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

5.2 Polarons in the amorphous hydrogenated silicon nitride $a$-Si${}_3$N${}_4$:H

5.2.1 Structure creation of $a$-Si${}_3$N${}_4$:H

The $a$-Si${}_3$N${}_4$ structures generated with the melt-and-quench method as described in Section 5.1.1 were used as a starting point for investigations in hydrogenated silicon nitride. Out of the 100 samples, 20 structures with low portions of under- and overcoordinated atoms ($<4$ %) were chosen due their comparably low number of localized states. Subsequently, the dangling bonds were passivated with H to remove any defect states in the band gap or at the band edges to create the $a$-Si${}_3$N${}_4$:H samples. For all the stoichiometric $a$-Si${}_3$N${}_4$:H samples, the number of Si–H bonds is the same as the number of N–H bonds. Cell relaxations with the hybrid functional were carried out after adding H to the structures to account for the impact of Si–H and N–H bonds already during the calculation of the cell parameters. The optimization of the cell parameters of the quenched structures with the hybrid functional is crucial in order to obtain accurate charge trapping kinetics for the studied precursor sites in DFT. Subsequently, the structures were further geometry optimized with the hybrid functional in different charge states to account for structural relaxations upon polaron formation. More details about computational setup are given in Appendix E.

Comparison with $a$-Si${}_3$N${}_4$:H thin films

The structures required between 2 and 8 H atoms to passivate all the dangling bonds, corresponding to a H concentration of $\sim {10}^{21}$ cm${}^{-3}$. This agrees with experimentally detected H concentrations in silicon nitride thin films, ranging from ${10}^{21}$ to ${10}^{22}$ cm${}^{-3}$ [71, 72, 70, 69]. The mass densities of the investigated sample structures vary from $\rho =$ 3.0 g cm⁻³ to 3.1 g cm⁻³, which is well withing experimentally determined values depending on the method used for sample production (2.6 to 3.2 g cm⁻³ [242, 243]). To validate the structural short- and long-range order, the structure factor $S$ is calculated from the model $a$-Si${}_3$N${}_4$:H structures as described in Section 2.3.2 and compared with $S$ obtained from scattering experiments of silicon nitride thin films [212]. $S$ is in good agreement over the whole range of the scattering vector magnitude $Q$ as shown for a sample structure in Fig. 5.8.

The coordination number of the sample structures is in good agreement with experimentally obtained values as discussed in Section 5.1.1. Motivated by the combined reasons given above, it is concluded that the amorphous model systems are excellent representatives of realistic amorphous Si${}_3$N${}_4$:H thin films.

5.2.2 Electronic structure of $a$-Si${}_3$N${}_4$:H

First, the localization of molecular orbital (MO) in the uncharged amorphous network is investigated. In most cases, charges added to the neutral system occupy semi-localized states at the band edges, leading to their collapse and full relaxation at particular sites during the subsequent atomic relaxation. The degree of localization of an electronic state in an amorphous network can be quantified by calculating the inverse participation ratio (IPR). The IPR was evaluated by taking advantage of the atom centered basis functions ${\varphi }_i$ utilized by the CP2K code. An electronic state $\psi$ is thereby described by a linear combination of the basis functions with ${\psi }_n=\sum _i^N{c}_{ni}{\varphi }_i$, where $N$ is the total number of the atom centered basis functions. The IPR can then be calculated as

$$\begin{array}{}\text{(5.1)}& \mathrm{IPR}({\psi }_n)=\frac{\sum _i^N{c}_{ni}^4}{(\sum _i^N{c}_{ni}^2{)}^2}.\end{array}$$

The IPR of a single model structure is depicted in Fig. 5.9.

This analysis shows that not only the electronic states at the band edges of $a$-Si${}_3$N${}_4$:H are localized, but also several states deeper in the bands, which agrees with previous results in $a$-Si${}_3$N${}_4$ [54] and $a$-Si${}_3$N${}_4$:H [74]. The states at the band edges are well localized for all structures, but the exact progression of the IPR for states close to the band edges can vary as demonstrated in the Supplementary Material of [53]. The intrinsic charge trapping sites that correspond to the localized states will be discussed in the following. It will be show that depending on the respective amorphous structure, additional charges can be trapped at both fully coordinated sites and in the vicinity of Si–H and N–H bonds.

Hole polarons

For eight of the model structures relaxed in the positive charge state, holes were found to (semi-)localize at fully coordinated N sites in the amorphous Si${}_3$N${}_4$:H network as shown in Fig. 5.10.

In the following, semi-localization refers to molecular orbitals (MO) that are almost evenly spread over more than one atom, but not delocalized over the whole structure. First, it is analyzed if all Si (N) around the trapping sites are actually bonded to four (three) N (Si) to exclude any dangling bond-like defects. To confirm the presence of covalent bonds, the maximally localized Wannier functions of the valence band are calculated as shown in Fig. 5.10(a). This method proved to efficiently detect bonds in disordered or amorphous materials [161]. Hereby, a Wannier center between a Si and a N indicates an existing bond between the two atoms. Wannier centers are located between each of the Si–N pairs with an additional Wannier center localized as a lone pair at the N. This shows that a covalent bond exists between all Si–N neighbors near the trapping site. A corresponding Wannier oribtal localized at a N of the trapping site is shown in Fig. 5.10(b). The HOMO is semi-localized over several adjacent N when plotted at an isovalue of 0.05 e/Å${}^3$ as shown in Fig. 5.10(c). When a hole is added to the system, it gets trapped at these N sites, following structural relaxations as shown in Fig. 5.10(d).

The PDOS of the system is plotted in Fig. 5.10 before (e) and after (f) trapping a single hole. Positive and negative values in the PDOS correspond to the spin majority and minority channels, respectively. Due to atomic relaxations after trapping the hole, the KS state at the HOMO, now unoccupied, is shifted towards the middle of the bandgap. Hereby, the charge does not merely fill an already existing state in the bandgap, but rather creates its own localized state. Therefore, it is referred to as a self-trapped hole or hole polaron.

Next, an attempt is made to identify structural features that may lead to the semi-localization of the HOMO. Finding structural predictors for polaron formation at a certain site in amorphous materials like HfO${}_2$ proved to be difficult in the past [76]. Here, the semi-localization of the orbitals makes linking the precursor sites to structural parameters in the random network challenging as well. Analyzing the Si–N bond lengths is not productive, as up to 9 different bonds near the trapping site have to be considered. Therefore, a different attempt to characterize the precursor sites in the amorphous network was made by calculating the partial charges of the atoms. First, the Mulliken charge analysis as implemented in the CP2K code is used to calculate the partial charges of the system. In Fig. 5.11(a), the partial Mulliken charge distribution of all N in the combined sample structures is shown.

The curves in Fig. 5.11(b) are normal distribution fits to the Mulliken charges of the N of each sample structure, where the largest portion of the hole localizes. The Mulliken charges for all N in the sample structures are distributed around $-0.75\text{±}0.07$ eV, while the Mulliken charges of the N of the trapping sites are distributed around $-0.72\text{±}0.07$ eV. This suggests that holes are likely to get trapped near less negatively charged N sites in the amorphous network. Still, the correlation is found to be too weak to give a clear tendency. The partial charges at the higher edge that could not be captured by the normal distribution correspond to N with an attached H. Furthermore, bond orders were analyzed from the ground state charge density with the DDEC6 method as implemented in [250] and are shown in Fig. 5.11(g). Fig. 5.11(h) shows the bond orders of the hole trapping sites, which are distributed comparably slightly higher. Similarly, as most of the bond orders are within the standard deviation of the total distributions, no clear correlation of the N of the hole trapping site with the bond order can be found.

When a second hole is added to the system, it is found that it mostly semi-localizes at different N sites in the amorphous network, introducing a second unoccupied defect state in the band gap. These additional trapping sites include fully coordinated N at a different position (50 % of the structures), as well as N–H bonds and N–N bond formation due to structural distortions near the trapping site. In most cases, it is energetically more favorable for a second hole to localize at one of these other sites than to form a hole bipolaron. These sites correspond to semi-localized states below the VBM with comparably high IPR values as shown in Fig. 5.9 and can therefore also act as precursor sites for hole trapping. Furthermore, as will be discussed in Section 5.2.3, structural relaxations upon trapping a hole are rather small. This is similar to polaron formation in crystalline solids, where charges can get trapped at many lattice sites with only small relaxation energies. Hence, it is concluded that hole polaron formation can occur on several different N sites in the amorphous system and therefore no clear information of the precursor site can be given. Still, it is found that spontaneous polaron formation at the analyzed precursor sites is dominant in the amorphous network. Only when a structure is driven out its equilibrium by random atomic displacements between 0 and 0.25 Å, the hole may form a polaron at different N sites with a higher formation energy. This is discussed in detail in Section 5.2.2.

Electron polarons

Two different types of electron polarons were detected in 16 of the 20 investigated structures. For the first type, the electron localizes at a Si and thereby breaks an already prolonged Si–N bond (12 structures). For the second type, the relaxation of the structure after trapping the electron is more homogeneously distributed to the surrounding amorphous network. An intrinsic electron trapping site with the localized MOs before and after trapping the electron is shown in Fig. 5.12.

Similarly as discussed in Section 5.2.2, it is analyzed if all Si (N) around the trapping sites are actually bonded to four (three) N (Si) by calculating the maximally localized Wannier functions of the valence band as visualized in Fig. 5.12(a). This shows that, indeed, a covalent bond exists between all Si–N neighbors near the trapping site, even for the strained Si–N bond. After adding an electron to the system, the corresponding additional Wannier center is attached to a Si in a dangling-bond like configuration. The corresponding localized electronic MOs before and after electron trapping are shown in Fig. 5.12(c) and (d). In the neutral system, the LUMO is mostly localized around the Si of a stretched Si–N bond. When an electron is added to the system, it localizes at this site, thereby stretching the Si–N bond by $0.43\text{±}0.32$ $\text{Å}$ on average ($\sim$25 % of the average Si–N bond length), with Si–N distance changes ranging from 0.08 to 1.06 Å. The related KS states in the PDOS are shown in Fig. 5.12(e) and (f). The trapped electron occupies the electronic state at the CBM, which is now shifted towards the middle of the band gap after atomic relaxations.

When a second electron is added to the system, it localizes at the same site as the first electron but with opposite spin, resulting in a doubly occupied defect state in the band gap. This is in contrast to polaronic hole trapping, where the second charge carrier mostly localizes at a different site as discussed in Section 5.2.2. By capturing the second electron, additional distortions near the trapping site occur. Most notably, the Si–N distance of the electron trapping site is further extended to 2.48$\text{±}$0.43 Å on average, with additional Si–N prolongations ranging from 0.1 to 0.89 Å. For eight of the structures, the Si of the prolonged Si–N bond becomes stabilized by another Si in the amorphous network, with the bipolaron mostly localized between the two Si. Similar to Ge and Sn [251], although not as pronounced, Si can form a $2+$ oxidation state upon charge trapping, which is in agreement with the formation of an electron bipolaron at a Si site.

Contrary to hole polarons, an obvious correlation between the formation of the electron polaron and structural properties of the precursor sites is found. As discussed above, electrons preferably localize at already strained Si–N bonds. This can be seen in Fig. 5.11, where the total Si–N bond length distribution of all structures is plotted (e) in comparison with the bonds which break upon electron trapping (f). The peak of the Si–N distribution is at 1.74 Å, which matches the peak (1.75 Å) of the radial distribution function from spectroscopy measurements of amorphous Si${}_3$N${}_4$ thin films [212]. The bond length distribution is asymmetric and follows a Weibull distribution as given in Eq. (C.1) with the parameters given in the plot. This asymmetry in the bond length distribution was also shown in previous studies of silicon nitrides with and without H [54, 245, 75]. The majority of the Si–N bonds of the electron trapping sites are at the higher edge of the bond length distribution. The stretched bonds of the trapping site in the neutral charge state have bond lengths of $1.87\text{±}0.06$ $\text{Å}$ on average as shown for comparison in Fig. 5.11(f).

The total Mulliken charge distributions of all N and Si of the combined sample structures are shown in Fig. 5.11(a) and (c), respectively. Normal distribution fitting parameters are given in the plots. Outliers on the lower end of the distribution in Fig. 5.11(c) correspond to Si atoms where a dangling bond was passivated with H. The partial charges of the Si and N atoms of the Si–N bonds that break or stretch most upon localization of an electron (see Fig. 5.12) are depicted in Fig. 5.11(b) and (d), respectively. Hereby, normal distributions were fitted to the Mulliken charges of these sites of all combined structures with the parameters of the fits given in the plots. The Si of the electron trapping sites tend to be more positively charged than the average (1.07$\text{±}$0.08), while the N that separate from the Si upon electron trapping are comparatively more negatively charged ($-0.81\text{±}0.05$). Furthermore, the bond order was investigated with the DDEC6 method as implemented in [250]. The distributions for the bond orders of Si and N of the neutral structures are shown in Fig. 5.11(g), with the bond order distributions of the Si and N of each electron trapping site plotted in Fig. 5.11(h). Both of them are at the lower edge of their distributions, emphasizing that Si and N of the intrinsic electron trapping sites are already comparably weakly bonded.

It it important to note that not all properties discussed above have to be at the maximum or minimum of the respective distributions of the system to indicate a precursor site for electron polaron formation. For example, an added electron does not necessarily localize at the Si of the longest Si–N bond in the structure. Still, a strong tendency can be observed that when a site has a certain combination of bond lengths, partial charges and bond orders on the outer edges of their respective total distributions, the formation of an electron polaron near this site is likely to occur. Similar to the hole polaron, when the structure is sufficiently perturbed, the electron may localize at a different Si–N site in the amorphous network to form a polaron with a higher formation energy. This is discussed in the following.

Hydrogen in $a$-Si${}_3$N${}_4$

Hydrogen is naturally incorporated in silicon nitride thin films as a result of the growth processes. The nitride layer of TANOS and SONOS devices for example is typically created by LPCVD using SiH${}_2$Cl${}_2$ and NH${}_3$ [252]. Other growing techniques involve plasma nitridation including NH${}_3$ and H${}_2$ [253]. While the H content of silicon nitride can be reduced by N${}_2$ annealing [254], this process also creates dangling bonds which might degrade the performance of electronic devices [255]. It is well known from theoretical calculations that MOs localized at the Si-dangling bonds correspond to electronic states at the CBM, while N-dangling bonds correspond to states at the VBM [75, 54]. These states were all passivated with H in the model structures in accordance with experimentally determined Si–H and N–H bond concentrations as discussed in Section 5.1.1. Typically, passivated Si and N-dangling bonds do not introduce states in the band gap but correspond to states inside the electronic bands of $a$-Si${}_3$N${}_4$ [256, 68]. The Mulliken charges of H at Si–H (N–H) bonds in the sample structures are always negative (positive) ($-0.170\text{±}0.081$ and $0.164\text{±}0.038$, respectively).

The impact of the dangling bond passivation on the band gap of the system can be observed by comparing the density of states (DOS) before and after passivation as shown in Fig. 5.13. Hereby, both structures with and without H were separately relaxed with DFT. It can clearly be seen that the band gap widens up due to elimination of these states. Averaged over 20 samples, the band gap increases from $4.08\text{±}0.25$ eV to $4.78\text{±}0.25$ eV as a consequence of the dangling bond passivation.

This is within the range of experimentally determined band gaps of $a$-Si${}_3$N${}_4$:H, ranging from 4.5 to 5.3 eV [58]. The DOS before and after passivation of three additional structures are shown in the Supplemental Material of [53]. The LUMO after passivation is not necessarily localized near the former Si dangling bond, but at a different fully coordinated Si site with e.g. a strained Si–N bond as described in Section 5.2.2. Furthermore, in pure $a$-Si${}_3$N${}_4$, the HOMO is either localized at N dangling bonds or near overcoordinated Si [54]. By eliminating the N dangling bonds with H, the HOMO is now localized at different sites, e.g. intrinsic sites as described in Section 5.2.2. For three of the investigated structures, the HOMO is semi-localized around a Si–H bond as shown in Fig. 5.14 with the corresponding DOS plotted below.

Similar to the hole polaron, the Si–H site can trap a hole, which shifts the states at the VBM, one of them now unoccupied, towards the middle of the band gap due to atomic relaxations. Most notably, the Si–H dimer rotates until the H gets stabilized by a nearby N or Si. Thereby, the Si–H bond length increases by 0.18 Å  on average.

Additional polaron sites

Charge trapping is dominated by the respective process with the lowest energy barrier. To search for possible additional polaron precursor sites, a typical approach for analyzing polaron formation in crystalline materials is applied to five different amorphous sample structures. Therefore, two relaxed neutral systems were perturbed by random atomic displacements between 0 and 0.15 Å for each atom in every direction of the Cartesian coordinate system. The other three samples were perturbed by larger atomic displacements between 0 and 0.25 Å each. The perturbed structures were then relaxed in the positive and negative charge state to search for additional polaron formation sites, which might differ from the ones that form after spontaneous localization. For the structures with small perturbations, the additional charge localized at the exact same site as for the unperturbed after atomic relaxation. Only for the structures with the larger atomic perturbations, the additional charges localized at different sites in the amorphous network due to larger atomic displacements. Still, it is found that the charge trapping mechanism is similar to the spontaneous polaron formation as discussed in the main text (hole semi-localized over several N or at Si–H bond, electron mostly localized at a Si site). The resulting configurations correspond to different local minima of the potential energy surface of the charged systems, which are between 0.25 and 1.5 eV higher in energy. Thus, these polarons are less likely to form than after spontaneous localization of the additional charge.

The two polaron formation sites of a single structure are shown in Fig. 5.15, one after spontaneous localization of an electron (a) and one after perturbing the structure (b).

Hereby, the highest occupied molecular orbitals of the negatively charged systems are plotted at an isovalue of 0.05 e/${\text{Å}}^3$. From these depictions it can also be seen that atomic reconfigurations upon charge trapping occur mostly near the respective trapping site. We thus conclude that the utilized methodology provides comprehensive identification of possible polaron formation sites in $a$-Si${}_3$N${}_4$:H. Even when the system is driven out of its thermal equilibrium due to an external perturbation, the added charge will preferably form a polaron at a comparable trapping site in the amorphous network. However, such charge trapping events are less likely to occur due to the higher formation energies of the respective polarons.

5.2.3 Charge transition levels of polarons

Formation energies in different charge states and related charge transition levels of polaron and hydrogen related defects were calculated according to Eq. (2.11) and Eq. (2.12). The term $\sum _i{\mu }_i{n}_i$ with the chemical potential ${\mu }_i$, which corresponds to the energy needed to add or remove an atom to form a defect, equals to 0 eV for polarons.

The formation energy diagrams for electron and hole polarons as a function of the Fermi level are shown in Fig. 5.16(a), (b) and (d). In these plots, the formation energy of the most stable charge state is drawn as a solid brown line, while the CTLs are depicted by dashed vertical lines. Depending on the respective sample structure, two possible progressions of the formation of the electron polaron were observed. Fig. 5.16(a) shows the formation energy diagram of an electron polaron where the singly charged state is not the most stable at any Fermi energy of the band gap of silicon nitride. Thus, for Fermi energies near the CBM, the formation of an electron bipolaron is energetically favorable. For the other class of polarons and for hole polarons (Fig. 5.16(b) and (d)), a region for the Fermi exists where the singly charged polaron is most stable. For the electron polaron, this corresponds to a 0/$q$/$2q$ transition.

Fig. 5.16(c) shows the statistics of occupied (unoccupied) Kohn-Sham states of electron (hole) traps that are introduced in the band gap after charge trapping as discussed in the previous sections. Electron bipolaron states are doubly occupied. Depending on the sample structure, the total energy of the VBM varies by $\text{±}0.27$ eV. In the following, the energies are plotted with reference to the respective VBM of each structure, which is set to arbitrary 0. The blue color bar depicts the deviation of the HOMO-LUMO gaps of the sample structures. The Gaussian fits serve as a guide to the eye. It is worth noting again that Kohn-Sham levels are electronic energy levels obtained from DFT calculations of a system with fixed atomic configurations and cannot directly be compared to experimental defect levels obtained from deep-level transient spectroscopy (DLTS) or other electrical measurements. To relate defect calculations to experimentally detected trap levels, the thermodynamic CTL, which also accounts for the relaxations upon trapping a charge at the defect site, has to be considered. The KS states, on the other hand, would correspond to optical excitations, when ignoring exciton effects and assuming the KS states have a physical meaning in the single particle picture.

The CTLs are extracted from the formation energy diagrams of each structure and are depicted in Fig. 5.16(e). Normal distributions were fitted to the CTLs of each charge trapping type with the fitting parameters given in the plot. The CTLs are shown in the context of a Si/$a$-Si${}_3$N${}_4$:H band diagram to visualize the energetic positions of the respective charge reservoirs of e.g. a SONOS device [8, 59]. We therefore use a VBM offset of 1.78 eV from x-ray photoelectron spectroscopy experiments on Si(111)/Si${}_3$N${}_4$ samples as reported in [128]. The calculated CTLs for the electron polarons agree well with experimentally determined electron trap levels of SiN thin films, ranging from 0.8 to 1.8 eV below the conduction band [257, 249, 258]. Hole trapping, on the other hand, has been only rarely considered in theoretical studies, as experiments suggest that the energy barrier for hole injection from metal gates is relatively high [63, 259]. Nevertheless, previous experimental works show that hole traps in $a$-Si${}_3$N${}_4$ contribute to the erase cycle of charge trap flash devices by trapping holes from the Si substrate [258, 60, 260] and can therefore not be disregarded.

The CTLs of the hole polarons are on the lower edge of experimental values, which range from 0.5 to 2.8 eV [248, 258, 260]. The CTLs of hole traps near Si–H sites are 0.47 eV higher on average (0.89$\text{±}0.18$ eV) and are therefore well within the range of experimentally determined hole trap levels. Even though the hole trap levels are quite far below the Si VBM, holes can likely be trapped in the nitride layer of e.g. SONOS devices. This is due to the thick tunneling and blocking oxide layers (3–14 nm [60, 8, 57]) employed in these devices, allowing for large shifts of the trap levels by an electric field according to an applied gate bias as for example discussed in [54].

Furthermore, relaxation energies $E_{\mathrm{Relax}}^{q_1/q_2}$ were calculated as described in Section 2.1.3. The results for hole polarons, electron (bi)polarons and hole traps at Si–H sites are presented in Table 5.1. Relaxation energies for electron polarons are in the range of structural defects in Si${}_3$N${}_4$, while the Si–H trapping sites are in the range of H related defects in $a$-SiO${}_2$ [4]. Following the discussion of Section 5.2.2, hole polaron formation is accompanied by relatively weak structural relaxations, comparable to small polarons in crystalline systems.