密度泛函理論是研究晶體材料的方法，近年來成為流行的一種。基本理想是電荷密度比電波函數收斂快。如果我們能找到一種通過電荷密度來表示哈密頓量的簡單方法，我們就可以證明該系統的性質。對科恩沈呂九方程和交換相關項的研究給出了模擬電子系統哈密頓量的路徑。此外，由於大衛范德比爾特的研究，密度泛函理論的哈密頓量可以很容易地轉換成具有瓦尼爾軌道（或玻以斯軌道）的跳躍模型。此外，藉由擴展密度泛函理論，密度泛函微擾理論可用於固態系統中的準粒子模擬，如聲子模擬。

本論文共分五章。在第一章中，我們將介紹在我們的研究中使用的基本理論，例如密度泛函理論，密度泛函微擾理論和瓦尼爾等。從第二章到第四章，我們研究了具有密度泛函理論和密度泛函微擾理論的固態系統的許多性質。在第二章中，我們介紹了二維繫統的研究，包括鍺銀合金表面合金和鍺烯。展示了莫爾格子展開方法的應用。第三章介紹了鍶錫磷化物和鉬碳化物的超導電性。此外，我們證明了自旋軌道耦合增強了鍶錫磷化物中的電子聲子耦合。另一方面，我們還展示了缺陷如何影響鉬碳化物中的聲子帶和科恩異常。在第四章中，我們研究了二氧化鉛的拓撲性質。給出了非費米液體和二氧化鉛的候選物。最後，在第五章中，我們提出了鍶釕氧量子阱狀態系統中的磁異向性。我們證明自旋軌道耦合可以看作是一個有效的塞曼項，並打破了dxz+dyz軌道的簡併性。此外，這種對稱性破壞會引發磁異向性並且這種磁異向性可通過柵極電壓和膜厚度調節。我們在這五章中的結果不僅展示了許多材料的新特性，而且還展示了第一原理模擬的新方法。

The Density functional theory (DFT), an ab initio (first-principle) method to study the crystal material become a popular recent year. The basic ideal is the electric charge densities are convergence faster than the electric wave function. If we can find a simple method to present the Hamiltonian by charge densities, we can show the properties of this system. The studies of Kohn-Sham equation and the exchange-correlation term give the route to simulate the Hamiltonian of electronic system. Moreover, owing to D. Vandibilt’s studies, the Hamiltonian of DFT could be easily transformed into the hopping model with Wannier orbitals (or Boys orbitals). Moreover, extending the DFT, the density functional perturbation theory (DFPT) can be used in quasiparticle simulations in solid state systems, such as phonon simulations.

This thesis has five chapters. In the first chapter, we will introduce the basic theory using in our studies, for example, DFT, DFPT, and Wannier basis, etc. From chapter two to four, we study many properties of solid state system with DFT and DFPT. In the second chapter, we present our studies of two-dimensional systems, including Ag2Ge surface alloy and germanene. The applications of unfolding method for Moire lattice are shown. In the third chapter, the studies of superconductivity of SrSnP and MoC are presented. Moreover, we demonstrate the spin-orbital coupling enhance the electron-phonon coupling in SrSnP. On the other hand, we also show how the defect influence the phonon band and Kohn anomaly in MoC. In fouth chapter, we study the topological properties in β−PbO2. The candidate of non-Fermi-liquid and in β−PbO2 are presented. Finally, in the chapter five, we presented the magnetic anisotropy (MA) in SrRuO3 quantum well state systems. We show the spin-orbital coupling can be seem as a effective Zeeman term and break the degeneracy of dxz dyz orbital. Moreover, this symmetry breaking induce the MA and the MA are tunable by gate voltage and film thickness. Our result in these five chapters not only present new properties of many materials but also show new methods in first-principle simulations.

1. Introduction 4
2. Ag2Ge and Germanene 16
3. SrSnP and MoC 23
4. PbO2 31
5. SRO 40
Bibliography 50
A. Publications 53

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