Abstract
An effective multi-element simulation methodology for quantum eigenvalue problems is investigated. The approach is derived from a reduced-order model based on a data-driven learning algorithm, together with the concept of domain decomposition. The approach partitions the simulation domain of a quantum eigenvalue problem into smaller subdomains that, referred to as elements, could be the building blocks for quantum structures of interest. In this quantum element method (QEM), each element is projected onto a functional space represented by a set of basis functions (or modes) that are generated from proper orthogonal decomposition (POD). To construct a POD model for a large domain, these projected elements can be combined together, and the interior penalty discontinuous Galerkin method is applied to achieve the interface continuity and stabilize the numerical solution. The POD is able to optimize the basis functions specifically tailored to the geometry and parametric variations of the problem and can therefore substantially reduce the degree of freedom (DoF) needed to solve the Schrödinger equation. To understand the fundamental issues of the QEM, demonstrations in this study focus on examining the accuracy and DoF of the QEM influenced by the training settings for generation of POD modes, selection of the penalty number, suppression of interface discontinuities, structure size and complexity, etc. It has been shown that the QEM is able to achieve a substantial reduction in the DoF with a high accuracy even beyond the training conditions for the POD modes if the penalty number is selected within an appropriate range.
Highlights
A wide range of engineering and scientific analysis and design in the areas of electronics, photonics, materials, physics, biology, medicines and chemistry involves quantum eigenvalue problems governed by the Schrödinger equation.1–20 These result from their small physical dimensions near the electron wavelength or molecular/nuclear scale, where quantum phenomena become essential and are primarily responsible for their photonic, electronic, and/or chemical characteristics
Demonstrations of the quantum element method (QEM) show that accurate wave functions (WFs) with a very small degree of freedom (DoF) can be achieved for the first several quantum state (QS) in QW structures if the penalty number Nμ > Nμ,min
This study reveals some encouraging capabilities in the proposed QEM
Summary
A wide range of engineering and scientific analysis and design in the areas of electronics, photonics, materials, physics, biology, medicines and chemistry involves quantum eigenvalue problems governed by the Schrödinger equation. These result from their small physical dimensions near the electron wavelength or molecular/nuclear scale, where quantum phenomena become essential and are primarily responsible for their photonic, electronic, and/or chemical characteristics. To minimize the time consuming training process of large domains, the quantum element method (QEM) based on a multi-element POD approach with domain decomposition was briefly presented and demonstrated at a conference.44 Such an approach makes the simulation and design of quantum structure more flexible and cost-effective. This study focuses on analysis of some crucial numerical and statistical aspects to improve the accuracy and the DoF needed for the solution These include (i) data quality as a result of training settings, (ii) interface discontinuities of the POD WFs associated with the number of modes and the penalty parameter, (iii) influences of the physical size and complexity of the elements on the number of modes needed to achieve a good accuracy, and (iv) robustness of the QEM beyond the training conditions.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call