Marulanda J.M. (ed.) Electronic Properties of Carbon Nanotubes

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Exciton Dephasing in a Single Carbon Nanotube Studied by Photoluminescence Spectroscopy

365

through exciton-phonon interactions. From the chirality dependence of the PL linewidth, we

concluded that exciton dephasing is caused by both the longitudinal acoustic and twisting

phonon modes. Saturation of the PL intensity and broadening of the PL linewidth in a single

SWNT occur simultaneously with an increase in excitation laser intensity. Our findings

show that the rapid exciton-exciton annihilation through multiparticle Auger recombination

broadens the homogenous linewidth.

5. Acknowledgment

The author thanks Y. Miyauchi, Y. Murakami, S. Maruyama, T. Inoue, K. Yoshikawa, R.

Matsunaga, and Y. Kanemitsu for collaborations and discussions. This study is partially

supported by Grant-in-Aid from MEXT, Japan.

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Electronic Properties of Carbon Nanotubes

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A Numerical Study o f the Vibration Spectrum for

a Double-Walled Carbon Nanotube Model

Marianna A. Shubov

and Matthew P. Coleman

Department of Mathematics & Statistics, University of New Hampshire, Durham

Department of Mathematics and Computer Science, Fairﬁeld University, Fairﬁeld

USA

1. Introduction

In this paper we offer a series of new results devoted to the numerical analysis of a

double-walled carbon nanotube model. This model is given in the form of two coupled

Timoshenko beams connected through the distributed Van der Waals force (Gibson et al.,

2007; Ru, 2000). Typically, nanotubes can be modeled as quantum systems and studied by

a molecular simulations approach, or as classical systems (such as ﬂexible beams, shells

membranes (Mahan, 2002; Pantano et al., 2003; 2004; Wang et al., 2004; 2005)), or as speciﬁc

hybrid models (Wang, 2005). The choice of model in any situation involves a tradeoff in that,

while molecular models may yield more accurate results, implementing them is extremely

time and labor intensive, which is not the case for models from continuum mechanics.

The scientiﬁc and engineering communities have acknowledged the very desirable properties

of carbon nanotubes (CNTs) and their potential use in wide-ranging applications. The author

of (Jamieson, 2000) argues that nanotechnology, mainly due to CNTs, may impact technology

more than did the silicon revolution. Depending on the atomic structure, CNTs have electrical

properties that can range from those of metals to those of semiconductors. The mechanical

properties of CNTs are also unique. They possess exceptionally high speciﬁc stiffness and

speciﬁc strength; they are extremely elastic, being able to bend through a complete 360

◦

without noticeable damage. The application potential for materials with these properties is

almost limitless.

Developing mathematical models for CNTs is of critical importance. Such models must be

veriﬁed and quantiﬁed by performing and analyzing experiments. As we have mentioned,

two groups of models exist: molecular simulation models and continuum mechanics models.

Continuum models are generally based on traditional engineering models such as beams,

shells, or membranes. The nanotubes are treated as continuous materials with deﬁnite

geometries and common material properties such as Young’s modulus. In contrast, molecular

models consider each atom, and mathematically deﬁne the interactions among the atoms.

Based on their work on atomic simulations of CNTs, the authors of (Jakobson et al., 1996)

provide a justiﬁcation for incorporating continuum mechanics models into CNTs study,

Corresponding author: Marianna A. Shubov, Department of Mathematics & Statistics, University of

New Hampshire, Durham

2 Will-be-set-by-IN-TECH

stating that “The laws of continuum mechanics are amazingly robust and allow one to treat

even intrinsically discrete objects only a few atoms in diameter.”

The most commonly used models are the following: the Euler–Bernoulli beam model,

Timoshenko beam model, and ﬂexible shell and membrane models. Typically, many models

for multi-walled nanotubes allow for independent wall movement, and the wall interaction is

a function of the local wall separation distance.

Vibration of a double-walled carbon nanotube (DWCNT) generated by a nonlinear interlayer

Van der Waals force is studied in (Xu et al., 2006). The results indicate that the nonlinear

factors of the Van der Waals force, on the one hand, have little effect on the coaxial free

vibrations. On the other hand, these nonlinear factors greatly affect noncoaxial free vibrations.

As is indicated in (Qian et al., 2002), although carbon nanotubes can have diameters only

several times larger than the length between carbon atoms, continuum models have been

found to describe their mechanical behavior very accurately, in many circumstances.

Our analysis of an initial boundary-value problem models small transversal vibrations of a

double-walled carbon nanotube. The system of equations is similar to the ones mentioned in

a number of papers (see references (Gibson et al., 2007; Jakobson et al., 1996; Pantano et al.,

2003; Qian et al., 2002; Ru, 2001; Wang et al., 2006; Xu et al., 2006; Yoon et al., 2003)). The

physical system consists of two nested nanotubes interacting through the distributed Van der

Waals force; each nanotube is modeled as a Timoshenko beam with speciﬁc parameters. As

pointed out in (Wang et al., 2006), “Unlike the Euler–Bernoulli beam model, the Timoshenko

beam model allows for the effects of transverse shear deformation and rotary inertia. These

effects become signiﬁcant for carbon nanotubes with small length-to-diameter ratios that are

normally encountered in applications.”

The model is given in the form of two coupled Timoshenko beams (i.e., in the form of

four coupled hyperbolic partial differential equations). The system is equipped with a set

of nonself-adjoint boundary conditions involving four independent complex parameters.

Indeed, all other articles treating the Timoshenko model consider only the traditional

energy-conserving boundary conditions, thus our treatment is a generalization of their work

(as these latter conditions are just limiting special cases of the nonself-adjoint conditions

treated herein). An asymptotic analysis of the eigenspectrum for this problem was performed

in (Shubov & Rojas-Arenaza, 2010a;b;c), under certain simplifying assumptions. We must

mention that the assumptions are somewhat restrictive—indeed, they cannot be satisﬁed by

a physical double-walled carbon nanotube system. However, even for this simpliﬁed case,

the necessary computations were extremely complex and cumbersome, and it is unclear if the

more general problem even is tractable.

Regardless, this special case is a valid and interesting mathematical problem whose behavior

should be quite similar to the more general physical problem. Thus, we feel that a study of the

vibration spectrum for this case certainly will shed light on the spectrum of the more general

problem, particularly by our choosing values for the physical parameters that are similar to

those for physical carbon nanotubes.

The paper is organized as follows. In Section 2, we introduce the general mathematical

model, perform separation of variables and rewrite the special case of the model treated

in (Shubov & Rojas-Arenaza, 2010a) in dimensionless form. In Section 3, we present the

asymptotic results derived in (Shubov & Rojas-Arenaza, 2010a). The Legendre-tau spectral

method is described in Section 4, and in Section 5 we present our numerical results and

comparison with the asymptotic results predicted by (Shubov & Rojas-Arenaza, 2010a).

370

Electronic Properties of Carbon Nanotubes

A Numerical Study of the Vibration Spectrum for a Double-Walled Carbon Nanotube Model 3

2. The mathematical model

We consider the system consisting of two Timoshenko beams coupled through the van der

Waals force, as given in (Shubov & Rojas-Arenaza, 2010a;b;c):

σA

1tt

(x, t)+k

[Φ

(x, t) − W

1xx

(x, t)] = −C[W

(x, t) − W

(x, t)] (1)

σI

1tt

(x, t) − EI

1xx

(x, t)+k

[Φ

(x, t) −W

(x, t)] = 0(2)

σA

2tt

(x, t)+k

[Φ

(x, t) − W

2xx

(x, t)] = C[W

(x, t) − W

(x, t)] (3)

σI

2tt

(x, t) − EI

2xx

(x, t)+k

[Φ

(x, t) −W

(x, t)] = 0. (4)

For boundary conditions, the left end of each beam is free, while the right end of each is subject

to the standard set of two-parameter boundary conditions:

(0, t) −Φ

(0, t)=Φ

(0, t)=0 (5,6)

(0, t) −Φ

(0, t)=Φ

(0, t)=0 (7,8)

[Φ

(L, t) −W

(L, t)] = σI

(L, t) (9)

EΦ

(L, t)=−σβ

(L, t) (10)

[Φ

(L, t) −W

(L, t)=σI

(L, t) (11)

EΦ

(L, t)=−σβ

(L, t). (12)

Here, 0

≤ x ≤ L where L is the length of each beam, and t ≥ 0. W

(x, t) is the transverse

displacement of beam i, Φ

(x, t) is the bending angle of beam i, i = 1, 2. The physical and

geometrical constants are as follows: σ is the mass per unit volume; E,Young’smodulus;G,

the shear modulus; A

, the uniform cross-sectional area of beam i; I

the uniform area moment

of inertia of beam i;andk

; the shear connection factor for beam i.WenotethatE = 2(1 + ν)G,

where ν is the Poisson’s ratio.

Further, we note the following:

= β

= 0 ⇒ right end of beam i is free (13)

= β

= ∞ ⇒ right end of beam i is clamped (14)

= ∞, β

= 0 ⇒ right end of beam i is simply-supported (15)

= 0, β

= ∞ ⇒ right end of beam i is roller-supported. (16)

We separate variables by letting

(x, t)=e

−iωt

(x),

(x, t)=e

−iωt

(x),

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A Numerical Study of the Vibration Spectrum for a Double-Walled Carbon Nanotube Model

4 Will-be-set-by-IN-TECH

j = 1, 2, and, following the notation in (Shubov & Rojas-Arenaza, 2010a), the system (1)–(12)

becomes

(x)=



[φ



(x) − w



(x)] + C

(x) − w

(x)] (17)

(x)=−



(x)+

[φ

(x) − w



(x)] (18)

(x)=



[φ



(x) − w



(x)] −C

(x) − w

(x)] (19)

(x)=−



(x)+

[φ

(x) − w



(x)] (20)



(0) −φ

(0)=0 (21)



(0)=0 (22)



(0) −φ

(0)=0 (23)

(0)=0 (24)

[φ

(L) − w



(L)] = −iωα

(L) (25)



(L)=iωβ

(L) (26)

[φ

(L) − w



(L)] = −iωα

(L) (27)



(L)=iωβ

(L). (28)

Here, we have

σI



, C

σA

, i = 1, 2.

Again, following (Shubov & Rojas-Arenaza, 2010a), we consider the special case



k. (29)

We must note that these conditions cannot hold for a physical double-walled carbon nanotube

(e.g., the shape factors must be different, k

= k

). However, without these assumptions, the

asymptotic treatment of the problem becomes extremely difﬁcult, and possibly intractable.

Thus, at this point in time, this particular special case is the only one for which there are

analytical results with which to compare. We now cast the problem in dimensionless form.

Following (Traill-Nash & Collar, 1953) and, more appropriately, (Coleman & Schaffer, 2010),

we introduce dimensionless quantities as follows:



(



(x),



(



)=φ

(x), i = 1, 2,





ω, γ



, γ

, (30)



σA





, β



σA





, i = 1, 2,





C, i = 1, 2.

372

Electronic Properties of Carbon Nanotubes

A Numerical Study of the Vibration Spectrum for a Double-Walled Carbon Nanotube Model 5

We abuse notation, and use x, w

, φ

instead of



and



, and the resulting dimensionless

system is

−γ

(x)=−φ



(x)+w



(x) − C



(x) − w

(x)] (31)

−γ

(x)=γ



(x) − φ

(x)+w



(x) (32)

−γ

(x)=−φ



(x)+w



(x)+C



(x) − w

(x)] (33)

−γ

(x)=γ



(x) − φ

(x)+w



(x),0< x < 1, (34)



(0) −φ

(0)=0 (35)



(0)=0 (36)



(0) −φ

(0)=0 (37)



(0)=0 (38)

(1) −w



(1)+iα



λw

(1)=0 (39)



(1) −iβ



λφ(1)=0 (40)

(1) −w



(1)+iα



λw

(1)=0 (41)



(1)+iβ



λφ(1)=0. (42)

3. Asymptotic estimation of vibration spectrum

The ﬁrst-order asymptotic estimation of the vibration frequencies for problem (31)–(42) is

given in Theorem 2.5, of (Shubov & Rojas-Arenaza, 2010a); we present the results here, but

in dimensionless form.

Theorem (Shubov, Rojas-Arenaza). Assume that the boundary parameters α



and β



,i ≥ 1, 2,

satisfy the following conditions



= α



, β



= β



, α



=

√

, β



=

√

, and





−

√



√



=





−

√



√



Then, the set of frequencies

−iλ of system (31)–(42) splits into the following four separate branches:

−iλ

(1)

√



log

−α



1 + α



+ 2nπi



+ O





, (43)

−iλ

(2)

√



log

− β



1 + β



+ 2nπi



+ O





, (44)

−iλ

(3)

√



log

−α



1 + α



+ 2nπi



+ O





, (45)

−iλ

(4)

√



log

− β



1 + β



+ 2nπi



+ O





, n

= ±1, 2, 3, . . . (46)

where



√



, β



√



, i = 1, 2. 

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A Numerical Study of the Vibration Spectrum for a Double-Walled Carbon Nanotube Model

6 Will-be-set-by-IN-TECH

We note that log represents the complex logarithm, log z = ln |z| + i arg z.Wenotealsothe

important fact that the Van der Waals force between the two tubes does not appear in the

ﬁrst-order approximation.

4. The legendre-tau spectral method

We compare the asymptotic results of the Theorem with a numerical approximation of

the spectrum using the Legendre-tau spectral method (Gottlieb et al., 1984). This entails

transforming problem (1)–(12) to one on the interval

−1 ≤ x ≤ 1 by letting x →

x − 1.

Assuming there will be no confusion, we keep the original variables x, w

, φ

, i = 1, 2, and the

resulting system is

(x)=





(x) −





(x)+C

(x) − w

(x)] (47)

(x)=−

σL



(x)+



kφ

(x) −





(x) (48)

(x)=





(x) −





(x) − C

(x) − w

(x)] (49)

(x)=−

σL



(x)+



kφ

(x) −





(x), −1 ≤ x ≤ 1, (50)



(−1) − φ

(−1)=0 (51)



(−1)=0 (52)



(1) −φ

(−1)=0 (53)



(−1)=0 (54)



kφ

(1) −





(1)=−iωα

(1) (55)

∂E

σL



(1)=iωβ

(1) (56)



kφ

(1) −





(1)=−iωα

(1) (57)

∂E

σL



(1)=iωβ

(1). (58)

We let

(x)=

∑

n=0

(x), φ

(x)=

∑

n=0

(x),

(x)=

∑

n=0

(x), φ

(x)=

∑

n=0

(x),

where P

is the Legendre polynomial of degree n.

We then compare coefﬁcients of x

,forn = 0, 1, . . . , N −2, in each of the equations resulting

from (47)–(50) and, including the 8 equations resulting from boundary conditions (51)–(58),

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Electronic Properties of Carbon Nanotubes