In heavily doped materials the swollen orbitals associated with the impurity atoms begin to overlap and as a result the discrete energy level associated with the impurities broadens to form a band of finite width. Potential fluctuations due to the random distribution of the impurities also lead to a broadening of the impurity band. The ionization level of the impurities is consequently reduced. Eventually the impurity band overlaps the conduction band or valence band, effectively narrowing the band gap. In this situation the density of states no longer have the same form as that derived in Equation 3-21 or Equation 3-22. Nonetheless it is common to model band gap narrowing using these equations, but assuming a band gap that varies as a function of doping level.

The Semiconductor interface allows for models of band gap narrowing to be defined using an arbitrary expression. To add band gap narrowing to a material, select User defined in the Band Gap Narrowing section of the Semiconductor Material Model. Then enter user-defined expressions for ΔEg (the amount of band gap narrowing) and for α (the fraction of the band gap narrowing taken up by the conduction band). For convenience the energy level is entered in units of volts and is converted to an energy behind the scenes by multiplying by the electron charge, q. When band gap narrowing is active the following equations apply:

where Eg is the band gap with band gap narrowing, Eg0 is the material band gap, Ec is the conduction band edge with band gap narrowing, Eci is the conduction band edge in the absence of band gap narrowing, Ev is the valence band edge with band gap narrowing, and Evi is the valence band edge in the absence of band gap narrowing.

The Slotboom model is frequently used to model band gap narrowing in silicon. It is an empirical model that calculates the narrowing as a function of the total doping concentration (Ref. 27). This empirical model combines all of the physical effects (random potential fluctuations, electron-electron, carrier-impurity, and electron-hole interactions) into one energy narrowing, and consequently the calculated narrowing is the same in neutral and depleted parts of the device. The Slotboom model computes the band gap narrowing according to the equation:

where NI = Nd + Na and the other parameters are material properties (Eref has the same units as the band gap itself and Nref has SI units of 1/m3). For silicon the Semiconductor Module material library uses the updated material properties due to Klassen and others (Ref. 28) rather than the original properties given in Ref. 27. Note that the fraction of the band gap narrowing taken up by the conduction band is also treated as a material property (0.5 for silicon).

The model developed by Jain and Roulston (Ref. 29) is a physics-based model in which the only empirical parameter is the fraction of the band gap narrowing taken up by the conduction band (once again this is defined as a material property, and a default of 0.5 is usually employed). The model can be rewritten in a simple form with three coefficients. Coefficients for the model are available for a wide range of III-V materials, as well as for silicon and germanium.

where An, Ap, Bn, Bp, Cn, and Cp are material properties (with the same units as the band gap), and Nref is a reference doping level (with SI units of 1/m3). The fraction of the band gap narrowing taken up by the conduction band is also treated as a material property (0.5 for silicon). Material properties for a number of materials are available in the material library, including the original materials described in Ref. 29 and several materials whose properties were obtained from Ref. 30.