4.2.3 Thermal Conductivity

The thermal conductivity is modeled by a power law:

$$\kappa(T_\mathrm{L}) = \kappa_{300}\cdot\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300 \ensuremath{\mathrm{K}}}^\alpha,$$

where $\kappa_{300}$ is the value at 300 K. From early experiments $\kappa_{300}$=130 W/mK for ``bulk'' GaN [292] was extracted. However, later measurements of epitaxial structures yielded higher values [293], and a strong dependence on the dislocation density was observed [294]. Based on various studies [292,293,295,296,297,298] we give two parameter sets in Table 4.4, applicable for different material quality. Fig. 4.1 compares the two model sets with other models and experimental data.

Table 4.4: Model parameters for the thermal conductivity.

Material	$\kappa_{300}$ [W/m$\cdot$K]	$\alpha$
GaN model 1	130	-0.43
GaN model 2	220	-1.2
AlN	350	-1.7
InN	45-176	0

Figure 4.1: GaN thermal conductivity as a function of temperature.

For AlN the variation of the measured values for the thermal conductivity is smaller (Fig. 4.2). We assume $\kappa_{300}$=350 W/mK, which is close to the value reported in [299]. The parameter $\alpha$, which models the decrease with temperature, is calibrated against measured data [299,300,301].

Figure 4.2: AlN thermal conductivity as a function of temperature.

As of today no studies of the temperature dependent thermal conductivity of InN are available. Based on [302] a $\kappa_{300}$=176 W/mK at 300 K is assumed. This is a theoretical estimation, while the measured value was only 45 W/mK due to phonon scattering by point-defects and grain-boundaries.

Several expressions exist for the thermal conductivity $\kappa^\ensuremath{\mathrm{ABC}}_{300}$ of semiconductor alloys. As an example, Adachi et al. [303] use one based on Abeles's complex model [304]. However, an even more straightforward approach is proposed in [305], where a harmonic mean is used to model the conductivity at 300 K, while the exponent $\alpha^\ensuremath{\mathrm{ABC}}$ is linearly interpolated as there is no experimental data for temperatures other than 300 K yet:

$$\kappa^\ensuremath{\mathrm{ABC}}_{300} = \left(\frac{1-x}{\kappa^\ensuremath{\mathrm{AC}}_{300}}+\frac{x}{\kappa^\ensuremath{\mathrm{BC}}_{300}}+\frac{(1-x)x}{C_\kappa}\right)^{-1},$$

$$\alpha^\ensuremath{\mathrm{ABC}} = (1-x) \alpha^\ensuremath{\mathrm{AC}} + x \alpha^\ensuremath{\mathrm{BC}}$$

Applying this expressions, a value of 3.1 W/mK is adopted for $C_\kappa$ of Al$_x$Ga$_{1-x}$N. This results in a fair agreement with the experimental data of Daly et al. [306] and Liu et al. [298] as depicted in Fig. 4.3 and corresponds to the value used in [303].

Figure 4.3: AlGaN thermal conductivity as a function of Al content.

Figure 4.4: InGaN thermal conductivity as a function of In content.

For In$_x$Ga$_{1-x}$N $C_\kappa$=1.5 W/mK is adopted, again matching the model in [303] (Fig. 4.4) and the experimental data of Pantha et al. [307]. For In$_x$Al$_{1-x}$N a fit to the only available experimental data [308] resulted in an $C_\kappa$=1.2 W/mK (Fig. 4.5).

Figure 4.5: InAlN thermal conductivity as a function of In content.