Generalized Messenger-Spratt Equation for Gain Degradation of Bipolar Transistors

As discussed in [Consolandi et al. (2006)] (see also Sections 7.4.1, 11.1–11.1.4 of [Leroy and Rancoita (2016)] and [Leroy and Rancoita (2007)]) defects introduce trapping energy states which degrade the minority carrier lifetime. As well known, the operation of bipolar transistors results from the physical and electrical behavior of majority and minority carriers in the junctions which constitute the transistor. In fact, minority carriers are injected into the base at the emitter-base junction, are transported to the collector and, finally, are retrieved as majority carriers across the base-collector junction. For instance, the effect of radiation on bipolar devices is the reduction of the common emitter current gain, which can be described in terms of the reduction of the minority carrier lifetime in the transistor base.

The relationship between the variation of the inverse of the gain and the fast neutron fluence is provided by means of the Messenger-Spratt equation ([Messenger (1966), Messenger (1972), Messenger (1992)] and references therein):

, (1)

where β and β_irr are the common emitter current gain before and after the irradiation, ω_T = 2πf_T , f_T (the so called- cut-off frequency) is the frequency at which the common emitter current gain is unity (e.g., see Section 3.10 in [Messenger and Ash (1982)], pages 67–69 of [Grebene (1984)] and [Messenger (1973)]), K is the relevant damage constant which depends also on the base resistivity (see for instance [Srour (1973)]) and Φ_n is the fast neutron fluence in n/cm²:

(2)

where E_min value is usually taken at about 10 keV. At large particle fluences where the damage is dominated by the atomic displacements, Δ depends linearly on the fluence also for particles other than neutrons (see Refs. [Colder et al. (2001), Colder et al. (2002), Consolandi et al. (2006)] and, for instance, Figure 5.3 at page 215 of Ref. [Holmes-Siedle and Adams (2002) and Figure 42 at page 124 of Ref. [Vavilov and Ukhin (1977)]). As discussed in [Consolandi et al. (2006), Leroy and Rancoita (2007)], Sections 7.4.1, 11.1–11.1.4 of [Leroy and Rancoita (2016)], in absence of saturation effects we can assume that the number of recombination centers produced in silicon is proportional to the energy released into atomic-displacement processes (i.e., on the energy deposited by NIEL processes). Therefore, we have that the concentration of recombination centers is proportional to the concentration of Frenkel-pairs created trough displacement processes and, as a consequence that, Eq. (1) can be generalized as (see [Consolandi et al. (2006), Leroy and Rancoita (2007)], Sections 7.4.1, 11.1–11.1.4 of [Leroy and Rancoita (2016)] and references therein):

(3)

where λ is almost independent of the incoming particle (although it may exhibit a slight dependence on the size of the generated point-defects cascade) but depends on the type and on the doping profile and compensation-level of the base.

Equation (3) is the expression for the common-emitter current gain degradation in bipolar transistors and generalizes the Messenger-Spratt equation (e.g., the so-called generalized Messenger-Spratt equation), originally derived for fast-neutron environment. It predicts a linear dependence of the inverse of the gain variation on the concentration of Frenkel pairs (FP) generated by incoming particles, i.e., it predicts an approximate NIEL scaling.

References