3.4.2 Thermal Properties

The thermal material properties are modeled by the tensors of the thermoelectric power $\overset{\sim }{P}_\nu$ for electrons and holes, the heat capacity $c$, and the tensor of the thermal conductivity $\overset{\sim }{\kappa }$. To date no measured data about the tensors of the thermoelectric power $\overset{\sim }{P}$ are available for SiC. Hence, corresponding values measured for Si have to be used [143,144].

The measured data which are available for the heat capacity of $\alpha$-SiC [145] can be adapted to (3.98). The model evaluates the specific heat capacity for the transient simulation with self-heating.

$$c(T) = A_{c}+B_{c}\cdot T_\mathrm{L}+C_{c}\cdot T_\mathrm{L}^{2}+D_{c}\cdot T_\mathrm{L}^{-2}.$$ (3.98)

The fitting parameters A$_c$, B$_c$, C$_c$ and D$_c$ are summarized in Table 3.7.

Similarly, published experimental data of the thermal conductivity available for $\alpha$-SiC [146,147] can be adapted to the model

$$\kappa (T_\mathrm{L}) = \frac{1}{A_{k}+B_{k}\cdot T_\mathrm{L}+C_{k}\cdot T_\mathrm{L}^{2}}.$$ (3.99)

The constants A$_k$, B$_k$ and C$_k$ are listed in Table 3.7 and  Fig. 3.12 illustrates the influence of the temperature on the heat capacity and the thermal conductivity in $\alpha$-SiC. Furthermore, the anisotropic ratio of 6H-SiC has been experimentally determined to $\kappa _{\perp }/\kappa _{\parallel}=0.7$ [146] independent of the temperature and the doping concentration.

Table 3.7: Model parameters for the heat capacity and the thermal conductivity in $\alpha$-SiC.

	A$_{c}$[Jkg$^{-1}$K$^{-1}$]	B$_{c}$[Jkg$^{-1}$K$^{-2}$]	C $_{c}$[Jkg$^{-1}$K$^{-3}$]	D$_{c}$ [Jkg$^{-1}$K]
c	1026	0.201	0	-3.66$\times$10$^{7}$
	A $_{\kappa }$[mKW$^{-1}$]	B $% _{\kappa }$[mW$^{-1}$]	C $_{\kappa }$[mK$^{-1}$W$^{-1}$]
$\kappa$	2.5$\times$10$^{-5}$	2.750$\times$ 10$^{-6}$	1.3$\times$10$^{-8}$

Figure 3.12: Influence of temperature on heat capacity and thermal conductivity in $\alpha$-SiC.

For the purpose of device simulation the thermal conductivity is often modeled by a power law expression given by

$$\kappa(T_\mathrm{L}) = \kappa_{0}\cdot{\left(\frac{T_\mathrm{L}}{\mathrm{300\ K}}\right)}^\alpha,$$ (3.100)

where $\kappa_{0} = 490$ W/mK is the value for the thermal conductivity at ${\mathrm{300\,K}}$ for $\alpha$-SiC, and $\alpha = -1.5$ is a fitting parameter.

The lattice thermal flux density between two boxes and its derivatives with respect to the input quantities, which are the temperatures $T_1$ and $T_2$, are calculated by

$$\int_{T_1}^{T_2} \kappa(T_\mathrm{L})\ dT_\mathrm{L}= \frac{\kapp... ...{\alpha+1} - \left(\frac{q\cdot T_1}{300\ \mathrm{K}}\right)^{\alpha+1}\right).$$ (3.101)

T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation