Emerging need for decrease in device size, the use of molecular level theories in device design and modeling becomes more and more important. One particular issue needs to be addressed in nano-electronic device modeling is the thermal transport properties of the components. Hence, the study of the thermal conductivity of nanotubes and its dependence on structure, defects, and strain is of critical importance. The anisotropic character of the thermal conductivity of the graphite crystal is naturally reflected in the carbon nanotubes. However, when the tube diameter decreases, the change from 2 dimensional planar structure to a quasi 1-dimensional tube plays a crucial role in the thermal conductivity. At the same time, the smaller diameter nanotubes have large strains due to increased curvature in contrast to the large diameter nanotubes. This particular strain effect can also explain the differences in thermal conductivity of small diameter nanotube and planar graphite.

We employed a newly modified empirical potential to carry out the calculation of the thermal transport properties for carbon nanotubes [6]. The effects of the structural defects in 1-dimensional systems are also studied.

Introduction

The race for reducing the size of electronic devices and integrated micro/nano-electro-mechanical systems (MEMS and NEMS) has provided the main driving force behind the scientific research and technological advancement in the field of nanotechnology. It is now widely accepted that the thermal management in nanosize devices becomes increasingly important as the size of the device reduces. Therefore, the thermal conduction of nanometer materials plays a fundamentally critical role that controls the performance and stability of nano/micro devices. Among various potential candidates for future MEMS/NEMS applications, carbon nanotubes stand on a unique position. Their remarkable properties, such as great strength, light weight, special electronic structures, and high stability, make carbon nanotubes the ideal material for a wide range of applications. Because of the bright future for carbon nanotubes, a great deal of effort has been devoted to understand and characterize their properties[1-4] since their discovery by Ijima[5] in 1991.

Carbon nanotubes has very unique electronic properties. It can be either metallic or semiconductor depending on its chirality (i.e. conformational variation). A lot of experiments and theoretical investigations have focused on electronic structures of carbon nanotubes in order to understand the origin of the remarkable phenomena. In addition, large effort has been given to characterize their mechanical properties, such as Young's modulus, energetics, etc. However, to finally assemble a fully functional NEMS/MEMS, the thermal management has to be addressed. To date, there is little progress made to characterize and to understand the thermal conduction in nanoscale materials. The purpose of our work is first to understand the lattice thermal transport properties for carbon nanotubes. The method employed here[6] is generally valid, and can be applied to different systems of interests.

Due to technological difficulties of synthesizing large quantity, high quality, and well ordered nanotubes, it is still challenging to perform experiments on thermal conduction measurements. It is therefore desirable for theory to predict the thermal conductivity and the influence of point defects. In general, there are two directions to calculate transport properties of materials. One is based on phenomenological Boltzmann equation (BE), and the other is based on fluctuation-dissipation relation from linear response theory. Usually, the parameters in BE are deduced from experimental measurements. For novel materials where no experimental results are available, BE can not be directly used to predict transport properties. On the other hand, molecular dynamics simulations (MD) model systems of interests at microscopic level. The motions of atoms are completely governed by interatomic interactions. In MD, fluctuation-dissipation relation can then be applied to calculate the transport coefficients in the linear response regime. It is clear that the only pre-required information is to know the interatomic interactions in the system. Due to the development of empirical classical potentials for a wide range of systems, it is relatively easy to obtain high quality empirical potentials from ab initio calculations for simple models and related experimental measurements. Therefore, MD has the unique advantage in predicting thermal transport properties of novel materials and materials that are difficult to perform experiments.

In this paper, we use equilibrium molecular dynamics simulations to calculate carbon nanotubes' thermal conductivities and its dependence on vacancies and defects. Green-Kubo (GK) relation derived from linear response theory is used to extract the thermal conductivity from heat current correlation functions. In next section, we briefly go over the theoretical background for thermal conductivity calculation.

Theoretical Background

The macroscopic thermal conductivity is defined from Fourier's law for heat
flow under nonuniform temperature. The steady state heat flow J_{q}
is obtained by keeping the system and reservoirs in contact,
(1)
where L is called thermal conductivity tensor, and J_{q} is the heat current produced by the temperature gradient T. Fourier's law of heat flow can be derived from linear response theory [7]. For isotropic systems, the conventional thermal conductivity l is given by the average quantity of different directions,
(2)
Note that the thermal conductivity calculated here does not include the electronic contribution and no net particle flow persists in the system. The contribution to the thermal conduction only comes from the atomic vibrations, so called lattice thermal conductivity. In MD simulations, the heat current is given by the following equations,
(3)
(4)
h_{i} is the total energy of particle i, which includes kinetic
and potential energies. For pairwise interactions, it is
(5)

Follow the fluctuation-dissipation theorem, the thermal conductivity
tensor can be expressed in terms of heat current correlation functions[8],
(6)
(7)
where C_{J}^{q} is called quantum canonical correlation
function, and < a;b > is defined as,
(8)
and r is the density matrix of the system at equilibrium.

C_{J}^{q}(t) is a quantum mechanical correlation function, it is usually difficult to evaluate directly. Here, we only briefly discuss its relationship with its classical counterpart. In the classical limit, Planck constant h approaches zero, the canonical correlation reduces to the usual classical correlation function. The classical counterparts of Eq.(6) and (7) are,
(9)
(10)
where C_{J}^{c}(t) is the classical correlation function given
by the phase space averaging,
(11)

Both classical and canonical correlation functions are symmetric. Naturally, quantum effects are not important when T >> T_{D}, where T_{D} is the Debye temperature. However, when the above inequality is not satisfied, certain cautions are needed for quantum effects. In order to estimate the quantum correction to the classical equation, we ought to find the relationship between the quantum canonical correlation function and its classical counterpart. Here, we employed a harmonic analysis to examine the quantum correlation function[7]. Symmetrized quantum mechanical correlation function can be defined based on quantum anticommutator, i.e. C^{'}(t)=<[O(t),O(0)]_{+}>/2. After some algebra[7], the relation between quantum canonical correlation function and symmetrized quantum correlation function can be obtained,
(12)
where the tilde represents the Fourier transform of respective functions. To further examine the quantum effects, it is useful to generalize the definition of thermal conductivity to include the frequency dependence. The generalized thermal conductivity is then written as,
(13)
Substitute Eq. (12) into (13), we arrive at,
(14)

Until now, all the derivations are exact. To establish the relationship
between C^{'} and classical correlation function C^{c},
we make use of harmonic mapping method, which is exact for harmonic
systems,
(15)
Substitute Eq. (15) into Eq. (14), we then have
(16)
It is just the classical thermal conductivity,
(17)
This result is expected since Eq. (15) is only exact for harmonic systems. In harmonic systems, each normal modes is decoupled from others. Therefore, the only contribution to thermal conductivity for a macroscopic system is from phonon modes that have wavelength equal to or longer than the macroscopic length scale, which translates into effectively zero frequency mode. As pointed out in Ref.[7], this is very similar to vibrational relaxation of a harmonic system bilinearly coupled to a harmonic bath, where both full quantum relaxation rate and full classical relaxation rate are same.

However, the major contribution to thermal conductivity of a system is
the anharmonicity in the interaction potential. Because of the anharmonicity,
phonons can scatter from each other, which gives rise to the limited
phonon mean free path and finite thermal conductivity. Although the
mapping equation (i.e. Eq.(15)) is only exact for harmonic systems,
the classical correlation function already includes the anharmonic
effects when it is evaluated during MD simulations. In this sense,
apart from the approximate nature of the prefactor in Eq. (15),
the quantum correlation function obtained from the mapping can still
capture the dominate anharmonic effects. Based on the above reasonings,
we carry out the thermal conductivity calculation using just classical
correlation functions.

Numerical Calculations

The empirical interatomic interaction used in our calculations is Brenner type of bond order dependent potential[7,9]. Brenner potential is parameterized for hydrocarbon systems, and is widely used in modeling carbon based systems, such as diamond, graphite sheet, fullerenes, and carbon nanotubes. In all MD simulations, 1 fs time step is employed, and 40 ps initial MD is used to equilibrate the systems. After equilibration, 400 ps constant energy (NVE) simulation is carried out, and the heat current is calculated every time step. The average temperature in all simulations is 300 K.

One main concern of using MD to calculate the thermal conductivity is the size effects of the simulation box due to periodic boundary condition. When the simulation is conducted in a small box, phonons will get scattered more frequently because they re-enter the simulation box before they can be dissipated. In other words, the mean free path of phonons is limited to the order of simulation box. This artifacts usually underestimate the thermal conductivity. In order to obtain correct thermal conductivity, we test the convergence of MD simulations on thermal conductivity by using different size systems. For (10,10) single walled nanotube, 4 different systems are investigated. They contain 400, 800, 1600, 3200 atoms, respectively. As we expected, phonon mean free path is the limited factor to obtain accurate results. For a small simulation system, the calculated thermal conductivity is smaller than the correct value because of the overestimation of phonon scattering. As the simulation system size gets larger, the theoretical value converges to a constant which is independent of simulation size. The convergence behavior can be seen from Figure 1. The theoretical predicted value approaches 29.8 W/cm/K along the tube axis, which is very high compared to conventional materials.

Figure 1. Convergence of thermal conductivity.

In Figure 2, we show the initial autocorrelation function of the heat current along the tube axis for the system containing 3200 atoms. The figure clearly shows the very fast decay at the beginning followed by a slow decay. The fast decay is due to the optical modes in the nanotube that has high vibrational frequency. This behavior also indicates that the high vibrational modes do not contribute the thermal conductivity in a significant way. The correlation function can be well characterized by a double exponential function with two time constants.

Figure 2. Current autocorrelation function.

The high value of thermal conductivity is for pure and defect free carbon nanotube. However, nanotubes can natural have defects and vacancies. It is therefore important to know how defects influence the thermal conduction properties in carbon nanotubes. Unlike macroscopic 3 dimensional materials, carbon nanotube can be thought as a quasi 1-dimensional wire. In Figure 3, the thermal conductivities are calculated for various vacancy concentrations. As we expected, the thermal conductivity decreases as the vacancy concentration increases. However, the rate of decreasing in thermal conductivity is quite unexpected. We have calculated the vacancy influence in thermal conductivity for diamond crystal previously [7], which showed inverse proportionality between thermal conductivity and vacancy concentration. It is natural to think that vacancies should have less severe effects in 3-dimensional materials than in 1-dimensional ones. However, the calculation results show that vacancies in nanotubes are even less influential than in 3-dimensional diamond. This is surprising, and it is probably due to the fact that the strong valence double bond network provides effective additional channels for phonon to bypass the vacancy sites. The detailed mechanism needs to be further studied.

Figure 3. Thermal conductivity as a function of vacancy concentration.

Similarly, conformational defects can also reduce the thermal conductivity significantly. One of the common conformation defect in nanotubes is (5,7,7,5) defect where 4 hexagons change into 2 pentagons and 2 heptagons. Figure 4 shows how this conformational defect affects the overall thermal conductivity of nanotubes. Compared with vacancies, the (5,7,7,5) defect is a milder form of point defect, since it does not change the basic bonding characteristic and causes much less overall structural deformation. So both the rate and absolute amount of decreasing of thermal conductivity here are less than in the case of vacancies.

Figure 4. Thermal conductivity as a function of defect
concentration.

Conclusions

We have reported the high thermal conductivity for single walled nanotubes based on MD simulations. The implication is very important since it promises efficient thermal management in carbon nanotube based MEMS/NEMS devices. The defect influence in thermal conductivity of carbon nanotube reveals an interesting phenomenon that has not been realized before. More detailed analysis is needed to understand the origin of this behavior. In addition, more accurate interaction potentials will be developed to quantitatively study the nanotube bundles and other nanostructures.