discussed later. In addition, we also ensured that hysteresis did not occur in our heater and sensor lines. This is clear from observing the temperature versus power lines in Figure S5a and Figure S5e.
Figure S6 shows a typical Seebeck coefficient measurement procedure. Similar to the thermal conductivity measurement, the heater current is ramped up from its calibration value, setting up a lateral temperature gradient along the 2DEG mesa which translates to a measurable Seebeck voltage (Figure S6d). The measured Seebeck voltage includes a minor contribution from the temperature drop across the Ti/Al/Pt/Au Ohmic contacts to the 2DEG (visualized in Figure S6f). At room temperature, we measured the Seebeck voltage across the 2DEG mesa and the Ohmic metal line for an identical temperature gradient. Using this, we estimated the contribution of the Ohmic metal line to be less than 2% of the overall Seebeck voltage, and thus neglected its effect in subsequent measurements. The Seebeck voltage of the 2DEG is given as $S_{2DEG}=V_{2DEG}/(T_1-T_2)$, as depicted in Figure S6e and Figure S6f. The temperature at the contact outside the suspended region ($T_2$) is assumed to be at the substrate temperature. The temperature drop in the silicon supported region is <1% of the total temperature drop ($T_1-T_2$) (Figure S4b), thus, the contribution to the Seebeck coefficient from the supported region can be ignored. $T_1$ is related to the heater temperature $T_H$ as:
where $R_F$ is calculated using the measured film thermal conductivity and a length of 30 µm ($D_S$, depicted in Figure S6e) and $R_{Al}$ is calculated as discussed earlier.
Finally, external losses from convection are significant at high temperatures which lead to errors in the thermal conductivity measurement. This can be seen in Figure S7b, which shows the temperature profile when no current is applied through the heater with bottom fixed at 200°C for the bulk GaN film. Notice the cooling in the suspended membrane due to external convection, leading a relative temperature difference between the heater and the