Carbon Nanostructures as an Electromechanical Bicontinuum

TL;DR

This paper introduces a comprehensive two-field model that unifies various physical phenomena in graphene and carbon nanotubes, including elasticity, lattice dynamics, and electromechanical effects, providing a broad theoretical framework.

Contribution

It presents a novel two-field model that unifies multiple aspects of carbon nanostructures, extending previous analytical and computational results into a single cohesive framework.

Findings

Describes optical phonons and nontrivial acoustic branches.

Explains strain-induced gap opening and phonon softening.

Accounts for doping-induced deformations and the hexagonal Brillouin zone.

Abstract

A two-field model provides an unifying framework for elasticity, lattice dynamics and electromechanical coupling in graphene and carbon nanotubes, describes optical phonons, nontrivial acoustic branches, strain-induced gap opening, gap-induced phonon softening, doping-induced deformations, and even the hexagonal graphenic Brillouin zone, and thus explains and extends a previously disparate accumulation of analytical and computational results.

Equations

$$V[u,v]=d[u]+d[v]+c[u,v].$$

$$d[u]={\mu }^{\prime }{u}^{ij}{u}_{ij}+\frac{{\lambda }^{\prime }}{2}{u}_i^i{u}_j^j.$$

$$c[u,v]$$

$$e_{ijk}=\frac{4}{3}\sum _{l=1}^3{\hat{e}}_i^{(l)}{\hat{e}}_j^{(l)}{\hat{e}}_k^{(l)}.$$

$$\left\{\begin{array}[]{l}\!\ \ddot{u}^{i}=\partial_{j}\sigma^{ij}_{\left(u\right)}-2\alpha\left(u^{i}-v^{i}\right)+\beta\!\ e_{lm}^{~{}~{}\!\ i}\!\ \left(v^{lm}+u^{lm}\right)\\ \!\ \ddot{v}^{i}=\partial_{j}\sigma^{ij}_{\left(v\right)}+2\alpha\left(u^{i}-v^{i}\right)-\beta\!\ e_{lm}^{~{}~{}\!\ i}\!\ \left(v^{lm}+u^{lm}\right)\end{array}\right.$$

$$\left\{\begin{array}[]{ll}\sigma^{ij}_{\left(u\right)}&=2\mu^{\prime}\!\ u^{ij}+\lambda^{\prime}\!\ \delta^{ij}u^{k}_{k}+2\mu\!\ v^{ij}+\lambda\!\ \delta^{ij}v^{k}_{k}\\ {}&-\beta\!\ e^{ij}_{~{}~{}k}\!\ \left(u^{k}-v^{k}\right)\\ \sigma^{ij}_{\left(v\right)}&=2\mu^{\prime}\!\ v^{ij}+\lambda^{\prime}\!\ \delta^{ij}v^{k}_{k}+2\mu\!\ u^{ij}+\lambda\!\ \delta^{ij}u^{k}_{k}\\ {}&-\beta\!\ e^{ij}_{~{}~{}k}\!\ \left(u^{k}-v^{k}\right)\end{array}\right.$$

$$2q^i=\ell e_{lm}^i{u}^{lm}=\ell e_{lm}^i{v}^{lm},$$

$$V_u[u,q]$$

$$V_{\perp }$$

$$\left\{\begin{array}[]{l}q^{\zeta}\left(\omega^{2}-4\alpha\right)+2\frac{\beta}{r}p^{r}=0\\ p^{r}\left(\omega^{2}-\frac{v_{L}^{2}+\beta^{2}/\alpha}{r^{2}}\right)+2\frac{\beta}{r}q^{\zeta}=0\\ q^{\xi}\left(\omega^{2}-4\alpha\right)=0\\ q^{r}\left(\omega^{2}-4\alpha_{\perp}+\frac{2\mu-2\mu^{\prime}+\lambda-\lambda^{\prime}}{r^{2}}\right)=0\\ \end{array}\right.~{}.$$

$$\mathrm{d}{t}^{(l)}=-\tau {\hat{e}}_i^{(l)}{\hat{e}}_j^{(l)}{u}^{ij}+\tau {\hat{e}}_i^{(l)}{q}^i/e$$

$${ϵ}_c(k{)}^2-{ϵ}_v(k{)}^2=\sum _l{t}^{(l)}+2\sum _{m>l}{t}^{(l)}{t}^{(m)}\cos (k\cdot a^{(n)}),$$

$$\frac{{\Delta }^2}{{\left(3\tau \right)}^2}$$

$$\Delta =3\tau \left|q^z/e+{\gamma }^{\prime }\cos (3{\theta }_c)+{\eta }^{\prime }\sin (3{\theta }_c)\right|,$$

$$\left\{\begin{array}[]{l}\mathrm{d}L/L=\frac{\rho_{e}\tau}{8m_{C}v_{T}^{2}}\left[\pm\left(1-\frac{\ell}{e}\right)+\frac{3\tau_{1}}{2\tau}\frac{2\mu_{R}+\lambda_{R}}{\mu_{R}+\lambda_{R}}\right]\\ \mathrm{d}b_{ax}=\pm\frac{\rho_{e}\tau}{2m_{C}\omega_{opt}^{2}e}\\ \end{array}\right.~{}.$$

Full text

Vibrations in carbon nanostructures such as tubes, fullerenes, or graphene sheets ji ; imperial ; Dresselhaus have a ubiquitous influence on electronic, optical and thermal response: scattering from optical phonons limits charge transport in otherwise ballistic nanotube conductors Tans ; Kane ; twist deformations gap metallic tubes Rochefort ; Yang ; ballistic phonons transport heat in nanotubes with great efficiency Berber ; Chiu ; Kim ; resonant Raman spectroscopy can unambiguously identify a tube’s wrapping indices (n,m) Richter ; Saito ; Jorio ; Jorio2 ; electron-phonon interactions may ultimately limit the electrical performance of graphene Novoselov ; Zhou . Computationally intensive atomistic models of lattice dynamics often lack simplified model descriptions that can facilitate insight, yet traditional analytical continuum models ji ; Mahan1 ; Suzu ; imperial , while very useful and important, cannot describe atomistic phenomena without phenomenological extensions Comas ; Chico ; Gartstein . Although continuum models are restricted to long-wavelength physics, they have been used to describe atomic-scale phenomena in bulk binary compounds by incorporating a separate continuum field for each sublattice mindlin : in graphene, two fields are necessary. Here we present an analytical “bicontinuum” model that represents the full atomistic detail of the graphenic lattice, including optical modes, nonlinear dispersion of in-plane phonons, electromechanical effects and even the hexagonal graphenic Brillouin zone, a construct generally held to be exclusively atomistic.

Graphene decomposes into the two triangular sublattices of Fig. 1. We describe in-plane deformations of the sublattices via two fields, $u^{i}(x)$, $v^{i}(x)$, $i=1,2$, and their strain tensors $u^{ij}=\partial^{(i}u^{j)}$ and $v^{ij}=\partial^{(j}v^{i)}$. The density of elastic energy contains direct and cross terms:

[TABLE]

Six-fold symmetry of the sublattices implies isotropy of the direct terms Landau :

[TABLE]

Symmetry dictates the form of the cross term

[TABLE]

The tensor $e_{ijk}$, which is invariant under $C_{3v}$, can be represented by the three unit vectors $\{\hat{e}^{(l)}\}$ of Fig. 1:

[TABLE]

Only the last term in Eq. Carbon Nanostructures as an Electromechanical Bicontinuum is not invariant under general rotation. (In nanotubes, it depends on the helical angle $\theta_{c}$: $e_{\phi\phi\phi}=-e_{\phi zz}=-\sin(3\theta_{c})$, $e_{zzz}=-e_{\phi\phi z}=-\cos(3\theta_{c})$, where $\phi$, $z$ are defined in Fig. 1). This elastic energy density, the lowest-order approximation in both derivatives and fields, contains six parameters: $\mu^{\prime}$ and $\lambda^{\prime}$, being confined to one sublattice, describe next-neighbor interactions; the cross terms $\mu$ and $\lambda$ describe nearest-neighbor interaction; $\alpha$ describes the stiffness against relative shifts of the sublattices; $\beta$ determines the strength of rotational symmetry breaking and so carries the point group symmetry of graphene. These parameters are normalized to the sublattice surface density $\sigma_{s}$, so that the elastic energy is $W=\int\sigma_{s}\!\ V\!\ d^{2}x$.

Taking $\frac{1}{2}\sigma_{s}\left(\dot{u}^{2}+\dot{v}^{2}\right)$ as the surface density of kinetic energy, the equations of motion read

[TABLE]

with the sublattice 2-D stress tensors

[TABLE]

As expected, $\alpha$ determines the frequency of two degenerate $k=0$ optical modes: ${\omega_{\Gamma}}^{2}=4\!\ \alpha$.

First, we briefly show that the usual macroscopic elastic energy of graphene and its Lamé coefficients can be obtained from $V$ by considering a static, uniform solution of Eqs. 5 with identical deformations on both lattices with an internal displacement $2q^{i}\equiv u^{i}-v^{i}$ :

[TABLE]

where $\ell=\beta/\alpha$ is a characteristic length. Anisotropic ($2\gamma=u^{xx}-u^{yy}$) and shear ($\eta=u^{xy}$) strains produce internal displacements $q^{x}=\ell\!\ \gamma$ and $q^{y}=-\ell\!\ \eta$ (Fig. 1). The elastic energy for uniform deformations $W_{u}=\int V_{u}\sigma_{g}\!\ d^{2}x$ then simplifies to

[TABLE]

where $\sigma_{g}=2\sigma_{s}=2.26$ g cm*-2* is the surface density of graphene, $\mu_{R}\equiv\mu+\mu^{\prime}-\frac{\!\ \beta^{2}}{\alpha}$, $\lambda_{R}\equiv\lambda+\lambda^{\prime}+\frac{\!\ \beta^{2}}{\alpha}$ the measurable Lamé coefficients Landau . Macroscopic problems do not distinguish between the two sublatices; eliminating $q^{i}$ in Eq. 8 through Eqs. 4 and 7 we obtain the familiar, isotropic, macroscopic energy for graphene, $V_{u}=\mu_{R}u^{ij}u_{ij}+\lambda_{R}u^{i}_{i}u^{j}_{j}/2$. In the long wavelength limit Eqs. 5 returns the familiar longitudinal and transverse speeds of sound in terms of the Lamé coefficients: $v_{L}^{2}=2\mu_{R}+\lambda_{R}$, $v_{T}^{2}=\mu_{R}$.

The out-of-plane displacements $u_{\perp}(x)$ and $v_{\perp}(x)$ do not couple with the in-plane $u^{i}$, $v^{i}$ in the harmonic limit: invariance under simultaneous sign change of $u_{\perp}$ and $v_{\perp}$ prevents it, for flat sheets. Introducing $2p_{\perp}(x)=u_{\perp}(x)+v_{\perp}(x)$ and $2q_{\perp}(x)=u_{\perp}(x)-v_{\perp}(x)$, $V_{\perp}$ must be invariant under $p_{\perp}\rightarrow p_{\perp}+L(x)$, $L(x)$ a linear function in the plane, and thus, can contain only second (and higher) derivatives in $p_{\perp}$. Symmetry dictates (cf. Appendix)

[TABLE]

The frequency of the $k=0$ out-of-plane optical mode is $2\sqrt{\alpha_{\perp}}$, and the out-of-plane acoustic branch is quadratic at small wave-vector, as expected.

The bicontinuum phonons are much more richly structured than in a traditional continuum model: they include all the optical branches, show nonlinear dispersion at large wavevector, and even display the main features of the Brillouin zone, all without sacrificing the advantages of a continuum framework. Plane-wave solutions of Eqs. 5 returns an analytically solvable fourth-order secular equation in $\omega(k)$, yielding two acoustic and two optical branches. The longitudinal branches cross at the vertices of a hexagon. Since the two-field elastic energy density respects the point group symmetry of the graphene lattice, this hexagon is oriented just as the graphene Brillouin zone; although the model, unlike in the envelope function approach Foreman , has no built-in length scale, the elastic parameters can be constrained so that the crossing point coincides with the $K$ point of graphene. A similar argument holds for the out-of-plane modes: strikingly one can construct the correct Brillouin zone within a continuum model. Fig. 2 shows the bicontinuum phonons fit to electron-energy-loss spectroscopy (EELS) data Oshima ; Siebentritt for parameters fitted either to the full Brillouin zone or just around $\Gamma$ BZ .

The bicontinuum provides a unified framework for nanotube mechanics which can describe all current computational results on the coupling of nanotube phonons to static structural distortions, to each other (e.g. breathing-to-Raman or longitudinal-to-transverse modes in helical tubes) and to the tube electronic structure. In a cylindrical geometry with coordinates $\{r,\phi,z\}$, a coupling between the tangential displacements $u^{i}$, and the radial $u^{r}=u_{\perp}$ appears in $V$ of Eq. 1 via $u^{\phi\phi}=\left(\partial_{\phi}u^{\phi}+u^{r}\right)/r$ (and similarly for $v$); this accounts for the emergence of the Radial Breathing Mode (RBM) ccorr . We consider uniform solutions: $u=u_{o}e^{-i\omega t}$, $v=v_{o}e^{-i\omega t}$. The tube’s helicity can be subsumed into new axes $\{\xi,\zeta\}$ ($\xi={\phi}\cos{3\theta_{c}}+{z}\sin{3\theta_{c}}$ $\zeta=-{\phi}\sin{3\theta_{c}}+{z}\cos{3\theta_{c}}$) rotated by an angle $3\theta_{c}$ with respect to the base of the tube. In terms of $p$, $q$ we obtain $p^{\xi},p^{\zeta}=0$ and

[TABLE]

Unlike standard elasticity Mahan1 , which cannot describe optical modes, or standard atomistic descriptions, which cannot be solved analytically, the two-field continuum model enables an exact analytical solution for the coupling between the RBM and the graphite-like optical mode through the first two of Eqs. in (10); the RBM induces a shear in the sublattices, $u^{\phi\phi}=v^{\phi\phi}=u^{r}/r$, which couples with the internal displacement through $\beta$, and vice versa. Thus, the RBM is not purely radial, but has a longitudinal component $q_{B}^{z}\sim\frac{\ell}{2r}\cos{3\theta_{c}}$, as previously seen in a numerical calculationserbians . Expansion of the RBM frequency in powers of $l/r$ reveals a correction to the the standard continuum result $v_{L}/r$ Mahan1 : $\omega_{B}=\frac{v_{L}}{r}\left[1-\frac{1}{8}\left(\frac{\ell}{r}\right)^{2}+O\left(\frac{\ell}{r}\right)^{4}\right]$. The graphite-like optical modes of chiral tubes are $\omega_{\xi}=\sqrt{4\,\alpha}$, $\omega_{\zeta}/\omega_{\xi}=1+\frac{1}{8}\left(\frac{l}{r}\right)^{2}+O\left(\frac{l}{r}\right)^{4}$, also of mixed longitudinal/transverse character except for armchair and zig-zag nanotubes, while the out-of-plane optical mode $\omega_{\perp}=\left(4\alpha_{\perp}-\frac{2\mu-2\mu^{\prime}+\lambda-\lambda^{\prime}}{r^{2}}\right)^{1/2}$ is purely radial. A density functional theory calculation of the breathing mode breathing reports different frequencies with ($\omega_{B}$) and without ($\tilde{\omega}_{B}$) coupling to optical modes. We predict $r^{2}\left(\tilde{\omega}_{B}^{2}-\omega_{B}^{2}\right)\rightarrow\beta^{2}/\alpha$ as $r\rightarrow\infty$: using ref breathing data for $\tilde{\omega}_{B}$, $\omega_{B}$ we obtain $\ell\equiv\beta/\alpha=0.25$ Å  ($0.27$ Å) for non metallic zig-zag (armchair) tubes, in good agreement with the parameters from our fit to the graphene phonons BZ .

The bicontinuum can also describe electron-lattice coupling to both acoustic and optical modes, by incorporating a tight-binding model whose nearest neighbor hopping integrals $t^{(1)},t^{(2)},t^{(3)}$ are modulated by the in-plane elastic deformations:

[TABLE]

where $e$ is the inter-atomic distance and $\tau$ a parameter to be determined hopping . For example, lattice deformations open gaps in metallic tubes, and these gaps in turn affect vibrational frequencies. If $\epsilon_{c}$, $\epsilon_{v}$ are the conduction and valence bands, we have to nearest neighbors

[TABLE]

where $a^{(n)}\equiv e^{(l)}-e^{(m)}$, $n(l,m)$ is cyclic in $\{1,2,3\}$ (e.g. $a^{(3)}\equiv e^{(1)}-e^{(2)}$) and $\{e^{(i)}\}$ connects nearest neighbors. From Eqs. 11,12 we find the band gap opened by strain in a metallic nanotube to be

[TABLE]

In the second line of equation (LABEL:gap) the symmetry of the honeycomb lattice is broken by the unit vectors $\hat{\phi}^{i},\hat{z}^{i}$ of the cylindrical coordinates. In terms of $2\gamma^{\prime}\equiv u^{\phi\phi}-u^{zz}$, $\eta^{\prime}\equiv u^{\phi z}$, $q^{z}$, equation (LABEL:gap) reads

[TABLE]

which corrects and extends a well known previous result within a one-field continuum model Yang that neglected the inner displacement (i.e. $q^{i}=0$).

Opening bandgaps in metallic nanotubes causes several shifts in observed quantities. The term proportional to $q_{z}^{2}$ in Eq. (LABEL:gap) show that longitudinal optical modes open a bandgap in metallic tubes of any helicity; the elastic energy lowers by a term proportional to the square of the bandgap, leading to a the softening of longitudinal optical frequency in metallic nanotubes, as revealed by a recent DFT study Dubay . Eq. (LABEL:gap) predicts also a softening of the RBM in metallic nanotubes $\frac{\delta\omega_{B}}{\omega_{B}}=-A\cos^{2}\left(3\theta_{c}\right)$, highest for zig-zag tubes as seen in DFT breathing , and relates it to the optical softening, with $A=(1-\ell/e)\omega_{opt}\delta\omega_{opt}e^{2}/4v_{L}^{2}$, $\omega_{opt}$ the graphite-like optical mode, and $\delta\omega_{opt}$ its softening in metallic tubes ($A\simeq 2\%$). Other shifts can be predicted: the speed of sound for the twist mode softens by $\frac{\Delta c_{t}}{c_{t}}=-\frac{v_{L}^{2}}{2v_{T}^{2}}A\sin^{2}\left(3\theta_{c}\right)$, or $\simeq 2.2\%$ in armchair tubes.

Doping-induced structural deformations can also be studied by minimizing the total energy (elastic plus doped electrons). Subtle phenomena absent in other models Verissimo can be accessed within the bicontinuum framework. Going to next-nearest-neighbor in the hopping integrals ($\mathrm{d}t_{1}^{(l)}=-\tau_{1}\!\ \hat{a}^{(l)}_{i}\hat{a}^{(l)}_{j}v^{ij}$ hopping ), we find that at first order in both $a/r$ and the number of dopant electrons per atom $\rho_{e}$, semiconducting $(n,0)$ nanotubes show doping-induced changes in tube length ($\mathrm{d}L/L=u^{zz}$) and axial bond-length ($\mathrm{d}b_{ax}=eu^{zz}-q^{z}$):

[TABLE]

where $m_{C}$ is the mass of the carbon atom. The sign is positive (negative) for $r=n\mod 3=2$ ($n\mod 3=1$). Recent DFT results Margine indeed show shrinking or stretching of $b_{ax}$ for $n=16,13$ or $n=14,11$ tubes respectively, as predicted by Eq. 15. In DFT, the overall tube lengthens in the second case ($n=14,11$), again in accord with the bicontinuum; the lengthening found for $r=2$, is less than for $r=1$, perhaps a consequence of the change in sign in Eqs. 15. Finally the shrinking of the axial bond determines an up-shift in the longitudinal graphite-like optical mode and might explain recent Raman results that point toward anomalous bond contraction under doping in semiconducting nanotubes Chen1 ; Chen2 .

In summary, a symmetrized two-field continuum model of graphene and carbon nanotubes provides the first unified analytical treatment for a wide range of vibrational and electromechanical phenomena including nonlinear dispersion of in-plane phonons, zone-edge degeneracies and optical modes. A full range of vibrational-electronic-mechanical couplings, which were absent from previous continuum models or happened upon in an ad hoc fashion in computational work, can now be understood within a single unified analytical framework. Extending the formalism to include higher-order effects arising from curvature or metallic character (i.e. symmetry breaking terms containing $\hat{\phi}^{i}$, $\hat{z}^{i}$, as in Eq. LABEL:gap), anharmonicity (terms higher order in $u^{ij}$, $v^{ij}$), or long-distance interactions (higher partial derivatives) is straightforward. An extension to boron nitride nanotubes, with different coefficients for each sublattice in the direct terms of Eq. 2, might prove useful to study their piezoelectricity.

0.1 Appendix: Derivation of Eq. Carbon Nanostructures as an Electromechanical Bicontinuum

The term $c[u,v]$ must be invariant under the combination of $2\pi/6$ rotations and the exchange of fields $u\leftrightarrow v$. Adding reflection through the $x$ axis (Fig. 1) then implies $C_{3v}$ invariance. There is also a field translation invariance: $u(x)\rightarrow u(x)+p$, $v(x)\rightarrow v(x)+p$. The objects $u^{i}$, $v^{j}$, $u^{ij}$, and $v^{ij}$ can be combined pairwise only into tensors of rank two, three and four; thus $c[u,v]$ decomposes into three parts. The first part has terms like $u^{i}v^{j}$; symmetry then implies the form $\alpha(u-v)^{2}$ with $\alpha>0$ to ensure an energy minimum. The second part has terms like $u^{ij}v^{kl}$; the only admissible form is $2\mu u^{ij}v_{ij}+\lambda u^{i}_{i}v^{j}_{j}$. The third part contains only rank three terms such as $u^{ij}v^{k}$ contracted with a $C_{3v}$ invariant tensor $e_{ijk}$, giving $e_{ijk}u^{ij}v^{k}$. By requiring invariance under $2\pi/6$ rotations conjugated with sublattice switching, and also the field translation invariance, we obtain the form $e_{ijk}u^{ij}(u^{k}-v^{k})+e^{*}_{ijk}v^{ij}(v^{k}-u^{k})$, where the star means a $2\pi/6$ rotation. Since $C_{3v}$ invariance implies $e^{*}_{ijk}=-e_{ijk}$ we finally obtain the third row of Eq. Carbon Nanostructures as an Electromechanical Bicontinuum.