Figure 3. a) Schrödinger–Poisson model of the energy band diagram for the bulk GaN structure with AlGaN thickness of 30 nm. The thickness of the 2DEG region, $t_{2D}$, is shown in the region where GaN is degenerate. b) Volumetric charge density, $n_v$, versus position, depicting the approximately triangular charge profile, with $t_{2D} \approx 6.1$ nm. c) Simulated energy band diagram for the thin GaN structure with AlGaN thickness of 30 nm. Note that the GaN layer is 100 nm. d) Volumetric charge density, $n_v$, versus position, depicting the approximately triangular charge profile, with $t_{2D} \approx 4.4$ nm for the thin GaN structure.
to be undoped, and the barrier height for the GaN capping layer is set to 1 eV.[20] For the bulk and thin GaN models, the GaN thickness and the Al0.2Ga0.8N layer thickness (first buffer layer below the GaN) were varied until 2DEG sheet density ($n_s$) convergence was observed. In both cases, the 2DEG region is visible as a triangular potential well near the AlGaN/AlN/GaN interface. From the simulation, we found $n_s = 1.06 \times 10^{13}$ cm-2 and $n_v = 0.91 \times 10^{13}$ cm-2 for the bulk and thin GaN heterostructures, respectively. It should be noted that a good match, within ≈10% of the theoretically calculated values, is observed when comparing these values with experimental data extracted from Hall-effect devices fabricated on the same platform, which supports the model (Section S4, Supporting Information). The physical thickness of the 2DEG region, $t_{2D}$, can be extracted as the region where GaN is degenerate.[9] From simulation, these thickness values were obtained to be ≈6.1 and ≈4.4 nm for the bulk and thin GaN heterostructures, respectively, which can be used to obtain the 2DEG conductivity $\sigma$ from the sheet resistance ($R_{sh}$) extracted via CTLM measurements. Finally, we note that an average 2DEG volumetric density can be estimated as $n_v = n_s/t_{2D}$ for the bulk GaN (1.73 × 1019 cm-3) and thin GaN (2.07 × 1019 cm-3) heterostructures. We note that the higher $n_v$ for the thin GaN sample reflects the smaller 2DEG quantum well thickness.
4. Electrical and Thermal Property Measurements
The measurements of $R_{sh}$ averaged over four samples up to 300 °C via CTLM measurements (Section S5, Supporting
Information) can be combined with the 2DEG thickness $t_{2D}$ to obtain the average electrical conductivity [$\sigma = 1/(R_{sh} \times t_{2D})$] of the electrons in the 2DEG. At room temperature, we obtained $R_{sh}$ values of ≈350 and ≈500 Ω sq-1 for the bulk and thin GaN samples, respectively. We note that these values are among the lowest reported $R_{sh}$ values for AlGaN/GaN 2DEGs, which highlights the quality of our samples.[16] The average conductivity in the thin GaN sample is observed to be similar to the bulk GaN sample due to simultaneous reduction in the sheet density and quantum well depth, as seen in Figure 4a. Sheet densities in this temperature range are approximately constant due to negligible strain relaxation in the heterostructure layers,[21] stable piezoelectric coefficients,[22] and minimal intrinsic carrier concentration change due to the wide bandgap. Thus, the decrease of $\sigma$ at high temperatures is mainly determined by the 2DEG mobility, $\mu$. The dependence is well described by a temperature power law ≈$T^{-2.5}$ that arises from electron–optical phonon scattering, which further supports this fact. We also note that our exponent is consistent with former exponent ranges (from −2.2 to −3.4) reported in the literature.[21]
Temperature-dependent in-plane thermal conductivity measurements for the bulk and thin GaN samples are shown in Figure 4b. Room temperature thermal conductivity dropped from ≈115 Wm-1 K-1 for the bulk GaN sample to ≈45 Wm-1 K-1 for the thin GaN sample due to phonon boundary scattering, i.e., the size effect.[15] The measurements for the bulk GaN sample follow a $T^{-1.18}$ fit. This is consistent with a similar temperature exponent observed in measurements of cross-plane thermal conductivity measurements of GaN films of thickness of ≈0.7 µm.[12,23] However, for the thin GaN sample, we note that the measured thermal conductivity values follow a $T^{-0.88}$