1. Introduction

As a result of the aggressive scaling of non-volatile memories, traditional floating-gate flash memory devices present several problems, such as the inability to scale the tunnel oxide sufficiently, cross-coupling between cells and long term reliability^[1]. Owing to the smaller module size and simpler integration processes, Si$_{3}$N$_{4}$-based charge trapping memories (CTMs) with the SONOS structure (silicon–oxide–nitride–oxide–silicon)^[2], as shown in Fig. 1, have attracted much attention. The CTM stores up charge carriers at the atomic scale, making the charges carriers spatially separated. The discrete storage has robust resistance against charge leakage. Due to the special property to localize electrons and holes for about 10 years, silicon nitride is widely used as the trapping layer in CTM. The electronic analysis and capturing property of intrinsic defects in silicon nitride have been widely studied in experiments and through theoretical work. Robertson and Powell have calculated the local density of states for each of the main defects by using a tight-binding recursion method^[3]; Gritsenko et al. have studied electron and hole trapping in Si$_{3}$N$_{4}$ by using photoluminescence experiments and quantum-chemical simulation^[4]; Vianello et al. have investigated the trapping properties of standard and Si-rich Si$_{3}$N$_{4}$^{[5, 6]}; Valentin et al. have explored the transition levels of intrinsic defects in Si$_{3}$N$_{4}$^[7]. However, most of these studies concentrate on the intrinsic defects in silicon nitride. In conventional fabrication processes, it has been experimentally observed that oxygen is an important heteroatom in a Si$_{3}$N$_{4}$ layer^[5]. Yet, there is no thorough understanding about the memory action of oxygen-related defects in Si$_{3}$N$_{4}$. In this paper, we theoretically investigate the properties of oxygen defects in the Si$_{3}$N$_{4}$ trapping layer in SONOS-type memories, especially their influence on trapping behaviors. Because the atomic radius of oxygen is close to that of nitrogen, we mainly consider substitutional defects, i.e., oxygen substituting nitrogen (O$_{\rm N})$.

2. Computational details

2.1 Models and methods

Ab initio DFT calculations are performed based on the projector augmented wave method, as implemented in the Vienna ab initio simulation package (VASP)^{[8, 9]}. The generalized gradient approximation (GGA) is used to describe the exchange-correlation functional^[10]. A plane-wave basis set with cutoff energy of 500 eV and a grid of 2 $\times$ 2 $\times$ 2 special points for Brillouin zone integrations is adopted; geometries of the supercells are fully relaxed until atomic forces are smaller than 0.015 eV/Å.

As a $\beta$ phase is more stable at all temperatures with respect to the other phases (the hexagonal $\alpha$ and the cubic $\gamma )$, we chose the hexagonal primitive cell of $\beta$-Si$_{3}$N$_{4}$ with group space P$_{\rm 63/m}$ as the basic computational model. The bulk lattice parameters have been optimized: $a=b=$ 7.60440 Å, $c=$ 2.90630 Å. Isolated oxygen-related defects are modeled by using a 280-atom supercell. The supercell is obtained by a 2 $\times$ 2 $\times$ 5 translation from the primitive unit cell. For $\beta$ phase Si$_{3}$N$_{4}$, two different types of N atoms exist, as shown in Fig. 2: type 1, two equivalent bonds of 1.73610 Å and a third one of 1.73597 Å; type 2, three equivalent bonds of 1.73014 Å. As a result, there are two kinds of substitutional oxygen defects: oxygen substitutes the nitrogen of type 1 (O$_{\rm N\_1})$ and type 2 (O$_{\rm N\_2})$.

2.2 Interaction between substitutional oxygens

Because of Coulomb force, the interaction between defects should not be ignored. Here we calculate the interaction energy between substitutional oxygens at different sites and distances using the formula:

$$\begin{equation} \label{eq1} E_{\rm int} =E_2 +E_0-2E_1, \end{equation}$$

(1)

where $E_{0}$, $E_{1}$, and $E_{2}$ are the total energy of the supercell containing zero, one and two substitutional oxygens respectively. Because of two different substitutional oxygens (O$_{\rm N\_1}$ and O$_{\rm N\_2})$ in $\beta$ phase silicon nitride, there are several situations for the calculation of $E_{2}$. Positive value of $E_{\rm int}$ indicates the interaction is repulsive, while negative value means attractive interaction^[11].

2.3 Capturing properties

2.3.1   Trap ability

To simulate the operating process of a CTM device, namely, defect traps and emission charge carriers (electrons or holes), we have simulated charge injection (program) and removal (erase) by changing the total number of electrons in the supercell. Neutral charge state D$^{0}$ stands for unprogrammed status of the defect and it turns as D$^{-}$ (negatively charged state) or D$^{+}$ (positively charged state) after program. A negative or positive sign depends on the operational charge carrier. Negative (positive) sign indicates that electron (hole) acts as operational charge carrier. The ability of different defects to trap charge carriers differ from each other. Here we apply three methods to study the distinct trap abilities: trap energy, modified Bader analysis, and charge density difference.

The trap energy is the difference of the electron affinities (or ionization energies in the case of hole capturure) in a perfect supercell and in a supercell containing defects. The definition is shown as follows^[4]:

$$\begin{equation} \begin{split} \label{eq2} {}& \Delta E_{\rm electron}=\chi_{\rm bulk}^{\rm e}-\chi_{\rm defect}^{\rm e}, \quad \Delta E_{\rm hole}=\chi _{\rm bulk}^{\rm h}-\chi _{\rm defect}^{\rm h} . \\[1mm]& \chi^{\rm e}=E^{q = -1}-E^{q = 0}, \quad \chi^{\rm h}=E^{q = +1}-E^{q = 0}. \end{split} \end{equation}$$

(2)

$E^{q, =, 0}$, $E^{q, =, -1}$, $E^{q, =, +1}$ are the total energy of the supercell or the supercell containing defects in neutral, negative, and positive charge states respectively. The subscript depends on different systems (bulk system or defect system). If the value of the trap energy is positive, then trapping an electron or hole will result in the release of energy. A negative value means absorption of energy. From the point of energy minimum, it is obvious that the negative value implies the incapacity to trap a corresponding carrier for a certain defect. The trap energy can also quantify the ability to trap different charge carriers. A higher value indicates a stronger capacity.

Bader charge analysis is a good method to approximate the total electronic charge of an atom by calculating the charge enclosed within the Bader volume^[12-14]. To charge trap memory, it is essential to discuss where the injected charge carrier locates in the storage cell. Hence we apply modified Bader charge, which is the subtraction of an atom's Bader charge in charged and neutral state at the geometry of a charged state (D$^-$ or D$^+$). By the subtraction, the injected operational charge carrier is extracted from the total Bader charge. Thus, modified Bader analysis can locate the atom which traps the injected charge carrier. Moreover, we can quantify the portion of the distribution by comparing different atoms' modified Bader charge.

Charge density difference is another novel method to visualize the distribution of trapped charge carriers around the defect^[4]. Compared with modified Bader analysis, deformation charge density provides a visualized spatial distribution. The image is obtained by subtracting the total charge density in the charged state from that in the neutral state at the geometry of the charged state (D$^{-}$ or D$^{+})$.

2.3.2   Endurance

After the program/erase cycles, defects lead to reversible and irreversible structural changes. The irreversible structural change is regarded as the microcosmic explanation for CTM's bad endurance performance^[15]. To distinguish the reversible change from the irreversible structural change of oxygen defects, firstly we compare the initial neutral structure ($S_{\rm init})$ and the structure after a P/E cycle ($S_{\rm final})$. If $S_{\rm init}$ and $S_{\rm final}$ are the same, then the structural change is reversible. Otherwise, it is irreversible. A configuration coordinate diagram is a good way to interpret the two types of structural change. The overview of configuration coordinate diagram is shown in Fig. 3. In the diagram, the energy in a specific charged state (D$^{-}$ or D$^{+})$ is plotted as a function of a generalized coordinate which is an overall measure of all the changes in the defect atom and its neighbor's coordinates^{[16, 17]}. Based on the configuration coordinate diagram, Figure 3 represents the program ($S_{\rm init}$ $\to$ $S_{1}$ $\to$ $S_{2})$ and erase ($S_{2}$ $\to$ $S_{3}$ $\to$ $S_{\rm final})$ processes. In Fig. 3, $S_{\rm init}$ is the initial unprogrammed structure; $S_{1}$ is the transient structure after carrier injection. They are the same from the point of the configuration coordinate. This is due to the fact that the movement of charge carriers is much faster compared with nuclear motion. After relaxation, structure $S_{1}$ turns to be $S_{2}$, which is the most stable status in a charged state. During the erase process, $S_{2}$ changes to be $S_{3}$ (transient structure) and $S_{3}$ will relax to be $S_{\rm final}$ eventually. If $S_{\rm init}$ and $S_{\rm final}$ are different at the $X$-axis, then the change is irreversible, just as the case of Fig. 3(a); While Figure 3(b) shows a reversible change, for $S_{\rm int}$ is equal to $S_{\rm final}$. For Fig. 3(a), of immediate concern is the energy barrier between $S_{\rm init}$ and $S_{\rm final}$. The barrier stops $S_{\rm final}$ relaxing back to $S_{\rm init}$, which is the nature of an irreversible structural change. Here, the nudged elastic band method (NEB) is used to calculate the barrier. The NEB is an efficient method for finding saddle points and minimum energy paths (MEP) between known reactants and products^[18]. In our calculation, $S_{\rm init}$ is regarded as the reactant and $S_{\rm final}$ as the product.

3. Results and discussion

3.1 Interaction energy

The interaction energy between O$_{\rm N}$s is calculated with respect to distance. The result is shown in Table 1. O$_{\rm N\_1}$–O$_{\rm N\_1}$ represents the interaction of two O$_{\rm N\_1}$s, and a similar meaning can be applied to O$_{\rm N\_1}$–O$_{\rm N\_2}$ and O$_{\rm N\_2}$–O$_{\rm N\_2}$. The definition of distance is according to the substituted N atoms. To show the trend of interaction energy value, we plot $E_{\rm int}$ as a function of the distance in Fig. 4. It can be seen that interaction energy depends both on the type of O$_{\rm N}$ and distance. Firstly, the interaction energy shows a relatively high value in 3 specific sites, i.e., O$_{\rm N\_1}$–O$_{\rm N\_1}$ at 2.808 Å (–2.7 eV), 2.906 Å (–2.82 eV) and O$_{\rm N\_1}$–O$_{\rm N\_2}$ at 2.846 Å (–2.75 eV). The values are greater than that of others by at least 0.44 eV. However, the value between O$_{\rm N\_2}$s varies around zero. The high negative value indicates strong attractive interaction between the substitutional oxygen defects, which implies O$_{\rm N}$ is most likely to form clusters at the three sites. The fact that O$_{\rm N}$ prefers to a cluster differs from the intuitive understanding that substitutional oxygens are separated from each other. In order to discuss the clusters clearly, we name the cluster at the three sites as clusters A, B and C correspondingly. Figure 5 shows a schematic of the clusters.

Based on the comprehensive grasp of Table 1, Figs. 4 and 5, we find the oxygen defect clusters favorably locate near the substituted N of type 1. In summary, clusters A and B originate from the nearest and second nearest O$_{\rm N\_1}$s, respectively; cluster C consists of O$_{\rm N\_1}$ and its nearest O$_{\rm N\_2}$.

3.2 Capturing properties

3.2.1   Trap ability

To find out about the distinct ability to trap charge carries of the 3 different clusters, we calculate trap energy ($\Delta E$) by Eq. (2). The result is shown in Table 2. Trap energy of clusters A and C for both charge carriers are positive, which confirms that the two clusters are able to trap both electrons and holes. As the processes are energetically favorable, i.e., release energy, they are amphoteric defects. For cluster B, only the hole trapping energy is positive and the electron trap energy is about zero, which means cluster B can only trap holes. Meanwhile, by making a comparison between electron trap energy and hole trap energy for a certain cluster, we find the value of trap energy for holes is always higher than that for electrons, which indicates that clusters are much stronger for trapping holes.

The special interest is the location of the trapped carriers around the defect. Here, we are the first to adopt modified Bader analysis to study the carrier location. By Bader analysis, we can locate and quantify the carrier distribution on certain atoms. Figure 6 shows the result of modified Bader charge on the 3 clusters. The amount of electrons (holes) is plotted as a function of atom number. For cluster A, the electron distribution peaks at atoms of No.194 (0.11), No.218 (0.1) and No.229 (0.22), yet the distribution on each of the rest atoms is almost zero; it indicates that the three atoms in cluster A (No.194, No.218, and No.229) have a special ability to trap electrons. And the hole distribution reaches a peak (about 0.3) at the atom of No. 229, it reveals that only No. 229 atom in cluster A can locate holes. As a whole, cluster A is an amphoteric defect. By a similar analysis on clusters B and C, we conclude that cluster B can only trap holes and cluster C is amphoteric.

As a supplement to the modified Bader analysis above, we also calculate the charge density difference and visualize it, as shown in Fig. 7. The left side of each section in Fig. 7 is the electron distribution and the right side is that of holes. The cloud is the isosurface level of the charge density difference. All the points with densities larger than the level lie inside the isosurfaces whereas those with densities smaller than the level are situated outside the isosurface. The green cloud stands for positive value, while purple is negative. After charge carriers are trapped by defects; charge density in some areas increases, yet it decreases in some other areas due to structural changes after trapping. We should point out that all the isosurface levels are at the same value for a certain carrier. Because the trapped carriers are around defects, here we emphasis the charge density difference around defective area. In these images, firstly we find that cluster B it not able to trap electrons, as there is no electron distribution in Fig. 7(b); and for the other cases, we can clearly distinguish which atom or atoms the charge carriers distribute on. Based on a comprehensive analysis on the increase and decrease of charge density around a certain atom, we label the atoms the charge carriers are most likely to distribute. By comparing the labeled atoms in Figs. 7 and 6, we find that the result of charge density difference analysis is consistent with that of modified Bader analysis.

Summing up the three methods discussing trap ability, i.e., trap energy, modified Bader analysis and charge density difference, we draw the conclusion that clusters A and C are amphoteric defects, and cluster B only has the ability to trap holes. In addition, for all the three clusters, the trapping capability for hole is much stronger than that for electrons.

3.2.2   Endurance

Owing to two different charge carriers, we should differentiate their influences on endurance.We first compare $S_{\rm init}$ with $S_{\rm final}$ for each cluster. The comparison result is shown in Table 3. The result demonstrates that there is no structural change after hole operation for all clusters; that is, all three clusters have a fine endurance against the existence or absence of holes. In other words, a hole is a good type of operation carrier for oxygen defects in terms of endurance. If an electron acts as an operation carrier, we find $S_{\rm init}$ and $S_{\rm final}$ is different for cluster C in Table 3, i.e., the structure undergoes distortion after a P/E cycle. The structural distortion leads to endurance degeneration. Obviously, for an oxygen cluster, an electron is a bad operation charge carrier in terms of endurance, which originates from the deterioration of cluster C.

To further study the mechanism of the bad endurance of cluster C against electron operation, we adopt the NEB method to plot a configuration coordinate diagram, as shown in Fig. 8. It displays a drastic atomic move after an electron program/erase cycle, as shown by the circled dashed line in Fig. 8. The atomic motion leads to an energy barrier of about 0.2 eV between $S_{\rm init}$ and $S_{\rm final}$. In order to relax to the initial structure, the final structure has to overcome the barrier. As a general rule, the barrier height is too high to be crossed. Thus, $S_{\rm final}$ cannot relax to $S_{\rm init}$, and then the structure of cluster C deteriorates after P/E cycles. In summary, the bad endurance of cluster C against electron operation stems from the high energy barrier between $S_{\rm init}$ and $S_{\rm final}$.

4. Conclusion

In summary, based on first principles calculations, oxygen defects in silicon nitride and their memory-related properties have been investigated intensively. We reveal that substitutional oxygen tends to form clusters at three specific sites due to the intense attractive interaction between defects. All three clusters have a special ability to trap holes, whereas only two of them can trap electrons. In addition, the capacity of trapping holes prevails over that of electrons in the two amphoteric oxygen clusters. From the aspect of endurance, hole operation in the three clusters has no influence on the structural change, indicating excellent endurance; for the two clusters capable of trapping electrons, one of them causes an irreversible structural change after electron program/erase cycles, which is the origin of device degeneration. In order to achieve a better capturing property, i.e., more powerful trapping ability and excellent endurance, we draw an insightful conclusion that holes are the optical operation carrier for oxygen defects in SiN-based charge trapping memories.

Acknowledgements: The authors acknowledge the support of the Supercomputing Center of USTC and the Supercomputing Center of Anhui University, as the centers performed all the numerical calculations in this paper.