Abstract
This chapter illustrates the main contributions to the transitions of the inter-band type, that give rise to the generation-recombination terms in the continuity equations for electrons and holes, and to those of the intra-band type, that give rise to the electron and hole mobilities in the current-density equations. The inter-band transitions that are considered are the net thermal recombinations (of the direct and trap-assisted type), Auger recombinations, impact-ionization generations, and net-optical recombinations. The model for each type of event is first given as a closed-form function of the semiconductor-device model’s unknowns, like carrier concentrations, electric field, or current densities. Such functions contain a number of coefficients, whose derivation is successively worked out in the complements by means of a microscopic analysis. The case of semiconductors having a distribution of traps within the gap, like, e.g., polycrystalline silicon, is treated as well. Some discussion is devoted to the optical-generation and recombination events to show how the concepts of semiconductor laser, solar cell, and optical sensor may be derived as particular cases of nonequilibrium interaction between the material and an electromagnetic field. The intra-band transitions are treated in a similar manner: two examples, the collisions with acoustic phonons and ionized impurities, are worked out in some detail; the illustration then follows of how the contributions from different scattering mechanisms are combined together in the macroscopic mobility models. The material is supplemented with a brief discussion about advanced modeling methods.
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- 1.
A more detailed example of calculations is given below, with reference to collisions with ionized impurities.
- 2.
- 3.
The result expressed by (20.16) is intuitive if one thinks that adding up all continuity equations amounts to counting all transitions twice, the first time in the forward direction (e.g., using the electrons), the second time in the backward direction (using the holes). The reasoning is similar to that leading to the vanishing of the intra-band contribution in (19.63).
- 4.
In a polycrystalline semiconductor with a large spatial concentration of traps it may happen that the traps’ current densities are not negligible; in fact, the whole system of equations (20.13) and (20.14) must be used to correctly model the material [26–28]. The conduction phenomenon associated with these current densities is called gap conduction.
- 5.
It is [α n, p ] = m3 s−1, [e n, p ] = s−1.
- 6.
This simplification is not applicable in a polycrystalline or amorphous semiconductor.
- 7.
Considering for instance the example in Sect. 18.4.1, one has n eq ≃ 1015 cm−3, p eq ≃ 105 cm−3, whence n eq∕p eq ≃ 1010.
- 8.
The units are [c n, p ] = cm6 s−1 and [I n, p ] = s−1.
- 9.
In fact, Auger recombination becomes significant in the source and drain regions of MOSFETs and in the emitter regions of BJTs, where the dopant concentration is the highest.
- 10.
In principle, high-energy electrons or hole exists also in the equilibrium condition; however, their number is negligible because of the exponentially vanishing tail of the Fermi-Dirac statistics.
- 11.
The high-field conditions able to produce a significant impact ionization typically occur in the reverse-biased p-n junctions like, e.g., the drain junction in MOSFETs and the collector junction in BJTs.
- 12.
As indicated in Sect. 17.6.6, among semiconductors this is typical of the direct-gap ones.
- 13.
In fact, LASER is the acronym of Light Amplification by Stimulated Emission of Radiation.
- 14.
In principle, a time dependence of the intensity is incompatible with the hypothesis that the radiation is monochromatic. However, the frequency with which the intensity may vary is extremely small with respect to the optical frequencies.
- 15.
It is implied that h ν ≥ E C − E V , and that two-particle collisions only are to be considered.
- 16.
As mentioned in Sect. 19.6.5, it is assumed that the different types of collisions are uncorrelated.
- 17.
The first-principle derivation of the scattering probabilities is carried out by applying Fermi’s Golden Rule (Sect. 14.8.3) to each type of perturbation, using the Bloch functions for the unperturbed states [73]. Examples are given in Sect. 14.8.6 for the case of the harmonic perturbation in a periodic structure, and in this chapter (Sect. 20.5.2) for the case of ionized-impurity scattering.
- 18.
- 19.
An example of derivation of the screening length is given in Sect. 20.6.4.
- 20.
An example of this procedure is given in Sect. 14.8.6 with reference to the case where the spatial part of the perturbation has the form of a plane wave.
- 21.
- 22.
The units of C n0 are [C n0 = J s−1 m−3].
- 23.
As shown by (A.118), the property ∂ϱ∕∂φ < 0 holds true also in the degenerate case.
- 24.
The electron charge is indicated here with e to avoid confusion with q c .
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Rudan, M. (2018). Generation-Recombination and Mobility. In: Physics of Semiconductor Devices. Springer, Cham. https://doi.org/10.1007/978-3-319-63154-7_20
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