狄拉克半金属是一种具有非平庸拓扑性质的新型量子态，它的体能带在费米面附近一般具有三维狄拉克点，在狄拉克点附近的三个动量方向都是线性色散关系，狄拉克点处的四重简并一般由晶格对称性与时间反演对称性共同保护。非平庸拓扑性质一般体现在三维布里渊区中的某个二维子平面具有非零的$\mathbb{Z}_2$拓扑不变量，或者在狄拉克点 处可以打开拓扑非平庸的带隙。基于第一性原理计算，我们主要预测并系统地研究了两种实验上 尚未发现的狄拉克半金属相材料。 \uppercase\expandafter{\romannumeral4}A族元素在传统半导体及微电子领域中有着 广泛的应用，所以如果能够在\uppercase\expandafter{\romannumeral4}A族元素中实现狄拉克半金属相，那么对于推广这种新型拓扑材料的应用将大有裨益。我们的研究发现了一种可能的狄拉克半金属材料是Ge和Sn的三维层状同素异形体，通过计算其体能带结构确定了三维狄拉克点位于三重旋转轴上，时间反演不变点处的宇称计算与表面态的计算证实了非平庸的拓扑性质。此外，我们的研究还表明在合适的厚度时，Ge的这种新型同素异形体的准二维薄膜可以表现出量子自旋霍尔效应，这个发现将促进人们对把非平庸拓扑材料融合于锗基电子器件中的探索。 近些年兴起了一种新型铁电材料，其六角LiGaGe晶格结构相对传统铁电材料（如BaTiO$_3$等）的 正八面体结构更为简单，同时又具有相同的极化强度。另一方面，将拓扑性质与传统的半导体性质（铁电、铁磁等）相结合一直是人们关心的问题，于是我们试图在这类新型铁电材料中去寻找狄拉克半金属相。在半金属材料LiZnBi中，我们发现其在$\Gamma$点处的低能能级的带序以及潜在的拓扑性质对于应力十分敏感。在合适应力的条件下，它会进入狄拉克半金属相，$\Gamma-K-M$平面的$\mathbb{Z}_2$拓扑不变量非零，表面态中明显存在一对费米弧连接体狄拉克点的投影。我们还对应力影响电子结构背后的物理机制进行了初步的探究，并得出结论面内压应力和垂直面的拉伸应力都有利于狄拉克半金属相在LiZnBi材料中出现。 除了狄拉克半金属外，我们也对其他拓扑非平庸相进行了研究。在石墨烯与三维拓扑绝缘体的异质结中，我们发现由于两个材料中狄拉克费米子的相互作用使得界面态产生了一个线性项特别小的无质量狄拉克费米子，利用平均场近似，我们预计这样的狄拉克费米子只需要很小的电子-电子相互作用就可以打开拓扑非平庸带隙，使体系进入半量子霍尔相。
Dirac semimetal is a new kind of quantum matter with nontrivial topological properties. There are three-dimensional Dirac nodes near the Fermi level in the bulk band structure, around which the dispersions are linear along three directions. The four-fold degeneracy at the Dirac node is often protected by time-reversal symmetry and crystal symmetry. As far as the nontrivial topology, it is related to either the non-zero $\mathbb{Z}_2$ topological invariant defined in certain two-dimensional plane, or the possible topologically nontrivial gap exactly at the Dirac nodes with symmetry-breaking perturbations. Based on the first-principles' calculations, we mainly predict and investigate two Dirac semimetal materials, which have not been identified in experiments. Group \uppercase\expandafter{\romannumeral4}A elements have been widely used in conventional semiconductors and microelectronics, so the realization of Dirac semimetal in group \uppercase\expandafter{\romannumeral4}A elements will prompt the commercial application of this new quantum matter. Our studies reveal that one layered allotrope of Ge and Sn could be a good candidate. We find the three-dimensional Dirac points along the $C_3$ rotation axis in the bulk band structure, and the parity of the occupied states at the time-reversal invariant points and the surface states demonstrate the nontrivial topology. In addition, our theoretical calculations show that the quasi-2D film of the new allotrope of Ge is a quantum spin Hall insulator when the thickness is properly tuned. This could be an important progress in the integration of topologically nontrivial materials into Ge-based electronic devices Recently, the hexagonal ABC compounds come into sight as a new kind of ferroelectrics. Their lattice structure is simple and they have similar polarizations as those of well-known ferroelectrics (such as BaTiO$_3$). On the other hand, combining nontrivial topology and traditional properties of semiconductors (such as ferroelectrics and ferromagnetism) has application potentials in spintronics and thus has attraced people's attention. In one of the hexagonal ABC compounds-LiZnBi, we find that its low-energy bands near the $\Gamma$ point and the related topological properties are strongly dependent on the strain. Under suitable strains, LiZnBi can be turned into Dirac semimetal, where the $\mathbb{Z}_2$ topological invariant is nonzero and there are Fermi arcs in the surface states connecting the projections of the bulk Dirac nodes. Furthermore, we study how the strain affects the low-energy band structure in detail and draw a conclusion that both biaxial compressive in-plane strain and tensile out-of-plane strain favor the Dirac semimetal phase in LiZnBi. Besides the Dirac semimetal phase, we also explore other topologically nontrivial phases.In the graphene/topological insulator heterostructure, we discover a gapless Dirac Fermion with a much depressed Fermi velocity in the interface. The anomalously small linear terms are attributed to the interaction between the Dirac Fermions in the two materials. Moreover, based on the mean-field approximations, we predict that with a weak electron-electron interaction, the Dirac Fermion with small Fermi velocity will open a topologically nontrivial gap and the half quantum Hall effect is expected.
