Impact of Charge Transitions at Atomic Defect Sites on Electronic Device Performance

7.2 Conclusions

Field effect transistors Section 4.1 offers a generalized non-radiative multi-phonon (NMP) defect model for the defect types oxygen vacancy (OV), hydrogen bridge (HB) and hydroxyl-E ′ (H-E′ ) center based on statistical DFT investigations of atomic defects in $a$ -SiO $_{2}$ . The final model combines both electron and hole transition processes in a unified manner by containing six different defect states, a stable and a metastable configuration in each charge state 0, $+ 1$ and $- 1$ .

The thermal energy barriers for all occurring charge conserving transitions in this model are calculated with the climbing image nudged elastic band (CI-NEB) method. Classical energy barriers for charge transitions were extracted from crossing points of differently charged potential energy curves (PECs), which were modeled based on the calculated relaxation energies and CTLs. Thereby, it is also revealed that a transition from charge state $- 1$  ( $+ 1$ ) to 0 is energetically always favored compared to the direct transition $- 1$  ( $+ 1$ ) to $+ 1$  ( $- 1$ ). Supported by the formation energy analysis, CTLs and experimentally determined oxygen vacancy concentrations, it is argued that the hydroxyl- $E^{'}$ center (H- $E^{'}$ ) is the most relevant defect type in $a$ -SiO $_{2}$ for causing device degrading phenomena in Si/SiO $_{2}$ MOSFETs. Furthermore, it is shown that OVs have $0 / + 1$ CTLs near the valence band maximum of SiC, making them suitable for hole trapping in SiC/SiO $_{2}$ devices.

The varying CTLs due to meta-stable configurations of the same defect give an explanation for vanishing defects and differing time constants of anomalous RTN observed in time dependent defect spectroscopy. In the negative charge state, the H- $E^{'}$ center was linked via transition barrier calculations to a different stable configuration, were a Si gets fivefold coordinated with four O and one H. This defect state, called H- $E^{'}$   $_{\min}^{-}$ , is on average 1.5 eV lower in energy compared to all other negatively charged H- $E^{'}$ configurations with charge transition levels distributed $> 1.6$  eV below the conduction band minimum (CBM) of Si and SiC substrates of MOS devices. The migration of the H from a H-E′ configuration in the negative charge state to the H- $E^{'}$   $_{\min}^{-}$ has a vanishing transition barrier and is thus considered to happen spontaneously. As the $0 / + 1$ CTLs are distributed around the valence band maximum of Si, it can be concluded that capturing holes from the substrate during operation is more pronounced in Si/SiO $_{2}$ MOSFETs than capturing electrons. Additionally, H- $E^{'}$   $_{\min}^{-}$ configurations have $0 / - 1$ CTLs far below the CBM of both Si and SiC. These observations are in good agreement with the reduced positive bias temperature instability (PBTI) effect compared to its negative counterpart (NBTI) observed in Si/SiO $_{2}$ MOS devices, as positive gate voltages shift the $0 / - 1$ trapping level even deeper below the band gap of the substrate.

In Section 4.2, the hole trapping properties of the tungsten vacancy ( $V_{W}$ ) and the selenium antisite ( ${Se}_{W}$ ) in two-dimensional monolayer tungsten diselenide (1L-WSe $_{2}$ ) were characterized. The DFT calculations reveal that both defect types have $0 / + 1$ CTLs close to the valence band maximum (VBM) with small relaxation energies below 0.2 eV. These observations are in good agreement parameters extracted from electrical measurements and technology computer aided design (TCAD) models in scaled FETs based on 1L-WSe $_{2}$ semiconductors. It is thus concluded that both defect types can be responsible for the generation of RTN signals as detected in these devices.

Charge trap flash In Section 5.1, intrinsic electron and hole trapping sites in $a$ -Si $_{3}$ N $_{4}$ were investigated. The amorphous sample structures were created by simulating a melt-and-quench procedure with MD, employing an MLIP. This MLIP was specifically trained for this purpose based on DFT calculations of 1000 different silicon nitride structures. Subsequently, the electronic structure and charge trapping properties of the amorphous samples were statistically analyzed. It was shown that electronic states at the CBM are introduced by overcoordinated N and undercoordinated Si, while states at the VBM correspond to overcoordinated Si and undercoordinated N.

When a charge carrier is added to the system, it localizes at one of these sites, thereby shifting the electronic state towards the middle of the band gap due to structural relaxations. The trapping sites are characterized by calculating relaxation energies and CTLs to model their PECs in different charge states. Subsequently, the energy barriers for charge emission and capture processes were extracted from the crossing points of the PECs to analyze NMP transitions in the classical limit. Thereby it is argued that intrinsic trapping sites can easily capture and store electrons from a Si substrate as, for example, realized in silicon–oxide–nitride–oxide–silicon (SONOS) stacks used in non-volatile flash memory devices. By applying a positive voltage to the gate of a SONOS device, which is typically done during the programming cycle, the energy barriers and thus the time constants for charge emission increase, while the barriers for charge capture decrease according to the NMP model. This shows that overcoordinated N and undercoordinated Si are suitable intrinsic defect candidates to contribute to the memory effect in non-volatile flash memory devices. Furthermore, it is shown that, although holes can be captured from the Si substrate when a negative gate voltage is applied, energy barriers for hole emission are significantly lower compared to the barriers for emitting electrons, resulting in a comparably reduced storage capability for holes in $a$ -Si $_{3}$ N $_{4}$ .

Section 5.2 presents a theoretical study on polaronic charge trapping sites in $a$ -Si $_{3}$ N $_{4}$ :H using first-principles methods in conjunction with a hybrid functional. The investigated sample structures are in excellent agreement with experimentally determined features of $a$ -Si $_{3}$ N $_{4}$ :H thin films, including mass density, H content, electronic band gaps and structural properties such as coordination number, structure factor and average bond lengths. Intrinsic charge trapping sites at over- and undercoordinated atoms as presented in the previous section do not appear to have sufficient concentrations to explain the much larger experimental values. Here, the formation of both hole and electron polarons near fully coordinated sites, which have far higher densities, is discussed.

To create the $a$ -Si $_{3}$ N $_{4}$ :H sample structure, the dangling bonds of the $a$ -Si $_{3}$ N $_{4}$ samples studied in the previous chapter were passivated with H. This substantially widens the band gap by $\approx 10$  % and occasionally introduces hole traps at Si–H sites in the amorphous network. It is found that holes may either localize near Si–H bonds or form a polaron at fully coordinated N sites. Structural relaxations upon hole polaron formation are rather weak ( $\sim 0.5 - 0.7$  eV), which is comparable to small polarons in crystals. When a second hole is added to the system, it is found that it mostly gets trapped at a different semi-localized state near a fully coordinated N site in the amorphous network. While the formation of hole bipolarons is possible, according to our calculations, it is more likely for a second hole polaron to form at a different site. The variety of different hole trapping sites can be related to the results of the inverse participation ratio analysis, showing that several states below the VBM are semi-localized at different fully coordinated N sites. The thermodynamic trap levels of the hole polarons and Si–H bonds lie closely above the valence band edge of silicon nitride as also reported in experimental literature.

It is also demonstrated that electrons preferably localize at fully coordinated, comparably positively charged Si near already strained, weakly bonded Si–N bonds. When an electron is trapped at these sites, the Si–N distance increases by 26 % on average, resulting in a dangling-bond-like configuration at the Si. In contrast to the hole trapping mechanism, a second added electron localizes at the same site as the first electron, effectively forming an electron bipolaron. The CTLs of the electron polarons and bipolarons are distributed below the Si CBM, in accordance with experimental electron trap levels from literature. The structural relaxations after trapping an electron are distributed around 1.3 eV, which is $\approx 0.7$  eV higher than for polaronic hole trapping.

During the program and erase cycle of CTF devices employing silicon nitride as the charge trapping layer in e.g. SONOS stacks, typically voltages above $p m 10$  V are applied. Due to band bending caused by the applied gate bias, the potential shift can easily bring the charge transition level for hole and electron polarons and bipolarons in the vicinity of the valence and conduction band edges of a Si substrate. Therefore, the fully coordinated polaronic sites are likely able to trap charges from a Si substrate under an applied bias, following considerable structural relaxations, and can thus be expected to contribute to the memory effect of charge trap flash devices. Furthermore, when silicon nitride is used as a dielectric in metal-oxide-semiconductor (MOS) devices, polaronic sites can trap electrons from the Si substrate under small applied positive gate voltages, which might deteriorate the reliability of these devices.

Vibrational Broadening In Section 6, single vacancies in $α$ -Al $_{2}$ O $_{3}$ are characterized with hybrid functional calculations in the supercell approach and optical emission and absorption processes at the defect sites are modeled. The formation energy and CI-NEB calculations indicate that the Al vacancy ( $V_{Al}$ ) is likely to form in a split-vacancy configuration $V_{Al, s}$ , where an Al atom occupies an interstitial position between two $V_{Al}$ sites in the corundum lattice along the $c$ -axis.

The optical broadening related to the exchange of an electron localized at the $V_{O}^{0}$ , $V_{O}^{+ 1}$ , $V_{Al}^{- 3}$ and $V_{Al, s}^{- 3}$ sites with the CBM is calculated. Hereby, the corresponding absorption and emission line shapes were modeled based on parameters extracted from effective phonon modes. It is confirmed that the origin of the $F$ center in Al $_{2}$ O $_{3}$ is an oxygen vacancy in the neutral charge state. The calculated absorption line shape for the $V_{O}^{0 / + 1}$ transition matches spectroscopic measurements of the $F$ center. Likewise, the calculated emission line shape of the $V_{O}^{+ 1 / 0}$ transition agrees with luminescence data of the $F$ center.

For the $F^{+}$ center, it is argued against the assignment in the literature of the three absorption bands at 4.8, 5.4 and 6.3 eV to internal transitions, since the energy difference between occupied and unoccupied Kohn-Sham states localized at $V_{O}$ are found to be far too high. Instead, based on the calculated line shapes, it is proposed to assign the observed absorption bands to transitions of an electron from the VBM to $V_{O}^{+ 1}$ or to $V_{O}^{+ 2}$ sites, and from $V_{O}^{+ 1}$ to the CBM. Furthermore, the calculated luminescence line shape for a transition of an electron from the CBM to $V_{O}^{+ 2}$ is in excellent agreement with the observed emission spectrum of the $F^{+}$ center. This confirms the identification of the $F^{+}$ center with the singly charged oxygen vacancy based on first-principles calculations.

Additionally, absorption and luminescence spectra are calculated for the transition of an electron at $V_{Al}^{- 3}$ and $V_{Al, s}^{- 3}$ defects with the CBM. The line shapes of both Al vacancy configurations are broader than for the $V_{O}$ and overlap, with a peak separation of $0.46$  eV.