Length Dependence of Carbon Nanotube Thermal Conductivity and the

NANO LETTERS

Length Dependence of Carbon Nanotube Thermal Conductivity and the “Problem of Long Waves”

2005 Vol. 5, No. 7 1221-1225

N. Mingo*,† and D. A. Broido‡ NASA Ames Center for Nanotechnology, 229-1, Moffett Field, California 94035, and Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467 Received April 18, 2005; Revised Manuscript Received May 24, 2005

ABSTRACT We present the first calculations of finite length carbon nanotube thermal conductivity that extend from the ballistic to the diffusive regime, throughout a very wide range of lengths and temperatures. The long standing problem of vanishing scattering of the “long wavelength phonons” (Pomeranchuk, I. J. Phys. (U.S.S.R.) 1941, 4, 259; Phys. Rev. 1941, 60, 820) manifests itself dramatically here, making the thermal conductivity diverge as the nanotube length increases. We show that the divergence disappears if 3-phonon scattering processes are considered to second or higher order. Nevertheless, for defect free nanotubes, the thermal conductivity keeps increasing up to very large lengths (10 µm at 300 K). Defects in the nanotube are also able to remove the long wavelength divergence.

In 1941 Pomeranchuk pointed out that the mean free path of long wavelength longitudinal phonons in a bulk solid diverges as their frequency tends to zero, resulting in an infinite thermal conductivity for infinitely large bulk solids.1 Ziman named this “the problem of long longitudinal waves”.2 In the case of finite size samples, where the low-frequency phonon mean free path is limited by boundary scattering, this problem implies that the thermal conductivity would keep increasing with the sample’s size. For three-dimensional solids, this issue was addressed by Herring, who showed that the existence of degeneracy points in the phonon dispersions of a material can reduce the divergence to just a logarithmic one, or even remove it completely, depending on the character of the degeneracy.2,3 In quasi one-dimensional systems, this old problem acquires a new light, which is especially dramatic in suspended single-walled carbon nanotubes (SWNTs). Unlike bulky nanowires, SWNTs do not have an outer surface to provide “boundary scattering”. Therefore, for phonons traveling through a defect-free nanotube, the only scattering mechanisms are due to lattice anharmonicity, most importantly 3- phonon processes. A crucial question immediately comes to mind: are 3-phonon processes enough to scatter the long wavelength longitudinal phonons and lead to a finite thermal conductivity? Or, on the contrary, will the thermal conductivity diverge as the carbon nanotube’s length increases? By comparing nanotubes with strictly one-dimensional systems, it has been suggested that the latter should be the case.4,5 In this paper we answer this question. For † ‡

NASA Ames Center for Nanotechnology. Boston College.

10.1021/nl050714d CCC: $30.25 Published on Web 06/17/2005

© 2005 American Chemical Society

this, we iteratively solve the linearized Boltzmann-Peierls (BP) phonon transport equation for finite length SWNTs, individually computing the 3-phonon scattering events with full observance of their selection rules. We will show that, if only first-order 3-phonon processes are considered, the thermal conductivity diverges with nanotube length (L), as LR, where R can range between 1/3 and 1/2 depending on the linear or quadratic character of the acoustic branches.6 However, if 3-phonon processes to the second-order or higher are included, the thermal conductivity saturates to finite values, but the long wavelength phonons still dominate its large L limit, and the saturated thermal conductivity can be very large. Finally, we will show that the presence of defects can efficiently scatter the long wavelength phonons and eliminate their otherwise dominant role in the heat conduction process. The BP equation fully describes the flow of interacting phonons in a solid.7 Its exact solution was considered intractable for many decades,2 and several methods were devised to obtain approximate solutions.2 Only very recently, and partly helped by the tremendous advances in computer speed and memory, was an algorithm proposed to obtain a virtually exact solution of the full BP equation via an iterative procedure.8 In the present case, we need to extend this iterative procedure to treat spatially varying systems, such as the finite length carbon nanotubes we are interested in. This can be done in the following way. We consider a SWNT with length L connected on either end to reservoirs at temperatures, Th and Tc such that an infinitesimal temperature difference ∆T ) Th - Th is established. In a space dependent representation, the BP equation is2,9

dnp -Vp ) ∂cnp dx

(1)

where p ) {q,R} denotes the phonon’s wave vector, q, and branch index, R, np is the distribution function, and Vp is the phonon group velocity. The coordinate x is measured along the nanotube’s axial direction. It is convenient to define the quantities g(x), such that np(x) ≡ n + n(n + 1)g(x), with n ≡ (epω/kBT - 1)-1 being the Bose distribution evaluated at temperature T ) (Th + Tc)/2. The small temperature gradient allows a linearized approximation to the BP equation for which the collision term, ∂cnq, is2,9 ∂cnp/(n(n + 1)) )

automatically recovered. To see this, we note that in the ballistic limit the distribution of phonons traveling to the right equals the undisturbed distribution of phonons exiting the left-hand side reservoir, and vice versa: npf ) n(Th, ωp) ) n +

dn ∆T dT 2

(4)

npr ) n(Tc, ωp) ) n -

dn ∆T dT 2

(5)

Using dn/dT ) (pω/kBT2)n(n + 1), and the definition of g, the above equations imply that

p′′ (g′′ - g′ - g) + ∑ (n′ - n′′)Q p,p′

gpf(r) ) + (-)

dir(p)

1

∑ (1 + n′ + n′′)Q p′,p′′ p (g′′ + g′ - g)

2 rev(p)

dn ∆T nc ) n dT 2 and dn ∆T dT 2

The boundary condition means that phonons coming out from the reservoir into the nanotube are emitted with an energy distribution correponding to the distribution in the reservoir they just exited. Imposing this boundary condition and making a linear approximation for the spatial dependence of the distribution, expressing it in terms of its value and first derivative at the nanotube’s middle point, L/2, we get, after some algebra, pωp 2|Vp| (∆T/L)Vp 2 = gp + ∂cnp/(n(n + 1)) L kBT

which is obviously what one obtains from eq 3 when L f 0. We solved eq 3 by an iterative method similar to that described in ref 8. We note that in this equation we are including both Normal and Umklapp processes explicitly, without having to rely on relaxation time approximations.9 After the converged distribution function is obtained, the thermal conductivity is computed as κ)

1 S

∑R ∫Vppωpg˜ pnp(np + 1) dq

(7)

where g˜ p ) lim∆Tf0gpL/∆T. S is the cross section of the nanotube, defined as the circumference times the graphitic interlayer separation. In defect-free SWNTs, the anharmonicity of the interatomic potential provides the only scattering mechanism for phonons. Before analyzing the results of iteratively solving eq 3, we will analytically show10 that the lowest order anharmonic scattering process, 3-phonon scattering, gives a thermal conductivity that diverges with increasing nanotube length. In a relaxation time approximation to eq 2, a relaxation time can be defined as τ-1 p ≡

1

p′′ (n′ - n′′)Q pp′ + ∑(1 + n′ + n′′)Q p′p′′ + ∑ p 2 rev dir

2|Vp| , L (8)

in terms of which the zeroth iteration in the solution of the phonon transport equation yields an approximation for g:

(3)

where all the quantities are evaluated at x ) L/2. We call the first element on the right-hand side the “ballistic term”. Let us understand the meaning of this equation. First of all, if L f ∞, the equation becomes the usual BP equation for an infinitely extended, homogeneous system. Now, if instead we take the small L limit, the ballistic regime is 1222

(6)

(2)

where dir(p) and rev(p) represent the collections of {p, p′, p′′} satisfying the energy-momentum conservation for direct, p + p′ T p′′, and reverse, p T p′ + p′′, processes, respectively. For a finite nanotube, we have boundary conditions at the interfaces with the warmer and cooler reservoirs, stating that the phonons emitted from either end must be thermalized. We consider the nanotube extending from x ) 0 to x ) L, and the warmer side on the left. The distributions at the reservoirs, nc or nh, correspond to the equilibrium distributions of a body at temperature Tc or Th. Therefore

nh ) n +

pωp ∆T kBT2 2

gp =

pωp

Vpτp

kBT2

∆T L

(9)

So the thermal conductivity is approximately κ=

1 2

kBT S

∑R ∫V2p(pωp)2τpnp(np + 1) dq

(10)

Nano Lett., Vol. 5, No. 7, 2005

Let us consider the case of only linear dispersions first. At low frequencies we can take Vp and (pωp)2np(np + 1) = (kBT)2 out of the integral. This leaves τp as the only quantity depending on q. It can be shown2,11 that, at low frequency, the 3-phonon scattering elements depend on frequency as ∂∆ω Qqq′q′′ ∼ ωω′ω′′/| | ∂q′

(11)

where ∆ω ) ω ( ω′ - ω′′, with different sign for direct or reverse processes. For ω belonging to the longitudinal branch, if all acoustic branches are linear, the gradient dividing the expression is independent of frequency. Therefore

quadratic,14 and not linear, as it had been earlier suggested.15 To ensure this, we use the interatomic potential proposed in ref 14. In first-order perturbation theory, the 3-phonon processes have to satisfy the energy and momentum conservation rules. They are also further restricted by selection rules arising from the symmetry of the eigenmodes involved. Since the divergence we are interested in is caused by the acoustic branches, it is possible to treat these branches exactly and use a relaxation time approximation for the rest of the spectrum. For the relaxation time of these upper branches, we use the parameters given in Ref 16 for graphene. We have explicitly computed the exact scattering amplitude of the 3-phonon processes among different branches. This involves a summation over the phonon modes involved as9,13

τ-1 ∼ dir

∑ pp′p′′

(n′ - n′′)ωω′ω′′ +

1

reV

∑ (1 + n′ + n′′)ωω′ω′′ + a/L 2 pp′p′′ (12)

where a is a constant. For ω f 0 we have n′ - n′′ ≡ (n′ + 1)n′′/n ∼ ω/(ω′ω′′), and 1 + n′ + n′′ ≡ n′n′′/n∼ω/(ω′ω′′). Consequently, τ-1 ∼ q2 + a/L, and κ∼

∫0(q2 + a/L)-1 dq ∝ L1/2 (when L f ∞)

(13)

In the case where some of the modes have quadratic dispersions, we can follow a similar argument. It can be verified that ∂∆ω ∼ xq ∂q′ in this case. Thus, we have that in the presence of some quadratic branches, τ-1 ∼ q3/2 + a/L, and the thermal conductivity would diverge as κ∼

∫0(q3/2 + a/L)-1 dq ∝ L1/3,

Lf∞

(14)

Similar behaviors have been predicted for pure onedimensional systems.4,12 However, the present “quasi one dimensional” case is in fact three-dimensional in the sense that the atomic vibrations are not constrained to be along one particular direction. Therefore, one should not try to establish a link between the pure one-dimensional cases and the present one. In the case of bulk materials, the presence of lines or surfaces of degeneracy, where two branches come in contact, can soften or remove the divergence.13 In confined systems this does not occur. This is because, in these quasi 1-D systems, a degeneracy can occur only at a single point, rather than a line or a surface, and therefore it does not change the dependence of the number of scattering processes with q. We have solved eq 3 iteratively including the three-phonon interactions to first order. We find that it is very important to correctly describe the acoustic modes. In particular, the phonon dispersions of the two flexural modes should be Nano Lett., Vol. 5, No. 7, 2005

bbq ,qb′,qb′′ ∝

∑ lmn

1

Blmn

xMlMmMn

ωω′ω′′

ele′me′′nei(qb′′∆B l -qb′∆B l n

m)

(15)

Here, e, e′, e′′ are the eigenstates of the three phonons, ∆ B nl is the vector joining the positions of the atoms associated to degrees of freedom l and n, and Blmn are the anharmonic terms in the Hamiltonian, Hanh ) ∑lmnBlmnaˆ laˆ maˆ n, where the aˆ ’s are the displacement operators. Anharmonicity was considered on the first nearest neighbor spring constants. From eq 15, we found that the only processes allowed between the acoustic modes of zigzag nanotubes are L T F1 + F1, L T F1 + F2, L T F2 + F2, L T S + S. All other processes either cancel by symmetry or are forbidden by the energy and momentum conservation rules. Here, L denotes longitudinal, S shearing, and F1(2) are the two flexural modes, which have quadratic dispersions. Especially surprising is the fact that L T L + S are not allowed in zigzag nanotubes. (These processes are generally allowed in three-dimensional bulk solids.) We have verified that the equivalent transition in graphene is also forbidden, involving the longitudinal and in-plane transverse modes, when all three phonons are aligned in the Γ - M direction. We also verified numerically that the approximate dependence for long wavelengths, eq 11, given by Klemens for linear dispersions, is valid in the case of quadratic dispersions as well. Given the large number of processes one needs to evaluate, and the fact that we are primarily interested in the effect of the long wavelength phonons, we opted to use the long wavelength approximation in the computations, together with the energy momentum conservation rules and the additional selection rules specified earlier. The calculated thermal conductivity is shown in Figure 1 for different temperatures as a function of the nanotube’s length, L, for a (10,0) nanotube. For short lengths, all phonons travel ballistically, resulting in a linear dependence of the thermal conductivity with L. The lower the temperature, the longer this ballistic regime holds. At 100 K, 1 µm long nanotubes are still ballistic. As the length is increased, the thermal conductivity increases more slowly, however, it never saturates. Although the contribution from the upper branches saturates, the long wavelength phonon contribution does not. This corroborates the qualitative results obtained 1223

Following ref 17, for the relaxation time due to these processes we have τ-1 2 )

∫

2 M4

( )

( )

|c2|2 kBT 2 a 2 ω δ(∆ω) dq′dq′′ (17) 2π ωω′ω′′ω′′′ p ω′ω′′ω′′′

where a is the length of the unit cell. A simple graph shows that ωi ω′ ∼ ∆ωi ω Figure 1. Calculated thermal conductivity versus nanotube length at different temperatures, for a (10,0) nanotube. Thin lines: including 3-phonon processes to first-order only. Thick lines: including 3-phonon processes to second order.

in eqs 13 and 14. The results show that it is true not just for the zeroth iteration (or relaxation time approximation), but also for the fully converged results. Normal processes do not equilibrate the transport, which remains superdiffusive in the long nanotube limit. The behavior is approximately proportional to L1/2, rather than the L1/3 expected in the presence of quadratic dispersions. The higher exponent is a consequence of the lack of S + L T L processes. These processes always allow the decay of a low-frequency S phonon by combining with a high-frequency L phonon to produce another high-frequency L phonon. In that case, the denominator in eq 11 tends to zero as q f 0, increasing the scattering rate, so that there is no divergence of the S phonon contribution to the thermal conductivity. However, if S + L T L are not allowed, then low frequency S phonons can only decay by a S + S T L process, in which case the denominator of eq 11 does not tend to zero, and the S phonon contribution to the thermal conductivity diverges as L1/2 (since S phonons do not directly interact with flexural phonons). We have explicitly checked that, if we allow S + L T L processes, the dependence becomes L1/3. The divergence of the thermal conductivity is a result of having considered only 3-phonon processes to the first order. We now show that the effect of considering 3-phonon processes to the second order effectively removes the divergence, and yields a finite thermal conductivity for carbon nanotubes. A second-order 3-phonon scattering process involves the virtual combination of two phonons of frequencies ω and ω′ into an intermediate one, ωi, and the splitting of that virtual phonon into two new phonons with frequencies ω′′ and ω′′′. The probability of these processes thus involves a summation over all possible intermediate virtual phonons. An approximation for the order of magnitude of this scattering amplitude was given as17 4 c2 ) M(γ/V)2 3

∑i ωω′ωiω′′ω′′′/∆ωi

(16)

where γ is the Gruneisen constant and ∆ωi ) ωi - ω - ω′. 1224

Substituting the expression for c2, we obtain that for the longitudinal branch of a quasi 1-D system such as a carbon nanotube, the scattering rate due to second-order 3-phonon processes is τ-1 2 ∼

( )

32 4 kBT 2 γ ωb 27 MV2

(18)

where ωb is the phonon branch frequency at the zone boundary. The second-order processes thus remove the singularity, by contributing a constant scattering rate at low frequencies. The second-order processes begin to dominate for frequencies below 2 γ 3

x

kBT

MV2

ωb

The result of including the second-order 3-phonon processes is shown in Figure 1. As we see, the thermal conductivity now saturates to a finite value when L f ∞. However, below room temperature this saturation takes place at rather long values of L, because of the low rate of scattering of the acoustic modes. The thermal conductivity consequently gets very large.18 At 316 K we get κ ∼ 4000 W/m - K. Due to the approximate treatment of the second order 3-phonon processes, the actual values of the saturated thermal conductivity at large L obtained here should be viewed as only qualitatively correct. Let us briefly discuss the effect of scattering by defects on the thermal conductivity. For a scattering process to remove the thermal conductivity divergence of the linear acoustic branches, it has to depend on frequency more slowly than ω. It can be shown that, for these quasi one-dimensional systems, the phonon scattering rate from the longitudinal or shearing branch into a flexural branch is proportional to ω3/2 in the case of isotopic impurities. Therefore, isotopic impurities cannot bring down the thermal conductivity by a lot. On the other hand, defects, in the form of random variations of the lattice constants, result in a scattering rate that is proportional to ω1/2, for phonons scattering out of a linear branch into a quadratic one.22 Therefore, the presence of lattice defects can completely eliminate the otherwise large Nano Lett., Vol. 5, No. 7, 2005

thermal conductivity contribution of the low-frequency acoustic modes. It is out of the scope of this paper to establish the defect concentrations necessary to counteract the divergence of the long waves. This will be presented elsewhere. It is interesting to comment on the case of nanotube bundles.23 Such systems are conceptually between SWNTs and graphite. Due to the large concentration of defects and the random nature of the bundle intertube scattering, it is expected that long waves would not give a large contribution to the thermal conductivity, for the reasons explained in the previous paragraph. Therefore, the thermal conductivity would be smaller than for SWNTs, and it would saturate at shorter lengths. If defects and intratube scattering are large, it is also expected that κ will always increase with T at low temperature and display a peak, no matter how long the bundle is, analogously as the effect of grain boundaries on the thermal conductivity of graphite. Modification of the phonon dispersions may also affect κ dramatically. The low T thermal conductivity of finite samples of graphite, graphene, and SWNTs, scale as T2.5, T1.5, and T, respectively, due to their different phonon dispersions.24 For nanotube bundles, the intertube interactions change the dispersions, from purely one-dimensional, into those of a complex 3-D system, and κ may therefore display a low T behavior between those of SWNTs and graphite, as it has been experimentally shown.23 In conclusion, we have presented the first calculations of the length dependence of lattice thermal conductivity of single-walled carbon nanotubes that extend from nanoscopic to macroscopic lengths. Low frequency phonons are found to disproportionately contribute to the thermal conductivity of SWNTs, in a dramatic manifestation of the long standing “problem of long waves”. Solution of the BP equation with 3-phonon processes to first order predicts hyperdiffusive heat conduction, in which the thermal conductivity diverges with length. However, inclusion of higher order processes removes the low frequency singularity and yields a finite though high thermal conductivity at infinite lengths. Despite the effect of higher order processes, transport still remains ballistic up to very long sample lengths. Lattice defects scatter the long

Nano Lett., Vol. 5, No. 7, 2005

wavelength phonons more efficiently, and would significantly reduce the thermal conductivity contribution of lowfrequency modes. References (1) Pomeranchuk, I. J. Phys. (U.S.S.R.) 1941, 4, 259. Pomeranchuk, I. J. Phys. (U.S.S.R.) 1942, 6, 237. Pomeranchuk, I. Phys. ReV. 1941, 60, 820. (2) Ziman, J. M. Electrons and Phonons; Oxford University Press: Oxford, 1960. (3) Herring, C. Phys. ReV. B 1954, 95, 954. (4) Livi, R.; Lepri, S. Nature 2003, 421, 327, and references therein. (5) An earlier attempt to calculate the length dependence of the thermal conductivity of SWNTs has been presented by Maruyama, S. Physica B 2002, 323, 193. However, those results, based on classical molecular dynamics, violate the quantum upper bounds to the maximum thermal conductance achievable through the nanotube. (Mingo, N.; Broido, D. A., submitted.) (6) When R ) 1 one speaks of diffusive transport, whereas R < 1 is referred to as superdiffusive. (7) Peierls, R. Ann. Physik 1929, 3, 1055. (8) Omini, M.; Sparavigna, A. Physica B 1995, 212, 101. (9) Callaway, J. Quantum Theory of the Solid State; Academic Press: New York, 1974. (10) A similar discussion for bulk solids can be found in ref 2. (11) Klemens, P. G. in Thermal ConductiVity; Tye, R. P., Ed.: Academic Press: London, 1969; Vol. 1, p 1. (12) Narayan, O.; Ramaswamy, S. Phys. ReV. Lett. 2002, 89, 200601. (13) Lifshitz, E. M.; Pitaevskii, L. P. Physical Kinetics; Pergamon Press: Oxford, 1981. (14) Mahan, G. D. Gun Sang Jeon, Phys. ReV. B 2004, 70, 075405. (15) Jishi, R. A.; Venkataraman, L.; Dresselhaus, M. S.; Dresselhaus, G. Chem. Phys. Lett. 1993, 209, 77. (16) Klemens, P. G.; Pedraza, D. F. Carbon 1994, 32, 735. (17) Ecsedy, D. J.; Klemens, P. G. Phys. ReV. B 1977, 15, 5957. (18) Very large values of κ ∼ 3000-6000 W/m - K at room temperature have been obtained for infinitely long SWNTs using other techniques [see refs 19, 20, 21]. (19) Berber, S.; Kwon, Y-K.; Toma´nek, D. Phys. ReV. Lett. 2000, 84, 4613. (20) Che, J.; C¸ agin, T.; Goddard, W. A. III Nanotechnology 2000, 11, 65. (21) Osman, M. A.; Srivastava, D. Nanotechnology 2001, 12, 21. (22) Glavin, B. A. Phys. ReV. Lett. 2001, 86, 4318. (23) Shi, L.; Li, D.; Yu, C.; Jang, W.; Kim, D.; Yao, Z.; Kim, P.; Majumdar, A.; J. Heat Transfer 2003, 125, 881. (24) Mingo, N.; Broido, D. A., submitted.

NL050714D

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