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Global transition path search for dislocation formation in Ge on Si(001)
Emile Maras, Oleg Trushin, Alexander Stukowski, Tapio Ala-Nissila, Hannes Jonsson
TL;DR
The paper addresses how to find minimum-activation-energy transition paths with multiple intermediate minima for misfit dislocation formation in strained Ge on Si(001). It globally optimizes candidate paths using generated minimum-energy structures and modified nudged elastic band calculations, identifying several 90° dislocation mechanisms, including one with a 26% lower activation energy than isolated 60° dislocation half-loop nucleation.
Problem
The challenge is selecting intermediate local minima and connecting them into a transition path with minimal overall activation energy.
Method
The approach generates intermediate minima through heredity transformations and local-minimum identification along minimum energy paths calculated with a modified nudged elastic band method.
Results
26% smaller activation energy is calculated for the detailed 90° misfit-dislocation mechanism than for half-loop nucleation of an isolated 60° dislocation.
Takeaways & Limitations
The method identifies several mechanisms connecting a homogeneously strained Ge film to one containing a misfit dislocation, including a previously unreported mechanism.
Takeaways & Limitations
The parameterizations did not include dislocations in their training sets, so the associated potentials should not be extrapolated to dislocation configurations.
Abstract
from arXiv · showhide
Global optimization of transition paths in complex atomic scale systems is addressed in the context of misfit dislocation formation in a strained Ge film on Si(001). Such paths contain multiple intermediate minima connected by minimum energy paths on the energy surface emerging from the atomic interactions in the system. The challenge is to find which intermediate states to include and to construct a path going through these intermediates in such a way that the overall activation energy for the transition is minimal. In the numerical approach presented here, intermediate minima are constructed by heredity transformations of known minimum energy structures and by identifying local minima in minimum energy paths calculated using a modified version of the nudged elastic band method. Several mechanisms for the formation of a 90° misfit dislocation at the Ge-Si interface are identified when this method is used to construct transition paths connecting a homogeneously strained Ge film and a film containing a misfit dislocation. One of these mechanisms which has not been reported in the literature is detailed. The activation energy for this path is calculated to be 26% smaller than the activation energy for half loop formation of a full, isolated 60° dislocation. An extension of the common neighbor analysis method involving characterization of the geometrical arrangement of second nearest neighbors is used to identify and visualize the dislocations and stacking faults.
1. Introduction
The paper frames 90° misfit-dislocation formation in strained Ge/Si(001) as a complex transition requiring global selection and ordering of intermediate minima. It introduces a global path-search procedure that identifies several mechanisms, including one with a lower activation energy than isolated 60° half-loop nucleation.
- Motivation: Transitions in complex atomic systems involve multiple intermediate minima connected by minimum energy paths.The optimization challenge is selecting and ordering minima to minimize the largest energy increase along the transition.
- Motivation: 90° misfit dislocations efficiently relax strain in Ge/Si(001) films and can support films with low threading-dislocation density.Experiments report regular arrays of 90° interface dislocations and high-performance device applications.
- Prior mechanisms: 90° misfit-dislocation formation is indirect because these dislocations are sessile while dislocations preferentially nucleate at the surface.Earlier mechanisms generally form two complementary 60° dislocations before their reaction into a 90° dislocation.
- Approach: The paper performs global rather than local transition-path optimization by generating candidate structures through heredity transformations and modified NEB calculations.Local minima identified along minimum energy paths are also used as intermediate candidates.
- Contributions: The method identifies several 90°-dislocation formation mechanisms and details one previously unreported path with a single large-scale barrier.Its activation energy is 26% smaller than half-loop nucleation of an isolated 60° dislocation; modified CNA identifies dislocations and stacking faults.
2. Methods
The study models Ge on Si(001) with an empirical Stillinger–Weber potential and searches transition paths by iteratively generating and connecting low-energy intermediate configurations. The resulting mechanism for 90° misfit-dislocation formation proceeds through half-loop nucleation, partial-dislocation reactions, and growth of the final interface dislocation.
- 2.1. Model system: The atomic interactions are described by the Stillinger–Weber potential, which is computationally practical for the large system but overestimates transition-state and energy-barrier quantities.The paper attributes this overestimate to the potential’s treatment of rehybridization and notes that the calculated formation mechanism has an activation energy of 39.8 eV.
- 2.2. Global optimization procedure: The path-search procedure combines heredity transformations with NEB calculations to generate low-energy intermediates and connect them through minimum-energy paths.NEB energy dips reveal additional minima, which are added to the candidate set and reused as parent configurations for subsequent transformations.
- 2.2. Global optimization procedure: Cutting planes are selected to generate configurations likely to occur between the initial commensurate film and final 90°-dislocation state; the best results use the (111)-type family.Automating this choice remains difficult because the optimization must handle both configurations and paths while accounting for energy and connectivity.
- 2.3. Results: A heredity-generated configuration yields a localized dislocation-and-stacking-fault seed, avoiding the overextended half-loop produced by direct NEB interpolation.The direct interpolation path exceeds 50 eV and depends on system size, whereas the localized seed has energy almost 120 eV lower than the initial configuration.
3. Discussion
The discussion identifies conditions that lower the activation energy for 90° misfit-dislocation formation, while noting limitations from empirical potentials, initial defects, stacking faults, and film thickness. The proposed mechanism may be especially relevant in thin Ge-rich films, although its quantitative accuracy and efficiency have boundaries.
- Defect-assisted pathways: Point defects such as impurities can facilitate dislocation nucleation, whereas the calculations began from a defect-free film.Including defects in the initial configuration could significantly lower the activation energy.
- Model limitations: The SW and Tersoff potentials omit dislocations from their training sets, so they may overestimate dislocation-core energies and poorly describe rehybridization.Preliminary Tersoff calculations lower the activation energy by about 10 eV relative to SW, while mechanism differences remain similar.
- Mechanism comparison: The new mechanism has an activation energy of 39.8 eV, compared with 54 eV for half-loop nucleation of an isolated 60° dislocation.It crosses only one large-scale energy barrier and is therefore more favorable unless a lower-energy 60° mechanism exists.
- Mechanism comparison: The mechanism could explain why Ge-rich GeSi films on Si(001) mainly show 90° misfit dislocations and few 60° dislocations.The authors expect it to be dominant when the film is very thin, during early Ge-film growth.
- Stacking-fault implications: A stacking fault predicted after 90° dislocation formation may disappear when two Shockley partials glide together and react into a 90° dislocation.Stacking faults are rarely observed experimentally in Ge/Si(001) films, creating a scope boundary for direct comparison.
- Thickness dependence: Because stacking-fault area and energy scale with h^2, a critical thickness exists above which this mechanism becomes less efficient.The authors also expect defects in the initial configuration to reduce the activation energy relative to defect-free calculations.
4. Summary
The paper presents a global transition-path optimization approach and applies it to 90° dislocation formation in Ge/Si(001), identifying a previously unreported mechanism. The mechanism has a lower activation energy than isolated 60° dislocation half-loop nucleation, while extended CNA identifies dislocations and stacking faults.
- 4. Summary: The approach generates intermediate configurations through heredity transformations and identifies low-energy intermediates from minima along revised-NEB paths.The revised NEB adds part of the spring force normal to the path to improve convergence for long, complex paths.
- 4. Summary: The Ge/Si(001) application reveals a 90° dislocation-formation mechanism unlike those previously reported.The mechanism begins with a 60° dislocation half loop, followed by splitting into Shockley-related partials and reaction with a complementary 60° dislocation.
- 4. Summary: The mechanism produces a 90° misfit dislocation when a complementary 60° dislocation reacts with the initial loop’s threading arm.A characteristic feature is the formation of a stacking fault.
- 4. Summary: 26% smaller activation energy is obtained for a 19 ML film than for half-loop nucleation of an isolated 60° misfit dislocation.The reported comparison is specific to the 19 ML film and the isolated 60° dislocation half-loop baseline.
- 4. Summary: The extended CNA method characterizes second-nearest-neighbor geometry to identify and visualize misfit dislocations and stacking faults.The method was implemented in OVITO.
Appendix A. Common Neighbor Analysis
The appendix extends common neighbor analysis for diamond-structure materials by classifying second-nearest-neighbor arrangements. This produces an efficient, parameter-free, perturbation-insensitive method that distinguishes cubic and hexagonal diamond environments for defect visualization.
- Appendix A. Common Neighbor Analysis: Standard CNA is unsuitable for diamond structures because nearest-neighbor pairs lack common neighbors and second- and third-neighbor shells are poorly separated.Small elastic strains or thermal displacements can therefore disturb CNA fingerprints and make the method unreliable.
- Appendix A. Common Neighbor Analysis: The extended method analyzes second-nearest neighbors by exploiting diamond’s two interleaved FCC lattices.Nearest neighbors are excluded from this analysis.
- Appendix A. Common Neighbor Analysis: A perfect diamond-lattice atom has 12 second-nearest neighbors, whose pairs with the central atom are classified using common-neighbor indices.The method identifies four nearest neighbors and three additional atoms per neighbor, yielding 12 second-nearest neighbors.
- Appendix A. Common Neighbor Analysis: FCC and HCP signatures mark atoms as belonging to cubic or hexagonal diamond local environments, respectively.Second-neighbor geometry enables discrimination between cubic and hexagonal diamond arrangements.
- Appendix A. Common Neighbor Analysis: The method is computationally efficient, has no adjustable parameters, and is insensitive to small perturbations in atomic positions.It has been implemented in OVITO and can support automated dislocation identification.
Appendix B. Modified NEB method
The modified NEB method relaxes discretized transition paths toward minimum energy paths while addressing failures caused by sparse images on highly curved, complex paths. It improves path straightness and convergence relative to regular NEB.
- Appendix B. Modified NEB method: NEB represents a transition path as images with position vectors and relaxes the chain toward a minimum energy path.Convergence requires enough images to resolve the path’s intermediate minima.
- Appendix B. Modified NEB method: Sparse images relative to path complexity can make NEB convergence problematic, especially when local curvature is high.An inaccurate tangent estimate can result when adjacent path segments form acute angles.
- Appendix B. Modified NEB method: After 30 000 steps, the modified NEB keeps the path relatively straight and converges better than regular NEB.The regular calculation remains unconverged, with maximum image forces reaching 10 eV/Å.
- Appendix B. Modified NEB method: The modified NEB adds a perpendicular spring-force contribution, switched on according to the angle between adjacent path segments.The switching function increases from zero for a straight path to unity at a right angle and remains unity beyond π/2.
- Appendix B. Modified NEB method: After 30 000 steps, the modified-NEB path rises to significantly lower energy than the unconverged regular-NEB path.A close-up near the maximum resolves the energy barriers of individual elementary steps.
- Appendix B. Modified NEB method: Long paths can be resolved by using many images or splitting the path into several parts.The calculations in this study continue until the maximum 3N-dimensional image-force norm falls below 0.01 eV/Å.