Recent Advances in Topological Quantum Materials by Angle-Resolved Photoemission Spectroscopy

Abstract

The research and application of topological quantum materials (TQMs) have been booming in the past decade. On one hand, TQMs have expanded from the quantum Hall effect to a huge family to date, including topological insulators, topological semimetals, and topological superconductors. On the other hand, new electronic devices based on TQMs are starting to emerge and show great application potential. In the process of discovering and studying novel TQMs, angle-resolved photoemission spectroscopy (ARPES) has played a crucial and irreplaceable role by directly mapping the electronic structures in momentum space. At the same time, ARPES itself has been considerably improved. Representative developments include laser-ARPES and spatial-/spin-/time-resolved ARPES, which greatly expands the research scope and capabilities of ARPES. In the future, ARPES will remain resonating with the new challenges and developments in the study of TQMs. In the past decade, the investigation of the electronic structures of topological quantum materials (TQMs) has not only provided important insights for fundamental science research but has also laid a solid foundation for designing and fabricating novel functional devices. Angle-resolved photoemission spectroscopy (ARPES), with its capability of directly visualizing the electronic structures of crystals in momentum space, has played a critical role in discovering and understanding many TQMs. On the other hand, the ARPES technique has also been greatly improved—including the much enhanced energy and momentum resolutions, and the importation of new detection degrees of freedom—which in turn further advanced the research on TQMs. In this review, we first give a brief introduction to the principle of ARPES, then focus on its application in different TQMs; we also review some recent advances in ARPES techniques with their representative applications in TQMs and finally present a brief perspective. In the past decade, the investigation of the electronic structures of topological quantum materials (TQMs) has not only provided important insights for fundamental science research but has also laid a solid foundation for designing and fabricating novel functional devices. Angle-resolved photoemission spectroscopy (ARPES), with its capability of directly visualizing the electronic structures of crystals in momentum space, has played a critical role in discovering and understanding many TQMs. On the other hand, the ARPES technique has also been greatly improved—including the much enhanced energy and momentum resolutions, and the importation of new detection degrees of freedom—which in turn further advanced the research on TQMs. In this review, we first give a brief introduction to the principle of ARPES, then focus on its application in different TQMs; we also review some recent advances in ARPES techniques with their representative applications in TQMs and finally present a brief perspective. The electronic structure of materials is important information that determines their electric, magnetic, optical, and even thermal properties. Since the development of band theory—one of the most important landmarks in condensed matter physics and materials science—many interesting physical phenomena have been understood. A representative example is the classification of metals, semiconductors, and insulators. At present, the study of electronic structures is no longer limited to the determination of the energy-momentum dispersions of electronic energy bands, but has been extended to the investigation of more subtle details, such as the spin/orbital degrees of freedom, the electronic distribution in real space and dynamics in time domain, the interaction between electrons and other elementary excitations, and the response of the electronic structures to external perturbations, which provides a solid foundation for the understanding of quantum materials and the exploration of their applications. As an example, the swift development of topological quantum materials (TQMs) in the past decade has benefited from the theoretical and experimental endeavors for the understanding of their unique electronic structures. In contrast to conventional materials, the electronic structures of TQMs show different properties with nontrivial topology, which cannot be described by the local order parameters in Landau's paradigm.1Qi X.-L. Zhang S.-C. Topological insulators and superconductors.Rev. Mod. Phys. 2011; 83: 1057-1110Crossref Scopus (7237) Google Scholar, 2Hasan M.Z. Kane C.L. Colloquium: topological insulators.Rev. Mod. Phys. 2010; 82: 3045-3067Crossref Scopus (10191) Google Scholar, 3Armitage N.P. Mele E.J. Vishwanath A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys. 2018; 90: 015001Crossref Scopus (906) Google Scholar, 4Yan B. Felser C. Topological materials: Weyl semimetals.Annu. Rev. Condens. 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Topological insulators and superconductors.Rev. Mod. Phys. 2011; 83: 1057-1110Crossref Scopus (7237) Google Scholar, 2Hasan M.Z. Kane C.L. Colloquium: topological insulators.Rev. Mod. Phys. 2010; 82: 3045-3067Crossref Scopus (10191) Google Scholar, 3Armitage N.P. Mele E.J. Vishwanath A. Weyl and Dirac semimetals in three-dimensional solids.Rev. Mod. Phys. 2018; 90: 015001Crossref Scopus (906) Google Scholar, 4Yan B. Felser C. Topological materials: Weyl semimetals.Annu. Rev. Condens. Matter Phys. 2017; 8: 337-354Crossref Scopus (317) Google Scholar, 5Bradlyn B. Cano J. Wang Z. Vergniory M.G. Felser C. Cava R.J. Bernevig B.A. Beyond Dirac and Weyl fermions: unconventional quasiparticles in conventional crystals.Science. 2016; 353: aaf5037Crossref PubMed Scopus (385) Google Scholar, 6Chang G. Wieder B.J. Schindler F. Sanchez D.S. Belopolski I. Huang S.-M. Singh B. Wu D. Chang T.-R. Neupert T. et al.Topological quantum properties of chiral crystals.Nat. Mater. 2018; 17: 978-985Crossref PubMed Scopus (56) Google Scholar9Fu Liang Topological Crystalline Insulators.Physical Review Letters. 2011; 106https://doi.org/10.1103/PhysRevLett.106.106802Crossref Scopus (910) Google Scholar These TQMs have not only provided rich scientific implications but have also served as an ideal platform to realize various exotic phenomena and practical applications, including quantum spin Hall (QSH) effect, topological magneto-electric effect, unusual magneto-optical effect, and topological quantum computation.1Qi X.-L. Zhang S.-C. Topological insulators and superconductors.Rev. Mod. Phys. 2011; 83: 1057-1110Crossref Scopus (7237) Google Scholar,2Hasan M.Z. Kane C.L. Colloquium: topological insulators.Rev. Mod. 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Springer Science & Business Media, 2013Google Scholar Based on these advantages, it is not surprising that even before TQMs,12Damascelli A. Hussain Z. Shen Z.-X. Angle-resolved photoemission studies of the cuprate superconductors.Rev. Mod. Phys. 2003; 75: 473-541Crossref Scopus (2466) Google Scholar, 13Chen Y. Studies on the electronic structures of three-dimensional topological insulators by angle resolved photoemission spectroscopy.Front. Phys. 2012; 7: 175-192Crossref Scopus (26) Google Scholar, 14Yang H. Liang A. Chen C. Zhang C. Schroeter N.B.M. Chen Y. Visualizing electronic structures of quantum materials by angle-resolved photoemission spectroscopy.Nat. Rev. Mater. 2018; 3: 341-353Crossref Scopus (10) Google Scholar ARPES has been successfully applied in the study of other quantum materials such as high-temperature superconductors,11Hüfner S. Photoelectron Spectroscopy: Principles and Applications. Springer Science & Business Media, 2013Google Scholar,15Lv B. Qian T. Ding H. 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Toward the end, we briefly introduce some recent developments of new ARPES techniques together with their representative applications in TQMs and present a short perspective. ARPES evolves from the photoelectric effect detected by Hertz44Smallwood C.L. Kaindl R.A. Lanzara A. Ultrafast angle-resolved photoemission spectroscopy of quantum materials.EPL. 2016; 115: 27001Crossref Scopus (22) Google Scholar in 1887 and explained by Einstein45Hertz H. Ueber einen Einfluss des ultravioletten Lichtes auf die electrische Entladung.Ann. Phys. 1887; 267: 983-1000Crossref Scopus (144) Google Scholar in 1905. In the photoemission process, the electrons absorb the incident photons with energy hν higher than the work function ϕ (a potential barrier between the Fermi energy EF and vacuum level Evac) of the crystal and emit out of the sample, known as photoelectrons. The electron analyzer then records the kinetic energy Ekin and emission angles θ,ϕ of the photoelectrons, as schematically illustrated in Figure 1A. From the energetic diagram of the photoemission process in Figure 1B, the electron binding energy EB inside the solid can be directly calculated from the energy conservation law:EB=hν−ϕ−Ekin.(Equation 1) The in-plane momentum k∥ of the electrons inside the crystal depends on the emission angles and can be calculated according to the momentum conservation law thanks to the translational symmetry parallel to the sample surface:k∥=K∥=1ℏ2mEkinsinθcosφxˆ+sinφyˆ,(Equation 2) where K∥ is the in-plane momentum of photoelectrons. However, it is not straightforward to obtain the out-of-plane momentum kz of electrons inside the solid due to the broken translational symmetry perpendicular to the sample surface,11Hüfner S. Photoelectron Spectroscopy: Principles and Applications. Springer Science & Business Media, 2013Google Scholar i.e., kz≠Kz, where Kz denotes the out-of-plane momentum of photoelectrons (Figure 1A). Under a priori assumption of free-electron final state,11Hüfner S. Photoelectron Spectroscopy: Principles and Applications. Springer Science & Business Media, 2013Google Scholar kz can be approximated bykz=1ℏ2mEkincos2θ+V0zˆ,(Equation 3) where V0 is a material-dependent parameter known as the inner potential whose value can be extracted by self-consistently fitting the periodic variation of photon-energy-dependent (or kz-dependent) ARPES spectra.11Hüfner S. Photoelectron Spectroscopy: Principles and Applications. Springer Science & Business Media, 2013Google Scholar One useful bonus of photon-energy-dependent measurement is to distinguish the surface and bulk states since the surface states show no kz dispersion, in contrast to the bulk states that usually show obvious kz dispersion.12Damascelli A. Hussain Z. Shen Z.-X. Angle-resolved photoemission studies of the cuprate superconductors.Rev. Mod. Phys. 2003; 75: 473-541Crossref Scopus (2466) Google Scholar,13Chen Y. Studies on the electronic structures of three-dimensional topological insulators by angle resolved photoemission spectroscopy.Front. Phys. 2012; 7: 175-192Crossref Scopus (26) Google Scholar In the following sections, we present how to identify characteristic surface states in various TQMs using this method. The above discussion has provided a good explanation of the key energy exchange and momentum relation in the photoemission process. In a realistic ARPES measurement, we collect the photoelectron intensity I(k, E) as a function of electron energy and momentum, which can be approximated as10Alicea J. New directions in the pursuit of Majorana fermions in solid state systems.Rep. Prog. Phys. 2012; 75: 076501Crossref PubMed Scopus (1493) Google Scholar,11Hüfner S. Photoelectron Spectroscopy: Principles and Applications. Springer Science & Business Media, 2013Google ScholarI(k,E)=∑f,i|Mf,ik|2A(k,E)f(T,E),where |Mf,ik| is the matrix element term that describes the transition between the initial and final states by photoexcitation. It depends on the electron momentum and incoming photon energy and polarization; the one-particle spectral function term A(k,E) can describe the probability that an electron with momentum k can have an energy of E, which directly reflects the dependence of ARPES intensity on the band structure; f(T,E) is the Fermi-Dirac distribution function at temperature T. The instrument-related effects such as finite energy and momentum resolutions are neglected in the above equation. The energy-momentum dispersion of the electrons, i.e., the electronic band structure of the system, can be directly extracted from the measured ARPES intensity distribution I(k, E) by tracing the maxima of ARPES intensity along energy or momentum axis. Figures 1C and 1D show the simulation of a typical set of ARPES data based on a linear Dirac band structure, i.e., the binding energy of the electron is a linear function of its momentum. By scanning the sample angles (polar or tilt as shown in Figure 1A), photoelectrons emitted from different angles can be collected, and therefore the electronic structure across the momentum space can be mapped according to Equation 2 (Figure 1C). The FS structure and band dispersion are shown in the 3D plot of ARPES intensity in the energy-angle or energy-momentum space (Figure 1D). As a prototypical example, the ARPES data collected on 2H-TaSe2 is presented in Figure 1E, showing its FS structure and band dispersions.46Einstein A. Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt.Ann. Phys. 1905; 322: 132-148Crossref Scopus (851) Google Scholar According to the universal mean free path (λ) curve for electrons in solid,47Li Y.W. Jiang J. Yang H.F. Prabhakaran D. Liu Z.K. Yang L.X. Chen Y.L. Folded superstructure and degeneracy-enhanced band gap in the weak-coupling charge density wave system 2H-TaSe2.Phys. Rev. B. 2018; 97: 115118Crossref Scopus (10) Google Scholar ARPES signal is dominated by photoelectrons emitted from the topmost 2–3 surface atomic layers if photons at typical energy range (20–200 eV) were used. Such intrinsic surface sensitivity of ARPES, combining with photon-energy-dependent measurements discussed above, is key for the identification of different types of TQMs by distinguishing their characteristic surface states. To ensure the high quality of ARPES spectra during the measurements, an atomically clean and flat sample surface is desirable, which can be prepared by in situ mechanically cleaving bulk single crystals or synthesizing thin films under ultrahigh vacuum conditions. On the other hand, in light of the universal curve, λ increases up to 100 Å if the kinetic energy of electrons is higher than several hundreds of eV or lower than several eV,47Li Y.W. Jiang J. Yang H.F. Prabhakaran D. Liu Z.K. Yang L.X. Chen Y.L. Folded superstructure and degeneracy-enhanced band gap in the weak-coupling charge density wave system 2H-TaSe2.Phys. Rev. B. 2018; 97: 115118Crossref Scopus (10) Google Scholar which promises X-ray ARPES and UV laser-ARPES an opportunity to investigate the bulk states of crystals.48Seah M.P. Dench W.A. Quantitative electron spectroscopy of surfaces: a standard data base for electron inelastic mean free paths in solids.Surf. Interfaces Anal. 1979; 1: 2-11Crossref Scopus (4797) Google Scholar The discovery of QH effect by Klaus von Klitzing in 1980 has challenged the classification of solid materials based on the concept of spontaneous symmetry breaking.7Klitzing K.v. Dorda G. Pepper M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance.Phys. Rev. Lett. 1980; 45: 494-497Crossref Scopus (3978) Google Scholar,8Thouless D.J. Kohmoto M. Nightingale M.P. den Nijs M. Quantized Hall conductance in a two-dimensional periodic potential.Phys. Rev. Lett. 1982; 49: 405-408Crossref Scopus (3151) Google Scholar In a QH state, a topological phase can be defined without symmetry breaking.1Qi X.-L. Zhang S.-C. Topological insulators and superconductors.Rev. Mod. Phys. 2011; 83: 1057-1110Crossref Scopus (7237) Google Scholar,2Hasan M.Z. Kane C.L. Colloquium: topological insulators.Rev. Mod. Phys. 2010; 82: 3045-3067Crossref Scopus (10191) Google Scholar,8Thouless D.J. Kohmoto M. Nightingale M.P. den Nijs M. Quantized Hall conductance in a two-dimensional periodic potential.Phys. Rev. Lett. 1982; 49: 405-408Crossref Scopus (3151) Google Scholar Here the mathematical terminology “topology” refers to the global properties of a system that are invariant under continuous deformation. For example, due to the topological property of the QH state, electrons at the edge with quantized Hall conductance are insensitive to the smooth change of certain material parameters such as the impurity concentration, and the QH state can be described by a topological invariant, the Chern number, which counts the number of edge modes. In the past decade, tremendous research efforts have been made to search for new TQMs that are characterized and classified by the topology of their electronic structures, as shown by the timeline of recent development of TQMs in Figure 2.7Klitzing K.v. Dorda G. Pepper M. New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance.Phys. Rev. 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