[17] Amekpewu M, Abukari SS, Adu KW, Mensah SY, Mensah NG. Effect of hot electrons on the electrical conductivity of carbon nanotubes under the influence of applied dc field. The

[18] Mensah SY, Allotey FKA, Mensah NG, Nkrumah G. Differential thermopower of a CNT chiral carbon nanotube. Journal of Physics: Condensed Matter. 2001;13(24):5653

[19] Jeon T-I, Son J-H, An KH, Lee YH, Lee YS. Terahertz absorption and dispersion of fluorinedoped single-walled carbon nanotube. Journal of Applied Physics. 2005;98(3):34316 [20] Hattori Y, Touhara H. Fluorination-defluorination and fluorine storage properties of single-wall carbon nanotubes and carbon nanohorns. New Fluorinated Carbons: Funda-

[21] Zhang KS, Pham D, Lawal O, Ghosh S, Gangoli VS, Smalley P, Kennedy K, et al. Overcoming catalyst residue inhibition of the functionalization of single-walled carbon nanotubes via the Billups-birch reduction. ACS Applied Materials & Interfaces. 2017;9(43):37972-37980

direct current electric field. Journal of Applied Physics. 2012;112:043707

degenerate energy dispersion. arXiv preprint arXiv. 2015;1503:07360

flow in graphene. Applied Physics Letters. 2012;100(13):133105

degenerate carbon nanotube. arXiv preprint arXiv. 2015;1504:05484

carbon nanotubes. Physical Review B. 2004;70(23):233401

European Physical Journal B. 2015;88(2):1-6

mentals and Applications. 2017:113-133

Applied Physics Letters. 2014;105(26):263106

6(19):3479

2015;1502:07636

Letters. 2000;85(13):2829

Nanostructures. 2003;19(3):257-262

## Author details

Daniel Sakyi-Arthur1 \*, S. Y. Mensah1 , N. G. Mensah2 , Kwadwo A. Dompreh1 and R. Edziah1

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department of Physics, College of Agriculture and Natural Sciences, University of Cape Coast, Ghana

2 Department of Mathematics, College of Agriculture and Natural Sciences, University of Cape Coast, Ghana

## References

[7] Mensah SY, Allotey FKA, Adjepong SK. The effect of a high-frequency electric field on hypersound amplification in a superlattice. Journal of Physics: Condensed Matter. 1994; 6(19):3479

stimulated absorption of acoustic phonons by electrically determined electrons experiencing intraminiband transport. The study is appropriate and furthermore considers a strong absorption of energized FCNT and SWCNT phonons. The Flourine doping affects the absorption properties of F-CNT, whereas SWCNT absorbs better than the F-CNT as was observed by Jeon et al. [19]. The phonons absorbed in this study have THz frequencies with wavelengths in the nanometer run, and takes into account examinations with high spatial determination, e.g., in phonon filters, spectroscopy (phonon spectrometer), microbiology, micro-nanoelectronic gadgets, tetrahertz adjustment of light, nondestructive testing of microstructures, and acoustic

, N. G. Mensah2

1 Department of Physics, College of Agriculture and Natural Sciences, University of Cape

2 Department of Mathematics, College of Agriculture and Natural Sciences, University of

[1] Shinokita K, Reimann K, Woerner M, Elsaesser T, Hey R, Flytzanis C. Strong amplification of coherent acoustic phonons by Intraminiband currents in a semiconductor Superlattice.

[2] Gokhale VJ, Shim Y, Rais-Zadeh M. Observation of the acoustoelectric effect in gallium nitride micromechanical bulk acoustic filters. In: Frequency Control Symposium (FCS),

[3] Mensah SY, Allotey FKA, Adjepong SK. Acoustomagnetoelectric effect in a superlattice.

[4] Dompreh KA, Mensah SY, Abukari SS, Edziah R, Mensah NG, Quaye HA. Acoustomagnetoelectric effect in Graphene Nanoribbon in the presence of external electric and magnetic fields. Nanoscale Systems: Mathematical Modeling, Theory and Applications. 2014;4(1) [5] Sharma GS, Skvortsov A, MacGillivray I, Nicole K. Acoustic performance of gratings of cylindrical voids in a soft elastic medium with a steel backing. The Journal of the Acous-

[6] Ge H, Yang M, Ma C, Ming-Hui L, Chen Y-F, Fang N, Ping S. Breaking the barriers:

Advances in acoustic functional materials. National Science Review. 2017

, Kwadwo A. Dompreh1 and R. Edziah1

examination.

Coast, Ghana

References

Cape Coast, Ghana

Author details

Daniel Sakyi-Arthur1

140 Phonons in Low Dimensional Structures

\*, S. Y. Mensah1

Physical Review Letters. 2016;116(7):075504

2010 IEEE International. IEEE; 2010. pp. 524-529

tical Society of America. 2017;141(6):4694-4704

Journal of Physics: Condensed Matter. 1996;8(9):1235

\*Address all correspondence to<EMAIL_ADDRESS>

[22] Sadykov NR, Kocherga EY, Dyachkov PN. Nonlinear current in modified nanotubes with exposure to alternating and constant electric fields. Russian Journal of Inorganic Chemistry. 2013;58(8):951-955

**Chapter 8**

Provisional chapter

**Mesoscopic Physics of Phonon Transport in Carbon**

DOI: 10.5772/intechopen.81292

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

Keywords: carbon nanotube, ballistic phonon, quantized thermal conductance, phonon

Heating of electronic devices is an unavoidable serious problem toward the realization of nextgeneration 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

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

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Mesoscopic Physics of Phonon Transport

**Materials**

Abstract

1. Introduction

in Carbon Materials

Kenji Sasaoka and Takahiro Yamamoto

Kenji Sasaoka and Takahiro Yamamoto

http://dx.doi.org/10.5772/intechopen.81292

scattered by isotope impurities.

the presence of defects in synthesized CNTs.

Anderson localization, phonon waveguide

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Mesoscopic Physics of Phonon Transport in Carbon Materials** Mesoscopic Physics of Phonon Transport in Carbon Materials

DOI: 10.5772/intechopen.81292

Kenji Sasaoka and Takahiro Yamamoto Kenji Sasaoka and Takahiro Yamamoto

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.81292

#### Abstract

[22] Sadykov NR, Kocherga EY, Dyachkov PN. Nonlinear current in modified nanotubes with exposure to alternating and constant electric fields. Russian Journal of Inorganic Chemis-

[23] Mensah SY, Allotey FKA, Mensah NG. Nonlinear acoustoelectric effect in a semiconduc-

[24] Hasan S. Electron phonon interaction in carbon nanotube devices [PhD dissertation]. West

[25] Chen J, Yan L. Recent advances in carbon nanotube-polymer composites. Advances in

[26] Bauschlicher CW, So CR. High coverages of hydrogen on (10, 0), (9, 0) and (5, 5) carbon

[27] Sharma B, Kar R, Pal AR, Shilpa RK, Dusane RO, Patil DS, Suryawanshi SR, More MA, Sinha S. Role of hydrogen diffusion in temperature-induced transformation of carbon nanostructures deposited on metallic substrates by using a specially designed fused hollow cathode cold atmospheric pressure plasma source. Journal of Physics D: Applied Physics.

tor superlattice. Journal of Physics: Condensed Matter. 2000;12(24):5225

try. 2013;58(8):951-955

142 Phonons in Low Dimensional Structures

Materials. 2017;6(6):129

2017;50(15):155207

Lafayette: Purdue University; 2007

nanotubes. Nano Letters. 2002;2(4):337-341

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 nextgeneration 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

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1–3], which is given by

$$\dot{Q}\_{\rm ph} = \sum\_{m} \int\_{0}^{\prime} \frac{dk}{2\pi} \hbar \omega\_{\rm m}(k) \upsilon\_{\rm m}(k) [\eta(\omega\_{\rm m}, T\_{\rm hot}) - \eta(\omega\_{\rm m}, T\_{\rm cold})] \zeta\_{\rm m}(k) \tag{1}$$

κα ph ¼ 2k 2 BT h

xα m ¼ ℏωα

κ0 ¼ π2k

2

knowing the values of ωmin

pendent of the geometry of SWCNTs.

optical modes. Copyright 2004 American Physical Society.

X m

m=kBT. In particular, an acoustic mode (ωmin

m and ωmax

Here, α denotes "min" or "max," ϕð Þ¼ z;s P∞

BT=3h to the thermal conductance.

ϕ 2;e �xα m � � þ xα

mϕ 1;e

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

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 ¼ 0 for a CNT with chiral vector Ch ¼ ð Þ 10; 10 , where ∣T∣ is the magnitude of the unit vector along the tube axis. Here, the chiral vectorð Þ n; m uniquely 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 (E2g Raman active) modes have an energy gap of ℏωop ¼ 2:1 meV at k ¼ 0. As shown in the inset of Figure 1, ℏωop depends only on the tube radius R and decreases approximately according to � 1=R2 [5, 7]. These modes always lie in low-energy dispersion relations, inde-

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

Figure 1. Low-energy phonon dispersion curves for a (10,10) SWCNT [6]. The inset shows the energy gap of the lowest

�xα m � � þ

xα m � �2 2 η xα m

m for all m. These values can be obtained from the

n¼1 sn=nz ð Þ is the Appel function, and

m ¼ 0) contributes a universal quantum of

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145

� � " # (4)

where ℏωmð Þk a phonon energy dispersion of wave number k and a phonon mode index m, vmð Þ¼ k dωmð Þk =dk a group velocity, η ωð Þ¼ m; Tα exp ðℏωmð Þk =kBTαÞ � 1 � ��1 the Bose-Einstein distribution function in heat baths, and ζmð Þk is 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 ωmð Þk and ζmð Þk as a function of m and k. However, transformation of the integration variable in Eq. (1) from k to ωmð Þk leads to a cancelation between vmð Þk and the density of state, dk=dωm, so that Eq. (1) is rewritten as

$$\dot{Q}\_{\text{ph}} = \sum\_{m} \int\_{\omega\_{m}^{\text{min}}}^{\omega\_{m}^{\text{max}}} \frac{d\omega\_{m}}{2\pi} \hbar \omega\_{m} [\eta(\omega\_{m}, T\_{\text{hot}}) - \eta(\omega\_{m}, T\_{\text{cold}})] \zeta\_{m}(\omega\_{m}) \tag{2}$$

Here ωmin m and ωmax m are the minimum and maximum angular frequencies of the mth phonon dispersion, respectively. It is noted that Eq. (2) depends on only ωmin m and ωmax m regardless of the energy dispersion. Furthermore, within the linear response limit, ΔT � Thot � Tcold ≪ T � ð Þ Thot þ Tcold =2, and the limit of adiabatic contact between the system and heat baths, ζmð Þ¼ ωm 1, the thermal conductance, κph ¼ Q\_ ph=ΔT, is simplified as

$$\kappa\_{\rm ph} = \frac{k\_{\rm B}^2 T}{2\pi \hbar} \sum\_{m} \int\_{x\_m^{\rm min}}^{x\_m^{\rm max}} d\boldsymbol{x} \, \frac{\boldsymbol{\chi}^2 e^{\boldsymbol{x}}}{\left(e^{\boldsymbol{x}} - 1\right)^2} \tag{3}$$

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 ¼ κmin ph � κmax ph :

Mesoscopic Physics of Phonon Transport in Carbon Materials http://dx.doi.org/10.5772/intechopen.81292 145

$$\kappa\_{\rm ph}^{a} = \frac{2k\_{\rm B}^{2}T}{h} \sum\_{m} \left[ \phi \left( 2, e^{-\mathbf{x}\_{m}^{a}} \right) + \mathbf{x}\_{m}^{a} \phi \left( 1, e^{-\mathbf{x}\_{m}^{a}} \right) + \frac{\left( \mathbf{x}\_{m}^{a} \right)^{2}}{2} \eta \left( \mathbf{x}\_{m}^{a} \right) \right] \tag{4}$$

Here, α denotes "min" or "max," ϕð Þ¼ z;s P∞ n¼1 sn=nz ð Þ is the Appel function, and xα m ¼ ℏωα m=kBT. In particular, an acoustic mode (ωmin m ¼ 0) contributes a universal quantum of κ0 ¼ π2k 2 BT=3h to the thermal conductance.

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

In the one-dimensional (1D) phonon system formed between heat and cold baths, the thermal

where ℏωmð Þk a phonon energy dispersion of wave number k and a phonon mode index m, vmð Þ¼ k dωmð Þk =dk a group velocity, η ωð Þ¼ m; Tα exp ðℏωmð Þk =kBTαÞ � 1 � ��1 the Bose-Einstein distribution function in heat baths, and ζmð Þk is the transmission probability between the

Analytically, performing the integration in Eq. (1) is, generally, very difficult, and it requires a knowledge of ωmð Þk and ζmð Þk as a function of m and k. However, transformation of the integration variable in Eq. (1) from k to ωmð Þk leads to a cancelation between vmð Þk and the

energy dispersion. Furthermore, within the linear response limit, ΔT � Thot � Tcold ≪ T � ð Þ Thot þ Tcold =2, and the limit of adiabatic contact between the system and heat baths,

> ðxmax m xmin m

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 ¼

ℏωmð Þk vmð Þk ½ � η ωð Þ� m; Thot η ωð Þ m; Tcold ζmð Þk (1)

ℏωm½ � η ωð Þ� m; Thot η ωð Þ m; Tcold ζmð Þ ωm (2)

m and ωmax

eð Þ x � 1 2 (3)

m regardless of the

m are the minimum and maximum angular frequencies of the mth phonon

ph=ΔT, is simplified as

dx x2ex

2. Coherent phonon thermal transport in carbon nanotubes

current density is described as the Landauer energy flux [1–3], which is given by

2.1. Quantized thermal conductance of carbon nanotubes

ð∞ 0 dk 2π

density of state, dk=dωm, so that Eq. (1) is rewritten as

ðωmax m ωmin m

dωm 2π

dispersion, respectively. It is noted that Eq. (2) depends on only ωmin

κph ¼ k2 BT 2πℏ X m

Q\_ ph ¼ X m

ζmð Þ¼ ωm 1, the thermal conductance, κph ¼ Q\_

Q\_

system and heat baths [1].

m and ωmax

Here ωmin

κmin ph � κmax ph : ph ¼ X m

interpretations.

144 Phonons in Low Dimensional Structures

The thermal conductance in single-walled carbon nanotubes (SWCNTs) can be obtained by knowing the values of ωmin m and ωmax m for 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 ¼ 0 for a CNT with chiral vector Ch ¼ ð Þ 10; 10 , where ∣T∣ is the magnitude of the unit vector along the tube axis. Here, the chiral vectorð Þ n; m uniquely 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 (E2g Raman active) modes have an energy gap of ℏωop ¼ 2:1 meV at k ¼ 0. As shown in the inset of Figure 1, ℏωop depends only on the tube radius R and 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

Figure 1. Low-energy phonon dispersion curves for a (10,10) SWCNT [6]. The inset shows the energy gap of the lowest optical modes. Copyright 2004 American Physical Society.

can also be derived analytically from Eq. (4). Only the first term contributes to the conductance

As another important finding, the different curves of κphð Þ T for various SWCNTs seen in Figure 2(a) exhibit a universal feature when a scaled temperature is introduced, τop ¼ kBT=ℏωop. Taking account of the four acoustic and two lowest optical modes and substituting

�1=τop 1 þ

The curves in Figure 2(a) are replotted against the curve of Eq. (5) with τop in 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-T limit. The curves turn upward at around τop ≈ 0:14 from 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 κphð Þ T of SWCNTs indicates that the optical phonon energy gap, which is decided only by R, characterizes lowtemperature 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

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

electron energy bands, Em, 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

The low-T behavior 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 T ! 0, having an energy gap of the order of 0.1 eV [13–15]. For metallic SWCNTs, two linear energy bands crossing the Fermi level at k > 0 [5] contribute toκel at low temperatures, resulting in a universal value of κel ¼ 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 [16–18].

The total thermal conductance of metallic SWCNTs is given by κ ¼ κel þ κph ¼ 8π2k

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

m for these branches at the k ¼ 0 into Eq. (2), the thermal conductance can be

1 τop þ 1 2τ2

BT=3h � � ¼ 4κ0. Here, 4 is the number of acoustic branches.

op ! (5)

Mesoscopic Physics of Phonon Transport in Carbon Materials

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147

h i and then substituting the

2 BT=3h at

at the low-T limit, leading to 4 π2k2

κph 4κ0

replacing η ωð Þ m; T in Eq. (1) with fð Þ¼ Em; T 1= eð Þ Em�μ =kBT þ 1

≈ 1 þ 3 π2 e

the values of ωmin

theory directly [8, 9].

low temperatures.

relation to particle statistics [1, 10].

given as

Figure 2. Thermal conductance as a function of temperature (a) in units of Kelvin and (b) scaled by the energy gap of the lowest optical mode [6]. Copyright 2004 American Physical Society.

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 ℏωop are 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-T limit, leading to 4 π2k2 BT=3h � � ¼ 4κ0. Here, 4 is the number of acoustic branches.

As another important finding, the different curves of κphð Þ T for various SWCNTs seen in Figure 2(a) exhibit a universal feature when a scaled temperature is introduced, τop ¼ kBT=ℏωop. Taking account of the four acoustic and two lowest optical modes and substituting the values of ωmin m for these branches at the k ¼ 0 into Eq. (2), the thermal conductance can be given as

$$\frac{\kappa\_{\rm ph}}{4\kappa\_0} \approx 1 + \frac{3}{\pi^2} e^{-1/\tau\_{\rm op}} \left( 1 + \frac{1}{\tau\_{\rm op}} + \frac{1}{2\tau\_{\rm op}^2} \right) \tag{5}$$

The curves in Figure 2(a) are replotted against the curve of Eq. (5) with τop in 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-T limit. The curves turn upward at around τop ≈ 0:14 from 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 κphð Þ T of SWCNTs indicates that the optical phonon energy gap, which is decided only by R, characterizes lowtemperature 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 η ωð Þ m; T in Eq. (1) with fð Þ¼ Em; T 1= eð Þ Em�μ =kBT þ 1 h i and then substituting the electron energy bands, Em, 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-T behavior 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 T ! 0, having an energy gap of the order of 0.1 eV [13–15]. For metallic SWCNTs, two linear energy bands crossing the Fermi level at k > 0 [5] contribute toκel at low temperatures, resulting in a universal value of κel ¼ 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 [16–18]. The total thermal conductance of metallic SWCNTs is given by κ ¼ κel þ κph ¼ 8π2k 2 BT=3h at low temperatures.

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 ℏωop are not excited (lowest gapped branch in Figure 2). The quantization

Figure 2. Thermal conductance as a function of temperature (a) in units of Kelvin and (b) scaled by the energy gap of the

lowest optical mode [6]. Copyright 2004 American Physical Society.

146 Phonons in Low Dimensional Structures

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 ℏωop, 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=κph for 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 ð Þ � 1=3 [5, 11] of the crystalline ropes of SWCNTs in the experiment will be metallic and contribute to κel.

temperature-controlled layers is taken to be LC ¼ L=2, where L is the length of the phononconduction 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

Figure 4. Schematic of the SWCNT in which different temperatures are assigned to the left- and right-end several layers

κ ¼ Jth TH � TC

j¼1 ½ � ΔQHðÞ�j ΔQCð Þj

where n represents the number of MD steps and �ΔQH Cð Þð Þj is 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 5ð Þ ; 5 SWCNT for compression lengths lcomp ¼ 0, 60, and 120 nm, 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 5ð Þ ; 5 SWCNT with L ¼ 100 nm as 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 ¼ 100 to 250 nm as calculated. For the L-dependence, we refer the reader to other published papers [26, 28–31].

The bending robustness of κ can 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

2nΔt (7)

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149

[27] with the molecular-dynamics (MD) time step of 0.5 fs. In this subsection, the thermal conductance κ is treated as

[23]. Copyright 2009 the Japan Society of Applied Physics.

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

Jth ¼

Pn

#### 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 (¼ 290 K) and TH (¼ 310 K), 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

Figure 3. Ratio of thermal conductance by electrons, to that by phonons for a (10,10) SWCNT [6]. The inset gives results at low temperatures on an expanded scale. Copyright 2004 American Physical Society.

Figure 4. Schematic of the SWCNT in which different temperatures are assigned to the left- and right-end several layers [23]. Copyright 2009 the Japan Society of Applied Physics.

temperature-controlled layers is taken to be LC ¼ L=2, where L is the length of the phononconduction 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 κ is treated as

energy for phonon transport at low temperature is ℏωop, 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=κph for 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 ð Þ � 1=3 [5, 11] of the crystalline ropes of SWCNTs in the

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 (¼ 290 K) and TH (¼ 310 K), 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

Figure 3. Ratio of thermal conductance by electrons, to that by phonons for a (10,10) SWCNT [6]. The inset gives results at

low temperatures on an expanded scale. Copyright 2004 American Physical Society.

experiment will be metallic and contribute to κel.

2.2. Carbon nanotube as phonon waveguide

148 Phonons in Low Dimensional Structures

$$\kappa = \frac{J\_{\text{th}}}{T\_{\text{H}} - T\_{\text{C}}} \tag{6}$$

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

$$J\_{\rm th} = \frac{\sum\_{j=1}^{n} \left[\Delta Q\_{\rm H}(j) - \Delta Q\_{\rm C}(j)\right]}{2n\Delta t} \tag{7}$$

where n represents the number of MD steps and �ΔQH Cð Þð Þj is 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 5ð Þ ; 5 SWCNT for compression lengths lcomp ¼ 0, 60, and 120 nm, 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 5ð Þ ; 5 SWCNT with L ¼ 100 nm as 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 ¼ 100 to 250 nm as calculated. For the L-dependence, we refer the reader to other published papers [26, 28–31].

The bending robustness of κ can 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

ð Þ 5; 5 SWCNT for lcomp ¼ 0, 60, and 120 nm, 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, κ at 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

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 [33–35]. 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 κð Þ¼ T

∂T h i ζ ωð Þ , where s ℏ is Planck's constant, T is the average temperature of the hot

ð Þ� ω , where Gð Þ¼ ω ω2M � D � ΣLð Þ� ω ΣRð Þ ω � ��1 is the retarded Green's func-

L Rð Þð Þ ω

� � is the level broadening func-

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and cold baths, f Bð Þ ω; T is the Bose-Einstein distribution function for a phonon with a frequency ω in the baths, and h i ζ ωð Þ is the phonon-transmission function averaged over an ensemble of samples with different isotope configurations. We adopted over 200 realizations

In the NEGF technique, the phonon-transmission function ζ ωð Þ is given by ζ ωð Þ¼ Tr½ΓLð Þ ω

tion due to the left (right) lead [33–35]. Here, D is a dynamical matrix in the central region, M a diagonal matrix with elements corresponding to the masses of the constituent atoms, and ΣL Rð Þð Þ ω a 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

Figure 7. Schematic of an isotope-disordered SWCNT [36]. Copyright 2011 American Physical Society.

tion in the central region and ΓL Rð Þð Þ¼ ω i ΣL Rð Þð Þ� ω Σ†

2.3. Phonon Anderson localization in isotope-disordered carbon nanotube

Chang et al. [32].

Ð ∞ 0 dω 2π ℏω ∂f Bð Þ ω;T

for each L at each ω.

Gð Þ ω ΓRð Þ ω G†

Figure 5. The thermal conductance of the (5,5) SWCNT with 200 nm length as a function of the compression length [23]. The right panels (a)–(c) represent the MD snapshots of a bended CNT. Copyright 2009 the Japan Society of Applied Physics.

Figure 6. The phonon dispersion relations of the (5,5) SWCNT with the compression length (a) 0, (b) 60, and (c) 120 nm [23]. Copyright 2009 the Japan Society of Applied Physics.

ð Þ 5; 5 SWCNT for lcomp ¼ 0, 60, and 120 nm, 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, κ at 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

Figure 5. The thermal conductance of the (5,5) SWCNT with 200 nm length as a function of the compression length [23]. The right panels (a)–(c) represent the MD snapshots of a bended CNT. Copyright 2009 the Japan Society of Applied Physics.

Figure 6. The phonon dispersion relations of the (5,5) SWCNT with the compression length (a) 0, (b) 60, and (c) 120 nm

[23]. Copyright 2009 the Japan Society of Applied Physics.

150 Phonons in Low Dimensional Structures

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 [33–35]. 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 κð Þ¼ T Ð ∞ 0 dω 2π ℏω ∂f Bð Þ ω;T ∂T h i ζ ωð Þ , where s ℏ is Planck's constant, T is the average temperature of the hot and cold baths, f Bð Þ ω; T is the Bose-Einstein distribution function for a phonon with a frequency ω in the baths, and h i ζ ωð Þ is the phonon-transmission function averaged over an ensemble of samples with different isotope configurations. We adopted over 200 realizations for each L at each ω.

In the NEGF technique, the phonon-transmission function ζ ωð Þ is given by ζ ωð Þ¼ Tr½ΓLð Þ ω Gð Þ ω ΓRð Þ ω G† ð Þ� ω , where Gð Þ¼ ω ω2M � D � ΣLð Þ� ω ΣRð Þ ω � ��1 is the retarded Green's function in the central region and ΓL Rð Þð Þ¼ ω i ΣL Rð Þð Þ� ω Σ† L Rð Þð Þ ω � � is the level broadening function due to the left (right) lead [33–35]. Here, D is a dynamical matrix in the central region, M a diagonal matrix with elements corresponding to the masses of the constituent atoms, and ΣL Rð Þð Þ ω a 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

Figure 7. Schematic of an isotope-disordered SWCNT [36]. Copyright 2011 American Physical Society.

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 D from the second derivative of the total energy of the optimized structures with respect to the atom coordinate. By using D and the recursion method, we can easily compute ΣL Rð Þð Þ ω . 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 L of the central region: the ballistic regime for L ≪ lMFPð Þ ω , the diffusive one for lMFPð Þ ω ≪ L ≪ ξ ωð Þ, and the localization one for L ≫ ξ ωð Þ. Here, lMFPð Þ ω is the mean free path and ξ ωð Þ the localization length. Before discussing the phonon-transmission histogram, we first determine lMFPð Þ ω and ξ ωð Þ for isotope-disordered SWCNTs. We adopt the procedure used in Ref. [37] to estimate these lengths. Figure 8(a) shows the average phonon transmission h i ζ ωð Þ of the (5,5) SWCNT with 15% 13C for various L up to 5 m. In the very low-frequency region, h i ζ ωð Þ does 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-ω region is much longer than L. The Landauer expression of thermal conductance eventually exhibits universal quantization of 4κ0 at low temperatures irrespective of the presence and absence of isotope impurities (the factor 4 reflects the number of acoustic phonon modes).

In contrast, h i ζ ωð Þ decreases rapidly in the higher frequency region with increasing L, as shown in Figure 8(a). There are two possible mechanisms for the reduction of h i ζ ωð Þ : diffusive scattering and phonon localization. For the former, h i ζ ωð Þ decreases with L according to h i ζ ωð Þ ¼ Mð Þ ω =ð Þ 1 þ L=lMFPð Þ ω , where Mð Þ ω means the number of phonon modes. On the other hand, for the latter mechanism, the phonon-transmission function decays exponentially with L according to the scaling law lnh i ζ ωð Þ ¼ �L=ξ ωð Þ. In other words, ξ ωð Þ is defined by the scaling law. To clarify these mechanisms for the phonontransmission reduction, the L-dependences of h i ζ ωð Þ and lnh i ζ ωð Þ are plotted in Figure 8(b) and (c) for the two mechanisms, respectively. As Figure 8(b) shows that the numerical data of h i ζ ωð Þ at ω ¼ 34 cm�1 and 391 cm�1 are well fitted by the dashed lines. In particular, the slope of the dashed line for ω ¼ 34 cm�1 is almost zero, implying that lMFP is very long and the phonon transport is ballistic at this frequency, as has been discussed above. For ω ¼ 391 cm�1, 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 ¼ 1071,1207, and 1513 cm�1), the calculated values deviate from the dashed lines with increasing L, although they are well fitted in the short-L region. This deviation means that the phonontransmission reduction for high-ω phonons of a long-L SWCNT cannot be explained by the diffusive scattering mechanism. As shown in Figure 8(c), the data for ω ¼ 1071,1207, and 1513 cm�1 are well fitted by the dashed lines in the lnh iζ plot. Thus, it can be concluded that phonon localization causes the phonon-transmission reduction for high-ω phonons in a long-L SWCNT.

lMFP and ξ ωð Þ can be estimated from the slope of dashed lines in Figure 8(b) and (c), respectively. The estimated lMFP and ξ ωð Þ for the (5,5) SWCNT with 15% 13C are presented in Figure 8(d). This result is in excellent agreement with the phenomenological Thouless relation, ξ ωð Þ¼ ð Þ Mð Þþ ω 1 lMFPð Þ ω =2, similar to electron systems with time-reversal symmetry [38]. Thus, the

Figure 8. (a) The average phonon transmission of the (5,5) SWCNT with 15% 13C [36]. (b) The length dependence of the transmission for estimating the mean free path and (c) for the localization length. (d) The mean free path and the

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Figure 9 shows the 13C-concentration dependence of κð Þ T in 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%, κð Þ T decreases by 80% in

three distinct regimes (ballistic, diffusive, and localization) could be clarified.

localization length as functions of frequency. Copyright 2011 American Physical Society.

comparison with the pristine (8,0) SWCNT.

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 computa-

To perform the NEGF simulations, we first optimized the structures of a pristine (5,5) metallic and (8,0) semiconducting SWCNTs, and then calculated D from the second derivative of the total energy of the optimized structures with respect to the atom coordinate. By using D and the recursion method, we can easily compute ΣL Rð Þð Þ ω . Moreover, we assume that the isotopes

Coherent-phonon transport is classified into three regions based on a relation among the length L of the central region: the ballistic regime for L ≪ lMFPð Þ ω , the diffusive one for lMFPð Þ ω ≪ L ≪ ξ ωð Þ, and the localization one for L ≫ ξ ωð Þ. Here, lMFPð Þ ω is the mean free path and ξ ωð Þ the localization length. Before discussing the phonon-transmission histogram, we first determine lMFPð Þ ω and ξ ωð Þ for isotope-disordered SWCNTs. We adopt the procedure used in Ref. [37] to estimate these lengths. Figure 8(a) shows the average phonon transmission h i ζ ωð Þ of the (5,5) SWCNT with 15% 13C for various L up to 5 m. In the very low-frequency region, h i ζ ωð Þ does 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-ω region is much longer than L. The Landauer expression of thermal conductance eventually exhibits universal quantization of 4κ0 at low temperatures irrespective of the presence and absence of isotope impurities (the factor 4 reflects the number of acoustic

In contrast, h i ζ ωð Þ decreases rapidly in the higher frequency region with increasing L, as shown in Figure 8(a). There are two possible mechanisms for the reduction of h i ζ ωð Þ : diffusive scattering and phonon localization. For the former, h i ζ ωð Þ decreases with L according to h i ζ ωð Þ ¼ Mð Þ ω =ð Þ 1 þ L=lMFPð Þ ω , where Mð Þ ω means the number of phonon modes. On the other hand, for the latter mechanism, the phonon-transmission function decays exponentially with L according to the scaling law lnh i ζ ωð Þ ¼ �L=ξ ωð Þ. In other words, ξ ωð Þ is defined by the scaling law. To clarify these mechanisms for the phonontransmission reduction, the L-dependences of h i ζ ωð Þ and lnh i ζ ωð Þ are plotted in Figure 8(b) and (c) for the two mechanisms, respectively. As Figure 8(b) shows that the numerical data of h i ζ ωð Þ at ω ¼ 34 cm�1 and 391 cm�1 are well fitted by the dashed lines. In particular, the slope of the dashed line for ω ¼ 34 cm�1 is almost zero, implying that lMFP is very long and the phonon transport is ballistic at this frequency, as has been discussed above. For ω ¼ 391 cm�1, 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 ¼ 1071,1207, and 1513 cm�1), the calculated values deviate from the dashed lines with increasing L, although they are well fitted in the short-L region. This deviation means that the phonontransmission reduction for high-ω phonons of a long-L SWCNT cannot be explained by the diffusive scattering mechanism. As shown in Figure 8(c), the data for ω ¼ 1071,1207, and 1513 cm�1 are well fitted by the dashed lines in the lnh iζ plot. Thus, it can be concluded that phonon localization causes the phonon-transmission reduction for high-ω phonons in a long-L

tion time in the NEGF technique.

152 Phonons in Low Dimensional Structures

are taken into account only in M.

phonon modes).

SWCNT.

Figure 8. (a) The average phonon transmission of the (5,5) SWCNT with 15% 13C [36]. (b) The length dependence of the transmission for estimating the mean free path and (c) for the localization length. (d) The mean free path and the localization length as functions of frequency. Copyright 2011 American Physical Society.

lMFP and ξ ωð Þ can be estimated from the slope of dashed lines in Figure 8(b) and (c), respectively. The estimated lMFP and ξ ωð Þ for the (5,5) SWCNT with 15% 13C are presented in Figure 8(d). This result is in excellent agreement with the phenomenological Thouless relation, ξ ωð Þ¼ ð Þ Mð Þþ ω 1 lMFPð Þ ω =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 κð Þ T in 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%, κð Þ T decreases by 80% in comparison with the pristine (8,0) SWCNT.

We now discuss the phonon-transmission fluctuation, defined by a standard deviation Δζ ωð Þ� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ζ ωð Þ2 D E � h i ζ ωð Þ 2 r . Figure 10 shows Δζ ωð Þ for (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 h i ζ ωð Þ varies depending on ω [see also Figure 8(a)]. The constant value is estimated to be Δζ ωð Þ¼ 0:35 � 0:02 and indicated by the dashed lines in Figure 10(a) and (b). Thus, Δζ ωð Þ in 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 Δζ ωð Þ¼ 0:35 � 0:02 is the same as the value of the universal conductance fluctuation (UCF) for coherent electron transport in disordered quasi-1D systems, ΔG=G0 ¼ 0:365, within the statistical error. Here, G0 and ΔG are 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

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In the final of this subsection, we discuss the phonon-transmission histogram Pð Þζ that contains information for every moment of ζ ωð Þ. In Figure 11(a) and (b), Pð Þζ for 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

Figure 11. Phonon-transmission histograms for several frequencies in the diffusive regime for (a) the (5,5) SWCNT and (b) the (8,0) SWCNT, and in the localization regime for (c) the (5,5) SWNT and (d) the (8,0) SWCNT [36]. The insets show

the variance. Copyright 2011 American Physical Society.

microscopic theories are required.

Figure 9. 13C-concentration dependence of thermal conductance of the (8,0) SWCNT with 2 μm length at 300 K.

Figure 10. The root-mean-square phonon transmission for (a) the (5,5) SWCNT and (b) the (8,0) SWCNT [36]. Copyright 2011 American Physical Society.

ΔG=G0 ¼ 0:365, within the statistical error. Here, G0 and ΔG are 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.

We now discuss the phonon-transmission fluctuation, defined by a standard deviation

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 h i ζ ωð Þ varies depending on ω [see also Figure 8(a)]. The constant value is estimated to be Δζ ωð Þ¼ 0:35 � 0:02 and indicated by the dashed lines in Figure 10(a) and (b). Thus, Δζ ωð Þ in 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 Δζ ωð Þ¼ 0:35 � 0:02 is the same as the value of the universal conductance fluctuation (UCF) for coherent electron transport in disordered quasi-1D systems,

Figure 9. 13C-concentration dependence of thermal conductance of the (8,0) SWCNT with 2 μm length at 300 K.

Figure 10. The root-mean-square phonon transmission for (a) the (5,5) SWCNT and (b) the (8,0) SWCNT [36]. Copyright

. Figure 10 shows Δζ ωð Þ for (a) 625 nm-long (5,5) SWCNT with

Δζ ωð Þ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� h i ζ ωð Þ 2

ζ ωð Þ2 D E

154 Phonons in Low Dimensional Structures

r

2011 American Physical Society.

In the final of this subsection, we discuss the phonon-transmission histogram Pð Þζ that contains information for every moment of ζ ωð Þ. In Figure 11(a) and (b), Pð Þζ for 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

Figure 11. Phonon-transmission histograms for several frequencies in the diffusive regime for (a) the (5,5) SWCNT and (b) the (8,0) SWCNT, and in the localization regime for (c) the (5,5) SWNT and (d) the (8,0) SWCNT [36]. The insets show the variance. Copyright 2011 American Physical Society.

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 N probes in series, with respective temperatures Tið Þ i ¼ 1; 2⋯; N is

Lνð Þ ω

where the characteristic length Lνð Þ� ω Tν=rð Þ 1 � Tν is expressed by the density of scatters in the conductor, r ¼ N=L and 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 Lνð Þ ω in Eq. (10) as the mean free path Λνð Þ¼ ω τνð Þ ω j j vνð Þ ω , where τνð Þ ω and vνð Þ ω are the backscattering time and group velocity of a phonon with f g ν; ω , respectively. For phonon propagation over the distance between neighboring probes dL � L=N ¼ 1=r, the reflection probability Rνð Þ ω is given by Rνð Þ¼ ω ð Þ dL=j j vνð Þ ω =τνð Þ ω ¼ 1=rΛνð Þ ω . Thus, the phonon's mean free path is Λνð Þ¼ ω 1=rRνð Þ ω , and Lνð Þ� ω Λνð Þ ω in the large-N (or small-dL) limit where the transmission probability of each small segment with

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

For a short conductor ð Þ L ≪ Λνð Þ ω , Eq. (11) reproduces the Landauer formula [1, 6] for coherent phonon transport with perfect transmission. In the other limit ðL ≫ Λνð Þ ω ), it reduces to the

We now apply the developed formula (11) to thermal transport in SWCNTs at room temperature. Instead of estimating Λνð Þ ω from Eq. (9), we use an phenomenological expression Λνð Þ¼ ω cνA=ω2T for three-phonon Umklapp scattering events in the low-frequency limit ℏω=kBT ≪ 1, where A ¼ 3:35 � 1023 mK=s2 is the coupling constant for graphene [42] and cν represents the curvature effect of a CNT (cν ¼ 1 corresponds 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

> ωmax ν Ων � �

for understanding the crossover between ballistic and diffusive phonon transport in the CNTs. The ν dependence of cν is neglected hereafter, i.e., the mode-dependent characteristic

� arctan

cνA=TL p is an L-dependent characteristic frequency, which is a key quantity

ωmin ν Ων � � � � (12)

∂fð Þ ω; T ∂T

� � Λνð Þ ω

L þ Lνð Þ ω � Λνð Þ ω

ν ð Þ ω propagating in a conductor of

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Mesoscopic Physics of Phonon Transport in Carbon Materials

L þ Λνð Þ ω (10)

L þ Λνð Þ ω (11)

introduced. For N probes, the transmission function Ttot

Ttot ν ð Þ¼ ω

κ ¼ X ν

κCNT ¼ kB 2π X ν

ðωmax ν ωmin ν

dω 2π ℏω

Ων arctan

length L can be written as

length dL is close to one (Tνð Þ� ω 1).

Boltzmann-Peierls formula [41].

expressed simply as:

where Ωνð Þ¼ L ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Table 1. Phonon-transmission histogram in ballistic, diffusive, and localization regimes [36]. Copyright 2011 American Physical Society.

Δζ ωð Þ¼ 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 L ≫ ξ ωð Þ regime, Pð Þζ becomes 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, Pð Þ ln ζ can be well described by a Gaussian distribution as shown in Figure 11(c) and (d). Unlike the other regimes, the variance Var ln½ �� ζ ð Þ Δ ln ζ 2 of Pð Þ ln ζ decreases with increasing lnh iζ according to Var ln½ ��� ζ 2 lnh iζ as 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 Var ln½ � ζ does 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.

## 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]:

$$\kappa = \sum\_{\nu} \int\_{\omega\_{\nu}^{\text{min}}}^{\omega\_{\nu}^{\text{max}}} \frac{d\omega}{2\pi} \hbar \omega \left[ \frac{\partial f(\omega, T)}{\partial T} \right] T\_{\nu}(\omega) \tag{8}$$

even when phonon-phonon scattering exists. Here, T is an averaged temperature as descried in the previous section, and Tνð Þ ω is a phonon transmission function, effectively including the phonon-phonon scattering given as

$$T\_{\nu}(\omega) = \zeta\_{\nu}^{\text{LR}}(\omega) + \frac{\zeta\_{\nu}^{\text{FL}}(\omega)\zeta\_{\nu}^{\text{FR}}(\omega)}{\zeta\_{\nu}^{\text{LF}}(\omega) + \zeta\_{\nu}^{\text{FR}}(\omega)}\tag{9}$$

where ζαβ ν ð Þ ω is the transmission function of a coherent phonon with a phonon mode ν and frequency ω flowing from α to β leads. Note that the inelastic component of thermal conductance in Eq. (8) is neglected, because it negligibly contributes to κ of 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 N probes in series, with respective temperatures Tið Þ i ¼ 1; 2⋯; N is introduced. For N probes, the transmission function Ttot ν ð Þ ω propagating in a conductor of length L can be written as

Δζ ωð Þ¼ 0:35 � 0:02. This is similar to the fact that the electrical conductance histogram in the

Table 1. Phonon-transmission histogram in ballistic, diffusive, and localization regimes [36]. Copyright 2011 American

In L ≫ ξ ωð Þ regime, Pð Þζ becomes 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, Pð Þ ln ζ can be well described by a Gaussian distribution as shown in Figure 11(c) and (d). Unlike the other regimes, the variance Var ln½ �� ζ ð Þ Δ ln ζ 2 of Pð Þ ln ζ decreases with increasing lnh iζ according to Var ln½ ��� ζ 2 lnh iζ as 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 Var ln½ � ζ does not depend on the tube geometry, isotope concentration, or the type of isotopes. The above-mentioned results for the ballistic, diffusive, and

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 for-

even when phonon-phonon scattering exists. Here, T is an averaged temperature as descried in the previous section, and Tνð Þ ω is a phonon transmission function, effectively including the

> ζFL ν ð Þ ω ζFR

ν ð Þ ω is the transmission function of a coherent phonon with a phonon mode ν and

frequency ω flowing from α to β leads. Note that the inelastic component of thermal conductance in Eq. (8) is neglected, because it negligibly contributes to κ of CNTs in the quasiballistic

ν ð Þþ ω ζFR

ζLF

ν ð Þ ω

ν ð Þ ω

∂fð Þ ω; T ∂T � �

Tνð Þ ω (8)

(9)

diffusive region is expressed by a Gaussian distribution function with the UCF [38].

localization regimes are summarized in Table 1.

mula for coherent phonon transport [1, 6]:

phonon-phonon scattering given as

where ζαβ

Physical Society.

156 Phonons in Low Dimensional Structures

regime.

3. Crossover from ballistic to diffusive phonon transport

κ ¼ X ν

Tνð Þ¼ ω ζLR

ðωmax ν ωmin ν

dω 2π ℏω

ν ð Þþ ω

$$T\_{\nu}^{\text{tot}}(\omega) = \frac{L\_{\nu}(\omega)}{L + L\_{\nu}(\omega)} \sim \frac{\Lambda\_{\nu}(\omega)}{L + \Lambda\_{\nu}(\omega)}\tag{10}$$

where the characteristic length Lνð Þ� ω Tν=rð Þ 1 � Tν is expressed by the density of scatters in the conductor, r ¼ N=L and 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 Lνð Þ ω in Eq. (10) as the mean free path Λνð Þ¼ ω τνð Þ ω j j vνð Þ ω , where τνð Þ ω and vνð Þ ω are the backscattering time and group velocity of a phonon with f g ν; ω , respectively. For phonon propagation over the distance between neighboring probes dL � L=N ¼ 1=r, the reflection probability Rνð Þ ω is given by Rνð Þ¼ ω ð Þ dL=j j vνð Þ ω =τνð Þ ω ¼ 1=rΛνð Þ ω . Thus, the phonon's mean free path is Λνð Þ¼ ω 1=rRνð Þ ω , and Lνð Þ� ω Λνð Þ ω in the large-N (or small-dL) limit where the transmission probability of each small segment with length dL is close to one (Tνð Þ� ω 1).

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

$$\kappa = \sum\_{\nu} \int\_{\omega\_{r}^{\text{min}}}^{\omega\_{\nu}^{\text{max}}} \frac{d\omega}{2\pi} \hbar \omega \left[ \frac{\partial f(\omega, T)}{\partial T} \right] \frac{\Lambda\_{\nu}(\omega)}{L + \Lambda\_{\nu}(\omega)} \tag{11}$$

For a short conductor ð Þ L ≪ Λνð Þ ω , Eq. (11) reproduces the Landauer formula [1, 6] for coherent phonon transport with perfect transmission. In the other limit ðL ≫ Λνð Þ ω ), 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 Λνð Þ ω from Eq. (9), we use an phenomenological expression Λνð Þ¼ ω cνA=ω2T for three-phonon Umklapp scattering events in the low-frequency limit ℏω=kBT ≪ 1, where A ¼ 3:35 � 1023 mK=s2 is the coupling constant for graphene [42] and cν represents the curvature effect of a CNT (cν ¼ 1 corresponds 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:

$$\chi\_{\rm CNT} = \frac{k\_B}{2\pi} \sum\_{\nu} \Omega\_{\nu} \left\{ \arctan \left( \frac{\omega\_{\nu}^{\rm max}}{\Omega\_{\nu}} \right) - \arctan \left( \frac{\omega\_{\nu}^{\rm min}}{\Omega\_{\nu}} \right) \right\} \tag{12}$$

where Ωνð Þ¼ L ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cνA=TL p is an L-dependent characteristic frequency, which is a key quantity for understanding the crossover between ballistic and diffusive phonon transport in the CNTs. The ν dependence of cν is neglected hereafter, i.e., the mode-dependent characteristic

frequency Ωνð Þ L is replaced by Ωð Þ L . In spite of the relative simplification, this works remarkably well to describe L dependence 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: κ�1 ¼ κ�1 CNT þ κ�1 int . The interfacial resistance κ�1 int can 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 abovedescribed theory. The L-dependence of thermal conductance was quantified for various tube lengths, up to micrometers at T ¼ 300 K (Thot ¼ 310 K and Tcold ¼ 290 K). 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 κint and c (e.g., κ�1 int ¼ 0:09 K=nW and c ¼ 0:65 for 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 Ωð Þ L with 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 Ωð Þ L relative 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, Ωð Þ L is much larger than the energies of the phonons, as shown in the left panel of

Figure 13. The relative position of the length dependent characteristic frequencies, 270 THz (left panel), 27 THz (middle

With an increase in L, up to micrometer length, the value of Ωð Þ L decreases, lying in the middle of the phonon dispersion relation, as shown in the central panel of Figure 13. In this situation,

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

Next, the case when Ωð Þ L is 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 lengthL reaches millimeters and the contribution of optical phonons to thermal conductance has a behavior as κ∝ 1=L, resulting in constant thermal conductivity, as λ ¼ ð Þ L=S κ ¼ const. Here, S is the cross-sectional area of a SWCNT. On the other hand, the acoustic modes show κ ∝L�1=2

leading to a power-law divergence λ∝ L1=2 of 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

reflecting a ballistic nature, whereas the high-frequency modes ωmin

panel), and 2.7 THz (right panel) [43]. Copyright 2009 the Japan Society of Applied Physics.

reflecting a diffusive nature. The intermediate-frequency phonon modes (ωmin

ν ≪ Ωð Þ L ) give L-independent thermal conductance

Mesoscopic Physics of Phonon Transport in Carbon Materials

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159

ν ≫ Ωð Þ L show κ∝1=L

ν < Ωð Þ L < ωmax

ν )

,

Figure 13.

low-frequency phonon modes (ωmax

in which ballistic and diffusive phonons coexist.

Figure 12. Length dependence of thermal conductance [43]. Copyright 2009 the Japan Society of Applied Physics.

frequency Ωνð Þ L is replaced by Ωð Þ L . In spite of the relative simplification, this works remarkably well to describe L dependence of thermal conductance in the quasi-ballistic regime, as will

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

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 abovedescribed theory. The L-dependence of thermal conductance was quantified for various tube lengths, up to micrometers at T ¼ 300 K (Thot ¼ 310 K and Tcold ¼ 290 K). 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 param-

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

We return to discuss the ballistic-diffusive crossover. The relative position of Ωð Þ L with 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 Ωð Þ L relative to the

Figure 12. Length dependence of thermal conductance [43]. Copyright 2009 the Japan Society of Applied Physics.

int . The interfacial resistance κ�1

int ¼ 0:09 K=nW and c ¼ 0:65 for the (3,3) SWCNT) excellently agree with

int can be decided by

CNT þ κ�1

be discussed below.

158 Phonons in Low Dimensional Structures

eters κint and c (e.g., κ�1

it by the following way: κ�1 ¼ κ�1

fitting experimental or numerical calculation data.

number of walls and their chiralities was not described.

Figure 13. The relative position of the length dependent characteristic frequencies, 270 THz (left panel), 27 THz (middle panel), and 2.7 THz (right panel) [43]. Copyright 2009 the Japan Society of Applied Physics.

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, Ωð Þ L is 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 Ωð Þ L decreases, 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 (ωmax ν ≪ Ωð Þ L ) give L-independent thermal conductance reflecting a ballistic nature, whereas the high-frequency modes ωmin ν ≫ Ωð Þ L show κ∝1=L reflecting a diffusive nature. The intermediate-frequency phonon modes (ωmin ν < Ωð Þ L < ωmax ν ) 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 Ωð Þ L is 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 lengthL reaches millimeters and the contribution of optical phonons to thermal conductance has a behavior as κ∝ 1=L, resulting in constant thermal conductivity, as λ ¼ ð Þ L=S κ ¼ const. Here, S is the cross-sectional area of a SWCNT. On the other hand, the acoustic modes show κ ∝L�1=2 , leading to a power-law divergence λ∝ L1=2 of 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].

Author details

Tokyo, Japan

References

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PhysRevLett.92.075502

New York: Academic; 1996

343. DOI: 10.1007/s003390201277

DOI: 10.1103/PhysRevB.59.13080

Kenji Sasaoka1 and Takahiro Yamamoto2

\*Address all correspondence to<EMAIL_ADDRESS>
1 W-FST Center, RIST, Tokyo University of Science, Tokyo, Japan

\*

Mesoscopic Physics of Phonon Transport in Carbon Materials

http://dx.doi.org/10.5772/intechopen.81292

161

2 Department of Liberal Arts, Faculty of Engineering, Tokyo University of Science,

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## 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.

## Acknowledgements

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).

## Author details

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

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

with the previously reported observation from Boltzmann's kinetic approach [30].

strain can also modulate the temperature dependence of the thermal conductivity.

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

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,

5. Conclusion

160 Phonons in Low Dimensional Structures

dependence of tube length.

Acknowledgements

15H03523 and 18H01816).

4. Thermal properties of graphene modulated by strain

Kenji Sasaoka1 and Takahiro Yamamoto2 \*

\*Address all correspondence to<EMAIL_ADDRESS>
1 W-FST Center, RIST, Tokyo University of Science, Tokyo, Japan

2 Department of Liberal Arts, Faculty of Engineering, Tokyo University of Science, Tokyo, Japan

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## *Edited by Vasilios N. Stavrou*

The field of low-dimensional structures has been experiencing rapid development in both theoretical and experimental research. *Phonons in Low Dimensional Structures* is a collection of chapters related to the properties of solid-state structures dependent on lattice vibrations. The book is divided into two parts. In the first part, research topics such as interface phonons and polaron states, carrier-phonon non-equilibrium dynamics, directional projection of elastic waves in parallel array of N elastically coupled waveguides, collective dynamics for longitudinal and transverse phonon modes, and elastic properties for bulk metallic glasses are related to semiconductor devices and metallic glasses devices. The second part of the book contains, among others, topics related to superconductor, phononic crystal carbon nanotube devices such as phonon dispersion calculations using density functional theory for a range of superconducting materials, phononic crystal-based MEMS resonators, absorption of acoustic phonons in the hyper-sound regime in fluorine-modified carbon nanotubes and single-walled nanotubes, phonon transport in carbon nanotubes, quantization of phonon thermal conductance, and phonon Anderson localization.

Published in London, UK © 2018 IntechOpen © BrianAJackson / iStock

Phonons in Low Dimensional Structures

Phonons in Low Dimensional

Structures

*Edited by Vasilios N. Stavrou*


*Article* **Apple (***Malus domestica* **Borkh.) Cultivar 'Majda', a Naturally Non-Browning Cultivar: An Assessment of Its Qualities**

**Anka Cebulj 1,\* , Andreja Vanzo 1 , Joze Hladnik 1 , Damijana Kastelec 2 and Urska Vrhovsek 3**

	- Hacquetova ulica 17, 1000 Ljubljana, Slovenia<EMAIL_ADDRESS>(A.V<EMAIL_ADDRESS>(J.H.)

**Abstract:** Browning of apple and apple products has been a topic of numerous research and there is a great number of methods available for browning prevention. However, one of the most efficient ways, and the one most acceptable for the consumers, is the selection of a non-browning cultivar. Cultivar 'Majda' is a Slovenian cultivar, a cross between 'Jonatan' and 'Golden Noble'. In this study, it was thoroughly examined and compared to the well-known cultivar 'Golden Delicious' with the aim to decipher the reason for non-browning. We have determined the content of sugars, organic acids, vitamin C, glutathione and phenolics in apple flesh, with the addition of phenolic content in apple peel and leaves. The change in color in halves and pomace was also measured and the activity of peroxidase (POX) and polyphenol oxidase (PPO) were determined. Additionally, the analyses of flesh were repeated post-storage. The most prominent results were high acidity (malic acid), low phenol content, especially hydroxycinnamic acid and flavan-3-ol content of cultivar 'Majda' in comparison to 'Golden Delicious', and no difference in PPO activity between cultivars. After the overview of the results, we believe that both low phenol content and high reduced glutathione content impact the non-browning characteristics of cultivar 'Majda'.

**Keywords:** non-browning; polyphenol oxidase; phenolics; vitamin C; glutathione