BAND GAP CALCULATOR
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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.
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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.
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 Eg. The decrease in Eg due to the dopant concentration (and carrier concentrations) is called band gap narrowing and it is quantified by ∆Eg.
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: ∆Eg = 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 ∆Eg,app = ∆Eslope∙ln(N/Nonset) for N > Nonset, and ∆Eg,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 ∆Eslope and Nonset 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', ∆Eg,app, and not the actual BGN, ∆Eg. 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 ∆Eoffset. It states that ∆Eg = ∆Eslope∙[ln(N/Nonset)]b + ∆Eoffset. 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 ∆Eg). Note that this equation is identical to del Alamo's when b = 1 and ∆Eoffset = 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.
We thank Pietro P. Altermatt (Univ. Hannover) for his assistance in deciphering [Sch98] and his many explanations about semiconductors over the years.
Please email corrections, comments or suggestions to firstname.lastname@example.org.
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Welcome to the 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.
New in this version:
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