2.3.4.1 Thermal Equilibrium

The four rate constants are not independent. Their relation can be found by examining the thermal equilibrium case. In thermal equilibrium the principle of detailed balance holds, which ensures that at $T_\mathrm{L}$ emission and capture processes are balanced

$$\mathrm{d}R_n = \mathrm{d}G_n \ ,$$ (2.204)

$$\mathrm{d}R_p = \mathrm{d}G_p \ .$$ (2.205)

Furthermore, all distribution functions have one FERMI level $\mathcal{E}_\mathrm{F}$ in common

$$F_n = F_p = F_t = \mathcal{E}_\mathrm{F} \ .$$ (2.206)

by applying eqns. (2.204) to (2.206) to eqns. (2.199) to (2.203) the following relationships between the rate constants are found

$$\frac{e_n}{c_n} = e^{- \frac{\mathcal{E}- \mathcal{E}_t}{\mathrm{k}_\mathrm{B}\, T_\mathrm{L}}} \ ,$$ (2.207)

$$\frac{e_p}{c_p} = e^{ \frac{\mathcal{E}- \mathcal{E}_t}{\mathrm{k}_\mathrm{B}\, T_\mathrm{L}}} \ .$$ (2.208)

Inserting eqns. (2.207) and (2.208) into eqns. (2.199) and (2.200) yields

$$\mathrm{d} R_n - \mathrm{d}G_n = c_n \, N_C \, N_t \, \mathrm{d}\mathcal{E}\, \Bigl( f_n \, (1 -... ...cal{E}_t}{\mathrm{k}_\mathrm{B}\, T_\mathrm{L}}} \, f_t \, (1 - f_n) \Bigr) \ ,$$ (2.209)

$$\mathrm{d} R_p - \mathrm{d}G_p = c_p \, N_V \, N_t \, \mathrm{d}\mathcal{E}\, \Bigl( f_p \, f_t ... ..._t}{\mathrm{k}_\mathrm{B}\, T_\mathrm{L}}} \, (1 - f_t) \, (1 - f_p) \Bigr) \ .$$ (2.210)

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF