3.7.1 Shockley-Read-Hall and Surface Recombination

Carrier generation in space charge regions and recombination in e.g. high injection regions is modeled using the well known Shockley-Read-Hall (SRH) equation

$$R^{\mathrm{SRH}}=\frac{n\cdot p - n_i^2}{\tau_p\cdot (n+n_1)+\tau_n\cdot (p+p_1)},\hspace{5mm} n_i^2=n_1\cdot p_1.$$ (3.142)

The auxiliary variables $n_{1}$ and $p_{1}$ are defined by

$$n_1 = N_{C}(T_{{\mathrm{L}}}) \cdot \exp\left(\frac{-E_{C}+E_{\mathrm{T}}}{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}\right),$$ (3.143)

$$p_1 = N_{V}(T_{{\mathrm{L}}}) \cdot \exp\left(\frac{-E_{\mathrm{T}}+E_{V}}{\mathrm{k_B}\cdot T_{{\mathrm{L}}}}\right).$$ (3.144)

For trap energy level $E_{\mathrm{T}}$ located in the mid gap $n_1=p_1=n_i$ and the recombination rate has its maximum. The variables $N_{C}$ and $N_{V}$ are the carrier effective densities of states (see Section 3.3). The dependence on the lattice temperature $T_{{\mathrm{L}}}$ is given by the variables $n_{1}(T_{{\mathrm{L}}})$ and $p_{1}(T_{{\mathrm{L}}})$ and the recombination lifetimes for electrons and holes, $\tau_{n}(T_{{\mathrm{L}}})$ and $\tau_{p}(T_{{\mathrm{L}}})$. The thermal carrier velocities at 300 K are calculated using

$$v_{n,p} = \sqrt{\frac{3\cdot\mathrm{k_B}\cdot T_{{\mathrm{L}}}}{m_{n,p}}}$$ (3.145)

to obtain the recombination times at 300 K

$$\tau_{n,300} = \frac{1}{\sigma_{\mathrm{T},n}\cdot N_{\mathrm{T}}\cdot v_{n,300} + S_n/y},$$ (3.146)

$$\tau_{p,300} = \frac{1}{\sigma_{\mathrm{T},p}\cdot N_{\mathrm{T}}\cdot v_{p,300} + S_p/y}.$$ (3.147)

The recombination times are modeled using traps of donor, acceptor, or neutral type, respectively, of trap density $N_{\mathrm{T}}$, and the trap capture cross sections for electrons and holes, $\sigma_{\mathrm{T},n}$ and $\sigma_{\mathrm{T},p}$. The effect of surface recombination is included by using non-zero surface recombination velocities for electrons $S_n$ and holes $S_p$, respectively. The effect is stronger with decreasing distance to the surface $y$.

Finally, the temperature dependence is included. The effect of trap assisted band to band tunneling (TBB) in Si is accounted for by a field enhancement factors $\Gamma_{n}$ and $\Gamma_{p}$, which are modeled after [187]. As TBB is of importance only for materials with indirect bandgap this effect is neglected in literature for technologically important III-V materials, which have a direct bandgap (see Section 3.3.1).

$$\tau_{n}(T_{{\mathrm{L}}}) = \left(\frac{{\mathrm 300K}}{T_{\mathrm{L}}}\right)^{3/2} \cdot \frac{\tau_{n,300}}{1 + \Gamma_n},$$ (3.148)

$$\tau_{p}(T_{{\mathrm{L}}}) = \left(\frac{{\mathrm 300K}}{T_{\mathrm{L}}}\right)^{3/2} \cdot \frac{\tau_{p,300}}{1 + \Gamma_p},$$ (3.149)

The default values recommended for the SRH recombination model are summarized in the following table:

Table 3.35: Parameter values for SRH recombination model

Material	$N_{\mathrm{T}}$ [cm$^{-3}$]	$E_{\mathrm{T}}$ [eV]	$\sigma_{T,n}$ [m$^2$]	$\sigma_{T,p}$ [m$^2$]	$S_{n}$ [m/s]	$S_{p}$ [m/s]
Si, Ge	1e13	0.0	1e-15	1e-15	0.0	0.0
III-Vs	2e16	0.4	1e-14	1e-13	0.0	0.0