3.5.1 Shockley-Read-Hall Recombination

The balance equation for each generation-recombination center yields a Shockley-Read-Hall (SRH) rate [148] within the quasi-static approximation. The individual characteristic properties of generation-recombination centers depend strongly on the technology. Therefore, they are usually lumped together in quantities like the effective electron and hole lifetimes, ending up with one effective single-level SRH rate

$$R^{\mathrm{SRH}}=\frac{n \cdot p - n_{i,e}^2}{\tau_{p}\cdot(n+n_{1})+\tau_{n}\cdot (p+p_{1})}.$$ (3.102)

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

$$n_{1} = N_\mathrm{c}\cdot \exp\left(\frac{-E_\mathrm{C}+ E_{\mathrm T}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right),$$ (3.103)

$$p_{1} = N_\mathrm{v}\cdot \exp\left(\frac{- E_{\mathrm T} + E_\mathrm{V}}{{\mathrm{k_B}}\cdot T_\mathrm{L}}\right).$$ (3.104)

For a trap energy level $E_{\mathrm T}$ located in the middle of the bandgap, $n_1=p_1=n_{i,e}$ and the recombination rate has its maximum. The variables $N_\mathrm{c}$ and $N_\mathrm{v}$ are the carrier effective densities of states (see section 3.2.2). 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 lifetime, $\tau_{\nu}(T_\mathrm{L})$. The thermal carrier velocities at ${\mathrm{300\,K}}$ are calculated using

$$v_{\nu,300} = \sqrt{\frac{3\cdot{\mathrm{k_B}}\cdot T_\mathrm{L}}{m_{\nu}}} \hspace{1cm}\nu = n,p$$ (3.105)

thus, the recombination lifetimes at ${\mathrm{300\,K}}$ read

$$\tau_{\nu,300} = \frac{1}{\sigma_{\mathrm{T},\nu}\cdot N_{\mathrm{T}}\cdot v_{\nu,300} + S_\nu/y}.\hspace{1cm}\nu = n,p$$ (3.106)

In (3.106), recombination lifetimes are modeled using traps of donor, acceptor, or neutral type, respectively, with the trap density $N_{\mathrm{T}}$ and the trap capture cross section, $\sigma_{\mathrm{T},\nu}$. The effect of surface recombination is included using non-zero surface recombination velocities for electrons $S_n$ and holes $S_p$, respectively. The effect becomes stronger with decreasing distance to the surface $y$.

The lifetime $\tau _{\nu}$ in (3.102) is a dominant recombination-generation process in a device. Various physical mechanisms may influence this process. Generally, a doping dependence of $\tau _{\nu}$ is experimentally observed in Si technology [149,150] and is empirically modeled by the so-called Scharfetter relation:

$$\tau _{\nu}(N,T_\mathrm{L})=\tau _{\nu}^{\min }+ \displaystyle\fr... ...u}^{\mathrm{SRH}}}\right) ^{\gamma _{\nu}^\mathrm{SRH}}}, \hspace{1cm}\nu = n,p$$ (3.107)

with

$$\tau^{\max} _{\nu}(T_\mathrm{L})=\tau_{\nu,300}\cdot\left(\displa... ...}}{\mathrm 300 \ K}\right) ^{\alpha _{\nu}^\mathrm{SRH}}. \hspace{1cm}\nu = n,p$$ (3.108)

Expression (3.107) can be basically regarded as a fit formula to account for experimental facts which strongly depend on process technology. There are no reliable experimental data available to extract the corresponding parameters for $\alpha$-SiC. Since SiC is a semiconductor with an indirect bandgap, long lifetimes comparable to Si should be observed in perfect crystals with low contamination. Also lifetime adjustment (with appropriate recombination centers in the middle of the band gap) should be possible [22]. Therefore, parameters typical for Si devices may be used in order to qualitatively account for those effects. T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation