# BAND GAP CALCULATOR

This calculator determines the band gap of silicon at any temperature, dopant concentration, and excess carrier concentration. It also determines the energy of the conductance band, valence band, electron Fermi level, and hole Fermi level, as well as the effective intrinsic carrier density at equilibrium and steady state.

# 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.

# PROCEDURE

The figure below presents a flow chart of the algorithms employed by the calculator. The tables below list the available models and define the symbols.

The outputs for equilibrium are determined by setting Δn = Δp = 0, whereas the outputs for steady state are determined with the input values for Δn and Δp.

# FLOW DIAGRAM OF ALGORITHMS

# OPTIONS FOR MODELS

# OPTIONS FOR BAND GAP NARROWING

The best choice for the band gap narrowing (BGN) model is not obvious and there exists disagreement over which to use. Moreover, the model that is selected for BGN dictates which model should be selected for carrier statistics, as is now explained.

There are currently four BGN models to choose from. Specifically, these models state how the dopant concentration affects the minimum energy gap between the conduction and valence bands E_{g}. The decrease in E_{g} due to the dopant concentration (and carrier concentrations) is called band gap narrowing and it is quantified by ∆E_{g}.

The simplest of the four models is 'none', i.e. no BGN. It is inaccurate for heavily doped silicon but it is simple to understand: ∆E_{g} = 0 for all conditions.

The second model is that devised by del Alamo *et al.* in 1985 [del85a, del85b]. It states that the 'apparent BGN' is given by ∆E_{g,app} = ∆E_{slope}∙ln(N/N_{onset}) for N > N_{onset}, and ∆E_{g,app} = 0 otherwise, where N is the dopant concentration. (In fact, we take N to be the absolute difference between the ionised donor and ionised acceptor concentrations.) As summarised in the table below, the constants ∆E_{slope} and N_{onset} have been determined in several studies for phosphorus-doped silicon at 300 K. This is the model used by PC1D.

Note that del Alamo's model incorporates degeneracy as well as BGN, which is why it determines the 'apparent BGN', ∆E_{g,app}, and not the actual BGN, ∆E_{g}. This means that the del Alamo BGN model should be used with Boltzmann carrier statistics rather than Fermi–Dirac statistics. Read more about it in [Alt02, Yan13].

A third model for BGN is that recently published by Yan and Cuevas [Yan13, Yan14]. It is similar to del Alamo's model but contains an exponent b and an offset ∆E_{offset}. It states that ∆E_{g} = ∆E_{slope}∙[ln(N/N_{onset})]^{b} + ∆E_{offset}. The table below gives values determined for these constants for crystalline silicon at 300 K. Yan and Cuevas find that with b = 3, they can simulate lifetime measurements of phosphorus-diffused silicon when using Fermi–Dirac statistics (i.e., when not incorporating degeneracy into the model for ∆E_{g}). Note that this equation is identical to del Alamo's when b = 1 and ∆E_{offset} = 0.

Neither the del Alamo or Yan–Cuevas models depend on temperature; it is therefore necessary to adjust the inputs for each temperature of interest.

A fourth and more extensive model for BGN is that published by Schenk in 1998 [Sch98]. It includes too many equations to present here, but its derivation is based on quantum mechanics and it accounts for dopant concentrations, carrier concentrations, and temperature. It does not incorporate degeneracy and should therefore be used with Fermi–Dirac statistics. It has been evaluated in [Glu01], [Alt02], [Yan13] and [Yan14], and its equations are summarised in Appendix A of [McI10]. It does not require the user to insert any input values.

# DEFINITION OF SYMBOLS

# REFERENCES

| |

[Alt02] | P.P. Altermatt, J.O. Schumacher, A. Cuevas, M.J. Kerr, S.W. Glunz, R.R. King, G. Heiser, A. Schenk, "Numerical modeling of highly doped Si:P emitters based on Fermi–Dirac statistics and self-consistent material parameters," *Journal of Applied Physics*, **92** (6), pp. 3187–3197, 2002. |

[Alt06a] | P.P. Altermatt, A. Schenck and G. Heiser, "A simulation model for the density of states and for incomplete ionization in crystalline silicon. I. Establishing the model in Si:P," *Journal of Applied Physics*, **100** 113714, 2006. |

[Alt06b] | P.P. Altermatt, A. Schenck, B. Schmithüsen and G. Heiser, "A simulation model for the density of states and for incomplete ionization in crystalline silicon. II. Investigation of Si:As and Si:B and usage in device simulation," *Journal of Applied Physics*, **100** 113715, 2006. |

[Blu74] | W. Bludau, A. Onton, and W. Heinke, "Temperature dependence of the band gap in silicon," *Journal of Applied Physics*, **45** (4), pp. 1846–1848, 1974. |

[Cou14] | R. Couderc, M. Amara and M. Lemiti, "Reassessment of the intrinsic carrier density temperature dependence in crystalline silicon," *Journal of Applied Physics*, **115** 093705, 2014. |

[Cue96] | A. Cuevas, P.A. Basore, G Giroult–Matlakowski and C. Dubois, "Surface recombination velocity of highly doped n-type silicon," *Journal of Applied Physics*, **80**, pp. 3370–3375, 1996. |

[del85a] | J. del Alamo, S. Swirhun and R. M. Swanson, "Simultaneous measurement of hole lifetime, hole mobility and band gap narrowing in heavily doped n-type silicon," *Proceedings of the 18th IEEE PVSC*, Las Vegas, pp. 290–293, 1985. |

[del85b] | J. del Alamo, S. Swirhun and R. M. Swanson, "Measuring and modeling minority carrier transport in heavily doped silicon," *Solid-State Electronics*, **28**, pp. 47–54, 1985. |

[Glu01] | S.W. Glunz, J. Dicker and P.P. Altermatt, "Band gap narrowing in p-type base regions of solar cells," *Proceedings of the 17th EU PVSEC*, Munich, pp. 1391–1395, 2001. |

[Gre90] | M.A. Green, "Intrinsic concentration, effective densities of states, and effective mass in silicon," *Journal of Applied Physics*, **67** (6), pp. 2944–2954, 1990. |

[Päs02] | R. Pässler, "Dispersion-related description of temperature dependencies of band gaps in semicondutors," *Physical Review B*, **6**, 085201, 2002. |

[McI10] | K.R. McIntosh and P.P. Altermatt, "A freeware 1D emitter model for silicon solar cells," *Proceedings of the 35th IEEE PVSC*, Honolulu, paper 531, pp. 2188–2193, 2010. |

[Sch98] | A. Schenk, "Finite-temperature full random-phase approximation model of band gap narrowing for silicon device simulation," *Journal of Applied Physics*, **84** (7), pp. 3684–3695, 1998. |

[Sen08] | SENTAURUS, User manual A-2008.09, Synopsys Inc. Mountain View, CA, www.synopsys.com/products/tcad/tcad.html, 2008. |

[Yan13] | D. Yan and A. Cuevas, "Empirical determination of the energy band gap narrowing in highly doped n+ silicon," *Journal of Applied Physics*, **114**, 044508, 2013. |

[Yan14] | D. Yan and A. Cuevas, "Empirical determination of the energy band gap narrowing in p+ silicon heavily doped with boron," *Journal of Applied Physics*, **116**, 194505, 2014. |

# ACKNOWLEDGEMENTS

We thank Pietro P. Altermatt (Univ. Hannover) for his assistance in deciphering [Sch98] and his many explanations about semiconductors over the years.

# FEEDBACK

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