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

Chapter 5 Charge Trap Flash Memories

In this chapter, the results of first-principles investigations of intrinsic charge trapping sites in amorphous silicon nitride are presented and linked to the localization of charge in charge trap flash (CTF) devices. The investigated $a$ -Si $_{3}$ N $_{4}$ sample structures were created with a machine learning interatomic potential (MLIP), which was generated according to the training process as described in Section 3.2.3 [194]. Additionally, $a$ -Si $_{3}$ N $_{4}$ :H structures were generated by subsequent passivation of dangling bonds with H. The resulting sample structures are in excellent agreement with experimental perceptions, including mass densities, coordination number, structural features and H concentration.

First, the localization of holes and electrons at over- and undercoordinated atoms in pure $a$ -Si $_{3}$ N $_{4}$ is investigated [54]. It will be shown that additional charges can localize near over- and undercoordinated atoms, which are present in this material in considerable portions. Namely, holes preferably localize near two-fold coordinated N or five-fold coordinated Si, while electrons localize near three-fold coordinated Si or four-fold coordinated N. The trapping sites are statistically investigated by analyzing defect states in the band gap after capturing a charge and relaxation of the structure. The charge trapping sites are characterized in the context of the non-radiative multi-phonon (NMP) model by calculating their relaxation energies $E_{Relax}$ and thermodynamic charge transition levels (CTLs) to construct the potential energy curves (PECs) of the systems in different charge states. Subsequently, the classical charge transition barriers are extracted from the crossing points of the PECs for phonon-driven charge capture and emission processes.

Additionally, hole and electron (bi)polaron formation in $a$ -Si $_{3}$ N $_{4}$ :H is investigated [55, 53]. The precursor sites for polarons are analyzed by determining bond lengths, partial charges, maximally localized Wannier centers and bond order of the amorphous silicon nitride systems. The influence of H on the electronic density of states is investigated and the defect states introduced in the band gap after the polaron formation is evaluated. The thermodynamic CTLs of hole and electron polarons are calculated and the structural relaxations upon charge trapping analyzed by calculating the relaxation energies of the trapping sites. It is found, that hole polarons and electron (bi)polarons are stable in wide energy ranges of the Fermi level within the band gap of $a$ -Si $_{3}$ N $_{4}$ :H. Based on the CTL and relaxation energy analysis, it is argued that polaronic sites as well as intrinsic charge trapping sites at over- and undercoordinated atoms are suitable candidates for trapping and storing charges from Si substrates and can thus contribute to the memory effect in CTF devices.

5.1 Structural defects – over- and undercoordinated atoms in $a$ -Si $_{3}$ N $_{4}$

The content presented in this section has been published in [54].
Both the text and the figures have been adapted from this publication.

5.1.1 Structure creation of amorphous silicon nitride

Due to the structural randomness of $a$ -Si $_{3}$ N $_{4}$ , investigations of the charge trapping properties have to be performed in a statistical manner, as each trapping site is expected to show different defect properties depending on the local environment. Therefore, 100 initial $a$ -Si $_{3}$ N $_{4}$ structures with 224 atoms each were created by simulating a melt-and-quench procedure, as has already been successfully used to model other amorphous compounds [241, 76, 4]. During the MD simulations, the interactions between the atoms were described with the MLIP, which was specifically trained for amorphous structure creation [194]. The volume of the cell was slightly enlarged in the beginning to construct a mass density of 2.93 g cm⁻³, which is well within experimentally determined values of amorphous Si $_{3}$ N $_{4}$ thin films (2.6 to 3.2 g cm⁻³ [242, 243]). The initial Si $_{3}$ N $_{4}$ samples were heated above their melting points to 5000 K and kept at this temperature for 60 ps until they lost all initial information. The samples were then slowly cooled down to room temperature (300 K) where they become amorphous solids. During the MD runs, the volume of the liquid was not equilibrated, which is typically done to achieve feasible simulation times and to reduce the computational costs for amorphous structure creation as discussed in [198]. The quenching rate for each structure was varied between 0.1 and 10 K ps $^{- 1}$ but no clear dependence of the coordination number and the structural properties on the cooling velocity was found within this range. Subsequently, the structures were geometry optimized with DFT in charge states $q$ = 0, $- 1$ and $+ 1$ to calculate the minimum energy configuration in each charge state. More details about the MD simulations, the training of the employed MLIP and the DFT setup are given in Appendix E.

Comparison with experiment

The model structures correctly reproduce the key structural properties of silicon nitride thin films. The structure factor $S$ and radial distribution function $g (r)$ of a model structure are compared to sample characterizations from scattering experiments [212, 244] in Fig. 5.1(a) and (b).

The average experimental Si–N bond length of 1.75 Å  (indicated by the green vertical line in Fig. 5.1 (c) coincides with the peak of the Si–N bond length distribution over all generated structures. The tail of the distribution at higher values can be attributed to slightly strained bonds in the amorphous network and is in agreement with previous theoretical studies [245, 246]. N–N bonds and Si–Si bonds were completely absent in all of the quenched structures due to the high energies associated with these structures in the training data set of the MLIP. It should be noted that for an accurate evaluation of the charge trapping energetics, also cell parameters should be relaxed to release any internal stress. This was additionally done for the realistic hydrogenated structures as presented in Section 5.2.

In ideal stoichiometric Si $_{3}$ N $_{4}$ , every Si in the network has four neighboring N atoms, while every N has three neighboring Si atoms. The average Si–N and N–Si coordination numbers are 3.91 and 2.93, respectively, which matches experimentally determined coordination numbers of samples with similar mass density (3.87 and 2.91 for amorphous Si $_{3}$ N $_{4}$ samples with $ρ =$ 2.87 g cm $^{- 3}$ [212]) very well. Occasionally, atoms are mutually overcoordinated, meaning that a fivefold coordinated Si is overcoordinated with a fourfold coordinated N. The energy gap between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) depends on the relative number of undercoordinated atoms, as shown in Fig. 5.1(d). With an increasing number of undercoordinated atoms in a single structure, the dangling bonds are more and more likely to introduce states in the band gap, resulting in a reduced HOMO–LUMO gap. In the following, only structures with a well-defined band gap above 3.8 eV, as denoted with a red horizontal line in Fig. 5.1(d), without any states in between are investigated to ensure sufficient quality of the structures and to allow for the analysis of specific single charge trapping processes. The band gaps of the further investigated 24 structures are distributed around $E_{GAP} = 4.08 ± 0.25$  eV, which agrees well with previous DFT calculations [75], but underestimates the experimentally determined band gap values from the literature, ranging from 4.5 to 5.3 eV [58]. This underestimation of the band gap is typical for DFT calculations employing hybrid functionals without explicitly tuning the mixing parameter $α$ to the experimental bandgap [228] as also discussed in Section 3.1.3. Passivation of the remaining dangling bonds with H in agreement with realistic H concentrations considerably widens up the band gap to match experimentally obtained values of silicon nitride thin films as will be discussed in Section 5.2.2.

5.1.2 Electronic structure of intrinsic defects

In this section, the localization of charge near over- and undercoordinated atoms and the projected density of states (PDOS) before and after a single charge capture event in $a$ -Si $_{3}$ N $_{4}$ are analyzed.

Hole traps

The PDOS of two different $a$ -Si $_{3}$ N $_{4}$ structures before and after trapping a hole are shown in Fig. 5.2(a) and (b) (bottom), with the trapping sites before and after a single hole is captured depicted above.

Positive and negative values of the PDOS correspond to the majority and minority spin channels respectively. We find that electronic states near the valence band maximum (VBM) are introduced either by overcoordinated Si or undercoordinated N, which is in agreement with previous theoretical and experimental studies [75, 74, 247]. In the first case, the HOMO is semi-localized and hybridized around the five adjacent N of the overcoordinated Si as shown in Fig. 5.2(a, top), with the isosurface of the orbital drawn at a value of 0.05 e/Å $^{3}$ . When a hole is introduced to the system, the structure relaxes to a new minimum energy configuration, thereby shifting the state at the VBM, now unoccupied, towards the middle of the band gap. Similarly in the second structure, where the HOMO localizes at an undercoordinated N, a hole can be trapped at the twofold coordinated N, thereby shifting the now unoccupied state towards the middle of the band gap as shown in Fig. 5.2(b).

Electron traps

The PDOS of two different $a$ -Si $_{3}$ N $_{4}$ structures before and after capturing an electron are shown in Fig. 5.3(a) and (b) (bottom), with the trapping sites plotted above the PDOS before and after a single electron is captured. Depending on the respective structure, the LUMO orbital is either localized at a Si near a fourfold coordinated N or hybridized between an undercoordinated and a fully coordinated Si, which also agrees with previous findings [75, 74]. After trapping an electron near a fourfold coordinated N, the Si–N distance increases, thereby shifting the state at the conduction band minimum (CBM) towards the middle of the band gap. For some cases, the LUMO is semi-localized between a threefold and a fully coordinated Si as shown in Fig. 5.3(b). After trapping an electron, these Si move closer together, thereby introducing two states in the band gap, one occupied and one unoccupied.

5.1.3 Kohn-Sham defect states

The energies of the Kohn–Sham (KS) states introduced in the band gap by adding a hole or an electron to the amorphous systems are shown in Fig. 5.4.

The energies are given with respect to the VBM of each structure. A normal distribution was fitted to the energies with the fitting parameters shown in the plot. The CBM is given as a band with the energetic distance to the VBM according to the distribution of HOMO–LUMO gaps of the analyzed structures. The energy distribution of the occupied states after electron capture is slightly broader and lower in energy compared to the distribution of unoccupied states after trapping a hole.

5.1.4 NMP characterization of intrinsic defect sites

In the following, the charge transitions at intrinsic sites are analyzed according to the NMP model in the classical limit as described in Section 2.1.4.

Charge transition level

The CTLs for hole and electron capture at the intrinsic trapping sites were calculated according to Eq. (2.12) and are shown in Fig. 5.5 in the context of a Si/Si $_{3}$ N $_{4}$ band diagram.

All CTLs are given with respect to the VBM of the respective $a$ -Si $_{3}$ N $_{4}$ model structure and are located inside the band gap of $a$ -Si $_{3}$ N $_{4}$ . In contrast to the KS states introduced by a trapped charge as shown in Fig. 5.4, the CTL is a thermodynamic property of a defect site as described in Section 2.1.3. The valence band offset between Si $_{3}$ N $_{4}$ and Si of 1.78 eV is taken from x-ray photoelectron spectroscopy experiments on Si(111)/Si $_{3}$ N $_{4}$ samples as reported in [128]. Normal distributions were fitted to the CTLs with the fitting parameters given in the plot. CTLs for hole transfer are narrowly distributed 1.1 eV below the VBM of the Si substrate and can therefore only be charged by applying a negative voltage to the gate of a memory device. This agrees well with experimentally determined hole trap levels of 0.5 and 1.1 eV, measured for low-pressure chemical vapor deposition (LPCVD)-generated Si $_{3}$ N $_{4}$ samples as reported in [248]. The distribution of the CTLs for electron transfer is broader, similar to the introduced defect states in the band gap as shown in Fig. 5.4 and located around the CBM of the Si substrate. The CTLs for trapping electrons compare well with theoretical literature values of intrinsic electron traps distributed around 1.5 eV below the CBM [63] and experimental values from trap spectroscopy by charge injection and sensing (TSCIS) measurements ranging between 0.8 and 1.8 eV below the CBM [249]. Hence, most of these sites can easily trap electrons from the substrate. Defect sites with CTLs above the CBM only efficiently capture charges with the Si substrate if the energy of the localized state is shifted towards the Si band edge by an external electric field as described in Section 2.1.4. The field can be generated by an applied voltage on the gate of, e.g., a silicon-oxide-nitride-oxide-silicon (SONOS) device [8, 59]. Thus, the absolute change in energy $Δ_{S}$ also depends on the position of the defect in the nitride.

Relaxation energy

Relaxation energies of charge transitions at intrinsic defects sites were calculated according to the notation in Section 2.1.3. Therefore, single point energy calculations were performed on relaxed configurations in different charge states and subsequently the energy difference of the relaxed system in the same charge state was evaluated. The relaxation energies for hole capture, hole emission, electron capture and electron emission at over- and undercoordinated atoms are shown in Fig. 5.6(a–d) with the parameters of fitted normal distributions given in the plots.

The distributions of the relaxation energies show that for charge transfer processes involving the emission or capture of electrons, the energies gained by structural relaxations of the systems are roughly 0.5 eV higher than for hole transitions. Furthermore, the energy distributions involving electron transfer are again broader compared to the energy distributions for hole transfer.

Energy barriers

The relaxation energies and CTLs presented in the previous sections are used to model the PECs in different charge states as described in Section 2.1.2 and Section 2.1.4. Subsequently, the energy barriers $Δ E$ are extracted from the minimum energy configurations to the classical barrier at the crossing points of the PECs. The resulting energy barriers are shown as a correlation plot on logarithmic scale for charge capture and charge emission in Fig. 5.7.

The energy barriers for electron transfer, which are shown in Fig. 5.7(a), are given with respect to an electron reservoir at the CBM of a Si substrate ( $Δ_{S} = 0$  eV). By applying a positive voltage to the gate of, e.g., a SONOS device, the Si $_{3}$ N $_{4}$ bands and thus the CTL with respect to the charge reservoir are shifted to lower values by $Δ_{S}$ as discussed in Section 2.1.4. The energy barriers are also plotted for a shift of $Δ_{S} = - 1$  eV in Fig. 5.7(a). Compared to the initial conditions, for which the energy barriers are shown as turquoise circles, the classical energy barriers $Δ E$ for electron emission significantly increase to higher values up to 2 eV, while the $Δ E$ for electron capture almost vanish. Thus, by applying a positive voltage, the intrinsic defect sites can easily trap and store electrons from the CBM of Si. Similarly, the $Δ E$ for hole transfer are shown in Fig. 5.7(b) with the VBM of Si acting as a hole reservoir. Initially, $Δ E$ for hole capture is clearly higher than for hole emission, showing that without an applied bias the intrinsic hole trapping sites are rather unlikely to trap a hole from the Si VBM. The barriers are also plotted for negative bias conditions, changing the energetic difference between CTL and Si VBM by $Δ_{S} = + 1$  eV as a result of the electric field in the oxide. $Δ E$ for hole capture is thereby reduced, while $Δ E$ for hole emission is slightly increased. Compared to electron transfer, the barriers for emitting holes are still significantly lower, resulting in reduced storage time of holes at the intrinsic trapping sites. Due to the small energy barriers and CTLs close to the VBM, the investigated intrinsic hole trapping sites could be related to the experimentally determined hole traps in [248]. In this work, the authors speculated that phonons do not play a significant role in the hole trapping mechanism, which corresponds to low energy barriers in the classical NMP model.