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 quasiFermi levels (sometimes called the implied opencircuit 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.
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 Δn_{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 Δn or hole Δp 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 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} + Δn and p = p_{0} + Δp.
When selecting to plot the separation of quasiFermi levels on the xaxis of the figure, the program sweeps the excess carrier concentration, Δn = Δp, and plots the results against (E_{Fn} – E_{Fp}) / kT. This separation of quasiFermi levels is often referred to as the semiconductor's implied opencircuit 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 Coulombenhanced 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 cSi: 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. 
[Dzi77]  J. Dziewior and W. Schmid, "Auger coefficients for highly doped and highly excited silicon," Applied Physics Letters 31, pp. 346–348, 1977. 
[Glu99]  S.W. Glunz, D. Biro, S. Rein and W. Warta, "Fieldeffect passivation of the SiO2–Si interfaceJournal 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.

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