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The Physics of Non–Equilibrium Reliability Phenomena

3.6 Hydrogen Migration & Passivation Dynamics

Building on top of the results of Sec. 3.2, that the Si–H bond breakage dynamics are determined by hydrogen moving into the next but one BC configuration, further migration pathways as well as the passivation kinetics can be investigated. In this context two important questions arise: What happens with the hydrogen after being released into the BC site and how exactly can Si–DBs be passivated again? As was shown in Sec. 3.2, the backward barrier for the H being in the BC\(_{2,3}\) position to re–passivate the Si–DB is around \(\SI {1.30}{eV}\). Assuming a thermally activated reaction following an Arrhenius law11 would, however, be incompatible with the conclusions by Brower [222] and Stesmans [221, 243245]. Rather, these studies suggest that the passivation of \(P_\mathrm {b}\) centers is inevitably linked to the presence and cracking of molecular H\(_2\). Although dedicated experiments investigated the effects of atomic H onto the passivation and depassivation kinetics of \(P_\mathrm {b}\) centers [246, 247], experimental studies conducted on a device level confirmed the prevailing opinion of H\(_2\) passivation [95, 248250].

11 assuming an attempt frequency of \(\nu _0\sim \SI {1e12}{}-\SI {1e13}{s^{-1}}\)

3.6.1 Migration Trajectories

First, the hydrogen in a Si–Si BC configuration at the interface and possible migration trajectories will be discussed. Well–tempered metadynamics (WTMD) in conjunction with ReaxFF has again been utilized to sample the free energy surface of interfacial H atoms. Five H atoms have been randomly inserted between Si–Si bonds at the interface layer. Subsequently, the simulations were run in parallel with a total number of \(\SI {1e9}{}\) timesteps, corresponding to a simulation time of \(\SI {0.5}{\mu s}\). During these simulations only the \(x\) and \(y\) coordinates of the respective hydrogen atoms were biased with no additional restrictions regarding their spatial position, i.e. the H atoms were allowed to move freely within the structure. The final result is summarized in Fig. 3.18.


Figure 3.18: Simulation results for hydrogen migration along the Si/SiO\(_2\) interface using the classical force field ReaxFF and the methods of metadynamics. Left: Top and side view of the (fully equilibrated) interfacial region in which the hydrogen remained during the simulations. Additionally, two individual migration pathways for a hydrogen atom extracted from dedicated simulations using the methods of steered MD are also shown. Right: The free energy surface extracted from the WTMD simulations. The yellow crosses mark the equilibrium positions of the interfacial Si atoms (see the left panels) and indicate that the various minima are Si BC sites. The highlighted MEPs are directly extracted from the potential landscape. Note that regions of higher energy are not properly sampled and explored within the simulations and therefore left white within the map.

Although the motion of the hydrogens was not limited in any way, they remained within the interfacial Si region and did not cross into the SiO\(_2\) side, see Fig. 3.18. Rather, the analysis suggests that once the H was released into a BC site, it can further diffuse along the interface by hopping to the next BC. The positions of the potential minima together with extracted minimum energy paths (MEPs) (right panel of Fig. 3.18) strongly indicate that the released hydrogen is mobile within the subinterfacial Si side (see upper left panel of Fig. 3.18), possessing a hopping barrier smaller than \(\SI {1}{eV}\). Furthermore, such an observation implies that the potential barrier into the oxide region is larger than the lateral hopping barrier for hydrogen along the interface. In order to be able to also quantify the barrier across the interface, additional simulations using Lammps and ReaxFF have been performed. An \(xz\) slice with \(\SI {5}{\angstrom }\) thickness has been chosen to sample the energy landscape, see Fig. 3.19. Placing the hydrogen at 20000 positions within this slice yields the potential map shown in the right panel of Fig. 3.19. One can clearly see the highlighted red isoline for \(E=\SI {1.8}{eV}\) at the Si/SiO\(_2\) interfaces, representing a barrier almost twice as large compared to the BC site hopping along the transition region.


Figure 3.19: Simulation results for hydrogen motion across the Si/SiO\(_2\) interface structure using the classical force field ReaxFF. Left: Front and side view of the (fully equilibrated) \(xz\) slices used for the sampling of the energy landscape. Furthermore, several individual hydrogen migration trajectories are shown which have been simulated using the method of steered MD within the full atomistic model. Right: The energy landscape extracted from sampling the reduced configuration space, i.e. the \(xz\) slice. One can clearly recognize the atomistic structure (see left panels) as well as potentially advantageous migration paths possessing low energy barriers. Furthermore, note the highlighted red isoline for \(E=\SI {1.8}{eV}\) at the interfacial transition regions for crossing into the SiO\(_2\) side.

However, due to the large configuration space for the above simulations even an excessive amount of spatial points or simulations steps does not guarantee a converged sampling which potentially leads to an overestimation of the involved transition barriers12. To further access the distribution of barriers, subsequently individual migration paths have been simulated. A technique particularly suited for the problem at hand is the so–called moving restraint bias, or steered MD method, implemented in Plumed [220]. It allows one to drag the system from an initial to a final state by adding a time–dependent, harmonic restraint on the CV(s). The simulations start from the equilibrated initial configuration and can be divided into three phases. First, the force constant for the restraint is slowly increased over 50000 timesteps to lock the system in its initial state without stressing it. Afterwards, the system is smoothly moved towards the final configuration using \(\SI {1e6}{}\) timesteps to allow the system to equilibrate and follow a relaxed trajectory. Finally, the force constant is released again to end up in a equilibrated and unconstrained final configuration. Various migration trajectories along and across the Si/SiO\(_2\) interface with different, and specifically chosen, initial and final configurations have been simulated, see the left panels of Fig. 3.18 and 3.19. All results and potential profiles are summarized in Fig. 3.20. Note that within these simulations the change of the systems’ free energy is connected to the work performed over time. However, the large noise and error bar associated with this quantity would require large statistics for a quantitative analysis. Nevertheless, the results shown in Fig. 3.20 allow for a qualitative understanding of the underlying mechanisms, particularly due to the large differences of the involved barriers.


Figure 3.20: Extracted migration paths and energy profiles using the method of steered MD. Left: Two trajectories, see Fig. 3.18, for H hopping between BC sites along the interface. Right: Five simulated migration pathways for H across the Si/SiO\(_2\) interface, see Fig. 3.19. The dashed lines indicate the Si/SiO\(_2\) transition region.

The resulting hopping barriers along the interface connecting the different BC configurations are between \(\SI {0.3}{}-\SI {1.3}{eV}\), see Fig. 3.20 (left panels). The rather large variation of the barriers can be explained by taking into account the deformation of the silicon lattice due to the residual strain at the Si/SiO\(_2\) interface. The MEPs across the structure, starting in the Si bulk and ending in the SiO\(_2\), see Fig. 3.19, yield a very narrow and consistent distribution of barriers in the bulk silicon side of around \(E_\mathrm {B,hop}=\SI {0.4}{eV}\). The diffusion trajectory of hydrogen in bulk crystalline silicon also tends to hop from one to the next BC site, which is indeed a well known stable position reported in the literature [212214]. On the other hand, approaching the transition region and crossing into the bulk oxide shows an increase of the barriers on the Si side and ultimately a large potential barrier between \(E_\mathrm {B,cross}=\SI {1.2}{}-\SI {1.9}{eV}\) for the H moving into the SiO\(_2\) side.

To establish a more accurate picture for the hopping barriers along the interface which is comparable to the results presented in Sec. 3.2, subsequent DFT simulations have been conducted. Five initial positions, relaxed BC configurations, of the hydrogen atom have been chosen and the respective barriers to the three closest BC sites were calculated using the CI–NEB method.


Figure 3.21: Left: Simulated distribution of barriers for hydrogen hopping between bond–center sites at the subinterfacial Si side of the Si/SiO\(_2\) interface using the method of CI–NEB in conjunction with the Pbe functional. Paths for connecting five initial BC configurations with the three next nearest BC sites have been calculated, yielding in total 30 barriers (forward and backward transition barriers). Right: Energy of various H configurations obtained from geometry optimizations. The mean value of hydrogen in a Si BC site serves as a reference for energy alignment.

The individual hopping barriers are summarized in Fig. 3.21. The calculated values range from \(\SI {0.16}{eV}\) to \(\SI {1.04}{eV}\) with two outliers possessing barriers of \(\sim \SI {1.35}{eV}\). The average barrier height is \(\SI {0.53}{eV}\) and, therefore, in very good agreement with the predicted values using the MEPs in the left panel of Fig. 3.18 and the respective energy paths shown in Fig. 3.20. Additionally, comparing the lateral H hopping barriers \(E_\mathrm {hop}\) to the reverse barrier \(E_\mathrm {backward}\) for H in the BC\(_{2,3}\) site re–passivating the Si–DB defect, see Fig. 3.12, shows that the hopping barriers are considerably lower (\(E_\mathrm {hop}\ll E_\mathrm {backward}\)).

Furthermore, the stability of H in various configurations within the interfacial region has been calculated. A hydrogen atom has been placed at different positions, i.e. a BC site in the Si side of the interface, interstitial within a void in the SiO\(_2\) network as well as close to O and Si atoms in the interfacial oxide region. The energy of the respective configurations has been determined using geometry optimizations within DFT. All results are summarized in the right panel of Fig. 3.21. The mean value of H being in a Si BC configuration has been used as the reference energy. One can see that the energy of such configurations shows a distribution of around \(\SI {1.5}{eV}\), which is consistent with the barriers calculated in the left panel of Fig. 3.21. On the other hand, the H in an SiO\(_2\) interstitial position possesses on average a \(\SI {1.8}{eV}\) higher energy, as indicated by the green bar. Additionally, it is worth noting that only around 30% of the hydrogens placed within a void actually stayed in an interstitial position. The remaining calculations showed that it is more likely for a neutral H to move back into the Si side (if the H was directly placed within an interfacial SiO\(_2\) void) or to form some defect by breaking Si–O bonds in the oxide. The third group shows the total energy of defect configurations formed by the hydrogen atom. Such defects can be either hydroxyl–E\(^\prime \) centers (HE\(^\prime \)) or [SiO\(_4\)/H]\(^0\) configurations [173175]13. While the average energy for forming a SiO\(_2\) defect is only \(\SI {0.2}{eV}\) higher in energy than the BC configuration, its distribution is rather broad with \(\SI {4}{eV}\). A thorough statistical analysis including the calculation of barriers using the CI–NEB method, is, however, beyond the scope of this work. Nevertheless, the results clearly suggest that in the majority of investigated cases the Si BC site provides an energetically favorable position for the H atom, although – in the most extreme variant – forming a defect in SiO\(_2\) can be up to \(\SI {3}{eV}\) lower in energy.

To summarize, one can conclude that once the Si–H bond is broken and the H is released into the next but one BC site, the hydrogen atom faces a rather small hopping barrier to a neighbouring BC configuration and is effectively mobile along the interface. Similar results have been obtained by the group of Pantelides et. al and others [251257]; however, they mainly focused on the migration of charged hydrogen species. They report a rather small migration barrier for H moving laterally within the subinterfacial Si region (\(\SI {0.3}{}-\SI {0.5}{eV}\)), while the potential barrier to cross over into the SiO\(_2\) side is at least twice as much. Due to the negative-U character of H [213, 214, 258, 259], it is indeed likely that the released hydrogen becomes charged (the BC provides a stable position for H\(^0\) and H\(^+\), whereas the negatively charged species prefers the AB configuration).

Additionally, the presented results potentially shed light on peculiarities of reliability issues. Once the initial Si–H bond is broken and the hydrogen released into a BC configuration, it potentially becomes charged. However, the H would be mobile along the interface and thereby possibly be able to trigger further reactions. Such a picture nicely fits into recent observations regarding reliability phenomena in MOSFETs. Occasional reports observe a peculiar feature of microelectronic devices: After removing the stress from a device, it tends to continue degrading, referred to as post–stress degradation build–up. Recent theoretical modeling approaches [17, 18] link this behaviour to the release of H during stress which is subsequently stored near the interface, assumed to be in a charged configuration. Such a theoretical description would be fully consistent with the conclusions drawn here.

12 as well as shifted MEPs

13 In around 10% of the simulations the H formed a Si–H bond by breaking an Si–O bond with a remaining oxygen dangling bond. However, these configurations have been discarded. This, however, could possibly be an artefact due to limited cell size as discussed in Appendix A

3.6.2 Passivation Reactions

The actual passivation mechanism of a \(P_\mathrm {b}\) center, however, can not be properly described by only taking into account atomic hydrogen at the interface. Instead Brower and later also Stesmans converged to the same conclusion by performing dedicated experiments: The passivation reaction is dominated by the cracking of molecular H\(_2\) [221, 222, 244]. The proposed mechanism reforms a Si–H bond by breaking a H\(_2\) molecule, \ch{+Si^. + H2 -> +Si-H + H^0}, having an activation barrier of \(E_\mathrm {B}=\SI {1.51}{eV}\) with a Gaussian spread of \(\sigma _\mathrm {B}=\SI {0.06}{eV}\). Recent experimental studies conducted on a device level, where MOSFETs were stressed under various HCD bias points and subsequently annealed at elevated temperatures, confirm the proposed mechanism and activation energy [248250]. The detailed atomistic dynamics, on the other hand, remain unclear with only a few ab initio based studies assessing the theoretical description [204, 206, 260]. The initial breaking of the extremely stable H\(_2\) molecule (\(E_\mathrm {bind}\sim \SI {4.5}{eV}\)) together with the remaining atomic hydrogen and its final configuration play a decisive role whether the passivation reaction would occur or not. Within this context interstitial H in Si and SiO\(_2\) has been investigated [204] as well as the interaction of H\(_2\) with defect free and defective SiO\(_2\) to identify possible H\(_2\) cracking sites [173, 174, 206, 260]. However, due to the multitude of different possible reactions, depending on the local environment of the amorphous oxide, no rigorous picture could be deduced so far.

In order to explore possible reactions DFT in conjunction with the NEB method has again been applied. Two different Si/SiO\(_2\) models have been used, with in total six different interface defect configurations. A H\(_2\) molecule has been placed in the direct vicinity and several final configurations for the remaining H including the respective barriers were calculated, see Fig. 3.22.


Figure 3.22: Passivation of an interface defect by H\(_2\). The cyan translucent profiles show the spin density of the respective snapshots. As an initial configuration the H\(_2\) molecule is placed in the direct vicinity of the Si–DB. Within the investigated reactions the remaining H can trigger different mechanisms leading to different final configurations. First, the neutral hydrogen becomes interstitial within the SiO\(_2\) network. Second, the remaining H breaks a Si–O–Si complex and forms a hydroxyl \(E^\prime \) defect. Furthermore, the last reaction shows that the H in its neutral state can also bind to a bridging oxygen and form a variant of the [SiO\(_4\)/H]\(^0\) configuration.

For the first reaction the remaining hydrogen moves into a void above the formed Si–H bond and becomes interstitial neutral H. The CI–NEB calculations for all different defect configurations properly converged and yield a broad distribution of forward reaction barriers between \(\SI {0.89}{}\) and \(\SI {2.24}{eV}\). The results show an apparent strong dependence on the local environment of the Si–DB defect and/or the final interstitial position. The backward barriers, on the other hand, for de–passivation reactions is much lower, suggesting that a \(P_\mathrm {b}\) center can be effectively recreated in the excess of H atoms. Nevertheless, interstitial hydrogen can undergo further reaction dynamics in the SiO\(_2\) network. Due to its negative U character [209, 261], H potentially becomes charged and binds within the oxide, e.g. a proton can attach to a bridging oxygen atom [61, 62, 174, 261, 262]. Furthermore, H\(^+\) can effectively move through SiO\(_2\) by hopping between bound states, as was shown in [62, 252, 263265]. Such a hopping mechanism was recently further extended and investigated for neutral hydrogen as well [61]. Despite H’s negative U character, in the same work it was found that H\(^0\) indeed exhibits possible metastable neutral configurations. Furthermore, theoretical studies demonstrate the possibility of H\(^0\) migration between voids [205, 266, 267] as well as interactions with the defect–free and defective SiO\(_2\) matrix [173, 174]. Overall, this suggests an extremely broad spectrum of possible reactions for the remaining hydrogen moving into the SiO\(_2\) region.

(-chemfig- diagram)
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Table 3.1: Possible \(P_\mathrm {b}\) center passivation reactions with H\(_2\). For the first reaction the remaining neutral hydrogen becomes interstitial in the SiO\(_2\) matrix, whereas reactions 2) and 3) represent defect transformations associated with the H. All energies are in eV.

In order to further investigate possible reactions of the remaining H atom, two test cases have been chosen to examine the interaction with the defect–free oxide. The first one accounts for the transformation of the remaining interstitial H\(^0\) into a hydroxyl–\(E^\prime \) center, see Fig. 3.22 and Reaction 2 in Table. 3.1. Out of the 15 configurations introduced into the Si/SiO\(_2\) structure only six converged to a realistic hydroxyl–\(E^\prime \) center configuration within a geometry optimization and have been used in subsequent NEB calculations. A common feature of the converged precursor sites is an elongated \ch{Si-O} bond within the \ch{Si-O-Si} unity, as indicated in Table. 3.1. The total transition barriers to break H\(_2\), passivate a \(P_\mathrm {b}\) center and create a hydroxyl–\(E^\prime \) center are between \(\SI {1.24}{}\) and \(\SI {2.09}{eV}\). Due to the relatively strong Si–H and O–H bonds, the backward transition barriers are of similar height, indicating that the induced final configuration would be indeed stable once the H\(_2\) is broken.

The second possibility is schematically illustrated as Reaction 3 in Table 3.1 and shown in Fig. 3.22. Thereby, the hydrogen binds to a bridging oxygen forming a so–called [SiO\(_4\)/H]\(^0\) center14. The structural relaxations are mainly associated with the opening of the \ch{O-Si-O} angle resulting in an electron trapping site at an adjacent Si atom [174]. Therefore, the defect actually resembles a proton bound to the oxygen atom where the additional electron is trapped at an adjacent Si site, see Table 3.1 and Fig. 3.22. For the simulations twelve initial defect configurations have been constructed with the H atom placed \(\SI {0.8}{\angstrom }\) away from a bridging O. Unfortunately, only three of the structures converged to a [SiO\(_4\)/H]\(^0\) defect within the geometry optimizations, the remaining ones transformed into either interstitial H, hydroxyl–\(E^\prime \) center like configurations or other odd defective formations, e.g. breaking of \ch{Si-O} bonds, which have been discarded, as mentioned above. The total energy of the optimized system suggests that the stability of [SiO\(_4\)/H]\(^0\) is comparable to interstitial H, whereas a hydroxyl–\(E^\prime \) defect is usually more stable being around \(\SI {0.8}{eV}\) lower in energy. The final activation energies for the full reaction starting with H\(_2\) are given in Table 3.1. The forward reaction is slightly shifted towards higher values compared to the hydroxyl–\(E^\prime \) center calculations, indicating a higher barrier to create the defect site, whereas the backward barrier to reform H\(_2\) seems to be lower with values ranging from \(\SI {0.85}{}\) to \(\SI {1.67}{eV}\). However, due to the rather small sample size the results can only provide a qualitative understanding. Additionally, the bound hydrogen, i.e. the [SiO\(_4\)/H]\(^0\), potentially becomes positively charged via a charge transfer reaction with a reservoir which could enhance its stability resulting in a proton sticking to an oxygen [17, 62, 173].

The presented results provide a qualitative understanding and insight of the \(P_\mathrm {b}\) center passivation kinetics. The complex and manifold reaction dynamics involving mechanisms related to charged hydrogen species allow a quantitative analysis and conclusions only at a much broader statistical level, which is beyond the scope of this work. However, already the transition barriers derived here clearly suggest that the efficiency of the H\(_2\) passivation process heavily depends on the remaining H and its final configuration. Interstitial hydrogen for example, would most probably immediately reform H\(_2\), particularly at elevated temperatures, due to the small backward barrier according to Reaction 1 in Table 3.1. On the other hand, due to the small migration barrier between voids in SiO\(_2\) it could effectively diffuse away or form H related defects in the oxide which provide a sufficiently large backward barriers. Thus, the interaction of H with defect–free or defective sites in the amorphous SiO\(_2\) system inevitably plays an important facet in understanding and describing the passivation mechanism of interface defects.

14 Note that in [174, 175] it was shown that the hydroxyl–\(E^\prime \) center is thermodynamically more stable than the [SiO\(_4\)/H]\(^0\) defect configuration and the transformation barrier is only around \(\SI {0.1}{eV}\).