Finite Element Framework for Nanomechanics and Electronic Structure Calculations of Semiconductors

The thesis presents a variational computational framework for nanomechanics and electronic structure calculations of semiconductors. In order to predict the coupled mechanical and electronic properties of semiconductor devices, especially in flexible electronics, a scalable computational framework is the need of the time. The thesis provides a step in that direction in carrying out such computations by providing a real space method for electronic structure computations embedded in a computationally efficient nanomechanics framework. However further study is required to combine mechanical and electronic computational framework. The first part of thesis provides a multiscale finite element computational framework for nanomechanics problems that combines discrete models like molecular structural mechanics models at nanometer or atomic scales and quasi-continuum mechanics models at micro meter scales. The quasi-continuum model uses material moduli defined via internal variables which are functions of local atomic configuration, while the molecular mechanics model incorporates interatomic potentials into its discrete model to derive nanoscale based material moduli. Point defects like vacancy perturb the local atomic configuration and induce forces locally. A homogenization scheme is used to evaluate the equivalent material moduli around the defect area by evaluation of defect formation energy and incorporating it into molecular structural mechanics model. The hierarchical multiscale finite element framework seamlessly combines both discrete and quasi-continuum models at each integration point and evaluates stress strain response and material properties for defective and non-defective material. Representative examples are provided. For electronic structure calculations the thesis presents a study of stabilized formulation of Schr\"{o}dinger wave equation and numerical studies are conducted with Lagrange basis functions for tetrahedral and hexahedral elements for three dimensional Kronig-Penney problem. It is followed by presentation of B-splines and NURBS based finite element formulation for linear Schr\"{o}dinger wave equation and non-linear, non-local Kohn-Sham equations. The higher order continuity and variation diminishing property of B-splines and NURBS basis functions offer significant advantage over C${}^{0}$ Lagrange basis functions for representing high gradient solutions with higher precision. In addition NURBS functions accurately represent geometries including conic sections with minimum parameters thus avoiding errors due to boundary conditions and/or geometries. Representative examples are provided.