The Physics of Non–Equilibrium Reliability Phenomena

A.2 Effects of Cell Size

The utilized interface models in this work have a cell size of $16\times 16\times 32\text{Å}$ and contain 473–475 atoms, depending on the number of defects which had to be passivated in the SiO${}_2$ and transition region. Furthermore, the oxide thickness is around $1.1nm$. Assuming the ideal scenario of one interface defect and one oxide defect, the resulting theoretical defect densities are around 1–2 orders of magnitude higher than the commonly assumed value of $N_{\mathrm{it}}=1\times {10}^{12}-1\times {10}^{13}/\mathrm{c}{\mathrm{m}}^2$ and $N_{\mathrm{ot}}=1\times {10}^{19}-1\times {10}^{20}/\mathrm{c}{\mathrm{m}}^3$. Moreover, taking into account that a H anneal during processing additionally leads to the relief of strain at the interface, an artificially large defect density can indeed introduce a higher strain in the structure.

Three different Si/SiO${}_2$ interface model variants have been investigated to explore the effects of cell size, and hence defect density, onto the residual strain at the interfacial transition region, see Fig. A.3. First, the structure chosen for this thesis with a cell size of $16\times 16\times 32\text{Å}$ including a $\sim 11\text{Å}$ thick SiO${}_2$ oxide (1) and $\sim 475$ atoms in total. Second, a model with an increased oxide thickness of $\sim 28\text{Å}$ but the same $xy$ dimensions, with a total cell size of $16\times 16\times 52\text{Å}$ and $\sim 792$ atoms (2). The increased number of oxide layers should facilitate the formation of bulk SiO${}_2$ properties and hence a strain reduction in the $z$ direction effectively allowing the interface to properly relax. Finally, the largest atomistic model used has a total size of $48\times 32\times 32\text{Å}$ which contains $\sim 2832$ atoms. The $x$ and $y$ dimensions were tripled and doubled, respectively, compared to the initial structure, allowing to reduce the density of interface states (assumed one $P_{\mathrm{b}}$ center) by almost one order of magnitude.

Again, to quantify the quality of the interface, the deviated of the highlighted silicon layers with respect to their positions in $c$–Si has been used as a measure of strain and distortion, see Fig. A.3. To provide a fair comparison, the results in Fig. A.3 show the mean lattice distortion of various different structures (6(1), 20(2), 12(3)). Furthermore, all models have been created in the exact same way (see Sec. 3.1) with a final cell optimization using DFT in conjunction with a Pbe functional. As already shown in Sec. 3.1, the smallest models exhibit some severely distorted atoms in the Si transition region with a mean distortion in the first three layers of $⟨\mathrm{\Delta }\mathbf{r}⟩=0.31\text{Å}$ and maximum values above $0.5\text{Å}$. Quite surprisingly, the structures 2) (extended $z$) and 3) (extended $xy$) yield very similar results in terms of average and maximum distortions, see Fig. A.3. The increase in $xy$ reduces the self–interaction of defect configurations at the Si/SiO2 interface which intuitively should reduce the remanent stress. On the other hand, a thicker oxide region, structure 3), exhibits a similar effect. Due to the flexibility of the (bulk) SiO${}_2$ network, residual strain associated with a defect configuration at the interface, can be effectively absorbed resulting in a smoother interface region. In order to be fully confident about the validity of the results, further, systematic investigations need to be performed. Nevertheless, the results suggest that for subsequent studies structures like 2), with an increased oxide thickness, should be used which provide the best tradeoff between credibility and computational efforts.