Ballistic and Diffusive Thermal Conductivity of Graphene

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As Published http://dx.doi.org/10.1103/PhysRevApplied.9.024017

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Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW APPLIED 9, 024017 (2018)

Ballistic and Diffusive Thermal Conductivity of Graphene

† Riichiro Saito,1,* Mizuno Masashi,1 and Mildred S. Dresselhaus2,3, 1Department of Physics, Tohoku University, Sendai 980-8578, Japan 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA 3Department of Electrical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA

(Received 2 August 2017; revised manuscript received 22 September 2017; published 20 February 2018)

This paper is a contribution to the Physical Review Appliedcollection in memory of Mildred S.Dresselhaus. Phonon-related thermal conductivity of graphene is calculated as a function of the temperature and sample size of graphene in which the crossover of ballistic and diffusive thermal conductivity occurs at around 100 K. The diffusive thermal conductivity of graphene is evaluated by calculating the phonon mean free path for each phonon mode in which the anharmonicity of a phonon and the phonon scattering by a 13C isotope are taken into account. We show that phonon-phonon scattering of out-of-plane acoustic phonon by the anharmonic potential is essential for the largest thermal conductivity. Using the calculated results, we can design the optimum sample size, which gives the largest thermal conductivity at a given temperature for applying thermal conducting devices.

DOI: 10.1103/PhysRevApplied.9.024017

I. INTRODUCTION many anharmonic terms are suppressed or give relatively small values compared to low-symmetry materials; (2) the Thermal conductivity of suspended monolayer graphene 5300 mass of a carbon atom is small, while large force constants has been reported to be very large up to W=mK at room 2 temperature (T ¼ 300 K) compared with copper of vibration exist due to the sp covalent bonding; and (400 W=mK) or diamond (2000 W=mK) [1], which is (3) the in-plane acoustic phonon modes have large phonon important for possible application of heat-transfer materials mean free paths due to the lack of interlayer coupling of the or low-dimensional thermoelectric materials [2,3]. Thermal phonons [5–9]. The small mass of a carbon atom with large conductivity of graphene occurs mainly by phonons, while force constants provides large group velocity for acoustic thermal conductivity of the conventional metal is mainly phonon modes with a large Debye temperature (2800 K) given by free electrons; this principle is known as the [10] of carbon compared to copper (343 K) [4]. Wiedemann-Franz law [4]. The thermal conductivity of a Within the harmonic term of the crystal potential, the phonon mode in the two-dimensional material is given by equation of motion for atoms can be diagonalized, from Cvl=2, where C, v, and l denote, respectively, the heat which we obtain phonon energy dispersion as a function capacity, the group velocity, and the mean free path of of the phonon wave vector q. When the anharmonic term of the phonon (MFP) [4]. The MFP l is given by l ¼ vτ, where the crystal potential that is expressed by the cubic term of τ is the relaxation time of phonons that is determined by the vibrational amplitude is introduced to Hamiltonian, the phonon-phonon scattering and phonon scattering by isotope. phonon wave vectors are no longer good quantum numbers Balandin and his co-workers have written comprehen- and phonon-phonon scattering occurs. By considering the sive review articles on the thermal conductivity of 2D anharmonic potential as a perturbation, the phonon relax- graphene, few-layer graphene, and one-dimensional (1D) ation time or MFP l can be calculated as a function of q graphene nanoribbon combined with their measurements, for each phonon mode, from which we can discuss the which contains comparison of theoretical methods and the diffusive thermal conductivity. The diffusive thermal con- difference of thermal conductivity between 2D or 1D graphene and 3D graphite [5–9]. Phonon thermal conduc- ductivity can be calculated by integrating the product l 2 tivity is large in graphene for the following reasons: Cv = over the Brillouin zone in which (1) the heat (1) because of the high symmetry of graphene crystal, capacity C of the material is calculated with the phonon density of states, (2) the group velocity v of the phonons is calculated with the phonon dispersion relation, and (3) the *[email protected] MFP is calculated with anharmonic terms. The total † Deceased. thermal conductivity is given as a function of T by

2331-7019=18=9(2)=024017(12) 024017-1 © 2018 American Physical Society SAITO, MASASHI, and DRESSELHAUS PHYS. REV. APPLIED 9, 024017 (2018) integrating Cvl=2 for all phonon modes and q over the control anharmonic terms for improving thermal conduc- Brillouin zone. tivity, it is important when considering possible applica- The MFP increases with decreasing T value, and tions to understand the microscopic picture of the diffusive thermal conductivity thus increases. When the anharmonicity of graphene. In this paper, we focus only MFP l of the sth phonon mode is larger than the sample on the contribution from each phonon mode for a wide size L, ballistic thermal conductivity for the sth phonon range of temperatures in which we predict a crossover from mode occurs in which the phonon-distribution difference diffusive to ballistic thermal transport with decreasing between the high-T side and the low-T side of the sample temperature which depends on the sample size L. generates the heat flux, which is proportional to the product The organization of this paper is as follows. In Sec. II, of the group velocity of the phonon and the heat capacity. we show how to construct anharmonic terms of graphene The ballistic thermal conductivity monotonically decreases with the tight-binding method. We will discuss the formu- with decreasing T value since the heat capacity below the lation of thermal conductivity for diffusive and ballistic Debye temperature monotonically decreases to zero with thermal transport of phonons. In Sec. III, calculated results decreasing T value to T ¼ 0 K. Thus, the maximum of thermal conductivity as a function of T and L will be 13 thermal conductivity appears at a crossover temperature shown for different concentrations of C isotopes. In from diffusive to ballistic thermal conductivity, which Sec. IV, a discussion and a summary are given. should depend on the sample size L. In this paper, we calculate thermal conductivity of graphene as a function of II. METHOD T and L by the so-called force-constant model [11,12] while considering anharmonic force constants. A. Anharmonic Hamiltonian Chen and co-workers reported [13,14] that the thermal A Hamiltonian for the lattice vibration of graphene conductivity of a suspended single crystal of graphene which includes third-order anharmonic terms is expressed monotonically increases with decreasing T value or with by [20] decreasing concentrations of an isotope of carbon, 13C, up 13 to 0%. The temperature dependence and the C concen- H ¼ K þ V2 þ V3 tration dependence of the thermal conductivity above X 2 X X jpij 1 ðijÞ 300 K can be explained, respectively, by diffusive thermal ¼ þ Φαβ uiαujβ 13 2m 2 conductivity and phonon scattering by C because of the i i ij αβ different mass of 12C. If we estimate the MFP, we can 1 X X ΨðijkÞ α β γ ∈ predict that the thermal conductivity has a maximum at the þ 3! αβγ uiαujβukγð ; ; fx; y; zgÞ; ð1Þ lower temperature and then decreases with further decreas- ijk αβγ ing T values because the heat capacity becomes zero at where K, V2,andV3 denote, respectively, the kinetic energy, T ¼ 0 K. Paulatto et al. calculated anharmonic properties the harmonic potential, and the third-order anharmonic of graphite and graphene by an ab initio calculation in m p u which the phonon lifetime is expressed by the spectral potential, and i, i,and i denote, respectively, the mass, i width of phonon energy dispersions [15]. Anharmonic the momentum, and the displacement of the th atom from its u 0 ΦðijÞ ΨðijkÞ terms are obtained by empirical potentials, too, by other equilibrium position ( ¼ ). and are, respec- groups [16,17], though some empirical potentials do not tively, the second- and third-rank tensors of forces that are reproduce the phonon energy dispersion and the potentials defined by the second and third derivatives of potential V at are not stable for a small displacement of atoms. Although the equilibrium position, which are given by [21] the first-principles calculation does not require any empiri- ∂2 cal parameters, it is not clear from the calculated results ðijÞ V Φαβ ¼ ; ð2Þ which anharmonic terms are important for given T and L ∂uiα∂ujβ 0 values. In this paper, using the symmetry analysis, we propose the dominant anharmonic terms of graphene whose ∂3 ΨðijkÞ V values are fitted to reproduce the results of the experiment αβγ ¼ : ð3Þ ∂uiα∂ujβ∂ukγ 0 and the first-principles calculations. Our method can be used for other unknown materials once we can obtain In a previous calculation of the phonon dispersion of anharmonic potential parameters by first-principles calcu- graphene [11,12,22], we adopted the so-called force- lation [15,18]. constant model, in which the harmonic potential between Saito et al. calculated temperature dependence of the ith and jth atoms is given using a spring model such as ballistic thermal transport of electrons and phonons [19] in which they showed that some phonons contribute to the X 1 VðijÞ − ΦðijÞ − 2 thermal conductivity diffusely, while other phonons con- 2 ¼ 2 αβ ðuiα ujβÞ : ð4Þ tribute ballistically. Although we do not discuss how to αβ

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When we consider the case of i ¼ j in the summation of i lattice of graphene. For example, a third-order anharmonic ΦðijÞ 2 term, ðu − u Þ3, is an odd function of ðu − u Þ. and j in Eq. (1), a quadratic term such as αβ uiα in Eq. (4) B1y Ay B1y Ay is included in V2 in Eq. (1). Thus, we can adopt the force However, since the potential V is invariant for the C2 constant set in the previous calculation for obtaining the (or 180°) rotation around the center of the bond, the coefficient of the odd function should vanish. The same phonon dispersion. For given nearest-neighborffiffiffi A and B1 p discussion can be taken for any pair of nth nearest-neighbor atoms that exist at Að0; 0; 0Þ and B1 (a= 3; 0; 0) (where carbon atoms. a 2.46 Å is the lattice constant of graphene [22]), ¼ For a given potential V, the potential and a third-order respectively, we can select the following anharmonic terms, tensor such as ΨðAAB1Þ; ΨðAB1AÞ; ΨðAB1B1Þ; …; ΨðB1B1AÞ that is defined by Eq. (3) are related to one another by ðAB1Þ ð1Þ 3 V −ψ − ðAAB1Þ 3 ¼ r ðuB1x uAxÞ symmetry. For example, the nonzero terms of Ψ ψ ð1Þ − − 2 are given by þ ti ðuB1x uAxÞðuB1y uAyÞ ð1Þ 2 ψ − − ðAAB1Þ ð1Þ þ to ðuB1x uAxÞðuB1z uAzÞ ; ð5Þ Ψxxx ¼ −6ψ r ; 1 ΨðAAB1Þ ΨðAAB1Þ ΨðAAB1Þ 2ψ ð Þ; ð6Þ ð1Þ ð1Þ ð1Þ xyy ¼ yxy ¼ yyx ¼ ti where ψ r , ψ , and ψ to denote, respectively, third order ti ΨðAAB1Þ ΨðAAB1Þ ΨðAAB1Þ 2ψ ð1Þ force-constant parameters for nearest-neighbor carbon xzz ¼ zxz ¼ zzx ¼ to . atoms in radial (r), in-plane tangential (ti), and out-of- plane tangential (to) vibrations. In Fig. 1, we show Since the potential V is an even function of z for monolayer graphene, all nonzero anharmonic terms have a factor of corresponding third-order displacements for each term of 2 Eq. (5). The first term of Eq. (5) represents the anharmonic either uz or 1 (not uz). Further, since the A and B1 atoms term for the bond length. Since the increase of the potential exist along the x axis, a component such as xxy for the energy is smaller for a case of expanding bond length (AAB1) pair should be zero. For a general pair of (ijk), we − 0 get the following 14 nonzero components by symmetry of (uB1x uAx > ) than for a case of shrinking bond length − 0 ΨðijkÞ: αβγ ¼ xxx, xxy, xyx, xyy, xzz, yxx, yxy, yyx, yyy, (uB1x uAx < ) in the hexagonal lattice, we can put the αβγ ð1Þ yzz, zzx, zzy, zxz, and zyz. negative sign for ψ r > 0 in Eq. (5). The second term of ðAAB1Þ ðAB1AÞ Eq. (5) represents the anharmonic effect of increasing Once we obtain Ψ , the other term—Ψ , energy with increasing interatomic distance by bending ΨðAB1B1Þ, etc.,—is given by the following relationship: − 2 0 the bond [ðuB1y uAyÞ > ] in the direction of y (in plane) − 0 ΨðiijÞ ¼ ΨðijiÞ ¼ ΨðjiiÞ ¼ −ΨðijjÞ ¼ −ΨðjijÞ ¼ −ΨðjjiÞ: ð7Þ when the bond shrinks (uB1x uAx < ). The third term of Eq. (5) represents the anharmonic effect of increasing 2 Furthermore, for obtaining the tensor for the other pair of i, energy by bending the bond [ðu − u Þ > 0] in the B1z Az j, k atoms that does not exist along the x axis, we can use direction of z (out of plane) when the bond shrinks the rotation matrix U which connects to the i0, j0, k0 atoms (u − u < 0). B1x Ax along the x axis to the i, j, k atoms, Other anharmonic terms between the two atoms can be X set to zero by considering the symmetry of the hexagonal ðijkÞ Ψαβγ uiαujβukγ αβγ X ði0j0k0Þ ¼ Ψαβγ ðUuiÞαðUujÞβðUukÞγ αβγ X ði0j0k0Þ ¼ Ψαβγ UαluilUβmujmUγnukn αβγ lmn XX ði0j0k0Þ ¼ Ψαβγ UαlUβmUγn uilujmukn: ð8Þ lmn αβγ

ω k 1 … 3 The phonon dispersion relation sð Þ,(s ¼ ; ; Natom, where Natom is the number of atoms in the unit cell) is calculated by solving the dynamical matrix [22] by taking the Fourier transform of uj and by considering only the harmonic term V2. Then we can obtain the phonon e κ q κ 1 … FIG. 1. Anharmonic terms for the pair of carbon atoms A and eigenvectors, ð j sÞ,(¼ ; ;Natom) for the wave B1 shown in Eq. (5). vector q.

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0 0 00 00 When we label uj as uðlκÞ at the κth atom in the unit cell Ψðqs; q s ; q s Þ l u lκ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with lattice vector , the Fourier transform of ð Þ and its X 3 p lκ 1 ℏ momentum ð Þ are given by ¼ pffiffiffiffi 8mκmκ0 mκ00 ωq ωq0 0 ωq00 00 N κ;κ0;κ00 s s s 1 X X X X qκ ffiffiffiffi u lκ − q l 0 0 00 00 0 0 00 00 ð Þ¼p ð Þ expð i · Þ; ð9Þ × exp½iðq · l þ q · l Þ Ψαβγð0κ; l κ ; l κ Þ N l l0;l00 αβγ κ q κ0 q0 0 κ00 q00 00 1 X × eαð j sÞeβð j s Þeγð j s Þ: ð18Þ PðqκÞ¼pffiffiffiffi pðlκÞ expðiq · lÞ; ð10Þ N l When we expand AqsAq0s0 Aq00s00 in Eq. (16) by using Eq. (17), we get the following processes: (1) three-phonon where N denotes the number of unit cells in the crystal. † † † creation (aqsaq0s0 aq00s00 ), (2) one-phonon splits into two Furthermore, XðqκÞ and PðqκÞ can be expressed by a linear † † combination of the phonon eigenvectors eðκjqsÞ, phonons (aqsaq0s0 a−q00s00 ), (3) two phonons merging into † X one phonon (aqsa−q0s0 a−q00s00 ), and (4) three-phonon annihi- 0 0 00 00 XðqκÞ¼ XqseðκjqsÞ; ð11Þ lation (a−qsa−q s a−q s ). Since we discuss the phonon s scattering process in thermal conductivity, we consider only X processes (2) and (3), which satisfy energy conservation. In Fig. 2, we show (a) the merging process and (b) the PðqκÞ¼ Pqse ðκjqsÞ; ð12Þ s splitting process, where each process consists of the normal process (G ¼ 0) and the Umklapp process (G ≠ 0) in the δ where Xqs and Pqs are, respectively, operators of the momentum conservation, qþq0þq00;G, in Eq. (16). amplitude and the momentum for the sth phonon energy In the case of the normal process, the total momentum of band. Defining the annihilation and creation operators for phonons or the heat flux does not change and thus does not the sth phonon mode as contribute to the thermal resistivity, while, in the Umklapp process, since the total momentum can be changed by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏ reciprocal lattice vectors, it can change the heat flux or − † − thermal resistivity [4]. The normal process contributes to Xqs ¼ i 2 ω ðaqs a−qsÞ; ð13Þ mκ qs the relaxation of the distribution of phonons only at thermal equilibrium states. rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ℏmκωqs † Pqs ¼ 2 ðaqs þ a−qsÞ; ð14Þ (a) Merging process (b) Splitting process the Hamiltonian for the harmonic oscillation, H0 ¼ KþV2, and the anharmonic Hamiltonian V3 are, respectively, written as X 1 H K V ℏω † 0 ¼ þ 2 ¼ qs aqsaqs þ 2 ð15Þ qs and

1 X V δ Ψ q q0 0 q00 00 3 ¼ 3! qþq0þq00;G ð s; s ; s Þ qsq0s0q00s00

× AqsAq0s0 Aq00s00 ; ð16Þ where Aqs is defined by FIG. 2. (a) Merging process and (b) splitting process in the ≡ † − anharmonic Hamiltonian of Eq. (16) that contributes to the Aqs aqs a−qs: ð17Þ δ thermal conductivity. The momentum conservation, qþq0þq00;G, is shown for both the hexagonal Brillouin zone in which it is the Ψðqs0q0s0; q00s00Þ in Eq. (16) is the Fourier transform of normal process (G ¼ 0, middle panels) and in which it is the Ψð0κ; l0κ0; l00κ00Þ that is written as Umklapp process (G ≠ 0, bottom panels).

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ℏω B. Diffusive thermal conductivity − ¯ − qs ¯ ¯ 1 τ v ∇ nqs nqs ¼ 2 nqsðnqs þ Þ qs qs · T: ð22Þ There are two kinds of thermal conduction by phonons. kBT One is diffusive conduction and the other is ballistic conduction. In diffusive conduction, the heat is transported The derivation of Eq. (22) is given in Appendix A. by phonons that are scattered many times, while in the We can define the specific heat of lattice Cvqs, per unit q ballistic conductance, the heat is transported by phonons volume, for the sth phonon mode at and the specific heat without scattering. If the sample size L is sufficiently larger of lattice Cv, per unit weight, as, respectively, than the MFP for each phonon mode, diffusive conduction ℏ2 occurs, while, when L is comparable to or smaller than the ω2 ¯ ¯ 1 Cvqs ¼ 2 qsnqsðnqs þ Þ; MFP, ballistic conductance occurs. VkBT The definition of thermal conductivity κ is given by V X Fourier’slaw, Cv ¼ 2 Cvqs: ð23Þ NmC qs

Q ¼ −κ∇T; ð19Þ Using Eq. (23), the diffusive thermal conductivity [Eq. (21)] is conventionally expressed by the product of Cvqs, the MFP Λqs ¼ τqsvqsð∥vqsÞ, and the group velocity in which Q and ∇T represent, respectively, the heat flux per vqs as follows [2]: unit time and per unit cross section, and the temperature gradient. Q is given by 1 X κdiff Λ v ¼ 2 Cvqs qs · qs: ð24Þ qs 1 X Q ¼ ℏωqsðnqs − n¯ qsÞvqs; ð20Þ V qs C. Ballistic thermal conductivity If L is smaller than the MFP, the phonon can propagate where nqs and n¯ qs denote, respectively, the number of q from one end to the other without scattering. In this phonons per unit volume for the wave vector and for the case, we can assume that the phonon moves in one direction sth phonon mode, as well as that for thermal equilibrium from the high-temperature side to the low-temperature side ℏω states. In Eq. (20), qs denotes the heat that the sth with the group velocity [19]. Making a comparison with − ¯ phonon carries, nqs nqs represents an effective number of Eq. (24), the ballistic thermal conductivity is given by [18] v the phonons that contribute to the thermal conduction, qs is X ball the group velocity of the phonon, and V is the volume of the κ ¼ L CvqsvqsxΘðvqsxÞ; ð25Þ sample. As for the volume V, we adopt the conventional qs Ω definition of volume for this problem as V ¼ N dgraphite, Θ 0 where N, Ω, and d ¼ 3.35 Å denote, respectively, the where ðvqsxÞ is either 1 for vqsx > (a right-going graphite 0 number of sampling points in the Brillouin zone, the area of phonon) or 0 for vqsx < (a left-going phonon). If we κdiff κball the unit cell of graphene, and the interlayer distance take the ratio qs = qs , we get between two graphene layers in graphite. κdiff π Λ Here, we assume that the heat flux is parallel to the qs j qsj ¼ : ð26Þ Q∥∇ ball 2 temperature gradient ( T) and that the thermal con- κqs L duction is isotropic in the two-dimensional plane. Then the diffusion thermal conductivity is given by In order to discuss the crossover of the diffusive and ballistic conduction, we define the MFP for the sth phonon 1 Q · ∇T mode and for all phonons, respectively, given as a function κdiff ¼ − of T, as follows: 2 j∇Tj2 P P 1 X κdiff Λ v − ℏω − ¯ v ∇ q qs qCvqs qs · qs ¼ 2 qsðnqs nqsÞ qs · T hΛ i¼P ¼ P ; ð27Þ 2Vj∇Tj s diff qs qκqs =jΛqsj qCvqsjvqsj 2 X ℏ 2 2 P P ωq τq n¯ q n¯ q 1 vq : 21 κdiff Λ v ¼ 2 2 s s sð s þ Þj sj ð Þ qs qs qsCvqs qs · qs VkBT q hΛ i¼P ¼ P : ð28Þ s All κdiff Λ qs qs =j qsj qsCvqsjvqsj

In the last line of Eq. (21), we use the following relationship Finally, using Eq. (26), we define the thermal conduc- from the Boltzmann equation: tivity κ that includes both κdiff and κball as follows:

024017-5 SAITO, MASASHI, and DRESSELHAUS PHYS. REV. APPLIED 9, 024017 (2018) X X π Λ q0s0 0 0 0 0 0 j qsj ball Mqs ¼ Δmlκe ðκjqsÞ · e ðκjq s Þ exp½−iðq − q Þ · xlκ: κ ¼ min ; 1 × κq 2 s lκ qs L X ball ð35Þ κqs Λqs > 2L=π ¼ : ð29Þ κdiff Λ 2 π qs qs qs < L= Using Fermi’s “golden rule,” the scattering probability from the qs phonon to the q0s0 phonon is given by In this definition, for each q and sth phonon mode, if the Λ 2 π κball q0 0 2π MFP qs is smaller than L= , we adopt , while, if the s − 1 1 2δ ω − ω Pqs ¼ 2 jhnqs ;nq0s0 þ jHmdjnqs;nq0s0 ij ð q0s0 qsÞ MFP is larger than 2L=π, we adopt κdiff. ℏ π q0s0 2 ¼ nq ðnq0 0 þ 1Þωq ωq0 0 jMqs j δðωq0 0 − ωq Þ: 2ðNm¯ Þ2 s s s s s s D. Phonon scattering by isotope impurities ð36Þ Finally, we briefly mention the phonon scattering by isotope impurities, which was discussed by Klemens [23]. The decreasing rate of the number of qs phonon is given by Since the mass of atom mlκ depends on the position of atom lκ, the momentum of the atom at lκ is written as ∂ X nqs q0s0 qs − Pq − P 0 0 ; 37 rffiffiffiffiffiffiffiffiffiffi ∂ ¼ ð s q s Þ ð Þ t md q0 0 ℏm X ffiffiffiffiffiffiffi s p lκ lκ pω e0 κ q ð Þ¼ 2 qs ð j sÞ N qs and the spectral width of phonon energy dispersion due to † the isotope impurity is expressed by × ðaqs þ a−qsÞ expð−iq · xlκÞ; ð30Þ X ¯ 1 −1 πg nq0s0 þ q0s0 τ ω ω 0 0 Eq δ ω 0 0 − ω where xlκ is an equilibrium position of an atom at lκ. When qsðmdÞ ¼ qs q s s ð q s qsÞ; 2N n¯ q þ 1 we define the averaged mass by q0s0≠qs s X ð38Þ m¯ ¼ fimi; ð31Þ i q0s0 where Eqs represents the inner products of phonon eigenfunctions defined by where fi and mi denote, respectively, the abundance 12 13 X and the mass of i ¼ Cor C. The perturbation q0s0 0 0 0 0 2 Eqs ¼ je ðκjqsÞ · e ðκjq s Þj : ð39Þ Hamiltonian for the mass difference between isotope atoms κ is written as [23] The derivation of Eq. (38) is shown in Appendix B. Since X 1 1 Δ p lκ 2 Δ − ¯ the formula for the spectra width contains the inner product Hmd ¼ mlκ 2 j ð Þj ð mlκ ¼ mlκ mÞ: ð32Þ 2 m¯ q0s0 lκ of phonon eigenvectors Eqs [Eq. (39)], the in-plane (or out-of-plane) phonon mode is scattered only to the same 2 Using the following equation for jpðlκÞj , in-plane (or out-of-plane) phonon mode by the isotope impurities. jpðlκÞj2 ¼ pðlκÞ · p†ðlκÞ It is useful to define the averaged phonon frequency for a ¯ ℏ ¯ X ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi general isotope concentration. When we denote mnat and m p 0 0 0 0 ¼ ωq ωq0 0 e ðκjqsÞ · e ðκjq s Þ ω as the mass and the frequency for natural abundance of 2N s s nat qs;q0s0 1.1% 13C, the average phonon frequency is given by † † − q − q0 x × ðaq þ a−qsÞðaq0 0 þ a−q0 0 Þ exp½ ið Þ · lκ; rffiffiffiffiffiffiffiffi s s s ¯ ω mnatω ð33Þ ¼ nat: ð40Þ m¯

Hmd of Eq. (32) is given as X III. CALCULATED RESULTS ℏ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi † † q0s0 H ¼ ωq ωq0 0 ðaq a 0 0 þ a−q a−q0 0 ÞMq ; md 4Nm¯ s s s q s s s s A. Phonon dispersion relations with spectral width qs;q0s0 In order to calculate phonon dispersion relations, we adopt ð34Þ the harmonic force-constant parameters that we obtained in previous work [12] up to the 14th nearest neighbors by q0s0 where the matrix element Mqs is expressed as fitting the experimental results. As for anharmonic force

024017-6 BALLISTIC AND DIFFUSIVE THERMAL CONDUCTIVITY … PHYS. REV. APPLIED 9, 024017 (2018) constants, we fit the force constants up to the fourth nearest neighbor (4NN) so as to reproduce the temperature dependence of experimental thermal conductivity for 1.1% natural abundance of 13C [13,14]. We also try to fit the phonon dispersion relation with its spectral width by the first- principles calculation given by Paulatto et al. [15] for justifying that our fitting procedure to the experiment is reasonable. In Table I, we show the anharmonic force constants up to 4NN in units of eV=Å3 fitted to the experimental results (the top row) and the first-principles calculations (the bottom row). When we compare two of the anharmonic FIG. 3. Phonon dispersion with its spectral width by using force-constant sets, we find that (1) the force constants anharmonic parameters fitted to the experimental results. The for 1NN and 2NN give similar values to one another, but spectral width is magnified by 100 times. that (2) the force constants for 3NN and 4NN are rather different from one another. For example, we cannot adopt adopt the anharmonic potentials, we can see some singu- ð4Þ ð4Þ the 4NN value of ψ to, ψ to since the ψ to > 0 gives the larly large spectra width at the Γ point or the crossing points opposite temperature dependence of thermal conductivity of two phonon dispersion relations (not shown in Fig. 3). as that of the experimentally observed temperature depend- This singularity comes from the fact that the lowest-energy ence, and thus we cannot fit well to the experimental phonon mode near the Γ point, Z-axis accoustic (ZA, where results. The difference between the fitted values for the Z indicates that the displacement vector is along the Z axis), 3NN and 4NN force constants and those by first-principles can be easily excited by phonon-phonon scattering. In order calculation is reasonable since we cut the range of anhar- to avoid such a singularity, we do not consider (for the sake monic potential up to 4NN, while they consider a much of simplicity) the scattering process with ZA phonons for longer range of anharmonic potential in the first-principles wave vectors smaller than 0.06jΓKj. In fact, an out-of- calculations. It is difficult for us to consider a long-distance plane phonon with such a long wavelength does not exist in anharmonic potential since we do not have sufficient the samples, in which we expect defects of the lattice and information on experiments such as those involving a structural fluctuation, as the phonon is known to not be spectral width of phonon dispersion as a function of q. completely flat of graphene. In Fig. 3, we plot phonon dispersion relations with their The phonons that contribute to thermal conduction are spectral width, which is magnified by 100 times. A large mainly ZA, transverse-acoustic (TA), and longitudinal- spectral width for a given s and q corresponds to large acoustic (LA) phonons that have relatively small energies. thermal resistivity. Furthermore, the phonon modes with Even if we plot the spectral width of the phonon mode by lower energies contribute to thermal conductivity at lower considering only ψ to, we can reproduce the spectral width values of T. Thus, as long as we consider T<700 K, the of phonon modes below 900 cm−1. Thus, we can say that spectral widths of the optical phonon modes such as the anharmonic force constant ψ to is essential of total longitudinal optical (LO), tangential optical (TO) and Z- thermal resistivity. Furthermore, when we adopt the anhar- axis optical (ZO, where Z indicates that the displacement monic force constants fitted to first-principles calculations, vector is along Z axis) do not contribute to thermal the calculated temperature dependence of thermal conduc- resistivity since such phonons rarely exist. If we simply tivity increases more slowly with decreasing values of T compared to that in the experiment. This is the reason we 3 adopt to fit the anharmonic force constants to the exper- TABLE I. Anharmonic force constant in units of eV=Å by imental results. fitting to the experimental data (the top row) and by fitting to the first-principles calculation (the bottom row). In the calculation of thermal conductivity, the scattering of isotope impurity is also taken into account when we ψ r ψ ti ψ to make a comparison to the experimental results. In Fig. 4, we plot the spectral width of the phonon dispersion relation 1NN 37.016 4.230 1.852 13 37.275 5.270 4.379 for the case with 1.1% C concentration, in which we 2NN −2.071 8.174 0.519 consider not anharmonic terms but rather isotope effects for −1.995 8.197 0.518 the spectral width, which is magnified by 1000 times. As is 3NN 0.533 −0.433 −0.041 seen in Fig. 4, the spectral width is large for optical phonon 0.331 −0.248 0.000 modes. However, when we compare the spectral width in Fig. 4 to that in Fig. 3, the former is relatively small. Since 4NN 0.142 −1.262 −0.244 −1.197 0.484 the spectra width for isotope scattering is proportional to the phonon frequencies at initial and final phonon states

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the experimental values for 300–750 K well—especially the fitted values for 1.1% and 99.2% 13C, where the values are related to each other in a manner similar to isotope impurity, which is consistent with the experimental results. However, we can see a non-negligible deviation of the calculated results from the experiments for 50% and 0.0% 13C for 300–500 K. In the case of 50% (or 0%) concentration, the scattering of phonons by a 13C isotope cannot be expressed by perturbation [or by an approximation of relaxation time represented by a single phonon mode; see Eq. (B6)]. Thus, the quantitative analysis for a general 13 FIG. 4. The spectral width of phonon dispersion by scattering concentration of C has some limitations for temperatures of a 1.1% 13C isotope. The spectral width is magnified by 1000 lower than 500 K. times. In this plot, the spectra width of the phonon taken by Although the experimental results are given only when anharmonicity is not taken into account. T>300 K, we extend the calculated results to the lower temperature region while using the same parameters. The [see Eq. (38)], the spectral width of ZA phonon is relatively calculated thermal conductivity shows a maximum at small. It is noted that the spectral width of ZO phonon around 100 K for all cases of concentration of 13C whose mode becomes small around the Γ point due to the fact that peak position appears at a lower temperature for a there are not many scattered phonon modes as the final state smaller concentration of 13C. Increasing the thermal Γ near the point. conductivity with decreasing values of T is explained by decreasing the phonon scattering events. If we further B. Thermal conductivity decrease T, thermal conductivity monotonically In Fig. 5, we plot the calculated thermal conductivity decreases to zero, which is explained by a decreasing as a function of temperature for 13C concentrations of 0.0%, number of phonons (or by a vanishing heat capacity), 1.1%, 50%, and 92.2%, using the fitting parameters that are which is known to be a general behavior of thermal fitted to the experimental results of 1.1% 13C for the conductivity of a solid [24]. – In Fig. 6, we show calculated thermal conductivity for temperature region 300 750 K. Open and solid symbols 13 denote, respectively, the calculated results and the exper- 1.1% C for different sample sizes L. The value of L can be imental results of Chen et al. [13] for comparison. As seen considered the size of a single crystal of graphene in the in Fig. 5, the calculated thermal conductivity reproduces polycrystalline sample. As shown in Fig. 6, the maxima of the thermal conductivity increases with increasing values of L at the lower temperature. It is for this reason that the criterion for selecting either ballistic or diffusive [Eq. (29)] depends on L. Once the MFP for any phonon mode is larger than L, the thermal conductivity becomes the ballistic thermal conductivity, κball, which is proportional to L

FIG. 5. Thermal conductivity as a function of temperature for several concentrations of 13C using anharmonic force constants that are fitted to the experimental data for a natural abundance of 1.1% 13C. For temperatures above 300 K, solid symbols denote FIG. 6. Thermal conductivity for sample sizes L ¼ 1, 2, 5, 10, 13 the experimental data by Chen et al. [13], while, below 300 K, and 20 μm. Here, we consider the isotope scattering by 1.1% C. only theoretical calculated results are shown. Below 100 K, The dashed line for each L corresponds to the ballistic thermal thermal conductivity decreases since the number of phonon conductivity, κball, while the dotted lines represent diffusive decreases with decreasing temperature. thermal conductivity, κdiff .

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[see Eq. (25)] or the square of the number of phonons. In the ZA phonon modes that have the lowest phonon energy Fig. 6, we show the κball by dashed line for each L as the among the phonon modes are important for determining ballistic limit. Furthermore, we also show κdiff without the average MFP or phonon occupation numbers. Thus, considering the crossover to κball shown by a dotted line as when we consider the ballistic and diffusive thermal conduc- the diffusive limit. It is interesting to see that the thermal tivity, the ZA mode gives a dominant contribution to both conductivity gives a different temperature dependence effects of thermal conductivity. However, the ZA mode is simply by changing L for the same sample quality of a sensitive to the substrate or stacking order of multilayer crystal. The L-dependent thermal conductivity at low graphene, especially for the case of long wavelength. If we temperature is evidence of the crossover of thermal consider the ZA phonons near the Γ point, a large phonon conductivity from diffusive to ballistic thermal conductivity scattering occurs for the crossing point of phonon dispersion, which should be observed in the experiment. For a temper- which would result in too much thermal resistivity to explain ature larger than 300 K, on the other hand, since the MFP the experimental results. Thus, it is not clear from the present for each phonon mode is smaller than L, the L-dependent calculation what kind of substrate is the best for graphene thermal conductivity does not appear anymore and the thermal conducting devices. If we can control the ZA phonon thermal conductivity is expressed only by diffusive thermal modes and their anharmonicity, the thermal properties would conductivity, in which phonon-phonon scattering is a be significantly improved, which is a problem to address in dominant contribution to thermal resistivity. the future. In conclusion, using a tight-binding description of harmonic and anharmonic vibration, we calculate the IV. DISCUSSION AND SUMMARY thermal conductivity in which not only phonon-phonon Finally, let us briefly comment on the application of scattering, but also isotope scattering of phonons, is taken graphene to thermal conductive devices. It is important to into account in the calculation. Fitting the harmonic and use the maximum thermal conductivity of graphene at anharmonic force-constant sets to the experiment and the relatively low temperature. However, as we can see in first-principles calculations, we calculate thermal conduc- Fig. 6, if we use graphene at 300 K, the sample size tivity as a function of temperature and the sample size L. L ¼ 2 μm is already sufficient since thermal resistivity by In the low-temperature region, we see the maximum of phonon-phonon scattering is dominant. Even if we pre- thermal conductivity, and the thermal conductivity goes to pare a better quality of sample whose L value is larger zero with a further decreasing temperature. The maximum than 2 μm, the value of thermal conductivity does not value of the thermal conductivity and the corresponding change at 300 K, which we can see in Fig. 6. On the other temperature depends on the sample size L for the same hand, when we increase L, we expect that the maximum sample. It is for this reason that the crossover of diffusive value of thermal conductivity increases at lower temper- thermal transport to ballistic thermal transport occurs with atures. Nevertheless, L ¼ 10 μm is already sufficiently decreasing temperature. large for the use of thermal conducting devices since the maximum of the diffusive limit has a maximum at around ACKNOWLEDGMENTS 100 K, as shown in Fig. 6.SuchL-dependent thermal conductivity should be observed in the experiment even R. S. acknowledges MEXT KAKENHI Grants though there is some quantitative limitation of calculation. No. JP25107005 and No. JP15K21722. We sincerely thank If we adopt a single crystal without any 13Cisotope,the the late Professor Mildred S. Dresselhaus for this collabo- maximum thermal conductivity becomes much larger ration; she passed away in February 2017 before finishing (6500 W=mK in our calculation; not shown in this paper) the project of thermal conductivity. than in the case of 1.1% 13C, as shown Fig. 6[18],andthe maximum temperature for 0% 13C decreases to 80 K. In APPENDIX A: NUMBER OF PHONONS THAT this case, L ¼ 20 μmisneededtoobtainthemaximum CONTRIBUTE TO κdiff ballistic thermal conductivity at 80 K. Such behaviors should be checked experimentally in the future, too. It is Here, we derive Eq. (22). Occupation numbers of important to note that the data in Fig. 6 do not mean that phonons, nqs, as a function of time, are calculated using polycrystalline graphene is sufficient for temperatures the Boltzmann equation, higher than 400 K. It is for this reason that we do not ∂nq ∂nq ∂nq consider the phonon scattering at the domain boundary of s s s 0 ∂ ¼ ∂ þ ∂ ¼ ; ðA1Þ the polycrystalline sample, which might affect the thermal t t diff t scatt conductivity if the MFP is larger than L.Whenwe consider the effect of phonon scattering at the boundary, where the subscripts diff and scatt denote, respectively, we need to discuss the interference effect of multiple diffusive and scattering terms. The diffusive terms can be reflected phonons. It is important to note, in addition, that given by

024017-9 SAITO, MASASHI, and DRESSELHAUS PHYS. REV. APPLIED 9, 024017 (2018) ∂nq ∂nq s −v ∇ s Adding Eqs. (A3) and (A7) to the Boltzmann ∂ ¼ qs · T ∂ ; ðA2Þ equation (A1), we get t diff T where vq ¼ ∇qωq is the group velocity of the phonons s s Xqs ¼ Γqsψ qs: ðA8Þ at qs and ∇T indicates the temperature gradient. ð∂nqs=∂TÞ∇T represents the spacial change of the phonon By solving the Boltzmann equation, we get ψ q as follows: occupation number caused by the temperature gradient. s ≃ ¯ ∂ ∂ ≃∂¯ ∂ When we assume nqs nqs and thus nqs= T nqs= T, −1 ψ qs ¼ Γqs Xqs Eq. (A2) can be written by defining Xqs, τq s ¼ ¯ ¯ 1 Xqs ∂nq ∂n¯ q nqsðnqs þ Þ s −v ∇ s ≡ ∂ ¼ qs · T ∂ Xqs: ðA3Þ ℏω t diff T − qs τ v ∇ ¼ 2 qs qs · T: ðA9Þ kBT Here, we adopt the relaxation-time approximation (RTA) as follows: Finally, using the definition of ψ qs [Eq. (A5)], we get

∂nq ∂ðnq − n¯ q Þ nq − n¯ q − ¯ − s − s s s s nqs nqs ¼ ¼ ; ðA4Þ ψ q ¼ ; ðA10Þ ∂t ∂t τq s s n¯ qsðn¯ qs þ 1Þ τ where qs is the relaxation time that determines how fast the and thus we get Eq. (22): nqs decays. We define a dimensionless parameter ψ qs in the Boltzmann distribution function for expanding nq [20], s ℏωqs nq − n¯ q ¼ − n¯ q ðn¯ q þ 1Þτq vq · ∇T: ðA11Þ s s 2 s s s s −1 kBT ℏωqs nq ¼ exp − ψ q − 1 s k T s B ∂ nqs ≃ nq jψ 0 þ ψ q s qs¼ s APPENDIX B: SPECTRAL WIDTH OF PHONON ∂ψ qs ψ 0 qs¼ BY ISOTOPE SCATTERING ¼ n¯ q þ n¯ q ðn¯ q þ 1Þψ q : ðA5Þ s s s s Here, we show that the spectral width by scattering of the phonons by isotope impurity is given by Eq. (38). Let us Using Eq. (A5), Eq. (A4) can be written as [20] start the discussion by rewriting Eq. (36):

0 0 2π ∂nqs n¯ qsðn¯ qs þ 1Þ q s 2 − ψ Pq ¼ jhnq − 1;nq0 0 þ 1jH jnq ;nq0 0 ij δðωq0 0 − ωq Þ ¼ qs: ðA6Þ s ℏ2 s s md s s s s ∂t τqs π q0s0 2 ¼ nq ðnq0 0 þ 1Þωq ωq0 0 jMqs j δðωq0 0 − ωq Þ: 2ðNm¯ Þ2 s s s s s s Furthermore, using the RTA in Eq. (A6) and defining Γqs, the scattering term is given as follows: ðB1Þ ∂nq n¯ q n¯ q 1 s − sð s þ Þ ψ ≡ −Γ ψ If we assume that the isotope impurities distribute uni- ¼ qs qs qs: ðA7Þ 0 0 ∂t τq q s 2 scatt s formly and randomly, we approximate jMqs j as follows,

q0s0 2 q0s0 q0s0 jMq j ¼ Mq ðMq Þ s Xs s 0 0 0 0 0 0 0 0 0 0 0 ¼ ΔmlκΔml0κ0 × ½e ðκjqsÞ · e ðκjq s Þ½e ðκ jqsÞ · e ðκ jq s Þ × exp½−iðq − q Þ · ðxlκ − xl0κ0 Þ lκ l0κ0 X; X 2 0 0 0 0 2 ¼ ðΔmlκÞ je ðκjqsÞ · e ðκjq s Þj þ ðsame as aboveÞ lκ l0κ0≠lκ X X 2 0 0 0 0 2 ≃ N fiðΔmiÞ je ðκjqsÞ · e ðκjq s Þj i κ 2 q0s0 ¼ Ngm¯ Eqs ; ðB2Þ

024017-10 BALLISTIC AND DIFFUSIVE THERMAL CONDUCTIVITY … PHYS. REV. APPLIED 9, 024017 (2018) where fi and Δmi denote, respectively, the concentration Chun Ning Lau, Superior thermal conductivity of single- and the differing masses of isotopes i ¼f12C; 13Cg. Here, layer graphene, Nano Lett. 8, 902 (2008). q0s0 [2] L. D. Hicks and M. S. Dresselhaus, Thermoelectric figure we define g and Eq as follows: s of merit of a one-dimensional conductor, Phys. Rev. B 47, X 16631 (1993). 2 g ≡ fið1 − mi=m¯ Þ ; ðB3Þ [3] L. D. Hicks and M. S. Dresselhaus, The effect of quantum i well structures on the thermoelectric figure of merit, Phys. Rev. B 47, 12727 (1993). and [4] C. Kittel, Introduction to Solid State Physics, 6th ed. X (John Wiley & Sons, New York, 1986). q0s0 0 0 0 0 2 [5] A. A. Balandin, Thermal properties of graphene and Eqs ≡ je ðκjqsÞ · e ðκjq s Þj : ðB4Þ κ nanostructured carbon materials, Nat. Mater. 10, 569 (2011). [6] D. L. Nika and A. A. Balandin, Two-dimensional phonon Then Eq. (B1) is given by transport in graphene, J. Phys. Condens. Matter 24, 233203 (2012). q0s0 πg q0s0 Pqs ¼ nqsðnq0s0 þ 1Þωqsωq0s0 Eqs δðωq0s0 − ωqsÞ: ðB5Þ [7] D. L. Nika, E. P. Pokatilov, and A. A. Balandin, Theoretical 2N description of thermal transport in graphene: The issues of phonon cut-off frequencies and polarization branches, The decreasing rate of phonons by isotope scattering per Phys. Status Solidi (b) 248, 2609 (2011). unit time is expressed by [8] S. Gosh, W. Bao, D. L. Nika, S. Subrina, E. P. Pokatilov, C. N. Lau, and A. A. Balandin, Dimensional crossover of ∂ X nqs q0s0 qs thermal transport in few-layer graphene, Nat. Mater. 9, 555 − Pq − P ∂ ¼ ð s q0s0 Þ (2010). t md q0 0 Xs [9] D. L. Nika and A. A. Balandin, Phonons and thermal ¯ q0s0 transport in graphene and graphene-based materials, Rep. ¼ Pqs ðψ q − ψ q0 0 Þ s s Prog. Phys. 80, 036502 (2017). q0s0 [10] M. S. Fuhrer, Tuning the area of the fermi surface of X πg ¼ n¯ q ðn¯ q0 0 þ 1Þðψ q − ψ q0 0 Þωq ωq0 0 graphene demonstrates the fundamental physics of elec- 2N s s s s s s q0s0 tron-phonon scattering, Physics 3, 106 (2010). [11] R. Saito, T. Takeya, T. Kimura, G. Dresselhaus, and M. S. q0s0 × Eqs δðωq0s0 − ωqsÞ Dresselhaus, Raman intensity of single-wall carbon nano- X πg tubes, Phys. Rev. B 57, 4145 (1998). ¼ n¯ q ðn¯ q0 0 þ 1Þψ q ωq ωq0 0 2N s s s s s [12] R. Saito, M. Furukawa, G. Dresselhaus, and M. S. q0s0≠qs Dresselhaus, Raman spectra of graphene ribbons, J. Phys. q0s0 Condens. Matter 22, 334203 (2010). × Eq δðωq0 0 − ωq Þ: ðB6Þ s s s [13] S. Chen, Q. Wu, C Mishra, J. Kang, H. Zhang, K. Cho, W. Cai, A. A. Balandin, and R. S. Ruoff, Thermal conductivity In the last line, we use the so-called single-mode relaxation- of isotopically modified graphene, Nat. Mater. 11, 203 time approximation. In this approximation, when we (2012). consider relaxation of a phonon mode, we postulate that [14] S. Chen, Q. Li, Q. Zhang, Y. Qu, H. Ji, R. S. Ruoff, and the other phonon modes are in the thermal equilibrium W. Cai, Thermal conductivity measurements of suspended states. This approximation makes the calculation simple graphene with and without wrinkles by micro-Raman since the relaxation of each phonon occurs independently. mapping, Nanotechnology 23, 365701 (2012). We believe that this approximation is justified as long as [15] L. Paulatto, F. Mauri, and M. Lazzeri, Anharmonic properties from a generalized third-order ab initio approach: nqs does not deviate much from n¯ qs. Finally, using the RTA [Eq. (A6)] for Eq. (B6), we get Eq. (38): Theory and applications to graphite and graphene, Phys. Rev. B 87, 214303 (2013). X ¯ 1 [16] L. Lindsay and D. A. Broido, Optimized Tersoff and −1 πg nq0s0 þ q0s0 τ ω ω 0 0 Eq δ ω 0 0 − ω Brenner empirical potential parameters for lattice dynamics qsðmdÞ ¼ qs q s s ð q s qsÞ: 2N n¯ q 1 q0s0≠qs s þ and phonon thermal transport in carbon nanotubes and graphene, Phys. Rev. B 81, 205441 (2010). ðB7Þ [17] V. K. Tewary and B. Yang, Parametric interatomic potential for graphene, Phys. Rev. B 79, 075442 (2009). [18] M. Mizuno, master’s thesis, Tohoku University, 2014 (in Japanese). [19] K. Saito, J. Nakamura, and A. Natori, Ballistic thermal [1] Alexander A. Balandin, Suchismita Ghosh, Wenzhong Bao, conductance of a graphene sheet, Phys. Rev. B 76, 115409 Irene Calizo, Desalegne Teweldebrhan, Feng Miao, and (2007).

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[20] G. P. Srivastava, The Physics of Phonons (CRC Press, [23] P. G. Klemens, The scattering of low-frequency lattice London, 1990). waves by static imperfections, Proc. Phys. Soc. London [21] A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Theory Sect. A 68, 1113 (1955). of Lattice Dynamics in the Harmonic Approximation [24] T. Takeuchi, N. Nagasako, R. Asahi, and U. Mizutani, (Academic Press, New York, 1963). Extremely small thermal conductivity of the Al-based [22] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Mackay-type 1=1-cubic approximants, Phys. Rev. B 74, Properties of Carbon Nanotubes (Imperial College Press, 054206 (2006). London, 1998).

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