3.2.6 Saturation Velocity

For the III-V materials the bulk saturation velocity used in the high field mobility is modeled as a function of lattice temperature ${\it T}_\mathrm{L}$ with the following two parameter model:

$${\it v}_{{sat}}({\it T}_\mathrm{L}) =  \frac{v_{sat 300}}{\big(1-... ...+ A_\mathrm {n}\cdot\bigg(\displaystyle \frac{{\it T}_\mathrm{L}}{300 K}\bigg)}$$ (3.47)

The model (3.50) allows a precise fit of the measurement data, while preserving computational efficiency. The model assumes that the saturation velocity is independent of the doping concentration which is sound in agreement with various publications [251]. The values for the parameters of the model are summarized in Table 3.18.

Table 3.18: Model parameters for the bulk saturation velocity.

Material	Valley	v $_{sat,n 300}$ [10$^7$ cm/s]	A$_n$	v $_{sat,p 300}$	A$_p$	References
		(for E [kV/cm])		[10$^7$ cm/s]
GaAs	$\Gamma$	0.72	0.56	0.9	0.41	[226]
	X	2.73	-	-	-	-
AlAs	X	0.85	0.55	0.8	0.3	[226]
	$\Gamma$	0.39	-	-	-	see Fig. 3.9
InAs	$\Gamma$	0.9	0.57	0.5	0.3	[226]
	X	-	-	-	-	-
InP		0.67	0.68	0.5	0.3	[226]
GaP		0.88	0.30	0.5	0.3	[226]
GaN	$\Gamma$	1.4	0.15	-	-	[17,38]
		(for 500)
AlN	$\Gamma$	1.6	0.135	-	-	[17,204]
		(for 1000)
InN	$\Gamma$	2.5	-	-	-	[17,203]
		(for 300)
Si	X	1.0	0.26	0.704	0.63	[226]

The saturation velocity is extracted for field values of 100 kV/cm and above for the GaAs and InP based materials. It is considered a material property here, and should not be mixed with effective velocities and related terms. For reference on the exact use of the term saturation velocity and related terms, see Section 3.6. It should be noted, that the X values for GaAs are only used for the material compositions $x$$\ge$ 0.45 to obtain a good fit, and have no meaning for other material compositions. For the saturation velocity in GaN and AlN there is a significant scatter. One reason for these discrepancies are the unusually high values for the peak drift velocity of the static overshoot, so that only at extremely high electric fields $>$ 500 kV/cm a saturated region is reached.

Figure 3.8: Saturation velocity versus temperature for GaN and AlN [17,204].

Fig. 3.8 shows the temperature dependence of the saturation velocity for GaN and AlN. The data are taken from [17,204] using MC data. For the ternary compounds significant experimental data for the drift mobility is missing to properly account for the alloy scattering [89].

Figure 3.9: Saturation velocity as a function of material composition for In$_x$Al$_{1-x}$As.

Figure 3.10: Saturation velocity as a function of material composition for In$_x$Ga$_{1-x}$As.

The composition dependence of the saturation velocity is modeled according to:

$${\it v}_{{sat}}^{AB} = x \cdot {\it v}_{{sat}}^A +(1-x)\cdot {\it v}_{{sat}}^B +x \cdot (1-x) \cdot C_{Bow}$$ (3.48)

The model values are found in Table 3.19. Fig. 3.9, Fig. 3.10, and Fig. 3.11 illustrate the saturation velocity as a function of material composition for In$_x$Al$_{1-x}$As, In$_x$Ga$_{1-x}$As, and Al$_x$Ga$_{1-x}$As. For Al$_x$Ga$_{1-x}$As a valley dependent concept is used to account for the cross-over transition. For In$_x$Al$_{1-x}$As, it is found that the one-valley approach models the available data relatively good, as can be seen in the comparison to the $\Gamma$ model. Experimental data for x$\leq$ 0.3 are not available.

Figure 3.11: Saturation velocity versus material composition for Al$_x$Ga$_{1-x}$As.

Table 3.19: Model parameters for the saturation velocity.

Mat.	C$_{n one}$	C$_{p one}$	C $_{n \Gamma}$	C $_{p \Gamma}$	C$_{n X}$	C$_{p X}$
	[10$^7$ cm/s]	[10$^7$ cm/s]	[10$^7$ cm/s]	[10$^7$ cm/s]	[10$^7$ cm/s]	[10$^7$ cm/s]
AlGaAs	-0.0512	-	0.94	-	-4.58	-
InGaAs	-0.196	-	-0.196	-	-
InAlAs	-2.13	-	-1.24	-	-	-
AlGaN	0.182	-	0.182	-	-	-

Figure 3.12: Saturation velocity versus temperature for

Figure 3.13: Saturation velocity versus material composition for Al$_x$Ga$_{1-x}$N [17].

Fig. 3.12 shows the temperature dependence of the saturation velocity in In$_{0.53}$Ga$_{0.47}$As. The bowing factor for Al$_x$G$_{1-x}$N was obtained by the MC data of [17], as shown in Fig. 3.13.