1.4.1 Hess Model

The main breakthrough in the area of HCD modeling associated with the Hess concept was the introduction of two competing mechanisms for Si-H bond-breakage, namely the single- and multiple-carrier processes, see Figure 1.2 [99,32,23]. A single-particle process is due to the interaction of a high-energy solitary carrier with the bond. During this interaction energy is transfered to the bond followed by its dissociation. Due to the large disparity of the electron mass and the mass of the hydrogen nucleus, the most probable way to deliver such an energy is via excitation of one of the bonding electrons to an antibonding state. As a consequence, a repulsive force acting on the H atom is induced followed by the release of hydrogen. The desorption rate of this process is [32]

where I(E) is the flux of carriers with energies in the range of [E;E + dE], σ(E) energy-dependent Keldysh-like reaction cross section, P(E) the desorption probability, while the integration starts from the threshold energy E_th.

**Figure 1.2:** Two competing processes of Si-H bond-breakage: the single- and multiple-carrier mechanisms. The bond is interpreted as a truncated harmonic oscillator.

The first success of the theory was achieved when hydrogen/deuterium desorption induced by subsequent bombardment by several ("cold") carriers from the tip of a STM was investigated on hydrogen- and deuterium-passivated Si surfaces [139,98,140,141,142]. These experiments showed that the D-passivated surfaces are much more resistant with respect to electron bombardment compared to hydrogenated ones. In other words, substantially higher densities of scanning tunneling microscope (STM) currents are required to release the same amount of D atoms vs. H atoms. The difference in depassivation rates (Figure 1.3) may be more than two orders of magnitude at high voltages, which gave rise to the name "giant isotope effect". The similarities between the dangling bonds at surfaces and interfaces lead to the application of the theory to H-passivated Si/SiO₂ interfaces subjected to HC stress [143,144,145,88,146].

This giant isotope effect was explained by the concept of multivibrational mode excitation by linking to the excitation of the phonon modes to a cascade of subsequent bombardments by interfacial carriers. It is assumed that the Si-H bond can be considered as a truncated harmonic oscillator characterized by a system of levels in the corresponding quantum well and sketched Figure 1.2. The interface bombardment by carriers leads to the multivibrational mode excitation accompanied also by the phonon mode decay (corresponding rates are designated as P_u and P_d). The bond excitation is eventually terminated when the last bonded level N_l is reached. Being settled in this level the H ion can overcome the potential barrier separating the last level N_l and the transport state thereby breaking the bond and becoming mobile. The reciprocal process when the hydrogen ion is transferred from the transport to the bonded state is linked to the bond passivation. This scenario corresponds to the multiple-carrier mechanism.

**Figure 1.3:** Disparity between H and D desorption rates induced by electrons tunneling from the STM tip on the passivated Si surface (data from [140]).

To obtain an expression for the phonon excitation and decay rates (P_u, P_d), the formalism described in [99,32] is applied. The electron flux can induce either phonon absorption (that is, bond heating) or phonon emission (related to the multivibrational mode decay). These absorption and emission rates are

where j_d is the electron flux and σ_ab/σ_emi are phonon absorption/emission reaction cross sections, Since the reaction cross section is energy-dependent, the electron flux is to be expressed over energies employing the carrier DF. Summarizing all these considerations one obtains the following expressions for (P_u, P_d) [99]

where I(E) is the carrier impact frequency on the surface per unit area per unit energy, ℏω phonon energy and phonon occupation numbers entering the expressions as f_ph(E). Finally, using (1.3), one obtains the bond-breakage rate corresponding to the MP-process as

with E_B being the energy of the last bonded level in the quantum well (Figure 1.2) and the phonon reciprocal life-time ω_e; k_B and T_L are the Boltzmann constant and the lattice temperature, respectively. It is worth emphasizing that the particle flux differential I(E) entering formulae (1.1,1.3) requires knowledge of the carrier DF. Thus, one of the main conclusions of the works by the group of Hess is the idea that for a proper description of HCD, the carrier energy distribution function is required. Another important achievement of this concept is that the isotope effect is essentially explained because different vibrational properties of Si-H and Si-D lead to different parameters of the corresponding quantum wells (see Figure 1.2), that is, to different positions of the last level E_B and phonon life-time.

**Figure 1.4:** The total degradation dose (cumulative N_it) as a function of stress time: experiment vs. theory obtained for a 180nm device under worst-case stress conditions, i.e. V_gs= 0.4 V_ds. Inset: distribution of Si-H bond-breakage activation energy. The data are borrowed from [24].
$\includegraphics[height=0.43\textwidth]{chapter_introduction/figures/Hess_degradation_my.eps}$

Another characteristic feature of the Hess model is the assumption that the activation energy E_a for the Si-H bond-breakage rate is statistically distributed, see Figure 1.2. This assumption is supported by ab initio calculations using density functional theory [99,147]. As a consequence, the dispersion of E_a leads to different power-law slopes during degradation, see Figure 1.4 [24,40,23]. This is essential as simple first-order kinetics of Si-H bond-breakage with a single-valued activation energy lead to an exponential transition between the bonded and broken states within about a single decade in time. However, experimental observations demonstrate a double-power law of degradation

where τ₁/τ₂ are characteristic times and α₁/α₂ are two different sublinear time slopes (~1/2). This time evolution has been explained assuming that two different types of traps (realized with the probabilities p₁ and p₂) contribute to HCD. These traps are similarly distributed and can be fit by the derivative of the Fermi-Dirac function with different mean values E_am,1/E_am,2 and standard deviations σ_a,1/ σ_a,2 [148], see Figure 1.4, inset

Despite the significant progress due to the work of Hess et al., the interface traps are considered on a microscopic level and remain unconnected to the device level. For instance, the device life-time is estimated as the time when the concentration N_it reaches a certain level. Also, the degradation of such parameters as transconductance, linear drain current and so forth, is not really addressed. Furthermore, although the necessity for the evaluation of the carrier DF is acknowledged, this information has not been incorporated into the approach. As a result, the model operates with a static N_it, thereby not considering its distributed nature and that the details in the N_it distribution follows the features found in the DF.