Topological insulators are materials that are electrically insulating in the bulk but can conduct electricity due to topologically protected electronic edge or surface states. Since 2013, the German Research Foundation (DFG) has been supporting the Priority Program “Topological Insulators: Materials – Fundamental Properties – Devices” (SPP 1666). The program has three areas of activity: (i) Understanding and improvement of existing topological insulator materials, regarding the size of the band gap and intrinsic doping levels, to enable room temperature applications, (ii) explore the fundamental properties necessary for the development of device structures, and (iii) discover new materials to overcome deficits of current materials and explore new properties.

A considerable number of collaborative works resulted from the activities of the SPP members and the scientific subject as a whole has gained momentum with many initiatives worldwide. In 2016 the Nobel Prize in Physics was awarded to D. J. Thouless, F. D. M. Haldane and J. M. Kosterlitz “for theoretical discoveries of topological phase transitions and topological phases of matter”. This led to a further increase of the awareness for this research field and reflected the fact that topology has nowadays entered and changed our perception of the solid state.

The present issue exemplifies and summarizes contributions from the priority program. We have 20 articles, 9 of which are Feature Articles and 11 are Original Papers. The three areas of activity are reflected in the present issue.

An example of area (i) is the work of Pereira et al. (DOI:10.1002/pssb.202000346) who optimized the molecular beam epitaxy (MBE) growth of Bi₂Te₃ achieving a low level of defects. A modified procedure enabled the growth of Bi₂Te₃ on ferrimagnetic oxides. In these systems, magnetotransport indicates a magnetic proximity effect on the topological surface state. Mussler (DOI:10.1002/pssb.202000007) suppressed bulk carriers further in (Bi,Sb)₂(Te,Se)₃ and minimized crystal dislocations and twin domains despite the large lattice mismatch imposed by Si(111) substrates. He further demonstrated the in situ growth of topological insulator‐superconductor Josephson junctions. Jafarpisheh et al. (DOI:10.1002/pssb.202000021) used physical vapor deposition to grow flakes of Bi₂Se₃ and (Bi,Sb)₂(Te,Se)₃ with varying thicknesses on several different substrates and employed Raman spectroscopy for characterization. They encapsulated these flakes in BN and performed a comparison of transport properties with bulk exfoliated samples.

Mesoscopic conductors such as nanowires and nanoribbons promise an enhanced effect of topological surface states in transport. Giraud and Dufouleur (DOI:10.1002/pssb.202000066) review quantum transport of single‐crystal nanostructures. In quantum confined nanowires, in which the transport length exceeds the diameter, quantum interference is investigated in the form of Aharonov–Bohm oscillations and, at 100 mK temperature, reproducible conductance fluctuations.

In area (ii), Morgenstern et al. (DOI:10.1002/pssb.202000060) perform angle‐resolved photoemission and scanning tunneling spectroscopy (STS) and point out the use of STS for the identification of weak topological insulators and of four‐tip scanning tunneling microscopy for in‐situ transport measurements without the need for extra contacts. In this way, Bi₁Te₁ consisting of Bi₂Te₃ and Bi₂ layers is identified as a dual topological insulator, combining a weak topological insulator and a topological crystalline insulator phase. A switchable topological phase is identified in the commercial phase‐change material Ge₂Sb₂Te₅: its conducting phase is a topological insulator. Another switchable topological phase is reported in angle‐resolved photoemission by Rader et al. (DOI:10.1002/pssb.202000371): Pb_1‐xSn_xSe, switches from topological crystalline to Z₂ topological insulator by adding 1 to 2% Bi. They also argue that the surface states of the topological Kondo insulator candidate SmB₆ are trivial. They further show that Bi₂Te₃ doped by Mn is actually a heterostructure consisting of MnBi₂Te₄ and Bi₂Te₃ layers and that this system gives rise to a magnetic gap at the Dirac point, required for the quantum anomalous Hall effect. In the same context, Riha et al. (DOI:10.1002/pssb.202000088) investigate vanadium‐doped BiTe_2.4Se_0.6 single crystals by angle‐resolved photoemission and transport experiments which both support that the surface states in this system are gapless. Weak antilocalization is observed and an enhanced phase coherence length is derived from the data.

Scanning tunneling microscopy and STS are valuable methods for topological insulator research. In all – except for the simplest layered – systems, the characterization and identification of surface terminations is crucial and for SmB₆ it is one of the aspects that render the assignment of SmB₆ to a topological insulator controversial. Wirth et al. (DOI:10.1002/pssb.202000022) performed therefore a direct comparison between the (100) surfaces of SmB₆ and EuB₆. They introduce a double‐bias technique to compare different bias voltages on the same probed area. Quasiparticle interference (QPI) in STS has been an important experimental method since the beginning of topological insulator research, revealing fundamental scattering properties of topological surface states. Rüßmann et al. (DOI:10.1002/pssb.202000031) derive a Green function‐based formalism for the ab initio computation of Fourier‐transformed QPI images and apply it to magnetic defects embedded in the surface of Bi₂Te₃.

The work of Braun and Ebert (DOI:10.1002/pssb.202000026) closes a gap in the theoretical description of direct and inverse photoemission experiments by studying the spin texture of surface‐barrier‐induced image states using one‐step inverse photoemission calculations for Bi₂Se₃ and metal surfaces. The spin texture and degree of spin polarization of topological surface states are important for the use of topological insulators in spintronic devices and a determination of these properties is desirable independently of photoelectron spectroscopy with possible complications from final state effects. Götte and Dahm (DOI:10.1002/pssb.202000032) derived a formalism to determine the out‐of‐plane spin from spin Hall effect tunneling spectra. They test the method using realistic tight‐binding models of Bi₂Se₃ and Sb₂Te₃.

HgTe is the material where the quantum spin Hall effect was experimentally confirmed for the first time. Cd_xHg_1‐xTe in the topological insulator phase has advantages in terms of group velocity and band gap size over HgTe and tuning of x allows comparing the normal and inverted state and also reaching a linear dispersion for the critical concentration in between these states. Ganichev et al. (DOI:10.1002/pssb.202000023) conducted terahertz cyclotron resonance in transmission, photocurrent, and photoconductivity and were able to experimentally distinguish 2D from 3D transport. They also found that a sharp interface in terms of the Cd concentration profile is crucial for the appearance of the topologically protected state. Photocurrent spectroscopy of topological materials for higher energies, from near‐infrared to the visible range, has been reviewed by Kiemle et al. (DOI:10.1002/pssb.202000033). Spacial resolution enables photocurrent images across the Hall bar. Photogalvanic effects within the surface states – such as a circular photogalvanic effect – are separated from other effects such as thermoelectric currents by pump‐probe spectroscopy. In Bi₂Te₂Se they find a lifetime of nonequilibrium charge and spin populations of several 100 ps at room temperature. The pump‐probe technique in angle‐resolved photoemission has been reviewed by Güdde and Höfer (DOI:10.1002/pssb.202000521). Mid‐infrared pump pulses cause in the topological surface state of Sb₂Te₃ a strong asymmetry in momentum space equivalent to the generation of macroscopic photocurrents. The strongest asymmetry is reached with linearly polarized light, a small helicity dependence can be observed in a particular experimental geometry. In contrast to this optical interband excitation, terahertz excitation of Bi₂Te₃ leads to a substantial intraband redistribution of electrons in momentum space. Comparable scattering times of ~1 ps are obtained in both experiments.

Several different directions have been explored in area (iii). Heterostructures enable various proxmity effects that influence topological phases. The ferromagnetic insulator EuS interfaced with Bi₂Se₃ showed a strong modification of magnetic structure and critical temperature that could not be clarified so far. Meyerheim et al. (DOI:10.1002/pssb.202000290) studied the growth of EuS on Bi₂Se₃ in detail by surface X‐ray diffraction and observed sharp interfaces on the atomic scale. Based on this structural characterization, ab initio calculations predict the magnetic structure of the EuS at the interface to depend on the extent of n‐doping from the Bi₂Se₃. Also the possibility of an enhanced Néel temperature was investigated in this way. Zollner and Fabian (DOI:10.1002/pssb.202000081) investigated heterostructures of graphene with Bi₂Se₃, Bi₂Te₃, and Sb₂Te₃ theoretically and predict that induced spin‐orbit coupling and charge doping can be strongly tuned by gating. These results are important for gate‐tunable spin‐charge conversion which is currently explored in experiments.

The existence of topological insulators based on strong electron correlation has not been confirmed yet. Na₂IrO₃ is one of the theoretically proposed correlated topological insulators for which experimental evidence is still outstanding. Dziuba et al. (DOI:10.1002/pssb.202000421) performed a combined electrical transport and STS study of in situ cleaved Na₂IrO₃. They observe an unusual surface conductivity that does not disappear at low temperatures.

Nodal semimetals are novel materials with a unique Dirac‐type bulk dispersion and comprise also topologically protected systems. Pronin and Dressel (DOI:10.1002/pssb.202000027) review optical conductivity of different types of nodal semimetals and point out the strengths of the method in this material class, such as bulk sensitivity and high energy resolution. Weyl semimetals are topologically protected but in contrast to nonmagnetic Weyl semimetals which depend on broken inversion symmetry, magnetic Weyl semimetals have remained elusive until recently. Dyck et al. (DOI:10.1002/pssb.202000067) grew the theoretically predicted ferromagnetic Weyl semimetal Co₂TiGe by sputter deposition as the growth method most relevant for applications. They observed that physical properties are close to those of bulk samples but not sufficient to conclude from the magnetotransport data on Weyl semimetallicity.

Trifunovic and Brouwer (DOI:10.1002/pssb.202000090) reviewed higher order topological phases, e.g., 3D crystals that show protected hinge or corner states, and their role for the classification of topological crystalline insulators as well as superconductors. They also pointed out that, as an alternative to full classification, searches by symmetry‐based indicators resulted in the discovery of many new topological insulator materials and the same can be expected for topological superconductors.

In the six years of this priority program topological insulators developed from a mere curiosity to a material class that entered many fields of applied research. According to theoretical databases, one out of four materials is topological, including novel variants like Weyl semimetals and Chern insulators. The rising field of two‐dimensional materials incorporates many of these compounds and opens new pathways to engineer topological properties, e.g. in twisted bilayer graphene or transition metal dichalcogenide heterostructures. The fabrication of magnetic topological insulators clears the way to realize the integer quantum Hall effect, being the basis for the electrical resistance in quantum metrology, without the need for high external magnetic fields. Finally, in combination with superconductivity, topological insulators are the basis for topological qubits that constitute a promising but demanding platform for quantum computing. These are just a few developments that support the viewpoint that topological insulators and other classes of topological materials that emerged over the years will remain an active field of research in the years to come.

We wish to thank our colleagues in the program committee of the SPP1666 who have helped to initiate and steer the program: Hartmut Buhmann, Hubert Ebert, Claudia Felser, Robin Klett, Laurens W. Molenkamp, Kornelius Nielsch, Philipp Rüßmann, and Björn Trauzettel. We want to thank Ellen Reister and Michael Mößle from the DFG for their support, the authors of the present issue for their contributions, Lourdes Marcano for organizing them, and Marc Zastrow for the expert editorial realization of this Special Issue.

Oliver Rader, Gustav Bihlmayer, and Saskia F. Fischer

Berlin and Jülich, November 2020

拓扑绝缘体是在整体上电绝缘但由于拓扑保护的电子边缘或表面状态而可以导电的材料。自2013年以来，德国研究基金会（DFG）一直在支持优先项目“拓扑绝缘子：材料–基本特性–器件”（SPP 1666）。该计划的活动范围包括三个方面：（i）了解和改进现有的拓扑绝缘体材料，涉及带隙的大小和固有掺杂水平，以实现室温应用；（ii）探索开发绝缘子所需的基本特性。器件结构；（iii）发现新材料以克服当前材料的不足并探索新特性。

SPP成员的活动和整个科学主题开展的大量协作工作在全球范围内采取了许多举措，从而获得了发展势头。2016年，DJ Thouless，FDM Haldane和JM Kosterlitz被授予诺贝尔物理学奖，“其原因是物质的拓扑相变和拓扑相的理论发现”。这导致人们对该研究领域的认识进一步提高，并反映出当今拓扑已经进入并改变了我们对固态的认识这一事实。

本期例证和总结了优先计划的贡献。我们有20篇文章，其中9篇是专题文章，而11篇是原始论文。本期反映了三个活动领域。

区域（i）的一个例子是Pereira等人的工作。（DOI：10.1002 / pssb.202000346）优化了Bi ₂ Te ₃的分子束外延（MBE）生长，从而实现了低水平的缺陷。一种经过改进的程序可以使Bi ₂ Te ₃在亚铁磁性氧化物上生长。在这些系统中，磁传输指示对拓扑表面状态的磁邻近效应。Mussler（DOI：10.1002 / pssb.202000007）进一步抑制了（Bi，Sb）₂（Te，Se）_3中的大体积载流子，并使晶体位错和孪晶域最小化，尽管Si（111）衬底施加了很大的晶格失配。他进一步现场演示绝缘子-超导体约瑟夫森结的增长。Jafarpisheh等。（DOI：10.1002 / pssb.202000021）使用物理气相沉积法在几种不同的基底上生长了厚度不同的Bi ₂ Se ₃和（Bi，Sb）₂（Te，Se）₃薄片，并采用拉曼光谱进行了表征。他们将这些薄片包裹在BN中，并与大量剥落的样品进行了运输性能的比较。

介观导体，例如纳米线和纳米带，有望在运输中增强拓扑表面态的作用。Giraud和Dufouleur（DOI：10.1002 / pssb.202000066）综述了单晶纳米结构的量子输运。在其中传输长度超过直径的量子约束纳米线中，以Aharonov-Bohm振荡形式以及在100 mK温度下可再现的电导波动形式研究了量子干扰。

在区域（ii）中，Morgenstern等人。（DOI：10.1002 / pssb.202000060）执行角度分辨光发射和扫描隧道光谱（STS），并指出使用STS识别弱拓扑绝缘体和使用四尖端扫描隧道显微镜进行现场运输测量而无需需要额外的联系。以此方式，由Bi ₂ Te ₃和Bi ₂层组成的Bi ₁ Te ₁被确定为双重拓扑绝缘体，结合了弱拓扑绝缘体和拓扑晶体绝缘体相。在商用相变材料Ge ₂ Sb ₂ Te _5中确定了可切换的拓扑相：其导电相是拓扑绝缘体。Rader等人在角度分辨的光发射中报告了另一个可切换的拓扑阶段。（DOI：10.1002 / pssb.202000371）：Pb _1-x Sn _x Se通过添加1-2％Bi从拓扑晶体转变为Z ₂拓扑绝缘体。他们还认为，拓扑近藤绝缘子候选SmB ₆的表面状态是微不足道的。他们进一步表明，Mn掺杂的Bi ₂ Te ₃实际上是由MnBi ₂ Te ₄和Bi ₂ Te ₃组成的异质结构。层，并且该系统在狄拉克点处产生了磁隙，这是量子异常霍尔效应所必需的。在相同的背景下，Riha等。（DOI：10.1002 / pssb.202000088）通过角度分辨光发射和传输实验研究了掺钒的BiTe _2.4 Se _0.6单晶，这均支持该系统的表面状态无间隙。观察到较弱的反局部化，并且从数据得出增强的相干长度。

扫描隧道显微镜和STS是拓扑绝缘子研究的有价值的方法。在所有系统中（最简单的分层系统除外），表面端接的表征和识别至关重要，对于SmB _6而言，这是使SmB ₆分配给拓扑绝缘子的方面之一。Wirth等。（DOI：10.1002 / pssb.202000022）执行了SmB ₆和EuB ₆的（100）个表面之间的直接比较。他们引入了双偏置技术来比较同一探测区域上的不同偏置电压。自拓扑绝缘子研究开始以来，STS中的准粒子干扰（QPI）一直是一种重要的实验方法，它揭示了拓扑表面状态的基本散射特性。Rüßmann等。（DOI：10.1002 / pssb.202000031）得出了基于绿色函数的形式主义，用于傅里叶变换QPI图像的从头算起，并将其应用于Bi ₂ Te ₃表面嵌入的磁性缺陷。

Braun和Ebert（DOI：10.1002 / pssb.202000026）的工作通过使用Bi的一步反光发射计算研究表面势垒诱导的图像状态的自旋纹理，从而填补了正反光发射实验的理论描述中的空白。₂硒₃和金属表面。拓扑表面状态的自旋织构和自旋极化程度对于在自旋电子器件中使用拓扑绝缘体很重要，并且独立于光电子光谱法确定这些特性是理想的，因为最终状态效应可能带来复杂性。Götte和Dahm（DOI：10.1002 / pssb.202000032）得出了一种形式主义，用以根据自旋霍尔效应隧穿光谱确定平面外自旋。他们使用Bi ₂ Se ₃和Sb ₂ Te _3的逼真的紧密结合模型测试了该方法。

HgTe是首次通过实验证实量子自旋霍尔效应的材料。与HgTe相比，拓扑绝缘体相中的Cd _x Hg _1-x Te具有群速度和带隙大小方面的优势，并且x的调整允许比较正态和反相状态，并且在这些状态之间的临界浓度也达到了线性弥散。Ganichev等。（DOI：10.1002 / pssb.202000023）在传输，光电流和光电导率方面进行了太赫兹回旋共振，并且能够从实验上区分2D与3D传输。他们还发现，就Cd浓度分布而言，清晰的界面对于拓扑保护状态的出现至关重要。Kiemle等人已经审查了拓扑材料在近红外到可见光范围内的更高能量的光电流能谱。（DOI：10.1002 / pssb.202000033）。空间分辨率可在大厅栏中显示光电流图像。表面状态内的光电流效应（例如圆形光电流效应）通过泵浦探针光谱法与其他效应（例如热电流）分开。在Bi ₂ Te _2中例如，他们发现了非平衡电荷的寿命，并且在室温下自旋种群为几百ps。Güdde和Höfer（DOI：10.1002 / pssb.202000521）审查了角度分辨光发射中的泵浦探测技术。中红外泵浦脉冲在Sb ₂ Te ₃的拓扑表面状态下导致动量空间中的强烈不对称性，等效于宏观光电流的产生。线性偏振光可以达到最强的不对称性，在特定的实验几何形状中可以观察到小的螺旋度依赖性。与这种光学带间激励相反，Bi ₂ Te _3的太赫兹激励导致相当大的带内电子在动量空间中的重新分布。在两个实验中均获得了〜1 ps的可比散射时间。

在区域（iii）中探索了几个不同的方向。异质结构可实现影响拓扑阶段的各种邻近效应。与Bi ₂ Se ₃交界的铁磁绝缘体EuS表现出对磁性结构和临界温度的强烈改变，目前尚不清楚。Meyerheim等。（DOI：10.1002 / pssb.202000290）通过表面X射线衍射详细研究了EuS在Bi ₂ Se ₃上的生长，并在原子尺度上观察到了清晰的界面。基于这种结构特征，从头算可以预测界面处EuS的磁性结构，取决于Bi ₂ Se中n掺杂的程度。₃。还以这种方式研究了提高尼尔温度的可能性。Zollner和Fabian（DOI：10.1002 / pssb.202000081）从理论上研究了Bi ₂ Se ₃，Bi ₂ Te ₃和Sb ₂ Te ₃石墨烯的异质结构，并预测通过门控可以强烈调节自旋轨道耦合和电荷掺杂。这些结果对于目前可在实验中探索的栅极可调自旋电荷转换非常重要。

尚未确认基于强电子相关性的拓扑绝缘子的存在。Na ₂ IrO ₃是理论上提出的相关拓扑绝缘子之一，其实验证据仍很出色。Dziuba等。（DOI：10.1002 / pssb.202000421）对原位裂解的Na ₂ IrO ₃进行了电传输和STS的研究。他们观察到异常的表面电导率，在低温下不会消失。

节点半金属是具有独特Dirac型本体分散体的新型材料，还包含受拓扑保护的系统。Pronin和Dressel（DOI：10.1002 / pssb.202000027）综述了不同类型的节点半金属的光导率，并指出了该方法在此类材料中的优势，例如体积敏感性和高能量分辨率。魏尔半金属在拓扑学上受到保护，但是与依赖于破坏的反演对称性的非磁性魏尔半金属相反，磁性魏尔半金属直到最近仍然难以捉摸。戴克等。（DOI：10.1002 / pssb.202000067）增长了理论预测的铁磁Weyl半金属Co ₂TiGe通过溅射沉积是最适合应用的生长方法。他们观察到物理性质接近于大块样品，但不足以根据Weyl半金属性的磁传输数据得出结论。

Trifunovic和Brouwer（DOI：10.1002 / pssb.202000090）审查了更高阶的拓扑相，例如显示受保护的铰链或拐角状态的3D晶体，以及它们在拓扑晶体绝缘体和超导体分类中的作用。他们还指出，作为完全分类的替代方法，通过基于对称性的指标进行搜索导致发现了许多新的拓扑绝缘体材料，并且拓扑超导体也有望获得同样的结果。

在这个优先计划的六年中，拓扑绝缘子从单纯的好奇心发展到进入许多应用研究领域的材料类别。根据理论数据库，四种材料中的一种是拓扑结构，包括Weyl半金属和Chern绝缘子等新型变体。二维材料的兴起领域融合了许多这些化合物，并开辟了工程拓扑特性的新途径，例如在扭曲的双层石墨烯或过渡金属二卤化异氰酸酯异质结构中。磁性拓扑绝缘体的制造为实现整数量子霍尔效应扫清了道路，该效应是量子计量学中电阻的基础，而无需高外部磁场。最后，结合超导，拓扑绝缘体是构成拓扑量子位的基础，拓扑量子位构成了一个有前途但要求很高的平台。这些只是支持以下观点的一些发展，这些观点是多年来出现的拓扑绝缘体和其他类别的拓扑材料仍将是未来几年的活跃研究领域。

我们要感谢SPP1666程序委员会中帮助启动和指导该程序的同事：Hartmut Buhmann，Hubert Ebert，Claudia Felser，Robin Klett，Laurens W. Molenkamp，Kornelius Nielsch，PhilippRüßmann和BjörnTrauzettel。我们要感谢DFG的Ellen Reister和MichaelMößle的支持，感谢本期杂志的作者的贡献，Lourdes Marcano的组织，以及Marc Zastrow对此特刊的专业编辑认识。

奥利弗·雷德（Oliver Rader），古斯塔夫·比尔迈耶（Gustav Bihlmayer）和萨斯基亚·菲舍尔（Saskia F.Fischer）

柏林和尤利希，2020年11月