本論文主要是利用密度泛函理論(density functional theory，DFT)和格林函數方法(Green’s function method)做外爾半金屬(Weyl semimetal)的能帶結構(energy band structure)的模擬與預測。DFT適合模擬塊材(bulk)和板材(slab)，格林函數適合模擬板材。然而用DFT方法模擬板材所需要的計算量很大，用格林函數方法模擬板材時所需的計算量較用DFT方法模擬板材時所需的計算量少很多，因此我們會需要格林函數方法。我們用這兩種方法模擬TaAs、TaP、NbAs和NbP等材料的表面態(surface state)，並將調整能量去觀察能帶結構，然而我們發現這兩種方法的模擬結果似乎會有差別。我們發現DFT的結果通常與實驗結果較相近，但格林函數方法有時會跟DFT方法有差異。而可能導致格林函數的計算跟DFT結果不同的可能原因為軌域選擇的方式、解格林函數的演算法…等。此外我們也發現了每一篇參考文獻對費米弧的判別結果有時有不一致的問題。儘管如此，外爾點(Weyl point)的拓樸電荷(topological charge)數對應費米弧(Fermi arc)的連結數這件事是正確的。

此外我們將TaAs的Ta替換為V、Nb，As替換為N、P、Sb、Bi時發現除了As被替換為N時不會是外爾半金屬，其餘的組合都有機會成為外爾半金屬。我們推測這可能與原子電負度(electronegativity)、電子游離能(Ionization energy)或其它原因有關，而我們也進一步驗證了原子量(atomic mass)影響外爾點的間距，其原因是自旋軌域耦合(spin orbital coupling，SOC)的強度導致。事實上SOC是外爾半金屬的形成因素之一。我們也發現除了As被替換為N的情形之外都有部分的點保有TaAs的外爾點分布，這代表晶格的結構主導了外爾點的分布。這些系統遵守時間反演對稱(time reversal symmetry)，但不遵守空間反向對稱(inversion symmetry)。

In this thesis, we mainly use the density functional theory(DFT) and the Green’s function method to do the simulations and the predictions of the energy band structures. DFT is suitable to simulate bulk and slab, and Green’s function method is suitable to simulate slab. However, Using DFT to simulate slab needs very large calculation, and Using Green’s function to simulate slab needs much less calculation than using DFT, so we need Green’s function method. We use these two methods to simulate the surface states of TaAs, TaP, NbAs, and NbP lattices and adjust the energy to observe their energy band structures. However, we find that the results of the two methods seem different. We also find that the results of DFT are usually similar to the ones of the experiments, but the results of the Green’s function are sometimes different from the ones of DFT. The reasons which cause the difference between the calculation results of the Green’s function and DFT probably are the way of choosing orbitals, the algorithm of solving Green’s function, …etc. Besides, the recognition results of the Fermi arcs sometimes may not be the same in each reference. Despite the fact, it is correct that the number of the topological charges of the Weyl points corresponds to the connection number of the Fermi arcs.

In addition, we replace Ta of TaAs with V and Nb and replace As with N, P, Sb, and Bi, and we find that all of the combinations have probability to cause Weyl semimetal except for the cases of replacing As with N. We infer that it may be related to the electronegativity of the atoms, the ionization energy of the electrons or the other reasons. We also verify the atomic mass can affect the distance between the Weyl points, and it is caused by the strength of the spin orbital coupling(SOC). In fact, SOC is one of the reasons which causes the Weyl semimetal. We also find that there are some Weyl points have the distribution behavior of the TaAs for each combination except for the cases of replacing As with N. It means that the lattice structure dominates the distribution of the Weyl points. These systems have time reversal symmetry but have no inversion symmetry.

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