3.2.3 Thermal Conductivity

The temperature dependence of $\kappa_{\mathrm{L}}$ of the basic materials and insulators is modeled by a simple power law

$$\kappa_{\mathrm{L}}(T_{\mathrm{L}}) = \kappa_{300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^\alpha$$ (3.55)

where $\kappa_{300}$ is the value for the thermal conductivity at 300 K. This approximation is in good agreement with experimental data [105,106,107,108], as presented in Fig. 3.1 and Fig. 3.2 where comparisons in the temperature range from 300 K to 800 K. The parameter values used are summarized in Table 3.5.

Table 3.5: Parameter values for thermal conductivity

Material	$\kappa_{300}$ [W/K m]	$\alpha$	Reported $\kappa_{300}$ [W/K m]	References
Si	148	-1.3	150	[85,86]
Ge	60	-1.25	60	[85,86]
GaAs	46	-1.25	45.5-46	[108,92,85,86]
AlAs	80	-1.37	80	[108]
InAs	27.3	-1.1	27.3-48	[108,92]
InP	68	-1.4	68	[108,92]
GaP	77	-1.4	77	[108,92]
SiO$_2$	1.38	0.33	1.4	[85,86]
Si$_3$N$_4$	18.5	0.33	15-27	[98]

In the case of alloy materials $\mathrm{A}_{1-x}\mathrm{B}_{x}$, $\kappa_{{\mathrm{L}}}$ varies between the values of the basic materials (A and B). A harmonic mean is used to model $\kappa_{300}$. An additional bowing factor $C_\kappa$ is introduced in order to account for the drastic reduction of the thermal conductivity with the increase of material composition $x$. The exponent $\alpha$ is linearly interpolated because of lack of experimental data at temperatures other than 300 K.

$$\kappa_{300}^{\mathrm {AB}} = \frac{1}{\displaystyle{\left(\frac{1-x} {\kappa_{300}^{\mathrm {A... ...{\kappa_{300}^{\mathrm {B}}}+ \frac{\left(1-x\right)\cdot x}{C_\kappa}\right)}}$$ (3.56)

$$\alpha^{\mathrm {AB}} = \left(1-x\right) \cdot\alpha^{\mathrm {A}} + x\cdot\alpha^{\mathrm {B}}$$ (3.57)

The parameter values used are summarized in Table 3.6.

Table 3.6: Parameter values for thermal conductivity bowing factor

Material	$C_\kappa$ [W/K m]
SiGe	2.8
AlGaAs	3.3
InGaAs	1.4
InAlAs	3.3
InAsP	3.3
GaAsP	1.4
InGaP	1.4

In Fig. 3.3 and Fig. 3.4 comparisons between data from [106,107,108,109,110,111] and the results obtained with our model are shown for the thermal conductivity in alloy materials at 300 K.

Figure 3.1: Temperature dependence of the thermal conductivity: Comparison between experimental data and the model for Si, Ge, and GaP

Figure 3.2: Temperature dependence of the thermal conductivity: Comparison between experimental data and the model for InP, GaAs, and InAs

Figure 3.3: Material composition dependence of the thermal conductivity: Comparison between experimental data and the model for SiGe and InGaAs

Figure 3.4: Material composition dependence of the thermal conductivity: Comparison between experimental data and the model for InAsP and AlGaAs