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[.]

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

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

geome-try generation and energy minimization tasks associated with ab initio and lattice

dynamics-based techniques A critical assessment of the technique is pursued to de-termine the validity range of the resulting dispersion calculations, and to identify any dispersion anomalies Two small anomalies in the dispersion curves are docu-mented, 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 veer-ing in nanotubes with nonzero chiral angle The twistveer-ing mode quickly restores its correct group velocity as wavenumber increases, while the branch veering is associ-ated 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 , 2The 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).6For 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 tech-niques, such as lattice dynamics,7 , 8or quantum mechanical techniques based on second quantiza-tion and creaquantiza-tion/annihilaquantiza-tion operators.9In 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 Author(s) 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 Fur-ther, 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 ap-proached used in the dispersion calculations Full details of the underlying model can be found

in,10which 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

Figure1illustrates 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 both its undeformed (reference)0⊂ R3and deformed ⊂ R3configurations Since the curvilinear CNT forms a two-manifold, only two coordinates of a pointξ =ξ1, ξ2

in a parametric body ¯

are required to locate0 and in R3using mappingsϕ0andϕ, respectively A third mapping 

takes position X on 0to position x on 

where U denotes the displacement field.

Due to the reduced dimensionality of the manifolds, basis vectors in the undeformed and

de-formed configuration lie not on the manifolds, but instead in their tangent spaces T X 0(undeformed)

and T x  (deformed) For particular locations, the tangent space at X is represented by T X 0while

that at x by T x  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 0referenced to the Euclidean unit vectors

(I 1 ,I 2 ,I 3 ), the (covariant) basis vectors for T X 0are denoted by G α,

where α = 1, 2 and repeated indices imply summation For a cylindrical nanotube, these basis

vectors are depicted in Fig.2 In general, the basis vectors G αare not mutually orthogonal nor are

FIG 1 Kinematics and mappings defining manifold configurations.

FIG 2 Cylindrical coordinate system.

of unit length Orthogonality is accomplished through introduction of a dual (contravariant) basis

G α (spanning the cotangent space T X∗0) such that G α (G β)≡ G α · G β = δ α

β whereδ α

βdenotes the

Kronecker delta (one ifα =β, zero otherwise) Similar to the above development, a basis for T x 

is given by

∂ξ α = ∂z a

where (i 1 ,i 2 ,i 3) 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,

∂ξ α = G α+∂U β

∂ξ α G β + U β ∂ G β

where U β is a displacement component relative to G βsuch that the physical component of

displace-ment in the G β direction is given by U βG β, no sum onβ intended.

B Deformation

The deformation gradient F gives rise to a stress tensor P (the 1stPiola 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≡ ∂x

∂ X ∈ 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

tangent spaces T X 0and T x  and not in the configurations 0and The deformation gradient can

be expressed in terms of displacement components U α referenced to G α- quantities which reference solely the undeformed configuration0,

F≡ ∂ X ∂x =



δ η β+∂U ∂ξ β η + U ρ η ρβ



where ρβ η 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,

δE tot = δ



0

W0(F)d 0+ δE ext =



0

∂W0

∂F :δFd0+ δE ext = 0, ∂W0

∂W0

∂F α β

G α ⊗ G β ,

(5)

where W0is the strain energy per unit undeformed area, andδE extholds external virtual work terms

In the d’Alembert sense, the external virtual workδE extterm includes virtual work done by surface and inertial forces,

δE ext= −



0

δU · t P

0d 0+



0

where t0Pdenotes the first-order surface traction andρ0denotes 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., t0P = 0 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 (Pβ α= ∂W0

∂F α) For two-manifolds, the absence of thickness results in stress tensors with units of force per unit length With the above considerations, the virtual work principle simplifies to



0

P :δFd0+



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 sp 2bonds 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 a1and a2need be considered, where each has an undeformed

length l0equal 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

√

n2+ nm + m2, 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 angleφ; C: (n,n)] and the zig-zag

tubes [zero degree chiral angleφ; 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 W , the continuum

FIG 3 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≡ E r ae

A H ex

r ae str etch + 2E r ae

angle

A H ex

where E r ae

str etch is calculated from the Modified Morse Potential (see App A) summing the stretch

energy from the three RAE bond lengths, E r ae

angle is calculated summing the angle-bending energy

using the three RAE bond angles, and A H exrepresents 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 r aerequires the positions of individual atoms in Letting R i jrepresent a position vector in0 of atom j relative to atom i, the position vector of atom j relative to atom i in  is

given by,

Note then that(4), the definition P=∂W0

∂F, and (7-9) yield a completed displacement-based weak

form referencing only quantities in the undeformed configuration0 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 interpo-lations In order to avoid difficulties capturing rigid body modes (not considered phonons), the Euclidean displacement components are first introduced Introducing a finite element

discretiza-tion in the usual sense, the field displacement vector U can be discretized using shape funcdiscretiza-tions

N I

ξ1, ξ2

which interpolate the Euclidean displacement terms ˆU i appearing in U β,

ξ1, ξ2 ˆ

such that ¯t i βis a conversion tensor from curvilinear to Euclidean bases, ˆU i

I denotes the ithEuclidean

component of the displacement at node I, I ranging over the number of nodes n, and the shape functions N I

ξ1, ξ2

are chosen such that they evaluate to 1 at their home location

ξ1

I , ξ2

I

 and 0 at other nodal locations

ξ1

J , ξ2

J



, J = I Substitution of(10)into(7)yields the stiffness matrix,

KσρIJ =



0



∂ P β α

∂ ˆU J ρ

∂F α β

∂ ˆU σ I

and a mass matrix,

MσρIJ =



0

ρ0N I N J ¯t i

σ ¯t j

such that the equations of motion are expressed as,

MσρIJU¨ˆρ J+ KσρIJUˆρ

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 0is 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 frequencyω2and its associated phonon mode shape,

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,

N1= 1

4



ς1− 1 ς2− 1, N2 = −1

4



ς1+ 1 ς2− 1, N3= 1

4



ς1+ 1 ς2+ 1,

N4= −1

4



ς1− 1 ς2+ 1 where the four nodes are located uniformly such that the element sides trace out curves of constant

ξ 1andξ 2 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 1 in the RAE is located at the Gauss point, and the three remaining atoms are located in the graphene sheet using the chiral angleφ and the undeformed bond length r 0 The geometry of the RAE in the undeformed CNT is generated by a cylindrical parameterization,

which defines the required relative position vectors between the ithand jthatoms Atomic energy is computed from bond lengths and angles as described in AppendixA Assembly of the global mass and stiffness matrix is accomplished using standard procedures

FIG 4 Node numbering and naming convention (left, internal, and right) for Bloch analysis procedure The parametric body ¯ is shown discretized by 3n nodes (top left) Note that nodes 1, n, 2n, and n+1 are connected to compose an element;

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 conditions Fig.4depicts an example unit cell in the parametric body ¯ discretized by 3n nodes 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.4are ‘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 uL, by the internals nodes as uI, and by the right nodes as uR, the Bloch theorem11

relates the total set u = [uLuI uR]Tto a reduced degree of freedom set ˜u = [uLuI]Tby ˜u = Su,

where

S=

⎡

⎢

⎣

e i μI 0

⎤

⎥

⎦

denotes the propagation matrix andμ 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.

FIG 5 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,



where ˜ M = SHMS, ˜ K = SHKS, and D (ω; μ) denotes the dynamical matrix The eigenvalue problem

(15), parameterized by the propagation constantμ, 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.12A single unit cell for each nanotube chirality is shown in Fig.5

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.13–15This 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 fourth-neighbor interaction terms and computes dispersion via lattice dynamics, is chosen for comparing the continuum-atomistic dispersion calculations presented in this section

Fig.6provides 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,

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 compar-isons 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 behav-ior 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 discrep-ancy 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 do 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, do show good agreement with the optical results of Dres-selhaus 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 How-ever, 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 6 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 0 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 7 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