This section deals with the basic properties of defects, especially the position of their corresponding trap levels. Previous theoretical investigations were focused on defects in crystalline [112, 113] as a substitute for amorphous materials. At this point it is emphasized that one has to consider the amorphous nature of since it strongly affects the defect properties: For instance, oxygen molecules in have only discrete values for the barriers to migrate from one void to the next one. By contrast in , they encounter a distribution of barriers whose average eventually determines the effective activation energy for diffusion [167, 168]. Another example is the stability of the center, which can transform to an oxygen vacancy by overcoming a thermal barrier. The height of this barrier and thus the stability of the center have been found to strongly depend on the the local surrounding silica network [19, 20]. These two examples suggest that small variations in the bond lengths and angles of the surrounding structure could result in a wide distribution of trap levels [162, 161] and thereby impact the trapping dynamics. Unfortunately most of the defect properties are difficult to determine experimentally so that theoretical methods, such as DFT, were chosen for the determination of defect levels. The empirical potential molecular dynamics was employed for the production of . The details of this procedure are described in Section 3.4.2. Pair-correlation functions, angle distributions, and the ring distribution have been evaluated in order to ensure that the obtained samples mimic real . The generated structures were used for DFT calculations, whose parameters are summarized in Section 3.3.3. The defect structures have been obtained by adding, shifting, or removing the silicon, the oxygen, or the hydrogen atoms. The switching trap levels for the created defects have been evaluated using the formulas (3.33)(3.36). The calculated trap levels have been aligned to the silicon bandgap using the procedure proposed in . The obtained valence band offset of for the interface has been found to be in good agreement with the values extracted from [80, 169, 170]. In the following, a study of several prominent defects in will be presented. They will be discussed based on their configuration in their various charge states in order to check the correctness of the produced defect structures. Furthermore, their expected trapping behavior will be inferred from the switching levels gained from DFT simulations.
In stoichiometric , two silicon atoms are always connected by one bridging oxygen atom. When this atom is removed, the two neighboring silicon atoms establish a common bond, the heart of the oxygen vacancy (see Fig. 5.1). In , the length of this bond is approximately [19, 149], which compares well with the - distance of in crystalline bulk silicon . In the DFT simulations of this thesis, the defects are embedded in an amorphous host material so that this characteristic length is distributed. Our set of structures covers - bond lengths between and . These values are in reasonable agreement with the range () obtained in the study of Mukhopadhyay et al. . Nicklaw  extended his defect calculations to highly strained oxygen vacancies whose bond lengths reach values up to . This kind of defects is associated with large formation energies , thus exists in small concentrations, and has a negligible contribution to the trapping kinetics of NBTI. The - bond is associated with the level, which is sharply peaked and situated far below the silicon valence band edge (cf. Fig. 5.1 and Table 5.1). For comparison, the corresponding defect calculations in  predict at approximately the same position.
In the context of EPR measurements, the positively charged counterpart of the oxygen vacancy is referred to as the center. The missing negative charge within its bond causes a repulsion between the two electropositive silicon atoms and results in a stretching but not in a breakage of the bond. In the reference, the - bond of center was found to extend from to [149, 19], which is large in comparison to the bond length of a neutral oxygen vacancy. In the DFT simulations of this thesis, this kind of weak bonds experiences large tensile and compressive forces due the amorphous host material. The corresponding bond lengths are found to be distributed within a range of , consistent with the values used in  and . This - bond is associated with a defect level close to the silicon valence band edge. Due to the amorphous nature of , the defect levels are spread widely over an energy range from to (cf. Fig. 5.1). By contrast in the case of the neutral oxygen vacancy the impact of the surrounding network can be neglected due to the strong bond so that the distribution of levels is narrow as shown in Fig. 5.1.
Regarding the tunneling dynamics, one has to differentiate between two cases: If the defect level is located below the silicon valence band edge, electrons from the substrate valence band can be captured by the defect via elastic tunneling. For the reverse process, the defect level is already shifted downwards, where the electron in the defect is unlikely to find a high energetic hole in the substrate. From this argumentation it is expected that oxygen vacancy remains neutral if it is discharged once. In case the level is located above the silicon valence band edge, electron capture into the defect is inhibited, which suggests that this defect acts as a ‘fixed positive charge’. However, keep in mind that these charges may be neutralized via interface states instead.
The existence of the center as an additional metastable configuration of the oxygen vacancy has been confirmed by a wide range of theoretical [20, 19] as well as experimental [15, 16] studies. Starting from the center, one side of the defect undergoes a transformation called ‘puckered’. During this, the dimer bond is disrupted and the ‘puckered’ silicon atom moves through the plane defined by its three oxygen neighbors where this new configuration is stabilized via a back bond to a nearby oxygen atom. In the DFT simulations of this thesis, the back bond has typically a length of , which is large compared to the - bond in bulk () [174, 155, 153, 154]. This indicates a weak bonding of the ‘puckering’ silicon atom to the back oxygen. On the other side of this defect, an unsaturated silicon dangling bond is left behind, which can carry up to two electrons and gives an EPR signal . Only if the center is neutralized, this defect is prone to a collapse into its oxygen vacancy configuration  again. The configuration of the center is depicted in Fig. 5.2.
In contrast to the oxygen vacancy, the defect levels and of the center have only a small spread of about since the silicon dangling bond along with the three other bonds almost preserve their tetrahedral symmetry during structural relaxation. The puckered side of the defect complex does not interact with the dangling bond and consequently does not impact its defect levels. The levels for tunneling into () or out of () the traps lie close to the silicon band edges (see Fig. 5.3). Therefore, only a small thermal excitation of the substrate charge carriers is required for a tunneling process. In this case, the band bending controls the concentrations of electrons in the silicon conduction band or holes in the silicon valence band, respectively, and consequently governs the tunneling rates. This means that the centers can be repeatedly charged and discharged by electrons by switching the MOSFET between strong inversion and accumulation. The configuration was already proposed by Lelis [15, 16] in the context of the temperature-dependent annealing behavior of oxide traps. The temperature dependence in his model was explained by a transition from a spin-triplet  to its corresponding spin-singlet state. The latter denotes the ground state of two electrons, which sit in the dangling bond of an center and have their spins aligned anti-parallel. The spin-triplet state is the excited counterpart, which is characterized by a parallel alignment of the electron spins and decays immediately to the spin-singlet ground state after it is occupied. According to the argumentation of Lelis, the concentration of thermally excited electrons is increased at elevated temperatures. This gives rise to an enhanced tunneling probability and thus an accelerated annealing of positively charged centers. Interestingly, the energy level of the excited spin-triplet state  coincides with the electron capture level obtained by the DFT simulations of this thesis. Thus, the level shift can explain the same trapping dynamics as in Lelis when replacing the spin-triplet state with the level. In this way, the defect levels responsible for the annealing behavior in the Lelis model has been theoretically confirmed but with the interpretation based on the level shift. As highlighted in , most of the centers immediately collapses into their oxygen vacancy configuration after neutralization. Then they feature defect levels, located down far below the silicon valence band edge. Therefore, once these defects are neutralized, they cannot be recharged again and will be permanently annealed out. However, this does not rule out the center as a cycling charge. A considerable fraction of the neutralized centers  have a large barrier for the relaxation to the oxygen vacancy configuration and thus may remain in the puckered configuration for time scales relevant for NBTI. In this configuration, they are capable of repeatedly exchanging electrons or holes with the interface.
Another variant of the center is the center (shown in Fig. 5.4), which has been extensively studied by Conley and Lenahan . Its structure can be visualized by replacing one of the nearby oxygen atoms with an hydrogen atom. The DFT simulations in this thesis have revealed that the neighboring hydrogen atom does not affect the position of the defect levels originating from the dangling bond. As a result, this defect features the same distribution of trap levels and therefore can be repeatedly charged and discharged similarly to the simple center.
In the context of reliability issues, the hydrogen atom [175, 176] is of special interest since it is available in appreciable amounts. Indeed it has been speculated in many investigations [177, 149] that hydrogen seriously affects the reliability of MOSFETs. Its configuration strongly differs with its charge state: The DFT simulations of this thesis predict the neutral atom in the middle of a void where it does not form a bond with any network atom, consistent with DFT investigations in [175, 176]. The positively charged atom weakly binds to one of the electronegative oxygen network atoms as shown in Fig. 5.5. The corresponding - bond length is found to be approximately , which compares perfectly well with the values for () in  and () in [149, 176]. The negatively charged atom attaches to an electropositive silicon atom with a bond distance of about , in agreement with [164, 176, 149].
The energy levels of the hydrogen atom are visualized in Fig. 5.6 and listed in Table 5.1. The trap levels , , and are located far away from the silicon band edges and consequently require highly excited substrate charge carriers for a tunneling transition. By contrast, is centered around the silicon conduction band edge, resulting in a high tunneling probability. This suggests that the defect is preferredly found in its negative charge state at a weak oxide field. At a first glance, this result seems to contradict the findings of other groups [176, 175, 149]. According to them, the proton has been predicted to be the most stable charge state of the hydrogen atom. However, the calculations are based on a thermodynamic transition level, which only applies to thermally-activated processes, such as NMP transitions for instance.
Hydrogen is frequently suspected to undergo reactions with oxide defects [178, 179]. Therefore, the present investigations include a hydrogenated variant of the center, also referred to as the hydrogen bridge (see Fig. 5.7). This defect can be constructed by placing a hydrogen atom inbetween the silicon dimer of an oxygen vacancy. In the positive charge state, the hydrogen atom forms a three-center bond involving the two silicon atoms from the dimer and the hydrogen atom in the central position. Its relaxed structure exhibits an asymmetry in the - distances with values of and . This is in qualitative agreement with the corresponding defect structure generated in , however, there are small deviations in the - distances attributed to variations in the atomic structure of the host structure. Note that although the aforementioned three-center bond is generally considered unusual in chemistry, the same -- bond chain has also been observed for in . In the neutral charge state, the three-center bond is disrupted on the side with the long distance. The breakage is accompanied by a large structural relaxation which detaches the - complex from the remaining dangling bond. The bond length reduces to its typical value of  for the unperturbed - bond while the other silicon atom is separated from the hydrogen atom by about and carries one electron in its dangling bond orbital. By contrast, Blöchl et al.  found that even in the neutral charge state the hydrogen atom interacts with both silicon atoms. This discrepancy may originate from the rigid network of the surrounding , which keeps both silicon atoms together and thus prevents the -- chain from bond breakage in the neutral charge state. In the negative charge state, the added electron sits on the dangling bond and causes a further repulsion between the dangling bond and the - complex ().
As shown in Fig. 5.8 and Table 5.1, all defect levels are located within a reasonable distance from silicon band edges on the energy scale. This suggests that none of the considered transitions can be ruled out based on the positon of its corresponding trap level. Thus the trapping dynamics are eventually governed by the band bending in the substrate and the operation state of the MOSFET.