Variational Problems Associated with Models for Both Orthodox and Unorthodox Materials

[00:72816]本提案中涉及非正统材料的部分的目标是发展对台阶结构和台阶运动在小台阶自由能极限下的理解，这些台阶是由台阶分隔的单原子高度台阶组成的表面缺陷晶体。这些缺陷是基本结构，因为步骤——作为原子的汇和源——影响表面的质量输运，因此影响基本的多维表面过程，如晶体的成核和生长、小面形成和热粗化及其应用。因此，自20世纪50年代以来，人们对这些缺陷进行了研究，并开发了相当成功的宏观模型来描述大多数材料的行为。这些模型将等温徘徊阶跃的能量表示为单位长度阶跃能量的积分（通常与方向有关）。然而，在阶跃自由能消失的极限下，新的物理学[如高阶弹性相互作用和阶间相互作用]进入了画面。这些高阶效应可以导致高周期阶跃结构的自发发展。这种新颖的阶跃形态是一类越来越重要的自组织系统的成员：自发形成原子尺度周期模式的系统。最近（1997）的一项关于掺硼硅的开创性实验表明，与所引用的简单模型相反，所讨论的等温漫游是周期性的，并且在一段时间内随着稳态温度的降低而增加，这对为晶体理论的这一分支建立适当的宏观模型提出了重大挑战。它揭示了目前对这种性质晶体缺陷的技术理解存在重大差距，因为对这些波动阶跃边缘的动力学的详细分析与当前阶跃能量学理论不相容。解决这个问题对于近期实现涉及小阶自由能晶体的非常规应用非常重要，因为它对理解它们发展的原子尺度结构有直接影响。反过来，这种理解对于具有均匀紧密间隔和几乎一致的自组织[量子点]缺陷的材料的开发是必要的——例如，随着器件尺寸的缩小，这种自组织材料对单色激光器和量子线的开发至关重要。本提案涉及正统材料的那一部分的目标集中在澄清非线性弹性材料理论中尚未解决的问题。这类材料在特定边界位移作用下的平衡对应于使存储能量积分最小化的变形，其被积函数与每个变形相关联，为非负实值函数。为了正确地表示可能的物理材料，这种存储的能量积分受到各种约束。在过去十年中，人们提出了一个以前未被注意到的问题。也就是说，考虑到一个鲜为人知的一维变分现象（最初建立于1926年），对于某些变分积分，最小化函数可以根据所考虑的变形类的平滑度而不同——即使更平滑的函数类在更大的类中是密集的。事实上，与这类相关的积分的无穷值之间可能存在非零差[Lavrentiev的间隙现象]。在三维非线性弹性中，连续变形类别之间的相应差距意味着，较大类别的整体平衡变形将是能量最小的，比较小类别的能量最小的整体变形更奇异——因此，基于较小奇异变形设计的结构可能会产生缺陷[断裂]，这与基于标准方法的计算提供的证据相反。总而言之，PI建议在以硼掺杂硅为例的非正统晶体材料的情况下，设计变化模型，以阐明这些材料的完全非标准物理行为。这样的模型将涉及到为这种材料的自由能设计出所谓的朗多-德-热纳型序参量，以反映控制这种晶体纳米原子结构的特别微妙的原子相互作用。另一方面，在正统材料的情况下，PI打算澄清在已建立的非线性弹性理论中，是否会出现在某些类别的边界条件下，在一类连续光滑变形的最小能量和较小类别的最小能量之间表现出能量缺口的材料。这种能量缺口的出现可能导致基于与较小类别相关的计算而设计的结构可能出现缺陷的情况，因为实际的能量最小变形比计算结果更奇异。

0072816MizelThe 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.