2.3.4 Generation/Recombination Processes

In indirect gap semiconductors, such as silicon and germanium, it was found experimentally that generation/recombination occurs primarily via trap centers. A theory of this effect has been established by SHOCKLEY and READ [41] and HALL [42].

Figure 2.5: Symbolic band diagram showing the four partial processes involved in indirect generation/recombination.

Interaction among the partial systems electrons, holes, and traps is described by four partial processes (Fig. 2.5)

$$e^- + T^0 \overset{c_n}{\underset{e_n}{\rightleftharpoons}} T^- \ ,$$ (2.197)

$$h^+ + T^- \overset{c_p}{\underset{e_p}{\rightleftharpoons}} T^0 \ .$$ (2.198)

$c_n$:

Electron capture. An electron from the conduction band is trapped by an unoccupied defect which becomes occupied.

$e_n$:

Electron emission. An electron from an occupied trap moves to the conduction band. The trap becomes unoccupied.

$c_p$:

Hole capture. An electron from an occupied trap moves to the valence band and neutralizes a hole. The trap becomes unoccupied.

$e_p$:

Hole emission. An electron from the valence band is trapped by a defect, thus leaving a hole in the valence band and an occupied trap.

Here $c_n$, $e_n$, $c_p$, $e_p$ are the respective rate constants. This description assumes acceptor-like traps which can exist in a neutral or a negatively charged state. Donor-like traps, which have a neutral and a positively charged state, lead however to exactly the same expression for the net recombination rate.

The generation- and recombination rates of electrons/holes within an energy interval $\mathrm{d\mathcal{E}}$ are described by the law of mass action which states that the rates are proportional to the concentration of the involved reactants [43, p.54]

$$\mathrm{d} R_n - \mathrm{d}G_n = c_n \, f_n \, N_C \, \mathrm{d}\mathcal{E}\, (1 - f_t) \, N_t - e_n \, f_t \, N_t \, (1 - f_n) \, N_C \, \mathrm{d}\mathcal{E}\ ,$$ (2.199)

$$\mathrm{d} R_p - \mathrm{d}G_p = c_p \, f_p \, N_V \, \mathrm{d}\mathcal{E}\, f_t \, N_t - e_p \, (1 - f_t) \, N_t \, (1 - f_p) \, N_V \, \mathrm{d}\mathcal{E}\ .$$ (2.200)

The occupation probability of an energy level is given by the FERMI-DIRAC statistics

$$f_n = \frac{1}{1 + e^{\frac{\mathcal{E}- F_n}{\mathrm{k}_\mathrm{B}\, T_n}}} \ ,$$ (2.201)

$$f_p = \frac{1}{1 + e^{- \frac{\mathcal{E}- F_p}{\mathrm{k}_\mathrm{B}\, T_p}}} \ , \textcolor{lightgrey}{.......}f_p = 1 - f_n \ ,$$ (2.202)

$$f_t = \frac{1}{1 + g \, e^{\frac{\mathcal{E}_t - F_t}{\mathrm{k}_\mathrm{B}\, T_\mathrm{L}}}} \,$$ (2.203)

with $F_n$, $F_p$, and $F_t$ being the respective quasi FERMI levels and $g$ the ground-state degeneracy of the trap [44, p.122] which is assumed to be 1 in the following.

Subsections

• 2.3.4.1 Thermal Equilibrium
• 2.3.4.2 Steady State
• 2.3.4.3 Higher Order Moments

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