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)

$\displaystyle e^- + T^0$ $\displaystyle \overset{c_n}{\underset{e_n}{\rightleftharpoons}} T^- \ ,$ (2.197)
$\displaystyle h^+ + T^-$ $\displaystyle \overset{c_p}{\underset{e_p}{\rightleftharpoons}} T^0 \ .$ (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]

$\displaystyle \mathrm{d} R_n - \mathrm{d}G_n$ $\displaystyle = c_n \, f_n \, N_C \, \mathrm{d}\mathcal{E}\, (1 - f_t) \, N_t$   $\displaystyle - e_n \, f_t \, N_t \, (1 - f_n) \, N_C \, \mathrm{d}\mathcal{E}\ ,$ (2.199)
$\displaystyle \mathrm{d} R_p - \mathrm{d}G_p$ $\displaystyle = c_p \, f_p \, N_V \, \mathrm{d}\mathcal{E}\, f_t \, N_t$   $\displaystyle - e_p \, (1 - f_t) \, N_t \, (1 - f_p) \, N_V \, \mathrm{d}\mathcal{E}\ .$ (2.200)

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

$\displaystyle f_n$ $\displaystyle = \frac{1}{1 + e^{\frac{\mathcal{E}- F_n}{\mathrm{k}_\mathrm{B}\, T_n}}} \ ,$ (2.201)
$\displaystyle f_p$ $\displaystyle = \frac{1}{1 + e^{- \frac{\mathcal{E}- F_p}{\mathrm{k}_\mathrm{B}\, T_p}}} \ , \textcolor{lightgrey}{.......}f_p = 1 - f_n \ ,$ (2.202)
$\displaystyle f_t$ $\displaystyle = \frac{1}{1 + g \, e^{\frac{\mathcal{E}_t - F_t}{\mathrm{k}_\mathrm{B}\, T_\mathrm{L}}}} \,$ (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.


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