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首页> 外文期刊>Journal of Basic and Applied Physics >Solution Procedure for a Class of Band Structures –Application of the Finite Element Method to Schr?dinger’s Equation
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Solution Procedure for a Class of Band Structures –Application of the Finite Element Method to Schr?dinger’s Equation

机译：一类能带结构的求解过程–有限元方法在薛定er方程中的应用

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Several publications used the finite element method to determine the band structures of periodic solids by solving the Schr?dinger equation (for example, Pask et al., Sukumar et al. [1-4]). The approaches used by these publications could basically be divided into two. The first approach (Pask et al. [1-3]) expresses the wave function ψ as the product of a harmonic function and a periodic function using Bloch’s theorem. The periodic function is then discretized over the domain. In contrast, the second approach (Sukumar et al. [4]) discretizes the wave function over the domain with its nodal values being complex in this case. This paper discusses a solution procedure for determining the band structures for a class of materials starting from the approach followed by Sukumar [4]. It assumes that one can obtain the discrete Hamiltonian and overlap matrices from a conventional finite element analysis program without reverting to a special program. The application of the boundary conditions and the solution of the band structures, for the defined class of material, are performed through matrix operations of well defined steps. The final complex eigenvalue problem, to determine the band energies of the system, is then solved by conventional methods. When solving the resulting system, two representations of the overlap matrix were tested in this work, namely, the consistent and lumped representations. Each of these representations displayed a different response when compared to the exact solution. The results from the lumped and consistent formulations as well as those from a simple averaging process are discussed in this paper.
机译：一些出版物使用有限元方法通过求解薛定？方程来确定周期性固体的能带结构（例如，Pask等人，Sukumar等人[1-4]）。这些出版物使用的方法基本上可以分为两种。第一种方法（Pask等人[1-3]）使用Bloch定理将波动函数ψ表示为谐波函数和周期函数的乘积。然后，周期性函数在域上离散化。相反，第二种方法（Sukumar等人[4]）在该情况下将波函数离散化到该域上，其波节值很复杂。本文讨论一种解决方法，该方法从Sukumar [4]的方法开始确定一类材料的能带结构。假设可以从常规的有限元分析程序中获得离散的汉密尔顿矩阵和重叠矩阵，而无需返回到特殊的程序。对于定义的材料类别，边界条件的应用和能带结构的求解是通过定义明确的步骤的矩阵运算执行的。然后通过常规方法解决最终的复特征值问题，以确定系统的带能量。在求解结果系统时，在这项工作中测试了重叠矩阵的两种表示形式，即一致表示和集中表示。与确切的解决方案相比，这些表示中的每一个都显示出不同的响应。本文讨论了集总和一致的公式以及简单的平均过程得出的结果。

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