此論文藉由第一原理計算研究籠目狀(Kagome)鐵磁材料RMn6Sn6和反鐵磁拓 樸絕緣體EuSn2P2，實驗上透過掃描穿隧電子顯微鏡發現在TbMn6Sn6中有很強烈 的反常量子霍爾效應證據，透過第一原理計算我們解釋了實驗上觀察到的狄拉克 錐的成因，並詳細解析其軌域組成，更進一步我們討論了此材料在塊材和薄膜下 的拓樸，在塊材下計算結果顯示TbMn6Sn6是一個磁性外爾半金屬，因此在薄膜情 況下很有可能實現反常量子霍爾效應。更進一步我們討論了GdMn6Sn6的角解析 光電子能譜中觀察到的破缺狄拉克錐，可以藉由考慮材料中的能帶開摺重現。此 外，透過磁結構的改變，我們驗證了此材料中Kane-Mele自旋軌道耦合是造成能隙 大小的關鍵。因此，此一系列材料能夠透過參雜來達到材料中貝里曲率的操控， 更進一步改變反常霍爾效應的值。論文的第二部分著重在反鐵磁拓樸絕緣體的預 測，EuSn2P2和傳統拓樸絕緣體Bi2Se3有相同的結構，透過第一原理計算，我們發 現在Γ點有能帶反轉，更驗證此材料的拓樸不變量Z2 = 1，在表面態計算上(001)面 的磷節理面能夠和實驗匹配，解釋了實驗上觀測到的行為，除了反鐵磁拓樸絕緣體 外，EuSn2P2和同類型材料EuSn2As2同時也是一個鏡像對稱保護得拓樸晶體絕緣 體(Topological crystalline insulator)，當材料中的Eu層數是偶數時，由於空間反映對 稱被破壞，透過計算，我們預期由量子度規(Quantum Metric)引起的非線性霍爾效 應將大於目前已知材料一個數量級。

This dissertation investigates the Kagome magnets RMn6Sn6 and the antiferromagnetic topological insulator EuSn2P2 through first-principle calculations. Experimentally, strong evidence of the quantum anomalous Hall effect was found in TbMn6Sn6 using scanning tunneling microscopy. Through first-principle calculation, we elucidated the origin of the Dirac cones observed experimentally and analyzed in detail their orbital composition. Additionally, the topology of this material in both bulk and thin film forms is discussed. In the bulk, the computational results show that TbMn6Sn6 is a magnetic Weyl semimetal, suggesting the potential realization of the quantum anomalous Hall effect in thin films. Additionally, we discussed the broken Dirac cones observed in the Angle-Resolved Photoemission Spectroscopy(ARPES) of GdMn6Sn6, which can be reproduced by considering the band unfolding in the material. Furthermore, alterations in the magnetic structure confirmed that the Kane-Mele spin-orbit coupling is vital for the magnitude of the energy gap in this material. Consequently, this series of materials can manipulate the Berry curvature through doping, thereby enhancing the anomalous Hall effect. The second part of the paper focuses on predicting the antiferromagnetic topological insulator, EuSn2P2, which has the same structure as the conventional topological insulator Bi2Se3. Through first-principle calculations, we found band inversion at the Γ point, verifying the topological invariant Z2 = 1 of this material. In the surface state calculations, the phosphorous termination surface on the (001) plane matched the experiments, explaining the behaviors observed experimentally. Besides being an antiferromagnetic topological insulator, EuSn2P2 and similar compound EuSn2As2 are topological crystalline insulator protected by mirror symmetry. The spatial inversion symmetry breaks when the number of Eu layers in the material is even while the PT symmetry is still preserved. Through calculations, we expect that the nonlinear Hall effect induced by the quantum metric will be an order of magnitude larger than in currently known materials.

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