Open access peer-reviewed chapter

# Mesoscopic Physics of Phonon Transport in Carbon Materials

Written By

Kenji Sasaoka and Takahiro Yamamoto

Submitted: February 23rd, 2018 Reviewed: September 4th, 2018 Published: November 6th, 2018

DOI: 10.5772/intechopen.81292

From the Edited Volume

## Phonons in Low Dimensional Structures

Edited by Vasilios N. Stavrou

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## 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

Q̇ph=m0dk2πωmkvmkηωmThotηωmTcoldζmkE1

where ωmka phonon energy dispersion of wave number kand a phonon mode index m, vmk=dωmk/dka group velocity,ηωmTα=expωmk/kBTα11the Bose-Einstein distribution function in heat baths, and ζmkis the transmission probability between the system and heat baths [1].

Analytically, performing the integration in Eq. (1) is, generally, very difficult, and it requires a knowledge of ωmkand ζmkas a function of mand k. However, transformation of the integration variable in Eq. (1) from kto ωmkleads to a cancelation between vmkand the density of state, dk/m, so that Eq. (1) is rewritten as

Q̇ph=mωmminωmmaxdωm2πωmηωmThotηωmTcoldζmωmE2

Here ωmminand ωmmaxare the minimum and maximum angular frequencies of the mth phonon dispersion, respectively. It is noted that Eq. (2) depends on only ωmminand ωmmaxregardless of the energy dispersion. Furthermore, within the linear response limit, ΔTThotTcoldTThot+Tcold/2, and the limit of adiabatic contact between the system and heat baths, ζmωm=1, the thermal conductance, κph=Q̇ph/ΔT, is simplified as

κph=kB2T2πmxmminxmmaxdxx2exex12E3

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, κph=κphminκphmax:

κphα=2kB2Thmϕ2exmα+xmαϕ1exmα+xmα22ηxmαE4

Here, αdenotes “min” or “max,” ϕzs=n=1sn/nzis the Appel function, and xmα=ωmα/kBT. In particular, an acoustic mode (ωmmin=0) contributes a universal quantum of κ0=π2kB2T/3hto the thermal conductance.

The thermal conductance in single-walled carbon nanotubes (SWCNTs) can be obtained by knowing the values of ωmminand ωmmaxfor all m. These values can be obtained from the diagonalization of the dynamical matrix, constructed with the scaled force-constant parameters [4, 5]. Figure 1 shows energy dispersion curves for the region near k=0for a CNT with chiral vector Ch=1010, where Tis the magnitude of the unit vector along the tube axis. Here, the chiral vectornmuniquely determines the geometrical structure of CNTs [5, 7]. Figure 1 shows four acoustic modes with linear dispersion: a longitudinal acoustic one, doubly degenerate transverse acoustic ones, and a twisting one. The lowest doubly degenerate optical (E2gRaman active) modes have an energy gap of ωop=2.1meVat k=0. As shown in the inset of Figure 1, ωopdepends only on the tube radius Rand decreases approximately according to 1/R2[5, 7]. These modes always lie in low-energy dispersion relations, independent of the geometry of SWCNTs.

Figure 2(a) shows the thermal conductances normalized to a universal value of 4κ0(as explained later) as a function of temperature. The calculated values approach unity in the low-temperature limit, meaning that the phonon thermal conductance of SWCNTs is quantized in unit of a universal value of 4κ0, independent of the chirality of SWCNTs. The origin of the quantization of thermal conductance is low-energy excitations of long wavelength acoustic phonons (four branches in Figure 1) at temperatures sufficiently low that the two lowest optical modes with ωopare not excited (lowest gapped branch in Figure 2). The quantization can also be derived analytically from Eq. (4). Only the first term contributes to the conductance at the low-Tlimit, leading to 4π2kB2T/3h=4κ0. Here, 4 is the number of acoustic branches.

As another important finding, the different curves of κphTfor various SWCNTs seen in Figure 2(a) exhibit a universal feature when a scaled temperature is introduced, τop=kBT/ωop. Taking account of the four acoustic and two lowest optical modes and substituting the values of ωmminfor these branches at the k=0into Eq. (2), the thermal conductance can be given as

κph4κ01+3π2e1/τop1+1τop+12τop2E5

The curves in Figure 2(a) are replotted against the curve of Eq. (5) with τopin Figure 2(b). It is evident that all curves (only three curves are shown for clarity) fall on a single curve coinciding with the curve of Eq. (5) in the low-Tlimit. The curves turn upward at around τop0.14from a linear region in this plot (quantization plateau), with the plateau width determined by the relation 1/R2(see result in the inset of Figure 1). This universal feature of κphTof SWCNTs indicates that the optical phonon energy gap, which is decided only by R, characterizes low-temperature phonon transport, as shown in the inset of Figure 1. This theoretical result supports both the experimental observations and the inferred tube-radius dependence of the width of the thermal conductance plateau, although the unknown extrinsic factors in the experiment makes it impossible to compare the absolute values between the experiment and theory directly [8, 9].

The contribution of electrons to thermal conductance can be determined in a simple manner by replacing ηωmTin Eq. (1) with fϵmT=1/eϵmμ/kBT+1and then substituting the electron energy bands, ϵm, into the formula. According to this formulation, all conduction bands crossing the Fermi energy level yield κ0, as that of phonons, even though electrons obey different statistics. Generally, the quantum of thermal conductance should be universal out of relation to particle statistics [1, 10].

The low-Tbehavior of the electronic thermal conductance in SWCNTs is dependent on whether the SWCNT is metallic or semiconducting, which is sensitive to radius and chirality [11, 12]. For semiconducting SWCNTs, the electronic thermal conductance, κel, should vanish roughly exponentially in the limit of T0, having an energy gap of the order of 0.1 eV [13, 14, 15]. For metallic SWCNTs, two linear energy bands crossing the Fermi level at k>0[5] contribute toκelat low temperatures, resulting in a universal value of κel=4κ0, where 4 is the number of two spin-degenerate channels crossing the Fermi level. This result also satisfies the Wiedemann-Franz relation between electrical conductance and electronic thermal one [16, 17, 18]. The total thermal conductance of metallic SWCNTs is given by κ=κel+κph=8π2kB2T/3hat low temperatures.

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 ωop, typically a few meV, as described in Figure 2(b), while that for electron transport is of the order of 0.1 eV, which corresponds to the energy at a Van Hove singularity measured from the Fermi level [19]. As a result, it is predicted that the quantized nature of electron thermal conductance survives up to room temperature, at which phonons already cease to exhibit thermal quantization, giving rise to high thermal conductance. In other words, the contribution from electrons to thermal conductance is negligible compared to that from phonons at moderate temperatures. In Figure 3, the temperature dependence of the ratio of thermal conductance κel/κphfor electrons and phonons is illustrated. The experimentally observed ratio [20] is 1 order of magnitude lower than the present value. The discrepancy is attributed to the theoretical treatment of SWCNTs as purely metallic, whereas only a certain fraction 1/3[5, 11] of the crystalline ropes of SWCNTs in the experiment will be metallic and contribute to κel.

### 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, TC(=290K) and TH(=310K), are assigned to several layers of the left- and right-hand sides of a SWCNT. This leads to a thermal current from the right to left through the SWCNT, as shown in Figure 4. The Nosé-Hoover thermostat is utilized to control the temperature of the left and right-hand side several layers [24, 25], and we impose the fixed boundary condition, so that the edge atoms of SWCNTs are fixed rigidly. The length of the temperature-controlled layers is taken to be LC=L/2, where Lis the length of the phonon-conduction region. The tunable parameters in the Nosé-Hoover thermostat method were optimized so as to minimize contact thermal resistance [26]. In our simulations, we solve Hamilton’s classical equations of motion using second-order operator splitting integrators [27] with the molecular-dynamics (MD) time step of 0.5 fs.

In this subsection, the thermal conductance κis treated as

κ=JthTHTCE6

Here, the steady-state thermal current, Jth, is calculated as follows:

Jth=j=1nΔQHjΔQCj2nΔtE7

where nrepresents the number of MD steps and −ΔQHCjis the amount of heat added to the right temperature-controlled layers (removed from the left ones) per unit time (see Figure 4).

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 55SWCNT for compression lengths lcomp=0,60,and 120nm, respectively. It can be seen that the CNT is severely bent as the edge-layer distance decreases. In the simulations, the bending deformation arises from stretching of carbon-carbon bond lengths, and the hexagonal network of carbon atoms in the SWCNT is not broken. The left panel in Figure 5 shows the thermal conductance of the 55SWCNT with L=100nmas a function of the compression length. Our simulations exhibits that the bending does not affect the thermal conductance. Although the value of thermal conductance depends on L, the L-dependence is not discussed here because the conclusion of the study does not change qualitatively within the range from L=100to 250nm as calculated. For the L-dependence, we refer the reader to other published papers [26, 28, 29, 30, 31].

The bending robustness of κcan be understood through a perspective of the phonon dispersion relations as shown in Figure 6, given by the power spectra of velocity fluctuations calculated by MD simulations [26, 28]. Figure 6(a)-(c) show the dispersion relations of the bent 55SWCNT for lcomp=0,60,and 120nm, respectively. Since the bending deformation does not break the hexagonal network of the SWCNT is not broken by the bending deformation, a change of the dispersion structure due to the bending is very small. More specifically, the dispersion structure in the low-energy region remains unchanged after the bending, whereas that in the high-frequency region is slightly changed as shown in Figure 6. Consequently, κat the room-temperature is unaffected by the local bond-length deformation due to the bending. The bending robustness obtained by our simulations supports the experimental results of Chang et al. [32].

### 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% 13C and a (8,0) semiconducting SWCNT with 9.4% 14C. 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 L. This region is connected to semi-infinite pristine SWCNT leads, not including any defects or impurities (Figure 7). In accordance with the Landauer theory within the linear response with the temperature difference between hot and cold baths [1], the phonon derived thermal conductance can be expressed as κT=02πℏωfBωTTζω, where s is Planck’s constant, Tis the average temperature of the hot and cold baths, fBωTis the Bose-Einstein distribution function for a phonon with a frequency ωin the baths, and ζωis the phonon-transmission function averaged over an ensemble of samples with different isotope configurations. We adopted over 200 realizations for each Lat each ω.

In the NEGF technique, the phonon-transmission function ζωis given by ζω=TrΓLωGωΓRωGω, where Gω=ω2MDΣLωΣRω1is the retarded Green’s function in the central region and ΓLRω=iΣLRωΣLRωis the level broadening function due to the left (right) lead [33, 34, 35]. Here, Dis a dynamical matrix in the central region, Ma diagonal matrix with elements corresponding to the masses of the constituent atoms, and ΣLRωa self-energy due to the left (right) lead. A merit of NEGF technique is that the phonon transport in micrometer-length nanotubes can be efficiently computed. We can easily calculate the statistical average of the phonon transmission for nanotubes within the wide range of tube length with respect to huge number of isotope configurations. On the other hand, consideration of many-body interactions such as phonon-phonon scattering requires much computation time in the NEGF technique.

To perform the NEGF simulations, we first optimized the structures of a pristine (5,5) metallic and (8,0) semiconducting SWCNTs, and then calculated Dfrom the second derivative of the total energy of the optimized structures with respect to the atom coordinate. By using Dand the recursion method, we can easily compute ΣLRω. Moreover, we assume that the isotopes are taken into account only in M.

Coherent-phonon transport is classified into three regions based on a relation among the length Lof the central region: the ballistic regime for LlMFPω, the diffusive one for lMFPωLξω, and the localization one for Lξω. Here, lMFPωis the mean free path and ξωthe localization length. Before discussing the phonon-transmission histogram, we first determine lMFPωand ξωfor isotope-disordered SWCNTs. We adopt the procedure used in Ref. [37] to estimate these lengths. Figure 8(a) shows the average phonon transmission ζωof the (5,5) SWCNT with 15% 13C for various Lup to 5 m. In the very low-frequency region, ζωdoes not decrease and is almost four, even in the presence of isotope impurities. Perfect transmission (i.e., ballistic transport) is realized because the wavelength of acoustic phonons in the low-ωregion is much longer than L. The Landauer expression of thermal conductance eventually exhibits universal quantization of 4κ0at low temperatures irrespective of the presence and absence of isotope impurities (the factor 4 reflects the number of acoustic phonon modes).

In contrast, ζωdecreases rapidly in the higher frequency region with increasing L, as shown in Figure 8(a). There are two possible mechanisms for the reduction of ζω: diffusive scattering and phonon localization. For the former, ζωdecreases with Laccording to ζω=Mω/1+L/lMFPω, where Mωmeans the number of phonon modes. On the other hand, for the latter mechanism, the phonon-transmission function decays exponentially with Laccording to the scaling law lnζω=L/ξω. In other words, ξωis defined by the scaling law. To clarify these mechanisms for the phonon-transmission reduction, the L-dependences of ζωand lnζωare plotted in Figure 8(b) and (c) for the two mechanisms, respectively. As Figure 8(b) shows that the numerical data of ζωat ω=34cm1and 391cm1are well fitted by the dashed lines. In particular, the slope of the dashed line for ω=34cm1is almost zero, implying that lMFPis very long and the phonon transport is ballistic at this frequency, as has been discussed above. For ω=391cm1, the slope is finite, which indicates that phonon transport at this frequency is in the diffusive regime. In contrast to the low frequencies, at higher frequencies (ω=1071,1207, and 1513cm1), the calculated values deviate from the dashed lines with increasing L, although they are well fitted in the short-Lregion. This deviation means that the phonon-transmission reduction for high-ωphonons of a long-LSWCNT cannot be explained by the diffusive scattering mechanism. As shown in Figure 8(c), the data for ω=1071,1207, and 1513cm1are well fitted by the dashed lines in the lnζplot. Thus, it can be concluded that phonon localization causes the phonon-transmission reduction for high-ωphonons in a long-LSWCNT.

lMFPand ξωcan be estimated from the slope of dashed lines in Figure 8(b) and (c), respectively. The estimated lMFPand ξωfor the (5,5) SWCNT with 15% 13C are presented in Figure 8(d). This result is in excellent agreement with the phenomenological Thouless relation, ξω=Mω+1lMFPω/2, similar to electron systems with time-reversal symmetry [38]. Thus, the three distinct regimes (ballistic, diffusive, and localization) could be clarified.

Figure 9 shows the 13C-concentration dependence of κTin the (8,0) semiconducting SWCNT with 2 μm length at 300 K. As seen in Figure 9, thermal conductance decreases rapidly as the concentration increases. When the concentration overs about 20%, κTdecreases by 80% in comparison with the pristine (8,0) SWCNT.

We now discuss the phonon-transmission fluctuation, defined by a standard deviation Δζωζω2ζω2. Figure 10 shows Δζωfor (a) 625 nm-long (5,5) SWCNT with 15% 13C and (b) 210 nm-long (8,0) SWCNT with 9.4% 14C. 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 ζωvaries depending on ω[see also Figure 8(a)]. The constant value is estimated to be Δζω=0.35±0.02and indicated by the dashed lines in Figure 10(a) and (b). Thus, Δζωin the diffusive regime is universal and is independent of the background phonon transmission, the tube chirality and length, the isotope concentration, and the type of isotopes. This universal fluctuation is realized only in the diffusive regime and not in the ballistic and localization regimes. Interestingly, the value of Δζω=0.35±0.02is the same as the value of the universal conductance fluctuation (UCF) for coherent electron transport in disordered quasi-1D systems, ΔG/G0=0.365, within the statistical error. Here, G0and ΔGare respectively the electrical conductance quantum and the electrical conductance fluctuation. This means that the universal phonon-transmission fluctuation is closely related to the UCF even though electrons and phonons obey different quantum statistics. Similar to the UCF, the reason for the macroscopically observable phonon-transmission fluctuation can be qualitatively understood as follows: the fluctuations of phonon-transmission channels cannot cancel each other because there are very few effective transmission channels due to isotope scattering. To obtain a quantitative and complete understanding of the universal phonon-transmission fluctuation, some sophisticated microscopic theories are required.

In the final of this subsection, we discuss the phonon-transmission histogram Pζthat contains information for every moment of ζω. In Figure 11(a) and (b), Pζfor several typical frequencies in the diffusive regime of (a) 625 nm-long (5,5) SWCNT with 15% 13C and (b) 210 nm-long (8,0) SWCNT with 9.4% 14C are shown. All the histograms in these figures are well described by a Gaussian distribution function with the universal fluctuation Δζω=0.35±0.02. This is similar to the fact that the electrical conductance histogram in the diffusive region is expressed by a Gaussian distribution function with the UCF [38].

In Lξωregime, Pζbecomes no longer a symmetric Gaussian distribution. By analogy with the electrical conductance histogram in the localization regime [39], one can easily expect that the asymmetric histogram is a lognormal function of ζ. In fact, Plnζcan be well described by a Gaussian distribution as shown in Figure 11(c) and (d). Unlike the other regimes, the variance VarlnζΔlnζ2of Plnζdecreases with increasing lnζaccording to Varlnζ2lnζas shown in the insets of these figures, similar to the situation for electrons [38]. The transmission fluctuation in the localization regime is material independent in the sense that the slope of Varlnζdoes not depend on the tube geometry, isotope concentration, or the type of isotopes. The above-mentioned results for the ballistic, diffusive, and localization regimes are summarized in Table 1.

### Table 1.

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

## 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]:

κ=νωνminωνmax2πωfωTTTνωE8

even when phonon-phonon scattering exists. Here, Tis an averaged temperature as descried in the previous section, and Tνωis a phonon transmission function, effectively including the phonon-phonon scattering given as

Tνω=ζνLRω+ζνFLωζνFRωζνLFω+ζνFRωE9

where ζναβωis the transmission function of a coherent phonon with a phonon mode νand frequency ωflowing from αto βleads. Note that the inelastic component of thermal conductance in Eq. (8) is neglected, because it negligibly contributes toκof CNTs in the quasiballistic regime.

Thus far, we have discussed the role of a single probe with temperature TF. Generally, a spatial distribution of temperature exists inside the conductor. In order to incorporate this distribution, a conductor attaching Nprobes in series, with respective temperatures Tii=12Nis introduced. For Nprobes, the transmission function Tνtotωpropagating in a conductor of length Lcan be written as

Tνtotω=LνωL+LνωΛνωL+ΛνωE10

where the characteristic length LνωTν/ρ1Tνis expressed by the density of scatters in the conductor, ρ=N/Land Tν. The derivation process of Eq. (10) is analogous to that of effective transmission for inelastic electronic transport in mesoscopic conductors [40]. Here, it is explained that we can regard Lνωin Eq. (10) as the mean free path Λνω=τνωvνω, where τνωand vνωare the backscattering time and group velocity of a phonon with νω, respectively. For phonon propagation over the distance between neighboring probes dLL/N=1/ρ, the reflection probability Rνωis given by Rνω=dL/vνω/τνω=1/ρΛνω. Thus, the phonon’s mean free path is Λνω=1/ρRνω, and LνωΛνωin the large-N(or small-dL) limit where the transmission probability of each small segment with length dLis close to one (Tνω1).

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

κ=νωνminωνmax2πωfωTTΛνωL+ΛνωE11

For a short conductorLΛνω, Eq. (11) reproduces the Landauer formula [1, 6] for coherent phonon transport with perfect transmission. In the other limit (LΛνω), it reduces to the Boltzmann-Peierls formula [41].

We now apply the developed formula (11) to thermal transport in SWCNTs at room temperature. Instead of estimating Λνωfrom Eq. (9), we use an phenomenological expression Λνω=cνA/ω2Tfor three-phonon Umklapp scattering events in the low-frequency limit ℏω/kBT1, where A=3.35×1023mK/s2is the coupling constant for graphene [42] and cνrepresents the curvature effect of a CNT (cν=1corresponds to a graphene). By using this expression we can perform integration in Eq. (11) analytically. Strictly speaking, this expression can apply only to acoustic phonon modes with linear dispersion, but it has been shown to be useful to represent other modes as well [30]. Consequently, the thermal conductance is expressed simply as:

κCNT=kB2πνΩνarctanωνmaxΩνarctanωνminΩνE12

where ΩνL=cνA/TLis an L-dependent characteristic frequency, which is a key quantity for understanding the crossover between ballistic and diffusive phonon transport in the CNTs. The νdependence of cνis neglected hereafter, i.e., the mode-dependent characteristic frequency ΩνLis replaced by ΩL. In spite of the relative simplification, this works remarkably well to describe Ldependence of thermal conductance in the quasi-ballistic regime, as will be discussed below.

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: κ1=κCNT1+κint1. The interfacial resistance κint1can be decided by fitting experimental or numerical calculation data.

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 L-dependence of thermal conductance was quantified for various tube lengths, up to micrometers at T=300K(Thot=310Kand Tcold=290K). We refer the detailed simulation procedure to Ref. [26]. The thermal conductances for (3,3) and (5,5) SWCNTs obtained from the NEMD simulations are shown by blue and red circles in Figure 12, respectively. The solid curves representing theoretical curves given the proper choice of two parameters κintand c(e.g., κint1=0.09K/nWand c=0.65for the (3,3) SWCNT) excellently agree with the MD data. Most recently, the L-dependent thermal conductance (or conductivity) of SWCNTs shown here has been measured in experiments [31, 44], although we cannot compare the theory with the experiments because the detailed information on tube structure such as number of walls and their chiralities was not described.

We return to discuss the ballistic-diffusive crossover. The relative position of ΩLwith respect to the phonon dispersion relation determines the thermal-transport properties of SWCNTs. As illustrated in Figure 13, the dashed blue line indicates the position of ΩLrelative to the dispersion relation. As seen in Figure 12, nanometer-length SWCNTs display the thermal conductance independent of L, reflecting purely ballistic phonon transport. At nanometer length, ΩLis much larger than the energies of the phonons, as shown in the left panel of Figure 13.

With an increase in L, up to micrometer length, the value of ΩLdecreases, lying in the middle of the phonon dispersion relation, as shown in the central panel of Figure 13. In this situation, low-frequency phonon modes (ωνmaxΩL) give L-independent thermal conductance reflecting a ballistic nature, whereas the high-frequency modes ωνminΩLshow κ1/Lreflecting a diffusive nature. The intermediate-frequency phonon modes (ωνmin<ΩL<ωνmax) cannot be described in terms of both Landauer and Boltzmann-Peierls formulae, and the thermal conductance exhibits nonlinear L-dependence described by Eq. (12). Thus, it is concluded that micrometer-length SWCNTs belong to the quasi-ballistic thermal transport regime in which ballistic and diffusive phonons coexist.

Next, the case when ΩLis much lower than the excitation frequency of the lowest optical phonons is discussed, as shown in the right panel of Figure 13. In this case, the tube lengthLreaches millimeters and the contribution of optical phonons to thermal conductance has a behavior as κ1/L, resulting in constant thermal conductivity, as λ=L/Sκ=const. Here, Sis the cross-sectional area of a SWCNT. On the other hand, the acoustic modes show κL1/2, leading to a power-law divergence λL1/2of thermal conductivity [29, 30]. This divergence closely relates to the long-standing problem pointed out by Pomeranchuk in the 1940s that the low-frequency acoustic phonon contribution to thermal conductivity diverges in the thermodynamic limit 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 4κ0, where the factor, 4 is the number of the acoustic modes in SWCNTs. As the temperature increases, the crossover from the ballistic transport to diffusive one occurs and the thermal conductance in the intermediate region between them indicates the non-linear dependence of tube length.

## 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).

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Written By

Kenji Sasaoka and Takahiro Yamamoto

Submitted: February 23rd, 2018 Reviewed: September 4th, 2018 Published: November 6th, 2018