Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, 09210-170 Santo André, SP, Brazil.
J Phys Condens Matter. 2023 Mar 24;35(22). doi: 10.1088/1361-648X/acc3e9.
We discretize the Schrödinger equation in the approximation of the effective mass for the two-dimensional electron gas of GaAs, without magnetic field and on the other hand, with magnetic field. This discretization leads naturally to Tight Binding (TB) Hamiltonians in the approximation of the effective mass. An analysis of this discretization allows us to gain insight into the role of site and hopping energies, which allows us to model the TB Hamiltonian assembly with spin: Zeeman and spin-orbit coupling effects, especially the case Rashba. With this tool we can assemble Hamiltonians of quantum boxes, Aharanov-Bohm interferometers, anti-dots lattices and effects of imperfections, as well as disorder in the system. The extension to mount quantum billiards is natural. We also explain here how to adapt the recursive equations of Green's functions for the case of spin modes, apart from transverse modes, for the calculation of conductance in these mesoscopic systems. The assembled Hamiltonians allow to identify the matrix elements (depending on the different parameters of the system) associated with splitting or spin flipping, which gives a starting point to model specific systems of interest, manipulating certain parameters. In general, the approach of this work allows us to clearly see the relationship between the wave and matrix description of quantum mechanics. We discuss here also, the extension of the method for 1D and 3D systems, for the extension apart from the first neighbors and for the inclusion of other types of interaction. The way we approach the method, has the objective of showing how specifically the site and hopping energies change in the presence of new interactions. This is very important in the case of spin interactions, because by looking at the matrix elements (site or hopping) we can directly identify the conditions that can lead to splitting, flipping or a mixture of these effects. Which is essential for the design of devices based on spintronics. Finally, we discuss spin-conductance modulation (Rashba spin precession) for the states of an open quantum dot (resonant states). Unlike the case of a quantum wire, the spin-flipping observed in the conductance is not perfectly sinusoidal, there is an envelope that modulates the sinusoidal component, which depends on the discrete-continuous coupling of the resonant states.
我们对没有磁场和有磁场的 GaAs 二维电子气的有效质量近似下的薛定谔方程进行离散化，这种离散化自然导致有效质量近似下的紧束缚（TB）哈密顿量。对这种离散化的分析使我们能够深入了解位置和跃迁能的作用，这使我们能够用自旋来模拟 TB 哈密顿量的组装：塞曼和自旋轨道耦合效应，特别是 Rashba 情况。有了这个工具，我们可以组装量子盒、Aharanov-Bohm 干涉仪、反点晶格和不完善以及系统无序的哈密顿量。扩展到量子台球是自然的。我们还在这里解释了如何在自旋模式下（除了横向模式外）适应格林函数的递归方程，以便在这些介观系统中计算电导。组装的哈密顿量允许识别与分裂或自旋翻转相关的矩阵元（取决于系统的不同参数），这为模型特定感兴趣的系统提供了起点，操纵某些参数。一般来说，这项工作的方法使我们能够清楚地看到量子力学的波和矩阵描述之间的关系。我们还在这里讨论了该方法在 1D 和 3D 系统中的扩展，以及除了第一近邻之外的扩展和其他类型相互作用的包含。我们方法的方法是为了展示在存在新相互作用时，位置和跃迁能是如何具体变化的。这在自旋相互作用的情况下非常重要，因为通过观察矩阵元（位置或跃迁），我们可以直接确定导致分裂、翻转或这些效应混合的条件。这对于基于自旋电子学的器件设计至关重要。最后，我们讨论了开放量子点（共振态）状态下的自旋电导调制（Rashba 自旋进动）。与量子线的情况不同，电导中观察到的自旋翻转不是完全正弦的，存在一个调制正弦分量的包络，它取决于共振态的离散连续耦合。
