Phonon thermal transport in strained and unstrained graphene from

PHYSICAL REVIEW B 89, 155426 (2014)

Phonon thermal transport in strained and unstrained graphene from first principles L. Lindsay,1 Wu Li,3 Jesús Carrete,3 Natalio Mingo,3 D. A. Broido,2 and T. L. Reinecke4 1

NRC Research Associate at Naval Research Laboratory, Washington, D.C. 20375, USA 2 Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA 3 LITEN, CEA-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France 4 Naval Research Laboratory, Washington, D.C. 20375, USA (Received 12 February 2014; published 24 April 2014)

A rigorous first principles Boltzmann-Peierls equation (BPE) for phonon transport approach is employed to examine the lattice thermal conductivity, κL , of strained and unstrained graphene. First principles calculations show that the out-of-plane, flexural acoustic phonons provide the dominant contribution to κL of graphene for all strains, temperatures, and system sizes considered, supporting a previous prediction that used an optimized Tersoff empirical interatomic potential. For the range of finite system sizes considered, we show that the κL of graphene is relatively insensitive to strain. This provides validation for use of the BPE approach to calculate κL for unstrained graphene, which has recently been called into question. The temperature and system size dependence of the calculated κL of graphene is in good agreement with experimental data. The enhancement of κL with isotopic purification is found to be relatively small due to strong anharmonic phonon-phonon scattering. This work provides insight into the nature of phonon thermal transport in graphene, and it demonstrates the power of first principles thermal transport techniques. DOI: 10.1103/PhysRevB.89.155426

PACS number(s): 63.20.kg, 63.22.Rc, 66.70.−f, 65.80.Ck

I. INTRODUCTION

Graphene is a two-dimensional sheet of carbon atoms with intriguing structural, electronic, and thermal properties. The lattice thermal conductivity of graphene, κL , has attracted intense interest in recent years [1–8]. Most measured values of κL for suspended single-layer graphene around room temperature (RT) fall in the 2000–4000 Wm−1 K−1 range [2,4–7]. Typically, such measurements employed Raman spectroscopy [1–7] techniques and have large experimental uncertainties [9]. On the theoretical side, the κL of graphene is often described in the context of a transport approach for phonons, which uses the so-called Boltzmann-Peierls equation (BPE). The intrinsic thermal conductivity is limited by phonon-phonon scattering, which arises from the anharmonicity of the interatomic potential between carbon atoms. The complex inelastic nature of this scattering mechanism has made the development of rigorous theoretical treatments of the phonon lifetimes and κL challenging. As a result, many of the approaches that have calculated the κL for graphene have been based on approximate solutions of the BPE within the single-mode relaxation time approximation (RTA) for which an averaged Grüneisen parameter is sometimes taken to determine the strength of phonon-phonon scattering [10–15]. Recently a rigorous BPE approach has been developed for the κL of graphene, and related systems [16–24] that (i) determines explicitly quantum mechanical phonon-phonon scattering matrix elements from anharmonic atomic interactions, (ii) includes both Normal and Umklapp scattering processes, and (iii) employs an iteration approach to determine a full solution to the BPE thus going beyond the RTA. For freestanding unstrained graphene, this approach has shown that the majority of the contribution to κL comes from the flexural, out-of-plane acoustic (ZA) phonons, partly due to a phonon-phonon scattering selection rule that gives increased ZA phonon lifetimes [17,18]. This finding was in stark contrast to previous theoretical results, which argued

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that the contributions to κL from flexural phonons should be negligible [10–13]. The previous BPE calculations of the κL of graphene [17–21,23] employed a Tersoff empirical interatomic potential [25,26] to calculate the harmonic and the anharmonic forces between carbon atoms. The original Tersoff model is a popular potential for simulating atomic interactions in carbon-based systems. However, it does not accurately describe the vibrational properties of graphene [27,28]. The Tersoff potential employed in Refs. [17–21,23] was optimized to better fit the acoustic phonon frequencies and velocities of graphene [27] than those given by the original parameterization. However, deviations from the measured dispersions still exist, especially for higher frequency phonon modes (see Fig. 1). Further, this optimization of the Tersoff potential did not involve the anharmonic interatomic force constants (IFCs), which are required to describe the intrinsic phonon-phonon scattering rates. As a result, the finding of dominant flexural mode contributions to the κL of graphene obtained using the optimized Tersoff model [17–21,23] has been called into question [29]. First principles approaches for thermal transport based on density functional theory (DFT) [30,31] and density functional perturbation theory (DFPT) [32] can more accurately represent the harmonic interatomic forces. A demonstration of this is given in Fig. 1, which shows the phonon dispersions for unstrained graphene calculated with the DFPT harmonic IFCs used in this work (black curves) and with the optimized Tersoff empirical potential used in the previous work (red curves) [17–20,27]. It is obvious that the first principles calculations describe the phonon frequencies better over the entire frequency range and without adjustable parameters. Such first principles approaches have been implemented to calculate κL for a number of bulk materials [35–46] and have demonstrated good agreement with experiment. They have also been used to describe the phonon lifetimes in graphene [47–49], carbon nanotubes [50], and monolayer MoS2 [51].

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the enhancement in the graphene κL with isotopic purification is relatively small, around 14% at RT, a consequence of strong low frequency anharmonic scattering rates.

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scaled wave vector FIG. 1. (Color online) Calculated phonon dispersions for unstrained graphene using DFPT harmonic IFCs (black curves) and optimized Tersoff IFCs (red curves). The squares and triangles give experimental data obtained from neutron scattering for in-plane graphite from Ref. [33] and Ref. [34], respectively.

A more serious point has been raised that calls into question the validity of the BPE approach for unstrained graphene. Ab initio calculations have found that the in-plane transverse acoustic (TA) and longitudinal acoustic (LA) phonon lifetimes in unstrained freestanding graphene approach constant values in the long wavelength limit, making these phonons ill-defined in this limit [49]. However, infinitesimal mechanical strain is found to make long-wavelength TA and LA phonons well defined again [49]. This raises the question of whether the κL of strained freestanding graphene calculated within the BPE converges to that of unstrained graphene in the limit of vanishing strain. If so, then thermal transport calculations for unstrained freestanding graphene should accurately represent κL in spite of the long-wavelength TA and LA anomalies. In this paper, we employ a first principles approach to calculate the κL of strained and unstrained freestanding graphene sheets. This approach combines an iterative numerical solution to the BPE with harmonic and anharmonic IFCs obtained from DFT and DFPT. We have compared thermal conductivity results obtained from this first principles approach with those using the optimized Tersoff IFCs, and we have found that the two sets of results give κL for unstrained graphene that are in good agreement over a wide range of temperatures. This work validates the previous findings obtained using the optimized Tersoff IFCs [17,18] and may be of interest for groups using this potential in molecular dynamics calculations to determine lattice properties of graphene. Also, for the range of system sizes considered, the calculated κL for strained graphene shows little variation up to 1% strain and remains essentially unchanged from the κL value for unstrained graphene. Thus, in spite of the long wavelength anomaly for TA and LA phonons, we find that the BPE approach still can accurately describe thermal transport in unstrained graphene. Finally, we find that

We discuss briefly certain aspects of the first principles BPE approach used to calculate the κL of graphene. Many details of the BPE solution and the calculation of the anharmonic IFCs have been described previously [16,18– 20,22,24,35,37,41,42,44]. The only inputs to the BPE calculation of the thermal conductivity are harmonic and anharmonic IFCs [52]. The harmonic IFCs enter the dynamical matrices that determine the phonon dispersions, velocities, and eigenvectors, while the anharmonic IFCs enter the phonon-phonon scattering matrix elements. The harmonic IFCs were determined using standard DFPT [32], while the anharmonic IFCs were calculated using DFT [30,31] and a real space supercell approach, described in Refs. [41,42,44]. Interatomic forces were calculated using the plane-wave QUANTUM ESPRESSO (QE) package [54,55] within the local density approximation and using a norm-conserving pseudopotential to represent the core carbon electrons [56]. To test the κL results using the unstrained graphene IFCs obtained from the above approach, we also made independent calculations of the unstrained harmonic and anharmonic IFCs and κL using a real space approach within the generalized gradient approximation and using a projector-augmented wave pseudopotential implemented with the Vienna Ab initio Simulation Package (VASP) [57,58]. The unstrained lattice constants, a, were determined by ˚ energy minimization. For the first approach a = 2.441 A, ˚ while for the second approach a = 2.468 A. We have also considered graphene systems under isotropic tensile strain, implemented by increasing the lattice constant from its energy minimized value. Harmonic and anharmonic IFCs were calculated directly for each strain value considered. Alternatively, the strained harmonic IFCs can be obtained in terms of the harmonic and third-order anharmonic IFCS of the unstrained system [49]. Using harmonic IFCs from these two approaches, we have found that the phonon dispersions are similar for the strains considered. The anharmonic IFCs are also strain-dependent, decreasing in magnitude with increasing tensile strain, as found previously in diamond [59]. Since the latter approach does not consider strained anharmonic IFCs, we employ the direct calculation of harmonic and anharmonic IFCs at each strain value to determine the κL of strained graphene in this work. Using the harmonic and anharmonic IFCs for each strain value, the graphene lattice thermal conductivity was calculated by solving the BPE with an iterative numerical approach [35,37,60,61]. For a small temperature gradient, ∂T /∂xα , taken along crystallographic direction, α, the BPE solution gives the steady-state distribution function, from which the phonon transport lifetime, τλα , in mode λ = (q,j ) is extracted, where q is the phonon wave vector and j is the branch index [62]. The κL is given by 2 Cλ vλα τλα , (1) κL =

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where vλα = dωλ /dqα is the velocity component in the αth direction, and ωλ is the angular frequency of mode λ. Cλ = ωλ (∂n0λ /∂T )/V is the mode specific heat, n0λ is a Bose factor at temperature T , and V = Aδ, where A is the crystal sheet area and δ = 0.335 nm is the separation of carbon planes in graphite. We consider finite unsupported graphene systems in the x-y plane and define κL = καα . In principle, for the finite hexagonal systems considered here, the thermal conductivity tensor, καβ , can be described by two diagonal components, κxx and κyy . For infinite graphene κxx = κyy , but finite systems can be anisotropic. However, for the relatively large graphene sheets considered here, the calculated anisotropy is within numerical uncertainty ( 1 is not satisfied, and the phonons become ill-defined in this limit. Interestingly, any amount of mechanical strain is found to make long-wavelength TA and LA phonons well-defined again [49]. This raises the question of whether the calculated κL of strained freestanding graphene approaches that of unstrained graphene in the limit of vanishing strain. If so, then thermal transport calculations for unstrained freestanding graphene should accurately represent κL in spite of the long-wavelength

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mean free path (μ μm) FIG. 5. (Color online) Calculated first-principles accumulated κL vs mean free path (black curve) for unstrained graphene with L = 10 μm, T = 300 K, and naturally occurring carbon isotopes. The per branch contributions to the accumulated κL are also given for the ZA (red), TA (blue), and LA (green) branches.

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TA and LA anomalies. Here we examine the effects of small amounts of isotropic tensile strain on the κL for graphene. The application of tensile strain gives three noticeable changes to freestanding graphene that can affect κL : (i) lowered TO and LO phonon branches, (ii) linearized zone-center ZA phonon branch (see Fig. 2 of Ref. [49]), and (iii) weakened anharmonic IFCs. Figure 6 shows the calculated ab initio κL of graphene as a function of isotropic strain (for L = 10 μm, T = 300 K, and naturally occurring isotopic concentrations). The κL from the full solution to the BPE is given by the solid black circles (solid black line gives zero strain value), and the κL from the RTA is given by the open black circles (dashed black line gives zero strain value). The full solution to the BPE gives κL that is relatively strain independent up to 1% isotropic strain, i.e., the results for small strain are little changed from the unstrained value. The in-plane acoustic (green circles) and ZA (red circles) contributions to κL are also shown (lines give zero strain values). With increasing tensile strain, the ZA branch near the zone center becomes linear with increasing slope. Thus, the near zone center ZA phonon velocities increase, the density of ZA phonons decreases, and the character of the acoustic phonon-phonon scattering changes [49,74]. In addition, the tensile strain weakens the anharmonic IFCs, which acts to increase κL , but this is balanced by the changes in the phonon dispersion that tend to decrease κL . The result is a κL that is almost independent of strain. The lowered TO and LO branches play only a minor role in reducing κL through increased acoustic + acoustic↔optic three-phonon scattering. Also shown are results obtained using the zero-strain anharmonic IFCs and strained harmonic IFCs following the approach of Ref. [49] (black squares). In this


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% strain FIG. 6. (Color online) First principles κL vs isotropic tensile strain (solid black circles) for graphene with L = 10 μm, T = 300 K, and naturally occurring carbon isotopes. Also shown are the ZA (red circles) and in-plane (green circles) contributions to κL . The horizontal lines give the corresponding unstrained values for reference. The hollow black circles give the RTA κL (dashed black line gives the unstrained RTA κL ). The hollow black squares show the κL calculated using strained harmonic IFCs and unstrained anharmonic IFCs.

case, κL shows a decrease with increasing strain, a result of not taking into account the weakening of the anharmonic IFCs with strain. The dotted green curve in Fig. 4 shows the L dependence of κL for 1% isotropic tensile strain. Comparison of this curve with that for unstrained graphene (solid green curve) shows that the strain independence of κL is well maintained over the wide range of L values considered. This provides further validation for the BPE approach for unstrained graphene. Though there is no experimental data for the strain dependence of κL of freestanding graphene, theoretical calculations using a Grüneisen-parameter-based RTA approach have predicted that κL decreases with strain for large strain (4%–16%) [15]. The first principles results presented here suggest that κL of graphene is fairly insensitive to small isotropic strain. Further, divergence with system size for any amount of isotropic strain is predicted in Ref. [49], while convergence with system size for small uniaxial strain (2%) and small isotropic strain (1%) is predicted in Ref. [75]. Our calculated first principles results for finite graphene, isotropically strained and unstrained, do not contradict these findings. The iterative approach to solve the BPE does not retain a stable solution for L significantly larger than 50 μm, thus the system sizes cannot be extended to explore this issue further. Higher order phonon-phonon scattering processes, though predicted to be weaker than first-order three-phonon scattering [76], may stabilize the solution to the BPE for larger system sizes, as

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discussed in Ref. [77] for single-walled carbon nanotubes. Including higher order scattering in the first principles BPE framework is a huge computational challenge and is beyond the scope of this work. IV. SUMMARY

We have presented first principles calculations of the thermal conductivity, κL , of strained and unstrained graphene using a rigorous BPE approach with IFCs determined from DFT. The first principles calculations of κL using two independent approaches are in good agreement with experimental data and validate earlier results and conclusions [17,18] that used an empirical potential to determine the IFCs. The ZA phonons are found to give the dominant contribution to κL for all temperatures, system sizes, and strains considered. The accumulated κL data presented here show much larger mean free paths for ZA phonons, a finding that could be probed by recently developed experimental techniques [71–73]. The κL of freestanding graphene is found to be insensitive to

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small isotropic tensile strain, and it matches the κL calculated for unstrained graphene, thus validating the BPE approach for this case in spite of ill-defined long wavelength TA and LA phonons. Changes in the phonon dispersion with increasing strain tend to decrease κL , but this is balanced by weakening anharmonic IFCs that tend to increase κL . This work further demonstrates the power of first principles techniques and provides insight into the nature of thermal transport in two-dimensional graphene systems. ACKNOWLEDGMENTS

This work was supported in part by the Office of Naval Research and the Defense Advanced Research Projects Agency (L.L. and T.L.R.). L.L. acknowledges support from the National Research Council/Naval Research Laboratory Research Associateship Program. D.A.B. acknowledges support from the National Science Foundation under Grant No. 1066634 and from ONR under Grant No. N00014-13-1-0234. Work at Grenoble was partly supported by project Carnot SIEVE.

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[57] [58]

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[72] [73]

[74]

[75] [76] [77]

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and eighth-nearest neighbor cutoffs are all within 5% of the κL value using a fourth-nearest neighbor cutoff. More details of the calculation of the anharmonic IFCs can be found in Refs. [41, 42, and 44]. Vienna Ab initio Simulation Package, available from www.vasp.at. The IFCs calculated using the VASP package [57] were determined with a Perdew-Burke-Ernzerhof exchange-correlation functional, a projector-augmented wave pseudopotential, and a 40 Ryd plane-wave cutoff. The ground state structure calculation employed a 6 × 6 × 1 k-point mesh, and the IFCs were determined from -point self-consistent calculations with 128 atom supercells. The anharmonic IFCs were calculated out to tenth-nearest neighbor to the unit cell atoms. A vacuum space ˚ was used between periodic graphene layers. The VASP of 17 A calculations of the unstrained IFCs were done independently of the QE calculations [56], thus different inputs were used. We note that despite differences in the calculations, both give similar κL , which demonstrates the robustness of the results. D. A. Broido, L. Lindsay, and A. Ward, Phys. Rev. B 86, 115203 (2012). M. Omini and A. Sparavigna, Phys. Rev. B 53, 9064 (1996). M. Omini and A. Sparavigna, Nuovo Cimento Soc. Ital. Fis., D 19, 1537 (1997). We note that the full details of the calculation of the transport lifetimes for graphene are given in Ref. [18]. T. Sun and P. B. Allen, Phys. Rev. B 82, 224305 (2010). J. R. Yates, X. Wang, D. Vanderbilt, and I. Souza, Phys. Rev. B 75, 195121 (2007). Wu Li, N. Mingo, L. Lindsay, D. A. Broido, D. A. Stewart, and N. A. Katcho, Phys. Rev. B 85, 195436 (2012). S.-i. Tamura, Phys. Rev. B 27, 858 (1983). N. Mingo, K. Esfarjani, D. A. Broido, and D. A. Stewart, Phys. Rev. B 81, 045408 (2010). The error bars shown in Fig. 3 for the experimental data [5] give a 25% error range. For the lower temperature data, this error is underestimated, and for the higher temperature data the error is slightly overestimated. The relatively large error arises from details of the Raman thermometry method used to measure κL [9]. Z.-Y. Ong and E. Pop, Phys. Rev. B 84, 075471 (2011). H. Zhang, G. Lee, and K. Cho, Phys. Rev. B 84, 115460 (2011). A. J. Minnich, J. A. Johnson, A. J. Schmidt, K. Esfarjani, M. S. Dresselhaus, K. A. Nelson, and G. Chen, Phys. Rev. Lett. 107, 095901 (2011). K. T. Regner, D. P. Sellan, Z. Su, C. H. Amon, A. J. H. McGaughey, and J. A. Malen, Nat. Comm. 4, 1640 (2013). J. A. Johnson, A. A. Maznev, J. Cuffe, J. K. Eliason, A. J. Minnich, T. Kehoe, C. M. Sotomayor Torres, G. Chen, and K. A. Nelson, Phys. Rev. Lett. 110, 025901 (2013). At 1% strain, the net effect is slightly smaller ZA mode contributions than at zero strain, but this is compensated by an increase in contributions from the in-plane phonon modes. L. F. C. Pereira and D. Donadio, Phys. Rev. B 87, 125424 (2013). D. J. Ecsedy and P. G. Klemens, Phys. Rev. B 15, 5957 (1977). N. Mingo and D. A. Broido, Nano Lett. 5, 1221 (2005).