3.2.3 Bandgap Narrowing

It is observed experimentally that a shrinkage of the bandgap occurs when the impurity concentration is particulary high. This effect is called the bandgap narrowing effect which is ascribed to the emerging of the impurity band formed by the overlaped impurity states. In devices containing adjecent layers or regions with different doping concentrations, doping-induced shifting of the conduction band minimum and the valence band maximum may greatly influence the device behavior. This is because the shifts in the band edges represent a potential barrier which influences the carrier transport across the junctions [119,120].

The bandgap narrowing and edge displacment effect has been modeled for Si based on measurements of the quantity ${\mu}_{n}n_{i}^{2}$ in npn-transistors [121,122]:

$$\Delta E_{\mathrm{g}}= C_{n,p}^{BGN}\cdot \left( F+\sqrt{F^{2}+0.5}\right),$$ (3.72)

$$F=\ln \left( \frac{N_{D}+N_{A}}{N_{n,p}^{BGN}}\right).$$ (3.73)

Calculated band edge displacement parameters for 4H- and 6H-SiC are listed in Table 3.4 [119]. Compared to Si, a larger $\Delta E_{g}$ is expected in n-type material for 4H- and 6H-SiC, respectively, whereas approximately the same displacements are expected in p-type material for both polytypes.

Since Boltzmann statistics is not valid at high doping levels $(N>10^{19}$cm$^{-3})$, where the aforementioned interactions are not negligible, the bandgap narrowing effects should be taken into account by an effective intrinsic carrier concentration $n_{i,e}$ [123].

$$n_{i,e}=n_{i}\cdot\gamma_{BGN}$$ (3.74)

where $\gamma_{BGN}$ is a correcting factor introduced to Fermi statistics, given by

$$\gamma _{BGN}=\displaystyle\frac{F_{1/2}\left[ x+\displaystyle\fr... ...g}}}{2\cdot{\mathrm{k_B}}\cdot T_\mathrm{L}}\right] }{F_{1/2}\left[ x\right] },$$ (3.75)

$$x=\displaystyle\frac{E_{i}-E_{c}+{\mathrm{q}}\cdot\left(\psi - \phi \right) }{{\mathrm{k_B}}\cdot T_\mathrm{L}}.$$ (3.76)

The effect of the bandgap narrowing on the intrinsic carrier concentration is shown in Fig. 3.8.

Table 3.4: Calculated bandgap narrowing parameters in n- and p-type $\alpha$-SiC.

	C $_{n}^\mathrm{BGN}$ [eV]	N $_{n}^\mathrm{BGN}$ [cm$^{-3}$]	C $_{p}^\mathrm{BGN}$ [eV]	N $_{p}^\mathrm{BGN}$ [cm$^{-3}$ ]
4H-SiC	$2.0\times 10^{-2}$	$1.0\times 10^{17}$	$9.0\times 10^{-3}$	$1.0\times 10^{17}$
6H-SiC	$9.0\times 10^{-3}$	$1.0\times 10^{17}$	$9.0\times 10^{-3}$	$1.0\times 10^{17}$

Figure 3.8: Bandgap narrowing in $\alpha$-SiC as a function of doping concentration, and its effect on the intrinsic carrier concentration.

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