Thermal transport in graphene nanoribbons supported on SiO${}_2$

We present a theoretical model for thermal transport in graphene nanoribbons (GNRs) on SiO${}_2$ based on solving the phonon Boltzmann transport equation. Thermal transport in supported GNRs is characterized by a complex interplay between line edge roughness (LER) and internal scattering, as captured through an effective LER scattering rate that depends not only on the surface roughness features, but also on the strength of internal scattering mechanisms (substrate, isotope, and umklapp phonon scattering). Substrate scattering is the dominant internal mechanism, with a mean free path (mfp) of approximately 67 nm. In narrow supported GNRs ($W<130$ nm, i.e., roughly twice the mfp due to substrate scattering), phonon transport is limited by LER and spatially anisotropic. For intermediate widths ($130\mathrm{nm}<W<1$ $\mu$m) a competition between LER and substrate scattering governs transport, while thermal transport in wide GNRs ($W>1\mu$m) is dominated by substrate scattering and spatially isotropic. Thermal transport in supported GNRs can be tailored by controlling the ribbon width and edge roughness. We conclude that narrow ribbons act as longitudinal heat conduits while wide ribbons act as good omnidirectional heat spreaders.

Figure 1

Schematic representation of a graphene nanoribbon, showing the ribbon width $W$, line edge rms roughness $\Delta$, and a phonon path through the ribbon with the pBTE solution $n_{\vec{q}}\left(y\right)$ and boundary conditions $n_{\tilde{q}}\left(W\right)$ and $n_{\vec{q}}\left(0\right)$ marked along that path. The phonon path inside the ribbon is interrupted by an internal scattering event. Thermal conductivities in the parallel (${\kappa }_{\left|\right|}$) and perpendicular (${\kappa }_{\perp }$) directions are also depicted.

Figure 2

Position dependence of the room-temperature lattice thermal conductivity of graphene nanoribbons in narrow [$W=15$ nm in (a) and (b)] and wide [$W=1.5$ $\mu$m in (c) and (d)]. Line edge roughness was set to $\Delta =1$ nm in all cases. The ${\kappa }^{+/-}$ components, as well as the total thermal conductivities, are plotted in the parallel [panels (a) and (c)] and perpendicular directions [(b) and (d)]. Thick lines are for supported ribbons, where the rapid spatial variation of ${\kappa }^{+}$ and ${\kappa }^{-}$ is caused by the smaller internal mean free path, limited to 67 nm by the substrate interactions. Thin lines are for suspended ribbons, showing a much slower spatial variation due to the larger internal mean free path, limited only by umklapp phonon-phonon and isotope scattering.

Figure 3

Comparison of lattice thermal conductivity of a wide ($W=2$ $\mu$m) graphene ribbon supported on SiO${}_2$ with experimentally measured data from Ref. 12. Dashed lines are calculated contributions by individual phonon branches and the solid line is the total, showing excellent agreement with experiment throughout the temperature range. At low temperatures, the dominant contribution is from the out-of-plane acoustic (ZA) mode, which gets suppressed by the strong substrate interaction above 100 K, where in-plane modes take over.

Figure 4

Effective lattice thermal conductivity of a 15-nm-wide graphene nanoribbon supported on a SiO${}_2$ substrate with varying line edge roughness, ranging from 0.1 nm up to 1 nm. The results agree well with the experimental data range reported for that same width of ribbon in Ref. 15. (b) Polar plot of directional thermal conductivity (radial coordinate, in units W/m$·$K) versus direction (polar angle with respect to the longitudinal $x$ axis), for temperatures from 100 K to 400 K, in 100 K steps. The spatial anisotropy of thermal conduction can also be depicted through the ratio of the parallel to the perpendicular component of the thermal conductivity tensor in (c). The ratio reaches a value of 3.5 at room temperature in the 15 nm wide ribbon with 0.25 nm edge roughness.

Figure 5

Dependence of the room-temperature parallel (a) and perpendicular (b) lattice thermal conductivities on the width $W$ of supported graphene nanoribbons. Contributions of each acoustic phonon branch, as well as the total, are shown. The longitudinal acoustic (LA) phonons are the least affected by LER scattering and have a dominant contribution to ${\kappa }^{\left|\right|}$. In contrast, ${\kappa }^{\perp }$ shows a steeper dependence on width $W$, with no clearly dominant branch. The thin lines show the values for the suspended GNR counterparts, demonstrating that the out-of-plane acoustic (ZA) mode is the most suppressed by scattering with the substrate. The ${\kappa }^{\perp }/{\kappa }^{\left|\right|}$ ratio is shown in panel (c) for line edge roughness rms values ranging from $\Delta =0.1$ nm to $\Delta =1$ nm. The anisotropy is stronger in narrower ribbons, and ribbons with larger LER values due to the strong dependence [shown in panel (d)] of the parallel and perpendicular components of the LER mfp on width $W$.