螺旋对称体系的从头算框架：理论、数值实现及其在纳米结构扭转变形中的应用
Ab initio framework for systems with helical symmetry: theory, numerical implementation and applications to torsional deformations in nanostructures
摘要：
我们制定和实施螺旋DFT -一个自洽的第一性原理模拟方法与螺旋对称性的纳米结构。这些材料在纳米技术、化学和生物学领域都有很好的代表性，预计将与前所未有的材料特性相关。我们严格证明了螺旋纳米结构，称为螺旋布洛赫波的单电子问题的特殊解决方案的存在性和完整性。我们描述了如何Kohn-Sham密度泛函理论方程的螺旋纳米结构可以减少到一个基本域的帮助下，这些解决方案。在我们的数学处理中的一个关键组成部分是螺旋布洛赫-弗洛奎变换的定义和使用执行块对角化的直接积分的意义上的哈密顿量。我们开发了一种螺旋坐标系下的自适应有限差分策略来离散控制方程，并获得了所提出的方法的工作实现。我们通过例子验证了我们的数值实现的准确性和收敛性。最后，我们利用螺旋密度泛函理论研究了锯齿形和手性单壁黑磷（即，磷烯）纳米管。我们使用我们的模拟来评估一个锯齿形纳米管从头计算的扭转刚度。此外，我们观察到绝缘体到金属的过渡，这种纳米管的电子特性，因为它受到扭曲。我们还发现，一个类似的过渡可以影响手性磷烯纳米管的轴向应变。值得注意的是，这种性质的自洽从头算模拟是前所未有的，远远超出了现有的任何其他系统的第一原理方法的范围。最后，我们讨论了各种未来的途径和应用。
英文摘要：
We formulate and implement Helical DFT -- a self-consistent first principles simulation method for nanostructures with helical symmetries. Such materials are well represented in all of nanotechnology, chemistry and biology, and are expected to be associated with unprecedented material properties. We rigorously demonstrate the existence and completeness of special solutions to the single electron problem for helical nanostructures, called helical Bloch waves. We describe how the Kohn-Sham Density Functional Theory equations for a helical nanostructure can be reduced to a fundamental domain with the aid of these solutions. A key component in our mathematical treatment is the definition and use of a helical Bloch-Floquet transform to perform a block-diagonalization of the Hamiltonian in the sense of direct integrals. We develop a symmetry-adapted finite-difference strategy in helical coordinates to discretize the governing equations, and obtain a working realization of the proposed approach. We verify the accuracy and convergence properties of our numerical implementation through examples. Finally, we employ Helical DFT to study the properties of zigzag and chiral single wall black phosphorus (i.e., phosphorene) nanotubes. We use our simulations to evaluate the torsional stiffness of a zigzag nanotube ab initio. Additionally, we observe an insulator-to-metal-like transition in the electronic properties of this nanotube as it is subjected to twisting. We also find that a similar transition can be effected in chiral phosphorene nanotubes by means of axial strains. Notably, self-consistent ab initio simulations of this nature are unprecedented and well outside the scope of any other systematic first principles method in existence. We end with a discussion on various future avenues and applications.
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