拓扑晶体绝缘体是一种新的物质相，它的拓扑性质受晶体对称性保护，并具有多个狄拉克表面态。基于第一性原理和量子输运计算，我们系统研究了拓扑晶体绝缘体SnTe的(111)表面和薄膜中的一些新奇性质。 由于极性，理想的SnTe(111)表面原则上是不稳定的。因此我们首先研究了SnTe(111)表面的稳定性，发现在不同的生长条件下可以形成三种稳定的表面结构。表面电子结构计算发现它们具有两种定性不同类型的拓扑表面态：未重构和(√3 × √3) 重构的表面具有第一种类型的表面态，即四个狄拉克点位于四个时间反演不变动量点；(2×1) 的表面重构引起表面布里渊区的折叠，使不同狄拉克谷产生相互作用，产生了新类型的表面态，即两个狄拉克点偏离了布里渊区中心的时间反演不变点。我们的研究表明，除了选择不同的表面方向还可以通过控制生长条件来产生不同类型的拓扑表面态。 拓扑晶体绝缘体在其表面能带中具有偶数个狄拉克锥（多个谷）。我们系统研究了SnTe(111)表面狄拉克谷在应变下的演化，发现压缩应变使¯Γ 和¯M 谷的狄拉克锥产生不同程度的移动，甚至相反的移动；拉伸应变可以增强上下表面间的耦合，甚至使¯Γ 和¯M 谷产生不同大小的能隙。在SnTe(111)表面上，我们设计了一种应变异质结构，并发现通过动态施加局部压力，可以实现强的狄拉克费米子的谷过滤效应。这些结果显示，拓扑晶体绝缘体中可以实现应变的功能化应用和狄拉克谷电子学应用。 拓扑材料薄膜的狄拉克费米子具有螺旋自由度。我们以拓扑晶体绝缘体SnTe的(111)薄膜为例，发现用适当的电场可以使薄膜的狄拉克费米子产生巨大的螺旋性劈裂。基于这些结果，我们对狄拉克费米子透过双栅极纳米结构的输运进行了计算，发现了一些螺旋性相关的特性，包括狄拉克费米子螺旋性的选择透射，螺旋性切换，螺旋性负折射以及双负折射。我们的结果为实现基于螺旋性的电子学应用提供了可能。

As a new matter phase, topological crystalline insulator (TCI) is protected by the crystal symmetry, and has multiple Dirac surface states. Based on the first-principles and quantum transport calculations, we have systematically studied the novel properties oftopological crystalline insulator SnTe on its (111) surface and thin film. Due to the polarity, the ideal (111) surface of SnTe is unstable in principle. So we first study the stability of SnTe (111) surface, and find that there are three stable surface structures under different growth conditions. The surface band calculations show that these structures support two qualitatively different types of topological surface states: the Te-terminated surface without reconstruction and the (√3 × √3)-reconstructed surface possess the first type of surface states, with four Dirac points at four time-reversalinvariant points; the (2×1)-reconstructed surface folds the surface Brillouin zone, effectively induces interactions between the different Dirac valleys and produce a new type of surface states, with two Dirac points nearby but not at ¯Γ. Besides selecting different surface orientations, our work suggests a promising alternative way to produce the different topological surface states of TCIs by controlling the growth conditions. Topological crystalline insulator has an even number of Dirac cones (i.e., multiple valleys) in its surface band structure. We systematically investigate the strain-induced evolution of topological surface states on the SnTe(111) surface. Our results show thatcompressive strain can shift the Dirac cones at the ¯Γ and ¯M valleys to different extents (even oppositely) in energy, while tensile strain can induce different band gaps at both valleys due to the enhanced intersurface coupling. Exploiting a strain-induced nanostructure with well-defined edges on the (111) surface, we demonstrate strong valley-selective filtering for massless Dirac fermions by dynamically applying local external pressure. Our findings may hold great promise for strain-engineered nanoelectronics and valleytronic applications in TCIs. The Dirac fermions in the thin films of topological materials have the helicity degree of freedom. Using topological crystalline insulator SnTe(111) thin films as an example, we find that giant helicity splitting in the band structures can be induced under moderate electric field. Based on this result, we perform the transport calculations of the Dirac fermions through a dual-gated nanostructure, and some helicity-resolved functionalities, including pronounced helicity-selective transmission, helicity switching, helical negative refraction and birefraction, are demonstrated, where the intra-helical scattering always dominates over the inter-helical one. Such intriguing control strategy for helical Dirac fermions may hold great promise for the applications of helicity-based electronics.