Phonon-transmission histogram in ballistic, diffusive, and localization regimes [36]. Copyright 2011 American Physical Society.

## Abstract

We give a theoretical review of recent development of the mesoscopic physics of phonon transport in carbon nanotubes, including the quantization of phonon thermal conductance, phonon Anderson localization, and so on. A single-walled carbon nanotube (SWCNT) can be regarded as a typical one-dimensional phonon conductor and exhibits various interesting phenomena originating from its one dimensionality. For example, a pristine SWCNT without any defects shows the quantization of phonon thermal conductance at low temperature. On the other hand, a defective SWCNT with randomly distributed carbon isotopes shows the phonon Anderson localization originating from the interference between phonons scattered by isotope impurities.

### Keywords

- carbon nanotube
- ballistic phonon
- quantized thermal conductance
- phonon Anderson localization
- phonon waveguide

## 1. Introduction

Heating of electronic devices is an unavoidable serious problem toward the realization of next-generation nanoscale devices. Carbon nanotube (CNT) is expected to be a potential material for removing the heat from heated devices because of its high thermal conductivity. However, concern has been raised that intrinsic high thermal conductivity of pure CNTs is lost because of the presence of defects in synthesized CNTs.

In this chapter, we give a review of recent progress of theoretical works on phonon transport in CNTs focusing on the quantization of phonon thermal conductance, phonon Anderson localization, and so on. The phonon transport in CNTs shows fully quantum behaviors at low temperatures and exhibits strong nonlinear behaviors due to phonon-phonon interaction at high temperatures. Therefore, traditional transport theories for bulk objects are not applicable to the thermal transport in CNTs. In the chapter, we will introduce a novel theory for mesoscopic phonon transport we developed and will describe various results and their physical interpretations.

## 2. Coherent phonon thermal transport in carbon nanotubes

### 2.1. Quantized thermal conductance of carbon nanotubes

In the one-dimensional (1D) phonon system formed between heat and cold baths, the thermal current density is described as the Landauer energy flux [1, 2, 3], which is given by

where

Analytically, performing the integration in Eq. (1) is, generally, very difficult, and it requires a knowledge of

Here

Carrying out the integration in Eq. (3), we can derive an analytical expression of the thermal conductance, which can easily apply to various 1D ballistic phonon systems,

Here,

The thermal conductance in single-walled carbon nanotubes (SWCNTs) can be obtained by knowing the values of

Figure 2(a) shows the thermal conductances normalized to a universal value of

As another important finding, the different curves of

The curves in Figure 2(a) are replotted against the curve of Eq. (5) with

The contribution of electrons to thermal conductance can be determined in a simple manner by replacing

The low-

Finally, in this subsection, a significant difference was recognized between the widths of the quantization plateau for phonons and those for electrons in metallic SWCNTs. The characteristic energy for phonon transport at low temperature is

### 2.2. Carbon nanotube as phonon waveguide

In this subsection, nonequilibrium molecular dynamics (NEMD) simulations are carried out with the Brenner bond-order potential for carbon-carbon covalent bonds [21] and the Lennard-Jones one for van der Waals interaction between the tube walls [22]. In our NEMD simulations, different temperatures,

In this subsection, the thermal conductance

Here, the steady-state thermal current,

where

First, the influence of bending deformation on the thermal conductance of SWCNTs is discussed. In our simulations, shortening the distance between the two ends of a SWCNT realizes bending. The right panels (a)-(c) in Figure 5 illustrate the bent

The bending robustness of

### 2.3. Phonon Anderson localization in isotope-disordered carbon nanotube

This subsection is focused on the interference effects of coherent phonons in SWCNTs. Here, we performed calculations for two typical examples: a (5,5) metallic SWCNT with 15.0% ^{13}C and a (8,0) semiconducting SWCNT with 9.4% ^{14}C. Our simulation is based on the Landauer theory of phonon transport combined with the nonequilibrium Green’s function (NEGF) technique [33, 34, 35]. We used the Brenner bond-order potential for the interaction between carbon atoms [21], as used in the previous subsection. It is assumed that isotope disorder exists only in a central region with a length

In the NEGF technique, the phonon-transmission function

To perform the NEGF simulations, we first optimized the structures of a pristine (5,5) metallic and (8,0) semiconducting SWCNTs, and then calculated

Coherent-phonon transport is classified into three regions based on a relation among the length ^{13}C for various

In contrast,

^{13}C are presented in Figure 8(d). This result is in excellent agreement with the phenomenological Thouless relation,

Figure 9 shows the ^{13}C-concentration dependence of

We now discuss the phonon-transmission fluctuation, defined by a standard deviation ^{13}C and (b) 210 nm-long (8,0) SWCNT with 9.4% ^{14}C. The fluctuation of a physical quantity generally decreases as its average value increases. However, the fluctuation of phonon transmission is constant within the frequency region in the diffusive regime although

In the final of this subsection, we discuss the phonon-transmission histogram ^{13}C and (b) 210 nm-long (8,0) SWCNT with 9.4% ^{14}C are shown. All the histograms in these figures are well described by a Gaussian distribution function with the universal fluctuation

In

## 3. Crossover from ballistic to diffusive phonon transport

This section discusses the crossover from ballistic to diffusive phonon transport in SWCNT using some basic arithmetic which follows from the fictitious-probe idea. In this idea, the thermal conductance was found to formally have the same expression as the Landauer formula for coherent phonon transport [1, 6]:

even when phonon-phonon scattering exists. Here,

where

Thus far, we have discussed the role of a single probe with temperature ~~,~~ a conductor attaching

where the characteristic length

According to the above discussion, a general expression of thermal conductance is given as

For a short conductor

We now apply the developed formula (11) to thermal transport in SWCNTs at room temperature. Instead of estimating

where *L*-dependent characteristic frequency, which is a key quantity for understanding the crossover between ballistic and diffusive phonon transport in the CNTs. The

In Eq. (10), effects of phonon scattering at interfaces between a CNT and the left/right leads were not included. One of simple treatments of the interfacial thermal resistance is to introduce it by the following way:

Now, we estimate the thermal conductance of SWCNTs by performing the NEMD simulations [26, 28] with Brenner’s bond-order potential [21], and compare the MD results to the above-described theory. The

We return to discuss the ballistic-diffusive crossover. The relative position of

With an increase in

Next, the case when *L* → ∞ [45]. However, it is known, in general, that the divergence disappears if we take into account higher-order phonon-phonon scattering events, although the possibility of the above-stated long-time tail in low-dimensional materials remains an open problem [46, 47]. In either case, the agreement between the current theory and MD simulation results indicates that the higher-order effects are negligible in the current length regime. This consists with the previously reported observation from Boltzmann’s kinetic approach [30].

## 4. Thermal properties of graphene modulated by strain

Finally, we shortly mention the recent theoretical work about the thermal property in another carbon material, a graphene [48]. In this work, the strain response of the phonon specific heat was investigated. The low temperature behavior of the specific heat is dominated by the three acoustic modes, i.e., the longitudinal acoustic (LA) mode, transverse acoustic (TA) mode, and out-of-plane acoustic (ZA) mode. It is well known that the LA and TA modes have a linear dispersion in the long wavelength region while the other has a quadratic one in the absence of the strain [49]. This means that the ZA mode is critical for low-temperature dependence of the specific heat. As a result, the specific heat has a linear dependence at low temperature. As the strain increases, the dispersion of the ZA mode drastically changes, so that this dispersion becomes linear in the same as the LA and TA modes [50, 51]. Due to the ZA mode linearized by the strain, the low-temperature dependence of the specific heat becomes quadratic. Therefore, since the specific heat directly relates the thermal conductivity, it is easily expected that the strain can also modulate the temperature dependence of the thermal conductivity.

## 5. Conclusion

This chapter reviewed recent progress of theoretical studies on phonon transport in SWCNTs focusing on the quantization of phonon thermal conductance, phonon Anderson localization, and so on. At low temperature, the phonon thermal conductance of SWCNTs has a quantized universal value of

## Acknowledgments

The authors would like to thank Kazuyuki Watanabe, Satoshi Watanabe, Shigeo Maruyama, Junichiro Shiomi, and Satoru Konabe for their useful discussions during this work. This work was supported, in part, by JSPS KAKENHI grants (nos. 19710083, 20048008, 22013004, 24681021, 15H03523 and 18H01816).