Calculation of phonon dispersion in carbon nanotubes using a continuum atomistic finite element approach

Calculation of phonon dispersion in carbon nanotubes using a continuum atomistic finite element approach Calculation of phonon dispersion in carbon nanotubes using a continuum atomistic finite element[.]

Calculation of phonon dispersion in carbon nanotubes using a continuum-atomistic finite element approach Michael J Leamy Citation: AIP Advances 1, 041702 (2011); doi: 10.1063/1.3675917 View online: http://dx.doi.org/10.1063/1.3675917 View Table of Contents: http://aip.scitation.org/toc/adv/1/4 Published by the American Institute of Physics AIP ADVANCES 1, 041702 (2011) Calculation of phonon dispersion in carbon nanotubes using a continuum-atomistic finite element approach Michael J Leamya School of Mechanical Engineering, Georgia Institute of Technology, 771 Ferst Drive, N.W., Atlanta, Georgia 30332-0405, USA (Received 26 October 2011; accepted 13 December 2011; published online 29 December 2011) Dispersion calculations are presented for cylindrical carbon nanotubes using a manifold-based continuum-atomistic finite element formulation combined with Bloch analysis The formulated finite elements allow any (n,m) chiral nanotube, or mixed tubes formed by periodically-repeating heterojunctions, to be examined quickly and accurately using only three input parameters (radius, chiral angle, and unit cell length) and a trivial structured mesh, thus avoiding the tedious geometry generation and energy minimization tasks associated with ab initio and lattice dynamics-based techniques A critical assessment of the technique is pursued to determine the validity range of the resulting dispersion calculations, and to identify any dispersion anomalies Two small anomalies in the dispersion curves are documented, which can be easily identified and therefore rectified They include difficulty in achieving a zero energy point for the acoustic twisting phonon, and a branch veering in nanotubes with nonzero chiral angle The twisting mode quickly restores its correct group velocity as wavenumber increases, while the branch veering is associated with a rapid exchange of eigenvectors at the veering point, which also lessens its impact By taking into account the two noted anomalies, accurate predictions of acoustic and low-frequency optical branches can be achieved out to the midpoint of the first Brillouin zone Copyright 2011 Author(s) This article is distributed under a Creative Commons Attribution 3.0 Unported License [doi:10.1063/1.3675917] I INTRODUCTION Quantized lattice vibrations, or phonons, play an important role in the electrical and thermal properties of crystalline materials Electrical resistance of a conductor can be strongly affected by scattering of electrons due to electron-phonon coupling.1, The specific heat and thermal conductivity of a crystalline structure are both directly related to phonon group velocities and phonon density of states,3 which can be obtained from phonon dispersion relationships As such, the thermal conductivity of a material is decreased by lowering either the phonon group velocities, the phonon density of states, or the phonon scattering length (i.e., mean free path) For very thin nanowires in which the wire diameters are on the order or lower than the phonon mean free path, the thermal conductance is completely determined by the phonon density of states via the Landauer formula,4 without the need to compute phonon scattering lengths Importantly, for these structures, only the phonon dispersion picture is required to compute conductance Beyond innate properties of natural materials, man-made materials such as multi-layered nanostructures5 are being actively studied to tailor phonon dispersion and enhance materials properties (e.g., thermoelectric figure of merit).6 For all of the above reasons, there is an important need to compute and understand phonon dispersion The prediction of phonon dispersion is typically accomplished using either semiclassical techniques, such as lattice dynamics,7, or quantum mechanical techniques based on second quantization and creation/annihilation operators.9 In both cases, the calculations can be tedious and require a Author to whom correspondence should be addressed Electronic mail: michael.leamy@me.gatech.edu 2158-3226/2011/1(4)/041702/14 1, 041702-1  C Author(s) 2011 041702-2 Michael Joseph Leamy AIP Advances 1, 041702 (2011) accounting for three degrees of freedom for every atom in the unit cell For carbon nanostructures, even pristine unit cells can be large – e.g., a relatively small diameter (17,6) carbon nanotube (CNT) has 1708 atoms in its unit cell If defects are to be studied, a supercell approach can easily exceed 10,000 degrees of freedom or more In addition to large size, these traditional phonon calculations require computing the location of each atom every time a new calculation is to be performed Further, the minimum energy state of each lattice should be computed before generating the phonon dispersion curves, which adds to the time and effort necessary to generate accurate curves These considerations make attractive an approach which greatly reduces the number of degrees of freedom and, importantly, reduces the burden of geometry generation This paper sets out to describe, and critically critique, a continuum-atomistic approach for quickly computing phonon dispersion in carbon nanotubes Due to the complexity of the CNT’s reduced dimensional geometry (a curved 2-manifold), this is a relatively ambitious undertaking with few to no parallels in the literature Reusable shell-like finite elements, developed in previous work to compute discrete phonon normal modes,10 are employed together with a Bloch analysis procedure to represent the CNT geometry and to compute the dispersion relationships Due to the use of an underlying continuum formulation, the dispersion curves are expected to be accurate at relatively low frequencies and long wavelengths This is not overly restrictive, however, since these branches (and in particular the acoustic variants) are of primary interest in many instances, such as in predicting thermal properties where only acoustic branches are typically considered due to their predominance in terms of excited numbers, and their much higher group velocities Finally, the validity of the technique, and the frequency and wavenumber range for which it is applicable, are quantified via comparisons of computed results to those found in the literature II OVERVIEW OF CNT CONTINUUM MODEL This section overviews the CNT continuum-atomistic model and finite element solution approached used in the dispersion calculations Full details of the underlying model can be found in,10 which also provides (and validates) discrete phonon spectra for example nanotubes Following presentation of the model, an accompanying Bloch analysis procedure is described which allows the finite element model to be used in dispersion calculations A Configurations, tangent spaces, and basis vectors Figure illustrates the general kinematics and mappings used to develop the curvilinear CNT model – a toroidal CNT is shown for the sake of generality The CNT is represented as a surface in Since the curvilinear both its undeformed (reference) 0 ⊂ R3 and deformed  ⊂ R3 configurations   ¯ CNT forms a two-manifold, only two coordinates of a point ξ = ξ , ξ in a parametric body  are required to locate 0 and  in R using mappings ϕ0 and ϕ, respectively A third mapping  takes position X on 0 to position x on   : X → x = (X ) = X + U(X ) − O, (1) where U denotes the displacement field Due to the reduced dimensionality of the manifolds, basis vectors in the undeformed and deformed configuration lie not on the manifolds, but instead in their tangent spaces TX 0 (undeformed) and Tx  (deformed) For particular locations, the tangent space at X is represented by TX 0 while that at x by Tx  The orientation of the basis vectors in each configuration relative to an underlying Euclidean system of basis vectors, not shown, depends on the chosen point Denoting by Z a the three components of the position vector X to point X in 0 referenced to the Euclidean unit vectors (I1 ,I2 ,I3 ), the (covariant) basis vectors for TX 0 are denoted by G α , Gα ≡ ∂X ∂ Za = I a, ∂ξ α ∂ξ α (2a) where α = 1, and repeated indices imply summation For a cylindrical nanotube, these basis vectors are depicted in Fig In general, the basis vectors G α are not mutually orthogonal nor are 041702-3 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Kinematics and mappings defining manifold configurations FIG Cylindrical coordinate system of unit length Orthogonality is accomplished through introduction of a dual (contravariant) basis G α (spanning the cotangent space TX∗ 0 ) such that G α (G β ) ≡ G α · G β = δβα where δβα denotes the Kronecker delta (one if α =β, zero otherwise) Similar to the above development, a basis for Tx  is given by gα ≡ ∂x ∂z a = i a, ∂ξ α ∂ξ α (2b) where (i1 ,i2 ,i3 ) denote Euclidean basis vectors present with  Note that in terms of the undeformed position vector, the expressions for the basis vectors in  are given by, gα = Gα + ∂ Gβ ∂U ∂U β = G + Gβ + U β α , α α α ∂ξ ∂ξ ∂ξ (3) where U β is a displacement component relative   to G β such that the physical component of displacement in the G β direction is given by U β  G β , no sum on β intended B Deformation The deformation gradient F gives rise to a stress tensor P (the 1st Piola Kirchhoff) via derivatives of a strain energy function W0 Later, a connection will be made between atomistic energy and the strain energy function Specifically, the deformation gradient takes points on the tangent space T 0 defined in the undeformed configuration and maps them to the tangent space T  defined on the deformed configuration F ≡ ∂∂Xx ∈ R3×3 : T 0 → T  Infinitesimal line elements in each configuration are then related via d x = F · d X where, strictly speaking, the line elements lie in the 041702-4 Michael Joseph Leamy AIP Advances 1, 041702 (2011) tangent spaces TX 0 and Tx  and not in the configurations 0 and  The deformation gradient can be expressed in terms of displacement components Uα referenced to G α - quantities which reference solely the undeformed configuration 0 ,   ∂U η ∂x η ρ η = δβ + + U ρβ G η ⊗ G β , (4) F≡ ∂X ∂ξ β η where ρβ denote Christoffel symbols for the chosen parameterization Z a (ξ α ) C Equations of motion The Principle of Virtual Work is used to derive the equations of motion governing the dynamic response of the CNT The Principle of Virtual Work for a finitely deformable two-manifold can be stated as,   ∂ W0 ∂ W0 ∂ W0 α tot ext G ⊗ Gβ , δ E = δ W0 (F)d0 + δ E = : δFd0 + δ E ext = 0, = ∂F ∂F ∂Fαβ 0 0 (5) where W0 is the strain energy per unit undeformed area, and δ E ext holds external virtual work terms In the d’Alembert sense, the external virtual work δ E ext term includes virtual work done by surface and inertial forces,   (6) δ E ext = U Ã t 0P d0 + Uă · δUd0 , 0 0 where t 0P denotes the first-order surface traction and ρ0 denotes the mass density per unit undeformed area In this application, a free response of the nanotube is sought and so no work is done by surface tractions, i.e., t 0P = In addition the derivative of the strain energy density with respect to F can be recognized to be the First Piola-Kirchhoff stress tensor P (Pβα = ∂∂FWα0 ) For two-manifolds, the β absence of thickness results in stress tensors with units of force per unit length With the above considerations, the virtual work principle simplifies to   P : δFd0 + Uă Ã Ud0 = (7) 0 0 D Atomic potential energy A carbon nanotube consists of a graphene sheet, shown in Fig 3, rolled into a tubular shape along the chiral vector C Graphene is a particular crystalline lattice form of carbon in which each carbon atom is bonded to three neighboring carbon atoms, forming a hexagonal arrangement In Fig 3, straight line segments depict the hybridized sp2 bonds between the carbon atoms, while the carbon atoms themselves (not shown) exist at the intersections of the line segments Since graphene is a planar crystal, only two base vectors a1 and a2 need be considered, where each has an undeformed length l0 equal to 2.46 Å These base vectors can be used √ to define the chiral vector C: (n,m) The length of the chiral vector is given by C = C = l0 n + nm + m , which when divided by 2π yields the nanotube radius Due to periodicity of the lattice, each choice of C defines a unit cell, which is defined to be the smallest rectangle defined by C, and a translate of C, such that all four corners of the unit cell coincide with an atomic lattice point The translation vector is given by T whose length is well documented, see for example [24] Note that many stable carbon nanotube configurations are known to exist which result in a variety of admissible radii and chiralities Two configurations in particular are the armchair tubes [30 degree chiral angle φ; C: (n,n)] and the zig-zag tubes [zero degree chiral angle φ; C: (n,0)] By sampling the atomic energy of Reference Area Elements (RAEs) lying on the surface of the nanotube, and connecting this energy with the continuum strain energy W0 , the continuum 041702-5 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Geometry of the graphene sheet and reference area element weak-form (7) can be completed and used for computation A four-atom RAE is chosen, as shown in Fig 3, consisting of three bond lengths and three bond angles covering completely the three bond length and angle varieties in each graphene hexagon Note that two RAEs cover the six total bond lengths and angles in the graphene hexagon However, as is evident in the figure, every bond length is shared by two hexagons, while each bond angle is unique to each hexagon As such, the three RAE bonds represent the three net bond lengths contained in a graphene hexagon, while the three RAE angles represent only half of the net bond angles in the same graphene hexagon The RAE then allows the atomic potential energy per unit area to be equated to the continuum strain energy per unit area, W0 ≡ rae rae E str E rae etch + 2E angle = , A H ex A H ex (8) rae where E str etch is calculated from the Modified Morse Potential (see App A) summing the stretch rae is calculated summing the angle-bending energy energy from the three RAE bond lengths, E angle using the three RAE bond angles, and A H ex represents the area of the graphene hexagon (3.605 Å2 ) Recall that the stress tensor appearing in (7) is related to derivatives of W0 with respect to the deformation gradient F To evaluate E rae requires the positions of individual atoms in  Letting R i j represent a position vector in 0 of atom j relative to atom i, the position vector of atom j relative to atom i in  is given by, r i j = F · Ri j (9) 041702-6 Michael Joseph Leamy AIP Advances 1, 041702 (2011) W0 Note then that (4), the definition P = ∂∂F , and (7-9) yield a completed displacement-based weak form referencing only quantities in the undeformed configuration 0 This weak form can then be discretized using a finite element formulation E Finite element formulation This section introduces a finite element discretization of (7) based on displacement interpolations In order to avoid difficulties capturing rigid body modes (not considered phonons), the Euclidean displacement components are first introduced Introducing a finite element discretization in theusual sense, the field displacement vector U can be discretized using shape functions N I ξ , ξ which interpolate the Euclidean displacement terms Uˆ i appearing in U β ,   β (10) U = N I ξ , ξ Uˆ Ii t¯i G β , β such that t¯i is a conversion tensor from curvilinear to Euclidean bases, Uˆ Ii denotes the ith Euclidean component of the displacement at node I, I ranging over the number of nodes n, and the   shape such that they evaluate to at their home location ξ and at , ξ functions N I ξ , ξ are chosen I I  other nodal locations ξ J1 , ξ J2 , J = I Substitution of (10) into (7) yields the stiffness matrix,   β α ∂ Pα ∂Fβ σρ KIJ = (11) d0 , ρ ∂ Uˆ ∂ Uˆ σ 0 and a mass matrix, σρ J I  MIJ = ρ0 N I N J t¯σi t¯ρj G i j d0 , (12) 0 such that the equations of motion are expressed as, σρ ρ σρ ρ MIJ Uă J + KIJ U J = (13) In the above, free indices I and σ yield the equation number Since each carbon atom is shared by three hexagons, see Fig 3, and each hexagon contains six carbon atoms, the per unit area density ρ is computed using the mass of two carbon atoms divided by the hexagonal area This holds for all chiral angles If a harmonic solution is assumed, (13) establishes a standard, general eigenvalue problem for frequency ω2 and its associated phonon mode shape, ˆ + KU ˆ = − ω MU (14) Results presented in this work are obtained using four-noded shell elements, which were chosen for their ease in generating structured meshes The Lagrange shape functions for this element are given as follows,       1 1 1 ς − ς − , N2 = − ς + ς − , N3 = ς + ς2 + , 4   1 N4 = − ς − ς + where the four nodes are located uniformly such that the element sides trace out curves of constant ξ and ξ Energy from each RAE is computed at the Gauss points of the element as follows, and Gauss quadrature is employed to compute the mass and stiffness matrices appearing in (13) The reference location of atom in the RAE is located at the Gauss point, and the three remaining atoms are located in the graphene sheet using the chiral angle φ and the undeformed bond length r0 The geometry of the RAE in the undeformed CNT is generated by a cylindrical parameterization, which defines the required relative position vectors between the ith and jth atoms Atomic energy is computed from bond lengths and angles as described in Appendix A Assembly of the global mass and stiffness matrix is accomplished using standard procedures N1 = 041702-7 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Node numbering and naming convention (left, internal, and right) for Bloch analysis procedure The parametric ¯ is shown discretized by 3n nodes (top left) Note that nodes 1, n, 2n, and n+1 are connected to compose an element; body  similarly for n+1, 2n, 3n, 2n+1 Also depicted (lower right) is a single element with Gauss point RAE’s F Bloch analysis procedure In order to predict dispersion, CNT unit cells are employed together with Bloch boundary ¯ discretized by 3n nodes conditions Fig depicts an example unit cell in the parametric body  and 2n elements Also depicted is an example element with RAEs located at the Gauss points, as described previously Note that when the same shell-like elements were used to compute phonon spectra,10 the number of unit cells covered by each element was large, and thus little error was introduced by assigning an RAE at the Gauss point In the present analysis approach, many elements are needed to cover a single unit cell, and thus locating an RAE at a Gauss point does not reflect the actual location of the RAE atoms This representation could be improved in future work, but as documented in the section on results, yields dispersion predictions with good accuracy at low frequency and wavenumbers Identified in Fig are ‘left’ nodes, ‘internal’ nodes, and ‘right nodes’ The right nodes are one translation vector T away from the left nodes Referring to the displacements (see (13)) owned by the left nodes as u L , by the internals nodes as u I , and by the right nodes as u R , the Bloch theorem11 relates the total set u = [u L u I u R ]T to a reduced degree of freedom set u˜ = [u L u I ]T by u˜ = Su, where ⎡ I ⎢ S=⎣ eiμ I ⎤ ⎥ I⎦ denotes the propagation matrix and μ denotes the propagation constant Note that at the outer edge of the first Brillouin zone, μ = π Although the parametric body lies on the plane, only the wavenumber associated with the T direction takes on continuous values – the wavenumber associated with the chiral direction C takes on multiples of 2π (or zero) Thus the propagation constant μ is associated with wavenumbers along T 041702-8 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Carbon nanotube unit cells Following introduction of the propagation matrix, a restated eigenvalue problem is established from (14) by introducing u˜ = Su and pre-multiplying by the Hermitian of S,   ˜ +K ˜ (μ) u˜ = ⇒ D (ω; μ) u˜ = 0, (15) −ω M ˜ = SH MS, K ˜ = SH KS, and D (ω; μ) denotes the dynamical matrix The eigenvalue problem where M (15), parameterized by the propagation constant μ, yields dispersion curves ω (μ) where the number ˜ of branches is determined by the size of u III DISPERSION RESULTS FOR EXAMPLE CARBON NANOTUBES In this section we present predicted phonon dispersion in example armchair (10,10), zigzag (10,0), and mixed CNTs composed of (8,0) and (7,1) heterojunctions.12 A single unit cell for each nanotube chirality is shown in Fig A (10,10) armchair nanotube The (10,10) armchair is one of the most commonly studied single-wall carbon nanotubes, and hence phonon dispersion results are presented in several works.13–15 This tube has a chiral angle of 30 degrees, a radius of 6.76 Angstroms, and a unit cell length of 2.46 Angstroms – these are the only input parameters necessary to specify the element properties in the continuum-atomistic tool, and to thus distinguish one nanotube from another Note that depending on the calculation approach (ab initio, tight binding, zone folding, lattice dynamics, etc.), the phonon dispersion calculations can show large differences However, for the low frequency acoustic branches, most methods are in close agreement The highly-cited work by Dresselhaus and Eklund,14 which employs fourthneighbor interaction terms and computes dispersion via lattice dynamics, is chosen for comparing the continuum-atomistic dispersion calculations presented in this section Fig provides three sets of dispersion curves computed using the continuum-atomistic approach (subfigures b, c and d) and a comparison set generated from data presented in Dresselhaus and Eklund (subfigure a) Three discretizations are explored in the continuum-atomistic approach: 10 nodes, 041702-9 Michael Joseph Leamy AIP Advances 1, 041702 (2011) 12 nodes, and 20 nodes per circumference All dispersion curves in the long wavelength limit (small μ) recover three acoustic branches and several optical branches The three acoustic branches include the longitudinal acoustic (LA), which exhibits the highest group velocity, or slope; the twisting acoustic (TW), which exhibits the next highest group velocity; and the transverse acoustic (TA), which yields bending solutions for the nanotube and has the slowest group velocity Note that the TA mode is a degenerate mode, with two overlapping branches present The optical branches combine longitudinal, twisting, and transverse motions to various degrees, and include atomic motions within the unit cell which may oppose each other The optical branches also typically come in degenerate pairs An optical mode of particular relevance is the radial breathing mode (RB), which appears in the Dresselhaus and Eklund dispersion curves at 156 cm-1 and μ = In the low frequency (up to approximately 200 cm-1 , or THz) long wavelength limit (see dashed boxes in Fig 6), acoustic dispersion branches show generally good agreement with those of Dresselhaus and Eklund Coarse discretizations – e.g., 10 nodes per circumference, or half the degrees of freedom present in the atomic system (see Fig 6(c)) – yield good acoustic comparisons to group velocities and overall branch evolution with increasing μ Some deviations are notable First, the TW mode starts above zero frequency due to the difficulty reduced-dimension curved elements can have recovering rigid body rotations16–18 – see Appendix B This behavior is corrected by finer meshes Second, branch veering is observed in the vicinity of 170 cm-1 when the LA branch starts to cross the TW mode Veering behavior is common in eigenvalue problems.19 Since at the point of veering the eigenvectors of the two branches swap, this discrepancy has little effect on the dispersion behavior since the branch now housing the LA mode bends sharply to the right, just as in the results of Dresselhaus and Eklund The optical branches of the 10 nodes per circumference discretization not show as good agreement as the acoustic, with the branches appearing at frequencies significantly shifted up from where they appear in Dresselhaus and Eklund’s results The finer discretizations, however, show good agreement with the optical results of Dresselhaus and Eklund First, a finer discretization (e.g., 12 nodes per circumference – see Fig 6(b)) allows the TW mode to start much closer to zero frequency Second, the starting frequencies of the optical branches decrease and show close agreement with those of Dresselhaus and Eklund However, as even finer discretizations are introduced (e.g., 20 nodes per circumference – see Fig 6(d)), spurious optical branches begin to pollute the spectrum and falsely increase the density of states Thus the desire to use fine meshes to achieve branch convergence must be balanced by the need to avoid spurious branches Based on the (10,10) results, a reasonable balance is achieved when the number of nodes per circumference is just over half that of the number of atoms per circumference in the atomic unit cell Note finally that the region of validity, as established by the dashed boxes in Fig 6, is up to a frequency close to 200 cm-1 (or THz) and halfway out to the edge of the Brillouin zone B (10,0) zigzag nanotube Dispersion behavior of a (10,0) zigzag nanotube is explored next using the continuum-atomistic tool This tube has a chiral angle of degrees, a radius of 3.91 Angstroms, and a unit cell length of 4.25 Angstroms The finite element input file used for this nanotube is identical to the one used for the (10,10) nanotube with the exception of specifying the three input parameters above This illustrates the high degree of model/element reuse possible when analyzing any chirality nanotube Dispersion results for this nanotube are given in Fig Note that for this case, the branch veering observed in the (10,10) case is absent In fact, if in the (10,10) analysis the chiral angle is set to zero, the veering also does not occur It is not readily apparent why veering is caused simply by the orientation of the RAE - this question may be answered in future work Other than the lack of veering, the (10,0) phonon dispersion curves resemble closely those predicted for the (10,10) with optical dispersion branches appearing at higher frequencies The bending of the LA branch also occurs at higher frequencies Both trends are consistent with other published works comparing (10,10) and (10,0) nanotubes.20 041702-10 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Computed dispersion curves for a (10,10) CNT: a) curves generated using data available in Dresselhaus and Eklund;14 b) best continuum model comparison using 12 nodes per circumference; c) continuum model with 10 nodes per circumference; d) continuum model with 20 nodes per circumference 041702-11 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Computed dispersion curves for a (10,0) CNT using the continuum model with 12 nodes per circumference C (8,0)/(7,1) mixed-chirality nanotube The final nanotube system explored is an assembly of nanotube heterojunctions composed of (8,0) and (7,1) tubes that repeat periodically, much like a bi-material system The heterojunction considered is the smallest topological defect with minimal local curvature and defect energy, and is composed of a single heptagon-pentagon pair.12 The (8,0) nanotube has a radius of 3.12 Angstroms, a chiral angle of degrees, and a unit cell length of 4.25 Angstroms, while the (7,1) nanotube has a radius of 2.95 Angstroms, a chiral angle of 6.59 degrees, and a unit cell length of 10.70 Angstroms The mixed system considered consists of one unit cell of each CNT type, for a periodically-repeating length of 14.95 Angstroms The continuum-atomistic tool is particularly convenient to use in this case since the heterojunction can be accomplished by easily assembling elements in a conventional manner, without the need to explicitly model the heptagon-pentagon defect For the acoustic and low-frequency optical branches, which are of interest herein, it is reasonable to expect the defect to not significantly alter the dispersion curves For higher frequency dispersion branches with smaller wavelengths, it would be necessary to employ a more scale-appropriate analysis tool, such as lattice dynamics Figure provides the computed dispersion curves for the (8,0), (7,1), and (8,0)/(7,1) mixedchirality nanotubes The (8,0) nanotube is a zigzag tube much like the (10,0), and thus the dispersion curves (see Fig 8(a)) are qualitatively similar to those of the (10,0) with the exception of the (8,0) nanotube exhibiting higher frequencies due to its smaller radius (and hence higher stiffness) The (7,1) nanotube has similar acoustic branches to the other tubes studied, but its optical branches differ significantly Evident in Fig 8(b) are optical branches which lose their degenerate character Moving away from the Brillouin zone edges, two distinct branches emerge – this is seen frequently in phonon dispersion analysis Note that the loss of degeneracy is not present in armchair and zig-zag tubes due to the axial symmetry of the RAE The mixed-chirality nanotube dispersion curves are presented in Fig 8(c) The first feature of the mixed-chirality dispersion curves observed is a less-pronounced loss of degeneracy in the optical branches The presence of the (8,0) tube mitigates the loss of degeneracy such that the gap between 041702-12 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Computed dispersion curves for a) (8,0) CNT, b) (7,1) CNT, and c) (8,0)/(7,1) tube two similar branches is decreased compared to the (7,1) dispersion curves A second feature of the dispersion curves is the presence of negative group velocity branches, which appear due to the well-known folding that occurs at the right edge of the Brillouin zone when mixing materials For example, three negative group velocity branches start at 200 cm-1 and proceed downwards, where one branch is associated with a folded TA branch and the others are associated with folded optical branches These negative group velocity branches suggest the (8,0)/(7,1) mixed-chirality nanotube, in a band around 200 cm-1 , could serve as a type of nanophononic metamaterial with a negative index of refraction IV CONCLUDING REMARKS A continuum-atomistic modeling approach for predicting phonon dispersion in single-walled carbon nanotubes has been detailed The approach employs reusable shell-like finite elements, which following development, lead to rapid phonon dispersion predictions requiring only a structured mesh and three chirality-specific inputs: nanotube radius, chiral angle, and unit cell length The approach has been used to generate dispersion curves for armchair, zigzag, and other nanotubes, to include a mixed-chirality (8,0)/(7,1) nanotube Dispersion results from the mixed-chirality tube reveal nanophononic metamaterial qualities in a band near 200 cm-1 For a (10,10) nanotube, the computed dispersion curves are compared to one set of dispersion results available in the literature While overall good agreement in branch numbers, type, and group velocity is documented below 200 cm-1 (i.e., THz), some small discrepancies are evident First, the continuum-atomistic results exhibit branch veering in tubes with nonzero chiral angle; however, at the veering point the branches swap eigenvectors (i.e., polarizations), and thus the veering does not alter the group velocities on either side of the veering point Second, the continuum-atomistic approach has difficulty locating the twisting acoustic branch at the origin due to its required use of a coarse number of curved elements Similar to veering, the branch behavior a short distance away from the origin corrects itself and yields an accurate group velocity In summary then, with proper attention to a few easily-recognized (and therefore rectified) deficiencies, the continuum-atomistic approach provides efficient and accurate predictions of phonon dispersion in the long wavelength limit Its reusable elements and rapid model specification should make it attractive for exploring nanophononic systems, such as mixed-chirality nanotubes composed of periodically-repeating heterojunctions, and for generating dispersion-based input information needed for follow-on studies of thermal transport and electron-phonon interactions 041702-13 Michael Joseph Leamy AIP Advances 1, 041702 (2011) ACKNOWLEDGMENTS The author would like to thank Dr Eduardo A Misawa and the National Science Foundation for partial support of this research under Grant No (CMMI 0926776) APPENDIX A: MODIFIED MORSE POTENTIAL The atomistic potential chosen for this study is a Modified Morse Potential [26] If the classical Morse Potential is to be used for modeling CNT’s, a three-body term accounting for angular position must be included in order to stabilize a tubular position As such, the modified potential then takes the form,   2 E = E str etch + E angle ; E str etch = De − e−β(r −r0 ) − ;   (A1) kθ (θ − θ0 )2 + ksextic (θ − θ0 )4 , where Estretch is the bond energy due to bond stretch, Eangle is the bond energy due to bond anglebending, r is the length of the bond, and θ is the current angle of the adjacent bond The parameters used in all simulations herein correspond to hybridized sp2 bonds and are given by: E angle = r0 = 1.39 × 10−10 m; De = 6.03105 × 10−19 Nm; β = 2.625 × 1010 m−1 ; θ0 = 2.094 rad; kθ = 0.9 × 10−18 Nm/rad2 ; ksextic = 0.754 rad−4 Performance of this potential for strains below 10% has been shown to compare very well to the more commonly accepted Brenner potential – the advantage of adopting the Modified Morse Potential is that the stretching and angular contributions are distinct, which is important when forming a reference area element The model herein requires the total energy in a reference area element (RAE) where E rae = rae rae The chosen RAE contains four atoms and therefore, E str etch + 2E angle    2   E rae = De − e−β(r1 j −r0 ) − + kθ (θ123 − θ0 )2 + ksextic (θ123 − θ0 )4 j=2,3,4     + kθ (θ124 − θ0 )2 + ksextic (θ124 − θ0 )4 + kθ (θ134 − θ0 )2 + ksextic (θ134 − θ0 )4 , (A2) where the bond lengths and bond angles are given by, r1i = r 1i = r i − r , θ1i j = cos  −1 r 1i · r j   r 1i  r j  (A3) APPENDIX B: CURVED ELEMENT RIGID BODY ROTATIONS This appendix briefly discusses the difficulty many curved elements have with rigid body rotations The issue is explored in Fig where a straight and a curved reduced-dimension element are depicted undergoing rigid body rotation These two elements are identical other than their curvatures In both, the displacements are horizontal (denoted by u) and vertical (denoted by v) The nodal displacements are stored by a vector U = [u1 v1 u2 v2 ]T For the straight element, the rigid body rotation vector is given by nodal displacements R = [0 -1 1]T , while for the curved element the rotation is R = [1 -1 1]T The interpolation scheme is assumed to be linear with respect to a natural coordinate measured along the arc length of the element For the straight element with R = [0 -1 1]T , interpolation yields center displacement components u C = 1/2 (u + u ) = and vC = 1/2 (v1 + v2 ) = These are consistent with the rotation depicted in Fig 9, and not incur element strain energy On the other hand, for the curved element 041702-14 Michael Joseph Leamy AIP Advances 1, 041702 (2011) FIG Comparison of rigid body rotations for straight and curved one-dimensional elements For rotation about the element center C, nodal displacements are perpendicular to lines connecting element center to node with R = [1 -1 1]T , interpolation yields center displacement components u C = 1/2 (u + u ) = and vC = 1/2 (v1 + v2 ) = 0, which are not consistent with the rotation depicted As a result, element stretching and contraction 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cylindrical carbon nanotubes using a manifold-based continuum- atomistic finite element formulation combined with Bloch analysis The formulated finite elements allow any... negative index of refraction IV CONCLUDING REMARKS A continuum- atomistic modeling approach for predicting phonon dispersion in single-walled carbon nanotubes has been detailed The approach employs