Morphological Development in Strained Alloy Films

This research is supported by the Applied Mathematics program of the National Science Foundation under Grant No. DMS-0072532 for the period 7/15/00-6/30/04

Summary:

The long term objective of this research is to develop mathematical models to predict and control morphology development in strained solid films, which are of importance in emerging semiconductor device applications.

The research program focuses on determining how stress-driven morphological changes in solid films are affected by effects inherent in alloy films. For example, does the possibility of compositional nonuniformity make the growth of planar films more or less unstable?   The goal of the research is to develop models which can be used to predict and control morphologies in strained solid films.  Of special interest is the description of "island" or quantum dot morphologies.

The mathematical challenge of the work is to develop solutions for a nonlinear free boundary problem in which the development of the film shape and composition nonuniformities is coupled to the elastic strain in the material.   In this research program we analyze a new model of alloy film growth using asymptotic, analytic and numerical methods.

Contents:

1. Stresses and Dislocation Energetics in Epitaxial Islands

2. The Shape of Small Three-Dimensional Strained Islands

3. An Evolution Equation for a Strained Thin Film

4. Morphological Instability in Alloy Films

5. Pattern Formation during Strained Alloy Film Growth

6. Asymptotic Solutions for the Equilibrium Crystal Shape with Small Corner Energy Regularization

1. Stresses and Dislocation Energetics in Epitaxial Islands

(collaboration with J. Tersoff of IBM Research Center)

Mathematics:

The numerical method developed in Spencer and Tersoff (1997) was used to solve the coupled free boundary elasticity problem for equilibrium island shapes was used to characterize the stresses and first-order dislocation energetics for epitaxially-strained single-component islands.   The stresses inside the island and substrate were calculated using analytic continuation of the complex Goursat elasticity functions from the boundary data obtained from the boundary integral problem.   The first-order dislocation energetics were obtained by integrating the Peach-Kohler force from the stress-free surface.

The figure below shows the distribution of misfit stresses in epitaxial islands of increasing size.  The shape of the island is determined by the numerical solution of a free boundary problem coupled to the elastic deformation in the solid.  Red regions correspond to high levels of stress and have implications for the formation of defects in the film and substrate.

Significant Findings:

These calculations reveal important generic characteristics and quantitative details of the stress state in equilibrium islands, including stress relaxation near the island peak and stress concentration near the island edges.  We also use the stresses to determine the first-order energy of introducing dislocations of different Burgers vectors into Stranski-Krastanow islands.  These results characterize the ``energy wells'' sought by dislocations to relieve the misfit stress, and suggest that misfit dislocations in islands are segregated by location and type.

Publications:

*B.J. Spencer and J. Tersoff, "Dislocation Energetics in Epitaxial Strained Islands," Applied Physics Letters, vol 77, pp 2533-2535 (2000).

*B.J. Spencer and J. Tersoff, "Stresses and first-order dislocation energetics in equilibrium Stranski-Krastanow islands," Physical Review B, vol 63, article 205424 (2001).

2. The Shape of Small Three-Dimensional Strained Islands

(collaboration with M.A. student L. Shanahan)

Mathematics:

We develop asymptotic solutions to the free boundary elasticity problem for the shape of small axisymmetric strained islands.  Using a small-slope approximation we determine the leading-order solution for the island shape as a solution to an integro-differential equation in which the island width appears as an eigenvalue.  The solution to the eigenvalue problem is developed in terms of a rapidly convergent Bessel series, giving the island width and shape corresponding to the typical Stranksi-Krastanow "bump".  Other eigensolutions correspond to exotic island shapes such as quantum rings and quantum molecules.


The figure above shows the equilibrium solutions for the island shapes corresponding to the first three eigenmodes corresponding to a quantum dot, quantum ring, and quantum molecule.  Below are observations of quantum ring and quantum molecule type morphologies in CdTe/ZnTe (courtesy of H. Luo).

                     


Significant Findings:

We find that the width of an axisymmetric island is almost a factor of two larger than the width of a small two-dimensional ridge.  Our predictions of the island width compare favorably with experimental data in the GeSi/Si system on the width of quantum dot islands.  Also of significant interest is the prediction of quantum ring and quantum molecule type structures as equilibrium solutions to the model.


Publications:

*L.L Shanahan and B.J. Spencer, "A codimension-two free boundary problem for the equilibrium shapes of a small three-dimensional island in an epitaxially-strained solid film,"  Interfaces and Free Boundaries, vol 4, pp 1-25, (2002).

3. An Evolution Equation for Strained Thin Films 

(collaboration with Ph.D. student W. Tekalign)

Mathematics:

Current simulations of the growth and evolution of strained thin films are difficult because the critical role of elasticity requires a full three-dimensional simulation.  We derive here a self-contained two-dimensional evolution equation for the system which is asymptotically valid in the limit of film thickness much smaller than the lateral length scale of morphological features.  The evolution equation contains an elasticity term which is nonlocal and includes the effect of different elastic constants in the film and substrate.  The effect of wetting appears as a nonlinear attraction term and is derived from a prototype "transition layer" model for wetting.  

Significant Findings:

We have demonstrated that the thin film equation possesses steady-state solutions which correspond to island/quantum dot morphologies.  Because of the reduced dimensionality of the evolution equation, it is amenable to large-scale simulations of many-island systems.

Publications:

W.T. Tekalign and B.J. Spencer, Evolution equation for a thin epitaxial film on a deformable substrate, to appear in Journal of Applied Physics.

4. Morphological Instability in Alloy Films

(collaboration with P.W. Voorhees of Northwestern University and J. Tersoff of IBM Research Center)

Mathematics:

We have derived a mathematical model for the growth of strained alloy films.  The resulting model is a nonlinear free boundary problem which is coupled to partial differential equations for the elastic state of the solid.  The basis for the dynamics of the surface is the diffusion of each component along the surface in response to gradients in chemical potentials.  The model enables us to describe the effect of misfit strain, surface energy, compositional stresses generated by composition gradients of different size species, as well as the effect of different surface mobilities for the different components.  From a linear stability analysis of the model we determine the stability of planar alloy film growth with respect to compositional and surface nonuniformities.

Significant Findings:

We find that if the mobilities of the alloy species are the same, then the coupling of compositional stresses and misfit stresses acts to destabilize planar film growth with respect to the effect of misfit stresses alone.  On the other hand, if the mobilities of the components are different, then the coupling between misfit strain, compositional stresses and mobility difference can either stabilize or destabilize planar film growth.  The effect on film stability depends on the sign of the misfit strain, compositional strain and mobility difference.  For sufficiently large mobility difference the linear instability of planar film growth can be completely suppressed.  This stabilization occurs for compressive misfits when one component is large and slow relative to the other; and for tensile misfits when one component is large and fast relative to the other.  A comparison of our theory to the growth of SiGe films indicates that many features of the instability in SiGe can be explained by approximating the surface diffusivity of Ge as being much faster than that of Si.

Publications:

*B.J. Spencer, P.W. Voorhees and J. Tersoff, "Morphological instability theory for strained alloy film growth: the effect of compositional stresses and species-dependent surface mobilities on ripple formation during epitaxial film deposition," Physical Review B, vol 64, article 235318 (2001).

5. Pattern Formation during Strained Alloy Film Growth

(collaboration with Ph.D. student M. Blanariu)

Mathematics:

We analyze the bifurcation to patterned growth morphologies in a complicated model of strained alloy film growth due to Spencer, Voorhees and Tersoff (2001).  We use weakly nonlinear analysis to determine the amplitude evolution equations for hexagonal and banded patterns.   Multiple scale analysis is required in the growth direction to isolate composition-driven and morphological-driven elastic response.  From the quadratic (hexagons) and cubic (bands) amplitude equations we have characterized the bifurcation types in terms of realistic materials parameters, including the behavior of GeSi alloys.  

The figure below is a representative bifurcation diagram illustrating the strength of the compositional segregation accompanying banded and hexagonal patterns as a function of the composition of Ge in the film.  The blue line corresponds to planar film growth; the dashed portion of the line indicates the range of compositions for which planar film growth is unstable.  Bands (green curve) correspond to sheets of alternating composition in the growth direction along
the growth front.  Hexagons (red curve) correspond to compositional "wires" extending in the growth direction.  The arrangement of the wires for the hexagonal growth mode can be either a large Ge core with smaller Si satellites or a large Si core with smaller Ge satellites as shown inside the figure.

Significant Findings:

It has been shown that weakly nonlinear analysis can be adapted to describe the formation of patterns in alloy film growth. We have specifically addressed the strength of morphological and compositional patterns possible in the important SiGe system.  

Publications:

M. Blanariu and B.J. Spencer, Weakly nonlinear theory for pattern formation in strained alloy film growth, submitted to SIAM Journal on Applied Mathematics.

6. Asymptotic Solutions for the Equilibrium Crystal Shape with Small Corner Energy Regularization

Mathematics:

Fully faceted surfaces have sharp corners which cause difficulty and/or ill-posedness in numerical simulations of surface motion.  One approach to circumventing this difficulty is to use a small regularization term which effectively rounds the corner and makes dynamical simulations tractable.  We consider here the "curvature-squared" regularization which is used frequently as a model of corner energy regularization.    This regularization is a nonlinear singular perturbation, and  the connection between the regularized and unregularized behavior has not been firmly established.

In this paper we consider how the classic equilibrium "Wulff shape" problem in two dimensions is modified by the corner energy regularization.  We use matched asymptotic analysis to construct a uniformly valid solution for the equilibrium shape.  Despite the fact that local problem for the rounded corner is nonlinear, we show that there is always a unique corner-rounding solution which precisely matches the Wulff angles of the unregularized problem.

Significant Findings:

Our results show that for a class of surface energy anisotropy models the regularized solution approaches the classic sharp-corner results as the regularization approaches zero.  The results validate the use of the regularization in numerical calculations of the equilibrium problem.  A byproduct of the analysis is an exact solution for the equilibrium shape of a semi-infinite wedge in the presence of the regularization. 

Publications:

B.J. Spencer, Asymptotic solutions for the equilibrium crystal shape with small corner energy regularization, Physical Review E, vol 69, article 011603 (2004).

* Contains research from grant DMS-0072532 (7/15/00-6/30/04) and grant DMS-9622930 (6/1/96-5/31/00).

Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

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