低维量子系统中电子态的数值解法

A Numerical Method to Solve Electron States in Low Dimensional Quantum Systems

在有效质量近似下,讨论GaN基低维量子系统(包括量子线和量子点)中电子态的数值求解方法.首先,在无限深势阱近似时,由薛定谔方程解得量子线和量子点中单电子波函数的解析解,分别为贝塞尔函数和球贝塞尔函数,并获得本征能级的解析表达式;然后,通过有限元差分法,数值求解单电子的本征能级和本征态.对比发现,数值求解方法与解析解获得的电子波函数和本征能量在误差允许范围内相等.这说明有限元差分方法求解低维量子系统中电子态的可行性,并可进一步推广到有限高势有限厚垒结构中电子态的求解.最后,计算和讨论不同组分下纤锌矿InxGa1-xN/GaN核壳结构量子线和量子点中的电子态.

Within the effective mass approximation, the numerical solution of the electronic states in GaN-based low dimensional quantum systems （including quantum wires and quantum dots） is discussed. First,the wave functions of a single electron in quantum wires and quantum dots under the approximation of infinitely deep potential barriers are solved analytically from SchrOdinger equation as Bessel functions and spherical Bessel functions respectively. The analytic expressions of corresponding eigen-energies are also obtained. Then,the eigen-energies and eigen-wave functions of a single electron are solved numerically by the finite element difference method. By comparison,the numerical and analytical wave functions and eigen energies of the electron are consistent within allowable deviation. It indicates that finite element difference method is valid for solving electron states in low dimensional quantum systems and can be extended to solve the structures with finitely wide barriers and finitely high potenticals. At last,the wave functions and eigen-energies of electrons in InxGa1-x N//GaN core-shell quantum wires and quantum dots with different x have been computed and discussed.