## Morphological Development in Strained Alloy Films

##### Abstract

Strained solid films are an important component of newly-developed<br/>electronic devices such as the quantum-well semiconductor and the<br/>bipolar heterojunction transistor. The strain in the film is<br/>essential to these devices because it modifies the electronic band gap<br/>in the device to generate the desired electronic properties. The<br/>strain in the film, however, is also responsible for the generation of<br/>instabilities during growth, resulting in nonplanar films with<br/>inhomogeneous microstructure, such as "quantum dots" and "quantum<br/>wires". Because of the strain localization associated with the<br/>inhomogeneous microstructure, the resulting electronic properties can<br/>be enhanced due to quantum electronic effects. Thus, there has been<br/>intense interest in natural "self-assembly" of quantum dot and quantum<br/>wire morphologies during the growth process. The research of the PI<br/>will focus on the analysis of mathematical models for the development<br/>of such inhomogeneous microstructures in strained alloy films. The<br/>theoretical description of morphological development in strained films<br/>is difficult because of the major role that elastic strain plays in<br/>the development of the morphology; the generic case is that of a free<br/>boundary or moving boundary elasticity problem. While there has been<br/>a significant amount of progress made on the mathematical modeling of<br/>the growth of single-component strained films, models for the growth<br/>of alloy strained films are still in their infancy. The proposed<br/>research will examine the microstructure generated from the new alloy<br/>film models as nonlinear solutions to the moving boundary elasticity<br/>problem using a combination of both analytical and numerical<br/>techniques. The research will focus on three areas relating to the<br/>formation of microstructure in strained alloy films. First, the<br/>possibility of generating a new type of microstructure based on<br/>self-assembled compositional modulations in thick films will be<br/>investigated by a bifurcation analysis of the nonlinear free boundary<br/>problem. Second the growth of inhomogeneous alloy quantum dots will<br/>be described using a hybrid asymptotic and numerical approach, and the<br/>theoretical predictions will be compared to experiments carried out in<br/>parallel by a collaborator. Finally the appropriate mathematical<br/>modeling of facet corners in strained alloy crystals will be examined<br/>through a model which incorporates atomic-scale behavior in a<br/>macroscopic model for film growth. The overall goal of the research<br/>is to develop mathematical approaches that enable a comprehensive<br/>theoretical description of nanostructure formation in strained alloy<br/>films.<br/><br/>In nanoscale electronic devices, strained solid layers play an<br/>important role because of the improved electronic properties of the<br/>strained material. While flat, planar strained films have been used<br/>successfully, they can be susceptible to instabilities during the<br/>growth process and develop nanoscale bumps ("dots"). These nanoscale<br/>"quantum dots" have been found to give superior electronic behavior<br/>because of quantum-physics electronic effects. There is thus interest<br/>in growing quantum dots with a controlled size and spacing to give a<br/>material with specified or optimum electronic properties. The<br/>research supported by this grant focuses on the development of<br/>mathematical models for describing the formation of quantum dots and<br/>other nanostructures from detailed modeling of the physics of the<br/>growth process. In particular, the work will focus on the growth of<br/>alloy films, for which theoretical understanding not well developed.<br/>The goal of the theoretical work will be to evaluate the effect of<br/>different growth parameters to guide the development of "optimum"<br/>quantum dot structures. To solve the mathematical problem, advanced<br/>mathematical techniques will be developed and applied to determine the<br/>characteristics of the solutions and how they depend on the material<br/>properties and growth conditions. In addition, the research will also<br/>address the difficult question of how to incorporate atomic-level<br/>behavior into a large scale model for strained film growth. The<br/>theoretical predictions will be compared to experiments conducted in<br/>parallel by a collaborator here at the University at Buffalo. The<br/>results of the work will be twofold. The understanding of how to<br/>treat the mathematical issues of strained alloy film growth well will be <br/>improved. Also, a useful "parameter map" describing the important <br/>physical processes that influence the development of nanostructures <br/>will be developed. This parameter map can be used as a<br/>guide to the design of strained alloy nanostructures with specified or<br/>optimum properties.