# RECOMBINATION CALCULATOR

This calculator determines the effective lifetime and recombination rate in crystalline silicon. It calculates radiative recombination, Auger recombination, and Shockley–Read–Hall recombination as a function of the dopant concentration, excess carrier concentration, or the separation of quasi-Fermi levels (sometimes called the implied open-circuit voltage).

# DISCLAIMER

Neither PV Lighthouse nor any person related to the compilation of this calculator make any warranty, expressed or implied, or assume any legal liability or responsibility for the accuracy, completeness or usefulness of any information disclosed or rendered by this calculator.

# WARNING

Although temperature is an input, its effect is not included in every recombination model – see the table below.

In particular, be aware that (i) the SRH model assumes constant lifetimes (or capture cross sections) even though they actually
depend strongly on temperature; (ii) the only Auger model that contains
a temperature dependence is Altermatt1997 (valid for 70–400 K), and (iii) the various radiative
recombination models are valid over different temperature ranges.

Hence, be wary of the results for any recombination rate, lifetime, or diffusion length when the temperature
lies outside the range specified by the recombination models (see table below).
When sweeping temperature, only the results for temperature-dependent models are plotted.

# EQUATIONS

The program first applies the band gap models to determine the effective intrinsic carrier concentration *n*_{i eff}, the conduction band energy *E*_{c}, the valence band energy *E*_{v}, the electron Fermi energy *E*_{Fn}, and the hole Fermi energy *E*_{Fp} following the procedure used in the band gap calculator. The intrinsic Fermi energy *E*_{i} is defined to be zero.

The recombination rate is then calculated for radiative *U*_{rad}, Auger *U*_{Aug}, and Shockley–Read–Hall *U*_{SRH} recombination mechanisms, which sum to give the total recombination rate:

The lifetime for each mechanism is then determined by

where Δ*n*_{m} is the excess minority carrier concentration, and where 'a' represents either 'rad', 'Aug' or 'SRH'. The effective lifetime *τ*_{eff} is calculated in the same way from *U*_{tot}.

Various options are given for determining *U*_{rad} and *U*_{Aug}, which all depend on the ionised dopant concentration *N*_{D+} or *N*_{A–}, the excess electron Δ*n* or hole Δ*p* concentration, and *n*_{i eff}. The options for the models are summarised in the table below.

The Shockley–Read–Hall recombination rate is calculated by the equation

where

and

and where *σ*_{n} and *σ*_{p} are the capture cross sections of electrons and holes, *v*_{th e} and *v*_{th h} are the thermal velocities of electrons and holes, *N*_{t} is the concentration of defect states, and *E*_{t} is the energy of the defect state. It is assumed that *N*_{t} << *p*, *n*, and that the semiconductor is not degenerate.

The calculator can also account for photon recycling. Photon recycling is when a fraction of the photons emitted via radiative recombination *f*_{pr} are re-absorbed via band-to-band absorption before they exit the silicon. Thus, photon recycling effectively reduces radiative recombination. When photon recylcing is non-zero, the calculator gives the effective radiative recombinaion rate [Fel21],

Models for *f*_{pr} are given in the table below; they depend on the optical properties of the silicon wafer.

The electron diffusion length is determined from the equation *L*_{e} = Sqrt(*D*_{e}⋅*τ*), where *D*_{e} is the diffusivity of electrons. An analogous equation is used to determine the hole diffusion length.

All equations also assume that the semiconductor is in steady state and that the steady state carrier concentrations relate to the equilibrium and excess carrier concentrations by *n* = *n*_{0} + Δ*n* and *p* = *p*_{0} + Δ*p*.

When selecting to plot the separation of quasi-Fermi levels on the x-axis of the figure, the program sweeps the excess carrier concentration, Δ*n* = Δ*p*, and plots the results against (*E*_{Fn} – *E*_{Fp}) / *kT*. This separation of quasi-Fermi levels is often referred to as the semiconductor's implied open-circuit voltage.

# OPTIONS FOR RECOMBINATION MODELS

# Definition of symbols

# REFERENCES

| |

[Alt97] | P.P. Altermatt, J. Schmidt, G. Heiser and A.G. Aberle, "Assessment and parameterisation of Coulomb-enhanced Auger recombination coefficients in lowly injected crystalline silicon," *Journal of Applied Physics* **82**, pp. 4938–4944, 1997. |

[Alt05] | P.P. Altermatt, F. Geelhaar, T. Trupke, X. Dai, A. Neisser and E. Daub, "Injection dependence of spontaneous radiative recombination in c-Si: experiment, theoretical analysis, and simulation," *Proc. 5th Conference on Numerical Simulation of Optoelectronic Devices (NUSOD)*, pp. 47–48, 2005. |

[Alt11] | P.P. Altermatt, "Models for numerical device simulations of crystalline silicon solar cells — a review," *Journal of Computational Electronics* **10** (3) pp. 314–330, 2011. |

[Bla22] | L.E. Black and D.H. Macdonald, "On the quantification of Auger recombination in crystalline silicon," *Solar Energy Materials and Solar Cells* **234** 111428, 2022. |

[Dzi77] | J. Dziewior and W. Schmid, "Auger coefficients for highly doped and highly excited silicon," *Applied Physics Letters* **31**, pp. 346–348, 1977. |

[Fel21] | A. Fell, T. Niewelt, B. Steinhauser, F.D. Heinz, M.C. Schubert and S.W. Glunz, "Radiative recombination in silicon photovoltaics: Modeling the influence of charge carrier densities and photon recycling," *Solar Energy Material and Solar Cells* **230**, 111198, 2021. |

[Glu99] | S.W. Glunz, D. Biro, S. Rein and W. Warta, "Field-effect passivation of the SiO2–Si interface*Journal of Applied Physics* **86**, pp. 683–691, 1999. |

[Hal52] | R. Hall, "Electron–hole recombination in germanium," *Physics Review* **87**, p. 387, 1952. |

[Ker02] | M.J. Kerr and A. Cuevas, "General parameterization of Auger recombination in crystalline silicon," *Journal of Applied Physics* **91**, pp. 2473–2480, 2002. |

[Nie22] | T. Niewelt, B. Steinhauser, A. Richter, B. Veith-Wolf, A. Fell, B. Hammann, N.E. Grant, L. Black, J. Tan, A. Youssef, J.D. Murphy, J. Schmidt, M.C. Schubert and S.W. Glunz, "Reassessment of the intrinsic bulk recombination in crystalline silicon," *Solar Energy Material and Solar Cells* **235**, 111467, 2022. |

[Ngu14b] | H.T. Nguyen, S.C. Baker-Finch and D.H. Macdonald, "Temperature dependence of the radiative recombination coefficient in crystalline silicon from spectral photoluminescence," *Applied Physics Letters* **104**, 112105, 2014. |

[Ric12] | A. Richter, S.W. Glunz, F. Werner, J. Schmidt and A. Cuevas, "Improved quantitative description of Auger recombination in crystalline silicon," *Physics Review B* **86**, 165202, 2012. |

[Sch74] | H. Schlangenotto, H. Maeder and W. Gerlach, "Temperature dependence of the radiative recombination coefficient in silicon," *Physica Status Solidi A* **21**, pp. 357–367, 1974. |

[Sho52] | W. Shockley and W.T. Read, "Statistics of the recombinations of holes and electrons," *Physics Review* **87**, pp. 835–842, 1952. |

[Sin87] | R.A. Sinton and R.M. Swanson, "Recombination in highly injected silicon," *IEEE Transactions on Electron Devices* **34**, pp. 1380–1389, 1987. |

[Tru03] | T. Trupke, M.A. Green, P. Würfel, P.P. Altermatt, A. Wang, J. Zhao and R. Corkish, "Temperature dependence of the radiative recombination coefficient of intrinsic crystalline silicon," *Journal of Applied Physics* **94** (8), pp. 4930–4937, 2003. |

[Vei18] | B.A. Veith–Wolf, S. Schäfer, R.Brendel and J.Schmidt, "Reassessment of intrinsic lifetime limit in n-type crystalline silicon and implication on maximum solar cell efficiency," *Solar Energy Materials and Solar Cells* **186**, pp. 194–199, 2018. |

# ACKNOWLEDGEMENTS AND FEEDBACK

We thank Tim Niewelt for sending us references models on updated models for Auger recombination, radiative recombination and photon recycling.

Please email corrections, comments or suggestions to support@pvlighthouse.com.au.