3.1.5 Quasi-Fermi Potential

Obviously, when the excess carrier concentration is small compared to the equilibrium carrier concentration, the quasi-Fermi level must be very close to the Fermi level. For device operation, we often use a low-level injection condition, meaning that while the minority carrier concentration is changed, the majority carrier concentration remains un-affected. Thus the quasi-Fermi level of the majority carrier is the same as the Fermi level. At thermal equilibrium, that is, the steady-state condition at a given temperature without any external excitation, the individual electron and hole currents flowing across the junction are identically zero. Thus, for each type of carrier the drift current due to the electric field must exactly cancel the diffusion current due to the concentration gradient.

In unipolar devices like MOSFETs, it is often possible to assume a constant quasi-Fermi potential for one carrier type. For a p-channel MOSFET the electrons in the bulk represent the minority carrier system. Assuming Maxwell-Boltzmann statistics, the equation for the electron concentration reads

$$n=N_\mathrm{c}\cdot \exp\left(-\frac{{\mathrm{q}}\cdot(\phi_n-\psi)+\ensuremath{E_\mathrm{c}}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right),$$ (3.28)

similarly, the equation for the hole concentration reads

$$p=N_\mathrm{v}\cdot \exp\left(\frac{{\mathrm{q}}\cdot(\phi_p-\psi)+E_\mathrm{v}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right).$$ (3.29)

Here, $\phi_n$ and $\phi_p$ are the quasi-Fermi potentials for electrons and holes, $N_\mathrm{c}$ and $N_\mathrm{v}$ are the effective density-of-states for electrons in the conduction band and for holes in the valence band, respectively. They are obtained from (3.66) and (3.67).

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