1.4.4 The Energy-Driven Paradigm of Rauch and LaRosa

A new paradigm, or underlying concept, of nMOSFET hot-carrier behavior is proposed in works of Rauch and LaRosa [8,17]. It is believed that the fundamental "driving force" is the available energy, rather than the peak lateral electric field, as in the lucky electron model. Two main issues associated with the approach of Rauch and LaRosa are:

Electron-electron scattering is of special interest in the case of ultra-scaled MOSFETs because in these devices the supply voltage is rather low and therefore the single-carrier mechanisms of Si-H bond-breakage were expected to be suppressed. This energy exchange mechanism, however, populates the "hot" fraction of the DF and modifies the shape of the DF, that is, results in a pronounced hump in the carrier distribution function, see Figure 1.8. Thus, the high-energy tail of the DF can expand deeper into energy than expected from the supply voltage. As a result, the contribution from the SP-mechanism is increased. Additionally, only electron-electron scattering defines the acceleration of HCD at elevated temperatures, which is pronounced in the case of extremely-scaled MOSFETs [74,75,76,41].

**Figure 1.8:** The impact of electron-electron scattering on the shape of the carrier energy distribution function. In the former case an additional hump in the distribution function high-energy tail appears. Data from [29].

The energy-driven paradigm presented by Rauch and LaRosa claims that beyond the 180nm node the driving force of HCD is the energy deposited by carriers, not the maximal electric field in the channel as it was in the "lucky electron model" [5]. Both the impact ionization rate as well as the rate of hot-carrier induced interface state generation is controlled by integrals of the form ∫f(E)S(E)dE, where f(E) is the carrier DF and S(E) the reaction cross section; compare this to the formula (1.1) used previously which has the same structure. The DF is a strongly decaying function of energy while S(E)grows as a power-law. Hence, this trade-off results in a maximum of the rate pronounced at a certain energy (Figure 1.9) determined according to the criterion $\mathrm{d}(\ln f)/\mathrm{d}E=-\mathrm{d}(\ln S)/\mathrm{d}E$ . This energy E_knee is called "knee" energy and is a weak function of the applied bias V_ds. Therefore, if the maximum of the product f(E)S(E) is sufficiently narrow, it can be approximated by a delta-function and instead of integration in the entire energy range one can only calculate the value of the integrand for this energy. To conclude, the main message of the energy-driven paradigm is that one may avoid time-consuming calculations of the carrier DF substituting it by the empirical parameter. This parameter is proportional to the reaction rate calculated for E=E_knee which is defined by the bias conditions. This dependence will be discussed in the next subsection devoted to the Bravaix model.

**Figure 1.9:** Schematic representation of the energy-driven paradigm. Knee energies shift depending on the applied voltage (the data borrowed from [29]).

Although this paradigm substantially simplifies the treatment of HCD, it suffers from some shortcomings. Indeed, one can see in Figure 1.9 that the maximum of the integrand f(E)S(E) is not necessarily narrow and in the particular case shown by the authors [29] has a width of 1.5-2.0eV. Therefore, the concept of a dominant energy sounds doubtful. Furthermore, such a treatment of HCD does not deal with N_it as a distributed quantity and thus one of the main features of HCD - its strong localization - is not captured. Finally, as it was in the Hess approach, the device life-time is estimated by the interface state generation rate. However, it would be more reasonable to define it as the time when the degradation of V_th or I_dlin, etc., has reached a critical value.