The atomistic Green's function method (AGF) has emerged as a useful tool to study phonon transport across interfaces. A comprehensive review of developments in the AGF method over the last decade is provided in this chapter. The content includes a discussion of the fundamentals of a Green's function starting from a continuum viewpoint and extending it to the atomistic regime. Comprehensive derivations of the AGF equations (within the harmonic framework) are presented along with intuitive physical explanations for the various matrices involved. The numerical issues in computational implementation of the various mathematical equations are illustrated with a one-dimensional atom chain example. The application of the AGF method to dimensionally mismatched and bulk interfaces and the process of obtaining polarization-specific transmission functions are illustrated with examples. Recent advancements such as integration of the AGF method with other tools (such as density functional theory and Boltzmann transport equation solvers) and extension of the AGF method to include anharmonicity are also presented. Comparisons of results from the AGF method to experimental measurements for superlattice and metal-graphene interfaces are provided.

We investigate the thermal transport in isolated single layer graphene (SLG) and SLG supported on Cu substrate using equilibrium molecular dynamics simulations and relaxation time approximation (RTA) method. We observe significant changes in the SLG dispersion curve in low frequency and low wave-vector region due to the interaction with Cu substrate. Several new phonon modes related to out-of-plane vibrations appear at the low frequency and small wave vector regions, but their contribution to graphene thermal conductivity is negligible. The thermal conductivity of graphene decreases by 44% due to the interactions with Cu substrate for high interaction strength parameter in Lennard-Jones potential formulation for graphene-Cu interaction. The phonon mode analysis through the RTA approach shows that the acoustic phonons dominate the thermal transport for both isolated and supported graphenes. The longitudinal acoustic (LA), transverse acoustic (TA), and out-of-plane acoustic (ZA) phonons contribute 654, 330, and 361 W/mK to the lattice thermal conductivity of isolated graphene, respectively. The phonon life time of ZA modes decreases by order of magnitude due to the interactions with Cu substrate and ZA mode contribution to SLG thermal conductivity decreases by 282 W/mK, while the contributions of LA and TA phonons reduce by 77.4 W/mK and 82.9 W/mK, respectively.

We investigate phonon transmission and thermal boundary conductance (TBC) across graphene/Cu interface using density functional theory and atomistic Green's function method. The analysis of phonon dispersions and density of states of single layer graphene (SLG) shows that even weak SLG/Cu interaction can soften the transverse optical and longitudinal optical modes and suppress low frequency out-of-plane acoustic (ZA) modes. Our calculations predict that a small stretching of the SLG lattice (∼1.6%) significantly enhances the SLG/Cu interaction which is reflected in larger band-gap for ZA/ZO phonon mode and a remarkable increase in TBC (∼59%).

We use density functional theory and the atomistic Green's function method (AGF) to study the effect of bonding on phonon transmission and thermal boundary conductance (TBC) at the interface of metals (Au, Cu, and Ti) and single layer graphene (SLG)/multi-layer graphene (MLG). Our analysis shows that the TBC across Ti/SLG/Ti interfaces (∼500 MW m⁻² K⁻¹) is significantly larger than the TBC across Cu/SLG/Cu (∼10 MW m⁻² K⁻¹) and Au/SLG/Au (∼7 MW m⁻² K⁻¹) interfaces. However, the TBC across Ti/MLG/Ti (∼40 MW m⁻² K⁻¹) is an order of magnitude lower compared to TBC at the Ti/SLG/Ti interface, whereas the TBC at Cu/MLG/Cu and Au/MLG/Au interfaces are similar to those of Cu/SLG/Cu and Au/SLG/Au, respectively. We find that this substantial decrease in TBC at the Ti/MLG/Ti interface is a result of phonon mismatch between the graphene layer bonded to Ti and the non-bonded graphene layers. The effect of number of graphene layers on TBC at Cu/MLG/Cu and Au/MLG/Au interfaces is relatively insignificant because of the weak interactions at these metal/graphene interfaces. It was observed that the moderate attenuation of Ti/C bonding strength can enhance the phonon coupling between the graphene layers bonded to Ti and non-bonded graphene layers, and can increase the TBC across Ti/MLG/Ti by ∼100%. This impact of interfacial bonding strength on TBC at metal/MLG interfaces, predicted by AGF calculations, is further confirmed by non-equilibrium molecular dynamics simulations which show the transition of thermal transport mechanism from metal/graphene dominated resistance to graphene/graphene dominated resistance as the metal/graphene bonding strength increases in the metal/MLG/metal structure.