The Physics of Non–Equilibrium Reliability Phenomena

Chapter D Si-H Bending Dynamics

In order to validate the methodology used to calculate the vibrational relaxation time as well the applicability of the ReaxFF force–field, the following section presents the results for a Si–H bending mode. The Si–H bending motion is a well studied system in the literature and offers comparability to recent theoretical studies [270, 273, 274]. The investigated atomistic model here is a H passivated reconstructed Si(100)–2 $\times$ 1 surface with 500 atoms. A normal–mode analysis has been carried out and one of the 10 normal modes which could be classified as a Si–H bending mode was selected for the study, see Fig. D.1 (left panel).

The theoretical concepts and calculation details are described in detail in Sec. 4.1.3. The results are summarized in Table. D.1. Again, the total lifetime $τ_{1}^{total}$ is governed by a two–phonon process due to the energy forbidden one–phonon dissipations¹.

Mode	T [K]	$τ_{1}^{total}$	$Γ_{1, 0}^{(1)}$	$τ_{1}^{(1)}$	$Γ_{1, 0}^{(2 a)}$	$Γ_{1, 0}^{(2 b)}$	$τ_{1}^{(2)}$
	0	0.407	0.149	6.675	2.304	0.0015	0.433
Si–H $_{bend}^{1 \to 0}$	300 ( $↓$ )	0.202	0.229	5.405	4.751	0.003	0.210
	300 ( $↑$ )		0.0451		0.0019	0.0001

Table D.1: Calculated vibrational lifetimes (

τ_{1}^{total}

τ_{1}^{(1)}

and

τ_{1}^{(2)}

) together with the corresponding rates (

Γ_{1, 0}^{(1)}

Γ_{1, 0}^{(2 a)}

and

Γ_{1, 0}^{(2 b)}

) for the first excited system–mode and two different temperatures. The units for the lifetimes and rates are ps and ps

^{- 1}

, respectively.

Analogous to the Si–H bond breaking mode in Sec. 4.1.3, $Γ_{1, 0}^{(1)}$ shows a strong $γ$ dependence ( $τ_{1}^{(1)} = 34.4 p s - 3.24 p s$ ), whereas $Γ_{1, 0}^{(2)}$ seems to be rather insensitive ( $τ_{1}^{(2)} = 0.435 p s - 0.428 p s$ ) due to changes between $1$ and $10 {cm}^{- 1}$ .

Two–phonon processes have been examined as well in detail. Fig. D.1 (right panel) shows the contributions of phonon pairs ${ω_{k}, ω_{l}}$ to the total rate. For the Si–surface system the energy is transferred along the diagonal $ℏ ω_{k} + ℏ ω_{l} = Δ E_{1, 0}$ , as to be expected, with the biggest contribution comprising one low– and one high–energy phonon in the range of $21 - 26 m e V$ and $52 - 57 m e V$ . Furthermore, also ratios of $ω_{k} / ω_{l} \sim 1 / 6, 1 / 1.5$ yield non–negligible contributions to the total rate $Γ_{1, 0}^{(2)}$ , whereas $ω_{k} / ω_{l} \sim 1 / 1$ ( $Γ_{1, 0}^{(2 b)}$ ) only plays a minor role.

In summary, the results are compatible with previously published results [270, 273, 274] and support the concept as well as the results presented in Sec. 4.1.3.

¹ Note that the collective Si–H bending modes on the surface have not been considered in the calculations. Only normal–modes associated with the Si lattice contribute to the dissipation to avoid an artificially large coupling.