Topological materials host robust surface states that could form the basis for future electronic devices. As such states have spins that are locked to the momentum, they are of particular interest for spintronic applications. Understanding spin textures of the surface states of topologically nontrivial materials, and being able to manipulate their polarization, is therefore essential if they are to be utilized in future technologies. Here we use first-principles calculations to show that pyrite-type crystals OsX₂ (X = Se, Te) are a class of topological materials that can host surface states with spin polarization that can be either in-plane or out-of-plane. We show that the formation of low-energy states with symmetry-protected energy- and direction-dependent spin textures on the (001) surface of these materials is a consequence of a transformation from a topologically trivial to nontrivial state, induced by spin orbit interactions. The unconventional spin textures of these surface states feature an in-plane to out-of-plane spin polarization transition in the momentum space protected by local symmetries. Moreover, the surface spin direction and magnitude can be selectively filtered in specific energy ranges. Our demonstration of a new class of topological materials with controllable spin textures provides a platform for experimentalists to detect and exploit unconventional surface spin textures in future spin-based nanoelectronic devices.

拓扑材料具有坚固的表面状态，这些表面状态可能构成未来电子设备的基础。由于这种状态的自旋被锁定在动量中，因此它们对于自旋电子学应用尤为重要。因此，如果要在未来的技术中使用它们，则了解拓扑非平凡材料的表面状态的自旋纹理并能够操纵其极化是必不可少的。在这里，我们使用第一性原理计算来显示黄铁矿型晶体OsX ₂（X = Se，Te）是一类拓扑材料，可以承载具有自旋极化的表面状态，该状态可以在平面内或平面外。我们表明，在这些材料的（001）表面上形成具有对称保护的能量和方向相关的自旋纹理的低能态是自旋轨道相互作用引起的从拓扑琐碎状态到非琐碎状态转换的结果。这些表面状态的非常规自旋纹理在受局部对称性保护的动量空间中具有平面内到平面外的自旋极化转变。而且，可以在特定的能量范围内选择性地过滤表面自旋方向和大小。