- 3.3.3.1 Dopant-Dependent Bandgap Narrowing
- 3.3.3.2 Physical background of the new model
- 3.3.3.3 Extending the new model to semiconductor alloys

3.3.3 Bandgap Narrowing

(3.73) |

gives the part of the total BGN which is contributed to the conduction band. The default parameter values are summarized in the following table.

Using the physically-based approach from [93], a new analytical BGN model was developed. It is applicable to compound semiconductors [134] and considers the semiconductor material and the dopant species for arbitrary finite temperatures. In this section a comparison with experimental data and other existing models is presented and study of BGN in III-V group semiconductors is performed.

Here
and denote the screened and unscreened
*Coulomb* potentials of the impurity, respectively. (3.76) represents the
change in the electrostatic energy of the impurity before and after the
electron gas redistribution. If the potential of a point-like impurity is
assumed, the Fourier transform of the scattering potential is expressed by
(3.78), where and are the atomic number and the number of
electrons of a given material, denotes the inverse *Thomas-Fermi* length,
and
is the *Fermi* integral of order [139]. This
approach leads to a simple BGN model [93] given by (3.80).

Removing the point-charge approximation yields a refined model. The charge density of the impurity can be accounted for by an atomic form factor . Following the work [140] the impurity potential takes the form

Solving (3.77) using (3.81) and (3.82), and then replacing V(r) in (3.76), leads to the final expression for the energy shift

(3.83) |

The subscripts and refer to a semiconductor and impurity, respectively. and are the atomic number and the number of electrons of a given material. can be interpreted as size parameters of the electron charge density and is the Bohr radius. They are expressed as

(3.84) | |||

(3.85) |

The size parameter uses , which is the most pessimistic estimation, since it is still not clarified which value for in the range between 1 and is valid at microscopic level. Even though the influence of the dopant type is reduced to minimum this way, our model still delivers appropriate results at 300 K in agreement with experiment [141] (see Fig. 3.22).

The temperature dependence of the BGN in Si is shown in Fig. 3.23. Neglecting of the stronger BGN at low temperatures, especially for doping levels of about cm, may result in an error of about 50%. Thereby, even larger errors might be introduced into the simulation results, with respect to the electrical device characteristics. In the case of III-V semiconductors our model delivers a comparatively weaker BGN temperature dependence (see Fig. 3.24). Similar observations were reported in the case of p-GaAs in [142], [143]. Thus, according to our knowledge, our BGN model is the first theoretically derived model predicting different shifts for various dopant species and taking temperature into account.

In the case of p-type GaAs good agreement with experimental data
[142], [143] is obtained. The few available experimental data for n-type GaAs
suggest sometimes lower [147] (open triangles in Fig. 3.27) values for BGN
and more often higher [148] (filled triangles) than our model
delivers. This confirms the importance of modeling BGN in III-V semiconductors,
instead of leaving this effect unconsidered, which is the case with most device
simulators.

Experiments showed higher BGN in n-InP than in n-GaAs [149]. Higher conduction band density of states and lower relative permittivity explain the expected higher values for BGN in AlAs and GaP (Fig. 3.28) than in InP, GaAs, and InAs. The parameter values are taken from [108]. The model is physics-based and contains no free parameters except the ratio used in (3.74).

2001-02-28