Electromechancial Properties of Carbon Nanotubes
a D. Phil. Thesis by
Zhao WANG 2008
Institut UTINAM U.F.R. des Sciences et Techniques Universit´e de Franche-Comt´e
Latex source files sample of this thesis format can be found in : wangwzhao.googlepages.com
Abstract
This thesis aims at modeling the deformation of carbon nanotubes by an electric field. We started with molecular mechanics and molecular dynamics simulations investigating the non-linear elasticity of nanotubes. Comparing our results with those found in the literature, we demonstrated the acuuracy and the efficiency of our computational codes. In a second step, we calculated the charge distribution in nanotubes in/without an electric field, for which an atomic charge-dipole model has been developed. We compared our theoretical results with those from our experimental collaborators and quantitative agreement has been achieved. Finally, we combined the methods used in two previous steps, simulating the deformation of nanotubes in an electric field. The interplay between the external field, the tube geometry and the electrostatic deformation was clearly demonstrated in this thesis. The results of the Ph. D work is of interest to the potential applications of nanotubes/nanowires in nanoelectronics and nanoelectromechanical systems.
Acknowledgment
I would like to thank my supervisors for their guidance to the “nano-world ” during my D.Phil. period. They created the freedom for me to do what I thought was best, while giving me opportunities to make the most out of my years in Besan¸con. I benefited a lot from the Ph. D work of R. Langlet. I want therefore to gratefully thank her. Besides my advisors, I would like to thank the reviewers of my thesis : A. Bosseboeuf, F. Torrens Zaragoz´ a and T. M´elin, for their detailed reports on this thesis. The quality of the manuscript is definitively improved thanks to their comments. My sincere gratitude also goes to the rest of my thesis committee : J. Dijon and S. Meunier. I am particularly grateful that T. M´elin and S. Meunier took the job of being my Postdoc promoters. People in my institute helped me during my scientific explorations. I feel privileged to thank all of them and particularly the following researchers for both their insightful comments and the enjoyable discussions that I had with them : J.-M. Vigoureux, P. Hoang, F. Moulin, C. Thomas, V. Pouthier, C. Ramseyer, B. Honvault, P. Sylvain, C. Girardet, E. Prunel´e de. I equally thank the administrative and technique supports from A.-M. Greset, C. H´eritier and A. Larbi. A special thank goes to Dr. S. J. Stuart in Clemson University. The high efficiency and accuracy of his AIREBO code saved a lot of time for our research. I have also had great pleasure to collaborate with M. Zdrojek, T. M´elin, S. Meunier and P. Hoang. Of course, this research would not have been possible without the financial support of the Region of Franche-Comt´e. I also enjoyed several international conferences and comfortable journeys in these three years. For these I would like to acknowledge the financial support from the CNRS GDR-E, the Doctoral School of Louis Pasteur in UFC and the European Commission. My family and friends are thanked for their love, understanding and friendship. v
In particular I am grateful for active supports from A. and J.-L. Cabanes and Y. Lanteri. Zhao WANG A Besan¸con 22 juin 2009
Table des mati` eres
Acknowledgements
v
1 Introduction 1.1 Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Scope and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 1 3
2 Carbon nanotubes 2.1 Basic definitions . . . . . . . 2.2 Electronic properties . . . . . 2.3 Mechanical properties . . . . 2.4 Electrostatic properties . . . 2.5 Electromechanical properties 2.6 NEMS based on CNTs . . . .
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5 6 8 10 11 12 13
3 Mechanical Properties 3.1 Energy minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Molecular dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17 18 21
4 Electrostatic properties 4.1 Computational techniques . . . . . . . . . . . . . . . . . . . . . . . . .
35 36
5 Electromechanical properties
53
6 Conclusions
71
Bibliography
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Table of abbreviations
• AFM : Atomic Force Microscope • AIREBO : Adaptive Intermolecular REactive Bond Order • CNT : Carbon NanoTube • CVD : Chemical Vapor Deposition • DDA : Dipole-Dipole Approximation • DFT : Density Functional Theory • DWCNT : Double-Walled Carbon NanoTube • EFM : Electrostatic Force Microscope • HOMO : Highest Occupied Molecular Orbital • HR-TEM : High-Resolution Transmission Electron Microscope • LUMO : Lowest Unoccupied Molecular Orbital • MC : methode of Monte-Carlo • MD : Molecular Dynamics • MEMS : MicroElectroMechanical Systems • MWCNT : Multi-Walled Carbon NanoTube • NEMS : NanoElectroMechanical Systems • NVT : Canonical ensemble ix
• REBO : Reactive Empirical Bond Order • STM : Scanning Tunneling Microscope • SWCNT : Single-Walled Carbon Nanotube • TD-DFT : Time-Depending Density Functional Theory • TEM : Transmission Electron Microscopy • UTR : Ultimate Torsion Rate
Chapitre 1
Introduction
1.1
Questions
The deformation of carbon nanotubes (CNTs) by an electric field has been observed in recent experiments. This discovery directly suggested the uses of CNTs in NEMS such as nanorelays, nanoswitches and nanoresonators, and is as well of interests to nanoscale electronics and photonics. A better understanding on the deformations of CNTs in an electric field is also fundamental for the fabrication, separation and electromanipulation of CNTs. How are CNTs deformed in an external electric field, and why ? What is the correlation between the fields and the tube’s deformation ? Does this deformation depend on the geometry of CNTs ? To answer these questions, we have developed theoretical approaches in this thesis, predicting the deformation of nanocarbon systems in an electric field. The interplay between the field-induced deformation, the external electric fields and the nanostructure geometry has been clearly demonstrated.
1.2 1.2.1
Basic Idea About the electric charges
When a CNT is brought into an electric field, it will be electrically polarized (see Figure 1.1). As an interesting phenomenon, the electric polarization has long been a subject of investigation, different theories have been used to emphasize different aspects. The atomic dipole theory of Applequist is one of those that has been used to investigate electric polarization since 1972 [1, 2, 3, 4]. In this (so-called dipole-only) model, every atom is treated as an interacting polarizable point. In recent studies, this model has been extended to include free charge (so-called charge-dipole model) and it has been parameterized for conjugated carbon systems. These models have had considerable success in predicting molecular electric properties of fullerenes and
2
1. Introduction
CNTs [5, 6, 7, 8, 9].
Figure 1.1 – Electric polarization of a neutral CNT in an external electric field E. Field tends to shift positive and negative charges in opposite directions. Colors are proportional to densities of induced electric charges.
1.2.2
About the chemical bonds
The electric polarization gives a driving force to product the deformation, which depends on the mechanical resistance of CNTs. It was not easy to describe the mechanical properties of such a tiny material, since macroscopic continuum mechanical laws are not always valid in the atomic scale. In such a case, the most common way is to model the chemical bonds in the nanotube, one by one. In a great deal of publications in the literature, we have chosen the adaptive intermolecular reactive bond order (AIREBO) model to calculate the potential energy of the covalent bond in CNTs [10]. This model is an extension of the well-known many-body chemical pseudopotential model (REBO) parameterized by Brenner [11] for conjugated hydrocarbons. An important feature of the AIREBO models is its ability to deal with large systems (actually up to 12500 atoms), since the axial periodic condition can hardly be applied for the issue of electrostatic bending of CNTs.
1.2.3
About the coupling of the charges and the chemical bonds
To study the electrostatic deformation, our idea is to use the combination of above electrostatic and mechanical models. This idea has been realized in three steps as follows : I : Modeling mechanical properties of CNTs without external electric fields using atomic simulations based on AIREBO potential, and comparing our results with those in the literature, in order to valid our computational code. II. Studying
1.3. Scope and Outline
3
electrostatic properties of CNTs using atomic charge-dipole theories and comparing results with experimental data in a framework of collaboration with experimentalists. III. Combining the approaches used in two previous steps to calculate deformations of CNTs in electric fields.
1.3
Scope and Outline
The three steps will be presented to you in following Chapters : Chapter 2 : A brief introduction to the physical and the mechanical properties of CNTs and a brief review of their applications in electromechanical systems recently reported in the literature. Chapter 3 : Molecular simulations of mechanical properties of CNTs. Two topics are represented : The nonlinear elasticity and the torsion of CNTs. Chapter 4 : Studies on the properties coupled with electric charges in CNTs in/without external electric fields. The static charge enhancement effects in CNTs are investigated. The results were compared with experimental data. Chapter 5 : Electric-field-induced deformation of CNTs using atomic simulations based on the combined method. The influence of the field strength, the field direction and the tube geometry on the electrostatic deflection are investigated for metallic and semiconducting CNTs.
Chapitre 2
Carbon nanotubes Si Christophe Colomb n’avait rien d´ecouvert, Kennedy serait toujours vivant. Francis Blanche
Since their identification in 1991 [12], carbon nanotubes (CNTs) have attracted considerable scientific interest because of their unique structure and their fascinating properties leading to a wide range of potential applications [13, 14]. Basically, there are two kinds of CNTs : single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs) (see Fig. 2.1). A number of techniques have been developed to fabricate CNTs in sizeable quantities, including chemical vapor deposition (CVD), electric arc discharge, laser ablation, high pressure carbon monoxide (HiPCO) [15].
Figure 2.1 – (a) Transmission electron microscope (TEM) image of a single-walled CNT (SWCNT) (Source : IBM research). (b) TEM image of a multi-walled CNT (SWCNT) (Source : Endo lab.).
CNTs are known to be ones of the strongest materials currently known to mankind.
6
2. Carbon nanotubes
This arises from the high covalent energy of their quasi-sp2 carbon-carbon bonds. Their young’s moduli are around 1 TPa and their ultimate tensile stress can reach to 60 GPa [16]. The thermal conductivity of CNTs is also very high (near 4000 W/m.K) [17, 18]. These properties make CNTs promising for ultra-strong, multifunctional composite materials [19]. Electron transport properties of CNTs strongly depend on the detailed atomic structures. They can be either metallic or semiconducting [20]. Theoretically, the electrical current density in metallic nanotubes can reach to 109 A/cm2 and the electron transport can be almost ballistic [21, 22, 23]. This makes CNTs to be potential candidates for fields effects transistors (FET) [23, 24, 25], (bio-)sensors [26, 27, 28], supercapacitors[29, 30] and other nanoelectronic devices [31]. Moreover, their sharp geometry is very interesting for realizing nanoscale field emission (FE) [32, 33, 34]. These above exceptional mechanical and electronic properties can give CNTs many advantages in the design of nanoeletromechanical systems (NEMS) [35].
2.1
Basic definitions
Figure 2.2 – Schematics of wrapping a graphene (a single layer of carbon) (left) into a SWCNT (right).
For demonstration, we can make a SWCNT by wrapping a single layer of graphite (the so-called “graphene”) into a cylinder. As show in Fig. 2.2, the elemental atomic structure of a SWCNT is a hexagonal honeycomb crystal lattice of carbon atoms in quasi-sp2 covalent bonds. Different ways in which the graphite layer is wrapped up define different structures of SWCNTs. There are 3 basic types of SWCNTs : armchair, zigzag and chiral, as shown in Fig. 2.3(a). The structure of a SWCNT can be denoted by two integer
2.1. Basic definitions
7
Figure 2.3 – (a) Atomic structures of three basic types of SWCNTs : armchair, zigzag and chiral. (b) Chiral vectors (n, m) in the honeycomb graphite layer, the shadow part shows one period that can be wrapped to a (5, 2) SWCNT. a1 and a2 stand for two unit vectors in the honeycomb lattice. θ is the so-called chiral angle. C stands for the circumference of tube’s cross section. T is the length of one period of the SWCNT.
numbers (n, m) (so-called “chiral vectors”). As shown in Fig. 2.3(b), n and m denote the number of unit vectors a1 and a2 along two directions in the graphite layer. The geometry parameters of CNTs can be calculated from their chiral vectors, e.g. the chiral angle θ can be calculated by using √
3m sin θ = √ 2 2 n + m2 + nm
(2.1)
or the circumference of tube can be written as
C=a
p n2 + m2 + nm
(2.2)
where a = 0.249nm is the length of the unit vector [14]. We can write the tube radius R as follows : R = C/2π. Differing from fabrication methods, radii of SWCNTs are usually smaller than 2 nm and those of MWCNTs can be up to 30 nm. The length of one period along the tube axis can be calculated as
T =
√
3C/dR
(2.3)
8
2. Carbon nanotubes
Where dR is the greatest common divisor of (2m + n) and (2n + m). The number of atoms in one period of a SWCNT is 4(n2 + m2 + nm)/dR . We note that MWCNTs have several carbon layers and CNTs are often obtained in the form of bundles in experiments [36] as shown in Fig. 2.4.
Figure 2.4 – (a) A MWCNT of 5 armchair carbon layers (5, 5)@(10, 10)@(15, 15)@(20, 20)@(25, 25). (b) A SWCNT bundle with 5 (10, 10) tubes.
Furthermore, the caps of CNTs can be either open or closed. Further details about the geometry-related properties can be found in a recent review of Barros et al. [37].
2.2
Electronic properties
In CNTs, there exists two types of chemical bond formed from sp2 hybridization : The σ bond and the π bond (see Fig. 2.5). The σ bond is also called in-plane bond, it is composed of two electrons in s orbital, two electrons in px orbital and two electrons in py orbital from two bonded carbon atoms. It is a strong covalent bond with high binding energy. The mechanical strength of CNTs is mainly contributed from σ bonds. On the contrary, the energy level of the π bond (composed of two electrons in pz orbital) is much lower than that of the σ bond. The π bond therefore does not have main contribution in mechanical strength of CNTs. However, for the issue of electron transport in CNTs, it is important to understand the electronic properties of the π bond, since the binding energy of the σ bond is too far from the Fermi level. Note that the π bond is also responsible for the weak long-range interactions between layers in MWCNTs and between tubes in bundles, because of its perpendicular orientation.
2.2. Electronic properties
9
Figure 2.5 – sp2 hybridization of two carbon atoms in a CNT (Source : S. Dohn Thesis).
Figure 2.6 – (a) Reciprocal space of the graphene. (b) Band diagram for a graphene [14].
Electronic transport properties of CNTs can be well understood regarding the band structure of graphene (a single graphite layer). Its electronic structure near the Fermi level is given by an occupied π band and an empty π ∗ band, which cross each other at the six tip points K in a Brillouin zone (see Fig. 2.6(a)). This system is metallic with zero band gap. However, in case of CNTs, due to their curved and chiral structure, the k states is not always cross at the K points like that in graphene. Thus, CNTs can be either metallic or semiconducting, depending on their chiralities. Theoretical calculations show that all of the armchair tubes are metallic, and for other tubes, the condition for metallic nanotube is that (n − m) is a multiple of 3. Further details about electronic and transport properties of CNTs can be found in a recent review [20].
10
2.3
2. Carbon nanotubes
Mechanical properties
Since theirs discovery, steady progress has been made in exploring the mechanical properties of CNTs [16, 38]. In recent experimental studies, the Young’s modulus of CNTs has been measured by using TEM [39, 40, 41], AFM [42] and other techniques [43, 44, 45]. The average values of Young’s modulus of CNTs found in these studies range from 0.40 TPa to 1.8 TPa for a definition of wall-thickness about 0.34 nm. Fig. 2.7 shows AFM tips holding a CNT, which is pulled for measuring its elastic constant.
Figure 2.7 – SEM image of AFM tensile experiment on a MWCNT (Source : Ref.[42]).
Young’s modulus of CNTs has been reported in theoretical studies using molecular dynamics (MD) simulations [46, 47, 48], ab initio first principle calculations [49, 50], tight-binding methods [51] and structural mechanics approaches [52]. Young’s moduli in these papers range from 0.3 to 1.5 TPa, with most of values close to that of graphite. We note here that the wall thickness of a CNT is usually assumed to be the interplanar spacing of two layers in graphite, which is about 0.339 nm. However, there is not yet general agreement of the definition of the thickness of one single carbon layer. In recent years, different values has been used from 0.0617 nm to 0.34 nm [53]. It might make more sense to define Young’s modulus per unit thickness [54].
2.4. Electrostatic properties
2.4
11
Electrostatic properties
Since their discovery, the polarizability of CNTs has been investigated by means of tight-binding models [55, 56], density-functional theories (DFT) [57, 58, 59, 60] electrostatic atomic interaction models [61, 62, 63, 64, 6, 7, 9, 65] and other methods.[66, 67] After these calculations, it has been known that : • Longitudinal polarizability α// is in general much larger than the transverse ones α⊥ . • α of metallic CNTs is much higher than that of semiconducting ones. • α// is proportional to the inverse square of the band gap energy. • In a MWCNT or a bundle, α// is nearly the sum of that of their constituent layers, with slight effects of electric depolarization. • α⊥ of a MWCNT depends mainly on that of the outer layer, due to electric screening. In experiments, it is reported [68] that the transverse dielectric constant of CNTs is about 10, irrespective of tube diameter and chirality. The longitudinal polarization can be used to distinguish metallic from semiconducting nanotubes. The electric permittivity of CNTs measured using absorption spectroscopy [69], transmission ellipsometry and resonant Raman scattering. Paillet et al. [70] and Zdrojek et al. [71, 72] studied charging and discharging processes of MWCNTs using charge injection from EFM tip (as shown in Fig. 2.8). Their results show that charges are distributed uniformly along the SWCNTs.
Figure 2.8 – (a) Schematic of a charge injection experiment [71]. (b) EFM image of a neutral CNT. (c) EFM image of this CNT after charging with electrons.
12
2.5
2. Carbon nanotubes
Electromechanical properties
Coupled electric and mechanical effects are key characteristics for the applications of NEMS. A better understanding on these properties can as well be useful for CNTs’ fabrication [73, 74, 75] and separation [76, 77]. A recent review has reported detailed physical properties of CNT in electronic devices [35]. Early experiments [78] reported the electric deflection of cantilevered MWCNTs using a TEM. Wei et al. [79] observed reversible deformations of CNTs orienting themselves parallelly to electric fields using scanning electron microscopy (SEM) (as shown in Fig. 2.9). Electrostriction in SWCNTs was reported by El-Hami and Matsushige [80] using an AFM. Bao et al. [81] reported that the microstructure of CNTs can be changed by an electric field during growth using a high-resolution transmission electron microscope. Electric-field-induced alignment [82, 83] of CNTs have also been observed in experiments. Tombler et al. [84] show how mechanical deformation affects the intrinsic electrical properties of CNTs under atomic force microscope local-probe manipulation. Furthermore, Hall et al. [85] report experimental evidence of an electromechanical effect of torsional strain in SWCNTs and piezoresistive response in a self-contained NEMS based on CNTs. In a theoretical study, Guo and Guo [86] carried out quantum mechanics calculations to investigate the electric field induced tensile breaking and the influence of external fields on the tensile stiffness of CNTs, and found that both the tensile stiffness and strength decrease with increasing intensity of electric fields. Torrens [64] reported that the polarizabilities of CNTs can be modified reversibly by external radial deformation. Li and Chou [87] studied the electric-charge-induced failure of SWCNTs. Verissimo-Alves et al. [88] performed ab initio calculations for charged graphene and SWCNTs and found that both CNTs and graphene expand upon electron injection. Gartstein et al. [89, 90] also reported changes of tube geometry induced by electric charge. Electrostrictive deformations in CNTs were reported by Tang and Guo [91, 92] using quantum mechanics simulations and Cabria et al. [93] using DFT calculation. Kang et al. [94] modeled a memory device based on bridged CNTs using classical electrostatics and elasticity theories. Maiti et al. [95] analyzed effects of structural deformation and chirality on electronic transport through CNTs by means of a combination of classical force field and DFT calculations. Jonsson et al. [96] calculated electromechanical response of CNTs when a current is applied, and found that a shuttle-like electromechanical instability can occur if the bias voltage exceeds a
2.6. NEMS based on CNTs
13
Figure 2.9 – Reversible deflection of a CNT induced by an external electric field. (Source : Ref. [79]
dissipation-dependent threshold value. Moreover, Mayer and Lambin [97] calculated electrostatic forces acting on CNTs placed in the vicinity of metallic protrusions for dielectrophoresis using an atomic model.
2.6
NEMS based on CNTs
Commercial microelectromechanical systems (MEMS) now reach the submillimeter to micrometer size scale, and there is intense interest in the creation of next-generation NEMS, which is holding promise for a number of applications. CNTs have been reported as ideal building blocks for NEMS [98, 35] since their special characteristics give them many advantages in the design of NEMS. Here we give a brief review of reported researches of CNT-based NEMS. In the literature, coupled electric and mechanical properties in CNTs have been exploited in nanodevices such as : nanotweezers [99, 100], nanorelays [101, 102, 103], nanoswitches [104, 105], nanoactuators [106, 107, 108], field emission oscillators [109], tunable oscillators [110, 111], nanoresonators [112, 113, 114], displacement sensors [115] and torsional pendulums [116]. The fabrication and integration of CNTs into
14
2. Carbon nanotubes
electromechanical devices using chemical methods were recently investigated [15].
Figure 2.10 – Three examples of NEMS based on CNTs : (a) : a nanorelay, (b) : a nanotweezer, (c) : a rotational actuator.
2.6.1
Cantilevered CNTs
As shown in Fig. 2.10(a) and (b), a CNT is clamped over electrodes at one of its ends. This cantilevered structure has been reported as a building block in a number of NEMS [99, 100, 101, 102, 103, 104, 109]. Theoretical and experimental studies have been carried out for understanding their fundamental electromechanical and vibrational properties [78, 79, 117, 118, 119, 120]. Reported applications of these devices include NEM switches, logic devices, NEM-dynamic random access memory (RAM), nano-capacitor, electron-counters, pulse generators, current or voltage amplifiers, nanomanipulators, biological reaction controls and artificial muscles. Note that free-standing CNTs can have very high integration density, which is one of the most important factors in nanodevices.
2.6.2
Bridged CNTs
A bridged CNT is a CNT doubly clamped on both sides to two supports. The main advantages of this structure is that it is free of interactions with bottom substrate, and that its fundamental vibrational frequency is very high (up to 1.5 THz [121]). These make it attractive for a number of electromechanical devices [104, 105, 108, 104, 105, 110, 113, 114], which can be prepared by chemical vapor deposition (CVD) growth [122, 123]. Studies of its basic properties have been reported in experiments [84] and theories [124, 95, 94, 96]. Reported potential applications of these devices include metal deposition monitors, chemical reaction monitors, biomedical sensors, ultrasensitive mass or charge detectors and RF signal processing.
2.6. NEMS based on CNTs
2.6.3
15
CNTs as rotation elements
MWCNT rotational elements (see Fig. 2.10(c)) were reported as nanoactuator [108] and torsional pendulum [116]. Their fabrication processes have been studied in Ref.[125, 126]. Note that the shear modulus of SWCNTs can reach to 500 − 600 GPa. Potential applications of these devices are reported as ultra-high-density optical sweeping, switching devices, bio-mechanical elements, general chemically functionalized sensors and transmitter of electromagnetic radiators.
Chapitre 3
Mechanical Properties L’imagination est plus importante que le savoir. Albert Einstein
In order to make full use of the potential mechanical properties of CNTs, it is necessary to fully understand their elastoplastic behaviors. The Young’s modulus of CNTs has been experimentally measured and found to range from 0.40 TPa to 1.8 TPa for a definition of wall-thickness about 0.339 nm [16, 38]. In small deformation ranges, the stress-strain relationship of CNTs is supposed to be linear so that the stiffness remains constant, like in most of the previous studies on the issue of elastic behaviors of CNTs. However, the stress-strain relationship of CNTs has been found to be nonlinear under large tensile strain, [127, 128, 129] Yakobson et al. analyzed the plasticity of CNTs using a dislocation theory [130] and calculated the energy of defect formation as a function of CNT type [131]. Belytschko et al.[132] studied the failure of CNTs by means of both molecular mechanical and MD methods. The predicted range of fracture stresses is about 65-93 GPa. Bozovic et al.[133] found that single-walled CNTs (SWCNTs) can sustain strain of 30% without breaking using antiferromagnetic manipulation. Huang et al.[134] even found that a SWCNT can be stretched at least 280% at about 2000◦ C, when it is submitted to a 2.3 V bias. In this work, we studied the mechanical properties of CNTs submitted to large tensile loading using MD simulation based on AIREBO potential.[135] The quantum conductance of CNTs depends strongly on their chirality, which can be changed in torsion [111]. Therefore, understanding torsional behaviors of CNTs is a fundamental issue for their uses in nanoactuators [108] and electromechanical quantum oscillators [111], as well as in ultra-high-density optical sweeping and switching devices, bio-mechanical and chemical sensors or electromagnetic transmitters [98]. On the one hand, the change of tube’s electronic properties due to torsion has been predicted in several theoretical studies [136, 137]. Recently, metal-semiconducting periodic transitions were reported in experiments [111]. On the other hand, ideal torsional
18
3. Mechanical Properties
strengths and stiffness of zigzag CNTs have been studied using first-principle calculations and the strength of torsion of MWCNTs has been found to be about 20 times larger than for an iron rod of the same size [138]. The shear modulus of CNTs and the mechanical integrity of SWCNTs was evaluated with MD simulations [139, 140]. in order to exploit further details, we carried out MD simulations to compute the ultimate torsion deformation of CNTs at room temperature. Related change in deformation energies is also studied.
3.1
Energy minimization
Energy minimization (energy optimization) is a common technique to compute the equilibrium shape of molecules. The basic idea is that a stable state of the molecules should correspond to a local minimum of their potential energy. This kind of calculation generally starts from an arbitrary state of molecules, then a mathematical procedure is performed to move the atoms in a way that reduces the net forces on them. Like in MD or Mont-Carlo (MC) approach, periodic boundary conditions can be applied in energy minimization, by which one can make small systems. A well established algorithm of energy minimization can be an efficient tool for calculating deformations or optimized structures of molecules. The MD method is based on Newtonian dynamic laws and allows calculating the atomic trajectory with kinetic energy. Conversely, the effect of temperature is not taken into account in the energy minimization, in which the trajectory of atoms during the calculation does not make real physical sense, one can only obtain a final state of the molecule that correspond to a local minimum of the potential energy. From physical point of view, this final state of the system corresponds to the configuration of atoms when the system is cooled down to zero degree, e.g. as shown in Fig. 3.1, if there is a cantilevered beam vibrating between positions 1 and 2 around an equilibrium position 0 with an initial kinetic motion, no matter we start with the state 1, 2 or any other state between these two positions, the result of energy minimization for this system will always be the state 0. The methods of gradient are the most popular methods of energy minimization. The basic idea of these methods is to move atoms by the total net force on them. The force on atoms is calculated as the negative gradient of the total potential energy.
~ r U tot F (ri ) = ∇ i
∀i = 1, . . . , N
(3.1)
3.1. Energy minimization
19
Figure 3.1 – Schematics of a cantilevered beam vibrating between 2 positions.
where ri is the position of atom i. In general, an analytical formula of the gradient of the potential energy will be preferentially required by the gradient methods. If not, we need to calculate the numerical derivatives of the total energy function of the system. In such a case, the Powell’s direction set method or the Downhill simplex method can generally be more efficient than the gradient methods.
3.1.1
Simple gradient (steepest descent)
If we have a system of N atoms and the total potential energy is known as U tot (r), which is a function of the position of atoms (ri ), we can calculate the net force on each atom Fi (ri ) at each iteration step t. With the steepest descent algorithm, we displace every atom in the direction of F with a multiple factor κ as follows.
ri (t + ∆t) = ri (t) + κ · F (ri )
∀i = 1, . . . , N
(3.2)
The value of κ is taken to 0.01 in our unit system, however, κ can be adjusted to smaller value at the beginning of calculation if we start with a high potential energy, in order to avoid the system explosion. Note that similar strategy can be used in MD for reducing the probability of divergence problems at the beginning of simulation. We repeat the step in Eq. 3.2 as t = 1, 2... until F ≈ 0 for each atom. The potential energy of system goes down in a long narrow valley in this procedure. Despite that it is as well called “steepest descent”, the simple gradient algorithm is in fact very time-consuming, compared to the conjugated gradient method. It is therefore known as a not very good algorithm. However, it has an advantage “irresistible”
20
3. Mechanical Properties
to some types of simulations : Its numerical stability. In our test, th epotential energy can almost never increase if an appropriate value of κ was chosen. Thus, the simple gradient method is combined with a conjugated gradient algorithm in our energy minimization for avoiding numerical divergence problems when atoms are too close to each other.
3.1.2
Conjugated gradient method
A standard conjugated gradient algorithm includes two basic steps : adding an orthogonal vector to the direction of research, and then move atoms in a nearlyperpendicular direction. These two steps are as well known as : step on the valley floor and then jump down. Fig. 3.2 shows a highly simplified comparison between the conjugated and the simple gradient methods on a 1D energy curve.
Figure 3.2 – Comparison between two gradient algorithms on a simple 2D energy curve.
In conjugated gradient algorithm, we minimize the energy function moving atoms in following ways : rit = rit−1 + κ · hti
∀i = 1, ..., N
(3.3)
where : hti = F (rit ) + γit−1 ht−1 i
(3.4)
3.2. Molecular dynamics
21
where γ is updated using the Fletcher-Reeves formula as :
γit−1 =
F (rit ) · F (rit ) F (rit−1 ) · F (rit−1 )
(3.5)
Here we note that γ can also be calculated by using the Polak-Ribiere formula, however, it is found to be less efficient than the Fletcher-Reeves one in our tests. We note that. at the beginning of calculation (when t = 1), we can make the search direction vector h0 = 0. This algorithm is very efficient. However, it is not quiet stable with our potential function, i.e. it sometimes can step so far into a very strong repulsive energy range (e.g. when two atom are too close to each other), where the forces acting on atoms are almost infinite. This could directly result a typical data-overrun error during calculations. For resolving this problem, we combine the conjugated gradient with the steepest descent. Fig. 3.3 shows the schematics of this combined predicting algorithm. We note that the steps 2 and 5 can be combined to one single step.
3.1.3
Boundary conditions
The atoms in this algorithm can have different degrees of freedom. For example, in case of a molecule clamped over two supports, a number of atoms N f ix at the molecule ends should be fixed during the simulation. In such a case, it is enough to make the positions of these atoms unchanged in the step 4 or 8 shown in Fig. 3.3, but we still calculate their interaction with other atoms in the steps 2 and 5. From mathematical points of view, in this procedure we change the total number of variables in an energy function from 3 × N to 3 × (N − N f ix ) using the boundary condition, by which these 3 × N f ix variables are taken as known constants. Moreover, we can equally adding other boundary conditions to the minimized energy function, such as adding external forces or external electric fields to the system. In such a case, the shape of the potential energy curve will be changed but the number of variables remains constant.
3.2
Molecular dynamics
Known as “virtual experiments”, molecular dynamics (MD) are ones of the most popular methods for molecular simulations. It has been used for decades to investigate dynamical properties of molecules. Compared to the methods of MC or energy minimization, the main advantage of MD is that it provides both time-dependent and
22
3. Mechanical Properties
Figure 3.3 – Schematics of our computational energy minimization procedure.
temperature-dependent informations. In MD, atoms are allowed to move for a period of time following Newton’s law of motion. Their trajectories during the simulation can be visualized using molecular graphics software (such as : VMD). The general computational procedure of MD using atomic models is shown in Fig. 3.4. MD has been developed so far with numerous computational techniques. For our MD simulations in canonical ensemble, we use the Verlet-Leapfrog and Nose-Hoover thermostat algorithms, in which the velocities v and the positions r of atoms are updated as follows :
vi and
∆t t+ 2
= vi
∆t t− 2
+ ∆t
Fi (t) mi
∀i = 1, . . . , N
(3.6)
3.2. Molecular dynamics
23
Figure 3.4 – Schematics of the basic computational procedure of MD. (Source : wikipedia)
ri (t + ∆t) = ri (t) + ∆t · vi
∆t t+ 2
∀i = 1, . . . , N
(3.7)
The temperature of system is related to the kinetic energy of system Ekin . N 2 Ekin 1 X ∆t 2 T = = mi vi t + 3 N kB 3N kB 2
(3.8)
i=1
where Ekin is calculated as follows : λ=
p T set /T
(3.9)
The potential function used for calculating net force on atoms in our system (with up to 12500 atoms) is the adaptive intermolecular reactive empirical bond order (AIREBO) potential.[10] It is a widely-used function for calculating the potential energy of covalent bonds. In this model, the total potential energy of system is a sum of nearest-neighbor pair interactions which depend not only on the distance between atoms but also on their local atomic environment. A parameterized bond order function was used to describe chemical pair bonded interactions. The early formulation and parametrization of REBO for carbon systems was done by Tersoff in 1988 [141, 142], based on works of Abell [143]. The Tersoff’s model could describe single, double and triple bond energies in carbon structures such as in hydrocarbons and diamonds. A significant step was taken by Brenner in 1990 [11, 144]. He extended Tersoff’s potential function to radical and conjugated hydrocarbon bonds by introducing two
24
3. Mechanical Properties
additional terms into the bond order function. Compared to classical first-principle and semi-empirical approaches, the AIREBO model is less time-consuming, since only the 1st- and 2nd-nearest-neighbour interactions were considered. This advantage of computational efficiency is especially helpful for large-scale atomic simulations [145]. In recent years, the AIREBO model has been widely used in the studies concerning mechanical and thermal properties of CNTs [16, 38].
Geometry dependent nonlinear decrease of the effective Young’s modulus of single-walled carbon nanotubes submitted to large tensile loadings Zhao Wang,1, ∗ Michel Devel,1, † Bernard Dulmet,2 and Steve Stuart3 1
Institut UTINAM, UMR CNRS 6213, Universit´e de Franche-comt´e, 25030 Besan¸con CEDEX, France. 2 ´ FEMTO-ST, UMR CNRS 6174, D´epartement de Chronom´etrie, Electronique et Pi´ezo´electricit´e, ´ Ecole Nationale Sup´erieure de M´ecanique et des Microtechniques, 25030 Besan¸con CEDEX, France 3 Department of Chemistry, Clemson University, Clemson, South Carolina 29634, USA (Dated: February 16, 2009) In this study, we use molecular simulations based on the AIREBO potential function to find new results on the effective Young’s modulus of single-walled carbon nanotubes. For large tensile loadings, this effective Young’s modulus is found to decrease significantly down to only 45% of its initial value, with a decrease rate which is found, here, to be nonlinear. The dependence with radius or chiral angle of this nonlinear decrease rate is studied. For all the tubes tested, we also find values of the initial Young’s modulus in good agreement with those of previous studies, and a very linear increase of the tubes’ surface area for large axial deformations. PACS numbers: 61.46.Fg, 81.40.Jj, 62.20.Dc, 31.15.Qg
I.
INTRODUCTION
Since their identification in 19911 , Carbon Nanotubes (CNTs) have attracted a lot of scientific interest due to their exceptional mechanical, thermal, and electronic properties leading to many potential applications2 . As one of the strongest materials currently known to mankind, CNTs hold great promise for ultra-strong composite materials and nanoelectromechanical systems (NEMS). In order to make full use of the potential of CNTs for their various promising applications, it is necessary to fully understand their elastic and plastic behavior. In the experimental studies reported in the past, the Young’s modulus of CNTs has been measured in transmission electron microscopes3–5 , with AFM tip manipulation6 , by pulling nanotube ropes7,8 and using inelastic light scattering9 . The average values of Young’s modulus of CNTs found by these authors range from 0.40 TPa to 1.8 TPa for a wall thickness of 0.34 nm. Several theoretical studies using molecular dynamics (MD) simulations10–12 , ab initio first principle calculations13,14 , tight-binding method15 and structural mechanics approach16 reported that the Young’s modulus of CNTs ranged from 0.3 to 1.5 TPa, with most of the values close to that of graphite. Yakobson et al analyzed the plasticity of CNTs by using a dislocation theory17 , and calculated the energy of defect formation as a function of CNT type18 . Belytschko et al.19 studied the failure of CNTs with both molecular mechanical and MD methods. The predicted range of fracture stresses is 65–93 GPa. In an experimental study using antiferromagnetic manipulation, Bozovic et al.20 found that single-walled CNTs (SWCNTs) can sustain strain of 30% without breaking. Huang et al.21 even found that a SWCNT could be stretched at least 280% at about 2000 ◦C, when submitted to a 2.3 V bias. However, theoretical studies
of the elastic properties of CNTs at large deformation are limited, despite their usefulness for engineering applications. Indeed, in most of the previous studies on the elastic behavior of CNTs, the stress-strain relationship is considered to be linear so that stiffness remains constant. However, Xiao and Liao22 , Liew et al.23 and Natsuki and Endo24 examined the nonlinear elasticity of CNTs by using atomic simulations based on empirical potential functions and continuum analysis. The stressstrain responses are found to be nonlinear under large tensile strain. In order to explore further this point, we used molecular simulations to compute the effective Young’s modulus of SWCNTs with an energy optimization technique. The mechanical response of SWCNTs subjected to different tensile loading is simulated. The decrease rate of the effective Young’s modulus is found to be nonlinear and somewhat geometry dependent. Furthermore, we also studied the change of the tube surface area during tensile loading. Section 2 briefly outlines some technical details of our computational method, while the simulation results will be discussed in section 3 and compared with results from other studies.
II.
COMPUTATIONAL METHOD AND POTENTIAL ENERGY FUNCTION
In order to obtain relaxed geometries for the CNTs submitted to various tensile loadings, energy minimizations were performed using a simple gradient algorithm. We start with the tubes relaxed in vacuum. Then they are loaded at both ends with a given force F (i.e. the n + m leftmost atoms are loaded with −F and the n + m rightmost atoms with +F , n and m being the indices defining the nanotube). The atoms are then relaxed to reach the new equilibrium state at different force level
2 chiral angles chiral vectors
0◦
10.89◦
(10,0) (14,0) (8,2) (12,3)
19.11◦
30◦
(8,4) (10,5)
(6,6) (8,8)
(18,0) (22,0) (16,4) (20,5) (12,6) (14,7) (10,10) (12,12)
TABLE I: list of SWCNTs studied in this work
until fracture occurs. In our simulations, 4 groups of CNTs with the same chiral angle but different radii are placed in vacuum and submitted to various tensile loadings (table I). The interatomic potential is determined using the AIREBO potential function formulated by S. Stuart et al 25 . This potential is an extension of Brenner’s second generation potential26 and includes long-range atomic interactions and single bond torsional interaction. In this potential the total interatomic potential energy is the sum of that of individual pair interactions:
FIG. 1: Stress-strain curves for (22,0),(14,7),(12,12) SWCNTs, compared with Lew et al 23 .
1 X X V R (r ) − b V A (r ) + E L−J (r ) + P P E T ORS ij ij ij ij ij kijℓ , E = k6=i,j ℓ6=i,j,k 2 i b
j6=i
(1) where V R and V A are the interatomic repulsion and attraction terms between valence electrons, for bound atoms. The bond order function bij provides the many body effect by depending on the local atomic environment of atoms i and j. The long-range van der Waals interactions are included thanks to the Lennard-Jones 12– 6 potential term E L−J . Finally, the torsion interactions are represented by E T ORS . Note that the long-range van der Waals interactions must be considered in the case of large deformation to avoid an artificial cut-off barrier, as discussed in Sammalkorpi et al.27 . The inter-atomic force is calculated by the negative gradient of system’s total potential energy.
III.
FIG. 2: Changes of SWCNTs’ radii during tensile loading versus axial strain.
RESULTS AND DISCUSSION
The continuum elasticity theory is used to model SWCNTs under tensile loading. The value of the wall thickness of SWCNTs is assumed to be 0.34 nm to ease the comparison with the results of most of the previous studies. Note however that there is still no general agreement about the exact value of the wall thickness to be used in continuum mechanics modeling of CNTs, as discussed by Vodenitcharova and Zhang28 . In our simulations, the stress is calculated as the applied force per deformed crossed section (i.e. wall thickness times 2/pi times mean radius) as shown in Fig. 1. Note that this is different to previous works22,23 in which the axial force is calculated as the first derivative of strain energy. The effective Young’s modulus Y is then simply computed as the ratio of stress over strain Y = σ/ε, where the stress is the tensile force per cross sectional area and the strain is given as a percentage of axial elongation compared to the origi-
nal length of the tube (the tubes lengths are computed as the distance between the geometric centers of the n + m lefmost and n+m rightmost atoms). The original lengths and radii are calculated from the corresponding relaxed SWCNTs in the absence of any imposed loading. Fig. 2 depicts the changes of mean tube radius evaluated in our simulations. It can be seen that the transversal deformation of SWCNTs is much less important than the axial one, even tending to 0 for strains greater than about 10 % so that, for high strains, the tube surface area of the nanotubes increases nearly linearly with the axial strain. In Fig. 3 we can see the significant decrease of Y with the axial deformation. Comparing our results to the effective Young’s modulus calculated using the inplane stiffness model proposed by by Xiao and Liao22 , we find that our values of the effective Young’s modulus decrease slightly more slowly than that in the inplane stiffness model, for the same type of CNTs in the
3
FIG. 3: Effective Young’s modulus of (10,0), (8,2), (8,4) and (6,6) SWCNTs under tensile loading versus axial strain, compared with the Young’s modulus calculated using an in-plane stiffness model22 .
large deformation range. These differences can be understood as being due to the fact that the long-range van der Waals interaction is not considered in the second generation Brenner’s potential26 , but instead, it is considered in this work within AIREBO potential. These differences can also be considered to be due to the fact that the effective Young’s modulus is computed by stress over strain ratio in our simulations and is not the outcome of an approximate polynomial fit. Furthermore, for a given set of tubes with almost the same radius, the slope ( dY dε ) depends on the strain and changes with chirality at lower strain. However, at high strain level (ε > 15%), the slope becomes independent of chirality and Y is approximately linear versus strain. Alternatively, it can be seen from Fig. 4 that for a given chiral angle, Y is almost independent of the radius for strains higher than 5%, suggesting that the chirality plays a greater role than the radius at intermediate strains of 5% to 15%. However we can also see that the bigger the radius, the longer the zone for which Y is approximately constant. The effective Young’s modulus then decreases nearly linearly with increasing strain until at least ε = 15%. To account for the decrease of tensile stiffness, we express the effective Young’s modulus Y as a function of a decrease rate D: Y = Y0 (1 − D(ε))
(2)
where Y0 is the linear elasticity limit of Young’s modulus, which is plotted in Fig. 5 as a function of the tube radius. It corresponds to the zero–strain end of curves of Figs. 3 and 4. The values are in the range of 0.85–1.08 TPa, decreasing with the tube radius and also decreasing with the chiral angle. This is well corresponding to the results reported by several previous works9,11,13,16 . Furthermore, it can be seen that the zigzag tubes are
FIG. 4: Effective Young’s modulus of one group of zigzag SWCNTs under tensile loading versus axial strain.
’harder’ than the chiral tubes of the same radius. This chirality effect is in good agreement with a theoretical study of van Lier et al 14 using ab-initio method.
FIG. 5: Initial Young’s modulus Y0 of 4 groups of CNTs with different chiral angles.
As defined in Eq. 2, D is a function of axial strain ranging from 0 to 1. It represents the strain-dependent attenuation of the effective Young’s modulus under tensile loading. Its value is plotted in Figs. 6 and 7. It can be seen that D depends on both the diameter and the chirality. It is almost linear over a wide strain range from 5% to 15% with a slope nearly independent of radius but dependent on chirality. This feature is in qualitative agreement with the in-plane stiffness model proposed by Xiao and Liao22 who consider an expansion of deformation energy which implies that the in–plane stiffness is linear vs strain. However here–presented curves for D show a smaller nonlinearity of the elastic behavior in the two regimes ε < 5% (as required by the fact that Y
4 should be almost constant at small strain) and ε > 15%.
FIG. 6: Decrease rate D of the effective Young’s modulus for (10,0), (8,2), (8,4) and (6,6) SWCNTs versus axial strain
In conclusion, we found that the effective Young’s modulus of single-walled carbon nanotubes decreases almost linearly under large tensile loading. This decrease rate depends on the tube geometry with chirality, which plays a greater role than the radius at intermediate strains of 5% to 15%. The values of the initial Young’s modulus are in good agreement to that of previous studies and depends on the tube radius and chiral angle. Furthermore, the tube surface area is found to increase nearly linearly with increasing axial strain, since the tube diameter increases only slightly with tensile loading. Acknowledgments
FIG. 7: Decrease rate D of the effective Young’s modulus for (10,0), (14,0), (18,0) and (22,0) SWCNTs versus axial strain.
∗ † 1 2
3 4
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Prof. J.-M. Vigoureux, Dr. P. Hoang, Dr. S. Picaud, Prof. C. Girardet and F. Moulin are gratefully thanked for fruitful discussions. This work was done as part of the CNRS GDR-E 2756. Z. W. acknowledges the support received from the region of Franche-Comt´e (grants Nb 051129–91 and 060914–10).
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Torsion of Carbon Nanotubes at Room Temperature Zhao Wang,1, ∗ Michel Devel,1, † and Bernard Dulmet2 1
Institute UTINAM, UMR 6213, University of Franche-Comt´e, 25030 Besan¸con Cedex, France. 2 ´ FEMTO-ST, UMR CNRS 6174, DCEPE, Ecole Nationale Sup´erieure de M´ecanique et des Microtechniques, 25030 Besan¸con CEDEX, France
In this work, the torsion of carbon nanotubes at room temperature is computed using molecular dynamics simulations based on an atomic reactive empirical potential AIREBO. The ultimate twist rate and the deformation energy are calculated for nanotubes with different geometries. We find surface transformation of nanotubes during twisting. Formation of structural defects is also observed before the fracture occurs. PACS numbers: 61.46.Fg, 81.40.Jj, 62.20.Dc, 31.15.Qg
INTRODUCTION
Carbon nanotube (CNT) is one of the strongest materials currently known to mankind. This results from the high covalent energy of its conjugated bounds between quasi-sp2 carbon atoms. Their Young’s moduli are nearly 1 TPa and their ultimate strain can be up to 60 GPa [1, 2]. The thermal conductivity of CNTs is also very high (about 4000 W/m.K) [3, 4]. This makes them promising for future nanoelectromechanical systems (NEMS). Recently, it has been reported that CNTs can be used as key rotational elements in a nanoactuator [5] and in an electromechanical quantum oscillator [6]. Their potential application in ultra-high-density optical sweeping and switching devices, bio-mechanical and chemical sensors or electromagnetic transmitters has been mentioned [7]. Furthermore, it was shown by Jiang et al. [8] and Zhang et al. [9] that multifunctional nanoyarns have been fabricated by twisting multi-walled CNTs (MWCNTs) together. Understanding the torsional behavior of CNTs for these promising uses is a fundamental issue. In recent experimental studies, Williams et al. [10] measured the torsional constants of MWCNTs using atomic force microscopy force distance technique and found that the MWCNTs become stiffer with repeated deflection. Clauss et al. [11] presented atomically resolved scanning tunneling microscopy images of twisted armchair single-walled CNTs (SWCNTs) in a crystalline nanotube rope. Papadakis et al. [12] characterized nanoresonators incorporating one MWCNT as a torsional spring, and found that inter-shell mechanical coupling varies significantly from one tube to another. Treister and Pozrikidis proposed a new method for constructing the equilibrium shape of nanotubes and calculated the stretching, twisting and expansion deformation using unconstrained energy minimization [13]. CNTs’ conductance is also an important characteristic for their applications. The quantum conductance of CNTs depends strongly on their chirality, which can be changed by torsion [6]. The change of tube’s electronic properties due to torsion has been predicted in
several theoretical studies [14, 15]. Recently, metalsemiconducting periodic transitions were reported in experiments [6]. Moreover, Ertekin et al. [16] studied the ideal torsional strengths and stiffness of zigzag CNTs using first-principle calculations and found that the strength of torsion of MWCNTs is about 20 times larger than for an iron rod of the same size. Wang et al. [17] calculated the shear modulus of CNTs with a molecular dynamics (MD) method. The mechanical integrity of SWCNTs was evaluated by Shibutani et al. [18] with MD simulations. In this paper, we carry out MD simulations to compute the ultimate torsion deformation of CNTs at room temperature. Related change in deformation energies is also studied. The outline of this paper is as follows. The details about our computational model will be presented in Section II. The results will be shown and discussed in Section III. Then, we draw some conclusions in Section IV. Analytical formulae for the interatomic force calculation using the AIREBO potential are given in Appendix.
METHODS
To simulate the torsion of CNTs, we start with tubes fixed at one end by a hypothetical substrate and relaxed in vacuum to reach equilibrium at 298 K. An imposed deformation of torsion is then applied at the other end by successive steps of 0.1 degree every 1000 fs. Trajectories of atoms are updated at each iteration step during the simulation by solving Newton’s laws of motions, with the leap-frog algorithm and a 1 fs time step. In the framework of the AIREBO potential [19], total potential energy U p of the system is the collection of that of individual atoms: U
p
=
P tor 1 X X V R (r ) − b V A (r ) + V L−J (r ) + P Vkijℓ ij ij ij ij ij k6=i,j ℓ6=i,j,k 2 i j6=i (1)
where V R and V A are the interatomic repulsion and attraction terms between valence electrons, for bound atoms i and j at a distance rij . The bond order func-
2 tion bij provides many body effects by depending on the local atomic environment of atoms i and j. It is the key quantity which allows including the influence of the atomic environment of the bond in the computation of its strength. It is derived from Huckel electronic structure theory. The long-range interactions are included by adding a parameterized Lennard-Jones 12-6 potential term V L−J . V tor represents the torsional interactions and depends on atomic dihedral angles. Note that the long-range van der Waals interactions between atoms in the same tube must be considered in the case of large deformation, to avoid an artificial cut-off energy barrier, as discussed in Ref. [2]. bij can be written as follows. 1 σ−π σ−π DH bij + bji + bRC bij = ji + bji 2
(2)
σ−π where bij depends on the local coordination of i and j, and the bond angles, bRC ji represents the influence of possible radical character of atom j and of the π bond conjugations on the bond energy. bDH depends on the ji dihedral angle for C-C double bonds. Note that the value of bij is larger for a stronger bond.
σ−π bij =
1+
P
k(6=i,j)
FIG. 1: (Color online) Schematic of the definition of the torsion angle θ. Imposed deformations are applied to one tube end while the other end is kept fixed.
tube radius. In our simulation, the time step between each deformation level is taken to be long enough (10000 step/degree) for letting the tubes have enough time to adapt to the deformation imposed to one end. However, if we would apply torsion with higher speed (e.g. some degrees per ps), the fracture could occur earlier and the buckling shape of the surface could be different.
FIG. 2: (Color online) Shape of three twisted chiral CNTs
c fik (rik )
× G(cos θijk ) exp(λijk ) + Pij
(3) where θijk is defined as the angle between vector rij and rik . Pij and G(cos θijk ) are cubic and fifth-order polynomial splines, respectively. The inter-atomic force is then calculated as the negative gradient of the total potential energy of the system. Relevant formulae are presented in Appendix. RESULTS AND DISCUSSIONS
In this paper we study the torsion of various SWCNTs and of some MWCNTs made with monochiral SWCNTs. The distance between the tube layers is about 0.34 nm. The torsion angle θ is the angle between the initial position of the outer wall and its deformed position, after the imposed rotation of the free end of the CNT by this angle θ, as shown in Fig. 1. Periodic twisting waves appear on the tube surface under large torsional deformation. The change of the helical shape of CNTs depends on the tube radius. Fig. 2. shows the different helical surface shapes of three twisted chiral CNTs of the same length, but different radii just before the fracture occurs. We can see that the helical pitch is longer for big tubes than for the smaller one. It is not constant but decreases during the deformation. It also increases with the tube length for a given radius and a given twisting angle. Furthermore, we find that the length of each helical pitch depends on the torsional angle and the
−1/2 with the same length L = 9.6nm and the same chiral angle
= 19.1◦ , just before the fracture occurs, at θ = 630◦ , 497◦ and 427◦ , respectively. Left: (6, 3), R = 0.31nm; middle: (14, 7), R = 0.72nm; right: (20, 10), R = 1.03nm.
As the deformation of the cross-section of a tube may have an interesting influence on its electric properties [20], we present in Fig. 3, different shapes of the cross section of a tube twisted to several torsional angles. It can be seen that the section remains circular when the deformation is relatively small. It deforms to an ellipse when deformations become important. Then, with increasing torsional angle, this ellipse section rotates around the tube axis with a motion following the direction of deformation applied to the tube end.
FIG. 3: (Color online) Cross section in the middle of a (5, 5) tube (L = 9.5nm) under torsional deformation. The green arrows denote the direction of rotation.
How much torsional deformation can the CNT sustain? This is an important issue for the CNTs used as torsional elements. In Fig. 4, we show the fracture of a SWCNT induced by torsional deformation. We can see that when θ = 596◦ , the honeycomb lattice of the tube is strongly deformed and does not keep anymore its regular hexag-
3 onal shape. Then vacancies appear and the fracture of the tube occurs very soon (some ps) after the appearance of more defects at the location of the first defect. Note that the atomic configuration during the formation of these defects is similar to the helical lattice while a Stone-Wales defect is inclined under large tensile strain [24].
value of deformation energy for the big tubes is lower than that for the smaller one. In Fig. 6 (b). we use tubes of similar radii and lengths to show that the deformation energy is almost independent of the tube chirality. The increase rate of deformation energy of the zigzag tube is slightly higher than that of the chiral and the armchair ones. This corresponds to the fact that the average axial bond strength of a zigzag SWCNT is slightly higher than that of other tubes with similar sizes but differing chiralities [21].
FIG. 4: (Color online) Fracture of a (5, 5) tube (L = 9.5nm) under torsional deformation.
In order to draw some general results from the short tubes studied here, we define the twist rate as the torsion angle θ per unit length of CNTs. We plot in Fig. 5 the ultimate value of the twist rate (UTR) for 9 SWCNTs with the same length but with different radii and chiralities. It can be seen that the UTR of the small tubes is clearly higher than that of the big ones. This study generalizes similar results found in the paper of Ertekin et al. [16] (Figure 2) for zigzag tubes. Thus, we can see that the UTR of zigzag tubes decreases faster than the UTR of armchair tubes when the tube radius is increased. Thus, we conclude that big armchair tubes can resist better to torsional deformation than big tubes of other chiralities but that it is the contrary for small ones.
FIG. 5: Ultimate twist rate versus the tube radius for 3 groups of tubes with different chiralities. Each group has 3 tubes with different radii. The length of all these tubes is fixed to 95 ˚ A.
We also study the effects of tube geometry on the deformation energy of the tube, which is defined here as the change of the total interatomic potential energy of the CNT. It is an important factor coupled to the tube’s elastic constant. We plot in Fig. 6 the torsional energy against the twist rate. We can see in Fig. 6 (a) that the deformation energy of the biggest tube increases faster than that of the smaller tubes. However, the ultimate
FIG. 6: Deformation energy versus the twist rate for: (a) 3 zigzag tubes with the same length but with different radii, and (b) 3 tubes with almost the same length and radius but with different chiral angles. The deformation energy presented here is the average value on all the atoms.
We study also the torsion of multi-walled CNTs (MWCNTs), as demonstrated in Fig. 7. It shows how an initially armchair MWCNT breaks under torsion from two directions of observation. We can see the appearance of twisting waves in both the inner and outer layers when the tube is deformed, then the fracture occurs first at the outer layer after the appearance of defects on its surface. We show the ultimate twist rate of MWCNTs in Table 1. It can be seen that the UTR decreases with the number of carbon layers. As for SWCNTs, it is higher
4
FIG. 7: The torsional deformation and fracture of an MWCNT (5,5)@(10,10) (L = 194.3˚ A, R = (3.39˚
[email protected]˚ A)) UTR(degree/˚ A) MWCNTs (5,5)@(10,10) 2.99 1.64 (5,5)@(10,10)@ (15,15) @ (20,20) (5,5)@(10,10)@ ... @ 30,30) 0.96 3.66 (0,9)@(0,18) 2.36 (0,9)@(0,18)@ (0,27)@(0,36) (0,9)@(0,18)@ ... @(0,54) 1.55 TABLE I: Ultimate twist rate of MWCNTs with the same length about 200 ˚ A.
for the zigzag tubes. The deformation energy of an initially zig-zag MWCNT is plotted in Fig. 8 (a) and the corresponding images of the failure of this MWCNT in Fig. 8 (b). against the twist rate. We can see that the energy per atom of the outer layer during the deformation is much higher than for atoms of the inner layer. This correlates well with the fact that the outer layer breaks first. Furthermore, we can also see that the change of the van der Waals interactions do not play a very important role in the total deformation energy.
CONCLUSIONS
FIG. 8: (a) Average deformation energy (per atom) vs. twist rate for each layer in a MWCNT (0,9)@(0,18)@ (0,27)@(0,36) L = 84.0˚ A. The deformation energy is the average value per atom. (b) Failure of this MWCNT under torsional deformation. Top: side view. Bottom: cross-section view.
twist rate is higher for zigzag MWCNTs than that for armchair ones.
APPENDIX: INTERATOMIC FORCES CALCULATIONS USING AIREBO POTENTIAL
the gradient of the AIREBO potential can be written as follows:
F (ri ) = −∇i U
p
= −
R A A 1 X X ∇i V (rij ) − ∇i bij V (rij ) − ∇i V (rij )bij + P tor ∇ V L−J (r ) + P ∇i Vkijℓ 2 i j6=i ij i ij k6=i,j ℓ6=i,j,k (4)
In summary, the torsional deformation of CNTs is simulated by using MD with the AIREBO potential. Surface transition from zigzag or armchair to chiral type and periodic twisting waves are observed in our simulations. We observed also the creation of defects and the fracture on the tube surface. The cross section of SWCNTs is found to become elliptic and rotates around the tube axis when the deformation is large enough. We calculated the ultimate value of the twist rate and the deformation energy for several types of CNTs with different geometries. We find that the small tubes can be twisted more than the big ones, even if bigger ones can initially store more deformation energy for a given twist angle. The ultimate
where ∇i stands for the gradient at the position of atom i. In the following equations, we write it as ∇ for short. It is relatively simple to calculate ∇i V R , ∇i V A , ∇i VijL−J tor and ∇i Vkijℓ . Hence we present here the formulation for ∇i bij . It can be written as: ∇bij =
1 σ−π σ−π DH ∇bij + ∇bji + ∇bRC ji + ∇bji 2
(5)
σ−π The first term of the bond-order function bij depends on the local coordination and the bond angles. Its gradient can be written as follows.
5
∇b
σ−π ij
=
σ−π 3 ) P ij n c [ 5 n=0 an (cos(θpij )) · exp(λpij )∇fip 2 P5 c + n=0 an (cos(θqij ))n · exp(λqij )∇fiq P n−1 · exp(λ c + 5 pij )fip ∇ cos(θpij ) n=0 nan (cos(θpij )) P n−1 · exp(λ c ∇ cos(θ + 5 na (cos(θ )) )f n qij qij iq qij ) + ∇Pij ] n=0 (6) −(b
Note that details about spline calculations and corresponding parameters can be found in the appendix of Ref. [19]
∗ †
where θijk is defined as the angle between the vectors rij and rik . Pij and G(cos θijk ) are cubic and fifth-order polynomial splines, respectively.
[1]
rip · rij rip · rij ∇(rip · rij ) − ∇rip 2 − ∇rij 2 rip rij rip rij rij rip (7) The switching function f c (so-called bond-weight term wij (rij ) in Ref. [19]) restricts the pair interaction to the 1st and the 2nd nearest neighbors, its gradient can be written as follows.
[4]
[2] [3]
∇ cos(θpij ) =
max if rij > Dij
∇bRC ji = ∇Fij
(8)
(9)
where F is a tricubic spline, which depends on f c . The gradient of the dihedral-angle term bDH can be ji written as follows.
P P c (r )f c (r ) + (1 − cos2 (Θijkl )fik ik jl jl k(6=i,j) l(6=i,j,k) i h P P c (r )f c (r ∇(1 − cos2 (Θijkl ))fik Tij ik jl jl + k(6=i,j) l(6=i,j,k) i h P P c (r )∇f c (r + (1 − cos2 (Θijkl ))fik Tij jl ik jl k(6=i,j) l(6=i,j,k) h i P P c (r )∇f c (r Tij (1 − cos2 (Θijkl ))fjl jl ik ik k(6=i,j) l(6=i,j,k)
∇bDH = ∇Tij ij
rik · rip rik rip
Hence its gradient can be written as:
∇ cos Θpijk =
+ cos(θpij ) cos(θijk )
[8] [9]
[10]
[11] [12]
[13] [14] [15] [16] [17]
(10)
where T is a tricubic spline and Θijkl is the dihedral angle around a double bond.
[7]
min < r max if Dij ij < Dij
max min where Dij and Dij are the cut-off radii between atom max min i and atom j. We take Dij = 0.2 nm and Dij = 0.17 nm. The second term of the gradient of bRC ji can be calculated as:
cos Θpijk = epij · eijk =
[6]
min if rij < Dij
0 min ) π(rij −Dij c π − sin ∇fij = ∇rij D max −D min 2(D max −D min ) ij ij ij ij 0
[5]
−1
[18] [19] [20] [21] [22] [23]
sin(θpij ) sin(θijk ) (11)
rik ·rip ∇ + ∇ cos(θpij ) cos(θijk ) + cos(θpij )∇ cos(θijk ) × rik rip −1 sin(θpij ) sin(θijk ) # " cos(θijk ) cos(θpij ) ∇ cos(θijk ) + ∇ cos(θpij ) + cos Θpijk sin(θijk )2 sin(θpij )2 (12)
[24]
Electronic address:
[email protected] Electronic address:
[email protected] R. Ruoff, D. Qian, and W. Liu, C. R. Physique 4, 993 (2003). H. Rafii-Tabar, Phys. Rep. 390, 235 (2004). J. Hone, M. Whitney, C. Piskoti, and A. Zettl, Phys. Rev. B 59, 2514 (1999). S. Berber, Y.-K. Kwon, and D. Tomanek, Phys. Rev. Lett. 84, 4613 (2000). A. Fennimore, T. Yuzvinsky, W.-Q. Han, M. Fuhrer, J. Cumings, and A. Zetti, Nature 424, 408 (2003). T. Cohen-Karni, L. Segev, O. Srur-Lavi, S. Cohen, and E. Joselevich, Nature Nanotechnology 1, 36 (2006). C.-H. Ke and H. Espinosa, Handbook of Theoretical and Computational Nanotechnology (American Scientific Publishers, 2006), chap. 121. Jiang K., Li Q., and Fan S., Nature 419, 801 (2002). M. Zhang, K. Atkinson, and R. Baughman, Science 306, 1358 (2004). P. Williams, S. Papadakis, A. Patel, M. Falvo, S. Washburn, and R. Superfine, Phys. Rev. Lett. 89, 255502 (2002). W. Clauss, D. J. Bergeron, and A. T. Johnson, Phys. Rev. B 58, R4266 (1998). S. Papadakis, A. Hall, P. Williams, L. Vicci, M. Falvo, R. Superfine, and S. Washburn, Phys. Rev. Lett. 93, 146101 (2004). Treister Y., and Pozrikidis C., Comp. Mater. Sci. 41, 383 (2008). A. Rochefort, P. Avouris, F. Lesage, and D. Salahub, Phys. Rev. B 60, 13824 (1999). B. Liu, H. Jiang, H. Johnson, and Y. Huang, J. Mech. Phys. Solids 52, 1 (2004). E. Ertekin and D. C. Chrzan, Phys. Rev. B 72, 045425 (2005). Y. Wang, X. Wang, and X. Ni, Modell. Simul. Mater. Sci. Eng. 12, 1099 (2004). Y. Shibutani and S. Ogata, Modell. Simul. Mater. Sci. Eng. 12, 599 (2004). S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472 (2000). F. Torrens, Future Generation Computer Systems 20, 763 (2004). G. van Lier, C. van Alsenoy, V. van Doren, and P. Geerlings, Chem. Phys. Lett. 326, 181 (2000). D. W. Brenner, Phys. Rev. B 42, 9458 (1990). D. Brenner, O. Shenderova, J. Harrison, S. Stuart, B. Ni, and S. Sinnott, J. Phys.: Condens. Matter 14, 783 (2002). C. Pozrikidis, Arch. Appl. Mech. Article in Press, 1-11 (2008).
Published as : Z. Wang et al. Phys. Rev. B, 78, 085425 (2008) and ...
Chapitre 4
Electrostatic properties
Les th´eories ont caus´e plus d’exp´eriences que les exp´eriences n’ont caus´e de th´eories. Joseph Joubert
Understanding of the properties of electric charges in carbon nanotubes (CNTs) is one of the fundamental issues for their promising applications in nanoelectromechanical systems [35, 32, 30, 146, 147, 148]. In experiments, electrostatic properties of CNTs have been addressed so far using electric force microscopy (EFM) charge injection techniques [70, 72] indicating that electric charges are distributed uniformly along the tube, however without theoretical support. In theoretical studies, density functional theory [149] and classical electrostatics [150] calculations have been performed to compute the charge distribution on single-walled CNTs (SWCNTs). U -like shapes of the charge distribution with a charge accumulation at the tube ends have been predicted for short CNTs (< 100 nm), which is however not in a length easily accessible in experiments. In this work, we study the static distribution of electric charges in CNTs in a framework of collaboration with experimentalists. In particular, we study the interesting effect of weak charge enhancement localized at the tube ends and simultaneously uniform charge density along the tube taking into account the oxide surface on which SWCNTs are deposited. Theoretical calculations are compared with experimental measurements. We demonstrate that the previously predicted U -shape charge distribution in CNTs is strongly modified [149] in conditions that can be accessed by experiments, i.e. for tubes with ∼ µm length deposited upon a SiO2 substrate, and that the field enhancement at the tube end strongly depends on the cap geometry for short tube lengths.
36
4.1
4. Electrostatic properties
Computational techniques
In our calculations, the interactions between the electric charges and the induced dipoles are described using the Gaussian-regularized atomic charge-dipole interaction model [65, 151], in which the atoms are treated as interacting polarizable points with free charges, and the distribution of charges and dipoles are determined by the fact that their static equilibrium state should correspond to the minimum value of the total molecular electrostatic energy. Compared with classical Coulomb-law-based models in which only the charges are considered, this model provides more accurate description of electrostatic properties of CNTs, since the charges, the induced dipoles and the atomic polarizabilities are taken into account. In order to achieve a valid comparison between experimental data and calculation results, the effect of a silica substrate close to CNT surface is taken into account in our calculations, by adding surface-induced terms to the vacuum electrostatic interaction tensors using the method of mirror images [152]. The dielectric constant of SiO2 is taken to be 4.0 as that of the material used in the experiment. The average distance between the bottom of the tubes and the SiO2 surface is set to d = 0.34 nm after the computed CNT-SiO2 long-range interacting configurations from Refs. [153, 154]. Furthermore, we note that d can slightly vary with the tube radius R, it is fixed to 0.34 nm in this work as an adjusted average value. The atomic structure of CNTs is optimized by energy minimization using the method of conjugated gradient based on a many-body chemical potential model AIREBO (adaptive interatomic reactive empirical bond order) [10].
4.1.1
Charge-dipole model
In the electrostatic charge-dipole model [65], each atom is associated with a free charge q and an induced atomic dipole p (as shown in Figure 4.1). The total electrostatic energy U elec for a system with N atoms can be written as follows :
4.1. Computational techniques
37
Figure 4.1 – Simplified schematics of the atomic charge-dipole model. E stands for the electric field.
U elec =
N X i=1
qi (χi + V0,i ) −
N X i=1
N
pi · E0,i + −
N X N X i=1 j=1
N
1 XX i,j qi Tq−q qj 2 i=1 j=1
N
i,j pi · Tp−q qj −
N
1 XX i,j pi · Tp−p · pj 2
(4.1)
i=1 j=1
where χ is the electron affinity, V and E stand for the external potential and electric field, respective. T and T s are the electrostatic interacting tensors. They can i,j i,j i,j i,j be written as Tq−q = (1/4π0 ) × (1/rij ), Tp−q = −∇ri Tq−q and Tp−p = −∇rj ⊗ i,j ∇ri Tq−q , where ri presents the coordinate of atom i, ri,j stands for the distance between atom i and atom j. We regularize T and T by a Gaussian distribution for the close atoms in order to avoid divergence problems, as discussed in Refs.[? 65], by assuming a Gaussian distribution of extra charges. qi 2 2 exp − |r − r | /R i π 3/2 R3
(4.2)
√ 1 erf ri,j / 2R = 4π0 ri,j
(4.3)
" # r 2 /2R2 ri,j 2 ri,j −ri,j 1 ri,j = erf √ − e 3 4π0 ri,j π R 2R
(4.4)
ρi (r) =
where ri stands for a position around the electric center of atom i. Based on this assumption, the regularized interacting tensors can be written as follows
i,j Tq−q
i,j Tp−q
38
4. Electrostatic properties
i,j Tp−p =
2 1 3ri,j ⊗ri,j −ri,j I 5 4π0 ri,j
1 − 4π 0
q
h
2 1 ri,j ⊗ri,j e 2 π R3 ri,j
r √i,j 2R 2 /2R2 −ri,j
erf
−
q
2 /2R2 2 ri,j −ri,j π R e
i
(4.5)
The value of R used in this work is about 0.06862 nm, which was fitted to reproduce the polarizability of metallic tubes [8]. Note that the value of Gaussian charge distribution width R used in our previous work [155] for the atoms at tube edges is about 0.09nm (about 1.3 time that of the carbon atom with three chemical bonds). This value is obtained by fitting to data in a previous study using DFT calculation [149] (see Figure 4.2).
Figure 4.2 – Distribution de charge le long de l’axe d’un SWCNT (5, 5) ouvert et relax´e (110 atomes). L’intensit´e de la couleur de coloration des atomes est proportionnelle `a la densit´e de charge. Les fl`eches repr´esentent les dipˆoles. La courbe en vert correspond au calcul de r´ef´erence de Keblinski et al., celle en noire au mod`ele charges-dipˆoles avec une largeur R ´egale pour tous les atomes et la courbe en rouge au mod`ele charges-dipˆoles avec une largeur R ajust´ee s´epar´ement pour les atomes poss´edant seulement 2 liaisons
The relationship between R and the polarizability of atoms can be presented by following equation :
4.1. Computational techniques
39
p 2/π qi2 1 1 i,i qi Tq−q qi = 2 4π0 R 2 i,i pi · Tp−q qi = 0 1 1 i,i pi · Tp−p · pi = pi · αi−1 · pi 2 2
(4.6a) (4.6b) (4.6c)
where αi stands for the polarizabilities of atom i. The static equilibrium state of charges and dipoles should correspond to the minimum value of U elec , and hence the derivatives of U elec with respect to q and p should be zero. Taking this boundary condition into account with the self-energy terms (when i = j), we can obtain the equilibrium configuration of charge and dipole by solving N linear vectorial equations and N linear scalar equations as follows :
∀i = 1, ..., N
N N P P i,j i,j Tp−p ⊗ pj + Tp−q qj = −Ei j=1
j=1
N N P P i,j i,j Tp−q · pj + Tq−q qj = −(χi + Vi ) j=1
(4.7)
j=1
Eq. 4.7 can also be written in the form of matrix as follows. Tq−q Tp−q −Tp−q −Tp−p
!
q p
!
=
−χ − V E
!
(4.8)
The above formulations can apply to molecules initially neutral. In case of molecules carrying net electric charges, the total net charge Qtot can be appended to the system as a boundary condition by adding a Lagrange multiplier λ as follows :
Tq−q Tp−q 1 q −χ − V −Tp−q −Tp−p 0 p = E 1 0 0 λ Qtot
(4.9)
Note that it is possible to add several Lagrange multipliers to the system, by which we can perform calculations for the system with several molecules. Details about the Fortran implementation of the numerical solution of these linear algebraic equations can be found in the Chapter 2 of Ref. [156] (electronic edition : www.nrbook.com). In case of CNTs placed upon substrates, as those in EFM experiments [71, 70, 157], the charge and dipole distribution is different from that in space. We take this
40
4. Electrostatic properties
condition into account by adding a surface-induced term T m and T m to the vacuum electrostatic interacting tensors using the method of mirror images.[152]
PHYSICAL REVIEW B 78, 085425 共2008兲
Electric charge enhancements in carbon nanotubes: Theory and experiments Zhao Wang,1,* Mariusz Zdrojek,2 Thierry Mélin,3 and Michel Devel1 1Institute
UTINAM, UMR 6213, University of Franche-Comté, 25030 Besançon Cedex, France of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland 3Département ISEN, Institut d’Electronique de Microélectronique et de Nanotechnologie, CNRS-UMR 8520, Avenue Poincaré, BP 60069, 59652 Villeneuve d’Ascq Cedex, France 共Received 20 March 2008; revised manuscript received 4 June 2008; published 20 August 2008兲 2Faculty
We present a detailed study of the static enhancement effects of electric charges in micrometer-long singlewalled carbon nanotubes, using theoretically an atomic charge-dipole model and experimentally electrostatic force microscopy. We demonstrate that nanotubes exhibit at their ends surprisingly weak charge enhancements which decrease with the nanotube length and increase with the nanotube radius. A quantitative agreement is obtained between theory and experiments. DOI: 10.1103/PhysRevB.78.085425
PACS number共s兲: 73.63.Fg, 41.20.Cv, 68.37.Ps, 85.35.Kt
Understanding of the properties of electric charges in carbon nanotubes 共CNTs兲 is one of the important issues for their promising applications in nanoelectromechanical systems,1 field emission,2 chemical sensors,3 and charge storage.4–6 A key aspect of the electrostatics of these one-dimensional systems is the knowledge of the distribution of electric charges along the nanotubes, because charges are likely to accumulate at the nanotube ends due to Coulomb repulsion. Theoretical predictions have been established for this effect, but not in the range of lengths accessible from experiments, so that no comparison has been established between theory and experimental observations so far. More precisely, the electrostatic properties of single-walled nanotubes 共SWCNTs兲 have been addressed on the one hand using electric force microscopy 共EFM兲 experiments7,8 coupled to charge injection techniques.9,10 Results obtained for micrometer-long nanotubes indicated that electric charges are distributed rather uniformly along the tube length, with, however, no theoretical support in this range of nanotube length. On the other hand, in theoretical studies, density-functional theory11 and classical electrostatics12 calculations have been performed to compute the charge distribution in SWCNTs, and have predicted U-like shapes due to a charge accumulation at the nanotube ends. These calculations, however, only hold for short 共⬍100 nm兲 nanotube lengths, which are not easily accessible from experiments. It is the scope of this paper to provide a combined experimental and theory work on this issue. We present a detailed study of the static enhancement effects of electric charges in SWCNTs, using theoretically an atomic charge-dipole model and experimentally electrostatic force microscopy. It is demonstrated that the U-like shape of the charge distribution expected for short nanotubes is replaced in the case of micrometer-long tubes by weak charge enhancements localized at the nanotube ends, in agreement with the experimental values for the enhancement factors 共up to few tens of %兲 observed from EFM and charge injection experiments. The dependence of the charge enhancement factors on the nanotube radius has also been measured from EFM experiments, and falls in quantitative agreement with theoretical predictions for micrometer-long tubes. The paper is organized as follows: we first describe the numerical calculations of the charge distribution along nano1098-0121/2008/78共8兲/085425共5兲
tubes using the atomic charge-dipole model and the results obtained for short nanotubes with open or closed caps, either considered in vacuum or on a SiO2 substrate. The extrapolation procedure to the case of micrometer-long nanotubes is then presented, and compared with experimental EFM measurements of charge enhancement factors on SWCNTs. We finally discuss the dependence of the enhancement factors as a function of the nanotube radius. In the theoretical calculations presented throughout this work, the interactions between the electric charges and the induced dipoles are described using the Gaussian-regularized atomic charge-dipole interaction model,13,14 in which the atoms are treated as interacting polarizable points with free charges, and the distribution of charges and dipoles are determined by the fact that their static equilibrium state should correspond to the minimum value of the total molecular electrostatic energy. Compared with classical Coulomb-lawbased models in which only charges are considered, this model provides a more accurate description of electrostatic properties of CNTs, since the charges, the induced dipoles and the atomic polarizabilities are taken into account. In order to achieve a valid comparison between experimental data and calculation results, the effect of a SiO2 substrate 共nanotubes are usually deposited on a SiO2 thin film in experiments兲 is also taken into account in our calculations, by adding surface-induced terms to the vacuum electrostatic interaction tensors using the method of mirror images.15 The dielectric constant of SiO2 is taken as 4.0. The average distance between the bottom of the tubes and the SiO2 surface is set to d = 0.34 nm after the computed CNT-SiO2 long-range interacting configurations from Refs. 16 and 17. Furthermore, we note that d can slightly vary with the tube radius R. It will, however, be fixed to 0.34 nm in this work as an average value. The atomic structure of CNTs is then optimized by energy minimization using the method of conjugated gradient based on a many-body chemical-potential model AIREBO 共adaptive interatomic reactive empirical bond order兲.18 Linear charge densities of 0.055e / nm have been used in calculations, so as to match linear charge densities observed experimentally.19 To illustrate the typical outputs of the atomic-scale calculations, we show in Fig. 1 the charge distribution at the end of a 共9, 0兲 CNT of length L ⬇ 11.5 nm and average charge
085425-1
©2008 The American Physical Society
PHYSICAL REVIEW B 78, 085425 共2008兲
WANG et al.
FIG. 1. 共Color online兲 Charge distribution at the ends of an open-ended and a closed-ended 共9, 0兲 SWCNTs 共L ⬇ 11.5 nm, ave = −6.6⫻ 10−4 e / atom兲 in free space 关共a兲 and 共b兲兴 and on the SiO2 substrate 关共c兲 and 共d兲兴. The color of the atoms is proportional to the local charge density. The green vectors stand for the induced atomic dipoles. The dark arrows stand for the local electric fields induced by the net charge, their length and color are proportional to the field intensity. 共a兲 The minimum and maximum atomic charge densities in this tube are: min = −5.3⫻ 10−4e / atom and max = −16⫻ 10−4e / atom, respectively. 共b兲 min = −5.2⫻ 10−4e / atom and max = −34⫻ 10−4e / atom. 共c兲 min = + 5.6⫻ 10−4e / atom and max = −34⫻ 10−4e / atom. 共d兲 min = + 5.8⫻ 10−4e / atom and max = −74⫻ 10−4e / atom.
density ave = −6.6⫻ 10−4e / atom. The color of the atoms is proportional to their charge in the figure. We represented here for sake of clarity the four distinct situations in which the nanotube exhibits either a closed 关Figs. 1共a兲 and 1共c兲兴 or an open 关Figs. 1共b兲 and 1共d兲兴 cap structure, and the tube is either considered in vacuum 关Figs. 1共a兲 and 1共b兲兴 or deposited on a SiO2 substrate 关Figs. 1共c兲 and 1共d兲兴. As seen from Fig. 1, the charge density at the tube ends is higher than that at other parts of the tubes in all situations. The maximum charge density on the opened cap is here about twice that on the closed one for this small-radius tube. Finally, when the tube is deposited on the SiO2 surface, electrons are attracted by their image charge toward the SiO2 surface, as a typical semispace effect. Since the nanotubes used in experiments have lengths in the micrometer range, and since this scale can hardly be directly addressed by calculations using atomic models due to the limit of computational resources, the issue about the relationship between the tube length L and the charge distribution needs to be carefully addressed, so as to later extrapolate charge enhancement factors to the length scales of interest in experiments. The length dependence of the charge enhancements at the nanotube ends is illustrated in Fig. 2, in which we plotted the local average charge density as a function of the position along the nanotube 共the x-axis origin in
FIG. 2. Charge profile along three 共9, 0兲 SWCNTs with different tube lengths L, in space 共hollow symbols兲 and upon a SiO2 surface 共solid symbols兲, using a separation distance d = 0.34 nm 共see text兲. The total net charge density on each tube ave is fixed to 6.4⫻ 10−4e / atom 共equivalently, 0.055e / nm兲. Each point is calculated as the average value of the charge carried by the nanotube atoms over 10%L.
Fig. 2 corresponds to the nanotube midpoint兲. The local average charge density is defined from the charge carried by individual CNT atoms, when averaged along the nanotube circumference and along a fraction of the length L of the CNT 共this fraction is taken as 10%L in Fig. 2兲. ave is the quantity which can be accessed experimentally from EFM techniques.19 The typical shape of the CNT charge distribution observed in Fig. 2 corresponds to the U-like shape expected for short nanotubes,11 but the charge enhancement at the tube ends is already seen to become less significant when the tube gets longer. Furthermore, we can also see that the charge enhancement is weaker when the ends of the tubes are closed and when the nanotubes are placed on the SiO2 surface. The latter effect can be understood by the fact that the net nanotube charge is located at the CNT side close to the substrate 关see Fig. 1共c兲 and 1共d兲兴, which leads to an effective reduction of the charge-distributed area in the nonaxial direction, similar to an effective decrease of the nanotube radius R, which will be discussed further in this paper 共see Fig. 6兲. We now focus on charge enhancements for micrometerlong nanotubes, and their comparison with experimental results. Since the spatial resolution in EFM experiments is about 100 nm 共this resolution is mostly limited by the tipsubstrate separation during EFM detection兲, we now consider the enhancement zone in our calculation as a zone of length 10%L at the tube end, and define the charge enhancement ratio as the ratio between the charge density end averaged in the zone of length 10%L at the end of the nanotube, and the charge density middle at the center of the nanotube. The influence of the tube length on the charge enhancement ratio = end / middle is shown in Fig. 3. is seen to decrease significantly with L for short tubes 共particularly for L ⬍ 10 nm兲, but the variations get smaller when the tube is longer. Note that is independent of ave, because the local charge densities should be proportional to the total one by
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FIG. 3. Ratio of charge enhancement as a function of tube length L 共in common logarithmic scale兲. This ratio is calculated for both an open-ended 共circles兲 and a closed-ended 共triangles兲 共5,5兲 SWCNTs 共radius R ⬇ 0.34 nm兲 placed upon a SiO2 surface 共solid symbols兲 with d = 0.34 nm, and is compared with that for the same tubes in space 共empty symbols兲. The symbols present the calculated points, and the lines stand for the extrapolation curves.
requiring a constant electric potential on the tube surface. Furthermore, we find that if the 共open or closed兲 cap structure plays an important role in the charge enhancement for short tubes 共L ⬍ 15 nm兲 共as seen in Fig. 1兲, this effect already becomes unsignificant for L ⬇ 30 nm, and will become negligible for micrometer-long nanotubes in experiments with ⬇100 nm resolution. Finally, it appears that the only parameter that needs to be properly taken into account is the presence of the SiO2 surface below the nanotube, which still effectively reduces the charge enhancement ration for L = 30 nm. In order to extrapolate these results toward micrometerlong nanotubes, we performed a fit of the data points of Fig. 3. Since the analytical formula of the exact distribution of charge on a hollow tube is not known in the literature, we used the equation: = ln共a1 ⫻ L + a2兲 / ln共a3 ⫻ L + a4兲, in which an共1 ⱕ n ⱕ 4兲 are four fitted parameters for each nanotube radius. This phenomenological equation has been chosen since it describes a ratio between two cylindrical capacitances, and is thus well suited to account for , which is the ratio between the linear charge densities at the end and at the middle of the nanotube. The lines in Fig. 3 correspond to the fits obtained independently for the nanotubes with either open or closed caps, vacuum environment, or SiO2 surface. The extrapolated values for are seen to converge for large L for open and closed cap structures, but to differ depending on the vacuum or SiO2 environment. This behavior is in full agreement with the trend observed on the atomic calculation points obtained for L ⬇ 30 nm, which already brings confidence at this stage about the validity of our extrapolation procedure. Our theoretical predictions are finally compared with electrostatic measurements performed by injecting and detecting charges in individual CNTs using electrostatic force microscopy. In these experiments, nanotubes grown by chemical vapor deposition are deposited from dichlo-
FIG. 4. 共Color online兲 共a兲 Schematics of the charge injection and detection with the EFM tip. Charging takes place when the biased tip is put in contact with the CNT. During the data-acquisition cantilever is lifted at distance z above the surface. 共b兲 Topography image of a SWCNT with 0.5 nm radius deposited on 200 nm oxide layer. 共c兲 EFM image acquired before injection. The dark feature corresponds to the uncharged tube. 共d兲 EFM scan after injection experiment. The bright feature corresponds to the charged tube with the uniform linear charge density of 0.055e / nm.
romethane solutions onto silicon wafers covered by a 200nm-thick thermal dioxide layers. Individual nanotubes are located by atomic force microscopy, and then charged 关see Fig. 4共a兲兴 by pressing the biased tip of an atomic force microscope on the nanotube 共typically with an injection bias Vinj = −5 V, pressing force of a few nano-Newton兲. The CNT charge state is then measured before and after injection by EFM, by recording electrostatic force gradients acting on the tip which is intentionally lifted at a distance z about 50– 100 nm above the sample surface to discard short-range surface forces. Figure 4共b兲 shows the topography image of a SWCNT. In Fig. 4共c兲, the EFM scan of the tube before charging is shown, as a dark footprint of the CNT topography associated with attractive forces due to the nanotube capacitance. It can be shown experimentally that the negative frequency shifts are here of capacitive origin, and not originating in a positive charge transferred from the substrate to the nanotube 共see details in Ref. 10兲. The nanotube EFM image after charge injection is shown in Fig. 4共d兲. The tube is seen here as a bright feature as a result of the negative charges injected in the tube. From previous EFM studies, we have shown that the charge imaged for SWCNTs mainly correspond to charge emitted from the tube and “printed” in the oxide layer in the vicinity of the nanotube.20,21 To compare these predictions with our calculation results, we show in Fig. 5 the charge distribution at the end of a SWCNT 共total length 2 m兲 after a charge injection experiment. A nonlinear color scale has been used in Fig. 5共b兲 in order to evidence the weak charge enhancement localized within 200 nm at the nanotube end. The charge distribution along the nanotube is shown in Fig. 5共c兲, in which we plotted the charge enhancement factor measured from EFM, defined as the ratio of the EFM signal at the tube over that measured at the middle of the nanotube. From these experimental data, one gets the maximum value = 1.17⫾ 0.05 for this tube 关see Fig. 5共c兲兴, in agreement with the numerical extrapolation from theoretical results predict-
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FIG. 6. Ratio of charge enhancement for a number of 共n , n兲 nanotubes with different radius R, on a SiO2 surface. The solid squares stand for derived from the experimental measurements of seven nanotubes 共with length L = 1 ⬃ 9 m兲 deposited on 200 nm silicon oxide layer. The symbols “+” stand for the extrapolation results for micrometer length tubes.
FIG. 5. 共Color online兲 共a兲 AFM topography image of single nanotube with R = 0.8 nm and L ⬇ 2 m deposited on silicon dioxide. 共b兲 EFM scan of the same tube made after charge injection. A nonlinear color scale has been used in order to clearly show the weak enhancement at the tube end. The black line is a “guide to the eye” for the physical end of the tube. 共c兲 Experimental charge enhancement along the axis of the nanotube, defined as the ratio of the EFM signal with that measured at the middle of the nanotube.
ing ⬇ 1.165 for a 2 m tube with R = 0.8 nm deposited on a SiO2 surface. To further validate the comparison of our theoretical predictions with experiments, we now focus on the dependence of the charge enhancement ratio as a function of the nanotube radius R. Such an analysis would not be possible for short nanotubes, because the charge enhancement ratio would then be strongly dependent on the nanotube cap structure, as discussed previously 共see Fig. 3兲, while this effect is not relevant for micrometer-long nanotubes. Intuitively, one can guess that the nanotube charge enhancement factor will increase with the tube radius R, because the enhancement factor decreases with the nanotube length L: increasing R at fixed length L reduces the nanotube anisotropy, and is qualitatively similar as decreasing the nanotube length L for a fixed radius R. Experimentally, we measured the charge densities along seven SWCNTs with lengths between 1 and 9 m in a similar way as in Fig. 5共c兲, and plotted the corresponding charge enhancement ratios in Fig. 6, as a function of the nanotube radius R measured from atomic force topography images. The ⫾0.05 error bars on correspond here to the accuracy of the EFM measurements. Experimental data points clearly
show that slightly increases as a function of the nanotube radius R. The possibility to observe this behavior also confirms that the values of on micrometer-long nanotubes do not critically depend on the tube length, nor on the nanotube cap structure. Numerical calculations for the charge enhancement ratio have also been performed using 共n , n兲 nanotubes with different radius R, and are shown in Fig. 6. Direct calculations of obtained from the atomic dipole-charge models and using an averaging over 10%L are given in Fig. 6 for two short nanotubes 共9 and 12 nm, solid circles and triangles兲, as well as calculation results obtained for micrometer-long nanotubes 共dotted line兲 using the extrapolation procedure described in Fig. 3. Theoretical predictions are seen to quantitatively agree with experimental data within experimental error bars, and confirm the increase of the charge enhancement ratio as a function of the nanotube radius. The values of computed with 5%L and 15%L for micrometer length tubes 共data not shown兲 also vary within experimental error bars. In summary, we have characterized the enhancement of net electric charge in SWCNTs by both atomic-model calculations and EFM experiments. We have demonstrated that the U-like shape of the charge distribution expected for short nanotubes is replaced for micrometer-long nanotubes by weak charge enhancements localized at the nanotube ends, while the nanotube charge densities are otherwise almost constant along the nanotubes. The dependence on the tube length, nanotube cap structure, and the influence of silica substrate has been investigated. It has been shown that the charge enhancement at the ends of CNTs depends strongly on the geometry of the cap only for short tubes 共⬍100 nm兲, but has an insignificant influence for nanotubes with lengths in the micrometer range. The increase of the charge enhancement ratio with the nanotube radius has been demonstrated experimentally, in quantitative agreement with theoretical predictions.
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We expect that the mapping and the understanding of the charge enhancement of CNTs are important for many applications, besides the fundamental character of this study, e.g., imaging of field and charge in CNTs electronic circuits22,23 or emission devices modified by the presence of surfaces. The electrostatic response of nanotubes appears to be strongly sensitive to its environment, which is of high impor-
This work is a part of the CNRS GDR-E No. 2756. Z. W. acknowledges the support from the region of Franche-Comté 共Grant No. 060914-10兲.
Mayer, Phys. Rev. B 75, 045407 共2007兲. Mayer and P.-O. Astrand, J. Phys. Chem. A 112, 1277 共2008兲. 15 J. D. Jackson, Classical Electrodynamics 共Wiley, New York, 1975兲, pp. 54–62. 16 L. Tsetseris and S. T. Pantelides, Phys. Rev. Lett. 97, 266805 共2006兲. 17 J. Wojdel and S. Bromley, J. Phys. Chem. B 109, 1387 共2005兲. 18 S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472 共2000兲. 19 M. Zdrojek, T. Heim, D. Brunel, A. Mayer, and T. Mélin, Phys. Rev. B 77, 033404 共2008兲. 20 M. Zdrojek, T. Mélin, H. Diesinger, D. Stiévenard, W. Gebicki, and L. Adamowicz, Phys. Rev. Lett. 96, 039703 共2006兲. 21 M. Paillet, P. Poncharal, and A. Zahab, Phys. Rev. Lett. 96, 039704 共2006兲. 22 A. Bachtold, M. S. Fuhrer, S. Plyasunov, M. Forero, E. H. Anderson, A. Zettl, and P. L. McEuen, Phys. Rev. Lett. 84, 6082 共2000兲. 23 P. De Pablo, C. Gomez-Navarro, A. Gil, J. Colchero, M. Martinez, A. Benito, W. Maser, J. Gomez-Herrero, and A. Baro, Appl. Phys. Lett. 79, 2979 共2001兲. 24 J. Robinson, E. Snow, S. Badescu, T. Reinecke, and F. Perkins, Nano Lett. 6, 1747 共2006兲. 13 A.
*
[email protected] M. Anantram and F. Leonard, Rep. Prog. Phys. 69, 507 共2006兲. 2 W. De Heer, A. Chatelain, and D. Ugarte, Science 270, 1179 共1995兲. 3 E. Snow, F. Perkins, E. Houser, S. Badescu, and T. Reinecke, Science 307, 1942 共2005兲. 4 K. An, W. Kim, Y. Park, Y. Choi, S. Lee, D. Chung, D. Bae, S. Lim, and Y. Lee, Adv. Mater. 共Weinheim, Ger.兲 13, 497 共2001兲. 5 J. Cui, R. Sordan, M. Burghard, and K. Kern, Appl. Phys. Lett. 81, 3260 共2002兲. 6 S.-W. Ryu, X.-J. Huang, and Y.-K. Choi, Appl. Phys. Lett. 91, 063110 共2007兲. 7 M. Bockrath, N. Markovic, A. Shepard, M. Tinkham, L. Gurevich, L. Kouwenhoven, M. Wu, and L. Sohn, Nano Lett. 2, 187 共2002兲. 8 T. Sand-Jespersen and J. Nygard, Appl. Phys. A: Mater. Sci. Process. 88, 309 共2007兲. 9 M. Paillet, P. Poncharal, and A. Zahab, Phys. Rev. Lett. 94, 186801 共2005兲. 10 M. Zdrojek, T. Mélin, H. Diesinger, D. Stiévenard, W. Gebicki, and L. Adamowicz, J. Appl. Phys. 100, 114326 共2006兲. 11 P. Keblinski, S. K. Nayak, P. Zapol, and P. M. Ajayan, Phys. Rev. Lett. 89, 255503 共2002兲. 12 C. Li and T.-W. Chou, Appl. Phys. Lett. 89, 063103 共2006兲. 1
tance for nanotube based sensors.3,24 This work can also have implication in the field of nanoelectromechanical systems and charge storage devices.
14 A.
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Effects of substrate and electric fields on charges in carbon nanotubes Zhao Wang* Institute UTINAM, Université de Franche-Comté, FR-25000 Besançon, France and EMPA (Swiss Federal Laboratories for Materials Testing and Research), CH-3600 Thun, Switzerland 共Received 13 January 2009; published 3 April 2009兲 In this paper, we study how the distribution of net charges in carbon nanotubes can be influenced by substrate and external electric fields using theoretical calculations based on an extension of the atomic chargedipole model. We find that the charge enhancement becomes less significant when the tube gets closer to substrate or when the dielectric constant of substrate increases. It is demonstrated that net charges can be shifted to one side of the tube by longitudinal electric fields and the polarity of charges can be locally changed, while transversal fields give much less influence on the charge enhancement. These properties could be generalized for other metallic or semiconducting nano/microwires and tubes. DOI: 10.1103/PhysRevB.79.155407
PACS number共s兲: 73.63.Fg, 68.37.Ps, 85.35.Kt, 41.20.Cv
I. INTRODUCTION
The distribution of electric charges in carbon nanotubes 共CNTs兲 is of interest for their future uses in nanoelectromechanical systems 共NEMS兲 共Ref. 1兲 such as field-emission devices,2,3 sensors,4 actuators,5 and charge storages.6–10 Recently, electric force microscopies 共EFMs兲 have been used to inject and to detect net charges in CNTs11,12 and electric charges are found to distribute uniformly along CNTs. However, charge accumulation 共so-called charge enhancement兲 at tube ends has been predicted by theoretical studies using density-functional theory 共DFT兲13 and classical electrostatics14 calculations. These predicted properties were then confirmed by Zdrojek et al.15 in EFM experiments. It was also shown that electric charges can be trapped in CNT loops during periods of time.16 Furthermore, charge-induced failures17 and structure changes of CNTs18 were reported. In one of our previous works, weak charge enhancement at the tube ends and its geometry dependence were demonstrated by the combination of theoretical calculations and EFM experiments.19 In this paper, we address the issue of the substrate and electric-field effects on the charge distribution in CNTs, since CNTs are usually deposited on substrate and driven by electric fields in a number of nanodevices.20,21 It is known that substrate can exert quite strong influence on the charge distribution, as discussed recently in Ref. 22 for the case of ions inside CNTs. Theoretical calculations have been performed due to the difficulties for accurately quantifying this effect in recent experiments. Our calculation results reveal that the charge enhancement becomes less significant when substrate gets closer to CNTs and that the enhancement ratio decreases with increasing dielectric constant of substrate. These effects on the charge distribution in radial directions are also discussed. Furthermore, we find that the charge distribution in CNTs can be significantly modified in external fields. The dependence of field strength is demonstrated for both singlewalled and multiwalled CNTs 共SWCNTs and MWCNTs兲. We note that the properties demonstrated in this paper could also apply to semiconducting CNTs because semiconducting and metallic nanotubes are both expected to accept extra charges, from theoretical23 and experimental11,15 points of view. 1098-0121/2009/79共15兲/155407共6兲
The charge distribution has been computed using a Gaussian-regularized atomic charge-dipole interacting model.24,25 It has been developed from the atomic dipole theory of Applequist et al.26 and has recently been parameterized for CNTs.27 In this model, each atom is treated as an interacting polarizable point with a free charge, the static equilibrium state of charges is determined by minimizing the total electrostatic energy of system. In this work, we have extended this model to take the substrate effect into account by including surface-induced terms to vacuum electrostatic interacting tensors using the method of mirror image.28 Compared to classical Coulomb-law-based models in which only the charge is considered, this model provides a more accurate description of electrostatic properties of CNTs because not only the net charges, but also the induced dipoles, atomic polarizabilities, and the image charges are taken into account. For the outline, our computational model is presented in Sec. II. Results for the effects of substrate and fields are discussed in Secs. III and IV, respectively. We draw a conclusion in Sec. V. The formulation of the surface-induced electrostatic interacting tensors is given in the Appendix.
II. COMPUTATIONAL MODEL
In our calculation, each atom is associated with an electric charge q and an induced dipole p as shown in Fig. 1. The total electrostatic energy Uelec for a CNT of N atoms can be written as follows:
FIG. 1. Schematic of the principle of the charge-dipole model, in which each atom is modeled as a net charge q with an induced dipole p.
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N
i=1
i=1
Uelec = 兺 qi共i + Vi兲 − 兺 pi · Ei + N
N
− 兺 兺 pi · Ti,j p−qq j − i=1 j=1
N
N
N
1 i,j qj 兺 兺 qiTq−q 2 i=1 j=1 N
1 兺 兺 pi · Ti,jp−p · p j , 2 i=1 j=1
共1兲
where is the electron affinity and V and E stand for the external potential and electric field, respectively. T and T are the electrostatic interacting tensors. They can be written as i,j i,j i,j = 共1 / 40兲 ⫻ 共1 / rij兲, Ti,j Tq−q p−q = −ⵜriTq−q, and T p−p = −ⵜr j i,j 丢 ⵜr Tq−q, where ri,j = 兩r j − ri兩. We have regularized T and T i by a Gaussian distribution in order to avoid divergence problems when atoms are too close to each other, as discussed previously in Refs. 24 and 29. Note that the value of Gaussian charge distribution width R used in this work for free-end atoms is fitted to 0.1273 nm 共about 1.3 times that of the carbon atom with three chemical bonds兲 from results in a previous study using DFT calculation.13 The equilibrium state of charges and dipoles should correspond to the minimum value of Uelec and hence the derivatives of Uelec with respect to q and p should be zero. Taking this boundary condition as well as total molecular net charge Qtot into account with the self-energy terms 共when i = j兲, we can obtain the equilibrium configuration of charge and dipole by solving N linear vectorial equations and N + 1 linear scalar equations as follows:
冦
N
兺 j=1
N
Ti,j p−p 丢
p j + 兺 Ti,j p−qq j = − Ei
N
j=1
N
i,j Ti,j 兺 p−q · p j + 兺 Tq−qq j+ = − 共i + Vi兲 j=1 j=1 N
q j = Qtot , 兺 j=1 ∀i = 1, . . . ,N,
冧
FIG. 2. L stands for the tube length and Lⴱ is the length of the enhancement zone in which the average charge density is higher than the average ave over the whole tube. The circles present the calculated points. Inset: net charge density along a freestanding tube in space.
quantify this enhancement effect. In Fig. 2, we can see that the length of charge enhancement zone 共Lⴱ兲 increases with the tube length 共L兲 if we define this zone as the part where the charge density is higher than the average over the whole tube 共ave兲. Considering that the CNTs used in experiments are usually longer than those used in our calculation, we define the length of charge enhancement zone as 20% of L 共10%L at each tube end兲. The ratio of charge enhancement19 is denoted as follows:
= end/middle ,
共2兲
where is a Lagrange multiplier,30 which is related to the chemical potential of the molecule. In case of a CNT close to a substrate, as in EFM experiments,11,12,16 the distributions of charges and dipoles are different from those in free space. We have taken this boundary condition into account by adding a surface-induced terms Tm and Tm to the vacuum electrostatic interacting tensors using the method of mirror images.28 The detailed formulation of Tm and Tm can be found in the Appendix. Note that the substrate surface is assumed to be infinitely plane in this work. The structures of CNTs are relaxed by means of energy optimization29 using the conjugated gradient method31 based on the adaptive interatomic reactive empirical bond order 共AIREBO兲 potential.32 III. INFLUENCE OF SUBSTRATE
Previous studies show static charge accumulations at tube ends13,33 共as shown in the inset of Fig. 2兲. A well-defined zone of the charge accumulation is required in order to well
共3兲
where end is the average charge density in the enhancement zone 共10%L at each tube end兲 and middle is that at the middle of the tube. We note that is independent of ave because the local charge densities are proportional to ave with respect to a constant electric potential on the tube surface. In case of a CNT in a semi-infinite space 共e.g., deposited on substrate兲, net charges will be attracted to the tube bottom by opposite image charges appearing on substrate surface, as shown in the inset of Fig. 3共a兲. This surface effect mainly depends on the tube-surface physisorption distance34,35 共d兲 and the dielectric constant of the substrate14,36 共2兲. Both of them vary with the type of substrate material. To demonstrate the influence of these two parameters on charge enhancement, we plot versus d and 2 in Figs. 3共a兲 and 3共b兲, respectively. We can see in Fig. 3共a兲 that the charge enhancement becomes less significant when the substrate surface gets closer to the tube 共when d decreases兲. From electrostatic point of view, the main mechanism of this effect is that the charge distributed area 共band兲 in radial direction has been effectively reduced since a part of net charges is attracted to the tube bottom, and hence the charge distribution along the tube axis gets closer to that along an infinite-long tube, in which the charge distribution is perfectly uniform 共 = 1兲. Similar behavior can be contrasted with the situation when
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FIG. 4. Average charge density vs d for y ⬎ 0 共the top half part兲 and y ⬍ 0 共the bottom half part兲, respectively, for a close-ended 共9,0兲 SWCNT 共L ⬇ 12 nm ave ⬇ 2.0⫻ 10−3e / atom兲 on a metallic surface. Inset: schematic of transversal charge distribution at a tube end.
The top of the tube even shows electrically positive since most of net charges 共negative兲 are attracted to the tube bottom by the surface images.38 IV. INFLUENCE OF EXTERNAL FIELDS
FIG. 3. 共a兲 vs d for a charged 共5, 5兲 SWCNT 共L ⬇ 27 nm兲 with ave = 0.55⫻ 10−3e / atom and 2 = ⬁. Inset: net charge density in a nanotube in a semi-infinite space. 共b兲 vs 2 for the same tube 共d = 0.34 nm兲.
2 increases, as shown by the plot in Fig. 3共b兲. This implies that can get higher if one uses small-dielectric constant material instead of metal 共2 = ⬁兲 in experiments and that the CNT exhibits the strongest charge enhancement in an infinite space 共2 = 0兲. The issue about the charge distribution in radial 共transversal兲 direction has rarely been discussed in the literature, although it is one of the main mechanisms of discharging phenomenon observed experimentally.11,12 When a CNT is horizontally deposited upon a substrate, charge migration is mainly caused in the direction perpendicular to the substrate surface, as a typical mirror effect 共see Fig. 4兲. Local charge accumulation at the bottom of tube ends 共open circles兲 can directly lead to enhanced electron emission.37 We can also see that the top part of the tube 共y ⬎ 0兲 even shows opposite electric sign when d ⱕ 0.8 nm 共solid circles兲. The issue about charge distribution in MWCNTs is more complicated due to depolarization, field screening, and electrostatic interactions between layers. To show further details about substrate effects, we depict the atomic charge distribution in a double-walled CNT 共DWCNT兲 electrically charged in its both inner and outer carbon layers in Fig. 5. The migration of atomic charges induced by the metallic surface is shown in this figure. We can see the enhanced local electric fields around the tube bottom due to the charge enhancement.
Recent works showed that external electric fields could induce alignments,39 deformations,40 field emission,41 and conductivity transitions42 of CNTs. Here we concern mainly on how electric fields influence the static distribution of net charges in CNTs. The charge distribution of a SWCNT in free space is compared to that in an external electric field Eext in Fig. 6. As expected, net charges are shifted to one side, around which local electric fields 共fields induced by net charges and dipoles+ external fields兲 are enhanced. The magnitude of this polarization effect is roughly proportional to the external field intensity E and the tube length L;43 this implies that for CNTs used in experiments 共usually L ⬃ m兲, Eext can be hundred times weaker for producing similar effects as those shown in Fig. 6. Moreover, we note that Eext used in this work is about 2 orders of magnitude weaker than that can lead to field emission from our short CNTs.44 To achieve a quantitative comparison, we plot in Fig. 7 charge distribution along a SWCNT in an external field Eext. It can be seen that the typical U-like distribution13 in vacuum 共solid circles兲 can be significantly modified by the axial external electric field. On the other hand, the influence of transversal electric fields is expected to be weak due to the strong anisotropy of the polarizabilities of CNTs. We can see in Fig. 7共b兲 that, even with very strong field intensities in the order of V/nm, the charge profile does not change a lot. In fact, the transversal field mainly influences the charge distribution in nonaxial direction. Moreover, it needs to mention that the average charge density depends on the value of unit length taken in the calculation, e.g., the value of charge density represented by the solid circles in Fig. 7共b兲 is lower than that in Fig. 7共a兲 because it is calculated as average on every 10%L, instead of that on every 5%L.
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FIG. 5. 共Color online兲 Atomic graph of the charge distribution in a 共9,0兲@共10,10兲 DWCNT 共L ⬇ 10.3 nm, ave = 0.55⫻ 10−3e / atom for both two layers兲 on a metallic surface. Color of the atoms is proportional to the local charge density. Dark arrows stand for the local electric fields induced by the net charge around the tube ends, their length and color are proportional to the field intensities. The maximum atomic charge densities are max = 8 ⫻ 10−3e / atom. The maximum strength of the local electric fields presented in this figure Emax = 4.3 V / nm.
For MWCNTs, it has been reported that electric screening plays an important role for the field effects.45 To demonstrate its influence on the charge enhancement, we compare the charge distribution in an inner layer of a DWCNT to that of a SWCNT with the same size in Fig. 8. In this comparison, it can be seen that the effect of external fields is much weaker on the charge distribution in the inner layer of the DWCNT due to electrostatic screening. Furthermore, by comparing the longitudinal charge distribution of the DWCNT to that of the SWCNT, we have found that the charge enhancement is lower in the DWCNT due to the electric repulsive interaction between the two carbon layers. It is a typical “1 + 1 ⬍ 2” effect.46,47
extending the charge-dipole polarization model. The results obtained are relevant for a better understanding of the distribution and stability of electric charges in CNTs in possible experimental situations 共e.g., CNTs deposited on a solid surface兲. Local charge enhancement at tube ends is studied as a particular effect. Our results reveal that the charge enhance-
V. CONCLUSION
In summary, influences of substrate and external electric fields on electric charges in CNTs have been investigated by
FIG. 6. 共Color online兲 Charge distribution in a charged 共9,0兲 SWCNT 共L ⬇ 12 nm, ave = 1.0⫻ 10−3e / atom兲 共a兲 in space and 共b兲 in an axial uniform external electric field Eext = 0.05 V / nm. Color of atoms is proportional to charge density. Dark arrows stand for local electric fields around tube ends and their length and color are proportional to field intensity.
FIG. 7. Charge profile along a charged 共5,5兲 SWCNT 共L ⬇ 14 nm, ave = 0.9⫻ 10−3e / atom兲. 共a兲 In longitudinal 共along the tube axis兲 electric fields Eext. Each point is calculated as the average of 5%L. 共b兲 In transversal 共perpendicular to the tube axis兲 electric fields Etrans. Each point is calculated as the average of 10%L.
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FIG. 9. Schematic of the charge image.
Vm共r兲 = − kqi
FIG. 8. Charge profile along the tube axis, for the inner tube of a DWCNT 共9,0兲@共10,10兲 and for a SWCNT 共9,0兲 with the same size.
ment becomes less significant when the substrate-tube separation decreases or when the dielectric constant of substrate increases. Charge delocalization in radial direction has been observed as a typical mirror effect in presence of substrate or transversal electric fields. Longitudinal external electric fields have been found to have much more influence on the charge enhancement than the transversal ones with same intensities. Electric screening in MWCNTs is found to influence charge profile in MWCNTs, especially in presence of electric fields. In general, these above conclusions could also qualitatively apply to other nanowires and tubes, from electrostatic point of view. ACKNOWLEDGMENTS
1 = q iT m 0, 兩r − rm兩
共A1兲
where rm is the coordinate of the mirror image qm i and k = 共2 − 1兲 / 关40共2 + 1兲兴 is an electrostatic constant. Tm 0 stands for the 0th order interaction tensor for the mirror image. It is the Green’s function for the vectorial variable Laplace equation. For our system shown in this Fig. 9 using Cartesian coordinate, the 0th order mirror-charge interaction tensor can be written as Tm 0 =
−k −k = , 2 兩r − rm兩 冑␦x + ␦y 2 + 2
共A2兲
where ␦x = x − xi, ␦y = y − y i, and = z + zi. The interaction tensors of the first 共charge-dipole兲 and the second 共dipoledipole兲 orders for the image charges can be derived from that of the 0th order 共charge-charge兲 Tm 1
=−
ⵜ rmT m 0
冤冥
␦x −k ␦y = 兩r − rm兩3 −
共A3兲 ux,uy,uz
and
M. Devel, M. Zdrojek, T. Mélin, and A. Mayer are gratefully acknowledged for useful discussion.
m Tm 2 = − ⵜ r 丢 ⵜ rmT 0 =
冤
冥
␦ y 2 + 2 − 2␦x2 − 3␦x␦ y 3␦x 2 2 2 − 3␦x␦ y ␦x + − 2␦ y 3␦ y ⫻ . 2 2 2 − 3␦x − 3␦ y 2 − ␦x − ␦ y
APPENDIX: SURFACE-INDUCED TERMS OF ELECTROSTATIC INTERACTION TENSORS
To take substrate effects into account, we have extended the charge-dipole model of Mayer24 by adding surfacem m induced terms 共Tm 0 , T1 , and T2 兲 into the vacuum electrostatic interaction tensors 共T0, T1, and T2兲, respectively, using the method of mirror image.28 In this method, the electric potential Vm on an arbitrary point 共x , y , z兲 induced by the mirror image of a point charge qi embedded in a semi-infinite medium 1 close to another medium 2 共see Fig. 9兲 can be written as follows:
k 兩r − rm兩5
共A4兲
We note that the vacuum interaction tensors used in present study are regularized by a normal distribution in order to avoid divergence problems with point charges when atoms get too close to each other.24 However, it is not necessary to regularize the surface-induced terms since the distance between the net charge and its images is generally large enough.
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Anantram and F. Leonard, Rep. Prog. Phys. 69, 507 共2006兲. 2 W. De Heer, A. Chatelain, and D. Ugarte, Science 270, 1179 共1995兲. 3 S. T. Purcell, P. Vincent, C. Journet, and V. T. Binh, Phys. Rev. Lett. 89, 276103 共2002兲. 4 E. Snow, F. Perkins, E. Houser, S. Badescu, and T. Reinecke, Science 307, 1942 共2005兲. 5 S. Roth and R. Baughman, Curr. Appl. Phys. 2, 311 共2002兲. 6 S. Akita, Y. Nakayama, S. Mizooka, Y. Takano, T. Okawa, Y. Miyatake, S. Yamanaka, M. Tsuji, and T. Nosaka, Appl. Phys. Lett. 79, 1691 共2001兲. 7 J. Cui, R. Sordan, M. Burghard, and K. Kern, Appl. Phys. Lett. 81, 3260 共2002兲. 8 J. Jang, S. Cha, Y. Choi, G. Amaratunga, D. Kang, D. Hasko, J. Jung, and J. Kim, Appl. Phys. Lett. 87, 163114 共2005兲. 9 X. Lu and J. Dai, Appl. Phys. Lett. 88, 113104 共2006兲. 10 S.-W. Ryu, X.-J. Huang, and Y.-K. Choi, Appl. Phys. Lett. 91, 063110 共2007兲. 11 M. Paillet, P. Poncharal, and A. Zahab, Phys. Rev. Lett. 94, 186801 共2005兲. 12 M. Zdrojek, T. Mélin, C. Boyaval, D. Stiévenard, B. Jouault, M. Wozniak, A. Huczko, W. Gebicki, and L. Adamowicz, Appl. Phys. Lett. 86, 213114 共2005兲. 13 P. Keblinski, S. K. Nayak, P. Zapol, and P. M. Ajayan, Phys. Rev. Lett. 89, 255503 共2002兲. 14 C. Li and T.-W. Chou, Appl. Phys. Lett. 89, 063103 共2006兲. 15 M. Zdrojek, T. Mélin, H. Diesinger, D. Stivenard, W. Gebicki, and L. Adamowicz, J. Appl. Phys. 100, 114326 共2006兲. 16 T. Jespersen and J. Nygard, Nano Lett. 5, 1838 共2005兲. 17 C. Li and T.-W. Chou, Carbon 45, 922 共2007兲. 18 Y. N. Gartstein, A. A. Zakhidov, and R. H. Baughman, Phys. Rev. Lett. 89, 045503 共2002兲. 19 Z. Wang, M. Zdrojek, T. Melin, and M. Devel, Phys. Rev. B 78, 085425 共2008兲. 20 S. J. Tans, A. R. M. Verschueren, and C. Dekker, Nature 共London兲 393, 49 共1998兲. 21 T. Rueckes, K. Kim, E. Joselevich, G. Tseng, C.-L. Cheung, and C. Lieber, Science 289, 94 共2000兲. 22 D. J. Mowbray, Z. L. Miskovic, and F. O. Goodman, Phys. Rev. B 74, 195435 共2006兲. 23 E. R. Margine and V. H. Crespi, Phys. Rev. Lett. 96, 196803 共2006兲. 1 M.
A. Mayer, Phys. Rev. B 75, 045407 共2007兲. Mayer and P.-O. Astrand, J. Phys. Chem. A 112, 1277 共2008兲. 26 J. Applequist, J. Carl, and K.-K. Fung, J. Am. Chem. Soc. 94, 2952 共1972兲. 27 A. Mayer, Phys. Rev. B 71, 235333 共2005兲. 28 J. D. Jackson, Classical Electrodynamics 共Wiley, New York, 1975兲, pp. 54–62. 29 Z. Wang and M. Devel, Phys. Rev. B 76, 195434 共2007兲. 30 J. Lagrange, Theorie des Fonctions Analytiques 共Imprimérie de la République, Paris, 1797兲. 31 M. Payne, M. Teter, D. Allan, T. Arias, and J. Joannopoulos, Rev. Mod. Phys. 64, 1045 共1992兲. 32 S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472 共2000兲. 33 M. Zdrojek, T. Heim, D. Brunel, A. Mayer, and T. Mélin, Phys. Rev. B 77, 033404 共2008兲. 34 H. Rafii-Tabar, Phys. Rep. 390, 235 共2004兲. 35 L. Tsetseris and S. T. Pantelides, Phys. Rev. Lett. 97, 266805 共2006兲. 36 K. Besteman, M. A. G. Zevenbergen, and S. G. Lemay, Phys. Rev. E 72, 061501 共2005兲. 37 J.-C. Charlier, M. Terrones, M. Baxendale, V. Meunier, T. Zacharia, N. Rupesinghe, W. Hsu, N. Grobert, H. Terrones, and G. Amaratunga, Nano Lett. 2, 1191 共2002兲. 38 N. Lang and W. Kohn, Phys. Rev. B 7, 3541 共1973兲. 39 E. Joselevich and C. M. Lieber, Nano Lett. 2, 1137 共2002兲. 40 P. Poncharal, Z. L. Wang, D. Ugarte, and W. A. De Heer, Science 283, 1513 共1999兲. 41 A. Rinzler, J. Hafner, P. Nikolaev, L. Lou, S. Kim, D. Tomanek, P. Nordlander, D. Colbert, and R. Smalley, Science 269, 1550 共1995兲. 42 A. Rochefort, M. Di Ventra, and P. Avouris, Appl. Phys. Lett. 78, 2521 共2001兲. 43 L. X. Benedict, S. G. Louie, and M. L. Cohen, Phys. Rev. B 52, 8541 共1995兲. 44 S. Jo, Y. Tu, Z. Huang, D. Carnahan, D. Wang, and Z. Ren, Appl. Phys. Lett. 82, 3520 共2003兲. 45 B. Kozinsky and N. Marzari, Phys. Rev. Lett. 96, 166801 共2006兲. 46 A. G. Marinopoulos, L. Reining, A. Rubio, and N. Vast, Phys. Rev. Lett. 91, 046402 共2003兲. 47 R. Pfeiffer, H. Kuzmany, T. Pichler, H. Kataura, Y. Achiba, M. Melle-Franco, and F. Zerbetto, Phys. Rev. B 69, 035404 共2004兲. 24
*
[email protected],
[email protected]
25 A.
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Published as : Z. Wang et al. Phys. Rev. B, 75, 205414 et Phys. Rev. B, 76, 195434 (2007)
Chapitre 5
Electromechanical properties
Si les faits ne correspondent pas ` a la th´eorie, changez les faits. Albert Einstein
The electric-field-induced deformations are key characteristics for the uses of CNTs in NEMS, by which CNTs transform mechanical energies to electrical ones (or conversely) (cite chapter1). Recently, electric deformations are observed in experiments (cite chapter1) and predicted theoretically. It is shown that CNTs can bend or be restricted in electric fields.
Figure 5.1 – Electric torque acting on a CNT induced in a uniform electric field E. F stands for the electric force.
The basic mechanism of electric-field-induced deflection is depicted in Figure 5.1. When a CNT is brought into a electric field, the field tends to shift the positive and negative charges in opposite directions. Thus, an induced molecular dipole is created on the tube. The torque τ induced by the interaction between this dipole and the electric field tends to make the tube aligned to the field.
54
Figure 5.2 –
5. Electromechanical properties
d´eflexion d’un CNT (5,5) par un champ ´electrique externe uniforme. Ici
l’amplitude du champ n’est pas suffisante pour aligner compl`etement l’extr´emit´e libre sur le champ [119, 120]
In case that one tube end is fixed into substrate, the CNT will bend to a certain angle by the electric torque τ , as shown in Figure 5.2. What will be the relation between the intensity of the electric field and the tube’s deformation ? What are the geometry effects on the response of CNTs to electric fields ? To answer these questions, we investigate the mechanisms of the electrostatic deflections of CNTs.[119, 120] In this work, the deformations of CNTs are investigated using energy optimization based on two combined physicochemical models. The influence of field strength, field direction and tube geometry is studied. The metallic CNTs are found to be much easier to be deflected than semiconducting ones. The deflection is not changed by reversing the field direction. The curve of alignment ratio versus field strength is found to change with field directions. The deflection is found to decrease with the increase of the tube radius ; conversely, it increases when the tube is longer. The metallic MWCNTs are found to be much harder to be bent in electric fields than SWCNTs. Furthermore, we find that the electrostrictive deformation of SWCNTs is proportional to the square of field strength. In the Chapter 4, we demonstrated how to calculate distribution of charges and induced dipoles in external electric fields. From this distribution, electrostatic force acting on each atoms can be calculated using simple Coulomb’s law. Based on this force calculation, deformation of CNTs in electric fields can be simulated using energy minimization techniques (introduced in Chapter 3).
PHYSICAL REVIEW B 75, 205414 共2007兲
Electrostatic deflections of cantilevered semiconducting single-walled carbon nanotubes Zhao Wang,1,* Michel Devel,1 Rachel Langlet,2 and Bernard Dulmet3 1Institut
UTINAM, UMR 6213, University of Franche-Comté, 25030 Besançon Cedex, France de Physique du Solide, FUNDP, Rue de Bruxelles 61, 5000 Namur, Belgium 3 FEMTO-ST, UMR 6174, DCEPE, ENSMM, 25030 Besançon Cedex, France 共Received 4 December 2006; revised manuscript received 9 February 2007; published 10 May 2007兲 2Laboratoire
How do carbon nanotubes behave in an external electric field? What will be the relation between the intensity of the electric field and the tube’s deformation? What are the geometry effects on the response of carbon nanotubes to electric fields? To answer these questions, we have developed a computational technique combining a dipole interaction model with the AIREBO potential to study electrostatic-field induced deformations of carbon nanotubes. In this work, we find that the deflection angle of cantilevered semiconducting single-walled carbon nanotubes is proportional to the square of the electric-field strength, and the tubes can be most bent when the field angle ranges from 45° to 60° for a given weak field strength. Furthermore, the deflection angle is also found to be proportional to the aspect ratio L / R. Our results provide a good qualitative agreement with those of a previous experimental study by Poncharal et al., 共1999兲. DOI: 10.1103/PhysRevB.75.205414
PACS number共s兲: 85.35.Kt, 32.10.Dk, 31.15.Qg, 41.20.Cv
I. INTRODUCTION
Carbon nanotubes 共CNTs兲 can be ideal building blocks for nanoelectromechanical systems 共NEMS兲 due to their unique electrical and mechanical properties.1 Thus, they have attracted much interest from both technical and scientific communities concerned with sensing, actuation, vibration, and laboratory-on-a-chip applications.2 Cantilevered semiconducting CNTs can be used as key elements in NEMS such as nanotweezers3 and nanorelays.4 Their electric-field-induced deformation is a key character for these promising applications, as well as their fabrication,5 separation,6 and electromanipulation.7 In a previous study, Poncharal et al.8 reported the electric deflections of cantilevered multiwalled CNTs observed using a transmission electron microscope. Electrostrictions in single-walled CNTs 共SWCNTs兲 were observed by El-Hami and Matsushige9 using an atomic force microscope. Bao et al.10 reported that the microstructure of CNTs can be changed by an electric field during growth using a highresolution transmission electron microscope. Y. Guo and W. Guo11 carried out quantum mechanics calculations to investigate the electric-field-induced tensile breaking and the influence of external field on the tensile stiffness of CNTs, and found that both the tensile stiffness and the strength of CNTs decrease with increasing intensity of electric field. Torrens12 reported that the polarizabilities of CNTs can be modified reversibly by external radial deformation. Mayer and Lambin13 calculated the electrostatic forces acting on CNTs placed in the vicinity of metallic protrusions for dielectrophoresis. W. Guo and Y. Guo14 found giant electrostrictive deformations in SWCNTs using quantum mechanics simulations. The calculations using traditional first-principles methods are very time consuming, so that the size of the systems considered is always very limited in past theoretical studies concerning the effects of external field. That is one of the main reasons why deformations of CNTs in electric fields and their geometric effects are still not fully understood up to date. 1098-0121/2007/75共20兲/205414共6兲
How CNTs behave in an external electric field? What will be the relation between the intensity of the electric field and the tube’s deformation? What are the geometry effects on the response of CNTs to electric fields? To answer these questions, we have developed a combined computational technique to study electrostatic-field-induced deformations. In this technique, the interatomic potential is described in an empirical way and the electrostatic interaction is calculated using a Gaussian renormalized point-dipole interaction model 共e.g., interactions between broaden dipoles兲. The main advantage of this method is its ability to deal with much larger systems with a reasonable computational requirement. This enabled us to carry out a series of simulations to study the induced deformations of various carbon structures in electrostatic fields. In this work, we study the electrostatic deflections of cantilevered semiconducting SWCNTs in a nonaxial homogeneous field. This study addresses the relation between the external field, the mechanical resistance of SWCNTs, and their electrostatic polarizabilities. In the last few years, the polarizability of CNTs has been intensively investigated.15–20 For example, it is known that the static polarizabilities of CNTs are very anisotropic and that the polarizabilities are proportional to the square of the radius in the plane perpendicular to the tube axis and almost independent of chirality. In this study, we use Gaussian regularized propagators21,22 to calculate the local polarizability of carbon atoms. In this model, the standard vacuum propagator is convoluted by a Gaussian function in order to avoid polarization catastrophes. This is the same as considering that the dipoles are not real point dipoles but rather due to Gaussian distribution of charges whose width is related to the polarizabilities.23,24 Compared to first-principles techniques such as ab initio calculations, the local autocoherent polarizabilities related to the geometrical structure of carbon atoms can be quickly evaluated in this model. It can therefore be practically used in dynamic simulations for larger systems. In this work, a simple gradient algorithm is used to calculate equilibrium configurations of the atoms by minimizing the total energy of the systems, which consists of both the
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interatomic and the induced electrostatic interactions. The interatomic potential is computed using an adaptive intermolecular reactive empirical bond-order hydrocarbon 共AIREBO兲 potential function formulated by Stuart et al.25 During the simulations, deflections of cantilevered CNTs are recorded dynamically as a function of the iteration step. Examples can be viewed in Ref. 26. The theory will be presented in Sec. II. In Sec. III, we examine the relations between the deflections and the intensities and orientations of the applied fields, respectively. The effects of tube chiralities, radii, and lengths are also studied. We draw a conclusion at the end of this section. II. THEORY
In this study, we examine cantilevered semiconducting SWCNTs submitted to an external homogeneous electrostatic field in free space with zero net charge and zero permanent dipole moment. At the beginning of the simulations, an openended tube, fixed at one end, is relaxed. When an electric field is applied, a dipole is induced on each atom. The induced dipole is modeled as an ideal dipole with anisotropic linear polarizabilities. The field intensity is supposed small enough so that any hyperpolarization effect can be neglected. A simple gradient algorithm is used to simulate the deformation of CNTs in the field by minimizing the total energy of the systems Utot. The motionless equilibrium configuration of the atoms corresponds to the minimum value in the energy curve. Classical molecular dynamics is not used in order to avoid the thermal vibration because we consider only the equilibrium states of tubes in this study. Utot is the sum of two terms, the induced electrostatic energy Uelec and the interatomic potential U p, as follows: 共1兲
Utot = Uelec + U p , in which U
elec
consists of the following three terms:
N
U
elec
N
1 1 = 兺 pi␣−1 i pi − 兺 2 i=1 2 i=1
N
N
piE0共ri兲, 兺 piE j共ri兲 − 兺 i=1
Green’s generalized function for the Laplace equation, convoluted by a Gaussian distribution to avoid polarizability catastrophes.21,22 The distribution of dipoles pi is determined by the fact that the equilibrium state of the dipole distribution should correspond to the minimum value of Uelec. This means that the partial derivatives of the total electrostatic energy with respect to the 3 ⫻ N components of the dipoles should be zero,
Uelec Uelec Uelec = = = 0, pi,x pi,y pi,z
N
pi = ␣iE0共ri兲 +
In this equation, El共ri兲 describes the local field at atom i and the regularized tensor T2 is the double gradient of
共5兲
j⫽i
The local fields and the dipole moments can be calculated by solving this system. By putting these expressions for the minimizing pi back into Eq. 共2兲, cancellation of various terms leaves only a simple expression for the minimum value of Uelec as a function of the pi as follows: N
elec =− Umin
1 兺 piE0共ri兲. 2 i=1
共6兲
The interatomic potential U p is determined using the AIREBO potential function.25 This potential is an extension of the Brenner’s second generation potential of Brenner’s et al.27 and includes long-range atomic interactions and single bond torsional interactions. In this type of potential, the total interatomic potential energy is the sum of individual pair interactions containing a many-body bond-order function as follows:
共2兲
共3兲
兺 ␣iT2共ri,r j兲p j . j=1
Up =
j⫽i
E j共ri兲 = T2共ri,r j兲␣ jEl共r j兲 = T2共ri,r j兲p j .
共4兲
where pi,x, pi,y, and pi,z are the three components of the dipole pi. Putting Eqs. 共2兲 and 共3兲 into Eq. 共4兲 gives a linear system of N equations for N vectorial variables as follows:
j=1
where N is the total number of atoms, pi is the dipole induced on atom i, E j共ri兲 represents the electric field created by another dipole p j around atom i, E0共ri兲 stands for the external field, and ␣i is the local anisotropic polarizability tensor of atom i adapted from graphite.22 In this expression, the first term is the self-energy term, the second term accounts for the dipole-dipole interaction, and the last term presents the interaction with the external field.23 The two basic parameters of the polarizability tensor are taken to be ␣储 = 2.47 Å3 and ␣⬜ = 0.86 Å3, with 储 and ⬜ meaning parallel and perpendicular to the plane of graphene sheet, respectively. Note that due to the use of point-dipole model, these parameters are not valid for metallic tubes. The field on atom i due to another induced dipole E j can be written as
∀ i = 1, . . . ,N,
冋
1 兺 兺 VR共rij兲 − bijVA共rij兲 + VL−J ij 共rij兲 2 i j⫽i +
兺 兺
k⫽i,j ᐉ⫽i,j,k
册
Vtor kijᐉ ,
共7兲
where VR and VA are the interatomic repulsion and attraction terms between valence electrons, respectively for bound atoms. The bond-order function bij provides the many-body effects by depending on the local atomic environment of atoms i and j. The long-range interactions are included by adding a parametrized Lennard-Jones 12-6 potential term VL−J. Vtor represents the torsional interactions. III. RESULTS AND DISCUSSION
In this study, we use Cartesian coordinates with the z axis along the principal axis of the tube. The tubes are fixed at one end in an imposed homogeneous electric field. The tube length is in the range of 2.0– 8.4 nm. In order to get enough measurable deflection of these short SWCNTs, the strength of applied external electric fields is in the range of 0.5– 3.0 V / nm. This is much stronger than the ordinary field
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ELECTROSTATIC DEFLECTIONS OF CANTILEVERED…
FIG. 1. 共Color online兲 Induced dipoles in a 共8,0兲 tube 共with nonoptimized structure兲 in an electrostatic field. Field angle = 45°. Field strength E = 0.71 V / nm. Tube length L = 1.988 nm. The spheres stand for atoms. The arrow with a dashed line on the left presents the electric-field vector in the y-z plane. The arrows with solid lines on the tube stand for the induced dipoles.
reported in the experiments. On the other hand, as will be discussed below, the deflection of SWCNTs is proportional to the square of tube length and it is also proportional to the square of field strength. For example, in order to get the same electrostatic deflection for a tube 1000 times longer, it is just enough to apply a field a thousand times smaller. Hence, considering that the lengths of CNTs studied in previous experimental works are in the range of some micrometers, the field emission effect caused by strong fields is neglected. Furthermore, we note that the actual external electric field created by the electrodes is usually inhomogeneous in real experiments; however, this inhomogeneous distribution varies from one experimental setup to another. For the sake of simplicity, we have therefore chosen to use a homogeneous external field distribution, as in previous theoretical studies. In this work, all applied electric fields are parallel to the y-z plane due to the symmetry of the system. The field angle is defined as the angle between the field direction and the z axis. The deflection at the tip of the tube is noted as w. We studied at first the relation between the intensity of the external field and the deflection of cantilevered SWCNTs. When a carbon atom is brought into an electric field, the field tends to shift the electrons and the nuclei in opposite directions. Thus, induced dipoles are created. Figure 1 shows dipoles induced on atoms by an electrostatic field in a zigzag tube 共8,0兲. It is selected because of its symmetric geometry and small radius. The positive and negative charges of CNT are brought about a relative displacement due to the electric force. The tube tends to be parallel to the direction of field 共in most cases兲 by dipole-dipole and dipole-field interactions. Tubes can therefore be bent by induced forces and moments to the field direction. Figure 2 shows the deflections of a CNT induced by various external electric fields corresponding to various field strengths E, but at a given = 45°. Note that these are the equilibrium configurations of the atoms, around which the tubes would vibrate at nonzero temperature.28 It can be seen that as expected, the tubes are bent toward the direction of the external electric field. Note that the tube is only curved at the part closed to the fixed end. The right side part of the tubes remains straight. We compared the atomic structure of this straight part in Figs. 2共b兲–2共d兲 to an unmoved tube as shown in Fig. 2共a兲. It is found that the average bond length in this part is slightly smaller than that in the original one. This means that the tube is not only bent
FIG. 2. 共Color online兲 Induced deflections of a 共8,0兲 cantilevered tube in electrostatic fields with different field intensities. The direction of the applied electric fields is parallel to the y-z plane with = 45°. The original length of the tube is about 6.54 nm. The arrow with a dashed line represents the electric-field vector. 共a兲 shows a tube relaxed in vacuum without any electric field. 共b兲–共d兲 show the deflected shape of the tubes in various applied external electric fields 共E = 1.0– 3.0 V / nm兲.
by induced moments but also compressed by induced forces at the same time, as the electrostriction effects found in Refs. 9 and 14. In fact, the distribution of electrostatic forces acting on the atoms is rather inhomogeneous as shown in Fig. 3 for example. In consequence, the macroscopic continuum mechanics models of beam structures subjected to simple loadings cannot be directly used to calculate internal strains for tubes in an electric field, without incorporating volume densities of torques. For convenience, we define a deformation angle to describe the deformations of the tubes. It is equal to the angle
FIG. 3. 共Color online兲 Vectors of induced forces in a 共8,0兲 tube bent by an electrostatic field. = 45°. E = 4.0 V / nm. L = 6.54 nm. The spheres represent the carbon atoms. The arrow with a dashed line represents the external electric-field vector. The arrows on the atoms stand for the induced forces, which are calculated as the negative gradients of the electrostatic energy.
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FIG. 4. 共Color online兲 Sine of deformation angles, sin共兲, versus the square of the electric-field strength E for a 共8,0兲 tube. The tube length is about 6.54 nm. = 45°. The round points stand for the simulation data and the dashed line represents the fitting line.
between the z axis and the neutral axis of the deformed CNTs at their free end. The relation between and the intensity of the external field is presented in Fig. 4. It can be seen that sin共兲 is almost proportional to the square of E as follows: sin共兲 = AE2 ,
共8兲
where A = 0.0219± 0.0002 for this 共8,0兲 tube with = 45°. Note that when the value of remains small, one can also find that the deflection at the tip of a long tube w = L sin共兲 is almost proportional to E2. This provides a good qualitative agreement with the experimental study of Poncharal et al.8 In Fig. 5, we plot the change of the potential U p versus the
FIG. 5. 共Color online兲 Change of potential energy ⌬U p of this 共8,0兲 tube versus the square of sin共兲. The tube length is about 6.54 nm. = 45°. The round points stand for the simulation data and the dashed line represents the fitting line.
FIG. 6. sin共兲 versus electric-field angles for a 共8,0兲 tube. The tube length is about 6.54 nm. E = 3.0 V / nm. The round points stand for the simulation data.
square of sin共兲. It shows that ⌬U p ⬀ sin2共兲. Furthermore, we also find that the change of the global polarizability in the direction of the incident field E0 · ⌬E0 / E2 is proportional to sin共兲, although the individual components of ⌬ are not proportional to sin共兲 关⌬yy ⬀ sin2共兲, while ⌬zz ⬀ −sin2共兲 and ⌬xx remains almost constant兴. Combined with the fact that sin共兲 is almost proportional to the square of E 关Eq. 共8兲兴, this means that we also have ⌬Uelec ⬀ sin2共兲; hence, ⌬Utot ⬀ sin2共兲. In a second series of simulations, we studied the influence of the orientation of the external field on the response of CNTs. Figure 6 presents the relation between sin共兲 for a 共8,0兲 SWCNT and the field angle . There is no obvious deflection effect found in CNTs when the applied electric field is either parallel or perpendicular to the main tube axis. In a global point of view, this is due to the fact that the induced molecular dipole of the tube is already aligned to the direction of the field in these two cases. This means that the induced torque acting on the molecule is zero. The tube is therefore in equilibrium with the external field. We can also see in Fig. 6 that the deformation angle gets its maximum values when the field angle varies between 45° and 60°. This means that this tube can be most “efficiently” bent in this field angle range. This range is biased toward 90° because the axial polarizability of CNTs is always greater than the radial one. Geometric effects are also examined. In several previous studies,15,18,19 the static transverse polarizabilities of semiconducting CNTs are found to be proportional to the square of the tube diameter and almost independent of their chirality. We can therefore expect that the electrostatic deflections of two semiconducting SWCNTs of the same length and diameter but different chiralities are almost the same. In order to prove this expectation, we computed the induced deformation of a chiral 共5,4兲 tube in a field with = 45° and E = 3.0 V / nm. The tube length and radius are about 6.72 and 0.31 nm, respectively. The deformation angle of this tube is about 11.85°. This value is close to that shown in Fig. 4 for
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FIG. 7. 共Color online兲 sin共兲 versus aspect ratio 共L / R兲. = 45°. E = 3.0 V / nm. The round points stand for the simulation data and the dashed line represents the fitting line.
a 共8,0兲 tube in the same external field: 11.32°. The small difference between these two values may be attributed mainly to the fact that the elastic modulus of a zigzag SWCNT is slightly higher than that of a chiral or armchair one with the same length and radius.29,30 As shown in Fig. 7, we found that sin共兲 is proportional to the aspect ratio of the tube defined as their length to radius L / R. For this external electric field, we obtain a relation almost linear between sin共兲 and the aspect ratio L / R from our simulation results as follows: sin共兲 = B共L/R兲 + C,
共9兲
where the values of B and C are about 0.0168± 0.0003 and −0.163± 0.008, respectively. The mechanism of this phenomenon is rather complex: the deformation of a CNT depends both on the polarization and on its local mechanical resistance, the polarization depends on the external electric field and on the atomic polarizability, and at the same time, the polarizabilities change following the deformation of the
*Electronic address:
[email protected] 1
CNT. In our current work for building a global semiphenomenological model for electrostatic deflection of CNTs, we are preparing some more simulations to try to get a better understanding on this point. In conclusion, we have demonstrated that a renormalized dipolar model coupled with the AIREBO potential can lead to realistic modeling of the deflection of various nanotubes under homogeneous external electric fields. The sine of the deflection angle is proportional to the square of the external electric field, which is coherent with the conservation of the total energy of the system and the experimental data. This study also reveals an optimum deflection angle for an electric field making an angle between 45° and 60° with the original tube axis. This result is directly applicable in nanoelectronics where the nanotubes are oriented by means of electric fields in order to realize contacts between conductive plots. Furthermore, we also demonstrate the strong link existing between the deflection angle and the aspect ratio L / R of the tubes. These results are interesting and are going to be studied in more detail in forthcoming publications. For example, we have seen that the local electrostatic forces are strongly inhomogeneous along the tube, but it seems that there exists a periodicity of its local variations that should be studied in more detail for a wide variety of tubes. Furthermore, we would like to use a regularized monopole-dipole interaction model as developed by Mayer et al.23 to be able to deal with metallic nanotubes containing free charges. Thus, it could be possible to get a better understanding of the experimental results which suggest that metallic nanotubes can have a better response to electric fields than semiconducting ones.
R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes 共Imperial College Press, London, 1998兲. 2 C.-H. Ke and H. Espinosa, Handbook of Theoretical and Computational Nanotechnology 共American Scientific, CA, 2006兲, Chap. 121. 3 S. Akita, Y. Nakayama, S. Mizooka, Y. Takano, T. Okawa, Y. Miyatake, S. Yamanaka, M. Tsuji, and T. Nosaka, Appl. Phys. Lett. 79, 1691 共2001兲. 4 J. M. Kinaret, T. Nord, and S. Viefers, Appl. Phys. Lett. 82, 1287 共2003兲. 5 J. Li, Q. Zhang, D. Yang, and J. Tian, Carbon 42, 2263 共2004兲. 6 R. Krupke, F. Hennrich, H. Lohneysen, and M. Kappes, Science
ACKNOWLEDGMENTS
J.-M. Vigoureux and A. Mayer are gratefully thanked for fruitful discussions. This work was done as part of both the CNRS GDR-E 2756 and the NATO SFP Project No. 981051. Z.W. acknowledges the support received from the region of Franche-Comté 共Grant No. 051129,91兲 and the Marie Curie conference and training courses grants.
301, 344 共2003兲. T. Hertel, R. Martel, and P. Avouris, J. Phys. Chem. B 102, 910 共1998兲. 8 P. Poncharal, Z. L. Wang, D. Ugarte, and W. De Heer, Science 283, 1513 共1999兲. 9 K. El-Hami and K. Matsushige, Ultramicroscopy 105, 143 共2005兲. 10 Q. Bao, H. Zhang, and C. Pan, Appl. Phys. Lett. 89, 063124 共2006兲. 11 Y. Guo and W. Guo, J. Phys. D 36, 805 共2003兲. 12 F. Torrens, FGCS, Future Gener. Comput. Syst. 20, 763 共2004兲. 13 A. Mayer and P. Lambin, Nanotechnology 16, 2685 共2005兲. 14 W. Guo and Y. Guo, Phys. Rev. Lett. 91, 115501 共2003兲. 15 L. X. Benedict, S. G. Louie, and M. L. Cohen, Phys. Rev. B 52, 7
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WANG et al. 8541 共1995兲. A. Mayer, Phys. Rev. B 71, 235333 共2005兲. 17 A. Mayer, Appl. Phys. Lett. 86, 153110 共2005兲. 18 G. Y. Guo, K. C. Chu, D. Sheng Wang, and Chun-Gang Duan, Comput. Mater. Sci. 30, 269 共2004兲. 19 B. Kozinsky and N. Marzari, Phys. Rev. Lett. 96, 166801 共2006兲. 20 D. S. Novikov and L. S. Levitov, Phys. Rev. Lett. 96, 036402 共2006兲. 21 R. Langlet, M. Arab, F. Picaud, M. Devel, and C. Girardet, J. Chem. Phys. 121, 9655 共2004兲. 22 R. Langlet, M. Devel, and P. Lambin, Carbon 44, 2883 共2006兲. 23 A. Mayer, P. Lambin, and R. Langlet, Appl. Phys. Lett. 89, 063117 共2006兲. 16
Mayer, Phys. Rev. B 75, 045407 共2007兲. S. J. Stuart, A. B. Tutein, and J. A. Harrison, J. Chem. Phys. 112, 6472 共2000兲. 26 http://lpm.univ-fcomte.fr/Personnel/wangz/index.html 27 D. Brenner, O. Shenderova, J. Harrison, S. Stuart, B. Ni, and S. Sinnott, J. Phys.: Condens. Matter 14, 783 共2002兲. 28 M. M. J. Treacy, T. W. Ebbesen, and J. M. Gibson, Nature 共London兲 381, 678 共1996兲. 29 G. van Lier, C. van Alsenoy, V. van Doren, and P. Geerlings, Chem. Phys. Lett. 326, 181 共2000兲. 30 Z. Wang, M. Devel, B. Dulmet, and S. Stuart, Fullerenes, Nanotubes and Carbon Nanostructure 共to be published兲. 24 A. 25
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Electrostatic deflections of cantilevered metallic carbon nanotubes via charge-dipole model Zhao Wang* and Michel Devel Institute UTINAM, UMR 6213, University of Franche-Comté, 25030 Besançon Cedex, France 共Received 20 April 2007; revised manuscript received 12 July 2007; published 20 November 2007兲 We compute electrostatic field induced deformations of cantilevered finite-length metallic carbon nanotubes using an energy minimization method based on a charge-dipole moment interaction potential combined with an empirical many-body potential. The influence of field strength, field direction, and tube geometry on the electrostatic deflection is investigated for both single- and double-walled tubes. These results could apply to nanoelectromechanical devices based on cantilevered carbon nanotubes. DOI: 10.1103/PhysRevB.76.195434
PACS number共s兲: 85.35.Kt, 32.10.Dk, 31.15.Qg, 41.20.Cv
I. INTRODUCTION
II. PHYSICOCHEMICAL MODEL
Cantilevered carbon nanotubes 共CNTs兲 can be used as key elements in nanoelectromechanical systems 共NEMS兲 such as nanorelays,1,2 nanoswitches,3 nanotweezers,4 and feedback device,5 which are designed for memory, sensing, or actuation uses. The electric field induced movements and deformations are key characteristics for these applications, as well as for CNTs’ fabrication6–8 and separation.9 Indeed, electric field induced deflection,10,11 alignment,12,13 and microstructure change14 of CNTs have been observed in experiments. Compared to the semiconducting CNTs, the metallic ones can generally be more sensitive to the presence of external electric fields due to their free charge distribution and higher polarizability.7 They can, therefore, be expected to play a more important role in NEMS. In this work, motionless equilibrium forms of CNTs in electric fields are computed by the method of energy minimization using the algorithm of conjugated gradient. The total potential energy of the system is the sum of an induced electrostatic potential and a many-body interatomic potential. In one of our previous studies, this method was used with a regularized dipole-only model15 to calculate the induced electrostatic potential of semiconducting single-walled CNTs 共SWCNTs兲.16 However, the change of the charge distribution of metallic CNTs cannot be properly described by this model. Thus, a regularized charge-dipole model parametrized for fullerenes and metallic CNTs17–19 is used in this work. The interatomic potential is computed using the adaptive interatomic reactive empirical bond order 共AIREBO兲 potential.20 This potential is an evolution of a many-body chemical pseudopotential model 共reactive empirical bond ordor兲 parametrized by Brenner21 for conjugated hydrocarbons, which has been widely used in theoretical studies on mechanical and thermal properties of CNTs. Electric polarization may change the strength of bonds, as discussed in Ref. 22. However, this effect can be neglected in our work since the field strengths used here are far lower than those used in Ref. 22. Thus, we use a separate potential energy to take into account the interaction with the field. The details about the models will be presented in Sec. II. The results for both SWCNTs and double-walled CNTs 共DWCNTs兲 are shown and discussed in Sec. III. We draw conclusions in Sec. IV.
At the beginning of the calculation, open-ended tubes with zero net charge and zero permanent dipole moment are fixed at one of their two ends on an substrate which is supposed to be insulating in order to allow us to neglect transfer of charges from the nanotube to the substrate. Each atom is associated with both an induced dipole pi and a quantity of induced charge qi when the tube is submitted to an electric field. The total energy of this system Utot is the sum of the induced electrostatic energy Uelec and the interatomic potential U p: Utot = Uelec + U p, in which Uelec can be written as follows:
1098-0121/2007/76共19兲/195434共5兲
N
U
elec
N
N
N
1 i,j = 兺 qi共i + Vi兲 − 兺 pi · Ei + 兺 兺 qiTq−q qj 2 i=1 i=1 i=1 j=1 N
N
− 兺 兺 pi · Ti,j p−qq j − i=1 j=1
N
N
1 兺 兺 pi · Ti,jp−p · p j , 2 i=1 j=1
共1兲
where N is the total number of atoms, i stands for the electron affinity of atom i, Vi is the external potential, and Ei represents the external electric field. T and T stand for the vacuum electrostatic propagators regularized by a Gaussian distribution in order to avoid the divergence problem when two atoms are too close to each other. They erf关ri,j/冑2R兴
i,j i,j can be written as Tq−q = 41⑀0 ri,j , Ti,j p−q = −ⵜriTq−q, and i,j i,j T p−p = −ⵜr j 丢 ⵜriTq−q, where ri represents the coordinate of atom i, ri,j stands for the distance between atom i and atom j, and R is the width of the Gaussian distribution of charge. The value of R used in this work is about 0.068 62 nm, which was fitted to reproduce the polarizability of metallic tubes.18 Taking the lim ri,j → 0, we obtain the self-energy terms 关when i = j in Eq. 共2兲兴 as follows:
1 冑2/ q2i 1 i,i , qiTq−qqi = 2 4⑀0 R 2
共2a兲
pi · Ti,i p−qqi = 0,
共2b兲
1 1 −1 pi · Ti,i p−p · pi = pi · ␣i · pi , 2 2
共2c兲
where ␣i stands for the polarizability of atom i.
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FIG. 1. 共Color online兲 Induced dipoles and charges on an openended metallic 共5,5兲 SWCNT 共L = 4.8 nm兲 subjected to a horizontal electric field E = 1.0 V / nm. The positive charges move to the right side and the negative ones move to the left 共the color scaling is proportional to the density of charge兲. The green vectors stand for the dipoles. The maximal amplitudes of these charges and dipoles are about 0.34 unit of e and 0.11 Debye, respectively.
The distributions of dipoles 兵pi其 and charges 兵qi其 are determined by the fact that the static equilibrium state of these distributions should correspond to the minimum value of Uelec. Thus, by requiring that the partial derivatives of the total electrostatic energy with respect to the 3 ⫻ N components of the dipoles and N components of the charges should be zero within Eq. 共1兲, and taking account of Eqs. 共2a兲–共2c兲, we obtain 兵pi其 and 兵qi其 by solving N linear vectorial equations and N linear scalar equations as follows:
冦
∀i = 1, . . . ,N
N
N
兺 Ti,jp−p 丢 p j + 兺 Ti,jp−qq j = − Ei j=1
j=1
N
N
i,j Ti,j 兺 p−q · p j + 兺 Tq−qq j = − 共i + Vi兲. j=1 j=1
冧
共3兲
In Fig. 1, we show the distribution of the dipoles and charges induced by an electric field on a metallic SWCNT. The interatomic potential U p is computed using the AIREBO potential function.20 This potential is an extension of Brenner’s second generation potential23 and includes longrange atomic interactions and single bond torsional interactions. In this type of potential, the total interatomic potential energy is the sum of individual pair interactions containing a many-body bond order function as follows: N
Up =
1 兺 2 i=1
N
兺 j=1
j⫽i
+
冤
FIG. 2. Electrostatic deflection of a 共5,5兲 SWCNT, with tube length L ⬇ 19.8 nm, field angle = 45°, and field strength E = 兩E兩 = 0.775 V / nm.
induced net charges and dipoles on each atom are updated at every step of the minimization procedure. III. RESULTS AND DISCUSSION
In this work, Cartesian coordinates are used with the z axis along the principal axis of the tube. The open-ended tubes are fixed at one end to an insulating substrate, and relaxed in free space before being submitted to a homogeneous electric field. All applied fields are parallel to the y-z plane. The field strengths are between 0.1 and 3.0 V / nm for the metallic tubes and between 0.1 and 1.0 V / nm for semiconducting tubes. We note that Li and Lin showed that a semiconductor-metal transition takes place in a 共16,0兲 CNT when electric fields reach about 3.0– 4.0 V / nm.25 Furthermore, in actual experiments, the field strengths needed to get comparable deflections are much weaker than those used here, since we use tubes at least 100 times shorter than in experiment and, as shown hereafter, the longer the tube, the weaker the field needed to get a given deflection. This is the same as in field emission experiments in which the shorter the tube, the stronger the field strength needed to produce a
R共ri,j兲 − bi,jA共ri,j兲 + LJ共ri,j兲
N
N
k=1
ᐉ=1
兺 兺
k⫽i,j ᐉ⫽i,j,k
tor kijᐉ
冥
,
共4兲
where and A are the interatomic repulsion and attraction terms between valence electrons, respectively, for bound atoms. The bond order function bi,j provides the many- body effects by depending on the local atomic environment of atoms i and j. The long-range interactions are included by adding LJ, a parametrized Lennard-Jones 共LJ兲 12-6 potential term. tor represents the torsional interactions. Energy optimization is performed to obtain the motionless equilibrium configurations of the atoms using the method of conjugated gradient.24 We note that during this process, the R
FIG. 3. sin共兲 vs E for two SWCNTs: a metallic one 共5,5兲, with L ⬇ 13.16 nm and radius R = 0.34 nm, and a semiconducting one 共6,4兲 with the same L and R. The fields are applied in both = / 4 and = 5 / 4 共in opposite directions兲.
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FIG. 4. sin共兲 / sin共兲 vs E for a metallic tube 共5,5兲 共L ⬇ 13.16 nm兲.
FIG. 5. Axial strain = ⌬L / L 共%兲 versus E2 for a metallic tube 共5,5兲 共L ⬇ 13.16 nm兲, when the electric fields are applied parallel to the tube axis.
given field emission intensity, owing to the decrease of the tip effect on the field enhancement factor 共see, e.g., Fig. 3 of Ref. 26兲. Figure 2 shows the equilibrium position of a SWCNT in a uniform electric field. It can be seen from this figure that the tube is only curved at the part close to the fixed end. We find that its right side part remains straight and that it is slightly compressed by electrostrictive effects27 by comparing its average bond lengths before and after deflection. We note that in real experiments, the tube would have thermal vibrations around this equilibrium position,28,29 and that this deformation is generally reversible.11 Furthermore, the bending of the fixed end would generally not lead to important changes of the tube conductivity.30,31 As shown in Fig. 2, the field angle is defined as the angle between the field direction and the z axis; the deformation angle is defined as the angle between the neutral axis of the deformed CNTs at the free end and the z axis. Figure 3 shows the relation between the external fields and the deformation angles for two SWCNTs. The deflection of the semiconducting tube is calculated using the dipole-only model with parameters given in Ref. 15, while we use the charge-dipole model with parameters from Ref. 18 for the metallic one. As expected, it can be seen that the deflection of the 共5,5兲 tube is much larger than that of the 共6,4兲 semiconducting one for a given electric field, and that
the tube deflection is the same regardless of whether the field direction is reversed. Furthermore, we note that the form of the curves of sin共兲 versus E is in qualitative agreement with the results of the experiment of Poncharal et al. 共Fig. 1 in Ref. 10兲, and we find sin共兲 ⬀ E2 when the deflection is relatively small 关sin共兲 ⬍ 0.15兴 for both of these two CNTs. For higher field strength, the alignment ratio is defined as sin共兲 / sin共兲. It is calculated for several field directions and plotted in Fig. 4. It stands for the relative deformation to the field direction and nearly attains its maximum value of 1 once the tube is well aligned to the field. We can see that when the value of E remains small 共⬍1.4 V / nm兲, the alignment ratio is larger for the smaller field angles . On the other hand, this tube can be more efficiently aligned to the field direction in stronger fields for larger field angles. No deflection is found when the field is perfectly perpendicular to the tube axis, because the induced molecular dipole is already aligned to the field. However, we note that this case can hardly happen in realistic experimental condition due to the thermal vibration of the tube and the fact that, generally, the CNTs are more or less naturally curved due to the presence of defects. As expected, there is no electrostatic deflection found when = 0. Since the induced molecular dipole of the tube is already aligned to the direction of the field, the total induced
FIG. 6. In an external electric field E = 0.775 V / nm and = 45°: 共a兲 sin共兲 vs the radii R of six metallic tubes with the same length L ⬇ 13.2 nm; 共b兲 sin共兲 vs the length of six 共5,5兲 CNTs 共L ⬇ 6.52, 13.16, 19.8, 26.44, 33.08, and 39.72 nm兲.
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FIG. 7. For two metallic DWCNTs: 共4,4兲@共9,9兲 with L ⬇ 12.2 nm, Rinner ⬇ 0.27 nm, and Router ⬇ 0.61 nm, and 共6,0兲@共15,0兲 with L ⬇ 12.2 nm, Rinner ⬇ 0.23 nm, and Router ⬇ 0.59 nm. 共a兲 sin共兲 vs E; = 45°. 共b兲 sin共兲 vs cos共兲; E = 2 V / nm.
torque acting on the tube is therefore zero. Nevertheless, slight electrostriction effects are found in the axial direction of the tube. The electrostrictive deformation = ⌬L / L is plotted in Fig. 5 versus the square of field strength. It can be seen that is nearly proportional to E2 for these field strengths. This numerical experiment also allowed us to estimate the nanotube Young’s modulus 共Y兲 by the stress over strain ratio, since the axial external electrostatic force acting on the tube can be directly computed in our program. Using the commonly adopted wall thickness value of 0.34 nm, we find that Y is about 0.95 TPa, which is in good agreement with the average of the values found in the literature for that wall thickness 共see, e.g., Sec. 2.1 of the recent review by Coleman et al.32兲. Turning back to the question of electrostatic deflection, we also study tube geometry effects. Figure 6共a兲 shows sin共兲 versus the radius R for several metallic CNTs with the same length. It can be seen that the bigger the tube radius, the smaller the induced deflection. It is well known that the polarization effects are more important when the tube is bigger. However, at the same time, the tube becomes harder to bend due to the increase of the moment of inertia of its cross section. From our results, it is obvious that the latter effect plays a more important role. Note also that the deflection of zigzag tubes is slightly smaller than that of chiral and armchair ones due to their larger elastic moduli.33 This curve of sin共兲 can be fitted as sin共兲 = 1 / 共2.5886R2 − 0.3839兲. Hence, sin共兲 is roughly proportional to 1 / R2 for this electric field. Figure 6共b兲 shows the relation between the deflection and the tube length. We can see that the deflection increases significantly with the increase of tube length when remains much smaller than . Then, it reaches a plateau. It can be seen that the form of the curve of sin共兲 versus L is very similar to that of sin共兲 / sin共兲 versus E. This is probably because L and E play two similar roles in the total induced torque T = 共L兲E2 sin共 − 兲cos共 − 兲 共where  is the molecular polarizability of CNTs兲.34 Hence, considering that the lengths of the CNTs studied in previous experimental works are in the range from hundreds of nanometers to some micrometers, the required field strength can be much lower than the fields used in this paper for a given deflection angle. Furthermore, to let the readers conveniently find the values of sin共兲 in Fig. 6共b兲, we give the best fitting function of this curve as sin共兲 = sin共兲 / 关1 + 共L / 15.3010兲−5兴. Figure 7共a兲 shows the relation between the deflection and the field strength for two DWCNTs. It is found that the de-
flection of DWCNTs remains small even in strong electric fields. For a metallic cylinder, the screening factor is very high, thus the inner layer is almost completely screened. On the other hand, their effective cross sections increase with increasing layer number. Thus, a multiwalled CNT can be much harder to bend by the electric field than a SWCNT with the same radius. sin共兲 is also plotted in Fig. 7共b兲 for several field directions. We can see the DWCNTs can be most efficiently bent at = 60°, like SWCNTs, for this field intensity. This value can be biased toward 90° because the axial polarizability of CNTs is always greater than the radial one.
IV. CONCLUSION
In this paper, we investigate the mechanisms of the electrostatic deflection of cantilevered metallic SWCNTs and DWCNTs. The equilibrium positions of CNTs in electric fields are calculated. The metallic CNTs are much easier to be deflected than semiconducting ones. The deflection is not changed by reversing the field direction. The curve of alignment ratio versus field strength is found to change with field directions. The deflection is found to decrease with the increase of the tube radius; conversely, it increases when the tube is longer. The multiwalled metallic CNTs are found to be much harder to bend in electric fields than SWCNTs. Furthermore, we find that the electrostrictive deformation of SWCNTs is proportional to the square of field strength. Uniform external fields are applied as a theoretical simplification. However, our scheme is able to deal with inhomogeneous fields such as those from real experiments. We believe that this paper could help develop a better understanding of recently designed NEMS based on cantilevered CNTs. We also wish that these results can be useful to open the path to some new nanoelectromechanical devices.
ACKNOWLEDGMENTS
This work was done as part of the CNRS GDR-E Nb 2756. Z.W. acknowledges the support received from the region of Franche-Comté 共Grant No. 060914-10兲.
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ELECTROSTATIC DEFLECTIONS OF CANTILEVERED…
1
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[email protected] J. M. Kinaret, T. Nord, and S. Viefers, Appl. Phys. Lett. 82, 1287 共2003兲. 2 S. W. Lee, D. S. Lee, R. E. Morjan, S. H. Jhang, M. Sveningsson, O. A. Nerushev, Y. W. Park, and E. E. B. Campbell, Nano Lett. 4, 2027 共2004兲. 3 J. Jang, S. Cha, Y. Choi, G. Amaratunga, D. Kang, D. Hasko, J. Jung, and J. Kim, Appl. Phys. Lett. 87, 163114 共2005兲. 4 S. Akita, Y. Nakayama, S. Mizooka, Y. Takano, T. Okawa, Y. Miyatake, S. Yamanaka, M. Tsuji, and T. Nosaka, Appl. Phys. Lett. 79, 1691 共2001兲. 5 C. Ke and H. Espinosa, Appl. Phys. Lett. 85, 681 共2004兲. 6 A. Srivastava, A. K. Srivastava, and O. N. Srivastava, Carbon 39, 201 共2001兲. 7 E. Joselevich and C. M. Lieber, Nano Lett. 2, 1137 共2002兲. 8 J. Li, Q. Zhang, D. Yang, and J. Tian, Carbon 42, 2263 共2004兲. 9 R. Krupke, F. Hennrich, H. V. Lohneysen, and M. M. Kappes, Science 301, 344 共2003兲. 10 P. Poncharal, Z. L. Wang, D. Ugarte, and W. A. De Heer, Science 283, 1513 共1999兲. 11 Y. Wei, C. Xie, K. A. Dean, and B. F. Coll, Appl. Phys. Lett. 79, 4527 共2001兲. 12 X. Q. Chen, T. Saito, H. Yamada, and K. Matsushige, Appl. Phys. Lett. 78, 3714 共2001兲. 13 M. S. Kumar, T. H. Kim, S. H. Lee, S. M. Song, J. W. Yang, K. S. Nahm, and E. K. Suh, Chem. Phys. Lett. 383, 235 共2004兲. 14 Q. Bao, H. Zhang, and C. Pan, Appl. Phys. Lett. 89, 063124 共2006兲. 15 R. Langlet, M. Devel, and P. Lambin, Carbon 44, 2883 共2006兲. 16 Z. Wang, M. Devel, R. Langlet, and B. Dulmet, Phys. Rev. B 75, 205414 共2007兲.
18 A.
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Deformation of Doubly Clamped Single-Walled Carbon Nanotubes in an Electrostatic Field Zhao Wang1,2,* and Laetitia Philippe1 1
EMPA (Swiss Federal Laboratories for Materials Testing and Research), Feuerwerkerstrasse 39, CH-3602 Thun, Switzerland 2 Institute UTINAM, Universite´ de Franche-Comte´, F-25000 Besanc¸on, France (Received 28 January 2009; published 26 May 2009) In this Letter, we demonstrate a strong dependence of the electrostatic deformation of doubly clamped single-walled carbon nanotubes on both the field strength and the tube length, using molecular simulations. Metallic nanotubes are found to be more sensitive to an electric field than semiconducting ones of the same size. For a given electric field, the induced deformation increases with tube length but decreases with tube radius. Furthermore, it is found that nanotubes can be more efficiently bent in a centeroriented transverse electric field. DOI: 10.1103/PhysRevLett.102.215501
PACS numbers: 62.25.g, 63.22.Gh, 85.85.+j
Carbon nanotubes (CNTs) can be used in nanoelectromechanical systems (NEMS) for uses in sensing, actuation, vibration, and laboratory-on-a-chip applications [1]. Particularly, doubly clamped (suspended) structures of CNTs have been reported to be used as key components in a number of nanodevices [2–7]. In most of these recent NEMS, nanotubes are usually suspended between two electrodes with an applied voltage. However, as we will demonstrate in this Letter, even without applying electric current and excess charges, doubly clamped CNTs can still be significantly bent in a static transversal electric field, as a result of electric polarization. Cantilevered CNTs have long been known to deform in an electric field [8–10]. The mechanism of this deformation relies on the fact that CNTs can be bent by a bending moment induced by interactions between the electric field and the molecular dipoles due to electric polarization. This property has been exploited in the design of nanorelays [11,12]. Compared with cantilevered CNTs, doubly clamped ones can have many advantages in electronic devices [13] (e.g., they can be integrated in nanodevices with a well-defined spatial structure and resonance frequency). In this Letter we investigate the deflection of neutral single-walled CNTs (SWCNTs) by an electrostatic field. Two semiclassical theories have been combined for characterizing chemical potential and electrostatic interactions in both metallic and semiconducting CNTs. Considering the general correlation between the conductivity and polarizabilities of CNTs, metallic CNTs are supposed to be more sensitive to electric fields than semiconducting ones, and hence, can be expected to play a more important role in NEMS. When an electric field is applied to a metallic CNT, an induced dipole can be created at each atomic site with a quantity of free charge, by shifting the electrons and the nuclei. In our calculation, each atom is therefore modeled as an interacting polarizable point with a free electric charge, while the chemical bonds are described by using a many-body potential function. Motionless equilibrium 0031-9007=09=102(21)=215501(4)
positions of carbon atoms in an electric field are computed by minimizing the total potential energy of systems Utot , which is the sum of two terms: Utot ¼ Uelec þ Up , where Up is the interatomic potential due to the C-C chemical bonds in the absence of an external field, including the long-range interactions. Uelec stands for the electrostatic energy from the interaction between charges, dipoles, and external fields. Uelec is calculated using a Gaussianregularized charge-dipole [10] and a dipole-only [14] model for metallic and semiconducting CNTs, respectively. Up is computed using an adaptive interatomic reactive empirical bond order (AIREBO) potential [15], which is an extension of the REBO potential [16]. Compared to first-principle and semiempirical methods, an important feature of our combined models is their ability to deal with large systems (up to 5000 atoms). This is particularly important for studying geometric effects on the electrostatic bending of CNTs, since the axial periodic condition can hardly be applied in this issue. The strength of electric field E used in this work is in the order of V=nm, because the tubes used in our calculation are too short (18–27 nm) to be bent in a weak electric field. As will be shown, the required field strength decreases with increasing tube length L for a given deformation. Considering that the CNTs used in experiments are usually of m length, field emission effects [17] and conductance switching due to strong transversal fields [18] are both neglected. These field strengths are not large enough to cause a significant change in the chemical bond strength [19]. Moreover, it can expected that an armchair CNT can exhibit more deformation by an electric field than a metallic zigzag tube of the same size, since previous studies concluded that bending deformation would have a negligible effect on the electronic transport properties of armchair CNTs [20,21], while the quantum conductance of zigzag CNTs can significantly decrease under large bending deformations [22]. In this work, the tubes are assumed to be suspended between two electrically insulated supports.
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Ó 2009 The American Physical Society
PRL 102, 215501 (2009)
PHYSICAL REVIEW LETTERS
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FIG. 1 (color online). Schematics of electrostatic deformation of a SWCNT. (a) Two arrows p1 and p2 stand for two induced molecular dipoles. Eext is represented by the arrows with dashed lines. 1 and 2 stand for the induced bending moments acting on the tube. The color scale of atoms is proportional to the density of induced charges. The vectors stand for the induced atomic dipoles. (b) Two possible deflection directions in bending (upward or downward).
The mechanism of electrostatic bending of a SWCNT is depicted in Fig. 1(a). When a CNT (initially electrically neutral) is submitted to a transversal electric field Eext , the field tends to shift negative and positive charges in opposite directions (in this figure we can see that positive charges move to the top and negative ones move to the bottom of the tube); two induced molecular dipoles (p1 and p2 ) are hence created in the tube. The tube is then bent by two bending moments (1 and 2 ), which is induced by the interaction between the molecular dipoles and Eext ( ¼ pm Eext ). Molecular dynamics (MD) simulations were performed using the AIREBO potential, for generating arbitrary initial configurations of CNTs at room temperature. The energy of these MD-generated configurations is minimized in order to calculate their equilibrium positions in an electric field. MD simulations showed that CNTs slightly oscillate around their initial positions due to thermal fluctuations, which have been observed in experiments [23,24]. We found that the orientation of electrostatic deformation of CNTs strongly depends on their initial position at the moment when Eext is applied. For example, as shown in Fig. 1(b), if Eext is applied when the tube is at position 1, its equilibrium state in electric field will be at position 3; conversely, if the tube is initially in position 2, it will be bent to position 4. These results also imply that no change of the deformation direction will take place if the field direction is totally reversed, due to the system symmetry. For measuring the amplitude of deflection, we define u as the displacement of the center of the tube middle from its initial position. For a small electric field, the bending of the CNT is elastically reversible; i.e., the tube will come back and oscillate around its initial position once Eext is removed [6,25]. In our MD simulations, the
average frequency of thermal oscillation (without electric field) of a ð5; 5Þ tube (length L 20:0 nm) is found to be 60 10 GHz for the several first harmonics. This value is comparable to oscillation frequency of doubly clamped CNTs recently reported in Ref. [26]. We plot in Fig. 2 computed values of u versus external field strength E for both a ð4; 4Þ and a ð5; 3Þ CNT. It can be seen that the metallic tube is clearly more sensitive to Eext than the semiconducting one due to their different polarizabilities [27]. When u 5:5 nm, it is observed that the
FIG. 2 (color online). Electrostatic deformation u versus field strengths E ¼ jEext j for two SWCNTs: a metallic ð4; 4Þ (L 24:5 nm) and a semiconducting ð5; 3Þ with almost the same length and radius. The tube ð4; 4Þ begins to break down when E > 11:0 V=nm, then failure occurs very soon thereafter (umax > 5:1 nm). The inset shows the position of fracture and the distribution of von Mises stress at the bottom (before the fracture occurs).
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ð4; 4Þ tube begins to fracture near the fixed end, in which the maximal tensile strain is localized (inset of Fig. 2). We can also see that there is no deformation when E remains small, since in such cases the moments of electric force are not large enough to exceed the mechanical resistance barrier of the tube. These results also imply that semiconducting tubes can sustain higher field strengths. The CNTs used in experiments are usually microns in length. However, their field-induced deformations can hardly be addressed by the calculations using atomic models in a direct way. Hence, it is necessary to address the influence of the tube length L. In Fig. 3(a), it can be seen that the relative displacement u=L increases with L; i.e., a weaker electric field is required for a given u in a longer tube. This increased displacement depends on the field strength. An extrapolation was done from the data E versus L for tube lengths up to 1 m, in order to make experimental verification possible [Fig. 3(b)]. It was found that lnðEÞ decreases almost linearly with lnðLÞ for two given u. This is related to the fact that the electric capacitance per unit length of a metallic cylinder is roughly proportional to lnðLÞ. Thus, we can conclude that, for the m-length tubes
used in experiments, the required field strength is on the order of V=m. To explore further geometry effects, we plot u versus E for the tubes of different radii R in Fig. 4. It is found that small tubes are more sensitive to Eext than large ones. This looks abnormal since it is known that the polarizability of a CNT increases with its radius. However, from a mechanical point of view, the tube becomes more difficult to bend due to the increase of its area moment of inertia (I ¼ 0:5mðR2int þ R2out Þ for a thick-walled cylindrical tube, where m is the mass). From our results it is clear that the latter effect plays a more important role. The ratio u=E is roughly the same for all of these tubes in the case of large deformation (u > 1 nm). In Figs. 2 and 4, we can see a threshold field of a few V=nm for each u versus E curve before one obtains a nonzero deformation. The existence of this threshold field relies on the fact that the electric bending moments remain weaker than the tube resistance when E is small. With increasing E, one reaches the limit of the threshold field until a value of u allows a moment balance in the nanotube (inset of Fig. 4). It is found that the threshold field increases with R but decreases with L. In this study, the applied external field is assumed to be uniform as a common theoretical simplification. However, in experimental situations, the external fields are usually not homogeneous and strongly depend on the experimental setup. By changing the field direction, we found that CNTs can be more efficiently bent in a centeroriented ( %- -like) electric field [see Fig. 5(a)]. Results of the deformation of a SWCNT by this field are plotted in Fig. 5(b). We can see that values of u roughly follow a linear relationship with E when u > 1 nm. It was also found that, when ¼ =4, the tube can be most efficiently bent and its u versus E curve has the straightest shape.
FIG. 3. (a) u=L versus the tube length L for ð4; 4Þ SWCNTs, in four different transverse Eext (which is perpendicular to the initial axis of the tubes). (b) E versus L on a logarithmic scale for two given u. The symbols present the calculated points and the lines stand for the extrapolation curves.
FIG. 4. u versus E for six individual armchair SWCNTs (L 24:5 nm) with different radii. The direction of Eext is perpendicular to the initial axis of tubes. Inset shows the force balance between an electric moment and internal stresses in a half of a deformed tube.
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Mayer, R. Langlet, M. Devel, and W. Ren are acknowledged for useful discussions.
FIG. 5 (color online). (a) Schematic (in y-z plane) of the deformation of a CNT subjected to an %- -like electric field Eext (dashed lines). The field angle is defined as the angle between the field lines and the axis z. The field vector E ¼ ðEx ; Ey ; Ez Þ ¼ ð0; E sinðÞ; þE cosðÞÞ when z < 0 and E ¼ ð0; E sinðÞ; E cosðÞÞ when z > 0. (b) u versus E for a ð4; 4Þ SWCNT (L 24:5 nm) in Eext with different . The simulation data are represented by the symbols with eye-guide lines.
In summary, deformation of doubly clamped SWCNTs in an electrostatic field has been simulated using a chargedipole polarization model combined with an empirical potential. The interplay between the mechanical resistance and electric polarization of CNTs was investigated for both metallic and semiconducting tubes. Metallic CNTs are found to be more sensitive to an electric field than semiconducting ones. This study reveals that the field-induced deflection increases with tube length while it decreases with tube radius. It was also found that CNTs can be more efficiently bent in a center-oriented electric field. From a theoretical point of view, graphene nanoribbons and other metallic nanowires or tubes should have similar properties, depending on their dielectric constants. This electrostatic deformation of doubly clamped nanostructures can be expected to open a path to designs of novel nanodevices. We gratefully thank S. J. Stuart for his help in the implementation of our computational code. W. Mook, A.
*
[email protected];
[email protected] [1] M. Anantram and F. Leonard, Rep. Prog. Phys. 69, 507 (2006). [2] A. Javey, Q. Wang, A. Ural, Y. Li, and H. Dai, Nano Lett. 2, 929 (2002). [3] A. Javey, J. Guo, Q. Wang, M. Lundstrom, and H. Dai, Nature (London) 424, 654 (2003). [4] K. Keren, R. Berman, E. Buchstab, U. Sivan, and E. Braun, Science 302, 1380 (2003). [5] S. Sapmaz, Y. M. Blanter, L. Gurevich, and H. S. J. van der Zant, Phys. Rev. B 67, 235414 (2003). [6] V. Sazonova, Y. Yalsh, I. Ustunel, D. Roundy, T. Arlas, and P. McEuen, Nature (London) 431, 284 (2004). [7] L. Jonsson, L. Gorelik, R. Shekhter, and M. Jonson, Nano Lett. 5, 1165 (2005). [8] P. Poncharal, Z. Wang, D. Ugarte, and W. De Heer, Science 283, 1513 (1999). [9] Y. Wei, C. Xie, K. A. Dean, and B. F. Coll, Appl. Phys. Lett. 79, 4527 (2001). [10] Z. Wang and M. Devel, Phys. Rev. B 76, 195434 (2007). [11] J. Kinaret, T. Nord, and S. Viefers, Appl. Phys. Lett. 82, 1287 (2003). [12] S. W. Lee, D. S. Lee, R. E. Morjan, S. H. Jhang, M. Sveningsson, O. A. Nerushev, Y. W. Park, and E. E. B. Campbell, Nano Lett. 4, 2027 (2004). [13] H. Dai, Acc. Chem. Res. 35, 1035 (2002). [14] Z. Wang, M. Devel, R. Langlet, and B. Dulmet, Phys. Rev. B 75, 205414 (2007). [15] S. Stuart, A. Tutein, and J. Harrison, J. Chem. Phys. 112, 6472 (2000). [16] D. W. Brenner, Phys. Rev. B 42, 9458 (1990). [17] S. Jo, Y. Tu, Z. Huang, D. Carnahan, D. Wang, and Z. Ren, Appl. Phys. Lett. 82, 3520 (2003). [18] Y.-W. Son, J. Ihm, M. L. Cohen, S. G. Louie, and H. J. Choi, Phys. Rev. Lett. 95, 216602 (2005). [19] Y. Guo and W. Guo, J. Phys. D 36, 805 (2003). [20] A. Rochefort, P. Avouris, F. Lesage, and D. R. Salahub, Phys. Rev. B 60, 13 824 (1999). [21] M. Buongiorno Nardelli, Phys. Rev. B 60, 7828 (1999). [22] A. Maiti, A. Svizhenko, and M. P. Anantram, Phys. Rev. Lett. 88, 126805 (2002). [23] M. Treacy, T. Ebbesen, and J. Gibson, Nature (London) 381, 678 (1996). [24] B. Babic, J. Furer, S. Sahoo, S. Farhangfar, and C. Schonenberger, Nano Lett. 3, 1577 (2003). [25] Y. Wei, C. Xie, K. Dean, and B. Coll, Appl. Phys. Lett. 79, 4527 (2001). [26] C. Li and T.-W. Chou, Phys. Rev. B 68, 073405 (2003). [27] E. Joselevich and C. Lieber, Nano Lett. 2, 1137 (2002).
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Published as : Z. Wang et al. Phys. Rev. B, 75, 205414 et Phys. Rev. B, 76, 195434 (2007)
Chapitre 6
Conclusions
I believe that no conclusion would be the best conclusion.
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