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Atomic Level Modelling of Structure and Growth of Materials

The present line of research is concerned with the study of structural properties and growth phenomena associated to solid metals and semiconductors. All studies rely on microscopic modeling, in which the inter-atomic interactions are described through pair and many-body model potentials. In generic studies we have used Lennard-Jones potentials. When studying specific materials special attention has been given to the Tersoff, Stillinger-Weber, VFF (Valence Force Field) and EDIP (Environment Dependent Interatomic Potential) potentials as regards to the study of semiconductors and Effective Medium (EMT) and the Embedded Atom (EAM) model potentials as regards to studies of metals. Large scale Molecular Dynamics (MD) and Monte Carlo (MC) simulations have been the standard tools employed in all these studies. The newly installed cluster computer CLUX has been used to perform MD simulations of systems containing more than 106 atoms.

The main topics of investigation in the case of semiconductor materials have been the growth on substrates, ideal or with imperfections, structure and mechanical properties of various nanostructures - e.g. quantum wells, quantum dots and carbon nanotubes, and development of semiempirical potentials for III-V materials.

As regards to metals, the focus has been on studies related to detailed microscopic structure of materials. Examples of these studies include modelling of brittle/ductile fracture, relaxation of grain boundaries, and structure and dynamics of dislocations.

In connection with the MD simulations, development of the graphical user interface (GUI) for the simulation programs has been continued. The interactive simulation programs have been used to study dislocation dynamics and strain relaxation in two-dimensional heteroepitaxial systems and mechanical and structural properties of carbon nanotubes.

In relation to all these simulation studies, parallel computation algorithms and methods are being developed, specifically targeted to computer cluster environments.

Strain Relaxation in Heterostructures

Researchers: Antti Kuronen, Kimmo Kaski, Leonel Perondi*, Jari Rintala and Marco Patriarca
*Current address: Instituto Nacional de Pesquisas Espaciais - INPE, Brazil

The exact mechanisms of the strain relaxation caused by the lattice mismatch in heterostructures are currently not well known. We have studied the mechanisms responsible for the formation of a misfit dislocation in a lattice-mismatched system by using Molecular Dynamics simulations of a two-dimensional (2D) Lennard-Jones system. Results show clearly how the strain due to the lattice-mismatched interface acts as a driving force for migration of dislocations in the substrate and the overlayer and nucleation of dislocations in the overlayer edges. In the hexagonal lattice used in the simulations dislocations that are able to glide from the overlayer or substrate towards the interface do not have optimal Burgers vector for strain relief. However, interaction between the dislocation at the interface and dislocations left in the overlayer or substrate causes changes in their Burgers vectors that result in an optimal strain for relieving dislocation at the interface. In figure 15 can be seen an example of strain relaxation by dislocation nucleation. In addition to these 2D systems we are currently studying strain relaxation phenomena in three-dimensional Si/Ge systems with realistic semiempirical potentials and large scale molecular dynamics simulations using the newly installed cluster computer CLUX.


  
Figure 15
Figure 15: Strain relief in a lattice-mismathced hetero-structure. On the left: The upper part of the figure shows the system in the beginning of the simulation. The colors of atoms are coded according to their potential energy (blue: low, green: moderate and red: high). The overlayer has a lattice constant that is 4% smaller than that of the substrate, i.e. the misfit is -4%. In the center part of the figure can be seen that a dislocation starts to nucleate from the overlayer edge when the simulation is started and a finite temperature is switched on. In the lower part of the figure can be seen the final state of the system with a -4% misfit. On the right: Results of the nucleation simulations. The number of dislocations nucleated from the overlayer edges during the 130 ps simulations is plotted for different misfits and simulation temperatures. The size of the circles denotes the number of dislocations. Open triangles designate runs where no dislocations were observed. The thin line separates pseudomorphic and relaxed regions.

Modelling of Compound Semiconductors: Analytical bond-order potential for GaAs

Researchers: Karsten Albe*, Bob Averback*, Kai Nordlund+, Janne Nord+, Juhani Keinonen+ and Antti Kuronen
*University of Illinois at Urbana-Champaign, IL, USA
+University of Helsinki, Accelerator Laboratory

Various ab initio methods can be used to model materials at atomic level. However, due to the large demand for computational power their use is currently restricted to small systems containing up to 103 atoms. Many phenomena like strain relaxation in heteroepitaxial systems, ion beam interaction with materials or dislocation dynamics involve much larger amount of atoms. In these cases atomic level modelling has to be done using semiempirical potentials and classical molecular dynamics simulations. The quality of the simulations using model potentials depends on the quality of the potential used.

For semiconductors reasonable interatomic potential energy models have been developed for column IV materials, i.e. silicon, germanium and carbon. For the III-IV compound materials there are very few potentials in the literature. We have assessed these potentials and found that their applicability for simulations in phenomena far from bulk equilibrium is poor.

Therefore we have developed a new analytical bond-order potential for GaAs, that allows to model a wide range of properties of the compound structures as well as the pure phases of gallium and arsenide including non-equilibrium configurations. The analytical form follows in principle the bond-order scheme as devised by Abell and Tersoff, but angular forces and mixed interactions are treated differently. A number of tests, that cover a wide range of structural geometries including the metallic phases of gallium and arsenide, point defect properties, elastic moduli, surface properties and melting behavior, have been performed in order to validate the accuracy and transferability of the potential model. Reference data were taken from experiments where available and have been computed by self-consistent total energy calculations within the local density functional theory (LDFT) otherwise.


  
Figure 16
Figure 16: Schematic figures of the reconstructions of the GaAs(001) surface tested with the new bond-order potential. Lighter spheres designate As atoms and darker spheres Ga atoms.

Atomistic calculation of strain in heterostructures

Researchers: Sebastian von Alfthan, Antti Kuronen and Kimmo Kaski

The calculation of strain in quantum well and quantum dot structures is an important topic because it has a profound impact on the electric properties of the system. Because dimension of these systems get smaller and smaller the effect of discrete atomic structure has to be taken into account.

We have calculated strain fields in various heterostructures using an atomic level modeling. Systems studied include capped InAs pyramids grown on an GaAs substrate. More recently we have concentrated on InGaAs quantum wells in AlGaAs. We have calculated the strain by modelling the system on an atomic level. Interaction between atoms is described by the valence force field (VFF) potential which gives a reasonable description of the III-V compound semiconductor materials near their equilibrium configuration. Strain fields are calculated by minimizing the potential energy of the system and by calculating the displacements of the atoms in this equilibrium state.

Strain data obtained from our calculations are utilized by prof. Tulkki's group to calculate the electronic properties of the quantum wells. Moreover, our results are compared with the strain field obtained from finite element calculations based on linear continuum elasticity performed by prof. Tulkki's group.

 
Figure 17
Figure 17: On the left: The system used in the calculation of the strain in InGaAs quantum wells (yellow and blue atoms) in a AlGaAs substrate (red and blue atoms). On the right: The strain component in the horizontal direction (epsilon_{xx}) showing the concentration of the strain in the corners of the InGaAs layer.

Modelling of Crystal Growth

Researchers: Laura Nurminen, Antti Kuronen, and Kimmo Kaski

The self-organized growth of nanoscale islands offers an attractive possibility to develop new techniques for fabricating quantum dots. In the Stranski-Krastanov (SK) mode of crystal growth 3D islands form spontaneously on top of a wetting layer through a strain-relief mechanism. Generic lattice models are used in connection with kinetic Monte Carlo simulations to investigate the detailed mechanism behind the SK growth mode. In addition, new simulation methods are developed to include the effect of lattice-mismatch induced strain directly into the kinetic model.

Precise control over the distribution of island sizes is essential for technological applications of nanostructures. We have studied the effect of a patterned substrate on island growth. The nanoscale patterning of the substrate promotes spatial ordering of the islands in their nucleation stage, which consequently results in a narrow size distribution. Figure 18 shows snapshots of islands that are growing on both a smooth and a patterned substrate. Experimental examples of such a patterned substrate are vertically correlated quantum dot superlattices and dislocation networks in some metal systems (e.g.  2ML of Ag on Pt).

  
Figure 18
Figure 18: Kinetic Monte Carlo simulation of crystal growth. The figure shows growing islands on (a) a homogeneous substrate and (b) a substrate with nanoscale patterning (a checkerboard structure). Blue denotes the substrate and green the deposited atoms.

Interactive routine to track atomic defects in 2-D molecular dynamics simulations

Researchers: Miguel Robles, Kimmo Kaski and V. Mustonen

The use of interactive simulation programs brings a lot of new qualitative information of the dynamics of deformation processes in materials at the atomic scale. The ability to use different color codings for visualizing the evolution on-line of different physical fields can enhance our appreciation of such phenomena. Although a visualization program can allow us to locate a defect in a perfect lattice by eye, to achieve a precise quantification of the trajectory may be a very tedious task. The develop of automatic tools to track the evolution of different defects may improve the performance of an interactive simulation by allowing the user to decide when to track the trajectory without storing a huge amount of data in fixed media. This has been the aim of this project.

  
Figure 19
Figure 19: Tracking of a single dislocation. Velocity profile is shown in the upper left corner.

When a 2-D lattice is not deformed or a very slow deformation has been applied the potential energy field shows a clear change very near the core of a defect such as a dislocation or the border of a crack. Then filtering is possible in extracting a subset of atoms which belong to all possible defects. The routine proposed is based in this property.

Through a clustering algorithm the program may find the number of defects present at a time and using a linked-list routine may store efficiently all the physical information of the constituent particles. In this way it is possible to extract the defect's center of mass, kinetic and potential energy, or even other geometrical properties, like the length of a crack, in a dynamic way. Finally, the routine may be controlled inside the simulation with a control window where the main parameters of the clustering algorithm may be customized on-line. For the specific case of moving dislocations, the trajectory of the center of mass of the atoms in the core describe quite well its movement. A further analysis of the trajectories may lead to an accurate determination of the instant velocity, essential for the study of dislocation dynamics. Figure 19 displays a snapshot of the routine, tracking a dislocation in an interactive simulation.

The application of this routine is wide, e.g. in the study of the dynamics of defects in 2-D systems. It is currently is being applied in the study of the dynamics of dislocation in deformed lattices and in the problem of crack propagation.

Dynamics of the dislocations in 2-D crystals under strain

Researchers: Miguel Robles, Kimmo Kaski and V. Mustonen

It is well known that the mechanisms of deformation of a crystal under stress are strongly dependent on the movement of dislocations. So far, the dynamics of dislocations is not well understood due to the complexity and the difficulties in modelling the phenomena. Presently, due to large improvements in computing power, large scale simulations are making it possible to study the dynamics of dislocations in a consistent way. Presently we have been studying the dynamics of dislocations in a 2-D solid, through a molecular dynamics simulations coupled to a graphical user interface previously done and reported. New tools have been added to the molecular dynamics simulation: a thermostat to simulate a thermal bath and a many-body term in the interaction potential. This term may be modulated by a coupling parameter in such a way that the system may be tuned from brittle to ductile behaviour. In this way the influence of the many-body forces in the dislocation dynamics has been studied. Stress-strain and strain-velocity relations for different coupling parameters were obtained allowing to link the atomic scale dynamics with macroscopic parameters. Figure 20 display, as an example, the variation in the terminal velocity of one dislocation responding to different strain rates and for different tuning factors.

The results of these numerical experiments are encouraging us to extend the work to a 3-D simulation, where is possible to make direct comparison with real experiments.

  
Figure 20
Figure 20: Change in the terminal velocity (Vt) of one dislocation, normalized by the transverse speed of sound (Vs), as a funtion of the many-body coupling parameter when different strains are applied for 20 ps.

Development of Semi-empirical EAM-potentials for Metals

Researchers: Péter Szelstey, Leonel Perondi, and Kimmo Kaski

Properties of simple metals (Au, Ag, Cu, Al, Ni) have been investigated by Molecular Dynamics simulations using Embedded Atom (EAM) potential. EAM-potentials constitute a major progress to more accurate and computationally efficient model potentials, they have a many-body nature, eliminate several problems of the sole use of pair-potentials (e.g. Cauchy discrepancy of elastic moduli). Although EAM potentials have ab initio origin and their construction is mostly done by fitting to certain material parameters. EAM-potentials are generally applied for metals with non-directional bonding.

We have successfully developed a new type of EAM-potential which takes into account the second- and third-order elastic moduli, which is a significant improvement in comparison with the previously used EAM potentials. The third-order elastic moduli became especially important in case of large lattice deformations; obviously in the vicinity of dislocations, cracks and grain boundaries. We have obtained good quantitative agreement with experiments in terms of surfaces relaxation, surface energy, formation of a vacancy and interstitial. The potential correctly favours the FCC lattice structure over the HCP structure and yields realistic value for the stacking-fault energy, which is indispensable if one uses the potential for studying dislocations and plasticity.

Edge Dislocations in FCC-metals

Researchers: Péter Szelstey, Leonel Perondi, and Kimmo Kaski

Dislocations play a fundamental role in a variety of phenomena, such as plastic deformation of materials, fast diffusion in solids, crystal growth, crack propagation, ductile-brittle behaviour. It is known that (110) edge dislocations in FCC structures splits into two Schockly partial dislocations in a way that the equilibrium distance between partials is determined by the stacking-fault energy of the material. The separation distance affects the dynamics of the dislocations and their ability to cross-slip. Also, it is expected that the separation distance indirectly affects phenomena going on near a crack tip, as far as the velocity of the emitted dislocations are concerned. These effects have a direct impact on the deformation characteristics of the material. We carried out 3-D Molecular Dynamics simulations using the previously described EAM potential to study dissociated edge dislocations in copper. The aim of our research has been to determine the structure of dislocations and the effect of different parameters on it.

 
Figure 21
Figure 21: Snapshot of a partial dislocation.

We studied the role of several kinds of boundary conditions (free, fixed, partly fixed) on the separation distance, because choosing proper boundary conditions proved to be crucial to obtain correct equilibrium separation distance. We observed the splitting of edge dislocation with the value of separation distance similar to that found in experiments. Different types of tracking methods and the graphical interface, developed in the previous years, were used to obsesrve dislocation structure and motion. In addition, we studied the effect of many types of temperature control (e.g. Nosé-Hoover thermostat and heating-cooling cycles) and applied varying external pressure on the dislocation. We found that increasing preassure can significantly increase the separation distance between partials. The research now focuses on the interaction between dislocations. One example is the interaction of two dislocations gliding on different planes. The other objective of our research is the study the formation of dislocation kinks and jogs.

 
Figure 22
Figure 22: Motion of two partial dislocations (dashed lines). Following the initial time the partials move to their equilibrium position. The separation distance is denoted by solid line.

Dynamics of Dislocations in FCC metals

Researchers: Péter Szelstey, Miguel Robles, Leonel Perondi, and Kimmo Kaski

Deformation of a crystal under stress is strongly dependent on the movement of dislocations. So far, the dynamics of dislocations is not well understood due to the complexity and the difficulties in modelling the phenomena. Presently, due to large improvements in computing power, large scale simulations are making it possible to study the dynamics of dislocations in a consistent way. The main objective of our study is to understand the mechanism of dislocation motion and how dislocation velocity depends on material properties.

We used full-scale 3-dimensional Molecular Dynamics simulation of copper with the previously described EAM-potential to investigate dislocation motion. A dissociated edge dislocation was initially created, and after the equilibrium was reached, in a very well controlled way, external shear stress was applied to accelerate the dislocation. Our preliminary results show that the viscous drag model may be a very good description and dislocation can be considered as an object with an effective mass. The research now concentrates on, how the parameters (effective mass, drag coefficien t, limiting velocity) can be compared to values obtained from 2-D simulations and experiments.

 
Figure 23
Figure 23: Motion of dislocation (two partials) under external shear stress. The two solid lines denote the partial dislocations. After the initial acceleration the dislocation moves with constant velocity.

Mechanical and Structural Properties of Carbon Nanotubes and Nanotori

Researchers: Maria Huhtala, Antti Kuronen, and Kimmo Kaski

Carbon nanotubes are highly elastic semiconducting or metallic all-carbon molecules. Single walled nanotubes can be visualized as a graphite layer rolled seamlessly into a tubular form. Nanotubes have roused much interest in materials physics research in the last few years because they are considered to be an extremely promising novel material both for nanoelectronics components and for structural properties based applications.

We have implemented an interactive 3D-graphical simulation program for studying structural and mechanical properties of carbon nanotubes. The program runs coupled to a classical molecular dynamics algorithm with a Tersoff-Brenner carbon potential. We have performed extensive simulations of the properties of carbon nanotori, i.e. toroidal structures of the nanotubes, that were recently observed experimentally.

We have calculated structural, mechanical, and energetics properties of the tori and the tubes.

Figure 24 shows a curvature induced structural change we have found in the minimum energy configuration of the tori.

  
Figure 24
Figure 24: The effect of curvature induced strain on the minimum energy configuration of nanotori. The outer diameter of the tori is held at 23 nm while the tube diameter is grown. The upper images show a minimum energy configuration of (6,6)-, (8,8)-, (10,10)-, and (11,11)-tori with the tube diameter growing from left (0.818 nm) to right (1.493 nm). The lower images show the cross sections of the tori presented above. As the tube diameter reaches a critical value, which is for tori of diameter 23 nm approximately 1.4 nm, a structural change is observed - the surface shows periodical undulation. For tube diameters smaller than the critical diameter the cross section of of the tube is seen to be deformed to an oval shape with the ellipticity increasing as the tube diameter grows. Beyond the critical diameter the tube cross section no longer shows axial symmetry.


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