Effect of substrate modes on thermal transport in supported graphene

We examine thermal transport in graphene supported on SiO${}_2$ using molecular dynamics simulations. Coupling to the substrate reduces the thermal conductivity (TC) of supported graphene by an order of magnitude, due to damping of the flexural acoustic (ZA) phonons. However, increasing the strength of the graphene-substrate interaction enhances the heat flow and effective TC along supported graphene, contrary to expectations. The enhancement is due to the coupling of graphene ZA modes to the substrate Rayleigh waves, which linearizes the dispersion and increases the group velocity of the hybridized modes. These findings suggest that the TC of two-dimensional supported graphene is tunable through surface interactions, providing an interesting route for controlled energy flow in nanomaterials.

Figure 1

Setup for simulation of thermal transport in free and supported graphene in the $z$ direction. The graphene is coplanar to the x-z plane, and $L_x$, $L_y$, and $L_z$ are the width, height, and length of the simulated structure, respectively. We simulate thermal transport in (a) SLG sheet ($L_y$ $=$ 3.4 Å), (b) SiO${}_2$ thin film ($L_y$ $=$ 48.0 Å), and (c) sandwich structure with the SiO${}_2$ film between two SLG sheets ($L_y$ $=$ 54.8 Å). $L_x$ $=$ 51.1 Å and $L_z$ $=$ 295.2 Å for all structures. We impose periodic boundary conditions in the $x$ and $z$ directions. In the NEMD simulations, the “hot” atoms at the center are thermostatted at 310 K, and the “cold” edges are at 290 K.

Figure 2

(a) Temperature profile from center to edge of free graphene and (b) integral of the heat flux autocorrelation function (ACF). (c) Temperature profile from center to edge of SiO${}_2$ film and (d) integral of the heat flux ACF [Eq. (2)].

Figure 3

(a) and (b) Long-wavelength ($k$ $=$ 1 to 4) vibrational spectrum of free graphene (taller, narrower peaks) and supported graphene (shorter, broader peaks). (a) Distinct LA phonon peaks show little effect of the substrate. (b) ZA phonon peaks are significantly broadened and upshifted after the graphene is supported on the substrate. (c) Plot of heat flux through two suspended (2$Q_G$) and supported ($Q$ –$Q_{\mathit{ox}}$) graphene layers versus χ, which scales the vdW interaction at the graphene-SiO${}_2$ surface; χ $=$ 1 is the default interaction strength. The heat flux through supported graphene is approximately one order of magnitude lower than that through the suspended SLG. (d) Zoom-in version of previous plot. The heat flux along supported SLG ($Q$ –$Q_{\mathit{ox}}$) increases by a factor of three as the graphene-SiO${}_2$ interaction $\chi$ increases from 0.1 to 10. This counterintuitive result can be explained by the coupling of graphene ZA modes with SiO${}_2$ surface waves, which increase the group velocity of the hybridized modes (see text and Fig. 4).

Figure 4

Calculated ($d$ /$\mathrm{d\omega }$ ) Re[1/$h$ ($q$ ,$\omega$ )] for different values of the graphene-substrate interaction χ, from (a) nearly free graphene (χ $=$ 0.0001), (b) χ $=$ 0.1, (c) χ $=$ 1 (default case), and (d) tenfold stronger interaction, χ $=$ 10. For nearly free graphene the ZA modes show the typical quadratic dispersion. As the graphene-substrate interaction increases, ZA graphene modes hybridize with SiO${}_2$ Rayleigh waves, leading to linearized dispersion, higher group velocity, and enhanced thermal transport.

Figure 5

Schematic of our system of indexing the atoms in the 1920-atom graphene sheet in our computation of the SED. The lattice is divided into 4-atom unit cells with translational symmetries in the $x$ and $z$ directions. The number of repeating units in the $x$ and $z$ directions are 12 and 40, respectively. The translational symmetries also imply that we can define the wave vectors $k_x$ (=1 to 12) and $k_z$ (=1 to 40). The reflection symmetry about the x-y plane implies that the $k_z$ $=$ $n$ and the $k_z$ $=$ 40-$n$ modes are degenerate.