3.4 Electrothermal Simulations

To model the thermal conductivity as a function of lattice temperature the following relation was used:

$$\kappa({\it T}_\mathrm{L}) = \kappa_{300} \cdot \bigg( \frac{{\it T}_\mathrm{L}}{300 K}\bigg)^{\alpha}$$ (3.123)

The values for the basic semiconductors are given in Table 3.34.

Table 3.34: Thermal conductivity coefficients in basic semiconductors and insulators.

Material	$\kappa_{300}$	$\alpha$	Reported Range	References
	[W m$^{-1}$ K$^{-1}$]		[W m$^{-1}$ K$^{-1}$]
GaAs	46	-1.25	37,44	[85,194]
AlAs	80	-1.37	91	[4]
InAs	27.3	-1.1	27,28.9	[85]
InP	68	-1.4	-	[85]
GaN	125	-0.43	130,150	[259,260]
AlN	285	-1.577	200	[259,267]
InN	80 $\pm$ 20 (estimate)	-	-	[290]
Si	148	-1.33,-1.65	150,154	[194,251]
SiO$_2$	1.38	+0.33	1.4	[63,194]
Si$_3$N$_4$	18.5 (amorph.)	+0.33	16	[63,194]
	27 (crystal)		-	[173]
BeO	248	-1.27	300	[169,173]
Al$_2$O$_3$ (sapphire)	23.1 $\parallel$, 25.2 $\perp$ c	-1	33	[81,173]
Al$_2$O$_3$ (alumina)	28	-1	15-40	[173]
AlN /ceramic	145.9	-1.84	180	[169,266]
Diamond	2500 (natural Type II)	-1.85	500-3200	[299]
SiC	330 (490)	-1.61	320-490	[81,114,266]

Comparing Si and the GaAs III-V materials, the latter have a reduced thermal conductivity by at least a factor of 2. This results in several investigations performed, e.g. [209], where similar values for the GaAs based III-V semiconductors were used for power HBTs. For the nitride materials available so far, Fig. 3.22 shows a comparison of model and measurements. GaN has a similar thermal conductivity as Si. The available value for InN is an estimate only, although repeatedly cited. Thus for the temperature dependence no measurement data are available. Unfortunately, many references published for the temperature dependence in GaN refer back to the measurements of [260], which may be outdated. For AlN and SiC the modeling is shown in Fig. 3.22. This shortage of caliometric measurements is especially relevant, since the AlGaN/GaN HEMTs are thermally limited. For SiC there is a broad experimental base for the thermal conductivity. Natural diamond represents the best heat conducting material and is thus supplied for reference.

Figure 3.22: Modeling of the temperature dependence of the thermal conductivity of GaN, AlN, and SiC [260,266,267].

For ternary semiconductors the values for $\alpha$ and $\kappa$ are composed as given in (3.127) and (3.128).

$$\kappa_{300}^{AB} =\frac{1}{\bigg(\displaystyle \frac{1-x}{\kappa_{300}^A} +\frac{x}{\kappa_{300}^B}+\frac{(1-x)\cdot x}{C_\kappa}\bigg)}$$ (3.124)

$$\alpha^{AB} = (1-x) \cdot \alpha^A + x \cdot \alpha^B$$ (3.125)

The bowing parameters are summarized in Table 3.35. For InAlAs, AlGaN, or InGaN there are, to the authors knowledge, no measurements on the thermal properties available. The heat capacities of bulk materials are modeled as follows:

$$c({\it T}_\mathrm{L})= c_{300}+ c_1 \cdot \frac{\bigg( \displayst... ...laystyle \frac{{\it T}_\mathrm{L}}{300 K} \bigg)^{\beta} + \frac{c_1}{c_{300}}}$$ (3.126)

The parameter values are given in Table 3.36. For ternary semiconductors the parameters are composed as follows:

$$c_{AB}(x) = (1-x)\cdot c_L^A + x\cdot c_L^B.$$ (3.127)

Table 3.35: Bowing parameters for the thermal conductivity of ternary bulk III-V semiconductors.

Material	$C_{\kappa}$	Material Composition	Reported $c_{300}$	References
	[W m$^{-1}$ K$^{-1}$]	$x$
Al$_{x}$Ga$_{1-x}$As	3.3	0.1	21	[4]
		0.23	15	[4]
		0.43	11	[4]
InGaAs	1.4	0.53	6.3	[250]
InAlAs	0	-	-	-
AlInAlAs	0	-	-	-
AlGaN	0	-	-	-

Table 3.36: Heat capacity parameters for bulk III-V semiconductors and insulators.

Material	c$_{300}$	c$_1$	$\beta$	Reported c$_{300}$	References
	[J/K kg]	[J/K kg]		[J/K kg]
GaAs	322	50	1.6	350	[194,284]
AlAs	441	50	1.2	490	[294]
InAs	394	50	1.95	394	[294]
InP	410	50	2.05	-	[294]
GaP	519	50	2.6	-	[294]
GaN	491	70	1	-	[9]
AlN (polycrys.)	748	482	2.29	-	[180]
InN	325	57	1	-	[291]
Si	711	255	1.85	700	[194,284]
Al$_2$O$_3$(sapphire)	796	-	-	-	[173]
SiO$_2$	709	696	1.5	782, 1000	[194,251]
Si$_3$N$_4$	787	820	-	800	[251]