As discussed in Sections 7.4–7.4.1 of [Leroy and Rancoita (2016)] (see also
[Leroy and Rancoita (2007)]), among the most important semiconductor
material-parameters for practical applications in electronic devices, we have
the (excess-) carrier lifetime (e.g. see [Schroder (1997)]), the equilibrium
majority-carrier concentration and the majority-carrier mobility. To a first
approximation^{1},
the rate at which the electrical properties of semiconductors degrade by
irradiation is often expressed in terms of a damage coefficient. For instance, the
minority carrier (recombination) lifetime (τ) is given by

| (1) |

where τ_{irr} and τ are the lifetimes after and before the irradiation with a fluence
Φ_{i} of particles, respectively; K_{τ,i} is the (recombination) lifetime damage
coefficient^{2},
which may depend on i) the type of substrate, ii) the dopant concentration, iii)
the level of compensation and iv) the type (“i”) and energy of irradiating
particles (e.g., see Section 5 in Chapter I of Part II of [Vavilov and Ukhin
(1977)], [Srour and McGarrity (1988)], Section 3.4 of [Srour, Long, Millward,
Fitzwilson and Chadsey (1984)] and references therein). Values of K_{τ,i} are
available in literature. For instance, for low-resistivity silicon experimental
results on the lifetime damage constants are presented in [Srour, Othmer
and Chiu (1975)] for 0.5, 1.0 and 2.5 MeV electrons with fluences up to
≃ 3 × 10^{15} e/cm^{2} and 10 MeV protons with fluences up to ≃ 1.2 × 10^{12} p/cm^{2}
(see also Section 3.4 of [Srour, Long, Millward, Fitzwilson and Chadsey
(1984)]).

To a first approximation (e.g., see discussion in Sections 7.4.1, 11.1.2,
11.1.4., 11.4.1.1 of [Leroy and Rancoita (2016)]), in absence of saturation
effects^{3},
(mostly) for low-resistivity silicon we can assume that the concentration of
recombination centers is proportional – although the proportionality constant may
be slightly varied, in particular, when the cascades of primary defects
induced by recoil nuclei are largely different – to the energy deposited
by non-ionizing energy-loss (NIEL) processes per unit volume E_{dis} and,
consequently, to the concentration of Frenkel-pairs (FP) introduced as
primary point-defects. Furthermore, as treated in Section 7.4.1 of [Leroy
and Rancoita (2016)], one can derive that Eq. (1) can be re-written as:

is almost – under the above mentioned assumptions – independent of the type
and energy of the incoming particle, but depends on i) the type of substrate, ii)
(slightly) the dopant concentration and iii) the level of compensation and, where,
v_{e} is the average speed of minority carriers, σ_{m} is the cross section for the
absorption of minority carriers by recombination centers and γ_{dis} expresses the
proportionality constant between the deep defect concentration and that of
Frenkel-pairs FP. Equations (2) indicates that an approximate NIEL scaling is
expected for the variation of the reciprocal of the minority-carrier lifetime in
low-resistivity silicon.

### References

[Leroy and Rancoita (2007)] C. Leroy and P.G. Rancoita (2007), Particle Interaction and Displacement Damage in Silicon Devices operated in Radiation Environments Reports on Progress in Physics 70, 493-625, doi:10.1088/0034-4885/70/4/R0; http://iopscience.iop.org/0034-4885/70/4/R01/.

[Leroy and Rancoita (2016)] C. Leroy and P.G. Rancoita (2016), Principles of Radiation Interaction in Matter and Detection - 4th Edition -, World Scientific. Singapore, ISBN-978-981-4603-18-8 (printed); ISBN.978-981-4603-19-5 (ebook); https://www.worldscientific.com/worldscibooks/10.1142/9167#t=aboutBook; it is also partially accessible via google books.

[Lint, Flanahan, Leadon, Naber and Rogers (1980)] V.A. van Lint, T.M. Flanahan, R.E. Leadon, J.A. Naber and V.C. Rogers (1980), Mechanisms of Radiation Effects in Electronic Materials (New York: Wiley Interscience).

[Schroder (1997)] D.K. Schroder (1997), IEEE Trans. on Electron Devices ED-44, 160.

[Srour, Long, Millward, Fitzwilson and Chadsey (1984)] J.R. Srour, D.M. Long, D.G. Millward, R.L. Fitzwilson and W.L. Chadsey (1984), Radiation Effects on and Dose Enhancement of Electronic Materials (Park Ridge: Noyes Publications).

[Srour and McGarrity (1988)] J.R. Srour and J.M. McGarrity (1988) Proc. of the IEEE, Vol. 76 (Issue 11), 1443.

[Srour, Othmer and Chiu (1975)] J.R. Srour, S. Othmer and K.Y. Chiu (1975), IEEE Trans. on Nucl. Sci. 22 (no. 6), 2656.

[Vavilov and Ukhin (1977)] V.S. Vavilov and N.A. Ukhin (1977), Radiation Effects in Semiconductors and Semiconductor Devices (New York: Consultants Bureau, a division of Plenum Publishing Press).

^{1}The linear dependence on the particle fluence [see Eq. (1)] is expected as long as the
steady-state Fermi level is not moved significantly.

^{2}A coefficient that is the reciprocal of that given in Eq. (1) is also found in literature
(e.g., see Section 7.2.1 of [Lint, Flanahan, Leadon, Naber and Rogers (1980)] and references
therein).

^{3}For high-resistivity silicon (see discussion in Section 7.4.5 of [Leroy and Rancoita
(2016)]), it was found that there are secondary defects whose concentrations are not linearly
dependent on fluence.