larger radii, hexagonal rings are readily formed on the edge of the pattern, and longitudinal growth is observed. Accordingly, it can be concluded that longer carbon nanotubes are more easily formed when patterns of larger radii are employed, while shorter carbon nanotubes tend to result from patterns of smaller radii. (2) The final morphologies are insensitive to pattern temperature when the patterns are of small radii. In this study, the morphologies grown from small radii (5,5) carbon nanotube patterns at three temperatures are similar. However, for larger radii patterns, the effect of pattern temperature is more pronounced. At lower temperatures, the deposited atoms aggregate to form a disordered structure. However, at higher temperatures, the energy of the deposited atoms is enhanced, which enables them to attain equilibrium positions and hence results in the formation of a more ordered structure. (3) The incident energy of the deposited atoms has a crucial influence on the growth of the carbon nanotubes.

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If the incident energy intensity exceeds the covalent bonding energy between carbon atoms, the deposited incident atoms not only form unstable bonds, but also destroy the existing bonds between previously deposited atoms. Acknowledgment: The authors gratefully acknowledge the support provided to this research by the National Science Council of the Republic of China under Grant No. NSC 92-2212-E-006-053.

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Received: 17 September 2005. Accepted: 14 October 2005.

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