3.3.1 Low-Field Carrier Mobility

The lattice scattering (acoustic phonons) and ionized impurity scattering, together with piezoelectric scattering are the most relevant mechanisms which limit the mean free path of carriers at low electric fields in SiC [124,125]. At low electric fields the electron velocity increases almost linearly with the field. A widely used empirical expression for the doping dependence of the low-field mobility is the Caughey-Thomas equation [126].

$$\mu_{\nu{\perp ,\parallel }}^\mathrm{low}= \mu_{\nu{\perp ,\paral... ...\mathrm{ref}}\right)^\mathrm{\alpha_{\nu}^\mathrm{\mu}}}, \hspace{1cm}\nu = n,p$$ (3.77)

where N $_\mathrm{D}$ and N $_\mathrm{A}$ are the doping concentration of the dopants and acceptors. The symbols $\mu_{\nu}^\mathrm{\min}$, $\mu_{\nu}^\mathrm{max}$, N $_{\nu}^\mathrm{ref}$, $\alpha_{\nu}^{\mu}$, $\gamma_{\nu}^{\mu}$ are fitting parameters. The parameter $\mu_{\nu}^\mathrm{max}$ denotes the mobility of undoped or unintentionally doped samples, where lattice scattering is the main scattering mechanism, while $\mu_{\nu}^{\min}$ is the mobility in highly doped material, where impurity scattering is dominant, given by

$$\mu_{\nu{\perp ,\parallel }}^\mathrm{max} = \mu_{\nu{\perp ,\para... ...T_\mathrm{L}}{\mathrm{300\ K}}\right)^{\gamma_{\nu}^{\mu}}\hspace{1cm}\nu = n,p$$ (3.78)

$$\mu_{\nu{\perp ,\parallel }}^\mathrm{min} = \mu_{\nu{\perp ,\para... ...{T_\mathrm{L}}{\mathrm{300\ K}}\right)^{\beta_{\nu}^{\mu}}\hspace{1cm}\nu = n,p$$ (3.79)

$N_{\nu}^\mathrm{ref}$ is the doping concentration at which the mobility is ( $\mu_{\nu}^\mathrm{\min}+\mu_{\nu}^\mathrm{max})$/2, $\alpha_{\nu}^{\mu}$ is a measure of how quickly the mobility changes from $\mu_{\nu}^{\min}$ to $\mu_{\nu}^\mathrm{max}$, $\beta_{\nu}^{\mu}$ is a constant temperature coefficient, and $\gamma_{\nu}^{\mu}$ specifies how the undoped mobility changes due to lattice scattering. It is obvious from the previous reasoning that the anisotropic mobility in $\alpha$-SiC is a tensor of second rank and has the same form as the representative tensor $\sigma$ in (3.2). However, a rigorous modeling of the anisotropic properties of $\alpha$-SiC is still a challenge to the semiconductor transport theory.

A first attempt to calculate the anisotropy of the Hall mobility in n-type $\alpha$-SiC based on the band structure theory has been reported in [127]. However, as the mobility parameters of semiconductors significantly depend on the process technology, the reported mobility data from the period before wafers in acceptable quality were available can hardly be used to investigate state-of-the-art devices. Hall measurements of the bulk epitaxial free carrier mobility tensor components of 4H- and 6H-SiC have been reported [128,129]. The measured electron mobility in 4H-SiC is about twice that of 6H-SiC for a total impurity concentration less than $10^{17}$cm$^{-3}.$ Additionally, the hole mobility in 4H-SiC is $20\%$ larger than in 6H-SiC over the entire measured impurity range.

Table 3.5: Model parameters for low field mobility in 4H/6H-SiC.

	SPECIES	$\mu _{\perp, 300}^\mathrm{max}$	$\mu _{\perp,300}^{\min}$	$N_{\nu}^{\mathrm{ref}}$	$\alpha_{\nu}^{\mu}$	$\beta_{\nu}^{\mu}$	$\gamma_{\nu}^{\mu}$
	$\nu$	[cm$^2$/Vs]	[cm$^2$/Vs]	[ $\mathrm{cm}^{-3}$]
4H-SiC	n	950	40.0	1.94$\times$10$^{17}$	0.61	-0.5	-2.40
	p	125	15.9	1.76$\times$10$^{19}$	0.34	-0.5	-2.15
6H-SiC	n	420	30.0	1.1$\times$10$^{18}$	0.59	-0.5	-2.50
	p	100	6.8	2.1$\times$10$^{19}$	0.31	-0.5	-2.15

Figure 3.9: The n-type (N) and p-type (Al) mobility in $\alpha$-SiC as a function of the doping concentration.

Schaffer et al. [128] investigated over $100$ 4H- and 6H-SiC wafers and analyzed in detail the temperature dependence ${\mu}(T)$ of both polytypes. They obtained constant temperature coefficients $\beta_{v}^{\mu}$ for temperatures larger than 300 K in both polytypes. Summary of the measured parameters for the low field mobility in 4H- and 6H-SiC are listed in Table 3.5, and the corresponding model fitting is depicted in Fig. 3.9.

The anisotropic characteristics of the Hall mobilities have been investigated independently using epitaxial layers grown on [1100] and [1120] surfaces. The results for N-doped (n-type) and Al-doped (p-type) 4H-SiC can be expressed as

$$\mu_{n\perp}=\mu_n,\hspace{1cm}\mu_{p\perp}=\mu_p,$$ (3.80)

$$\mu_{n\parallel}=1.2\mu_n,\hspace{1cm}\mu_{p\parallel}=\mu_p,$$ (3.81)

similarly for 6H-SiC,

$$\mu_{n\perp}=5\mu_n,\hspace{1cm}\mu_{p\perp}=\mu_p,$$ (3.82)

$$\mu_{n\parallel}=\hspace{1cm}\mu_{p\parallel}=\mu_p,$$ (3.83)

for directions orthogonal ($\perp$) and parallel ($\parallel$) to the hexagonal c-axis. Thus, the largest Hall mobility is related to a current flow parallel to the c-axis in 4H-SiC and perpendicular to the c-axis in 6H-SiC. No dependence on the impurity concentration has been reported for these ratios. On the other hand, a rather large dependence on temperature for these ratios has been reported for both polytypes for $100\,\mathrm{K} <T< 600\,\mathrm{K}$ [128]. T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation