NANO LETTERS
First-Principles Calculation of the Isotope Effect on Boron Nitride Nanotube Thermal Conductivity
2009 Vol. 9, No. 1 81-84
Derek A. Stewart,*,† Ivana Savic´,‡ and Natalio Mingo‡,§ Cornell Nanoscale Facility, Cornell UniVersity, Ithaca, New York 14853, LITEN, CEA-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France, and UniVersity of California, Santa Cruz, Santa Cruz, California 95064 Received August 15, 2008; Revised Manuscript Received November 24, 2008
ABSTRACT Isotopic composition can dramatically affect thermal transport in nanoscale heat conduits such as nanotubes and nanowires. A 50% increase in thermal conductivity for isotopically pure boron (11B) nitride nanotubes was recently measured, but the reason for this enhancement remains unclear. To address this issue, we examine thermal transport through boron nitride nanotubes using an atomistic Green’s function transport formalism coupled with phonon properties calculated from density functional theory. We develop an independent scatterer model for 10B defects to account for phonon isotope scattering found in natural boron nitride nanotubes. Phonon scattering from 10B dramatically reduces phonon transport at higher frequencies and our model accounts for the experimentally observed enhancement in thermal conductivity.
As electronics continue to shrink in scale, efficient heat removal stands as a key limiting factor to further device miniaturization. Power densities in integrated circuits have risen exponentially over the last fifteen years1 and a clear need exists for nanoscale structures that can transfer heat efficiently. Carbon and boron nitride nanotubes are two potential structures for future nanoscale heat conduits. Carbon nanotubes are well known for their exceptionally high thermal conductivity.2 The measured thermal conductivity of boron nitride nanotubes is comparable to carbon nanotubes;3 however boron nitride nanotubes offer a practical advantage for fabrication. The chirality of carbon nanotubes cannot be controlled during growth, and both metallic and semiconducting nanotubes are formed. In contrast, boron nitride nanotubes are always semiconducting and should not interfere with electronic components. Manipulating the isotope distribution within a material offers one route to enhance the thermal conductivity of a structure. Natural isotopes in materials cause phonon scattering that limits thermal conductivity. This scattering dominates in midrange temperatures where boundary scattering is limited and umklapp scattering is minimal. Isotopically enriched materials can have considerably enhanced thermal conductivities. Removing 1.1% 13C from diamond leads to a 35% thermal conductivity enhancement,4 the record for bulk materials. * To whom correspondence should be addressed. E-mail: stewart@cnf. cornell.edu. † Cornell University. ‡ LITEN, CEA-Grenoble. § University of California, Santa Cruz. 10.1021/nl802503q CCC: $40.75 Published on Web 12/17/2008
2009 American Chemical Society
In 2006, Chang et al. grew isotopically pure 11BN nanotubes that had a 50% enhancement in thermal conductivity compared to boron nitride nanotubes with natural isotope concentration.3 This enhancement is significantly greater than that found in diamond (35%) and the enriched thermal conductivity values are comparable to those in carbon nanotubes. However, the origin of this large enhancement is unclear. Natural boron has a large concentration (19.9%) of 10B isotopes which could lead to high scattering rates. Klemens5 treated isotopes as Rayleigh point scatterers where scattering increases with phonon frequency ω as ω4. The phonon relaxation rate in this case is inversely proportional to the mass fluctuation, Γ, in the system Γ ) ∑i ci[(mi - maVg)/maVg]2 where maVg is the average natural isotope mass, mi is the mass of isotope i, and ci is the corresponding concentration. Boron has the highest mass fluctuation among atoms because boron is a light element and there is a large natural component of 10B isotopes. Even with the addition of nitrogen, which has a low mass fluctuation, the average Γ-value is still one of the highest and could contribute to the large difference observed in the enriched and natural boronnitridenanotubes.6 ApreviousstudyusingaDebye-Callaway model estimated that bulk boron nitride in the zincblende structure could have a thermal enhancement that exceeds 100%; however, this has not been observed experimentally.7 In this letter, we examine thermal transport in boron nitride nanotubes with isotope scattering in an effort to resolve the reason for the observed thermal conductivity enhancement. We model thermal transport in boron nitride nanotubes using an atomistic Green’s function approach based on interatomic
force constants determined from first principles.8 We treat the effect of isotope scattering on the thermal conductivity with a simple cascade scattering model. Using this approach, we can reproduce the thermal enhancement observed by Chang et al.3 We show that a 50% enhancement appears naturally for this system, even in a completely diffusive regime where scatterers act independently. We consider a single walled (8,0) zigzag boron nitride nanotube that links two thermal baths where the temperature difference between the two baths is maintained at ∆T. The nanotube diameter and chirality should have little effect on the predicted thermal conductivity, because single wall carbon nanotubes converge to common thermal conductivity values for temperatures greater than 150 K.9 In the linear response regime, the thermal conductance, σ, across the boron nitride nanotube is given by σ ) JQ/∆T, where JQ is the thermal current. The thermal conductance, σ, can be expressed in terms of the phonon transmission across the nanotube, Ξ(ω) σ)
∫
∞
0
dω
df(ω, T) pω Ξ(ω) 2π dT
(1)
where p is Planck’s constant, f is the phonon distribution, and ω is the phonon frequency. The transmission, Ξ(ω), is given as a trace over retarded and advanced phonon Green’s function terms, GR,A, that describe the ability of phonons to transverse the nanotube and coupling terms, ΓL,R, that describe the ability of phonons to enter and leave the left and right thermal leads at a given frequency Ξ(ω) ) Tr[ΓL(ω)GR(ω)ΓRGA(ω)]
(2)
R
The Green’s function, G , for the central region can be expressed in terms of the phonon Hamiltonian for the system as well as two self-energy terms, ΣL and ΣR, that account for phonon coupling to the left and right leads, respectively. GR ) [ω2I - H - ΣL - ΣR]-1
(3)
The full derivation of the Green’s functions and self-energies are given in a previous work.10 One of the key aspects of the current work is that the interatomic force constants that form the basis of the Hamiltonian in the Green’s function formalism are calculated directly from first principles. Previous studies of thermal transport in nanotubes and nanowires have primarily relied on empirical interatomic potentials.10,11 While these potentials do a reasonable job of reproducing static properties of materials (i.e., lattice constant, bulk modulus), they have problems modeling thermal transport in nanostructures accurately.8 Care must also be used in describing multiatom systems with empirical potentials. Defining bonding parameters between different atoms like boron and nitrogen can be difficult and they are often only valid for a single crystal configuration. Density functional approaches, in contrast, are well suited for modeling bonding between different atom types and have provided robust interatomic force constants for modeling thermal conductivity in both bulk materials12 and nanostructures.8 We use a standard density functional approach that takes advantage of a numerically truncated localized basis set for 82
efficient calculations of large supercells.13,14 All calculations were done in the local density approximation. A double-ζ polarized basis set with a 0.01 Ryd energy shift was used to describe the orbitals. Brillouin zone integration was done with a 1×1×15 Monkhorst-Pack k-point grid. The nanotube atomic coordinates were relaxed based on Hellmann-Feynman forces to their equilibrium positions (maximum force tolerance 2.5 × 10-3 eV/Å). Once this had been achieved, supercells consisting of 3, 5, and 7 unit cells were constructed. Real space force constants were extracted by shifting individual atoms in the central unit cell and determining the force induced on all other atoms in the supercell. The larger supercell calculations were done to ensure that sufficient nearest neighbor interactions were included in the interatomic force constants in order to obtain accurate phonon dispersions. Unlike empirical approaches that are inherently shortrange, ab initio approaches have no well-defined cutoff for interatomic interactions. This can lead to interatomic force constants that do not satisfy translational and rotational invariance. Since this is related to long-range interactions, associated errors in the phonon dispersion are found mainly in low frequency acoustic waves near the Γ-point. We recently developed an efficient symmetrization technique8 based on Lagrangian multipliers that restores the translational and rotational invariance of the interatomic force constants. This process was applied to boron nitride interatomic force constants and the calculated phonon dispersions agree with previous studies for boron nitride nanotubes.15 In the present case, the symmetrization procedure was improved by the addition of a small strengthening matrix s onto the symmetrized force constants that ensures the stability of the system after truncation of the interactions. The simplest form for this strengthening term is sij ) a(δ(ri - rj - R) + δ(ri - rj + R) - 2δ(ri - rj))
(4)
where ri is the lattice site of the atom to which the ith degree of freedom belongs, and R is the nanotube unit cell’s translation vector. Obviously, the addition of this matrix does not alter the phonon eigenfrequencies at the Γ-point. Constant a is adjusted to enforce a zero group velocity for the two flexural modes at the Γ-point. The flexural modes in carbon nanotubes behave quadratically at low frequencies without a need for this extra correction. However, for BN nanotubes, this term was required to enforce quadratic flexural modes and avoid a small range of imaginary frequencies which would otherwise occur close to the Γ-point. The need for this correction term could be due to long-range Coulomb interactions in boron nitride nanotubes. The effect of the s correction is largest at the Brillouin zone edge and mainly affects the acoustic modes. In the case presented, the largest induced changes were always less than 10%. To describe phonon scattering due to the effect of isotopes, we use a simple cascade scattering model.16,17 In this approach, scattering events are treated as independent and all events can be directly added together. Because of this, the transmission depends on the total number of scatterers Nano Lett., Vol. 9, No. 1, 2009
Figure 1. The phonon transmission through a (8,0) BN nanotube is shown as a function of angular frequency (ω) for 10 (panel a), 100 (panel b), 1000 (panel c), and 10000 (panel d) 10B impurities. In each panel, the phonon transmission for isotopic pure 11BN nanotube is given by a solid black line.
along the suspended section and not on the length. The total transmission for N scatterers in this model is given by 1 N N-1 ) TN Ti T0
(5)
where T0 is the phonon transmission in the presence of no scatterers and Ti is the phonon transmission in the presence of a single scatterer. The transmission can be calculated directly based on Green’s functions derived from first principles force constants. In the case of the single isotope scatterer, it is sufficient to take into account only the change in mass, since the force constants are not affected by isotopes. It should be noted that this approach neglects wave interference effects due to multiple scattering. Wave interference effects will become important for high concentrations of 10B isotopes. To rigorously take interference effects into account, it is necessary to consider phonon transmission averaged over an ensemble of nanotubes where 10B isotopes are randomly distributed over the nanotube. Through the use of eq 5, the total transmission through a collection of 10B isotope scatterers ranging from 0 to 10000 was calculated (Figure 1) for a (8,0) BN nanotube. In the case of no isotope scatterers, the phonon transmission exhibits a steplike structure and is merely a measure of the number of phonon modes available at a given frequency. For small numbers of isotope scatterers (N e 100), there is little change in the overall phonon transmission spectrum with the addition of isotopes. However for 100 10B isotopes (Figure 1b), the transmission through optical modes above 225 THz is clearly reduced. Since isotopes can be treated as point scatterers that do not distort the atomic configuration of a boron nitride nanotube, we expect scattering to be more prevalent from short wavelength (high frequency) phonons. A similar reduction in high phonon transmission was observed with substitutional nitrogen defects in carbon nanotubes.8 When the number of isotope scatterers increases to 1000 (Figure 1c), the phonon transmission reduction is much more broadband. Phonon transmission above 200 THz Nano Lett., Vol. 9, No. 1, 2009
Figure 2. The thermal conductivity for a (8,0) boron nitride nanotube, 1 µm long, is shown as a function of temperature. The calculated thermal conductivity with scaling factor for isotopically pure 11B and natural abundance boron nitride nanotubes without anharmonic effects are given by solid black and blue lines, respectively. The calculated thermal conductivity with anharmonic scattering are denoted respectively by dashed red and green lines for isotopically pure and natural abundance cases. Black squares and black triangles respectively denote experimental values for isotopically pure 11B and natural abundance boron nitride nanotubes from ref 3.
has been essentially eliminated. This trend in the reduction of phonon transmission with isotope content presents an interesting approach to phonon engineering nanostructures. A great deal of work has gone into tailoring the surfaces of nanowires in order to scatter low frequency acoustic phonons and reduce thermal conductivity.18,19 Our results indicate a simple technique to reduce phonon transmission due to high frequency phonons. A combination of these two approaches in a single nanostructure could provide an unique phonon band-pass filter for midrange phonon frequencies. Let us now consider the thermal conductivity through an isotopically pure (8,0) BN nanotube and one that contains the natural distribution of 10B isotopes. We take each nanotube to be 1 µm in length which corresponds to roughly 7400 10B isotopes based on the natural boron isotope distribution. The calculated thermal conductivity as a function of temperature is shown in Figure 2, rescaled by a factor of 0.118. The fact that experimental nanotube thermal conductivities are nearly proportional to the ballistic limit, but can be several times smaller, has been reported before.2,3,9 There is still controversy as to whether this is due to contact resistance, inefficient heat transfer to inner shells, or some other unidentified mechanism. The experimental data from the work of Chang et al.3 is also shown in the graph for comparison. For the natural distribution nanotube, there is good agreement between the predicted and measured thermal conductivity values over much of the temperature range. The predicted and measured thermal conductivities for pristine boron nitride nanotubes agree for temperatures up to 150 K. Above 150 K, the predicted thermal conductivities continue to increase nearly linearly with temperature. The measured 83
thermal conductivity for the natural distribution boron nanotubes by contrast has a much more gradual increase in thermal conductivity with temperature. The overestimation of the thermal conductivity at high temperatures clearly indicates that we are excluding a potential scattering mechanism. At higher temperatures, umklapp scattering will become important. Using a parametrized mean free path for umklapp scattering of the form given by Klemens and Pedraza,20 λU ) Aω-2T-1 with A ) 1025 m-K/s2, we can fit the high temperature portion of the measured thermal conductivity for both natural and isotopically enriched boron nitride nanotubes. From Figure 2, umklapp scattering leads to a significant reduction in the thermal conductivity at high temperatures for the isotopically enriched boron nitride nanotube. However, for the natural boron nitride nanotube, isotope scattering dominates and the thermal conductivity reduction due to umklapp scattering is much smaller. It is important to note that the past experimental study of thermal transport in enriched and natural boron nitride nanotubes considered multiwall nanotubes. Our model calculations presented here assume a single wall boron nitride nanotube. However, despite this fact we are able to provide good agreement with the experimental values. This could indicate that heat conduction in multiwall boron nitride nanotubes is carried primarily by the outer nanotube shell. A previous experimental2 study of multiwall carbon nanotubes indicated that heat conduction could be confined to the outer nanotube shell. In this letter, we have examined thermal transport in isotopically enriched and natural distribution boron nitride nanotubes. Using an atomistic Green’s function approach based on force constants derived from first principles, and treating isotope scattering in terms of a simple cascade scattering model, we find good agreement with the measured enhancement of thermal conductivity of isotopically pure BN nanotubes as compared to nanotubes with natural isotopic distribution. Our phonon transmission calculations indicate that isotope scattering strongly suppresses phonon transmission in high frequency optical phonon bands.
84
Acknowledgment. First principles boron nitride nanotube structural relaxations and raw ab initio force constants were determined by D.A.S. at the Cornell Nanoscale Facility and symmetrized by N.M. at UC Santa Cruz with support from NSF Grants 0651310 and 0651427. Transport calculations and associated theory were developed by I.S. and N.M. at CEA-Grenoble with support from European Union Grant MIRG-CT-2006-039302. References (1) Pop, E.; Goodson, K. E. J. Electron. Packag. 2006, 128, 102–108. (2) Kim, P.; Shi, L.; Majumdar, A.; McEuen, P. L. Phys. ReV. Lett. 2001, 87, 215502. (3) Chang, C. W.; Fennimore, A. M.; Afansasiev, A.; Okawa, D.; Ikuno, T; Garcia, H.; Lu, D.; Majumdar, A.; Zettl, A. Phys. ReV. Lett. 2006, 97, 085901. (4) Onn, D. G.; Witek, A.; Qiu, Y. Z.; Anthony, T. R.; Banholzer, W. F. Phys. ReV. Lett. 1992, 68, 2806–2809. (5) Klemens, P. G. Proc. Phys. Soc., Sect. A 1955, 68, 1113–1128. (6) Morelli, D. T.; Slack, G. A. In High Thermal ConductiVity Materials; Shinde´, S. L., Goela, J. S., Eds.; Springer-Verlag: New York, 2006; p 37-68. (7) Morelli, D. T.; Heremans, J. P.; Slack, G. A. Phys. ReV. B 2002, 66, 195304. (8) Mingo, N.; Stewart, D. A.; Broido, D. A.; Srivastava, D. Phys. ReV. B 2008, 77, 033418. (9) Mingo, N.; Broido, D. A. Phys. ReV. Lett. 2005, 95, 096105. (10) Mingo, N; Yang, L. Phys. ReV. B 2003, 68, 245406. (11) Yamamoto, T.; Watanabe, K. Phys. ReV. Lett. 2006, 96, 255503. (12) Broido, D. A.; Malorny, M.; Birner, G.; Mingo, N.; Stewart, D. A. Appl. Phys. Lett. 2007, 91, 231922. (13) Soler, J.; Artacho, E.; Gale, J. D.; Garcia, A.; Junquera, J.; Ordejo´n, P.; Sa´nchez-Portal, D. J. Phys: Condens. Matter 2002, 14, 2745–2779. (14) Sa´nchez-Portal, D.; Artacho, E.; Soler, J. M.; Rubio, A.; Ordejo´n, P. Phys. ReV. B 1999, 59, 12678. (15) Wirtz, L.; Rubio, A.; de la Concha, R. A.; Loiseau, A. Phys. ReV. B 2003, 68, 045425. (16) Datta, S.; Cahay, M.; McLennan, M. Phys. ReV. B 1987, 36, 5655– 5658. (17) Markussen, T.; Rurali, R.; Jauho, A.-P.; Brandbyge, M. Phys. ReV. Lett. 2007, 99, 076803. (18) Hochbaum, A. I.; Chen, R.; Delgado, R. D.; Liang, W.; Garnett, E. C.; Najarian, M.; Majumdar, A.; Yang, P. Nature 2008, 451, 163–167. (19) Boukai, A. I.; Bunimovich, Y.; Tahi-Kheli, J.; Yu, J.-K.; Goddard, W. A.; Heath, J. R. Nature 2008, 451, 168–171. (20) Klemens, P. G.; Pedraza, D. F. Carbon 1994, 32, 735–741.
NL802503Q
Nano Lett., Vol. 9, No. 1, 2009
