3.3.3 High-Field Mobility and Velocity Saturation

When strong electric fields prevail, the electron velocity is no longer proportional to the field, and can thus no longer be described by a field-independent mobility (3.77). The heating of free carriers at high electric fields results in a saturation of the drift velocity

$${\mathbf{v}}=\mu \cdot{{\mathbf{E}}}$$ (3.91)

originating from various scattering mechanisms such as optical phonon scattering, phonon dispersion, phonon absorption as well as emission, and the energy band non-parabolic-ity [123]. The high-field mobility model is based on the low-field mobility with further extensions to address the high-field phenomenon.

Little is known about the high-field mobility of SiC. The only experimental data was published by Khan and Cooper [134,135], where the drift velocity (n-doped at about 10$^{17}$cm$^{-3}$) was measured as a function of electric field using standard n-type and p-type 4H and 6H-SiC epilayers at different temperatures (see Table 3.6). All measured data refers to a current flow perpendicular to the $c$-axis.

The field dependence of the mobility can be modeled by the widely used expression of Canali et al. [136]

$$\mu _{\nu{\perp ,\parallel}}^\mathrm{high}=\displaystyle\frac{\mu... ...t] ^{\displaystyle\frac{1}{^{\alpha _{v}^\mathrm{sat}}}}} \hspace{1cm}\nu = n,p$$ (3.92)

where ${E_{\nu\parallel}}$ is the electric field component in the direction of the current flow as driving force, $v_{\nu{\perp, \parallel}}^\mathrm{sat}$ is the saturation velocity, and $\alpha _{\nu}^\mathrm{sat}$ is a constant specifying how abruptly the velocity goes into saturation.

The standard Si model [104] is used to describe the temperature dependence of the saturation velocity $v_{\nu}^\mathrm{sat}$, expressed by

$$v_{\nu{\perp ,\parallel }}^\mathrm{sat}=v_{\nu,300,\perp,\paralle... ...mathrm{L}}{\mathrm{300\ K}}\right) ^{\delta _{\nu}^{sat}} \hspace{1cm}\nu = n,p$$ (3.93)

and

$$\alpha _{\nu}^\mathrm{sat}=\alpha _{\nu,300}^\mathrm{sat}\cdot\le... ...L}}{\mathrm{300\ K}}\right) ^{\beta _{\nu}^\mathrm{sat}}. \hspace{1cm}\nu = n,p$$ (3.94)

A fit through the experimental high-field data by Khan and Cooper [134] and the MC results by Nilsson et al. [137,138] is shown in Fig. 3.10.

Recent MC simulations for 4H-SiC [139] by including more precisely the non-parabolic band structure excellently agree to the measured data. In addition, these investigations reveal a lower electron saturation velocity $v_{n_{\parallel }}^{sat}=1.8\times 10^{7}$ cm/sec parallel to the $c$-axis in 4H-SiC with an anisotropic factor of $v_{n_{\perp }}^{sat}/v_{n_{\parallel }}^{sat}=1.16$.

Table 3.6: Parameters of the electron saturation velocity in 4H- and 6H-SiC.

	$v_{n,300}^{\mathrm {sat}}$ [cm/s]	$\alpha _{n,300}^\mathrm{sat}$	$\delta _{n}^\mathrm{sat}$	$\beta _{n}^\mathrm{sat}$
4H-SiC	2.2$\times$10$^{7}$	1.2	-0.44	1.0
6H-SiC	1.9$\times$10$^{7}$	1.7	-1.0	1.25

Figure 3.10: Drift velocity of electrons parallel to the c-axis as a function of the electric field in n-doped $\alpha$-SiC at different temperatures.

The mobility commonly measured and modeled in (3.92) is perpendicular to the c-axis, $\mu_{n_{\perp}}$. The mobility parallel to the c-axis, $\mu_{n_{\parallel}}$ is different. The ratio between the two mobilities has been studied both experimentally [129,128] and through MC calculations [138,140]. These studies seem to agree on a ratio $\mu_{n_{\perp }}/\mu_{n_{\parallel}}= 0.8$ for 4H-SiC and 5 for 6H-SiC.

Fig. 3.11 illustrates the n-type 4H- and 6H-SiC mobility for different temperatures with increasing electric field.

Figure 3.11: Temperature dependence of the mobility of n-type ( $N_\mathrm{D}=10^{17}$ cm$^{-3}$) $\alpha$-SiC with increasing electric field.

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