Randomness-Induced Phonon Localization in Graphene Heat

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Surfaces, Interfaces, and Catalysis; Physical Properties of Nanomaterials and Materials

Randomness-Induced Phonon Localization in Graphene Heat Conduction Shiqian Hu, Zhongwei Zhang, Pengfei Jiang, Jie Chen, Sebastian Volz, Masahiro Nomura, and Baowen Li J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b01653 • Publication Date (Web): 03 Jul 2018 Downloaded from http://pubs.acs.org on July 3, 2018

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The Journal of Physical Chemistry Letters

Randomness-Induced Phonon Localization in Graphene Heat Conduction Shiqian Hu 1,2,3#, Zhongwei Zhang 1,2,3#, Pengfei Jiang1,2,3, Jie Chen1,2,3*, Sebastian Volz2,4,5*, Masahiro Nomura6, and Baowen Li7* 1

Center for Phononics and Thermal Energy Science, School of Physics Science and

Engineering, and Institute for Advanced Study, Tongji University, Shanghai 200092, People's Republic of China 2

China–EU Joint Lab for Nanophononics, School of Physics Science and Engineering, Tongji University, Shanghai 200092, People’s Republic of China

3

Shanghai Key Laboratory of Special Artificial Microstructure Materials and

Technology, School of Physics Science and Engineering, Tongji University, Shanghai 200092, People’s Republic of China 4

Laboratoire d’Energétique Moléculaire et Macroscopique, Combustion UPR CNRS

288, Ecole Centrale Paris, Grande Voie des Vignes F-92295 Chatenay-Malabry, France 5

Laboratory for Integrated Micro and Mechatronic Systems, CNRS-IIS UMI 2820,

University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan 6

Institute of Industrial Science, The University of Tokyo, Meguro-ku, Tokyo 153-8505,

Japan 7

Department of Mechanical Engineering, University of Colorado, Boulder, CO 80309,

USA *Corresponding author: Email: [email protected] (J.C.); Tel: +86-21-65985331 Email: [email protected] (S. V.); Tel: +81-3-5452-6305 Email: [email protected] (B.L.); Tel: (303) 492-0553

Abstract Through non-equilibrium molecular dynamics simulations, we report the direct 1

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numerical evidence for the coherent phonons participating in thermal transport at room temperature in graphene phononic crystal (GPnC) structure, and evaluate their contribution to thermal conductivity based on the two-phonon model. With decreasing period length in GPnC, the transition from the incoherent to coherent phonon transport is clearly observed. When a random perturbation to the positions of holes is introduced in a graphene sheet, the phonon wave packet simulation reveals the presence of notable localization of coherent phonons, leading to the significant reduction of thermal conductivity and suppressed length dependence. Finally, the effects of period length and temperature on the coherent phonon contribution to thermal conductivity are also discussed. Our work establishes a deep understanding of the coherent phonons transport behavior in periodic phononic structures, which provides effective guidance for engineering thermal transport based on a new path via phonon localization.

TOC Graphic

In recent years, efficient modulation of the thermal transport has become more

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and more urgent due to the imperative need to enhance the heat dissipation rate in the continuously miniaturized electronic devices, and the acuter demand to improve energy conversion efficiency based on thermoelectrics. Since phonons dominate heat transfer in crystalline dielectrics and semiconductors, the manipulation of the thermal conductivity (κ) can be realized through controlling phonons by two types of mechanisms: the incoherent phonon scattering mechanism and the coherent mechanism. Here, the coherent mechanism means the manipulation of phonons via the modification of phonon dispersion arising from the phonon wave effects, such as the opening of phononic band gaps and reduction of group velocity, which reflects the coherent nature of phonons. In contrast, the incoherent mechanism means to treat phonon scatterings similar as the particle collisions originated from the diffusive phonon scatterings by the boundary, impurity, or the intrinsic anharmonic phonon-phonon interaction. In this case, the coherence/correlation of phonons is lost, i.e., incoherent effect. Traditionally, the thermal conductivity is manipulated through impurities nanoparticles

3-6

, defects

1-2

,

7-12

, ion-intercalation13, encapsulation14, inter-particle

constriction15 and static electricity16 in materials by manipulating the incoherent phonon scattering17-18. Recently, a new strategy focuses on managing phonons by superlattice and nanostructured phononic crystals (PnCs)

4, 19-28

, which allows for

controlling heat by engineering the phonon band structures (coherent phonon transport)

29-33

. The coherent phonon transport behavior has been observed in several

theoretical34-36 and experimental37-39 investigations on superlattice structures, such as

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in Si/Ge34, graphene/boron nitride35-36 and GaAs/AlAs28 superlattice. Moreover, the manipulation of the coherent phonons in PnCs which have minimal influence on electrons is considered as the key for improving thermoelectric efficiency19. Although extensive efforts have been made to investigate the reduction of κ by constructing PnC structures and coherent phonon transport has been predicted by theoretical studies40-41, only few experimental works25,

38

could directly observe

coherent phonon transport in PnCs due to the difficulty of avoiding the phonon phase breaking events caused by phonon-phonon and phonon-roughness interactions. In most experiments, κ was found to increase monotonically with the increase of the lattice period, and the measured42-44 κ value of PnC samples are significantly lower than those of theoretical predictions after taking into account the combination of material removal and incoherent phonon scattering. And the role played by coherent and incoherent phonons in PnC has not been systematically investigated yet and remains unclear, which requires further investigation to separate these two types of phonons and quantify their contributions to thermal transport.

In this work, using non-equilibrium molecular dynamics simulations (NEMD), we report the direct evidence for the coherent phonon in periodic atomic structure and further quantify the contributions of coherent and incoherent phonons to the total thermal conductivity of graphene phononic crystals (GPnC) via the construction of a disordered graphene phononic crystal (DGPnC) structure. Firstly, the signature for coherent phonon thermal transport is observed in GPnC structures by varying the period length. Furthermore, we find that the thermal conductivity of DGPnC is 4


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significantly reduced compared to that of GPnC, showing negligible length dependence for the thermal conductivity of DGPnC. Based on the two-phonon model45, we then reveal the two different transport behaviors of coherent phonons in GPnC and DGPnC. Based on the spatial distribution of pressure and the phonon wave packet simulation, the direct numerical evidence for the emergence of the coherent phonons is obtained. And the underlying physical mechanism is uncovered by analyzing the spatial energy distribution of the localized phonons and the phonon transmission. Finally, the physical parameters affecting coherent phonon thermal transport are studied. A clear picture of the coherent phonons transport properties in GPnC is obtained in this work. As shown in Figure 1a, the GPnC structure is composed of the holey unit cells constructed by creating holes in the pristine graphene, ensuring simultaneously that the hole edge is zigzag, simply to remove the effect of different defect types on the thermal transport of graphene11. The unit cells are characterized by the period L0 and porosity. Here, the porosity is defined as NR/NP, where NR and NP are the number of removed atoms and the total number of atoms in the pristine graphene, respectively (see supporting information Part I). Thermal conductivity of graphene is in-principle composed of electron and phonon contributions. Before we start our systematic investigation, we first estimate the electron contribution to the total thermal conductivity in GPnC structure by using the non-equilibrium Green's function method (see supporting information Part II for calculation details). Our calculation results (Figure S2 in supporting information)

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show that the electron contributes only ~13% of the total thermal conductance in the undoped GPnC structure at room temperature, which is in good agreement with the calculation result from previous theoretical study on graphene46. Therefore, we only consider the lattice thermal conductivity in the following discussion. NEMD simulations in this paper are performed by using LAMMPS package47. Brenner48 potential is used to simulate the covalent bonding interaction between carbon atoms. Fixed and periodic boundary conditions are adopted along the length and width direction, respectively. To establish a temperature gradient, two Langevin thermostats49 with different temperatures are applied to the two ends of the simulation system. The thermal conductivity κ is calculated based on Fourier’s Law, κ = −

J ∇T ,

(1)

where ∇ T and J are, respectively, the temperature gradient and the heat flux. Here the effective thermal conductivity for the finite-size system is obtained through Eq. (1), as the thermal transport behavior in our study is not necessarily diffusive. Similar treatment has been widely adopted in the literature studies50-54 on the anomalous heat conduction in low-dimensional lattices and materials. More simulation details can be found in the supporting information Part III. We first study the thermal conductivity of GPnCs at 300 K. The width of GPnC is chosen as 8 nm, which is large enough to eliminate the numerical size effect stemming from the periodic boundary conditions set in the width direction. The system length L is fixed to 82 nm. Figure 1b shows the thermal conductivity κ of GPnCs as a function of the period L0 for different porosities at 300 K. The results 6


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show that κ first decreases and then increases with decreasing L0. Using another empirical force field for carbon (the optimized Tersoff

55

potential), we have verified

that this non-monotonic dependence of κ on L0 is generic (see supporting information Figure S4), regardless of the specific force field used. The critical period, at which the minimum κ is located, can be regarded as the dividing line between the incoherent phonon dominated regime and the coherent phonon dominated regime. This crossover phenomenon of the incoherent and coherent regimes was verified experimentally in a superlattice structure38. Above the critical period, large L0 provides enough space (higher probability) for the occurrence of phonon-phonon scatterings before a phonon traverses the holes in each unit cell, which can prevent the formation of coherent phonon modes. In this case, incoherent phonons dominate the thermal transport in the GPnCs. As a result, with a fixed total length and decreasing L0, the number of the unit cells (holes) increase, leading to the stronger edge scatterings and the decrease of the thermal conductivity of the GPnCs. In contrast, as L0 further decreases below the critical period, more phonons can be transmitted ballistically before getting reflected at the holes edge, which is beneficial to form coherent phonons due to the phonon interference caused by the multiple reflections at the periodic holes edge. In this coherent region, a smaller L0 corresponds to a larger Brillouin zone, leading to less band folding in the phonon dispersion. This results in a weaker flattening of the phonon band (see supporting information Figure S5), and thus leads to the increase of the phonon group velocity,

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which is responsible for the increase of the thermal conductivity.56-57 Moreover, at the same period L0, κGPnC is significantly reduced when increasing porosity from 10% to 25% due to the stronger edge scattering (Figure 1b), which is consistent with previous works 24, 32. Figure 1c shows κ of GPnCs versus L0 at a fixed porosity of 25% with the temperature varying from 300 K to 1000 K. One interesting finding is that the non-monotonic behavior and the κ minimum become less distinct at elevated temperature. This is because higher temperature causes stronger anharmonic phonon– phonon scattering, leading to the suppression of the coherent phonons. As shown in Figure 1b, the length scale for the critical period is very small, on the order of a few nanometers at room temperature. Defects are inevitably introduced during the experimental fabrication of such a fine PnC structure, which may affect the observation of such coherent phonon transport in real experiment. Furthermore, as the incoherent phonons only depends on the period L0 and the porosity, while the coherent phonons depends additionally on the exact periodicity, i.e., the distribution of these holes in unit cells42, the introduction of randomness provides us an opportunity to distinguish the contributions from the coherent and incoherent phonons in the thermal conductivity of GPnC as studied in two-dimensional phononic crystal and disordered nanostructures25. In this regard, we construct the DGPnC structure (Figure 2a) by randomly distributing the position of the hole in each unit cell (see supporting information Part V) while keeping the period L0 and porosity the same as that in GPnC.

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Figure 2b shows the length dependence of thermal conductivity for both GPnC (κGPnC) and DGPnC (κDGPnC) with the same porosity of 10% at 300 K. Here, L0 is fixed at 1.24 nm, which falls into the coherent phonons dominated regime. NEMD simulation results show that κGPnC (rectangle) is always higher than κDGPnC (triangle) for the same length, except for the sufficiently short length (L

Randomness-Induced Phonon Localization in Graphene Heat

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