Atomistic simulation of formation of misfit dislocations if f.c.c. heterostructures

Acta metall, mater. Vol.41, No. 12, pp. 3455-3462, 1993

0956-7151/93 $6.00+ 0.00 Copyright © 1993PergamonPress Ltd

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ATOMISTIC SIMULATION OF F O R M A T I O N OF MISFIT DISLOCATIONS IN F.C.C. H E T E R O S T R U C T U R E S A. S. NANDEDKARt IBM, Technology Modeling Department, Semiconductor Research and Development Center, East Fishkill, NY 12533, U.S.A. (Received 12 October 1992; in revised form 22 April 1993)

Abstraet--Atomistic simulations were performed to study formation and characteristics of misfit dislocations in heterostructures. The f.c.c. Au/Ni (15.9% mismatch) system was used for analysis. When the strain energy of the system, computed using Lennard-Jones potentials [Physica status solidi (a) 30, 619 (1975)], (or potentials based on embedded atom method [Phys. Rev. B 41, 9717 (1990)]) was minimized, misfit dislocations were generated. The dislocation type depended upon the orientation of the substrate, being 90° type for a (001) interface and 60° type for a (111) interface. The latter orientation also resulted in greater relaxation of the strain energy. Multiple dislocations were generated at appropriate spacings in large computational cells relaxed using rigid boundaries. When infinitelylarge computational cells were simulated using periodic boundaries, the size of the periodic unit affected the dislocation spacing and the energy relaxation. Based on the structure of the intermediate defect configurations, it is hypothesized that the nucleation of misfit dislocations starts with formation of vacancies at the film surface. As the vacancies migrated towards the substrate, a dislocation loop moved to the interface.

INTRODUCTION Growth of thin films on lattice mismatched substrate is a very widely used technique in today's technology. The film incorporates a strain during initial growth due to coherency with the substrate. With the formation of misfit dislocations, the strain energy is reduced. In order to control the dislocation density in epilayers, the formation of misfit dislocations has been investigated extensively [3-7]. It is known from experimental observations that free surfaces are preferred sites for nucleations [3]. Continuum simulations demonstrated that nucleation and growth of dislocations has an activation barrier which can be overcome from the residual coherent strain energy in the film [4]. This barrier may differ for different types of dislocations such as 60° glide or 90 ° climb dislocations [4]. The nucleation of a dislocation loop has been considered in planar epitaxial films [5, 6] to investigate the stability of coherent multilayer structures. It was concluded that the susceptibility to damage of the epitaxial layer increased with increasing lattice mismatch and with increasing layer thickness [5]. Continuum simulations have been extensively utilized for studying nucleation of misfit dislocations [3~i] and a more detailed and localized information about the same may also be obtained by studying the nucleation by atomistic simulations. In this paper, transformation of a thin film from a coherent epilayer to a semicoherent film (with misfit dislocations) is analyzed by observing formation of tPresent address: CASA Services, 15 Dartantra Dr., Hopewell Junction, NY 12533, U.S.A.

misfit dislocations in a coherent thin film using atomistic simulations on a computer. It will be shown that the coherent epilayer went through a series of intermediate defect configurations (IDC) when the dislocation was generated, Analysis of atomic structure and energetics of the IDCs was useful in understanding the nucleation phenomenon of misfit dislocations. This approach did not assume or influence formation of any particular type of defect but rather the defect was characterized as it was forming as a result of strain energy minimization. This study can also give information about the dislocation structure and energy, including the core [7, 8] by analyzing the final configuration. METHODS We used an Au/Ni heterostructure system to study formation of misfit dislocations. Due to the large lattice mismatch (15.9%), the system had a high strain energy and thus formed misfit dislocations with no activation energy barrier [7]. Furthermore, the size of the computational cell was small enough to conduct the experiment within a reasonable computational time. These factors effectively made the computations simpler. The qualitative information gained for this system can be used to extrapolate the nucleation mechanism of misfit dislocations in other systems of interest. So as to ensure that the formation of defects was not a simulation artefact, we conducted experiments by changing parameters of the models, e.g. the number of atoms, the boundary conditions (rigid and

3455

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SIMULATION OF FORMATION OF MISFIT DISLOCATIONS

periodic), substrate orientation, inter-atomic potential (LJ and embedded atom method), minimization algorithms and by conducting independent trials using the same algorithm. The final configurations were analyzed for the type of dislocations and for the energetics of formation of misfit dislocations. Simulations were performed using an IBM RISC 6000 workstation. As an input cell, coherent film was simulated on lattice mismatched substrate with the film compressed along the directions parallel to the interface. The compression was 15.9% for Au film in the Au/Ni system. The interplaner distance along the direction perpendicular to the interface was maintained according to the film lattice constant. This maintained a one-to-one correspondence between atoms across the interface and a high coherent strain energy in the film. Cells were simulated corresponding to a (001) and (111) interface. The energy was computed using an LJ potential [1] and by potentials based on the embedded atom method [2]. The driving force for the formation of misfit dislocation was the minimization of the above strain energy. Energy minimization was carried out using three methods. The first method tested 27 different positions of an atom and moved it to the position of least energy thus reducing the strain energy. The next atom in the computational cell was relaxed in a similar way. When all atoms were thus relaxed, one iteration of the process was complete. The second method calculated the conjugate gradient to define the direction of atom displacement and atom was moved along that direction to a site of lower energy. The third method was based on Monte Carlo simulations. F o r all methods the sequence in which individual atoms were relaxed within an iteration was defined in a random as well as systematic fashion. The minimization procedure was stopped when the energy of the computational cell changed by less than 0.01 eV after 30 successive iterations. Intermediate cell configurations were stored for analysis. Energy was minimized under two different boundary conditions. Periodic and rigid boundary conditions were utilized to simulate an infinite film (with finite thickness) on infinite substrate. For the rigid boundary, all atoms inside a rectangular well were allowed to relax (Fig. 1). It included some atomic layers of the substrate as well. The width of the rigid wall, w (Fig. 1), was greater than the cut-off radius used in the potential. For the periodic boundary, all of the film atoms and a few layers of the substrate near the interface were relaxed. F o r the Au/Ni system, computational cells with (001) and (111) substrates were constructed. For (001) substrate, two cells were relaxed using rigid boundary conditions. The small cell (Fig. 3) contained 6000 atoms (50 × 50 x 2 4 ~ along [1 1 0], [ i 1 0 ] and [001] directions respectively), and the larger cell [Fig. 5(a)] had 12000 atoms (100 x 50 × 24/~). Both cells had a 9 A thick Au film

(4 layers of atoms), along [00 1] direction. For the periodic boundary conditions, four cells were simulated containing 6, 7, 8 and 10 vertical planes [Fig. 2(a-d) respectively]. For the (111) substrate, a small cell [Fig. 4(a, b)] contained 4400 atoms (42 x 48 x 26 along [1 1 ]], IT 1 0], and [1 1 1] directions respectively), and a large cell [Fig. 5(b)] contained 8960 (68 x 48 x 26 ~ ) atoms. The resultant dislocations were analyzed for the type of Burgers vector, the direction of dislocation line, and the plane in which they lay.

RESULTS For the Au/Ni system, the final configuration of the relaxed cell was not sensitive to the method of energy minimization used, or the sequence in which the atoms were selected for relaxation. Similar results were obtained using the LJ potential and potentials based on the embedded atom method. Therefore, we have shown results obtained using LJ potential. In all of the figures, a section of two consecutive atomic planes within the three dimensional computational cell are shown. For periodic boundary conditions, the distance between neighboring dislocations (dislocation in the computational cell and its images in the image cells), varied considerably depending on the size of the input cell (Fig. 2). The Burgers vector was a/2(110) type, and the dislocation line was along (110) direction. All dislocations and Burgers vectors were in the plane of the interface. For (001) orientation and rigid boundary conditions, a cross of two single dislocations was observed in the small cell [Fig. 3(b)8]. For this orientation, a single dislocation was observed edge on, (it was lying in the (001) plane and was perpendicular to the plane of the paper), while the second dislocation was lying in the (001) plane and in the plane of the paper [Fig. 3(b)8]. The large cell contained three dislocations [seen edge on in Fig. 5(a)], with neighboring dislocations separated at about 17/~. All dislocations for (001) substrate were of 90 ° type, and a cross grid was observed. With (111) orientation and rigid boundaries, the dislocations were 60 ° type. One 60 ° dislocation is seen edge on for a small cell [Fig. 4(b)6]. The distance between the neighboring dislocations was about 12 [Fig. 5(b)] for a large cell. The average reduction in energy for the relaxed atoms, computed as the ratio of change in configurational energy per atom, was 3.2eV for (111) orientation and 1.7eV for (001) orientation [Fig. 4(c)]. The cell energy during relaxation along with some intermediate configurations of the small cells are shown in Figs 3(b) and 4(b) for (001) and (111) substrate orientations respectively.

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SIMULATION OF FORMATION OF MISFIT DISLOCATIONS Ca )

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Fig. 1. Schematic cross sections of a central portion of a small cell (a) and a large cell (b) with (001) interface show rigid boundary (hatched region) and relaxed area (light region). A completely relaxed small cell (a) produced a cross of 90° dislocations. One dislocation is seen edge on marked with "T'. A dislocation loop is drawn to illustrate the formation of intersecting dislocation loops. For the large cell (b), multiple dislocations (edge on, marked by a "'T") and the position of the dislocation loop intersecting them are shownt.

DISCUSSION Regardless of the method of relaxation or the sequence of atoms during relaxation, the final outcome of the minimization process was the same, i.e. formation of misfit dislocations. This study demonstrated that although the misfit dislocations formed by energy minimization, the nature of dislocations depended on the orientation of the substrate. The dislocation density for (111) substrate is larger than (001) substrate which may explain the greater reduction in average energy/atom for (111) substrate [Fig. 4(c)]. The distance between the dislocations was smaller for the (111) substrate since the 60 ° dislocations are mixed type, and only the edge component tThe figures in this paper are recreations of the originals. AM41/12

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of the Burgers vector reduces the mismatch between the substrate and the film. Our calculations were performed on a three dimensional computational cells up to 60/~ on a side. The biggest cell needed about 10 days of CPU time on a RISC 6000 workstation. We observed small cross grid of 90 ° dislocations and a small triangular grid of 60 ° dislocations for (001) and (111) substrates respectively. Grids of dislocations have been investigated in f.c.c, materials [9, 10]. It was suggested that hexagonal grids may form as a reaction product of triangular grids by lowering the net energy due to reduction in dislocation line length. In this paper, misfit dislocations were formed as a result of reduction in coherent strain energy by accommodating the lattice mismatch. The edge component of the Burger's vector accommodated the mismatch. For

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SIMULATION OF FORMATION OF MISFIT DISLOCATIONS

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Fig. 2. Calculated configurations of Au/Ni heterostructures with periodic boundaries for (001) interface. A computational cell and its images are marked. The cells with 6, 7, 8, and 10 planes of atoms in the computational cell produced misfit dislocations with the corresponding spacing of 6, 7, 8, and 10 planes (between the dislocation in the computational cell and its images) in (a~d) respectively. alleviating the mismatch to full extent, a triangular grid of parallel 60 ° dislocations would be necessary for a (111) substrate. This would create an energy balance between reduction of coherent strain energy by triangular grid (minimizing the mismatch) and reduction of dislocation energy by formation of

hexagonal grid (reduction of dislocation line length). Final grid would depend on which energy reduction is predominant. Furthermore, it is rather impractical to address this issue by direct calculations, since C P U time puts restrictions on a n u m b e r of atoms that can be processed in a reasonable timeframe.

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SIMULATION OF FORMATION OF MISFIT DISLOCATIONS

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Fig. 3. (a) A plot of iterative energy minimization for a small Au/Ni cell with (001) interface. Three simulation studies with three different sequence to select atoms for energy minimization are illustrated. Number of iterations are indicated and corresponding intermediate configurations are shown in (b). (b) Evolution of a 90~ dislocation from a coherent Au film on Ni substrate (iteration No. 0) to form a semicoherent interface (iteration No. 550). The initial changes in the film configuration are equivalent to formation of vacancies with subsequent migration of vacancies towards the interface.

Multiple dislocations with spacing appropriate for the mismatch were formed when rigid boundary conditions and large computational cells were used. While one could simulate large infinite cells using periodic b o u n d a r y conditions, we found that the size of the periodic element can affect the resulting configuration and energy. For example, ideally in Au/Ni system (15.9% lattice constant mismatch), six relatively unstrained planes of Au, would match with

seven unstrained planes of Ni. Therefore relaxing a coherent (Au/Ni) heterostructure with seven planes should yield a dislocation with the extra half plane in Ni side. Equilibrium spacing between misfit dislocations is constant for a given misfit between lattice constants of substrate and film. However, when we used periodic boundary conditions to simulate infinite film (along the directions parallel to the substrate) inaccurate results were obtained. For periodic

3460

NANDEDKAR: SIMULATION OF FORMATION OF MISFIT DISLOCATIONS (c)

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[lil] [~10] Fig. 4. (a) A plot of energy vs interations for a small Au/Ni cell with (11 I) interface. The three different simulation studies are illustrated with three different selection sequence of atoms during energy minimization. (b) Evolution of a 60° dislocation in case of a (111) interface system in (a). Atomic configurations illustrated here are for the successive iterations as indicated by the numbers on E vs iterations plot in (a). (c) Average energy change for relaxed atoms in case of (001) and (111) substrates.

boundary conditions, the spacing between the misfit dislocations depended on the original cell size. Configurational energy was higher for cells with nonequilibrium spacing between the misfit dislocations. This is a rather important but subtle observation e.g. periodic boundaries can produce inaccurate results when defect configurations are investigated atomistically. The reasons for this inaccuracy are as follows. (1) Periodic boundary conditions make use of lattice periodicity, by creating image cells. If the original cell forms a defect during calculations, image cell also forms the same defect. (2) The spacing between the defects (misfit dislocations)

at the edges of original cell and image cells is dependant on the choice of original cell size. Hence interaction energies between these defects are also a function of original cell size. We observed that using periodic boundaries, relaxing eight planes of heterostructure tried to match eight planes of Ni with seven planes of Au to form a dislocation in an original cell [Fig. 2(c)], thus retaining some extra strain energy. Thus the cell size affected the distance and interaction between the dislocations (i.e. interaction between the dislocation in the computational cell and its images in image cells) when an infinite cell is

NANDEDKAR: SIMULATION OF FORMATION OF MISFIT DISLOCATIONS (a)

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3461

towards the interface. The final result is formation of dislocations at the interface and formation of a new layer of atoms at the top [Fig. 3(b)8]. Conceptually, for the small cell shown in Fig. 3(b), if we removed one atom from each row of atoms in the film, we would end up with an extra half plane of the dislocation in the substrate. If this atom were to be relocated at the top of the film, this process would be the same as forming a vacancy in each row., The aforementioned migration of vacancies towards the interface should then correspond to movement of the dislocation from the surface to the interface. When several sections of the three dimensional cell were examined, we found no dislocation within the rigid boundaries as expected. Sections close to the rigid boundary showed an extra half plane which terminated inside the film [Fig. 6(b, c)], Near the center of the relaxed portion, [Fig. 6(a)], the extra half

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Fig. 5. (a) A large Au/Ni cell with (001) interface showing multiple misfit dislocations of 90° type. (b) Atomistic configuration of a relaxed, large Au/Ni cell with (111) interface showing multiple 60° dislocations. The dislocation line positions are marked by a "T" symbol and extra half planes are shown by arrows. constructed using periodic boundary conditions. Rigid boundary conditions did not interfere with the defect formation process as periodic boundaries did. Based on these considerations, we find the use of rigid boundaries and large computational cells to be a more robust approach to study formation of misfit dislocations. As the computational cell was relaxed, we stored intermediate configurations to investigate the formation of the misfit dislocations. Initially, the computational cell had a coherent interface [Fig. 3(b)l] with high strain energy [Fig. 3(a)]. After 10 iterations, the strain energy was reduced by about 200 eV [Fig. 3(a)], and the major configurational change was seen at the surface of the cell [Fig. 3(b)2]. Essentially several atoms from the top surface had migrated upwards almost by a lattice spacing. The atoms near the interface still maintained a good registry with the substrate. With atoms at surface moving out and atoms near interface maintaining their positions, vacancies must have formed. With more iterations, we find that the atoms in the top layer have rearranged, but without further significant migration away from the starting surface. We also find rearrangement of atoms near the interface [Fig. 3(b)3-8]. This would indicate migration of vacancies

(b)

(c)

Fig. 6. Three sections in a small computational cell with (001) interface. (a) In the middle of the cell, the dislocation is in plane of interface with an extra half plane terminating at the interface. For the sections closer to the rigid boundary the extra half plane is terminating inside the film with the terminating point rising closer to the surface as seen in (b) and (c) respectively.

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SIMULATION OF FORMATION OF MISFIT DISLOCATIONS

plane was confined to the substrate thus establishing the dislocation at the interface. This suggests that a dislocation loop has formed in the cell. Similar structural changes were also observed for the (111) substrate, although the dislocation type and the change in energy were different for this substrate orientation. These observations suggest that the nucleation of misfit dislocations in compressed films may begin by formation of vacancies near the film surface. As the vacancies migrate towards the interface, a dislocation loop will form that will move inside the film and eventually reach the interface. This will produce an array of misfit dislocations and reduce the strain energy in the lattice. Thus controlling the film surface may be of importance in growing heterostructure films. For example, use of surfactants may alter the energy of formation of vacancies in these sites and thus affect the formation of dislocations [11-13]. Our simulations demonstrate that the site of nucleation of dislocation is near the free surface. This is consistent with experimental observations [3]. The differences in energetics for the 60 ° vs 90 ° dislocations [for (111) and (001) interfaces respectively] were observed in our model. Similar observations have been made using continuum approach [4]. The activation energy barrier for misfit dislocations may then be interpreted as the energy required to generate aforementioned vacancies. Clearly this will vary with lattice strain, substrate orientation, film material type etc. This may explain why the presence and type of surfactant [11-13] affected defect introduction in coherent film during epitaxial growth. In conclusion, using atomistic simulations, we have investigated nucleation and formation of misfit dislo-

cations in coherent heterostructures. The study showed that the dislocation types depended on the interface orientation. It is hypothesized that the nucleation of misfit dislocations started with formation of vacancies at the film surface. As the vacancies migrated towards the substrate, a dislocation loop moved to the interface. Therefore, controlling the free surface of the film will be of great importance during growth of the thin film as far as defect densities are concerned.

REFERENCES

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