量子力學給予我們在微觀尺度的物理圖像，讓我們了解古典物理所無法解釋 的現象，像是波粒二象性等，我們可以藉由解決薛丁格方程的解來描述物質的行 為。然而，對於固態的多電子問題十分難解，因此直到 1970 年代，密度泛函理論 的發展成熟，才得以開拓量子化學計算等第一原理的研究。
本篇碩論將改變堆積排序來研究堆積排序對 RuI3 雙層的鐵磁性的影響，再 者，調控雙層間距來模擬單軸壓力對鐵磁性的影響。另一方面，在引入 SOC 後， 我們發現了 SOC 誘發的絕緣能隙，因此我們也進行了拓樸絕緣體的研究，發現藉 由改變堆積排序，可以得到至多 C = −1 的陳絕緣體 (Chern insulator) 且在一個高 對稱性平移方向，發現多個排序有 C = −1 的拓樸態。除此之外，我們藉由壓縮 的雙軸應變發現至多 C = −2 的陳絕緣體。

Quantum mechanics gives us a picture of physics at the microscopic scale, allow- ing us to understand phenomena that classical physics cannot explain, such as wave- particle duality, etc. We can describe the behavior of matter by solving the solution of the Schrödinger equation. However, the many-body problem is very difficult to solve, so it was not until the 1970s that the development of density functional theory matured, and it was possible to open up first-principles research such as quantum chemical calculations.
This master thesis will change the stacking order to study the effect of the stacking order on the ferromagnetism of RuI3 bilayers, and then adjust the bilayer spacing to sim- ulate the effect of uniaxial pressure on the ferromagnetism. On the other hand, after the introduction of SOC, we found the insulating energy gap induced by SOC, so we also con- ducted research on topological insulators and found that by changing the stacking order a Chern insulator with C=-1 can be obtained,and in a high-symmetry translation direction multiple topological states with C=-1 are found. In addition, we find Chen insulators up to C=-2 by compressive biaxial strain.

一、引言 .................1
二、計算方法與理論 .........2
三、DFT+U ................10
四、磁交互作用 .............14
五、拓樸絕緣體 .............20
六、霍爾效應 ...............28
七、瓦尼爾函數與陳數 .........42
八、計算細節 ................53
九、RuI3雙層的AB堆積結構 .....54
十、RuI3雙層的鐵磁性 .........56
十一、RuI3雙層的拓樸 .........68
十二、總結 ..................87
參考文獻 ....................88
附件 .......................97

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