Symmetry-Adapted Phonon Analysis of Nanotubes (to appear in J. Mech. Phys. Solids)

Phonons, i.e. normal modes, are extremely important to understand the properties of crystals. For instance, phonon analysis provides insight into thermodynamic properties and mechanical stability [Dov93, BH98, ETS06, EST06]. In this paper, we use the framework of Objective Structures (OS) introduced by James [Jam06] to extend the notion of phonon analysis to noncrystalline but symmetric nanostructures.

OS generalizes the notion of a crystal or periodicity by using ideas from frame-indifference. In brief, [Jam06] and following works have shown that many highly symmetric but non-crystalline nanostructures have close analogies to crystals. These analogies have led to some important practical methods: e.g., a generalization of periodic boundary conditions for classical molecular dynamics and tight binding to enable the analysis of chiral nanostructures as well as the ability to apply torsional loads [ZAD11, ZJD09b, ZJD09a, ZDS10, DJ07, NZJD10, DJ10, DJ11, AD12, AD11]. All of these exploit the fact that the symmetries of the nanostructure, together with frame-indifference, imply that the first derivative of the potential energy has certain symmetries. This symmetry in the first derivative is a generalization of the fact that forces on image atoms are identical in a periodic crystal.

In this paper, we exploit in an essential way the symmetry in the second derivative of the potential energy, i.e. the Hessian matrix has various submatrices that are related to each other. While this was noted by James [Jam06], it has not been exploited in practical calculations. Here, we find that this property implies that a preliminary linear transformation renders the Hessian matrix block-circulant as in periodic crystals, thus enabling the use of standard Fourier techniques after the preliminary transformation. While a significant part of our analysis is general and applies broadly to all structures that belong to the family of OS, we also specialize the analysis to carbon nanotubes and other one-dimensional systems to do numerical calculations. Except where stated as a model system, we use the well-characterized Tersoff interatomic potential for carbon that provides a balance between bond-order accuracy and computational efficiency [Ter88].

We emphasize that many researchers have studied phonons in carbon nanotubes for over a decade now. For instance, an early example is [YKV95]; a more recent review is [JDD08]. An important feature of all of these studies is that they directly use methods from periodic systems. This often requires the use of very large unit cells that require expensive calculations to compute the phonon analysis, and more importantly the computed information is extremely complex and difficult to analyze for new physics. Recent papers that exploit the symmetry of nanotubes are [GLW08, PL06].

We go beyond these methods in some significant ways. First, our approach based on OS provides a tight link to deformation of the nanotubes. This is critical to go beyond exclusively axial-load-free and twisting-moment-free nanotubes; in fact, as recent work shows, the load- and moment- free structure of chiral nanotubes likely does not correspond to the assumed highly-symmetric configuration [VBSF⁺10, ZAD11, AD12]. Relaxing or applying such loads is readily accomplished using the OS framework [Jam06, DJ07].

Second, our analysis exposes the close analogies to periodic crystals; additionally, it is general and applicable to a wide variety of OS that go beyond carbon nanotubes. In addition to providing a conceptual unity, this can potentially enable important practical advances such as integrating our method with other techniques developed for periodic crystals. For instance, we provide a demonstration of phonon soft-mode stability analysis to detect instabilities under torsion. Such analyses, in combination with OS methods for bending and other deformations, can conceivably provide insight into complex phenomena such as nanotube rippling [AA08]. Further, multiscale atomistic-to-continuum numerical methods have recently been used to predict structural phase transformations in crystals using phonon stability as a critical component [DELT07, Ell07]; conceivably, similar methods for complex OS nanostructures can be built on the phonon approach provided here. Another potentially important application of our phonon approach is, e.g., as a basis to construct effective Hamiltonian models that have been effective in predicting structural phase transformations in crystals based on soft modes.

The paper is organized as follows:

• •

  Section 2 describes the notation used in the paper.

• •

  Section 3 outlines the relevant aspects of OS, and the properties of the Hessian matrix in an OS. In crystals, each row of the Hessian is simply a shifted copy of the previous row, i.e. it is block-circulant; in OS, each row of the Hessian is related to the previous row but in a more complex way that involves the symmetry parameters of the structure.

• •

  Section 4 sets up the linearized equations of motion, presents the transformation to the intermediate Objective space as a similarity transform, presents the standard Discrete Fourier Transform as a similarity transform, and uses a composition of these transforms to block-diagonalize the Hessian matrix.

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  Section 5 shows examples of phonon spectra computed by the OS framework and contrasts these with standard phonon spectra. The OS framework provides important conceptual simplifications in understanding these curves. We also demonstrate the computation of the density of states.

• •

  Section 6 examines modes that are long-wavelength (in the Fourier space), and rigid-body and uniform deformation modes in real space. Unlike crystals, rigid-body modes in nanotubes are not always at long-wavelength, and long-wavelength non-optic modes do not always correspond to zero energy even in the limit.

• •

  Section 7 examines the stability information provided by phonons, in particular contrasting nanotubes with crystals in regards to analogies to “non-rank-one modes” in crystals that are not tested by phonon stability. We also demonstrate a numerical example of torsional buckling and examine the predictions of phonon soft-mode analysis.

• •

  Section 8 studies energy transport in nanotubes. We find characteristic oscillations in the phonon spectra that motivate a simple one-dimensional geometric nonlocal model of energy transport. The model provides insight into the balance between energy transport along axial and helical directions. The geometrically-motivated nature of the model gives it universal applicability to helical structures of all kinds.

$\mathbb{Z}$ denotes the set of all integers and $\mathbb{Z}^{3}$ the set of triples of all integers.

For a quantity ${\mathbf{A}}$, the Fourier transform is denoted by $\tilde{{\mathbf{A}}}$.

To avoid ambiguity, the summation convention is not used and sums are always indicated explicitly.

Throughout the paper, $M$ and $N$ are used for the number of atoms per unit cell and the number of unit cells in the OS respectively. Note that $M$ is always finite but $N$ can be infinite. The unit cells are labeled by multi-indices denoted by boldface, i.e. ${\mathbf{i}}=(i_{1},i_{2},i_{3})$, and atoms within a given unit cell are labeled by regular non-bold indices. For example, the position of the $k$ atom in the ${\mathbf{i}}$ unit cell is denoted ${\mathbf{x}}_{{\mathbf{i}},k}$.

Bold lower case and upper case letters represent vectors and matrices, respectively. The rectangular Cartesian component and the exponent of a vector or matrix are shown respectively by Greek letters and lower case Latin letters as superscripts.

The subscripts of vectors and matrices are used to convey information about the structure of the quantity in addition to denoting components. We will often deal with matrices of size $3MN\times 3MN$ corresponding to an OS with $3MN$ degrees of freedom (DoFs). Such a matrix ${\mathbf{A}}$ can be divided into $N\times N$ blocks, with each block further sub-divided into $M\times M$ sub-blocks of size $3\times 3$. Then, ${\mathbf{A}}_{({\mathbf{i}},k)({\mathbf{j}},l)}$ denotes a $3\times 3$ sub-block, typically corresponding to the pair of atoms labeled by $({\mathbf{i}},k)$ and $({\mathbf{j}},l)$. Also, ${\mathbf{A}}_{{\mathbf{i}}{\mathbf{j}}}$ denotes a $3M\times 3M$ matrix, typically corresponding to atoms in the unit cells labeled ${\mathbf{i}}$ and ${\mathbf{j}}$.

Similarly, for a vector ${\mathbf{b}}$ of size $3MN\times 1$, writing ${\mathbf{b}}_{\mathbf{i}}$ denotes the ${\mathbf{i}}$-th block of ${\mathbf{b}}$ of size $3M\times 1$ typically corresponding to the unit cell labeled by ${\mathbf{i}}$. Further, ${\mathbf{b}}_{{\mathbf{i}},k}$ denotes a sub-block of ${\mathbf{b}}_{\mathbf{i}}$ of size $3\times 1$, typically corresponding to the atom $k$ in the unit cell labeled by ${\mathbf{i}}$.

For Fourier quantities, the correspondences for sub-blocks in vectors and matrices are not to unit cells but rather to wave numbers.

James [Jam06] defined an objective atomic structure as a finite or infinite set of atoms in which every atom sees the same environment up to translation and rotation. Similarly, an objective molecular structure is defined as a structure with a number of identical molecules, each molecule consisting of a number of atoms, arranged such that corresponding atoms in every molecule see the same environment up to translation and rotation. We note that the molecules in an objective molecular structure need not correspond to standard physical molecules as usually understood. Bravais (multi) lattices are special cases of objective atomic (molecular) structures in which each atom (molecule) has the same environment up to translation and the rotation is trivial.

Following recent works that build on James’ original formulation, e.g. [DEJ, DJ11, DJ10, AD11, AD12], we can define OS equivalently in the language of group theory. The group theoretic approach enables practical calculations. Let $G=\{g_{\bf 0},g_{\bf 1},\cdots,g_{{\mathbf{N}}}\}$ be a set of isometries indexed by a multi-index. Each element of $G$ has the form $g_{{\mathbf{j}}}=({\mathbf{Q}}_{{\mathbf{j}}}|{\mathbf{c}}_{{\mathbf{j}}})$ where ${\mathbf{Q}}_{{\mathbf{j}}}\in O(3)$ is orthogonal and ${\mathbf{c}}_{{\mathbf{j}}}\in\mathbb{R}^{3}$ is a vector.

The action of an isometry on a point ${\mathbf{x}}\in\mathbb{R}^{3}$ is

Composition of mappings then provides:

This motivates a definition for multiplication of isometries:

From this definition, it follows that the identity element is $g_{\bf 0}:=({\mathbf{I}}|\mathbf{0})$ and the inverse of $g_{\mathbf{i}}$ is defined by $g_{\mathbf{i}}^{-1}:=({\mathbf{Q}}_{\mathbf{i}}^{T}|-{\mathbf{Q}}_{\mathbf{i}}^{T}{\mathbf{c}}_{\mathbf{i}})$.

If the set $G$ is additionally a group with respect to the multiplication operation above, then placing an atom at each of the points given by the action of elements of $G$ on a given point ${\mathbf{x}}_{0}$ gives an objective atomic structure. In addition, placing an atom of species $k$ at each of the points given by the action of elements of $G$ on a given set of points ${\mathbf{x}}_{0,k}$ gives an objective molecular structure.

In this paper, we will consider OS described by groups of the form

Here, $g_{1}=(\bar{\mathbf{Q}}_{1}|\bar{\mathbf{c}}_{1})$, $g_{2}=(\bar{\mathbf{Q}}_{2}|\bar{\mathbf{c}}_{2})$ and $g_{3}=(\bar{\mathbf{Q}}_{3}|\bar{\mathbf{c}}_{3})$ are the generators of the group. We assume that they commute, i.e. $g_{1}g_{2}=g_{2}g_{1}$ and so on. This immediately implies that $G$ itself is Abelian. As shown in [DEJ], $G$ of this form does not describe all possible OS; however, as also shown there, those OS that cannot directly be described can nevertheless be described by such a $G$ by enlarging the unit cell and neglecting certain intra-unit cell symmetries.

Denoting the atomic positions in the unit cell by ${\mathbf{x}}_{(0,0,0),k}:=\mathbf{x}_{\mathbf{0},k},k=1,\ldots,M$, the OS is described by

Using (3.2), we have that

for positive exponents $i_{1},i_{2},i_{3}$. Negative exponents are defined through the inverse.

An important property of OS is that elements of $G$ map images of the unit cell $\mathbf{x}_{\mathbf{0},k}$ to each other. For instance, consider the ${\mathbf{i}}$ and ${\mathbf{j}}$ images of the unit cell:

These relations follow directly from the closure and commuting properties of the Abelian group $G$.

Consider the orthogonal part of $g_{1}^{i_{1}-j_{1}}g_{2}^{i_{2}-j_{2}}g_{3}^{i_{3}-j_{3}}$:

This provides the important relation:

We first derive the standard linearized equations of motion in an OS. We then use the properties of the Hessian from the previous section to achieve a block-diagonalization of the Hessian such that each block is of size $3M\times 3M$. Each block is related to the frequency in Fourier space, but the transformation is not directly from real to Fourier space but goes through an intermediate linear transform.

While the OS framework can be used for nanotubes with any chirality, in this section we focus on $(6,6)$ carbon nanotubes to illustrate some typical features of the phonon curves. This particular chirality also has a small translational unit cell that enables comparisons with standard periodic calculations.

We compare the effect of using four different unit cells.

Choice 1:

We use a periodic unit cell with 24 atoms, the smallest number required for periodicity. In OS terms, the group generators are $g_{1}=\text{identity}$ and a translation $g_{2}=({\mathbf{I}}|0.246\text{nm }{\mathbf{e}})$. The phonon dispersion curves are plotted in Fig. 1a.

Choice 2:

We use 24 atoms in the unit cell as in Choice 1, but in this case the images are related not by periodicity but by both translation (along ${\mathbf{e}}$) and rotation (around ${\mathbf{e}}$). The generators are $g_{1}=\text{identity}$ and a screw $g_{2}=({\mathbf{R}}_{\pi/3}|0.246\text{nm }{\mathbf{e}})$. The phonon dispersion curves are plotted in Fig. 1b.

Choice 3:

We use 12 atoms in the unit cell, with generators closely related to Choice 2: $g_{1}=\text{identity}$ and a screw $g_{2}=({\mathbf{R}}_{\pi/6}|0.123\text{nm }{\mathbf{e}})$ The phonon dispersion curves are plotted in Fig. 2.

Choice 4:

We make full use of the OS framework and use 2 atoms in the unit cell⁴⁴4OS constructed by non-commuting groups can describe carbon nanotubes with 1 atom per unit cell, but the complexity introduced by the non-commuting elements is formidable.. The generators are $g_{1}=({\mathbf{R}}_{\pi/3}|\mathbf{0})$ and $g_{2}=({\mathbf{R}}_{\pi/6}|0.123\text{nm }{\mathbf{e}})$. In Choices 1, 2, 3, the wavevector was one-dimensional because the structure was indexed by a single index (not a triple-valued multi-index). In this case, the wavevector is two dimensional, but takes only $6$ discrete values in the direction corresponding to the rotation. This is because once we raise the rotation to the $6$-th power, we start over; conceptually, this is similar to a finite ring of atoms that has a finite set of normal modes. The phonon dispersion curves are plotted in Fig. 3.

Comparing the plots in Fig. 1 shows, as expected, that there is a mapping between the plots obtained from Choices 1 and 2. Any eigenvalue in one is also present in the other, though typically at a different wavevector. Further, comparing Figs. 1b and 2, shows that if we unfold the curves of Fig 1b we will recover Fig. 2. Similarly, Fig. 3 contains all the information, but in a much simpler description.

The OS description with 2 atoms per unit cell provides a useful perspective to examine the deformations. First, consider the case when the component of the wavevector in the discrete direction is $0$. Each unit cell in the cross-section has the same displacement (in Objective Space) in the direction that corresponds to the discrete component of the wavevectors. Roughly, this corresponds to “cross-sections” that retain their “shape” and remain circular. The lowest three modes corresponding to $k_{2}=0$ and $k_{1}\approx 0$ are plotted in Fig. 4.

Next, consider the case when $k_{2}$, the component of the wavevector in the discrete direction, is non-zero. In this case, the cross-sections no longer remain circular. Fig. 5 shows examples of these modes. Notice the relation between $k_{2}$ and the symmetry of the cross-section.

Fig. 6 shows an assortment of generic phonon modes at finite wavevectors.

In a 3D periodic crystal lattice, the lowest three eigenvalue branches tend linearly to $0$ as ${\mathbf{k}}\rightarrow{\bf 0}$. These acoustic modes correspond to uniform deformations with rigid body translation modes as the limit deformation. We find unusual contrasts with the crystal case when we apply this to carbon nanotubes. We find rigid body (zero energy) motions at both zero and finite wave vectors; in addition, we find that the long-wavelength deformation corresponding to uniform radial expansion costs finite energy in real-space even in the limit of ${\mathbf{k}}\rightarrow{\bf 0}$, thereby giving only two eigenvalue branches that tend to $0$. The essential explanation for these observations is that long-wavelength is now defined with respect to objective space and not real space, whereas rigid-body modes are posed in real space for physical reasons.

We first outline this issue using as an example the choices of unit cell from Section 5.

In Choice 1, four branches start from the origin, Fig. 1a. These correspond to (i) axial stretch / translation with uniform motion along ${\mathbf{e}}$, (ii) twist / rotation with uniformly tangential motion, and (iii) bending / translation in the plane normal to ${\mathbf{e}}$. The bending mode is characterized by two degenerate branches with zero slope. The degeneracy is due the subspace of translations in the plane being two-dimensional [ADE].

In Choice 2, Fig. 1b, only two branches start from the origin. One corresponds to axial stretch / elongation, and the other one corresponds to twist / rotation. There is a branch that has zero frequency at $kL_{0}/\pi=1/3$. This corresponds to the rigid translation modes in the plane normal to ${\mathbf{e}}$, as we examine below.

Choice 3, Fig. 2, is very similar to Choice 3, except that the branch with zero frequency at finite wavevector now goes to zero at $kL_{0}/\pi=1/6$. We recall that that $L_{0}$ differs in Choices 2 and 3 precisely by a factor of 2, therefore this shift is simply because of unfolding the band diagram.

Choice 4, Fig. 3, shows the rigid translation modes at $k_{2}P/\pi=1/3$. In addition, for $k_{2}P/\pi=0$, we see only two branches that go to $0$ as the wavevector tends to zero; the lowest non-zero branch corresponds to uniform radial motion of the atoms. Because every unit cell has precisely the same deformation (in Objective space), this appears at ${\mathbf{k}}\rightarrow{\bf 0}$. In addition, because the atoms within the unit cell do not move relative to each other, this corresponds to the acoustic modes that are zero energy at zero wavevector in crystals. The three higher branches at zero wavevector have the atoms in the unit cell moving with respect to each other, i.e. optic modes, and these are expected to have finite energy at zero wavevector.

Phonon analysis provides important insights into the stability of crystals through identifying soft modes, i.e., non-rigid deformations that cost no energy [Dov93]. In addition, the phonon framework provides important insights and enables systematic identification of the appropriate larger unit cells at instabilities [ETS06, EST06]. As discussed in [ETS06], phonon analysis does not provide information about stability with respect to certain deformation modes; in particular, phonon stability does not test against non-rank-one modes. The analogy in linear continuum elasticity is that strong ellipticity tests only that waves speeds are real in all directions and all polarizations. In terms of the stiffness tensor, this does not test positive-definiteness of the stiffness against all tensors in the 6-dimensional strain space; rather it tests only against the subspace of strains that are symmetrized rank-one tensors. While phonons do test if solids are stable to uniform uniaxial extensions in every direction, they do not test if they are stable to superpositions of these, such as biaxial and triaxial stretch. Because the Fourier transform does not exist for the limit deformation, superposing modes and taking the limit is not equivalent to taking the limit and then superposing.

We also note that phonons test only the material stability but not against structural instabilities such as buckling [GMT93]. Structural instabilities are typically very sensitive to boundary conditions, e.g. the elementary Euler buckling loads. Testing the linear stability of an atomic structure against structural modes requires, in general, the brute-force solution of the full eigenproblem with a very large number of degrees of freedom.

In this section, we discuss two findings relevant to the role of phonons and stability. First, we discuss why there do not exist the analog of non-rank one modes in carbon nanotubes. That is, assuming that all phonon branches are positive, and in addition those branches that tend to $0$ at long wavelength have positive slope (in the case of twisting and axial extension) or have positive second derivative (in the case of bending), then the nanotube is stable under any combination of these. In other words, a positive torsional modulus, extensional modulus, and bending modulus, do imply, unlike crystals, that they are stable under any combination of torsion-extension-bending. The second finding that we discuss is a numerical study of torsional buckling using two unit cells, one with $456$ atoms and another with $24$ atoms. As the former choice has much more freedom in deforming, we see torsional instabilities. With the latter choice, we find a signature of this instability in terms of zero phonon frequencies; in addition the eigenmode corresponding to the zero frequency predicts the nature of the instability. This also displays an important calculation that is enabled by the OS framework: torsion is simply not possible with periodic boundary conditions.

Motivated by the features of the computed phonon curves in nanotubes, in this section we present a simplified geometric model that aims to capture the key physics of energy transport. The model is based on a balance between energy transport along a helical path and energy transport along an axial path. In an “unwrapped” helix, the former corresponds to transport through short-range interactions, i.e. the interactions are between neighbors that are nearby in terms of the labeling index, and the latter corresponds to long-range interactions, i.e. the interactions are between distant atoms in terms of the labeling index. Of course, in physical space, both these types of neighbors are at comparable distances, and interactions are therefore of comparable strength.

We begin by examining the phonon curves of nanotubes $(m,n)$ where $m$ and $n$ are relatively prime. As noted in Appendix A, this implies that a single screw generator is sufficient to describe the nanotube with 2 atoms per unit cell. Figs. 13 and 14a show the dispersion curves of unloaded (11,9) and (7,6) nanotubes respectively. The rotation angles of the screw generator are $\theta_{1}=\frac{271\pi}{301}\approx 0.9003\pi$ and $\theta_{1}=\frac{39\pi}{127}\approx 0.307\pi$, respectively. As discussed in previous sections, two branches corresponding to torsion and axial elongation start from the origin, and one branch touches the $k$ axis at precisely $\theta_{1}$. We mention that if we had used the periodic description for these nanotubes, we would require at least $1204$ and $508$ atoms in the unit cell for the $(11,9)$ and $(7,6)$ nanotubes respectively. Besides the significantly larger computational expense, it would imply that Figs. 13 and 14a contain $3612$ and $1524$ curves respectively! Physical interpretation would be impossible.

The key features of interest here are the “wiggles” in the phonon curves in Figs. 13 and 14a. There exist certain distinguished wavevectors at which the group velocity (i.e. slope of the dispersion curve) becomes zero in all branches. These wavevectors are primarily selected by geometry: Fig. 14 compares the curves for a $(7,6)$ nanotube both with no load as well as with compressive axial force and nonzero twisting moment. In addition, the phonon curves depend on the specific interatomic potential, but we have found that the distinguished wavevectors have a very weak dependence. Similar wiggles, though not as prominent are also visible in Fig. 3 for a $(6,6)$ nanotube. Since the group velocity gives the speed of energy transport, there is no energy transport at these distinguished wavevectors. These observations motivate a geometric model for the energy transport that neglects much of the complexity of the interatomic potential.