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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 \(\tau _1^\mathrm {total}\) is governed by a two–phonon process due to the energy forbidden one–phonon dissipations1.

Mode T [K] \(\tau _1^\mathrm {total}\) \(\Gamma _{1,0}^{(1)}\) \(\tau _1^{(1)}\) \(\Gamma _{1,0}^{(2\mathrm {a})}\) \(\Gamma _{1,0}^{(2\mathrm {b})}\) \(\tau _1^{(2)}\)
0 0.407 0.149 6.675 2.304 0.0015 0.433
Si–H\(_\mathrm {bend}^{1\rightarrow 0}\) 300 (\(\downarrow \)) 0.202 0.229 5.405 4.751 0.003 0.210
300 (\(\uparrow \)) 0.0451 0.0019 0.0001
Table D.1: Calculated vibrational lifetimes (\(\tau _1^\mathrm {total}\), \(\tau _1^{(1)}\) and \(\tau _1^{(2)}\)) together with the corresponding rates (\(\Gamma _{1,0}^{(1)}\), \(\Gamma _{1,0}^{(2\mathrm {a})}\) and \(\Gamma _{1,0}^{(2\mathrm {b})}\)) 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, \(\Gamma _{1,0}^{(1)}\) shows a strong \(\gamma \) dependence (\(\tau _1^{(1)}=\SI {34.4}{ps}-\SI {3.24}{ps}\)), whereas \(\Gamma _{1,0}^{(2)}\) seems to be rather insensitive (\(\tau _1^{(2)}=\SI {0.435}{ps}-\SI {0.428}{ps}\)) due to changes between \(1\) and \(10~\mathrm {cm}^{-1}\).

Two–phonon processes have been examined as well in detail. Fig. D.1 (right panel) shows the contributions of phonon pairs \(\{\omega _k,\omega _l\}\) to the total rate. For the Si–surface system the energy is transferred along the diagonal \(\hbar \omega _k+\hbar \omega _l=\Delta E_{1,0}\), as to be expected, with the biggest contribution comprising one low– and one high–energy phonon in the range of \(\SI {21}{}-\SI {26}{meV}\) and \(\SI {52}{}-\SI {57}{meV}\). Furthermore, also ratios of \(\omega _k/\omega _l\sim 1/6,1/1.5\) yield non–negligible contributions to the total rate \(\Gamma _{1,0}^{(2)}\), whereas \(\omega _k/\omega _l\sim 1/1\) (\(\Gamma _{1,0}^{(2\mathrm {b})}\)) 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.


Figure D.1: Left: The phonon mode spectrum of a fully reconstructed Si(100) 2\(\times \)1 surface passivated with H atoms. The normal–modes associated with the Si–H bending modes are highlighted. Right: The contributions of phonon pairs \(\{\omega _k,\omega _l\}\) to the transition rate \(\Gamma _{1,0}^{(2)\downarrow }\) using a 2D histogram with a bin size of \(\SI {5}{\per \cm }\times \SI {5}{\per \cm }\).

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.