A Lagrangian approach for quantum-mechanical electrostatic analysis of deformable silicon nanostructures

Semiconductor mechanical components of nanoelectromechanical systems (NEMS) typically undergo deformations when subjected to electrostatic forces. Computational analysis of electrostatic NEMS requires an electrostatic analysis to compute the electrostatic forces acting on the nanomechanical structures and a mechanical analysis to compute the deformation of the nanomechanical structures. Typically, the mechanical analysis is performed by a Lagrangian approach using the undeformed position of the structures. However, the electrostatic analysis is performed by using the deformed position of the nanostructures. The electrostatic analysis on the deformed position of the nanostructures requires updating the geometry of the structures during each iteration. In this paper, based on a recently proposed hybrid BIE/Poisson/Schrödinger approach, we propose Lagrangian formulations for the BIE/Poisson/Schrödinger equations and solve the coupled Lagrangian BIE/Poisson/Schrödinger's equations self-consistently using the undeformed position of the semiconductors to compute the charge distributions on the deformed semiconductors. The proposed approach eliminates the requirement of updating the geometry and, consequently, significantly simplifies the procedure of coupled electromechanical analysis of NEMS.