5.5.2 Shockley-Read-Hall Recombination

Trap-assisted generation and recombination is modeled using the Shockley-Read-Hall model [289]. The rate dependence on the carrier concentration is described by the expression

$$R^{\ensuremath{\mathrm{SRH}}} = \frac{\ensuremath{n}\ensuremath{p... ..._1) + \ensuremath{\tau_{\ensuremath{n}}}(\ensuremath{p}+ \ensuremath{p}_1)} \,,$$ (5.47)

where $\ensuremath{\tau_{\ensuremath{n}}}$ and $\ensuremath{\tau_{\ensuremath{p}}}$ depict the generation/recombination lifetimes for electrons and holes, respectively. The auxiliary variables $\ensuremath{n}_1$ and $\ensuremath{p}_1$ are defined as

$$\ensuremath{n}_1 = \ensuremath{N_\mathrm{c}}(\ensuremath{T_{\mathrm{L}}}) \exp \le... ...mathrm{c}}}}}{k_\ensuremath{\mathrm{B}}\ensuremath{T_{\mathrm{L}}}} \right) \,,$$ (5.48)

$$\ensuremath{p}_1 = \ensuremath{N_\mathrm{v}}(\ensuremath{T_{\mathrm{L}}}) \exp \le... ...mathrm{T}}}}}{k_\ensuremath{\mathrm{B}}\ensuremath{T_{\mathrm{L}}}} \right) \,.$$ (5.49)

There, the effective densities of states for electrons and holes are denoted by $\ensuremath{N_\mathrm{c}}$ and $\ensuremath{N_\mathrm{v}}$ and the conduction and valence band edges by $\ensuremath{\ensuremath{\mathcal{E}}_{\ensuremath{\mathrm{c}}}}$ and $\ensuremath{\ensuremath{\mathcal{E}}_{\ensuremath{\mathrm{v}}}}$ , respectively. The most effective generation/recombination centers are those with an energy $\ensuremath{\ensuremath{\mathcal{E}}_{\ensuremath{\mathrm{T}}}}$ close to the mid of the band-gap. The generation/recombination lifetimes $\ensuremath{\tau_{\ensuremath{n}}}$ and $\ensuremath{\tau_{\ensuremath{p}}}$ at room temperature can be expressed as

$$\ensuremath{\tau_{\ensuremath{n},300}} = \frac{1}{\ensuremath{\sigma_{\ensuremath{\mathrm{T}},\ensuremath{n}}}\ensuremath{N_\mathrm{T}}v_{\ensuremath{n},300}}$$ (5.50)

$$\ensuremath{\tau_{\ensuremath{p},300}} = \frac{1}{\ensuremath{\sigma_{\ensuremath{\mathrm{T}},\ensuremath{p}}}\ensuremath{N_\mathrm{T}}v_{\ensuremath{p},300}} \,,$$ (5.51)

where they are dependent on the trap concentration $\ensuremath{N_\mathrm{T}}$ as well as the trap capture cross sections $\ensuremath{\sigma_{\ensuremath{\mathrm{T}},\ensuremath{n}}}$ and $\ensuremath{\sigma_{\ensuremath{\mathrm{T}},\ensuremath{p}}}$ for electrons and holes, respectively. The thermal velocities for electrons and holes are expressed as

$$\ensuremath{v_\ensuremath{\nu}}= \sqrt {\frac{3 k_\ensuremath{\mathrm{B}}\ensuremath{T_{\mathrm{L}}}}{\ensuremath{m^*_\ensuremath{\mathrm{c,v}}}} }$$ (5.52)

in order to formulate the generation/recombination lifetimes dependent on the trap concentration $\ensuremath{N_\mathrm{T}}$ as well as the trap capture cross sections $\ensuremath{\sigma_{\ensuremath{\mathrm{T}},\ensuremath{n}}}$ and $\ensuremath{\sigma_{\ensuremath{\mathrm{T}},\ensuremath{p}}}$ for electrons and holes, respectively. Furthermore, the temperature dependence of the generation/recombination lifetimes is described by the empirical power law

$$\ensuremath{\tau_{\ensuremath{n}}} = \left( \frac{300\,\ensuremath{\mathrm{K}}}{\ensuremath{T_{\mathrm{L}}}} \right)^{3/2} \ensuremath{\tau_{\ensuremath{n},300}}\,,$$ (5.53)

$$\ensuremath{\tau_{\ensuremath{p}}} = \left( \frac{300\,\ensuremath{\mathrm{K}}}{\ensuremath{T_{\mathrm{L}}}} \right)^{3/2} \ensuremath{\tau_{\ensuremath{p},300}}\,.$$ (5.54)

Additionally, doping dependent generation/recombination lifetimes can be introduced using the Scharfetter relation

$$\ensuremath{\tau_{\ensuremath{n}}}(\ensuremath{N_\ensuremath{\mathrm{tot}}}) = \ensuremath{\tau_{\ensuremath{n},\ensuremath{\mathrm{min}}}}\fr... ...nsuremath{\mathrm{ref}},\ensuremath{n}}} \right)^{\gamma_{\ensuremath{n}}}} \,,$$ (5.55)

$$\ensuremath{\tau_{\ensuremath{p}}}(\ensuremath{N_\ensuremath{\mathrm{tot}}}) = \ensuremath{\tau_{\ensuremath{p},\ensuremath{\mathrm{min}}}}\fr... ...nsuremath{\mathrm{ref}},\ensuremath{p}}} \right)^{\gamma_{\ensuremath{p}}}} \,,$$ (5.56)

which can be used to calibrate the model to experimental data. Since the lifetimes strongly depend on the process technology as well as material quality, the parameters have to be determined from sample to sample. While in single crystals relatively long lifetimes can be expected, grain boundaries within sintered samples have a reducing effect.

M. Wagner: Simulation of Thermoelectric Devices