interface. X-Ray diffraction data for the AlGaN/GaN/buffer layers in available in our former work.[8] After suspension, the total thickness of the heterostructure layers was obtained as ~1.695 µm for the thin GaN heterostructure and ~2.85 µm for the bulk GaN heterostructure from the SEM cross-section images.
Choice of buffer layer thicknesses:
The thermal conductivity of the AlGaN/GaN/buffer hetero-structure is typically determined by the thermal conductivity of the GaN buffer layer. The thickness of the GaN ($t_{GaN}$) layer in the thin GaN sample was designed to lower the thermal conductivity of the buffer layer structure due to size effect, while preserving the charge density $n_s$ of the 2DEG. This effect is shown in Figure S2b, which depicts the rapid decline in $n_s$ with $t_{GaN}$ due to band bending in the AlGaN and GaN layers from the Schrödinger Poisson model. As the GaN layer thickness decreases below ~100 nm, the decline is much sharper as the strain difference in the AlGaN/GaN layers decreases, leading to the GaN layer becoming pseudomorphic with the buffer layers beneath.
Supplementary Note 2: Test Setup and Calibration
Figure S3a shows the test setup used to measure the in-plane thermal conductivity of the AlGaN/GaN hetero-structure. In order to ensure accuracy in the thermal conductivity measurements, we performed careful resistance versus temperature calibration for the Ti/Pt heater and sensor lines. For the heater line, a DC current source (Keithley 2400) and a DC voltage source (Agilent 34401) were used to measure the resistance of the Ti/Pt trace. To measure the resistance of the sensor line, we used an AC voltage lock-in amplifier (Zurich Instruments HF2LI) with a lock-in frequency of 97 Hz. AC voltage measurement across a fixed resistor (1 kΩ, ultra-low TCR of less than 1 ppm) was used to infer the AC current from the applied AC voltage. The lock in-amplifier was chosen for the sensor side to minimize self-