Elsevier

Carbon

Volume 125, December 2017, Pages 384-390
Carbon

Kapitza thermal resistance across individual grain boundaries in graphene

https://doi.org/10.1016/j.carbon.2017.09.059Get rights and content

Abstract

We study heat transport across individual grain boundaries in suspended monolayer graphene using extensive classical molecular dynamics (MD) simulations. We construct bicrystalline graphene samples containing grain boundaries with symmetric tilt angles using the two-dimensional phase field crystal method and then relax the samples with MD. The corresponding Kapitza resistances are then computed using nonequilibrium MD simulations. We find that the Kapitza resistance depends strongly on the tilt angle and shows a clear correlation with the average density of defects in a given grain boundary, but is not strongly correlated with the grain boundary line tension. We also show that quantum effects are significant in quantitative determination of the Kapitza resistance by applying the mode-by-mode quantum correction to the classical MD data. The corrected data are in good agreement with quantum mechanical Landauer-Bütticker calculations.

Introduction

Graphene [1], the famous two-dimensional allotrope of carbon, has been demonstrated to have extraordinary electronic [2], mechanical [3], and thermal [4] properties in its pristine form. However, large-scale graphene films, which are needed for industrial applications are typically grown by chemical vapor deposition [5] and are polycrystalline in nature [6], consisting of domains of pristine graphene with varying orientations separated by grain boundaries (GB) [7], [8], [9]. They play a significant or even dominant role in influencing many properties of graphene [10], [11].

One of the most striking properties of pristine graphene is its extremely high heat conductivity, which has been shown to be in excess of 5000 W/mK [4], [12]. Grain boundaries in graphene act as line defects or one-dimensional interfaces which leads to a strong reduction of the heat conductivity in multigrain samples [13], [14]. The influence of GBs can be quantified by the Kapitza or thermal boundary resistance R. The Kapitza resistance of graphene grain boundaries has been previously computed using molecular dynamics (MD) [15], [16] and Landauer-Bütticker [17], [18] methods, and has also been measured experimentally [19]. However, these works have only considered a few separate tilt angles, and a systematic investigation on the dependence of the Kapitza resistance on the tilt angle between any two pristine grains is still lacking. The relevant questions here concern both the magnitude R for different tilt angles and possible correlations between the structure or line tension of the GBs and the corresponding value of R.

Modelling realistic graphene GBs has remained a challenge due to the multiple length and time scales involved. Recently, an efficient multiscale approach [20] for modelling polycrystalline graphene samples was developed based on phase field crystal (PFC) models [21], [22]. The PFC models are a family of continuum methods for modelling the atomic level structure and energetics of crystals, and their evolution at diffusive time scales (as compared to vibrational time scales in MD). The PFC models retain full information about the atomic structure and elasticity of the solid [22]. It has been shown [20] that using the PFC approach in two-dimensional space one can obtain large, realistic and locally relaxed microstructures that can be mapped to atomic coordinates for further relaxation in three-dimensional space with the usual atomistic simulation methods.

In this work, we employ the multiscale PFC strategy of Ref. [20] to generate large samples of tilted, bicrystalline graphene with a well-defined GB between the two grains. These samples are then further relaxed with MD at T = 300 K. A heat current is generated across the bicrystals using nonequilibrium MD (NEMD) simulations, and the Kapitza resistance is computed from the temperature drop across the GB. We map the values of R(θ) for a range of different tilt angles θ and demonstrate how R correlates with the structure of the GBs. Finally, we demonstrate that quantum corrections need to be included in R to obtain quantitative agreement with experiments and lattice dynamical calculations.

Section snippets

PFC models

PFC approaches typically employ a classical density field ψ(r) to describe the systems. The ground state of ψ is governed by a free energy functional F[ψ(r)] that is minimized either by a periodic or a constant ψ, corresponding to crystalline and liquid states, respectively. We use the standard PFC modelF=dr12ψε+q2+22ψ+13τψ3+14ψ4,where the model parameters ε and τ are phenomenological parameters related to temperature and average density, respectively. The component q2+22 penalizes for

Results and discussion

It is well known [15], [16], [33] that the calculated Kapitza resistance depends on the sample length in NEMD simulations. Fig. 4 shows the calculated Kapitza resistance R in the 2θ=9.43 case as a function of the sample length Lx. Using fixed boundary conditions as described above, R saturates at around Lx = 400 nm. On the other hand, using periodic boundaries as described in Ref. [15], R converges more slowly. To this end, we have here used fixed boundary conditions and a sample length of

Summary and conclusions

In summary, we have employed an efficient multiscale modelling strategy based on the PFC approach and atomistic MD simulations to systematically evaluate the Kapitza resistances in graphene grain boundaries for a wide range of tilt angles between adjacent grains. Strong correlations between the Kapitza resistance and the tilt angle, the grain boundary line tension, and the defect density are identified. Quantum effects, which have been ignored in previous studies, are found to be significant.

Acknowledgements

This research has been supported by the Academy of Finland through its Centres of Excellence Program (Project No. 251748). We acknowledge the computational resources provided by Aalto Science-IT project and Finland's IT Center for Science (CSC). K. A. acknowledges the financial support from Iran Ministry of Science and Technology. P.H. acknowledges financial support from the Foundation for Aalto University Science and Technology, and from the Vilho, Yrjö and Kalle Väisälä Foundation of the

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