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The Physics of Non–Equilibrium Reliability Phenomena

3.4 Resonances & Energies

The proposed multiple vibrational mechanism, see Section 2.1, where an incident carrier scatters into an available resonance state and upon inelastic electronic relaxation triggers the transition between vibrational states in the ground state potential, requires knowledge about the associated excited states. Unfortunately, excited state properties and dynamics are rather difficult to calculate and hardly accessible within standard DFT. A popular method of approximating the anionic or cationic PECs is to apply Koopmans theorem. Within this approach the negative of the energies of the highest occupied and lowest unoccupied molecular orbitals (HOMOs and LUMOs) is related to the ionization potentials and electron affinities. The respective orbital energies can, therefore, be used to approximate the excited PECs. The utilized cluster model here consists of 10 Si atoms and 23 H atoms, where 22 are needed to passivate the Si atoms and one represents the Si–H bond of interest. Assuming the same reaction coordinate (RC) as for the bond breakage trajectory in the neutral ground state, 150 single point calculations along the path have been carried out. The resulting 10 highest occupied and lowest unoccupied \(\alpha \) and \(\beta \)–spin states were used to construct adiabatic resonance PECs.


Figure 3.15: Potential energy curves along the reaction coordinate for the neutral bond breakage pathway, see Sec. 3.2 and [MJJ3]. Two approximations were employed to construct the potential profile for the excited complexes: Koopmans theorem using the 10 lowest (un–) oc- cupied \(\alpha \) and \(\beta \)–spin orbitals, and the method of constrained density functional theory (CDFT) to directly assess the diabatic states. Left: The results for the excited anionic complex \(V^-(q)\) together with the ground state potential \(V(q)\). Right: The constructed adiabatic PECs as well as the diabatic PEC for the positively charged Si–H bond.

The constructed PECs for the negatively and positively charged complex show a similar behaviour, see Fig. 3.15. The approximated resonance energy (at \(q=\SI {0}{a_0\sqrt {m}}\)) of the anionic state is between \(\SI {2.7}{eV}\) and \(\SI {4.2}{eV}\), whereas the cationic state is about \(\SI {0.5}{eV}\) higher, which is in good agreement with previous findings [106, 114, 115, 236, 237]. Furthermore, all excited energy profiles share that their minimum is shifted with respect to the ground state potential \(V(q)\) as well as a lower transition barrier. While for the anionic PECs the barriers are lowered by \(\sim \SI {0.9}{}-\SI {1.2}{eV}\), the cationic charged states possess activation energies of around \(\SI {2.0}{eV}\). On the other hand, the left well minima of the constructed \(V^-(q)\) are shifted by \(\Delta q=\SI {0.74}{}-\SI {0.96}{a_0\sqrt {m}}\) towards the barrier, whereas the positive PECs are only shifted by \(\sim \SI {0.45}{a_0\sqrt {m}}\). However, also the transition state along the trajectory seems to change for \(V^+(q)\).


Figure 3.16: Details of the CDFT calculations. Left and Middle: The difference of charge density \(\Delta \rho =\rho _\mathrm {DFT}-\rho _\mathrm {CDFT}\) for the negatively (left) and positively (middle) charged Si–H complex. Blue (red) indicates a higher (lower) electron density. One can see that for both states the additional charge is mainly localized on the Si atom. Right: The effect of structural relaxations for the charged Si–H bond. For the negative charge state the Si–H distance as well as the Si–Si–H angle increase, forcing the H in the direction of the bond breaking trajectory. The positive charge state does not show such a pronounced reconfiguration where only the central Si atom slightly moves out of its plane.

Nevertheless, these calculations only provide a qualitative understanding of the excited potential curves. A more rigorous approach is given by the method of CDFT [238, 239] which allows one to directly construct diabatic potential energy curves. CDFT as implemented in the CP2k package [240, 241] has been used where an additional charge (negative and positive) has been restricted to be localized on the Si–H bond. Single point calculations along the neutral trajectory with fixed ionic positions have been performed. The effect of lattice relaxations was explicitly neglected due to the short resonance lifetime, which is on the order of a few femtoseconds [114, 115, 236]. Furthermore, the calculations were restricted up to the transition state to ensure that the charge can indeed be localized on the Si–H complex. Investigations using the method of Bader charge analysis show that the Si–H bond is indeed negatively (positively) charged (\(\Delta (q_\mathrm {SiH}-q_\mathrm {SiH^-})=-0.93\,e\), \(\Delta (q_\mathrm {SiH}-q_\mathrm {SiH^+})=0.86\,e\)) and the additional charge is mainly localized around the Si atom, see Fig. 3.16.

The CDFT results for both charge states are shown in Fig. 3.15 (symbols). The diabatic PECs are in good agreement with the results obtained from Koopmans theorem and again yield shifted minima configurations as well as altered transition state. Furthermore, at the equilibrium position of the neutral state \(V(q_\mathrm {eq})\) the charge–localized states are \(\SI {3.64}{eV}\) (negative) and \(\SI {3.96}{eV}\) (positive), respectively, higher in energy. In order to qualitatively understand the new charged equilibrium configurations, full geometry optimizations within the CDFT simulations have been performed, see Fig. 3.16. Interestingly, for the negatively charged Si–H complex the bonding distance changes from \(\SI {1.48}{\angstrom }\) (\(V(q_\mathrm {equ})\)) to \(\SI {1.75}{\angstrom }\) (\(V^-(q_\mathrm {equ})\)) and the Si–Si–H angle opens from \(\SI {109}{\degree }\) to \(\SI {141}{\degree }\). Such a structural reconfiguration forces the H atom in the direction of the dissociation path, which explains the shifted minimum of \(V^-(q)\)10. The positive charge state does not undergo such pronounced relaxations. The reconfigurations are mainly determined by the Si atom slightly moving out of its plane and thereby stretching the Si–Si bonds in the vicinity. This, however, suggests a potentially different bond breakage trajectory in the positive charge state.

10 Qualitatively, this result can be compared to intrinsic electron traps in SiO\(_2\), where wide O–Si–O bond angles serve as efficient electron trapping sites [242].