PHONON THERMAL PROPERTIES OF GRAPHENE ON HEXAGONAL BORON NITRIDE

Transcription

1 Proceedings of the Asian Conference on Thermal Sciences 217, 1st ACTS March 26-3, 217, Jeju Island, Korea PHONON THERMAL PROPERTIES OF GRAPHENE ON ACTS-P146 HEXAGONAL BORON NITRIDE Ji-Hang Zou, Bing-Yang Cao * Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 184, China *Corresponding author: Tel/Fax: ; caoby@tsinghua.edu.cn ABSTRACT Graphene on hexagonal boron nitride (h-bn) has demonstrated great promise for future field effect transistors, and it is essential to understand the thermal dissipation in such nanoelectronics. Despite previous work concerning the thermal transport of graphene on Cu or SiO2, the phonon thermal properties of graphene remain elusive when interacting with the substrate h-bn. In this work, molecular dynamics simulations are performed to extract the phonon dispersion curves and lifetimes of graphene on h-bn. The mode thermal conductivity of graphene is calculated based on the Boltzmann transport equation with relaxation time approximation. It is found that the dispersion curves have minor changes for supported graphene because the interlayer coupling is too weak to shift the harmonic phonon properties. The flexural phonon lifetimes have significant reduction, due to the breakdown of the selection rule in supported graphene as well as the mismatch of the out-of-plane PDOS for graphene and h-bn. The thermal conductivities estimated for LA and TA branches have small contraction because of a large overlapping of in-plane PDOS for graphene and h-bn. The dominant MFP of graphene is reduced from 9-8 nm to 6-5 nm. The thermal conductivity of supported graphene decreases by about 37.4% owing to the large reduction of flexural phonon lifetimes, and the relative contribution of flexural modes decreases from 35.% to 16.7%. KEYWORDS: graphene, h-bn, molecular dynamics, phonon lifetime, thermal conductivity 1

2 1. INTRODUCTION Graphene has received enormous attention in thermal management fields owing to its extremely high thermal conductivity of 3-5 W/ (m K). [1], [2] However, the substrate interaction is usually inevitable in applied situations, which may significantly affect the thermal properties of graphene. The measured thermal conductivity of graphene on SiO 2 is about 6 W/ (m K), which is one order of magnitude lower compared to suspended graphene. [3] Boltzmann transport equation (BTE) calculations [3] revealed that the thermal conductivity reduction of graphene on SiO 2 is mainly due to the shrinkage of flexural phonon contribution. Molecular dynamics (MD) predictions found that the phonon lifetime of graphene deceases a lot when coupling with the substrate Si, SiO 2 or Cu. [4]-[6] Tuning the thermal properties of graphene is useful for thermal management and thermoelectric energy conversion, and available methods include isotope doping, chemical functionalization, strain engineering, and arranging periodic vacancies or pores. [7]-[12] Substrate interaction is also an effective and convenient route to control the thermal conductivity of graphene, [13] hence it is crucial to investigate the thermal properties of graphene in contact with various materials. Despite the recent progress on the interfacial thermal transport across graphene and hexagonal boron nitride (h-bn), [14]-[16] the effect of the substrate h-bn has not been fully elucidated. It is appealing to choose h-bn as the substrate, because h-bn has similar twodimensional honeycomb structure and exhibits excellent mechanical and thermal properties. [17], [18] Moreover, it was uncovered that graphene on h-bn displays the highest electronic mobility reported on the substrate, [19], [2] indicating that graphene/h-bn heterostructure has remarkable advantages for nanoscale electronic devices such as field effect transistors and integrated circuits. Therefore, it is of great interest to analyze the phonon thermal properties of graphene supported on h-bn, which facilitates the thermal design for nanodevices. 2. METHODS In this letter, the phonon properties of free and supported graphene are extracted from equilibrium MD (EMD) simulations combined with the lattice dynamics calculations. The LAMMPS package [21] is applied to perform all MD simulations. The lattice constants of graphene and h-bn are a C = 2.46 A and a BN = 2.51 A, respectively. Therefore, a lattice mismatch strain is φ = (a BN a C ) a C = 2.% for h-bn while no strain is employed for graphene. The supported graphene is a 5.4 A 59.1 A nanosheet, stacked 3.35 A above the h-bn monolayer. Periodic boundary conditions are applied for the in-plane directions while free vibrations are allowed for the out-of-plane direction. The optimized Tersoff potential [22] is adopted for C-C interactions, which gives a sound description of phonon dispersion and group velocities for graphene. [23] The Tersoff potential developed by Sevik et al. [18] is used to model atomic interactions in h-bn. The interlayer couplings are described as the van der Waals type using the Lennard-Jones potentials V LJ ij = 4ε [( σ 12 ) ( σ 6 ) ], (1) r ij r ij with the parameters calculated from the universal force field (UFF), [24] ε BC = mev, 2

3 σ BC = A ; ε NC = mev, σ NC = A. The time step is.5 fs. First, the system is equilibrated at 3 K for 1 ns using the Nosé-Hoover thermostat. [25] Then, the system is relaxed in the NVE (constant mass, volume, and energy) ensemble for 3 ns. At last, another 1 ns NVE ensemble is performed to obtain the atomic velocities of supported graphene. The phonon dispersion and phonon lifetime can be calculated from the spectral energy density (SED) analyses derived by Thomas et al., [26] m c Φ(q, ν) = 4πnτ α 1 b= τ n u α (l, b, t) exp (iq r l 2πiνt) l=1 2 dt, (2) where m c is the mass of carbon atom, n is the number of unit cells, u α is the atom velocity in the α direction, r l is the equilibrium position of each unit cell, and τ is the total integration time. The SED Φ(q, ν) is a function of wave vector q and frequency ν, and the SED peaks can be fitted to the Lorentzian function as, I k Φ k (q, ν) = 1 + [2(ν ν max ) γ] 2, (3) where I k denotes the peak magnitude of the branch k, ν max is the peak center frequency, and γ is the half-width of the peak. Thus, the phonon dispersion is derived from the relation of ν max and q, and the phonon lifetime is defined by τ = 1 γ. 3. RESULTS AND DISCUSSIONS 3.1 PHONON DISPERSION AND LIFETIME Frequency ( z) (a) Free graphene Graphene on h-bn Phonon lifetime (ps) (b) ZA, f TA, f LA, f ZO, f TO, f LO, f ZA, s TA, s LA, s ZO, s TO, s LO, s Frequency (THz) Fig. 1 (a) Dispersion curves of free and supported graphene along the Γ-M direction. (b) Frequency-dependent phonon lifetime. The subscripts f and s are short for free and supported, respectively. As seen in Fig. 1 (a), the substrate interactions have negligible influence on the dispersion curves of graphene, because the weak interlayer coupling is not able to change the phonon harmonic properties. Similar computation result has been reported by the Refs. [4] and [5] when graphene is supported on SiO 2 or Cu. Ong et al. [13] found that the quadratic dispersion for ZA branch would be altered into linear dispersion if the coupling strength becomes very strong. However, it is difficult and inconvenient to change the coupling strength between the layers in 3

4 practical situations. The phonon lifetimes of free and supported graphene are shown in Fig. 1 (b). The ZA mode lifetimes are in the range of 3-4 ps for the supported graphene, which are largely suppressed due to the phonon-substrate scattering. Particularly for the low-frequency (<1 THz) ZA phonons, the lifetimes are reduced by a factor of about four compared to the isolated graphene. The lifetimes of the ZO phonons averagely decrease by more than 6% for graphene on h-bn. One reason for the significant reduction of flexural phonon lifetimes is probably the breakdown of the selection rule proposed by Lindsay et al. [27], [28] The substrate coupling breaks the reflection symmetry of the single-layer graphene, which allows more scattering channels for acoustic modes, especially the flexural modes. [28], [29] The TA and LA phonon lifetimes have small reduction for the frequencies lower than 8 THz, while other in-plane acoustic mode lifetimes increase slightly in supported graphene. The TO and LO optical modes have small lifetime values, and are not obviously affected by the substrate. 6 5 (a) In-plane Graphene h-bn 6 5 (b) Out-of-plane Graphene h-bn Freuqency ( z) Freuqency ( z) PDOS (arb. units) PDOS (arb. units) Fig. 2 (a) In-plane and (b) out-of-plane PDOS for graphene and h-bn To further analyze the substrate effect, the phonon density of states (PDOS) of h-bn and graphene are calculated by the formula, τ D(υ) = < u α(t)u α() > (4) exp ( iυt)dt, < u α()u α() > where < u α(t)u α() > is the correlation function, and D(υ) is the PDOS as a function of frequency υ. The overlapping of PDOS between two combined materials is widely used to determine the phonon transport properties across the interfaces. [14], [16], [3]-[32] From the view of lattice dynamics, the phonon lifetime for supported graphene may be related with the PDOS overlap of graphene and h-bn. As shown in Fig. 2 (a), the in-plane PDOS peak frequencies are 48 THz and 46 THz for graphene and h-bn, respectively. At the acoustic frequencies (<4 THz), there is significant PDOS overlap, giving rise to a good maintenance of TA and LA phonon lifetimes despite the increased acoustic phonon scattering rates in 4

5 supported graphene. From Fig. 2 (b), it can be seen that the out-of-plane PDOS of graphene is notably different from that of h-bn. The out-of-plane PDOS displays two peaks of about 15 THz and 24 THz for graphene, while the peaks occur at 9 THz and 18 THz for h-bn. In addition, the peak intensities of graphene PDOS are lower than that of h-bn PDOS. The significant reduction of out-of-plane phonon lifetimes may be associated with the mismatch of the PDOS for graphene and h-bn. 3.2 PHONON THERMAL CONDUCTIVITY Accumulated (normalized) (a) Free graphene Graphene on h-bn MFP (nm) Mode contribution to (W/mK) (b) ZA Free graphene Graphene on h-bn TA LA ZO LO+TO SUM Fig. 3 (a) Accumulative thermal conductivity as a function of MFP. (b) Mode thermal conductivities of free and supported graphene. The phonon thermal conductivity of graphene is calculated by the BTE with the relaxation time approximation (RTA), κ = c ph v g 2 τ q ν where c ph is the specific heat, and v g is the group velocity. As the phonon Bose-Einstein distribution does not exist in the classic system, the specific heat is expressed as c ph = k B /V, where k B is the Boltzmann constant and V is the system volume. [33] The group velocities are derived from v g = ω q. The phonon mean free path (MFP) is defined as MFP = v g τ. As indicated in Fig. 3 (a), we compare the thermal conductivity accumulation of supported graphene with that of free graphene. The MFP in the range of 9-8 nm is demonstrated to contribute 8% of the total thermal conductivity for free graphene. In comparison with experimental data, the measured effective MFP is about 2-8 nm for graphene. [2], [34] It should be noted that the BTE calculations predicted an average MFP of several micrometer or even millimeters for suspended graphene, [27], [34], [35] which is consistent with theoretical results. [37], [38] When graphene is supported on h-bn, the dominant MFP is reduced to 6-5 nm. Bae et al. [39] experimentally found that the large MFP is around 1 nm in supported graphene on SiO 2, which is relatively smaller than that of graphene on h-bn. Feng et al. [11] studied the effect of structural flaws on the phonon MFP of graphene, and found that the MFP of dominant phonons could be suppressed to lee than 1 nm by only introducing 1.1% defects. Therefore, the phonon MFP is more sensitive to structure defects than the substrate h-bn. From Fig. 3 (b), the mode thermal conductivities are compared between free and supported graphene. The ZA, TA and LA branches contribute about 23.1%, 22.4% and 42.3% to the total thermal 5 (5)

6 conductivity of suspended graphene, respectively. The high-frequency optical modes are trivial for graphene thermal conductivity. The calculated results indicate that the in-plane acoustic phonons have major contribution the heat conduction in free graphene, which agrees well with theoretical predictions and MD analyses. [5], [11], [38] Regarding the graphene on h-bn, the flexural mode thermal conductivity decreases from 1232 W/ (m K) to 367 W/ (m K), and the in-plane phonon thermal conductivity is only reduced by 2.6%. The flexural mode contribution to the total thermal conductivity is suppressed from 35.% to 16.7% owing to the phonon-substrate scattering. The phonon thermal conductivities for free and supported graphene are 3517 W/ (m K) and 22 W/ (m K), respectively. The thermal conductivity contraction percentage is about 37.4% for graphene on h-bn, which is mainly due to the significant reduction of flexural phonon lifetimes. The thermal conductivity of graphene on h- BN is nearly as four times as that of graphene on SiO 2, [3] and is about 37.% higher than that of graphene on Cu. [5] Hence, the substrate h-bn could terrifically conserve the thermal properties of graphene. In summary, the phonon thermal properties of graphene on h-bn is investigated by the EMD simulations and lattice dynamics calculations. The phonon dispersion and lifetime are obtained from the SED analyses. It is found that the dispersion curves of graphene are not obviously affected by the substrate h-bn due to the weak van der Waals interactions between the layers. The flexural phonon lifetimes in supported graphene have significant reduction owing to the invalidity of the selection rule as well as the mismatch of the PDOS of graphene and h-bn. The TA and LA phonon lifetimes are slightly affected by the substrate coupling due to the large overlap of the PDOS of graphene and h-bn. The mode thermal conductivity is computed from the BTE with the RTA. The dominant MFP of graphene decreases from 9-8 nm to 6-5 nm. The thermal conductivity of graphene is suppresses from 3517 W/ (m K) to 22 W/ (m K), which is mainly because of the significant reduction of flexural mode contribution. Our calculated results imply that the thermal properties of graphene on h-bn maintains high performance for thermal management and nanoscale electronics. ACKNOWLEDGMENT This work was financially supported by the National Natural Science Foundation of China (Grant Nos , , , , and ), Tsinghua National Laboratory for Information Science and Technology, and Science Fund for Creative Research Group (No ). REFERENCE [1]A. A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C. N. Lau, Superior thermal conductivity of single-layer graphene, Nano Lett. 8(3) (28) [2]S. Ghosh, I. Calizo, D. Teweldebrhan, E. P. Pokatilov, D. L. Nika, A. A. Balandin, W. Bao, F. Miao, C. N. Lau, Extremely high thermal conductivity of graphene: Prospects for thermal management applications in nanoelectronic circuits, Appl. Phys. Lett. 92(15) (28) [3]J. H. Seol, I. Jo, A. L. Moore, L. Lindsay, Z. H. Aitken, M. T. Pettes, X. S. Li, Z. Yao, R. Huang, 6

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