Simple Algebraic Expressions for the Prediction and Control of High-Temperature Annealed Structures by Linear Perturbation Analysis
Abstract
:1. Introduction
2. Materials and Methods
2.1. Phenomena Formulation
2.2. First Order Perturbation Analysis
- Volume stays constant.
- The initial geometry is an unperturbed, infinitely long void cylinder, and the surface normal points inwards, which is important for the curvature calculation and its sign.
- The analyzed transformation is from an initial cylinder to a set of spheres. The evolution of the geometry is driven by an increasing sinusoidal perturbation of wavelength λ in the longitudinal direction. The evolution is schematically shown in Figure 2.
- The analysis is performed on the first seconds of the transformation. This is already provides if the initial sinusoidal disturbance shrinks or grows.
2.3. Analytical Model of the Void Shape Evolution for Finite Structures
- The first transformation process is obtained through the competition between the effects at the top and at the bottom of the trench.
- The top transformation has the priority. However, a balance between both extremes must be maintained, and this is used for obtaining the corresponding expressions (see Figure 5).
- Once the wavelength at the top has been calculated, the pinch-off occurs at half the maximum wavelength, having the top surface as the reference (see Figure 5).
- The void enclosed below the closing point transforms into one or several final spheres, depending on the aspect ratio of the void. The simplest transformation is shown in Figure 5, where the complete enclosed void volume transforms into a spherical void geometry with the same volume.
- The volume left over the closing point transforms into a step, i.e., a difference of surface level with respect to the initial surface (see Figure 5).
- The only considered final equilibrium geometries are structures with constant curvatures: spheres and planes. No intermediate states can be predicted without a proper process simulation.
- Only the formation of one trench is considered, and no coalescence is assumed. This simplifies the expression of the size of the final void. The distance of the neighboring trenches is sufficient to have a void spherical transformation dominated by the aspect ratio instead of the distance to other trenches. However, the latter effect is not completely neglected, as the distance between trenches affects the step and SON-layer sizes.
3. Results and Discussion
- l1 = l2 = 0.35 µm is obtained from Figure 4 of Sato et al. [3]. In that diagram, the value “a” is given which, according to Figure 2, is the side of the base squared side.
- The length of the trench would be estimated from the parameter “b” of Figure 4 [3], as the x-axis yields the aspect ratio “b/a”. In this case, it is estimated to be LC = 1.98 µm.
- The next step is to calculate which of the proposed models must be used. That comes from the aspect ratio calculation LC/RC, which yields about 8.14. This value lies between 2π (6.28) and 4π (12.57) and thus is the model case 2. In other words, Equations (27)–(31) must be used for the calculation of the final parameters.
- The last unknown parameter is A. A is the surface area of the repeated pattern of the trench as shown in the colored area in Figure 4 of this work. The top trench area is a squared area as shown in Figure 2 of Sato et al. [3]. The information provided by Figures 2 and 3 and the work of Mizushima et al. [1] suggest that the distance between trenches “a + c” would be about 2a (which is equivalent to Dt of Table 1) and the area A would be 2a × 2a or 2l1 × 2l1 = 0.49 µm2.
- As the base of the trench is a square, the equivalent cylindrical radius is given by Equation (25) and equals RC,eq = 0.1975 µm.
- TSTEP is calculated with Equation (29) and the values of RC and A. The obtained value equals TSTEP = 0.16 µm.
- RS is calculated with Equation (30) and the values RC and LC. The obtained value equals RS = 0.34 µm.
- TSON is calculated with Equation (31) and the values LC, RC, and A. The obtained value equals TSON = 0.80 µm.
- Calculated data are then added in Table 3 and compared to the results of Sato et al. [3], Figure 4, which, after an annealing time of 10 min at 1100 °C and 10 Torr, yields the diameter of the final equilibrium spherical void. The estimated measured radius can be inferred from such Figure 4 and is RS = 0.34 µm.
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Number | l1 [µm] | l2 [µm] | LC [μm] | LC/RC | Model Case | Dt,ave [μm] | Ref |
---|---|---|---|---|---|---|---|
1 | 0.25 | 0.55 | ~1.11 | ~5.29 | 1 | ~1.06 | [1,3] |
2 | 0.30 | 0.30 | ~1.24 | ~7.29 | 2 | ~0.60 | [3] |
3 | 0.26 | 0.26 | ~1.21 | ~8.07 | 2 | ~0.52 | [3] |
4 | 0.35 | 0.35 | ~1.28 | ~8.14 | 2 | ~0.70 | [3] |
5 | 0.30 | 0.30 | ~1.92 | ~8.14 | 2 | ~0.60 | [3] |
6 | 0.35 | 0.35 | ~1.98 | ~9.90 | 2 | ~0.70 | [3] |
7 | 0.22 | 0.22 | ~1.19 | ~9.92 | 2 | ~0.44 | [3] |
8 | 0.17 | 0.17 | ~1.13 | ~11.30 | 2 | ~0.34 | [3] |
9 | 0.26 | 0.26 | ~1.81 | ~12.07 | 2 | ~0.52 | [3] |
10 | 0.22 | 0.22 | ~1.73 | ~14.42 | 3 | ~0.44 | [3] |
11 | 0.17 | 0.17 | ~1.63 | ~16.30 | 4 | ~0.34 | [3] |
Number | LC [μm] | RC [μm] | LC/RC | Model Case | Dt,ave [μm] | Ref |
---|---|---|---|---|---|---|
12 | ~2.72 | ~0.36 | ~7.56 | 2 | 1.38 | [16] |
13 | ~3.50 | ~0.30 | ~11.67 | 2 | 1.00 | [51] |
Number | Source | RS [μm] | TSON [μm] | TSTEP [μm] | Ref |
---|---|---|---|---|---|
1 | Literature | - | - | - | [1,3] |
Model | - | - | 0.14 | ||
Abs. Error | - | - | - | ||
Rel. Error | - | - | - | ||
2 | Literature | ~0.29 | ~0.53 | - | [3] |
Model | 0.25 | 0.51 | 0.13 | ||
Abs. Error | 0.04 | 0.02 | - | ||
Rel. Error | 14% | 4% | - | ||
3 | Literature | ~0.26 | ~0.51 | - | [3] |
Model | 0.23 | 0.49 | 0.12 | ||
Abs. Error | 0.03 | 0.02 | - | ||
Rel. Error | 12% | 4% | - | ||
4 | Literature | ~0.30 | - | - | [3] |
Model | 0.27 | 0.53 | 0.16 | ||
Abs. Error | 0.03 | - | - | ||
Rel. Error | 10% | - | - | ||
5 | Literature | ~0.32 | ~0.77 | - | [3] |
Model | 0.31 | 0.78 | 0.13 | ||
Abs. Error | 0.01 | 0.01 | - | ||
Rel. Error | 4% | 2% | - | ||
6 | Literature | ~0.34 | ~0.80 | - | [3] |
Model | 0.34 | 0.80 | 0.16 | ||
Abs. Error | 0.00 | 0.00 | - | ||
Rel. Error | 0% | 0% | - | ||
7 | Literature | ~0.22 | ~0.49 | - | [3] |
Model | 0.21 | 0.48 | 0.10 | ||
Abs. Error | 0.01 | 0.01 | - | ||
Rel. Error | 5% | 3% | - | ||
8 | Literature | ~0.18 | ~0.47 | - | [3] |
Model | 0.18 | 0.46 | 0.08 | ||
Abs. Error | 0.00 | 0.01 | - | ||
Rel. Error | 0% | 3% | - | ||
9 | Literature | ~0.28 | ~0.74 | - | [3] |
Model | 0.28 | 0.74 | 0.12 | ||
Abs. Error | 0.00 | 0.00 | - | ||
Rel. Error | 0% | 0% | - | ||
10 | Literature | ~0.25 | ~0.72 | - | [3] |
Model | 0.24 | 0.73 | 0.12 | ||
Abs. Error | 0.01 | 0.01 | - | ||
Rel. Error | 4% | 2% | - | ||
11 | Literature | ~0.21 | ~0.67 | - | [3] |
Model | 0.20 | 0.72 | 0.11 | ||
Abs. Error | 0.01 | 0.05 | - | ||
Rel. Error | 5% | 8% | - |
Number | Source | RS [μm] | TSON [μm] | TSTEP [μm] | Ref |
---|---|---|---|---|---|
12 | Literature | ~0.55 | ~0.87 | - | [16] |
Model | 0.54 | 1.15 | 0.24 | ||
Abs. Error | 0.01 | 0.28 | - | ||
Rel. Error | 2% | 33% | - | ||
13 | Literature | ~0.58 | ~1.32 | ~0.10 | [51] |
Model | 0.56 | 1.40 | 0.27 | ||
Abs. Error | 0.02 | 0.08 | 0.17 | ||
Rel. Error | 4% | 7% | 170% |
Number of ESS | This Work Analytical Model | Sato et al. [3] | Müller et al. [51] | Nichols et al. [52] | Grau et al. [17] | McLean et al. [53,54] | Garín et al. [50] | Stapley and Beevers [55] |
---|---|---|---|---|---|---|---|---|
0 | ≤π ≈ 3.14 | ≤3.00 | ≈2.76 | - | ≤2.22 | - | ≈5.00 | ≤3.00 |
1 | ≤(3·21/2 + 2)π/2 ≈ 9.81 | ≤9.50 | ≈8.28 | ≤7.20 | ≤6.66 | ≤8.00 | ≈10.00 | ≤6.00 |
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Grau Turuelo, C.; Breitkopf, C. Simple Algebraic Expressions for the Prediction and Control of High-Temperature Annealed Structures by Linear Perturbation Analysis. Math. Comput. Appl. 2021, 26, 43. https://doi.org/10.3390/mca26020043
Grau Turuelo C, Breitkopf C. Simple Algebraic Expressions for the Prediction and Control of High-Temperature Annealed Structures by Linear Perturbation Analysis. Mathematical and Computational Applications. 2021; 26(2):43. https://doi.org/10.3390/mca26020043
Chicago/Turabian StyleGrau Turuelo, Constantino, and Cornelia Breitkopf. 2021. "Simple Algebraic Expressions for the Prediction and Control of High-Temperature Annealed Structures by Linear Perturbation Analysis" Mathematical and Computational Applications 26, no. 2: 43. https://doi.org/10.3390/mca26020043