Theory of substrate-directed heat dissipation for single-layer graphene and other two-dimensional crystals

We present a theory of the phononic thermal (Kapitza) resistance at the interface between graphene or another single-layer two-dimensional (2D) crystal (e.g., ${\mathrm{MoS}}_2)$ and a flat substrate, based on a modified version of the cross-plane heat transfer model by Persson, Volokitin, and Ueba [J. Phys.: Condens. Matter 23, 045009 (2011)]. We show how intrinsic flexural phonon damping is necessary for obtaining a finite Kapitza resistance and also generalize the theory to encased single-layer 2D crystals with a superstrate. We illustrate our model by computing the thermal boundary conductance (TBC) for bare and ${\mathrm{SiO}}_2$-encased single-layer graphene and ${\mathrm{MoS}}_2$ on a ${\mathrm{SiO}}_2$ substrate, using input parameters from first-principles calculation. The estimated room temperatures TBC for bare (encased) graphene and ${\mathrm{MoS}}_2$ on ${\mathrm{SiO}}_2$ are 34.6 (105) and 3.10 (5.07) ${\mathrm{MW}\mathrm{K}}^{-1}{\mathrm{m}}^{-2}$, respectively. The theory predicts the existence of a phonon frequency crossover point, below which the low-frequency flexural phonons in the bare 2D crystal do not dissipate energy efficiently to the substrate. We explain within the framework of our theory how the encasement of graphene with a top ${\mathrm{SiO}}_2$ layer introduces new low-frequency transmission channels, which significantly reduce the graphene-substrate Kapitza resistance. We emphasize that the distinction between bare and encased 2D crystals must be made in the analysis of cross-plane heat dissipation to the substrate.

Figure 1

Schematic representation of the bare single-layer 2D crystal (graphene or ${\mathrm{MoS}}_2)$ and the substrate. The springs represent the weak van der Waals interaction at the interface. The simulation parameters are also displayed.

Figure 2

Schematic of the single-layer 2D crystal (graphene or ${\mathrm{MoS}}_2)$ with the superstrate and the substrate.

Figure 3

Top and side views of the graphene/${\mathrm{SiO}}_2$ heterostructure used in our density function theory (DFT) calculations to estimate $K$ at the ${\mathrm{SiO}}_2$ (001) surface with (a) OH-termination and (b) H termination. The C, H, Si, and O atoms are colored in gray, white, yellow, and red, respectively. The change in total energy $\left(\mathrm{\Delta }E\right)$ as a function of $d$ and the parabolic fit used to find $K$ are also shown.

Figure 4

Plot of the thermal boundary conductance $G$ at 300 K at different values of ${\alpha }^{-1}$ for bare graphene (circle) and ${\mathrm{MoS}}_2$ (diamond) on OH-terminated ${\mathrm{SiO}}_2$ surface. A decrease in the intrinsic damping of the flexural phonons leads to lower thermal boundary conductance (or higher Kapitza resistance). This effect is more pronounced for graphene.

Figure 5

Temperature dependence of the thermal boundary conductance (TBC) for bare single-layer graphene (Gr) and ${\mathrm{MoS}}_2$ on OH- and H-terminated ${\mathrm{SiO}}_2$ surface. The data point for bare graphene from Mak et al. [15] is represented by the open circle, while the data point for bare ${\mathrm{MoS}}_2$ from Taube et al. [19] is represented by the open triangle.

Figure 6

Temperature dependence of the thermal boundary conductance (TBC) for ${\mathrm{SiO}}_2$-encased graphene (Gr) and ${\mathrm{MoS}}_2$ on OH- and H-terminated ${\mathrm{SiO}}_2$ interfaces. The open circles correspond to temperature-dependent TBC data for single-layer encased graphene taken from Chen et al. [14].

Figure 7

Plot of the transmission spectrum $\mathrm{\Xi }(\mathbf{q},\omega )$ [see Eq. (16)] for (a) bare and (b) encased graphene on OH-terminated ${\mathrm{SiO}}_2$ at $T=300$ K. The dashed and dotted lines correspond to the graphene flexural $\left(\omega =\sqrt{\kappa /\rho }{q}^2\right)$ and substrate transverse acoustic $\left(\omega =c_{T}q\right)$ phonon dispersion curves, respectively. The insets show a schematic representation of the bulk substrate phonon scattering with the graphene flexural phonon. (c) Comparison of the total transmission per unit area spectrum ${\left(2\pi \right)}^{-2}\int d^{2}q\mathrm{\Xi }(\mathbf{q},\omega )$ for bare (solid red line) and encased graphene (dashed blue line) on OH-terminated ${\mathrm{SiO}}_2$ at $T=300$ K. The inset shows the same spectra but with a logarithmic scale. The small fuzzy peaks are an artifact of the numerical integration in $q$.

Figure 8

Schematic representation of the physical model of heat dissipation in bare and covered single-layer 2D crystals. The flexural phonons in the bare 2D crystal absorb energy from the nonflexural phonons and electrons via anharmonic and direct electron-phonon coupling, and dissipate it into the substrate. The mechanical attachment of a superstrate results in additional resonant channels for energy to be absorbed by the flexural phonons and then transferred to the substrate, leading to a higher thermal boundary conductance.