3.2.2 Effective Masses and Intrinsic Carrier Density

A model for the intrinsic carrier concentration requires both the electron and the hole density-of-states masses. As aforementioned, the conduction band minimum in 4H-SiC is at the M-point in the 1BZ, thus giving rise to three equivalent conduction band minima. The effective mass components $m_{M\Gamma}$, $m_{MK}$ and $m_{ML}$ in the principal directions measured by the optically detected cyclotron resonance are listed in Table 3.3 [114,118] (in units of the free electron mass). The ML direction corresponds to electron motion along the hexagonal c-axis, while the directions MG and MK correspond to electron motion in a plane perpendicular to the c-axis in 1BZ (see Fig. 3.4). This leads to an electron density-of-states mass [118]

Table: Electron and hole effective mass in $\alpha$-SiC at 300 K in units of the free electron mass. ($a$ corresponds to the transversal mass, $b$ to theoretical values.)

	m $_{M\Gamma }$	m$_{MK}$	m$_{ML}$	m $_{n}^{\ast }$	$M_{\mathrm{c}}$	m $_{p\perp }$	m $_{p\parallel }$	m $_{p}^{\ast }$
4H-SiC	0.58	0.31	0.33	0.39	3	0.59	1.60	0.82
6H-SiC	0.42$^{a}$		2.0	0.71	6	0.58$^{b}$	1.54$^{b}$	0.90$^{b}$

$$m_{n}^{\ast }=\left( m_{M\Gamma}\cdot m_{MK}\cdot m_{ML}\right) ^{\frac{1}{3}}$$ (3.65)

Thus, the effective density-of-states of electrons in the conduction band is calculated from [37]:

$$N_\mathrm{c}= 2\cdot M_{\mathrm{c}}\cdot\left( \frac{2\cdot\pi\cd... ..._{n}^{\ast}\cdot{\mathrm{k_B}}\cdot T_\mathrm{L}}{h^{2}}\right) ^{\frac{3}{2}},$$ (3.66)

where $M_{\mathrm{c}}$ represents the number of equivalent energy minima in the conduction band.

Similarly, the effective density-of-states for holes in the valence band is obtained from:

$$N_\mathrm{v}= 2\cdot\left( \frac{2\cdot\pi\cdot m_{p}^{\ast}\cdot{\mathrm{k_B}}\cdot T_\mathrm{L}}{h^{2}}\right) ^{\frac{3}{2}}.$$ (3.67)

Fig. 3.6 shows the effective density-of-states as a function of temperature for $\alpha$-SiC.

Figure 3.6: Temperature dependence of the effective density-of-states in $\alpha$-SiC.

The Fermi level of an intrinsic $\alpha$-SiC is given by [37]:

$$E_{\mathrm{F}}=E_{\mathrm{i}}=\frac{\left(\ensuremath{E_\mathrm{c... ...\mathrm{L}}{2}\right)\cdot \ln{\left(\frac{N_\mathrm{v}}{N_\mathrm{c}}\right)}.$$ (3.68)

At room temperature, the second term is much smaller than the bandgap. Hence, the intrinsic Fermi level $E_{\mathrm{i}}$ of intrinsic $\alpha$-SiC generally lies very close to the middle of the gap.

At electrothermal equilibrium the quasi-Fermi potentials $\varphi$ are zero. Hence, the temperature dependent intrinsic carrier density $n_{i}$ is determined by the fundamental energy gap $E_{\mathrm{g}}$ and by the effective density-of-states $N_\mathrm{c}$ and $N_\mathrm{v}$ for the conduction and valency band, respectively.

Combining (3.29), (3.28) and (3.68) gives

$$n\cdot p = n^2_{i},$$ (3.69)

$$n^2_{i} = N_\mathrm{c}\cdot N_\mathrm{v}\cdot \exp \left( -\frac{E_{\mathrm{g}}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right),$$ (3.70)

$$n_{i} =\sqrt{N_\mathrm{c}\cdot N_\mathrm{v}}\cdot \exp \left( -\frac{E_{\mathrm{g}}}{2\cdot{\mathrm{k_B}}\cdot T_\mathrm{L}}\right),$$ (3.71)

Figure 3.7: Temperature dependence of the intrinsic carrier concentration in $\alpha$-SiC.

where $E_{\mathrm{g}}=\ensuremath{E_\mathrm{c}}-E_\mathrm{v}$.

The intrinsic carrier density of 4H/6H-SiC is smaller by the order of 16-18 compared to Si due to the wide bandgap. Furthermore, the exponential dependence of $n_{i}$ on $E_{g}$ introduces a rather large uncertainty, so that the overall error range can hardly be estimated. Fig. 3.7 illustrates the intrinsic carrier densities in $\alpha$-SiC as a function of temperature. T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation