Finite Element Solution Of Multidimensional Poisson-Schrodinger Equation In Quantum Devices

In recent years,with the continuous development of semiconductor technology,researchers have proposed and implemented a large number of prototypes of new semiconductor devices.The working principle of these semiconductor devices is related to the quantum characteristics inside the device.At present,the development of epitaxial growth technology makes the advantages of quantum devices more prominent,quantum devices have been widely used in infrared detectors,laser development,wireless communications systems and other production and research areas.Therefore,it is very important to choose an effective method to analyze the physical properties of quantum devices.In the fields of physics and other scientific research,Poisson equation and Schrodinger equation are very important partial differential equations in the study of modern quantum physics.By solving the Schrodinger equation in the microscopic system,the wave function and energy can be obtained,and the distribution of particles in the quantum system can be analyzed.By solving the Poisson equation,the potential distribution of the device can be derived.In the quantum device theory calculation,the exact solution of the Poisson-Schrodinger equation needs to be used to analyze the physical properties of the quantum device.Therefore,finding a more effective method to solve these two equations becomes an important issue in this study.From a multidimensional perspective,this paper proposes the use of the finite element method to study Poisson-Schrodinger equations,and gives concrete ideas and steps for the finite element method to solve equations.The content of this article is as follows:(1)The FEni CS library based on the finite element method can only solve the variational form of partial differential equations.This paper first needs to change the Poisson-Schrodinger partial differential equation into a variational equation.(2)Using open source FEniCS library,write Python program to define the variational form of Poisson equation and Schrodinger equation.(3)By setting up the boundary conditions,the one-dimensional Poisson-Schrodingerequation is self-consistently solved in an iterative manner.The Al Ga As/GaAs quantum well device is used as the research object to perform related calculations.The wave functions and electrons in the quantum system are demonstrated.density.(4)Based on the open source FEni CS library,two-dimensional and three-dimensional Poisson equations and Schrodinger equations are solved by finite element method.The computational results of the Poisson equation are applied to the photoconductive antenna model,and the potential distribution of the photoconductive antenna is shown.The results of the Schrodinger equation are applied to a novel AlGa As/Ga As quantum well device.Unlike the traditional Al Ga As/GaAs quantum device,the device consists of only two layers of semiconductor material.The structure of the device is to embed Ga As material in the middle of the Al Ga As material.Finally,the potential energy and wave function of the Al Ga As/GaAs quantum well devices are plotted.Through Python code development,this paper has studied the multidimensional Poisson-Schrodinger equation finite element solver.Compared to the solvers developed by other methods,the finite element solver in this paper is more flexible and easier to solve for partial differential equations.It can be applied to quantum devices in different situations.For a different dimension of the quantum device,as long as the dimension is set,the solver will automatically solve the Poisson-Schrodinger equation and analyze the physical characteristics of the device.For a quantum device with different structures,only a part of the code needs to be adjusted,and it can be applied to different types of new quantum devices.This study provides an important research basis for analyzing different types of quantum devices.