## 2 Theoretical Methods for TBR Prediction and Experimental Methods for TBR Measurement
### Experimental Methods for TBR Measurement  
The following experimental methods are conventionally employed for measuring TBR in thin film samples: the 3\(\omega\) method, Raman thermometry, and transient thermoreflectance (TTR) techniques, including frequency-domain thermoreflectance (FDTR) and time-domain thermoreflectance (TDTR) methods. The 3\(\omega\) method is based on the principle of analyzing the temperature-dependent electrical resistance subjected to periodic heating. The TBR can be extracted with known thermal properties adjacent to a thin film of interest. As shown in Figure 4a, a micro-metal strip heater was fabricated on the sample to function as a resistive heater and resistance temperature detector. The heater, driven by an alternating (AC) current with angular frequency \(\omega\), induces a temperature wave of frequency 2\(\omega\) to diffuse into the substrate. The third harmonic voltage (3\(\omega\)) carries information about thermal transport within the sample. The 3\(\omega\) voltage signal is exceptionally weak and usually approximately three orders of magnitude smaller than the amplitude of the applied 1\(\omega\) voltage, and a lock-in amplifier is usually employed in this measurement technique [58; 59]. Figure 4b shows the experimental setup of the Raman thermometry method. The Raman thermometry is based on the principle that the intensity, frequency, and width of  
Figure 4: **(a)** Schematic of the experimental setup (**left**) and the raw data (**right**) of the 3\(\omega\) method (reprinted from Ref. [60]. Reproduced with permission from AIP Publishing. All rights reserved). **(b)** Schematic of the experimental setup of the Raman thermometry method (**left**), the Raman peaks (**upper right**), and the temperature distribution (**lower right**) of the Graphene/SiO\({}_{2}\)/Si sample (reprinted from Ref. [61]. Reproduced with permission from Elsevier Ltd. All rights reserved). **(c)** Schematic of the experimental setup (**left**) and raw data (**upper right**) of the FDTR method, and the comparison between diffusive transport and quasi-ballistic transport (**lower right**) (reprinted from Ref. [20]. Reproduced with permission from Springer Nature publishing. All rights reserved). **(d)** Schematic of the experimental setup and the raw data (**lower right**) of the TDTR method (reprinted from ref. [62]. Reproduced with permission from Springer publishing. All rights reserved). Raman peaks are temperature dependent. Among these variations, frequency shift features high sensitivity and is often employed for temperature measurements. When joule heating is used to heat the sample on one side, the temperature of the materials on both sides of the interface can be measured simultaneously by Raman mapping, after which the TBR can be determined from TBR = \(\Delta T/l^{2}R\), where \(\Delta T\), \(I\), and \(R\) are the temperature difference, applied current, and electrical resistance, respectively [63; 64; 65]. Both FDTR and TDTR are noncontact optical methods for measuring thermal properties using the thermoreflectance phenomenon. A thin metal layer deposited on top of the sample is used as a transducer, with its reflectance changing with increasing temperature at the laser wavelength. This occurrence facilitates the detection of the thermal response by monitoring the change in reflectance. In FDTR, as shown in Figure 4c, a modulated pump laser beam is incident on the surface of the sample, causing periodic heating and resulting in temperature oscillation. A probe laser beam is used to detect changes in the amplitude and phase of the temperature oscillations as a function of the modulation frequency of the pump beam, after which the TBR is extracted [66; 67; 68]. A pump-probe configuration is also utilized in TDTR. As demonstrated in Figure 4d, a pump laser beam is incident on the sample surface and heats the sample. After a short delay, the probe laser beam is directed onto the surface of the sample following the pump laser.