# Computational Analysis of Low-Energy Dislocation Configurations in Graded Layers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Methods

## 3. Results and Discussion

#### 3.1. Constant Composition Layer

#### 3.2. Step Graded Profiles

#### 3.3. Linear Grading

#### 3.3.1. Single Dislocation

#### 3.3.2. Pileup Shapes as Function of Dislocation Number

#### 3.3.3. Effect of Grading Rate

#### 3.3.4. Comparison with Tersoff Model

#### 3.3.5. Different Burgers Vector

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stress Field Expressions

#### Appendix A.1. Periodic Stress Functions for Dislocations near a Free Surface

#### Appendix A.1.1. Contributions from b_{x}

#### Appendix A.1.2. Contributions from b_{y}

#### Appendix A.1.3. Periodic Stress Functions

#### Appendix A.2. Non-Singular Periodic Stress Functions

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**Figure 1.**Minimization procedure applied to a random dislocation distribution inside a constant composition layer. Simulation plane is perpendicular to dislocation line. Starting condition of 14 dislocations with the same Burgers vector is shown in (

**a**). In (

**b**) the final result of the minimization shows the full relaxation of the SiGe layer with a Ge concentration of 6.25%. In (

**c**,

**d**) starting condition and final results are shown for 14 dislocations with equal distribution among the two possible Burgers vectors, coalescing into an array of 7 edge dislocations.

**Figure 2.**Evolution of the typical low-energy dislocation configurations with increasing number of concentration steps. (

**a**) Two-step profile showing equispaced arrays of dislocations at the interfaces. (

**b**) Local energy minima obtained from the unconstrained steepest descent procedure applied to 100 initial random dislocation configurations. In the inset is shown the energy variation for a rigid shift of the upper dislocation array with respect to the lower one. (

**c**) Four-step profile, dislocations are still placed at the interfaces but start to organize themselves, losing the perfect spacing of the arrays. (

**d**) Eight-step profile, pileup organization is more pronounced and dislocations begin to place outside the interfaces (red-circled dislocation). (

**e**) Energy as function of the angle. The plot is obtained by considering two dislocations and moving one in circle around the other one as sketched in the inset.

**Figure 3.**Position of the first dislocations introduced in the system (

**a**). Comparison between the analytical model and the result of the steepest descent procedure. (

**b**) Contour plot showing the minimum energy position for the second dislocation when periodic stress field expressions are used.

**Figure 4.**Effect of the dislocation number on the typical pileups shapes. Grading rate has been kept fixed at $83.3\%\phantom{\rule{3.33333pt}{0ex}}\mathsf{\mu}$m${}^{-1}$ in a 1200 nm simulation cell (half of the simulation cells is shown). Film thickness is 600 nm. The number of dislocations per simulation cells are 7 (

**a**), 28 (

**b**), 56 (

**c**), and 90 (

**d**). Colormaps show ${\epsilon}_{xx}$ component of the strain field.

**Figure 5.**Comparison between different grading rates of 20.83% $\mathsf{\mu}{\mathrm{m}}^{-1}$ (

**a**), 41.67% $\mathsf{\mu}{\mathrm{m}}^{-1}$ (

**b**), and 83.33% $\mathsf{\mu}{\mathrm{m}}^{-1}$ (

**c**). Respective final concentrations of Ge are 12.5%, 25%, and 50%. Ratios between dislocation number in the unit cell and grading rate have been kept fixed. Colormaps show the values of ${\epsilon}_{xx}$ component of the strain field.

**Figure 6.**Comparison with Tersoff mean field model. The energies of the 500 collected local minima (relative to the lowest) (

**a**). Total energy of the obtained minimum configurations plotted against the value of our Tersoff predictor and the variance of the dislocation number per pile-up (

**b**). Results naturally cluster into subsets of minima depending on the number of pileups that remain in the cell. The insets (

**c**) show examples of the dislocation configurations belonging to each of the subsets found, with the corresponding colors.

**Figure 7.**Energy of configurations as a function of the number of edge dislocations present at the end of the steepest descent procedure (

**a**). Initial conditions (different Burgers vector or all edge dislocations) have been differentiated by the color of the point. (

**b**) Strain map, ${\epsilon}_{xx}$ component of the minimum energy configuration found.

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**MDPI and ACS Style**

Lanzoni, D.; Rovaris, F.; Montalenti, F.
Computational Analysis of Low-Energy Dislocation Configurations in Graded Layers. *Crystals* **2020**, *10*, 661.
https://doi.org/10.3390/cryst10080661

**AMA Style**

Lanzoni D, Rovaris F, Montalenti F.
Computational Analysis of Low-Energy Dislocation Configurations in Graded Layers. *Crystals*. 2020; 10(8):661.
https://doi.org/10.3390/cryst10080661

**Chicago/Turabian Style**

Lanzoni, Daniele, Fabrizio Rovaris, and Francesco Montalenti.
2020. "Computational Analysis of Low-Energy Dislocation Configurations in Graded Layers" *Crystals* 10, no. 8: 661.
https://doi.org/10.3390/cryst10080661