# 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

## References

- Kazior, T.E. Beyond CMOS: Heterogeneous integration of III-V devices, RF MEMS and other dissimilar materials/devices with Si CMOS to create intelligent microsystems. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2014**, 372, 20130105. [Google Scholar] [CrossRef] [PubMed][Green Version] - Matthews, J.; Blakeslee, A. Defects in epitaxial multilayers: I. Misfit dislocations. J. Cryst. Growth
**1974**, 27, 118–125. [Google Scholar] [CrossRef] - Hull, R. Equilibrium theories of misfit dislocation network in the SiGe/Si system. In Properties of Silicon Germanium and SiGe: Carbon; Number 24 in EMIS Datareviews Series; Inspec: London, UK, 2000. [Google Scholar]
- Fitzgerald, E.A. Dislocations in strained-layer epitaxy: Theory, experiment, and applications. Mater. Sci. Rep.
**1991**, 7, 87–142. [Google Scholar] [CrossRef] - Hull, R.; Stach, E.A.; Tromp, R.; Ross, F.; Reuter, M. Interactions of Moving Dislocations in Semiconductors with Point, Line and Planar Defects. Phys. Status Solidi (a)
**1999**, 171, 133–146. [Google Scholar] [CrossRef] - Stach, E.A.; Schwarz, K.W.; Hull, R.; Ross, F.M.; Tromp, R.M. New Mechanism for Dislocation Blocking in Strained Layer Epitaxial Growth. Phys. Rev. Lett.
**2000**, 84, 947–950. [Google Scholar] [CrossRef][Green Version] - Schwarz, K.W. Discrete Dislocation Dynamics Study of Strained-Layer Relaxation. Phys. Rev. Lett.
**2003**, 91, 145503. [Google Scholar] [CrossRef] - Sakai, A.; Taoka, N.; Nakatsuka, O.; Zaima, S.; Yasuda, Y. Pure-edge dislocation network for strain-relaxed SiGe/Si(001) systems. Appl. Phys. Lett.
**2005**, 86, 221916. [Google Scholar] [CrossRef] - Van Der Merwe, J.H. Crystal Interfaces. Part I. Semi-Infinite Crystals. J. Appl. Phys.
**1963**, 34, 117–122. [Google Scholar] [CrossRef] - Romanov, A.E.; Pompe, W.; Beltz, G.; Speck, J.S. Modeling of Threading Dislocation Density Reduction in Heteroepitaxial Layers I. Geometry and Crystallography. Phys. Status Solidi (b)
**1996**, 198, 599–613. [Google Scholar] [CrossRef] - Wang, G.; Loo, R.; Simoen, E.; Souriau, L.; Caymax, M.; Heyns, M.M.; Blanpain, B. A model of threading dislocation density in strain-relaxed Ge and GaAs epitaxial films on Si (100). Appl. Phys. Lett.
**2009**, 94, 102115. [Google Scholar] [CrossRef] - Skibitzki, O.; Zoellner, M.; Rovaris, F.; Schubert, M.; Yamamoto, Y.; Persichetti, L.; Di Gaspare, L.; De Seta, M.; Gatti, R.; Montalenti, F.; et al. Reduction of Threading Dislocation Density beyond the saturation limit by optimized reverse grading. 2020; Submitted. [Google Scholar]
- Taylor, P.J.; Jesser, W.A.; Benson, J.D.; Martinka, M.; Dinan, J.H.; Bradshaw, J.; Lara-Taysing, M.; Leavitt, R.P.; Simonis, G.; Chang, W.; et al. Optoelectronic device performance on reduced threading dislocation density GaAs/Si. J. Appl. Phys.
**2001**, 89, 4365–4375. [Google Scholar] [CrossRef] - Guha, S.; Madhukar, A.; Rajkumar, K. Onset of incoherency and defect introduction in the initial stages of molecular beam epitaxial growth of highly strained In
_{x}Ga_{1-x}As on GaAs(100). Appl. Phys. Lett.**1990**, 57, 2110–2112. [Google Scholar] [CrossRef] - Mo, Y.; Savage, D.; Swartzentruber, B.; Lagally, M. Kinetic pathway in Stranski-Krastanov growth of Ge on Si(001). Phys. Rev. Lett.
**1990**, 65, 1020–1023. [Google Scholar] [CrossRef] [PubMed] - Shchukin, V.; Bimberg, D. Spontaneous ordering of nanostructures on crystal surfaces. Rev. Mod. Phys.
**1999**, 71, 1125–1171. [Google Scholar] [CrossRef] - Chaparro, S.; Zhang, Y.; Drucker, J. Strain relief via trench formation in Ge/Si(100) islands. Appl. Phys. Lett.
**2000**, 76, 3535–3536. [Google Scholar] [CrossRef] - Capellini, G.; De Seta, M.; Evangelisti, F. SiGe intermixing in Ge/Si(100) islands. Appl. Phys. Lett.
**2001**, 78, 303–305. [Google Scholar] [CrossRef] - Gatti, R.; Uhlik, F.; Montalenti, F. Intermixing in heteroepitaxial islands: Fast, self-consistent calculation of the concentration profile minimizing the elastic energy. New J. Phys.
**2008**, 10, 083039. [Google Scholar] [CrossRef] - Bergamaschini, R.; Tersoff, J.; Tu, Y.; Zhang, J.; Bauer, G.; Montalenti, F. Anomalous Smoothing Preceding Island Formation During Growth on Patterned Substrates. Phys. Rev. Lett.
**2012**, 109, 156101. [Google Scholar] [CrossRef] - Medeiros-Ribeiro, G.; Bratkovski, A.; Kamins, T.; Ohlberg, D.; Williams, R. Shape transition of germanium nanocrystals on a silicon (001) surface from pyramids to domes. Science
**1998**, 279, 353–355. [Google Scholar] [CrossRef][Green Version] - Costantini, G.; Rastelli, A.; Manzano, C.; Songmuang, R.; Schmidt, O.; Kern, K. Universal shapes of self-organized semiconductor quantum dots: Striking similarities between InAs/GaAs(001) and Ge/Si(001). Appl. Phys. Lett.
**2004**, 85, 5673–5675. [Google Scholar] [CrossRef] - Zhang, J.; Montalenti, F.; Rastelli, A.; Hrauda, N.; Scopece, D.; Groiss, H.; Stangl, J.; Pezzoli, F.; Schaeffler, F.; Schmidt, O.; et al. Collective Shape Oscillations of SiGe Islands on Pit-Patterned Si(001) Substrates: A Coherent-Growth Strategy Enabled by Self-Regulated Intermixing. Phys. Rev. Lett.
**2010**, 105, 166102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rovaris, F.; Bergamaschini, R.; Montalenti, F. Modeling the competition between elastic and plastic relaxation in semiconductor heteroepitaxy: From cyclic growth to flat films. Phys. Rev. B
**2016**, 94, 205304. [Google Scholar] [CrossRef] - Alivisatos, A. Semiconductor clusters, nanocrystals, and quantum dots. Science
**1996**, 271, 933–937. [Google Scholar] [CrossRef][Green Version] - Abrahams, M.S.; Weisberg, L.R.; Buiocchi, C.J.; Blanc, J. Dislocation morphology in graded heterojunctions: GaAs
_{1-x}P_{x}. J. Mater. Sci.**1969**, 4, 223–235. [Google Scholar] [CrossRef] - Fitzgerald, E.A.; Xie, Y.; Green, M.L.; Brasen, D.; Kortan, A.R.; Michel, J.; Mii, Y.; Weir, B.E. Totally relaxed Ge
_{x}Si_{1-x}layers with low threading dislocation densities grown on Si substrates. Appl. Phys. Lett.**1991**, 59, 811–813. [Google Scholar] [CrossRef] - Kubin, L.P. Dislocations, Mesoscale Simulations and Plastic Flow; Oxford University Press: Oxford, UK, 2013. [Google Scholar]
- Rudolph, P. Dislocation cell structures in melt-grown semiconductor compound crystals. Cryst. Res. Technol.
**2005**, 40, 7–20. [Google Scholar] [CrossRef] - Horton, M.K. 3D Imaging of Dislocations. Physics
**2017**, 10, 126. [Google Scholar] [CrossRef][Green Version] - Hänschke, D.; Danilewsky, A.; Helfen, L.; Hamann, E.; Baumbach, T. Correlated Three-Dimensional Imaging of Dislocations: Insights into the Onset of Thermal Slip in Semiconductor Wafers. Phys. Rev. Lett.
**2017**, 119, 215504. [Google Scholar] [CrossRef][Green Version] - Rovaris, F.; Zoellner, M.H.; Zaumseil, P.; Schubert, M.A.; Marzegalli, A.; Gaspare, L.D.; Seta, M.D.; Schroeder, T.; Storck, P.; Schwalb, G.; et al. Misfit-Dislocation Distributions in Heteroepitaxy: From Mesoscale Measurements to Individual Defects and Back. Phys. Rev. Appl.
**2018**, 10, 054067. [Google Scholar] [CrossRef] - Rovaris, F.; Zoellner, M.H.; Zaumseil, P.; Marzegalli, A.; Gaspare, L.D.; Seta, M.D.; Schroeder, T.; Storck, P.; Schwalb, G.; Capellini, G.; et al. Dynamics of crosshatch patterns in heteroepitaxy. Phys. Rev. B
**2019**, 100, 085307. [Google Scholar] [CrossRef] - Tersoff, J. Dislocations and strain relief in compositionally graded layers. Appl. Phys. Lett.
**1993**, 62, 693–695. [Google Scholar] [CrossRef] - Tersoff, J. Erratum: Dislocations and strain relief in compositionally graded layers [Appl. Phys. Lett. 62, 693 (1993)]. Appl. Phys. Lett.
**1994**, 64, 2748. [Google Scholar] [CrossRef][Green Version] - Bertoli, B.; Suarez, E.N.; Ayers, J.E.; Jain, F.C. Misfit dislocation density and strain relaxation in graded semiconductor heterostructures with arbitrary composition profiles. J. Appl. Phys.
**2009**, 106, 073519. [Google Scholar] [CrossRef] - LeGoues, F.K.; Meyerson, B.S.; Morar, J.F. Anomalous strain relaxation in SiGe thin films and superlattices. Phys. Rev. Lett.
**1991**, 66, 2903–2906. [Google Scholar] [CrossRef] [PubMed] - LeGoues, F.K.; Meyerson, B.S.; Morar, J.F.; Kirchner, P.D. Mechanism and conditions for anomalous strain relaxation in graded thin films and superlattices. J. Appl. Phys.
**1992**, 71, 4230–4243. [Google Scholar] [CrossRef] - Schwarz, K.W. Simulation of dislocations on the mesoscopic scale. II. Application to strained-layer relaxation. J. Appl. Phys.
**1999**, 85, 120–129. [Google Scholar] [CrossRef] - Sidoti, D.; Xhurxhi, S.; Kujofsa, T.; Cheruku, S.; Reed, J.; Bertoli, B.; Rago, P.B.; Suarez, E.N.; Jain, F.C.; Ayers, J.E. Critical Layer Thickness in Exponentially Graded Heteroepitaxial Layers. J. Electron. Mater.
**2010**, 39, 1140–1145. [Google Scholar] [CrossRef] - Hirth, J.P.; Lothe, J. Theory of Dislocations, 2nd ed.; Krieger Publishing Company: Malabar, FL, USA, 1982. [Google Scholar]
- Lubarda, V.A. Dislocation Burgers vector and the Peach-Koehler force: A review. J. Mater. Res. Technol.
**2019**, 8, 1550–1565. [Google Scholar] [CrossRef] - Kuykendall, W. Investigating Strain Hardening by Simulations of Dislocation Dynamics. Ph.D. Thesis, Stanford University, Stanford, CA, USA, 2015. [Google Scholar]
- Head, A.K. Edge Dislocations in Inhomogeneous Media. Proc. Phys. Soc. B
**1953**, 66, 793–801. [Google Scholar] [CrossRef] - Cai, W.; Arsenlis, A.; Weinberger, C.R.; Bulatov, V.V. A non-singular continuum theory of dislocations. J. Mech. Phys. Solids
**2006**, 54, 561–587. [Google Scholar] [CrossRef] - Rovaris, F.; Isa, F.; Gatti, R.; Jung, A.; Isella, G.; Montalenti, F.; von Känel, H. Three-dimensional SiGe/Si heterostructures: Switching the dislocation sign by substrate under-etching. Phys. Rev. Mater.
**2017**, 1, 073602. [Google Scholar] [CrossRef][Green Version] - Gavazza, S.D.; Barnett, D.M. The self-force on a planar dislocation loop in an anisotropic linear-elastic medium. J. Mech. Phys. Solids
**1976**, 24, 171–185. [Google Scholar] [CrossRef] - Lardner, R.W. Mathematical Theory of Dislocations and Fracture; University of Toronto Press: Toronto, ON, Canada, 1974. [Google Scholar]
- Kaganer, V.M.; Ulyanenkova, T.; Benediktovitch, A.; Myronov, M.; Ulyanenkov, A. Bunches of misfit dislocations on the onset of relaxation of Si
_{0}.4 Ge_{0}.6/Si(001) epitaxial films revealed by high-resolution X-ray diffraction. J. Appl. Phys.**2017**, 122, 105302. [Google Scholar] [CrossRef] - Marzegalli, A.; Brunetto, M.; Salvalaglio, M.; Montalenti, F.; Nicotra, G.; Scuderi, M.; Spinella, C.; De Seta, M.; Capellini, G. Onset of plastic relaxation in the growth of Ge on Si(001) at low temperatures: Atomic-scale microscopy and dislocation modeling. Phys. Rev. B
**2013**, 88, 165418. [Google Scholar] [CrossRef] - Bolkhovityanov, Y.; Deryabin, A.; Gutakovskii, A.; Sokolov, L.; Vasilenko, A. Dislocation interaction of layers in the Ge/Ge-seed/Ge
_{x}Si_{1-x}/Si(001) (x ∼ 0.3–0.5) system: Trapping of misfit dislocations on the Ge-seed/GeSi interface. Acta Mater.**2013**, 61, 5400–5405. [Google Scholar] [CrossRef] - Sidoti, D.; Xhurxhi, S.; Kujofsa, T.; Cheruku, S.; Correa, J.P.; Bertoli, B.; Rago, P.B.; Suarez, E.N.; Jain, F.C.; Ayers, J.E. Initial misfit dislocations in a graded heteroepitaxial layer. J. Appl. Phys.
**2011**, 109, 023510. [Google Scholar] [CrossRef] - Zhang, T. A dislocation in a compositionally graded epilayer. Phys. Status Solidi (a)
**1995**, 148, 175–189. [Google Scholar] [CrossRef]

**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