Figure 4. a) Temperature-dependent electrical conductivity of the 2DEG for the thin and bulk GaN samples, extracted via CTLM measurements (Section S5, Supporting Information). b) Temperature-dependent thermal conductivity measurements for the bulk and thin GaN samples. Fits are shown with black, dotted lines. c) XRD rocking curve scan of the (0002) lattice plane in GaN, to investigate the density of screw-type dislocations. d) Modeled in-plane thermal conductivities of the layers in our composite stack as a function of thickness using a dislocation density of $10^9$ cm⁻². e) Thermal conductivity reduction due to size effect at 25 °C. The model corresponds to the dashed lines with increasing dislocation densities ($N_{dis}$), and the data points are the experimental measurements. f) Measured Seebeck coefficient versus temperature for thin and bulk GaN samples. The model uses $\eta_v \approx 2 \times 10^{19}$ cm⁻³, which is estimated from the Schrödinger-Poisson simulation.
fit, indicating that Umklapp scattering is less prominent for long-wavelength phonons which are suppressed due to the size effect.
In-plane thermal conductivity data in these films are limited, with little data available on the size effect and temperature dependence.[12] Since our suspended film is a composite consisting of an AlN layer, AlₓGa₁-xN transition layers, and a GaN layer, the overall thermal conductivity (k) can be estimated as $\sum k_i t_i / \sum t_i$, where $k_i$ and $t_i$ refer to the thermal conductivities and thicknesses of individual layers. For each multilayer, we used a Boltzmann Transport Equation (BTE) model to quantify $k_i$ with layer thickness ($t_i$). Using a simple Debye approximation for the phonon dispersion with an average velocity over the acoustic phonon modes ($v_{ac}$), the in-plane thermal conductivity for each layer can be written as[24]
where $k_B$ is the Boltzmann constant, $\theta_D$ is the Debye temperature for the multilayer,[25] T is the temperature, $\hbar$ is the reduced Planck's constant, $c_{ph}$ is the mode-specific volu-
metric heat capacity, evaluated as $3k_B \left(\frac{T}{\theta_D}\right)^3 \int_0^{\theta_D/T} \frac{x^4 e^x}{(e^x - 1)^2} dx$,
and $x = \hbar\omega/k_B T$, where $\omega$ is the phonon frequency. The integration is performed over the angular directions ($\theta$ and $\phi$) using a direction-dependent group velocity $v_g = v_{ac}\sin(\theta)\cos(\phi)$.
The total scattering time $\tau_C$ is calculated by Mathiessen's rule with contributions from Umklapp ($\tau_U$), impurity ($\tau_I$), alloy ($\tau_A$), boundary ($\tau_B$), and defect scattering ($\tau_D$), respectively. The Umklapp scattering term is evaluated via the Callaway relationship, $\tau_U = A/\omega^2$. We evaluated the constant A in the bulk limit as $2\pi^2 v_{ac} k_{\infty} / (c_{ph} \omega_D)$, where $k_{\infty}$ in the bulk thermal conductivity of the layer and $\omega_D$ is the Debye frequency. For instance, $k_{\infty}$ values of 240 and 285 Wm⁻¹ K⁻¹ are used for GaN and AlN at room temperature, respectively.[26,27] The Debye frequencies are