Variational Problems Associated with Models for Both Orthodox and Unorthodox Materials

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

0072816 Mizel The goal of the part of the present proposal involving unorthodox materials is to develop an understanding of step structure and step motion in the limit of small step free energy for crystals with surface defects consisting of monatomic height steps separated by terraces. Such defects are fundamental structures since steps--as sinks and sources for atoms-- influence mass transport at surfaces and therefore influence fundamental multidimensional surface processes such as nucleation and growth, facet formation and thermal roughening in crystals and their applications. Consequently these defects have been studied since the 1950's and fairly successful macroscopic models to describe behavior in most materials have been developed. These models express the energy of an isothermally wandering step as the integral of the (generally orientation dependent) step energy per unit length. However in the limit of vanishing step free energy, new physics [e.g. higher-order elastic interactions and step-to-step interactions] enters the picture. These higher-order effects can lead to the spontaneous development of highly periodic step structures. Such novel step morphologies are members of an increasingly important class of self- organizing systems: systems that spontaneously form atomic scale periodic patterns. A recent (1997) pathbreaking experiment on Boron- doped Silicon in which the isothermal wanderings in question, contrary to the simple models cited, are periodic and increase with a decrease of the steady state temperature in an interval presents a significant challenge to the development of appropriate macroscopic models for this branch of crystal theory. It reveals that there is a major gap in the current technical understanding of defects in crystals of this nature, since detailed analysis of the dynamics of these fluctuating step edges is incompatible with current theories of step energetics. Resolving this issue is important for near term attainment of non-routine applications involving crystals with small step free energy, since it has direct impact on understanding the atomic-scale structures they develop. In turn, this understanding is needed for the development of materials with uniformly closely spaced and nearly congruent self-organized [quantum dot] defects--such self-organizing materials being crucial to the development of monochromatic lasers and quantum wires, for example, as device scales shrink. The goal of that part of the present proposal involving orthodox materials is focussed on clarifying an as yet unresolved issue in the theory of nonlinearly elastic materials. Equilibria for such materials subjected to specified boundary displacements correspond to deformations which minimize a stored energy integral whose integrand associates to each deformation a nonnegative real valued function. Such stored energy integrands are subject to various constraints in order to correctly represent possible physical materials. In the last decade a previously unnoticed issue has been raised. Namely in view of a little known one-dimensional variational phenomenon (originally established in 1926) that for certain variational integrands the minimizing functions can differ depending on the smoothness of the class of deformations under consideration -- even though the smoother class of functions is dense in the larger class. In fact there can be a nonzero difference between the infima of the integrals associated with such classes [Lavrentiev's gap phenomenon]. A corresponding gap between classes of continuous deformations in three dimensional nonlinear elasticity would imply that the global equilibrium deformation in the larger class would be energy minimizing and more singular than the energy minimizing global deformation in the smaller class--whereby structures devised on the basis of the less singular deformations could develop flaws [fractures], contrary to the evidence provided by calculations based on standard methods. To summarize, the PI proposes in the case of the unorthodox type of crystalline material exemplified by Boron doped Silicon to devise variational models that will shed light on the entirely nonstandard physical behavior of such materials. Such models will involve devising what are known as Landau-de Gennes type order parameter terms for the free energy of such materials to reflect the particularly delicate atomic interactions governing the nanatomic structure of such crystals. On the other hand, in the case of orthodox materials the PI intends to clarify whether in the well-established theory of nonlinear elasticity there can occur materials which for certain classes of boundary conditions can exhibit an energy gap between the minimum energy on one class of continuous smooth deformations and the minimum energy on a smaller class. The occurrence of such an energy gap could lead to cases in which structures devised on the basis of computations associated with the smaller class could develop flaws because the actual energy minimizing deformation is more singular than the computations suggest.