3.2.3.1 The Band Gap

The band gap is the most important parameter for the description of semiconductors. It is modeled as a function of lattice temperature according to:

$${\it E}_\mathrm{g}(T) = {\it E}_\mathrm{g,0}- \frac{\alpha \cdot T^2_L}{\beta + {\it T}_\mathrm{L}}$$ (3.21)

where ${\it E}_\mathrm{g,0}$ is the extrapolated band gap at ${\it T}_\mathrm{L}$= 0 K. The parameter values and values at ${\it T}_\mathrm{L}$= 300 K are given in Table 3.5.

Table 3.5: Band gap parameters for elementary and binary semiconductors.

Material	Minimum	${\it E}_\mathrm{g,0}$	$\alpha$	$\beta$	ReFported Range	Rep.Range	References
		[eV]	[eV/K]	[K]	${\it E}_\mathrm{g,0}$ [eV]	${\it E}_\mathrm{g,300}$ [eV]
GaAs	$\Gamma$	1.521	5.58 $\cdot 10^{-4}$	220	1.517-1.55	1.424	[284,222]
AlAs	X	2.239	6.0 $\cdot 10^{-4}$	408	2.22-2.239	2.14, 2.163, 2.168	[91,265,110]
InAs	$\Gamma$	0.420	2.5 $\cdot 10^{-4}$	75	0.414-0.43	0.36	[284]
	also	0.43	2.76-3.16 $\cdot 10^{-4}$	83-93	0.42-0.43		[315]
InP	$\Gamma$	1.421	3.63 $\cdot 10^{-4}$	162	1.42-1.432	1.35	[284]
GaN	$\Gamma^{dir}$	3.50	7.32 $\cdot 10^{-4}$	700	3.51	3.2, 3.39,3.43	[224,280,303]
AlN	$\Gamma^{indir}$	6.312	1.7999 $\cdot 10^{-3}$	1462	6.118	6.02-6.2	[224,255]
InN	$\Gamma$	2.01	1.2 $\cdot 10^{-3}$	627	2.00	1.89-1.95	[206,291]
Si	X	1.1695	4.73 $\cdot 10^{-4}$	636	1.17	1.124	[194,265,284]

The significant scatter for the values of nitride based semiconductors is due to the high unintentional background concentrations, for InN ${\it N}_{\mathrm{D}}>$ 10$^{19}$ cm$^{-3}$. For the ternary semiconductors the band gap is modeled following a k-valley dependent approach. For HEMTs used in this work the material compositions are sometimes close to the cross over transition in k-space, e.g. for Al$_x$Ga$_{1-x}$As at $x$= 0.45, or for In$_x$Al$_{1-x}$As at $x$= 0.3. To achieve a better modeling, instead of using a simple bowing over the cross-over transition, two sets of parameters are given.

Table 3.6: Additional band gap parameters for binary semiconductors.

Material	Minimum	${\it E}_\mathrm{g,0}$	$\alpha$	$\beta$	Rep. Range	Rep. ${\it E}_\mathrm{g,300}$	References
		[eV]	[eV/K]	[K]	${\it E}_\mathrm{g,0}$ [eV]	[eV]
GaAs	$X$	1.981	4.6 $\cdot 10^{-4}$	204	-	1.899 -1.91	[2,213]
							[264]
AlAs	$\Gamma$	2.891	8.78 $\cdot 10^{-4}$	332	2.907-3.02	2.95	[2,152]
							[213,263]
InAs	$X$	2.278	5.78 $\cdot 10^{-4}$	83	-	2.142	[152,257]
InP	$X$	2.32	7.66 $\cdot 10^{-4}$	327	2.32-2.38	2.21	[4]
AlN	$\Gamma^{dir}$	7.0	1.7999 $\cdot 10^{-3}$	(1462)	-	6.9	[96]

Table 3.6 introduces this two-valley modeling, which includes independent values e.g. for X-valley AlAs and $\Gamma$-valley AlAs for the different minimum valleys. These values are only used for those material compositions, where they are physically relevant, i.e., $\Gamma$-AlAs only for Al$_x$Ga$_{1-x}$As with $x$$\leq$ 0.45, while for $x$$\geq$ 0.45, the value of X-AlAs is chosen. The split into two different parameter sets allows for a more precise modeling than the conventional one valley bowing approach, as shown in the following. The band gap of an alloy A$_x$B$_{1-x}$ is calculated as:

$$E_{g,i}^{AB} = x \cdot E_{g,A} + (1-x) \cdot E_{g,B} + x \cdot (1-x) \cdot C_{g,i}$$ (3.22)

where i stands for either $\Gamma$ or $X$. The valid band gap is used according to:

$$E_{g,i}^{AB} = \big(E_{g \Gamma}^{AB}, E_{g X}^{AB} \big)$$ (3.23)

The parameters for both valleys are given in Table 3.6. As was also shown in [207] also the conventional approach neglecting the cross over transition is presented in Table 3.7 and (3.25).

$$E_{g}^{AB}  = x \cdot E_{g,A} + (1-x) \cdot E_{g,B} + x \cdot (1-x) \cdot C_{g}$$ (3.24)

Table 3.7: Band gap parameters for ternary semiconductors.

Material	$C_{g,\Gamma}$	$C_{g,X}$	$C_{g one valley}$	Reported Range	References
	[eV]	[eV]	[eV]	[eV]
Al$_{x}$Ga$_{1-x}$As	0	-0.143	0.7	-	[222,250]
In$_{x}$Ga$_{1-x}$As	-	-	-0.475	-	[265,315]
In$_{x}$Al$_{1-x}$As	-0.3	-0.713	1.2	1.45 ($x$=0.52)	[163]
Al$_{x}$Ga$_{1-x}$N	-1.65 (first valley)	-	-1.2596	3.88 ($x$=0.2)	[255,303]
In$_{x}$Ga$_{1-x}$N	-2.05	-	-2.05	-	[303]

Figure 3.2: Bowing parameters for the band gap in

The significant scatter for the bowing parameter in AlGaN was resolved in [303] for $x$$<$ 0.16 and is attributed to a systematic discrepancy of the two experimental methods applied. From this data a specific $\Gamma$ (first)-valley value for the bowing parameter was obtained. Data from [255] were evaluated in a one valley model for a consistent description of the cross over transition, as shown in Fig. 3.2. For In$_x$Ga$_{1-x}$N, as a first approach, the strained data were compiled into the model for $x$$<$ 0.15.