sensor line. This underestimates the ΔT between the heater and the sensor line, leading a higher measured thermal conductivity than the true value. The actual temperature gradient can be expressed as $\Delta T_{\text{real}} = \Delta T_{\text{meas}} + \Delta T_{\text{corr}}$, where $\Delta T_{\text{corr}}$ is the temperature gradient from the heater to the sensor with no current applied. The prominence of this effect is seen in Figure S7c and Figure S7d, which shows the simulated true and estimated measurements of the thermal conductivity for the normal GaN and thin GaN film. The error in the measured thermal conductivity is estimated to be ~4% and ~10% for the normal and thin GaN films from simulations, using an external convection coefficient of 10 Wm⁻²K⁻¹. The correction factor $\Delta T_{\text{corr}}$ can be estimated from the natural convection coefficient ($h$). At progressively higher temperatures, a non-zero Seebeck voltage is observed when no current is applied in the heater line due to the effect shown in Figure S7b. Then, using the most recently corrected thermal conductivity value, we estimated the value of $h$ required to produce the observed non-zero Seebeck voltage using our knowledge of the Seebeck coefficient from the COMSOL model. Following this, we obtained the correction factor $\Delta T_{\text{corr}}$ for the current thermal conductivity value. Typical values for $h$ estimated using this procedure are in the range of 12-14 Wm⁻²K⁻¹, which are reasonable coefficients for natural convection. The correction factors ($\Delta T_{\text{corr}}$) we obtained for the normal GaN and thin GaN films for the different substrate are tabulated in Table S1.
Supplementary Note 4: Schrödinger-Poisson Model Notes and Validation
The models for the bulk and thin GaN heterostructures were made using a commercially available Schrödinger-Poisson device physics simulator (NextNano Inc.).[14] In both models, we set the barrier height for the GaN capping layer to 1 eV, based on the assumption that the surface is exposed to air.[15] The entire structure is simulated using a 1-D grid size of 0.5 nm, except in the region where the 2DEG quantum well forms, where we used a finer grid of 0.1