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Atomic Level Modeling of Structure and Growth of Materials

The present line of research is concerned with the study of the mechanical properties and growth phenomena associated to solid metals and semiconductors. All studies rely on microscopic modelling of the solid, in which the inter-atomic interactions are described through many-body model potentials. So far, special attention has been given to the Effective Medium (EMT) and the Embedded Atom (EAM) model potentials as regards the study of metals and Stillinger-Weber, Tersoff and EDIP (Environment Dependent Interatomic Potential) potentials (etc.) as regards to the study of semiconductors. Large scale Molecular Dynamics (MD) and Monte Carlo (MC) simulations have been the standard tools employed in all studies.

As far as semiconductors are concerned, the main topics of investigation have been the growth on substrates, ideal or with imperfections, the structure of layered structures, the elasticity, fracture and crack propagation, and detailed crack stuctures. Regarding metals, the focus has been on studies related to the deformation and strength of materials. Examples of these studies include modelling of brittle/ductile fracture, relaxation of grain boundaries, and structure and dynamics of dislocations.

Over the past two years, a great deal of effort has been devoted to the development of interactive molecular dynamics simulations. This is a quite recent concept, in which a molecular dynamics algorithm is coupled to a graphical interface, allowing the user to follow a run-time graphical representation of the simulated system. In this way, systems with great complexity may be studied interactively. Several ongoing projects in the present research line, most of them described below, make use of such interfaces. Both two- and three-dimensional simulation models have been implemented. The general characteristics of all simulations may be summarized as follows. All parameters defining the initial system (dimensions, parameters of the potential and so on) as well as dynamical variables (such as temperature, pressure and so on) are set through the graphical user interface (GUI). A run-time representation of the system is displayed on a graphics window. This feature allows the user to follow at run-time the effect of changes in the dynamical parameters. The GUI also provides a number of visualizations tools, such as zooming, scrolling, rotation, perspective view and so on. The programs run on UNIX-X11 window system platform and the graphical part relies either on the MOTIF or on the OPEN-GL library.

Growth and Properties of Semiconductor Heterostructures

In today's micro-electronics technology there is a trend towards smaller structures. Because the desired physical properties of the new components depend critically on atomic scale irregularities due to processing, thorough understanding of the crystal growth and atomic level structure is important.

Lattice mismatched heteroepitaxy is a versatile way to tailor the electronic properties of semiconductor materials to fulfill the needs of device developers. One example is the semiconductor quantum dots (QDs), i.e., clusters having a typical diameter of 5-50 nm. Remarkable improvements in optoelectronic device characteristics have been predicted to arise due to the quantum confinement of the charge carriers into a zero-dimensional (0-D) QD. This confinement depends on the atomic structure of the cluster and the deformed substrate. In the studies, described below, we have focused on the properties of various semiconductor heterostructures.

Atomic level structure of quantum dots

Researchers: Antti Kuronen, Teppo Hakkarainen, and Kimmo Kaski

Strain relaxation phenomena via misfit dislocations in Si/Ge systems are studied using molecular dynamics simulations and semi-empirical interatomic potential energy models. The atomic level structure of clusters used to manufacture quantum dots are calculated using optimization methods and potential models. Figure 17 shows a Ge island on Si substrate. The calculations are currently performed for Si/Ge and will be extended to III-V compound semiconductors. The results for the III-V systems will be utilized by prof. Tulkki's group to calculate the electronic properties of these clusters.

Figure 17
Figure 17: On the left: Ge pyramid and a Ge wetting layer grown on the (001) surface of a Si substrate (not shown). On the right: Strain component epsilon_{zz} of the Ge pyramid and Si substrate. The strain in percents (using colour coding) is plotted in the (110) plane crossing through the center of the pyramid.  

Strain relaxation in heterostructures

Researchers: Jari Rintala, Antti Kuronen, Leonel Perondi, and Kimmo Kaski

The mechanisms of the strain relaxation caused by the lattice mismatch in heterostructures are currently not well known. The objective of our studies is to investigate the dynamics and properties of the lattice mismatched interfaces and misfit dislocations. Presently we are studying interfaces in two-dimensional Lennard-Jones material. An example of the interaction between dislocation and the interface and between dislocations themselves are shown in figure 18. The study of strain relaxation phenomena via misfit dislocations in real materials has been initiated by investigating Si/Ge systems using MD simulations and semi-empirical potential models.

Figure 18
Figure 18: Two-dimensional lattice mismatched Lennard-Jones system. Atoms at the upper part of the system (19 atomic layers) have 4% smaller lattice constant than the lower substrate. (a) Initially, two dislocations were introduced into the system. (b) One of these glides to the interface. (c)-(f) Later the dislocation at the interface dissociates into two. One of them combines with the lower dislocation, which now is able to glide to the substrate edge. The colours of the atoms are coded according to their potential energy (blue: low, green: moderate and red: high). The black lines denote the extra atomic rows forming the dislocations and the arrows show the position of the interface.

Modelling of crystal growth

Researchers: Laura Nurminen, Antti Kuronen, Leonel Perondi, and Kimmo Kaski

One method to grow small clusters, for the purpose of creating quantum dots, is the Stranski-Krastanow (SK) mode of crystal growth. Crystal growth and particularly the SK growth mode are studied using generic lattice models in connection with kinetic Monte Carlo simulations. In addition, the development of off-lattice Monte Carlo simulation of growth phenomena is going on. The basic mechanisms in growth of lattice mismatched heterostructures are also studied using MD simulations of a two dimensional Lennard-Jones system.

Another possibility to grow nanostructures is deposition on patterned surfaces. Examples of this are vertical correlation of quantum dots and some metal systems (e.g. 2ML of Ag on Pt). We have studied the effect of inhomogeneous substrate on adatom nucleation using KMC simulations and simple lattice models. In figure 19 is illustrated the confining effect of the spatial variation in the adatom-substrate interaction energy on the island nucleation.

An interactive simulation program for studying crystal growth phenomena has been developed. Figure 19 shows snapshots of a growing surface simulated using this program.

Figure 19
Figure 19: Kinetic Monte Carlo simulation of crystal growth. The adatom-substrate interaction energy varied as a function of the adatom position on the substrate forming a checkerboard domain structure. The effect of this patterning of the substrate can be seen as a confinement of adatoms inside the domains at a narrow temperature range. Blue denotes the substrate atoms and red the deposited adatoms. (a) Simulation at T=390 K using homogeneous substrate. (b) Simulation using patterned surface at T=340 K. (c) As (b) but T=390 K. (d) As (b) but T=420 K.

Model potentials of semiconductor materials

Researchers: Karsten Albe*, Kaj Nordlund+, Antti Kuronen, Jura Tarus+
*University of Illinois at Urbana-Champaign, USA
+University of Helsinki, Accelerator Laboratory

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 and their applicability for simulations in phenomena far from bulk equilibrium (e.g. surfaces, high temperatures and crystal defects) still has to be tested. In collaboration with the Accelerator Laboratory of the University of Helsinki we have assessed potentials for III-V compound semiconductors. Because none of the potentials for these materials found in the literature seem to describe well non-equilibrium properties a project to develop new potentials has been initiated.

Fracture and Grain Boundaries in Crystalline Materials

The understanding of fracture phenomena is of great relevance from both the scientific and technological points of view. The way a macroscopic fracture forms and evolves in a strained sample is strongly dependent on the `brittleness' of the solid. In very brittle materials a crack with a sharp tip propagates at very high speeds, invariably resulting in the cleavage of the sample. At the other extreme, in very ductile materials, due to the emission of dislocations, a crack may be completely stationary. In addition, understanding the effect of grain boundaries on fracture mechanisms is both scientifically and technologically challenging.

Fracture in crystalline materials

Researchers: L. Perondi, M. Huhtala, K. Kaski, and A. Kuronen In this study we focus on the modelling of the ductile/brittle behaviour of the crack tip of a sample in terms of an atomistic model potential. Studies, so far, have been carried out through molecular dynamics simulations, making use of pair-wise and many-body model potentials. Figure 20(left) shows an instance of the simulation of a ductile crack, with two dislocations emitted from the crack tip and in their way to the borders of the system. Figure 20(right) displays the instantaneous microscopic velocity field corresponding to the system configuration shown in the figure, on the left.

Figure 20
Figure 20: Graphical representation of fracture phenomena obtained with the interactive simulation described in the report. The figure on the left shows a configuration of the particles after the emission of two dislocations while the figure on the right shows the correspondent microscopic velocity field.  

Grain boundaries in metals

Researchers: L. Perondi, P. Szeleste, K. Kaski, and A. Kuronen Understanding the structure and dynamics of grain boundaries (GB) and their interaction with defects is one of the major challenges in setting up realistic models for the mechanical behaviour of materials. Departing from atomistic modelling of simple metals (such as Cu and Al), the following topics are being addressed in the present study: solute segregation and diffusion at GBs, structure of low angle tilt grain boundaries and the effect of GBs on fracture formation and progression. In Figures 21 and 22 symmetric and asymmetric tilt boundary, with their characteristic dislocation structure, are shown.
Figure 21
Figure 21: An example of grain boundary movement. Relaxed configuration of a 12degreessymmetric tilt grain boundary.
Figure 22
Figure 22: Another example of grain boundary movement. Relaxed configuration of a 9degreesasymmetric tilt grain boundary. Line OA indicates the initial position of the boundary while line OB indicates its final position, after relaxation.

2-D interactive simulation of grain boundary and fracture phenomena

Researchers: J. Merimaa, L. Perondi, K. Kaski, and A. Kuronen

An interactive simulation program illustrating grain boundary and fracture phenomena in solids has been developed. The dynamical behaviour of a two-dimensional Lennard-Jones model system under stress, with either a grain boundary or an initial crack, is simulated through a molecular dynamics algorithm. The program has been devised for illustrative purposes. It displays the main elements of an interactive simulation and may be regarded as giving an illustration of the concept of interactivity. Due to the power of run-time animations in conveying ideas and concepts, it may prove to be useful, as well, as an instructional tool. Modified versions of this interface are being used in actual research.

3-D interactive molecular dynamics simulation of split edge dislocations

Researchers: L. Perondi, M. Robles, K. Kaski, and A. Kuronen

In the theory of plasticity, the mobility of dislocations is considered as the main factor in determining whether materials will display ductile or brittle behaviour. Several factors influence the mobility of a dislocation. The interactions with point defects, either native or externally generated, as well as with extended defects, such as boundaries, and with other dislocations constitute main examples. It may be said that a dislocation also interacts with the lattice that holds it. Geometrical constraints, dependent on lattice symmetry, allow conservative movement only along given planes at particular directions while the Peierls stress, a result of the discreteness of the lattice, gives rise to a periodic barrier for dislocation motion. The successful modelling and description of these interactions depend, ultimately, on the accurate description of the structure of the dislocation, including the splitting into partials and the formation of a stacking fault ribbon, a task that is beyond the reach of continuum theories such as linear elasticity theory. Molecular dynamics simulations with many-body model potentials have found widespread application in such situations, i. e., in the investigation of microscopic phenomena at scales where the discreteness of the lattice and non-linearities in the force-displacement relation play a significant role.

The general objective of this study is the investigation of dislocation structure and interactions in fcc metals. Making use of an EAM model potential, we have so far concentrated our studies to the splitting of edge dislocations in copper. Figure 23 displays an edge dislocation in copper, showing in detail its splitting into partials.

Figure 23
Figure 23: Splitting of an edge dislocation in copper into partials. The light coloured half planes indicate the approximate position of the core of the Shockley partials.

3-D steroscopic interface for molecular dynamics simulations

Researchers: V. Mustonen, M. Huhtala, L. Perondi, K. Kaski, and A. Kuronen

In this project a graphical interface for visualizing three-dimensional data has been developed. It takes as input the coordinates of a set of objects, which are represented as particles in a 3-D structure on a graphics screen. The image thus generated may be subjected to a number of operations, which include zooming, rotation, clipping and perspective view. It is also possible to select a subgroup of atoms from the original set and have them displayed alone on the screen. Other important feature is the display of stereo images, which may be visualized as real 3-D images through the use of appropriate equipment (shutter-glasses and special high refresh-rate monitor), thus enhancing the visualization of e.g. defects and other complex structures in crystals.

The most important feature of the interface is the provision of resources for the coupling of dynamical simulations, which may then be run in an interactive way. Presently, this interface has been used for running a molecular dynamics simulation with an EAM model potential. Figure 24 displays a rendered image of particles in an fcc lattice. The present interface is also being used in conjunction with simulations which make use of the Born-Maxwell model described below (see Figure 25).

Figure 24
Figure 24: Main window of the 3-D stereo graphics interface. The graphics window displays a rendered image of particles in an fcc lattice. The interface provides resources for studying isolated parts of the system, as illustrated by the the box at the upper right edge of the figure.

Figure 25
Figure 25: Three layers of material modelled through the Born-Maxwell model for the study of fracture.

Development of EMT and EAM semi-empirical potentials for metals

Researchers: P. Szeleste, L. Perondi, K. Kaski, and A. Kuronen

Properties of simple metals (like copper) have been investigated making use of Molecular Dynamics simulations. EMT (Effective Medium) and three different types of EAM (Embedded Atom) model potentials have been implemented. These semi-empirical potentials include long-range forces, have a many-body nature and are computationally efficient. In addition, the EAM potentials may be used in simulations of alloys. In parallel, we have been developing an EAM potential which takes into account elastic constants up to third order.So far, we have obtained a good description of relaxed surfaces, relaxed energy formation of vacancy and interstitials, energy of stacking fault defects and structure of edge dislocations. The developed potential seems to reproduce experimental data well, making it suitable for use in accurate large scale molecular dynamics simulations.

Dynamics of the dislocations in 2-D crystals under strain

Researchers: Miguel Robles, Leonel Perondi, and Kimmo Kaski

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 computer processing capacity, 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 Lennard-Jones solid, through molecular dynamics simulations coupled to a graphical user interface.

The main challenge has been the search for a method to track efficiently the position of one dislocation. At this moment, two numerical methods have been implemented successfully. The first one is based on the displacement field of each atom along the gliding line. The position of the dislocation is given by the atom that last moved more than half of one lattice spacing along the gliding line. The second method relies on computing the force between adjacent atoms in the gliding line. When a dislocation is present the value of this force increases considerably, indicating the position of the dislocation with good accuracy.

Both methods have been tested in tracking the dislocations in two different experimental situations: i) the emission of a dislocation from a crack tip in a strained crystal at a constant rate and ii) the movement of an isolated dislocation interacting with a stress pulse. As an example of case i), Figure 26 displays some snapshots of dislocations tracked after the emission from a crack tip.

The numerical experiments so far accomplished have stimulated research along several lines. The investigation of different dynamical regimes, of the interactions between dislocations and of border effects are just few examples of a large set of interesting and not yet well-explored topics.

Figure 26
Figure 26: Tracking the dislocations, emitted from a crack tip in a 14124 atoms system simulation, when the strain of 5.5 % in 36 ps has been applied.

Stacking fault energy of fcc metals

Researchers: Leonel Perondi, Peter Szeleste, and Kimmo Kaski

In fcc metals, it is a well known fact that energy balance favours the splitting of dislocations into Shockley partials, which may lie several lattice spacings apart. Between the partials, a stacking fault ribbon is formed in the gliding plane. The energy cost of increasing the size of the stacking fault ribbon is what prevents dislocations from completely dissociating. The equilibrium distance between partials is thus dictated by the stacking fault energy (SFE) of the material. The distance between partials, in turn, affects the dynamics of the dislocations and their ability to cross-slip. These effects have a direct impact on the deformation characteristics of the material. For instance, low SFE materials show strain hardening coefficients noticeably larger than those materials with a high SFE. Also, it is expected that the SFE indirectly affects phenomena going on near a crack tip, as far as the velocity of the emitted dislocations are concerned. These facts, together with others, make it important to predict with high accuracy the value of the SFE of various materials, under varied conditions.

In the present project (in collaboration with the Physics Laboratory of HUT), we have been calculating the stacking fault energy of noble metals making use of an ab initio Full Potential Linearized Augmented Plane Wave method (FPLAPW), as implemented in the scientific package, WIEN97. One of the specific objectives of these computations is to obtain an accurate value of the SFE as a function of the state of stress in the material. As an example of the applicability of these results, it will then be possible, for instance, to assess the impact of the state of stress on the dynamics of dislocations.

Crack initiation in silicon

Researchers: Matti Mäki-Jaskari, Antti Kuronen, and Kimmo Kaski

The applicability of semiempirical potential energy models for describing evolution of crack initiation in covalently bonded silicon has been studied, using Molecular Dynamics method. For describing interatomic interactions we have tested Stillinger-Weber and Tersoff potentials, but used mainly the recently developed Environment Dependent Interatomic Potential (EDIP) with two- and three-body terms, the role of which we have studied by modifying them. In this study we focused the main interest in the crack tip structures preceding the actual fracture. Our results indicate formation of fairly stable (or metastable) local structures of 5- and 7-fold rings, as pairs from the usual 6-fold ring-structure, near the crack tip before and during crack initiation. In our simulations, under tensile loading condition crack propagation from an initial cut in (110) and (111) crystal plane can be observed; for crack propagation in (111) plane see figure 27 (left). The stability of the ring-structures was studied by using quantum-mechanical tight binding calculations, which shows that these structures are at least long living, see figure 27 (right).

Figure 27
Figure 27: Left: Brittle crack propagation along (111) cleavage plane of silicon. Right: Crack-tip structure in silicon generated first by using the molecular dynamics method and then relaxed by using ab-initio tight binding calculations.  

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Next: Research on Semiconductor Optics Up: Computational Materials Research Previous: Computational Materials Research