## 2 Theoretical Methods for TBR Prediction and Experimental Methods for TBR Measurement
### Theoretical and Computational Methods for TBR Prediction  
Various theoretical and computational methods have been employed to predict TBR. The theoretical methods conventionally employed are the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM). In AMM and DMM, the mismatch in the acoustic impedance and phonon density of states (DOS) on both sides of the interface inhibits phonon transmission across the interface [28]. In the AMM, phonons are considered plane waves and the materials through which they propagate are treated as continuous media. In the AMM, a geometrically perfect interface is assumed, and the phonon transport across it is entirely elastic. The transmission probabilities of phonons are determined by calculating the acoustic impedances on both sides of the interface. By assuming the wave nature of phonon transport and specular scattering at the interface, the AMM becomes applicable for predicting the TBR at low temperatures, where long-wavelength phonons dictate the thermal spectrum. In comparison, in the DMM, complete diffuse scattering is assumed to occur at the interface, implying that once a phonon undergoes scattering, it loses all memory of its original direction, polarization, and material of origin, with only its frequency remaining constant. The transmission probabilities of phonons are determined based on the mismatch in the phonon DOS between the two sides of the interface. That is, the transmission probability of phonons with a specific frequency through an interface is significantly higher when matching phonons with the same frequency on the opposite side of the interface. Compared with the AMM, the DMM is more suitable for non-cryogenic temperatures and interfaces with rough surfaces. This suitability is attributed to most acoustic phonons at temperatures of 300 K and above having relatively short wavelengths comparable to the interatomic spacing and surface roughness. In certain studies, instances of agreement have occurred between the TBR values predicted by the AMM or DMM and the experimental results. However, this agreement has been regarded as coincidental because of the simplified assumption regarding the phonon scattering mechanisms occurring at the interfaces [17]. Interfacial properties, such as interfacial disorder, interfacial roughness, interdiffusion, interfacial microstructure, and interfacial bonding [29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40], which are not considered in either model, have been shown to have noteworthy effects on the TBR. Therefore, modifications to both models are required to improve their prediction accuracy. For example, Figure 2a shows the effects of surface roughness on the TBR of Al/Si interfaces. Al films were grown on Si substrates subjected to various pre-Al-deposition surface treatments. The measured TBC was found to decrease with increasing Si surface roughness. Figure 2b shows the increase of TBC induced by the bonding enhancement using organic nanomolecular monolayer (NML). The TBC of the Cu/SiO\({}_{2}\) interface was found to increase with increasing interfacial toughness, which is an indirect measure of bond strength. A fourfold increase in the TBC was observed by using SH-terminated NMLs. The MD simulation results demonstrate that the large NML phonon DOS at low frequencies of <-2 THz form a broad band with an excellent overlap with the SiO\({}_{2}\) phonon DOS, which increased phonon transmission that leads to a high TBC. Figure 2c shows the effects of chemical bonding on heat transport across interfaces. A gold film was transfer-printed to a self-assembled monolayer (SAM) with systematically varied termination chemistries. The Au/SH-C11-Si\(\equiv\)Qz interface has a higher TBC than the Au/CH\({}_{3}\)-C11-Si\(\equiv\)Qz interface. Furthermore, varying the density of covalent bonds within the SAM modulates both interfacial stiffness and the TBC. The TBC increases with increasing methyl:thiol end-group ratio.