This research was supported by the Applied Mathematics program of the National Science Foundation under Grant No. DMS-9622930 for the period 6/1/96-5/31/00.
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 the consequences of the stress-driven morphological instability which occurs during film growth. In particular, the formation of the "island" morphology will be explained in terms of mathematical models for the morphological instability. The goal of the research is to develop models which can be used to predict and control island morphologies in strained solid films.
The mathematical challenge of the work is to develop solutions for a nonlinear free boundary problem in which the position of the free boundary is coupled to the state of elastic strain in the material. To date, we have developed successful approaches to finding solutions to this problem based on asymptotic and numerical methods.
Contents:
Shapes and Energetics of Epitaxial Islands
Stresses and Dislocation Energetics in Epitaxial Islands
Asymptotic Derivation of the Glued Wetting Layer Model and Contact Angle Condition for Strained Islands
The Effect of Island Separation on the Shape of Small Strained Islands
The Shape of Small Three-Dimensional Strained Islands
Morphological Instability in Alloy Films
(collaboration with J. Tersoff of IBM Research Center)
(collaboration with J. Tersoff of IBM Research Center)
The figure below shows the distribution of 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.
*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).
Significant Findings:
Our results provide an explicit derivation of the zero contact angle condition for strained islands with isotropic properties. A consequence of our results is the equivalence of island morphologies calculated from boundary layer models in the limit of vanishing boundary layer thickness. Thus, our analysis validates the use of boundary layer models as a computational tool for calculating island morphologies: if the boundary layer thickness is small relative to the island size, the details of the wetting layer do not affect the macroscopic island shape.
Publications:
C. D. Rudin and B. J. Spencer, "Equilibrium ridge arrays in strained solid films," Journal of Applied Physics, vol 86, pp 5530-5536 (1999).
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).
(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, "Stabilization of strained alloy film growth by a difference in atomic mobilities," Applied Physics Letters, vol 76, pp 3022-3024 (2000).
B.J. Spencer, P.W. Voorhees and J. Tersoff, "Enhanced instability of strained alloy films due to compositional stresses," Physical Review Letters, vol 84, pp 2449-2452 (2000).
*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).
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