Topological Insulators, Topological Crystalline Insulators, Topological Semimetals and Topological Kondo Insulators
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Topological Insulators, Topological Crystalline Insulators, and Topological KondoInsulators (Review Article)
M. Zahid Hasan,
1, 2
Su-Yang Xu, and Madhab Neupane Joseph Henry Laboratories: Department of Physics,Princeton University, Princeton, NJ 08544, USA Princeton Institute for the Science and Technology of Materials,School of Engineering and Applied Science, Princeton University, Princeton NJ 08544, USA
In this Book Chapter we briefly review the basic concepts defining topological in-sulators and elaborate on the key experimental results that revealed and establishedtheir symmetry protected (SPT) topological nature. We then present key experimentalresults that demonstrate magnetism, Kondo insulation, mirror chirality or topologicalcrystalline order and superconductivity in spin-orbit topological insulator settings andhow these new phases of matter arise through topological quantum phase transitionsfrom Bloch band insulators via Dirac semimetals at the critical point.CONTENTS
I. Introduction 1II. Z topological insulators 3III. Topological Kondo Insulator Candidates 8IV. Topological Quantum Phase Transitions 11V. Topological Dirac Semimetals 14VI. Topological Crystalline Insulators 17VII. Magnetic and Superconducting dopedTopological Insulators 20References 27 INTRODUCTION
Topological phases of matter differ from conventionalmaterials in that a topological phase of matter featuresa nontrivial topological invariant in its bulk electronicwavefunction space which can be realized in a symmetry-protected condition [1–14]. The experimental discoveriesof the 2D integer and fractional quantum Hall (IQH andFQH) states [15–19] in the 1980s realize the first twotopological phases of matter in nature. These 2D topo-logical systems are insulators in the bulk because theFermi level is located in the middle of two Landau levels.On the other hand, the edges of these 2D topological insu-lators (IQH and FQH) feature chiral 1D metallic states,leading to remarkable quantized charge transport phe-nomena. The quantized transverse magneto-conductivity σ xy = ne /h (where e is the electric charge and h isPlanck’s constant) can be probed by charge transportexperiments, which also provides a measure of the topo-logical invariant (the Chern number) n that characterizesthese quantum Hall states [20, 21]. In 2005, theoretical advances [22, 23] predicted a third type of 2D topolog-ical insulator, the quantum spin Hall (QSH) insulator.Such a topological state is symmetry protected. A QSHinsulator can be viewed as two copies of quantum Hallsystems that have magnetic field in the opposite direc-tion. Therefore, no external magnetic field is required forthe QSH phase, and the pair of quantum-Hall-like edgemodes are related by the time-reversal symmetry (Fig. 1).The QSH phase was experimentally demonstrated in themercury telluride quantum wells of using charge trans-port by measuring a longitudinal (charge) conductanceof about 2 e /h (two copies of quantum Hall states) atlow temperatures [24]. No spin polarization was mea-sured in this experiment thus spin momentum lockingwhich is essential for the Z topological physics was notknown or proven from experiments [24].It is important to note that the 2D topological (IQH,FQH, and QSH) insulators are only realized at buried in-terfaces of ultraclean semiconductor heterostructures atvery low temperatures [24]. Furthermore, their metallicedge states can only be probed by the charge transportmethod [24]. These facts hinder the systematic stud-ies of many of their important properties, such as theirelectronic structure, spin polarization texture, tunnelingproperties, optical properties, as well as their responsesunder heterostructuring or interfacing with broken sym-metry states. For example, the two counter-propagatingedge modes in a QSH insulator is predicted to feature a1D Dirac band crossing in energy and momentum space[22, 23]. And edge mode moving along the + k direction isexpected to carry the opposite spin polarization as com-pared to that of moving to the − k direction [22, 23].However, neither the Dirac band crossing nor the spin-momentum locking of the edge modes in a QSH insulatorare experimentally observed, due to the lack of experi-mental probe that can measure these key properties for a1D edge mode at a buried interface at mK temperatures,which is challenging. In 2007, it was also theoretically re-alized that the Z topological number can have nontrivialgeneralization into three-dimensions [11, 12, 25, 26]. Inthree-dimensions, there exist four (not three) Z topo-logical invariants that define the topological property ofa 3D bulk material, namely ( ν ; ν ν ν ), where ν is thestrong topological invariant, and ν − ν are the weaktopological invariants, respectively [11, 12, 25, 26]. Ifthe strong topological invariant is nonzero ( ν = 1), thesystem is a 3D strong Z topological insulator. It is im-portant to note that the generalization from a 2D topo-logical insulator (QSH) to a 3D strong Z topologicalinsulator is not a trivial generalization, because a 3Dstrong topological insulator cannot be adiabatically con-nected to multiple copies of 2D QSH insulators stackedalong the out-of-plane ˆ z direction [11, 12, 25, 26]. There-fore, the Z topological order ( ν = 1) in a 3D strongtopological insulator represents a new type of genuinelythree-dimensional (symmetry protected) topological or-der, which is fundamentally distinct from its 2D analogs(IQH, FQH, and QSH phases). The new topological or-der ( ν = 1) leads to the existence of an odd numberof gapless topological surfaces states at all surfaces of astrong topological insulator, irrespective of the choice ofthe surface termination [1, 11, 12, 25, 26]. These sur-face states are expected to be spin-momentum lockedand their Fermi surfaces enclose the Kramers’ points foran odd number of times [1, 11, 13]. Moreover, they areprotected by the time-reversal symmetry, which meansthat the topological surface states are robust against non-magnetic disorder and cannot be removed (gapped out)from the bulk band gap unless time-reversal symmetryis broken [1, 11, 13]. Symmetry protected topologicalorder is distinct from the fractional quantum Hall typetopological order.It turns out that the experimental discovery of the 3Dtopological insulator phase in 2007 [13] opened a newexperimental era in fundamental topological physics. Incontrast to its 2D analogs, (1) a 3D topological insula-tor can be realized at room temperatures without mag-netic fields. Their metallic surface states exist at baresurfaces rather than only at buried interfaces [1, 13].(2) The electronic and spin groundstate of the topo-logical surface states can be systematically studied bythe spin- and angle-resolved photoemission spectroscopy(spin-ARPES) [1, 6, 13], which provides a unique andpowerful methodology for probing the topological or-der in three-dimensional topological phases. (3) Dueto the relaxed conditions (room temperature, no mag-netic field, bare surface), it is also possible to study theelectrical transport, tunneling, optical, nanostructured,and many other key properties of the topological surfacestates [1, 6]. (4) The 3D topological insulator materialscan be doped or interfaced to realize superconductivityor magnetism [1, 6]. (5) Since its discovery in 2007, therehave been more than a hundred compounds identified as3D topological insulators [1, 6].More importantly, the experimental discovery of 3D(Z ) topological insulator [27–33] has led to a surge ofresearch in discovering other types of new topologicalorder in three-dimensions [34–47]. The spin-resolvedangle-resolved photoemission spectroscopy technique to- FIG. 1.
Topological insulators. a,
In the quantum Halleffect, the circular motion of electrons in a magnetic field, B ,is interrupted by the sample boundary. At the edge, elec-trons execute “skipping orbits” as shown, intimately leadingto perfect conduction in one direction along the edge. b, Theedge of the “quantum spin Hall effect state” or 2D topologicalinsulator contains left-moving and right-moving modes thathave opposite spin and are related by time-reversal symmetry.This edge can also be viewed as half of a quantum wire, whichwould have spin-up and spin-down electrons propagating inboth directions. c, The surface of a 3D topological insulatorssupports electronic motion in any direction along the surface,but the direction of electron’s motion uniquely determines itsspin direction and vice versa. The 2D energy-momentum re-lation has a “spin-Dirac cone” structure but with helical ormomentum space chiral spin texture with Berry’s phase π (spins go around in a closed loop in momentum space) [1, 2].. day constitutes a standard experimental methodology fordiscovering and probing new topological order (non-Z )in bulk solids [34–47]. These fertile research frontiersinclude: (1) the topological crystalline insulator (TCI),where space group symmetries replace the time-reversalsymmetry in a 3D Z TI [48, 49]. The discovery of TCI[36–39] following theoretical predictions [48, 49] leadsto novel crystalline symmetry protected topological sur-face states. (2) the topological Kondo insulator (TKI),where the topological surface states in a TKI exist inthe bulk Kondo gap rather than a simple Bloch gap ina Z TI [50]. Demonstration of TKI [44–47] provides aplatform for testing the interplay between topological or-der and strong electron correlation. (3) the topologicalDirac/Weyl semimetals [40–43, 51–55], where new topo-logical order (not ν ) can exist even if there is no globalbulk energy gap, leading to multiple Dirac/Weyl nodesin the bulk and Fermi arc surface states on the surface[55]. (4) Superconducting [56–60] and magnetic [61–65]TIs and the topological phase transitions [34, 35, 66, 67],which are the keys for a wide range of quantum phenom-ena such as Majorana fermion excitation [68], topologicalmagneto-electrical effect [69], quantum anomalous Hallcurrent [70], as well as supersymmetry SUSY state [71].In this chapter, we review the experimental discover-ies of symmetry-protected topologically (SPT) orderedphases in three-dimensions. We first review the discoveryof 3D Z topological insulator, which serves as the firsttopologically ordered phase of matter in 3D bulk materi-als. We elaborate the way of measuring the 3D Z topo-logical variant ( ν = 1) by the spin- and angle-resolvedphotoemission spectroscopy (spin-ARPES) [27–33]. Inthe following sections, we review the experimental effortsin discovering new topological order (non-Z ) and newtopological phenomena including Topological Kondo In-sulators, Topological Quantum Phase Transition, Topo-logical Dirac Semimetals, Magnetic and SuperconductorTopological Insulators, and Topological Crystalline In-sulators, respectively. The 3D topological materials arealso experimentally studied by many groups world-wideusing various techniques such as ARPES [27–38, 40–47, 57, 59, 60, 62, 63, 66, 72–91], scanning tunnelingspectroscopies (STM) [39, 92–99], transport [64, 65, 100–111], and optical methods [67, 112–118]. Discovering andunderstanding topological ordered phases of matters inthree-dimensions constitutes one of the most active re-search areas in condensed matter physics today. Z TOPOLOGICAL INSULATORS
In this section, we review the experimental discoveryof 3D Z topological insulator, and elaborate the wayof measuring the 3D Z topological variant ( ν = 1) bythe spin- and angle-resolved photoemission spectroscopy(spin-ARPES). It was theoretically realized that strongspin-orbit coupling strength is one of the keys for real-izing the 3D TI phase [25, 119] since it leads to inver-sions between the bulk conduction and valence bands.The first 3D topological insulator is experimentally real-ized in the bismuth-antimony alloy system (Bi − x Sb x ).Bi − x Sb x is believed as a possible realization of 3D topo-logical order for the following reason as predicted in bandstructure calculation [25, 119–124]: Antimony (Sb) is asemimetal with strong spin-orbit interactions. Its bulkband electronic structure features one band inversion (anodd number) between the valence band maximum at the T point of the bulk Brillouin zone (BZ). This fact makesantimony Z topologically nontrivial ( ν = 1) but anti-mony is a semimetal, which means there does not exist afull bulk band gap irrespective of the choice of the Fermilevel. Substituting Sb by Bi is expected to change the rel-ative energy levels of the bands at T and L points, andat antimony composition of x ≃ .
1, a full bulk energygap is realized. Furthermore, it is also important to notethat increasing Bi composition also effectively enhancesthe spin-orbit coupling. Thus for the system with verylarge bismuth composition (0 ≤ x ≤ . L points are inverted. Thus there are in-total four (an even number of) bulk band inversions inbismuth for the system with very large bismuth compo-sition (0 ≤ x ≤ . ) topologically trivial.Therefore, theoretical band structure calculation predictsthe 3D topological insulator phase in Bi . Sb . ( x = 0 . . Sb . sample, we pre-form high-momentum-resolution angle-resolved photoe-mission spectroscopy (Fig.2) with varying incident pho-ton energy (IPEM-ARPES). The incident photon energydependent ARPES studies allow us to measure the en-ergy dispersion along the out-of-plane momentum spacedirection ( E − k ⊥ ), which can distinguish between thethree-dimensional bulk bands and the two-dimensionalsurface states. As shown in Fig.2 b, d, and e, a Λ-shaped dispersion whose tip lies less than 50 meV belowthe Fermi energy ( E F ) is observed. Additional featuresoriginating from surface states that do not disperse withincident photon energy are also seen in Fig.2 d and e.Our data are consistent with the extremely small effec-tive mass of 0 . m e (where m e is the electron mass)observed in magneto-reflection measurements on sampleswith x = 11% [125]. Studying the band dispersion per-pendicular to the sample surface provides a way to differ-entiate bulk states from surface states in a 3D material.To visualize the near- E F dispersion along the 3D L-X cut(X is a point that is displaced from L by a k z distanceof 3 π/c , where c is the lattice constant), in Fig.2a weplot energy distribution curves (EDCs), taken such thatelectrons at E F have fixed in-plane momentum ( k x , k y )= (L x , L y ) = (0.8 ˚A − , 0.0 ˚A − ), as a function of pho-ton energy ( hν ). There are three prominent features inthe EDCs: a non-dispersing, k z independent, peak cen-tered just below E F at about − − hν = 29 eV. To understand which bands these features orig-inate from, we show ARPES intensity maps along anin-plane cut ¯ K ¯ M ¯ K (parallel to the k y direction) takenusing hν values of 22 eV, 29 eV and 35 eV, which cor-respond to approximate k z values of L z − − , L z ,and L z + 0.3 ˚A − respectively (Fig.2b). At hν = 29 eV,the low energy ARPES spectral weight reveals a clear Λ-shaped band close to E F . As the photon energy is eitherincreased or decreased from 29 eV, this intensity shiftsto higher binding energies as the spectral weight evolves FIG. 2.
The first 3D topological insulator (2007): Topological Surface States and electronic band dispersionalong the k z -direction in momentum space. Surface states are experimentally identified by studying their out-of-planemomentum dispersion through the systematic variation of incident photon energy. a, Energy distribution curves (EDCs) ofBi . Sb . with electrons at the Fermi level ( E F ) maintained at a fixed in-plane momentum of ( k x =0.8 ˚A − , k y =0.0 ˚A − ) areobtained as a function of incident photon energy. b, ARPES intensity maps along cuts parallel to k y taken with electronsat E F fixed at k x = 0.8 ˚A − with respective photon energies of hν = 22 eV, 29 eV and 35 eV. c, Projection of the bulk BZ(black lines) onto the (111) surface BZ (green lines). Overlay (enlarged in inset) shows the high resolution Fermi surface (FS)of the metallic SS mode, which was obtained by integrating the ARPES intensity (taken with hν = 20 eV) from −
15 meV to10 meV relative to E F . EDCs corresponding to the cuts A, B and C are also shown; these confirm the gapless character of thesurface states in bulk insulating Bi . Sb . . d,e, ARPES dispersion cuts of Bi . Sb . . The cuts are along d , the k y direction, e , a direction rotated by approximately 10 ◦ from the k y direction. [Adapted from D. Hsieh et al. , Nature , 970 (2008)submitted in 2007 [27]]. from the Λ-shaped into a ∪ -shaped band. Therefore, thedispersive peak in Fig.2a comes from the bulk valenceband, and for hν = 29 eV the high symmetry point L =(0.8, 0, 2.9) appears in the third bulk BZ. In the mapsof Fig.2b with respective hν values of 22 eV and 35 eV,overall weak features near E F that vary in intensity re-main even as the bulk valence band moves far below E F .The survival of these weak features over a large photonenergy range (17 to 55 eV) supports their surface ori-gin. The non-dispersing feature centered near − . − .
02 eV reflects thelow energy part of the surface states that cross E F awayfrom the ¯ M point and forms the surface Fermi surface(Fig.2c).We now discuss the topological character of surfacestates in Bi . Sb . (Fig.2c), focusing on their key dif-ferences with respect to surface states in a conventional(topologically trivial) insulator. In general, surface statesare allowed to exist within the bulk energy gap owing to the loss of space inversion symmetry [ E ( k, ↑ ) = E ( − k, ↑ )]. However, there is a key distinction between surfacestates in a conventional insulator and a topological insu-lator, which is that along a path connecting two TRIMin the same BZ, the Fermi energy inside the bulk gapwill intersect these singly degenerate surface states ei-ther an even or odd number of times. If there are aneven number of surface state crossings, the surface statesare topologically Z trivial because disorder or correla-tions can remove pairs of such crossings by pushing thesurface bands entirely above or below E F . When thereare an odd number of crossings, however, at least onesurface state must remain gapless, which makes it non-trivial [25, 26, 119]. The existence of such topologicallynon-trivial surface states can be theoretically predictedon the basis of the bulk band structure only, using the Z invariant. Materials with band structures with Z = +1( ν = 0) are ordinary Bloch band insulators that aretopologically equivalent to the filled shell atomic insu-lator, and are predicted to exhibit an even number (in-cluding zero) of surface state crossings. Materials withbulk band structures with Z = − ν = 1) on the otherhand, which are expected to exist in rare systems withstrong spin-orbit coupling acting as an internal magneticfield on the electron system [127], and inverted bands atan odd number of high symmetry points in their bulk 3DBZs, are predicted to exhibit an odd number of surfacestate crossings, precluding their adiabatic continuationto the atomic insulator [12, 23–26, 119]. Such topologi-cal surface states that enclose the Kramers’ points by anodd number of times [12, 26] cannot be realized in anypurely 2D electron gas system, such as the one realizedat the interface of GaAs/GaAlAs systems.The nontrivial Z topological number ( ν = 1) in a3D topological insulator requires the terminated surfaceto have a Fermi surface (FS) that supports a non-zeroBerry’s phase (odd as opposed to even multiple of π ),which is not realizable in an ordinary spin-orbit material.More specifically, for the Z TI phase in Bi − x Sb x , ac-cording to Kramers theorem, they must remain spin de-generate at four special time reversal invariant momenta( ~k T = ¯Γ, ¯ M ) in the (111) surface BZ of Bi − x Sb x [seeFig.3(A)]. If a Fermi surface pocket does not enclose ~k T (= ¯Γ, ¯ M ), it is irrelevant for the Z topology [25, 128].Because the wave function of a single electron spin ac-quires a geometric phase factor of π [129] as it evolves by360 ◦ in momentum space along a Fermi contour enclos-ing a ~k T , an odd number of Fermi pockets enclosing ~k T in total implies a π geometrical (Berry’s) phase [25]. Inorder to realize a π Berry’s phase the surface bands mustbe spin-polarized and exhibit a partner switching [25]dispersion behavior between a pair of ~k T . This meansthat any pair of spin-polarized surface bands that aredegenerate at ¯Γ must not re-connect at ¯ M , or must sep-arately connect to the bulk valence and conduction bandin between ¯Γ and ¯ M . The partner switching behavior isrealized in Fig. 3(C) because the spin down band con-nects to and is degenerate with different spin up bandsat ¯Γ and ¯ M .We, for the first time, investigated the spin propertiesof the topological insulator phase [28], in order to exper-imentally demonstrate the non-zero Berry’s phase andthe nontrivial Z topological invariant. Spin-integratedARPES [130] intensity maps of the (111) surface states ofinsulating Bi − x Sb x taken at the Fermi level ( E F ) [Figs3(D)&(E)] show that a hexagonal FS encloses ¯Γ, whiledumbbell shaped FS pockets that are much weaker inintensity enclose ¯ M . By examining the surface band dis-persion below the Fermi level [Fig.3(F)] it is clear that thecentral hexagonal FS is formed by a single band (Fermicrossing 1) whereas the dumbbell shaped FSs are formedby the merger of two bands (Fermi crossings 4 and 5) [27].This band dispersion resembles the partner switching dis-persion behavior characteristic of topological insulators.To check this scenario and determine the topological in-dex ν , we have carried out spin-resolved photoemissionspectroscopy. Fig.3(G) shows a spin-resolved momentumdistribution curve taken along the ¯Γ- ¯ M direction at abinding energy E B = −
25 meV [Fig.3(G)]. The data re- veal a clear difference between the spin-up and spin-downintensities of bands 1, 2 and 3, and show that bands 1and 2 have opposite spin whereas bands 2 and 3 have thesame spin (detailed analysis discussed later in text). Theformer observation confirms that bands 1 and 2 form aspin-orbit split pair, and the latter observation suggeststhat bands 2 and 3 (as opposed to bands 1 and 3) are con-nected above the Fermi level and form one band. This isfurther confirmed by directly imaging the bands throughraising the chemical potential via doping. Irrelevance ofbands 2 and 3 to the topology is consistent with the factthat the Fermi surface pocket they form does not encloseany ~k T . Because of a dramatic intrinsic weakening of sig-nal intensity near crossings 4 and 5, and the small energyand momentum splitting of bands 4 and 5 lying at theresolution limit of modern spin-resolved ARPES spec-trometers, no conclusive spin information about thesetwo bands can be drawn from the methods employedin obtaining the data sets in Figs 3(G)&(H). However,whether bands 4 and 5 are both singly or doubly de-generate does not change the fact that an odd numberof spin-polarized FSs enclose the ~k T , which provides ev-idence that Bi − x Sb x has ν = 1 and that its surfacesupports a non-trivial Berry’s phase. This directly im-plies an absence of backscattering in electronic transportalong the surface (Fig.4), which has been re-confirmedby numerous scanning tunneling microscopy studies thatshow quasi-particle interference patterns that can onlybe modeled assuming an absence of backscattering [92–94]. More importantly, the spin-ARPES method that wedeveloped in Ref. [28] becomes a standard experimen-tal methodology for discovering and probing topologicalorder (non-Z ) in bulk solids [37, 40, 45].It is worth noting that the bulk band gap in Bi − x Sb x is rather small ( ≤
50 meV) and its surface states is quitecomplex with multiple pieces of surface Fermi surfacesboth near the ¯Γ and the ¯ M points. Therefore, it is im-portant to find a topological insulator consisting of a sin-gle surface state for the purposes of both studying theirphysical properties in fundamental physics and utilizingthem in devices. This motivated a search for topolog-ical insulators with a larger band gap and simpler sur-face spectrum. A second generation of 3D topological in-sulator materials, especially Bi Se , offers the potentialfor topologically protected behavior in ordinary crystalsat room temperature and zero magnetic field. Startingin 2008, work by the Princeton group used spin-ARPESand first-principles calculations to study the surface bandstructure of Bi Se and observe the characteristic signa-ture of a topological insulator in the form of a single Diraccone that is spin-polarized (Fig.5) such that it also carriesa non-trivial Berry’s phase [29, 31]. Concurrent theoret-ical work by [30] used electronic structure methods toshow that Bi Se is just one of several new large band-gap topological insulators. These other materials weresoon after also identified using this ARPES technique wedescribe [32, 33]. The Bi Se surface state is found fromspin-ARPES and theory to be a nearly idealized single FIG. 3.
Spin texture of a topological insulator encodes Z topological order of the bulk (2008) (A) Schematicsketches of the bulk Brillouin zone (BZ) and (111) surface BZ of the Bi − x Sb x crystal series.(B) Schematic of Fermi surfacepockets formed by the surface states (SS) of a topological insulator that carries a Berry’s phase. (C) Partner switchingband structure topology. (D) Spin-integrated ARPES intensity map of the SS of Bi . Sb . at E F . Arrows point in themeasured direction of the spin. (E) High-resolution ARPES intensity map of the SS at E F that enclose the ¯ M and ¯ M points.Corresponding band dispersion (second derivative images) are shown below. The left right asymmetry of the band dispersionsare due to the slight offset of the alignment from the ¯Γ- ¯ M ( ¯ M ) direction. (F) Surface band dispersion image along the ¯Γ- ¯ M direction showing five Fermi level crossings. (G) Spin-resolved momentum distribution curves presented at E B = −
25 meVshowing single spin degeneracy of bands at 1, 2 and 3. Spin up and down correspond to spin pointing along the +ˆ y and -ˆ y direction respectively. (H) Schematic of the spin-polarized surface FS observed in our experiments. It is consistent with a ν = 1 topology (compare (B) and (H)). [Adapted from D. Hsieh et al. , Science , 919 (2009)[28], Submitted in 2008].
Dirac cone as seen from the experimental data in Fig. 6.An added advantage is that Bi Se is stoichiometric (i.e.,a pure compound rather than an alloy such as Bi − x Sb x )and hence can be prepared, in principle, at higher purity.While the topological insulator phase is predicted to bequite robust to disorder, many experimental probes of thephase, including ARPES of the surface band structure,are clearer in high-purity samples. Finally and perhapsmost important for applications, Bi Se has a large bandgap of around 0.3 eV (3600 K). This indicates that in itshigh-purity form Bi Se can exhibit topological insulatorbehavior at room temperature and greatly increases thepotential for applications. Now, Bi Se has become theprototype TI that features a single-Dirac-cone topologi-cal surface state, which is widely used for many transport,tunneling, optical, nanostructured studies.Besides the Z topological variant ν , there is anothertopological number that can be uniquely determined byour spin-resolved ARPES measurements, the topologicalmirror Chern number n M . For example, we now deter-mine the value of n M of antimony surface states fromour data. As shown in figure 3A, the vertical plane alongthe ¯Γ − ¯ M direction (yellow plane in figure 3A) is a mir-ror plane for the bulk BZ of antimony. Therefore, the electronic states within this mirror plane are the eigen-states of the Mirror operator, which defines a topologicalnumber that is the topological mirror Chern number n M .The absolute value of the mirror Chern number | n M | isdetermined by the number of surface states moving to+ k (or − k ) along the ¯Γ − ¯ M direction (the ¯Γ − ¯ M is theprojection of the mirror plane onto the (111) surface).From figure 3, it is seen that a single (one) surface band,which switches partners at ¯ M , connects the bulk valenceand conduction bands, so | n M | = 1 . The sign of n M is related to the direction of the spin polarization h ~P i of this band [128], which is constrained by mirror sym-metry to point along ± ˆ y . Since the central electron-likeFS enclosing ¯Γ intersects six mirror invariant points, thesign of n M distinguishes two distinct types of handed-ness for this spin polarized FS. Figure 3(F) shows thatfor both Bi − x Sb x and Sb, the surface band that formsthis electron pocket has h ~P i ∝ − ˆ y along the k x direc-tion, suggesting a left-handed rotation sense for the spinsaround this central FS thus n M = −
1. We note that sim-ilar analysis regarding the mirror symmetry and mirroreigenvalues n M = − Se material class. In fact,a nonzero (nontrivial) topological mirror Chern number FIG. 4.
Helical spin texture naturally leads to ab-sence of elastic backscattering for surface transport:No “U” turn on a 3D topological insulator surface. (a)Our measurement of a helical spin texture in both Bi − x Sb x and in Bi Se directly shows that there is (b) an absence ofbackscattering. (c) ARPES measured FSs are shown withspin directions based on polarization measurements. L(R)HCstands for left(right)-handed chirality. (d) Spin independentand spin dependent scattering profiles on FSs in (c) rele-vant for surface quasi-particle transport are shown whichis sampled by the quasi-particle interference (QPI) modes.[Adapted from S.-Y. Xu et al. , Science
560 (2011). [34] does not require a nonzero Z topological number. Or inother word, there is no necessary correlation between amirror symmetry protected topological order and a time-reversal symmetry protected topological order. However,since most of the Z topological insulators (Bi − x Sb x ,Bi Se , Bi Te and etc.) also possess mirror symmetriesin their crystalline form, thus topological mirror order n M is typically “masked” by the strong Z topologicalorder. One possible way to isolate the mirror topologi-cal order from the Z order is to work with systems thatfeature an even number of bulk band inversions. Thisapproach naturally exclude a nontrivial Z order whichstrictly requires an odd number of band inversions. Moreimportantly, if the locations of the band inversions coin-cide with the mirror planes in momentum space, it willlead to a topologically nontrivial phase protected by themirror symmetries of the crystalline system that is irrel-evant to the time-reversal symmetry protection and theZ (Kane-Mele) topological order. Such exotic new phaseof topological order, noted as topological crystalline in-sulator [48] protected by space group mirror symmetries,has very recently been theoretically predicted and exper-imentally identified in the Pb − x Sn x Te(Se) alloy systems[36–38, 49]. An anomalous n M = − − x Sn x Te, distinct from the n M = − topological insulators, has also beenexperimentally determined using spin-resolved ARPESmeasurements as shown in Ref. [37]. Moreover, the mir-ror symmetry can be generalized to other space groupsymmetries, leading to a large number of distinct topo-logical crystalline insulators awaited to be discovered,some of which are predicted to exhibit nontrivial crys-talline order even without spin-orbit coupling as well astopological crystalline surface states in non-Dirac (e.g.quadratic) fermion forms [48].It can be seen from our ARPES data that as grownBi Se is in a doped semiconductor, where the chemi-cal potential cuts into the bulk conduction band. Theobserved n -type behavior is believed to be caused by Sevacancies. However, many of the interesting theoreti-cal proposals that utilize topological insulator surfacesrequire the chemical potential to lie at or near the sur-face Dirac point. This is similar to the case in graphene,where the chemistry of carbon atoms naturally locatesthe Fermi level at the Dirac point. This makes its den-sity of carriers highly tunable by an applied electricalfield and enables applications of graphene to both basicscience and microelectronics. We demonstrated [31] thatappropriate chemical modifications both in the bulk andon the surface of, which does not change its topologi-can nature, can achieve the condition with the chemicalpotential at the surface Dirac point, however, the Fermienergy of both the bulk and the surface can be controlled.This was achieved by doping bulk with a small concentra-tion of Ca, which compensates the Se vacancies, to placethe Fermi level within the bulk band gap. The surfacewas the hole doped by exposing the surface to NO gasto place the Fermi level at the Dirac point, and has beenshown to be effective even at room temperature (Fig.6).These results collectively show how ARPES can be usedto study the topological protection and tunability prop-erties of the 2D surface of a 3D topological insulator.In summary, in this section, we have reviewed the ex-perimental discovery of the 3D Z topological insula-tor phase in Bi − x Sb x , Bi Se and other related com-pounds. Utilizing spin-resolved angle-resolved photoe-mission spectroscopy (spin-ARPES), we experimentallyprobed the nontrivial topological order ( ν = 1) in 3DZ TIs by measuring the surface state Fermi surfacetopology (number of surface state Fermi pockets enclos-ing the Kramers’ points) and the spin-polarization tex-ture that determines the π quantum Berry’s phase. Fur-thermore, we demonstrated that the topological surfacestates are realized and stable even at room temperatures,their chemical potential can be tuned and engineeredto achieve the charge neutrality (Dirac) point, and fur-thermore their unique spin-momentum locking propertyleads to the prohibition of backscatting. These impor-tant experimental observations via spin-ARPES, not onlydemonstrate the 3D Z topological insulator phase andits topological order ( ν = 1), but also provide a powerfuland unique methodology that is now used to discover andstudy new topological order in three-dimensions, such FIG. 5.
First detection of Z (symmetry protected) Topological-Order: spin-momentum locking of spin-helicalDirac electrons in Bi Se and Bi Te using spin-resolved ARPES. (a) ARPES intensity map at E F of the (111) surfaceof tuned Bi − δ Ca δ Se (see text) and (b) the (111) surface of Bi Te . Red arrows denote the direction of spin around the Fermisurface. (c) ARPES dispersion of tuned Bi − δ Ca δ Se and (d) Bi Te along the k x cut. The dotted red lines are guides to theeye. (e) Measured y component of spin-polarization along the ¯Γ- ¯ M direction at E B = -20 meV, which only cuts through thesurface states. Inset shows a schematic of the cut direction. (f) Measured x (red triangles) and z (black circles) componentsof spin polarization along the ¯Γ- ¯ M direction at E B = -20 meV. (g) Spin-resolved spectra obtained from the y component spinpolarization data. (h) Fitted values of the spin polarization vector P . [Adapted from D. Hsieh et al. , Nature , 1101 (2009).Some data are adapted from Y. Xia et al. , Nature Physics , 398 (2009)]. as the topological Kondo insulator, the topological crys-talline insulator, the topological Dirac semimetal phasesthat we will discuss in the following sections. TOPOLOGICAL KONDO INSULATORCANDIDATES
Materials with strong electron correlations often ex-hibit exotic ground states such as the heavy fermion be-havior, Mott or Kondo insulation and unconventional su-perconductivity. Kondo insulators are mostly realized inthe rare-earth based compounds featuring f -electron de-grees of freedom, which behave like a correlated metalat high temperatures whereas a bulk bandgap opens atlow temperatures through the hybridization [132–134] ofnearly localized-flat f bands with the d -derived dispersiveconduction band. With the advent of topological insula-tors [1, 11, 13, 29] the compound SmB , often categorizedas a heavy-fermion semiconductor [132–134], attractedmuch attention due to the proposal that it may possiblyhost a topological Kondo phase (TKI) at low tempera-tures where transport is anomalous [50, 135, 136]. Theanomalous residual conductivity is believed to be associ-ated with electronic states that lie within the Kondo gap[137–146]. Following the prediction of a TKI phase, there havebeen several surface-sensitive transport measurements,which include observation of a three-dimensional (3D)to two-dimensional (2D) crossover of the transport car-riers below T ∼
7K [110, 111, 147]. However, due tothe lack of the critical momentum resolution for thetransport probes, neither the existence of in-gap surfacestates nor their Fermi surface topology (number of sur-face Fermi surfaces and enclosing or not enclosing theKramer’s points) have been experimentally studied. Bycombining high-resolution laser- and synchrotron-basedangle-resolved photoemission techniques in Ref. [45], wepresent the surface electronic structure identifying thein-gap states that are strongly temperature dependentand disappear before approaching the coherent Kondohybridization scale. Remarkably, the observed Fermi sur-face for the low-energy part of the in-gap states keepingthe sample within the transport anomaly regime ( T ∼ are also reported inRefs. [44, 46].SmB crystallizes in the CsCl-type structure with theSm ions and the B octahedra being located at the cor- FIG. 6.
Observation of room temperature (300K) topological order (symmetry protected) without appliedmagnetic field in Bi Se : (a) Crystal momentum integrated ARPES data near Fermi level exhibit linear fall-off of density ofstates, which, combined with the spin-resolved nature of the states, suggest that a half Fermi gas is realized on the topologicalsurfaces. (b) Spin texture map based on spin-ARPES data suggest that the spin-chirality changes sign across the Dirac point.(c) The Dirac node remains well defined up a temperature of 300 K suggesting the stability of topological effects up to theroom temperature. (d) The Dirac cone measured at a temperature of 10 K. (e) Full Dirac cone. (f) The spin polarizationmomentum-space texture as a function of energy with respect to the Dirac point. [Adapted from D. Hsieh et al. , Nature ,1101 (2009). [31]]. ner and at the body center of the cubic lattice, respec-tively (Fig. 8 a ). The bulk Brillouin zone (BZ) is a cubemade up of six square faces. The center of the cubeis the Γ point, whereas the centers of the square facesare the X points. Due to the inversion symmetry ofthe crystal, each X point and its diametrically oppositepartner are completely equivalent. Therefore, there ex-ist three distinct X points in the BZ, labeled as X , X and X . It is well-established that the low energy physicsin SmB is constituted of the non-dispersive Sm 4 f bandand the dispersive Sm 5 d band located near the X points[110, 140, 141, 147, 148]. Figs. 8 d and e show ARPESintensity profiles over a wide binding energy scale mea-sured with a synchrotron-based ARPES system using aphoton energy of 26 eV. The dispersive features origi-nate from the Sm 5 d derived bands and a hybridizationbetween the Sm 5 d band and Sm 4 f flat band is visibleespecially around 150 meV binding energies confirmingthe Kondo features of the electronic system in our study(Figs. 8 d and e ).In order to search for the predicted in-gap states within5 meV of the Fermi level, a laser-based ARPES systemproviding ∆ E ∼ T ≃ X points (Fig. 8 f ) inthe bulk BZ and the X points project onto the ¯ X , ¯ X ,and the ¯Γ points at (001) surface (Fig. 8 b ), the Kramers’ points of this lattice are ¯ X , ¯ X , ¯Γ and ¯ M and one needsto systematically study the connectivity (winding) of thein-gap states around these points. Fig. 9 c shows experi-mentally measured ARPES spectral intensity integratedin a narrow ( ± .
15 ˚A − ) momentum window and theirtemperature evolution around the ¯ X point. At temper-atures above the hybridization scale, only one spectralintensity feature is observed around E B ∼
12 meV in theARPES EDC profile. As temperature decreases below30 K, this feature is found to move to deeper binding en-ergies away from the chemical potential, consistent withthe opening of the Kondo hybridization gap while Fermilevel is in the insulating gap (bulk is insulating, accordingto transport, so Fermi level must lie in-gap at 6 K). Atlower temperatures, the gap value of hybridized states atthis momentum space regime is estimated to be about 16meV. More importantly, at a low temperature T ≃ E B ∼ T ≃ FIG. 7.
Surface gating : Tuning the density of helical Dirac electrons to the spin-degenerate Kramers pointand topological transport regime. (a) A high resolution ARPES mapping of the surface Fermi surface (FS) near ¯Γ ofBi − δ Ca δ Se (111). The diffuse intensity within the ring originates from the bulk-surface resonance state [12]. (b) The FS after0.1 Langmuir (L) of NO is dosed, showing that the resonance state is removed. (c) The FS after a 2 L dosage, which achievesthe Dirac charge neutrality point. (d) High resolution ARPES surface band dispersions through after an NO dosage of 0 L,(e) 0.01 L, (f) 0.1 L, (g) 0.5 L, (h) 1 L and (i) 2 L. The arrows denote the spin polarization of the bands. We note that due toan increasing level of surface disorder with NO adsorption, the measured spectra become progressively more diffuse and thetotal photoemission intensity from the buried Bi − δ Ca δ Se surface is gradually reduced. [Adapted from D. Hsieh et al. , Nature , 1101 (2009). [31]]. ilar spectra with the re-appearance of the in-gap statefeatures (Re 6K in Fig. 9 c ). The observed robustnessagainst thermal recyclings counts against the possibilityof non-robust (trivial) or non-reproducible surface states.We further performed similar measurements of low-lyingstates focusing near the ¯Γ point (projection of the X )as shown in Fig. 9 d . Similar spectra reveal in-gap statefeatures prominently around E B ∼ − T ≃
6K which clearly lie within the Kondo gap and exhibitsimilar (coupled) temperature evolution as seen in thespectra obtained near the ¯ X point.We further study their momentum-resolved structure or the k-space map for investigations regarding theirtopology: 1) The number of surface state pockets thatlie within the Kondo gap; 2) The momentum space loca-tions of the pockets (whether enclosing or winding theKramers’ points or not). Fig. 8 f shows a Fermi sur-face map measured by setting the energy window tocover E F ± d,e at a temperature of 6 K inside the 2D transport anomalyregime under the “better than 5 meV and 7 K combined1resolution condition”. Our Fermi surface mapping re-veals multiple pockets which consist of an oval-shaped aswell as nearly circular-shaped pockets around the ¯ X and¯Γ points, respectively. No pocket was seen around the¯ M -point which was measured in a synchrotron ARPESsetting. Therefore the laser ARPES data captures all thepockets that exist while the bulk is insulating. This resultis striking by itself from the point of view that while weknow from transport that the bulk is insulating, ARPESshows large Fermi surface pockets (metallicity of the sur-face) at this temperature. Another unusual aspect is thatnot all Kramers’ points are enclosed by the in-gap states.Our observed Fermi surface thus consists of 3 (or oddnumber Mod 2 around each Kramers’ point) pockets perBrilluoin zone and each of them wind around a Kramers’point only and this number is odd (at least 3). There-fore, our measured in-(Kondo) gap states lead to a veryspecific form of the Fermi surface topology (Fig. 8 f ) thatis remarkably consistent with the theoretically predictedtopological surface state Fermi surface expected in theTKI groundstate phase despite the broad nature of thecontours.Since for the laser-ARPES, the photon energy is fixed(7 eV) and the momentum window is rather limited (themomentum range is proportional to √ hν − W , where hν is the photon energy and W ≃ . e,f show the energy-momentum cuts mea-sured with varying photon energies. Clear E − k disper-sions are observed within a narrow energy window nearthe Fermi level. The dispersion is found to be unchangedupon varying photon energy, supporting their quasi-two-dimensional nature (see, Fig. 9 g ). The observed quasi-two-dimensional character of the signal within 10 meV ofthe gap where surface states reside does suggest consis-tency with the surface nature of the in-gap states. Dueto the combined effects of energy resolution (∆ E ≥ f -part of the cross-section and the strong band tails, thein-gap states are intermixed with the higher energy bulkbands’ tails. In order to isolate the in-gap states fromthe bulk band tails that have higher cross-section at syn-chrotron photon energies, it is necessary to have energyresolution (not just the low working temperature) betterthan half the Kondo gap scale which is about 7 meV orsmaller in SmB . Our experiment reports of Fermi sur-face mapping covering the low-energy part of the in-gapstates keeping the sample within the transport anomalyregime reveals an odd number of pockets that enclosethree out of the four Kramers’ points of the surface Bril-louin zone strongly suggesting the the topological originof the in-gap state. TOPOLOGICAL QUANTUM PHASETRANSITIONS
A three-dimensional topological insulator is a newphase of matter distinct from a conventional band in-sulator (semiconductor) in that a TI features a nontriv-ial topological invariant in its bulk electronic wavefunc-tion space [1, 2, 4, 10–13, 27–29, 71, 119, 149–153]. Thenonzero topological invariant in a TI leads to the exis-tence of spin-momentum locked gapless Dirac electronson its surfaces [13, 27–29]. It has been theoreticallyknown that a TI can be tuned from a conventional insula-tor by going through an adiabatic band inversion processin the bulk [1, 11, 119]. Such a quantum phase tran-sition from a conventional band insulator to a TI thatinvolves a change of the bulk topological invariant is de-fined as a topological phase transition. The topologicalphase transition is of great interest because its criticalpoint (the topological-critical-point) is expected to notonly realize new groundstates such as higher dimensionalDirac/Weyl fermions [119, 149–151] and supersymmetrystate [71], but also show exotic transport and optical re-sponses such as chiral anomaly in magnetoresistence [152]or the light-induced Floquet topological insulator state[153]. To achieve these novel phenomena, it is of impor-tance to study the electronic and spin groundstate acrossa topological phase transition. Studying the electronicand spin groundstate across a topological phase transi-tion also serves as the key to understanding the formationof the topological surface states across the topologicalphase transition. It is well established that the topologi-cal Dirac surface states and their spin-momentum lockingare the signature that distinguishes a topological insula-tor from a conventional insulator. However, an interest-ing and vital question that remains unanswered is howtopological surface states emerge as a non-topologicalsystem approaches and crosses the topological criticality.Therefore, in order to realize these proposed new topo-logical phenomena and also to understand the fate of thetopological surface states across the topological-critical-point, it is critically important to realize a fully tunablespin-orbit real material system, where such topologicalphase transition can be systematically realized, observed,and further engineered.Such a fully tunable topological phase transition sys-tem is first realized by our ARPES and spin-resolvedARPES studies on the BiTl(S − δ Se δ ) [34]. In Ref. [34],by studying the electronic and spin groundstate of theBiTl(S − δ Se δ ) samples with various δ compositions, abulk band inversion and a topological phase transitionbetween a conventional band insulator and a topologi-cal insulator is, for the first time, systematically demon-strated and visualized. Such study [34] serves as a cor-ner stone for realizing new topological phenomena basedon the topological phase transition as discussed above[71, 119, 149–153]. This work was therefore followed andexpanded by many later works (e.g. [35, 66, 79, 154]),which not only studied the BiTl(S − δ Se δ ) system in2 FIG. 8.
Brillouin zone symmetry, and band structure of SmB . a, Crystal structure of SmB . Sm ions and B octahedron are located at the corners and the center of the cubic lattice structure. b, The bulk and surface Brillouin zones ofSmB . High-symmetry points are marked. c, Resistivity-temperature profile for samples used in ARPES measurements. d,e,
Synchrotron-based ARPES dispersion maps along the ¯ M − ¯ X − ¯ M and the ¯ X − ¯Γ − ¯ X momentum-space cut-directions.Dispersive Sm 5 d band and non-dispersive flat Sm 4 f bands are observed, confirming the key ingredient for a heavy fermionKondo system. f, A Fermi surface map of bulk insulating SmB using a 7 eV laser source at a sample temperature of ≃ E F ± X reflect low-lying metallic states near the Fermi level, which is consistent with thetheoretically predicted Fermi surface topology of the topological surface states. [Adapted form M. Neupane et al., NatureCommun. , 2991 (2013)]. greater details and depth [35, 79] but also expanded therealization of the topological phase transition into otherclasses of topological materials [66, 154].Figure 10A presents systematic photoemission mea-surements of electronic states that lie between a pair oftime-reversal invariant points or Kramers points (¯Γ and¯ M ) obtained for a series of compositions of the spin-orbitmaterial BiTl(S − δ Se δ ) . As the selenium concentrationis increased, the low-lying bands separated by a gap ofenergy 0 .
15 eV at δ = 0 . .
05 eVat δ = 0 .
4. The absence of surface states (SSs) withinthe bulk gap suggests that the compound is topologi-cally trivial for composition range of δ = 0 . δ = 0 . δ = 0 .
4, a linearly dispersive band con-necting the bulk conduction and valence bands emergeswhich threads across the bulk band gap. Moreover, the Dirac-like bands at δ = 0 . δ = 0 . δ = 0 .
6. While the systemapproaches the transition from the conventional or no-surface-state side ( δ = 0 . δ = 0 .
4) show that thespectral weight at the outer boundary of the bulk con-duction band continuum which corresponds to the lociwhere the Dirac SSs would eventually develop becomesmuch more intense; however, that the surface remainsgapped at δ = 0 . δ varies from 0.0 to 1.0 (Fig. 10C), thedispersion of the valence band evolves from a “Λ”-shapeto an “ M ”-shape with a “dip” at the ¯Γ point ( k = 0);3 FIG. 9.
Temperature dependent in-gap states and its two-dimensional nature. a,
Cartoon sketch depicting the basicsof Kondo lattice hybridization at temperatures above and below the hybridization gap opening. The blue dashed line representsthe Fermi level in bulk insulating samples such as SmB (since the bulk of the SmB is insulating, the Fermi level must liewithin the Kondo gap). The theoretically predicted topological surface states within the Kondo gap are also shown in thiscartoon view (black dash lines) based on Refs [135, 136]. The black dash rectangle shows the approximate momentum windowof our laser-ARPES measurements between k = 0 . − to k = 0 . − . b, Partially momentum-integrated ARPES spectralintensity in a ± .
15 ˚A − window (∆ k defined in panel a ) above and below the Kondo lattice hybridization temperature ( T H ). c, Momentum-integrated ARPES spectral intensity centered at the ¯ X point at various temperatures. d, Analogous measurementsas in Panel c but centered at the ¯Γ pocket (∆ k = 0 . − ). ARPES data taken on the sample after thermally recycling (6K up to 50 K then back to 6 K) is shown by Re 6K, which demonstrates that the in-gap states are robust against thermalrecycling. e, Synchrotron based ARPES energy momentum dispersion maps measured using different photon energies along the¯ M − ¯ X − ¯ M momentum space cut-direction. Incident photon energies used are noted on the plot. f, Momentum distributioncurves (MDCs) of data shown in a . The peaks of the momentum distribution curves are marked by dashed lines near the Fermilevel, which track the dispersion of the low-energy states. g, Momentum distribution curves in the close vicinity of the Fermilevel (covering the in-gap states near the gap edge) integrated within the energy window of [ E F - 8 meV, E F ] are shown as afunction of photon energy which covers the k z range of 4 π to 5 π at 7 K. [Adapted form M. Neupane et al., Nature Commun. , 2991 (2013)]. δ = 0 . δ = 1 . − δ Se δ ) system. The bulkband inversion process in the BiTl(S − δ Se δ ) system isshown in Fig. 11. TOPOLOGICAL DIRAC SEMIMETALS
The relativistic (Dirac) fermions of solid-state bandstructure has been known since 1947 [155] wheregraphene is considered. Graphene is a two dimensionalsemimetal and its electrons are effectively relativisticwith velocity (1 / c , where c is the velocity of lightin vacuum. The Dirac nature of the band structure isprotected by symmetries of the graphene lattice. Recentrealizations of two-dimensional massless Dirac electronsin graphene and surfaces of the 3D topological insulatorhave generated enormous interest in condensed matterphysics [1, 13, 29, 51–54, 119, 156–162]. Many inter-esting phenomena such as exotic integer quantum Halleffect [156] has been observed in graphene. It is knownthat a minimum model for a 2D Dirac electronic systemis H = v F ( p x σ x + p y σ y ), where p is momentum and σ arePauli matrices. It is obvious that a mass term mσ z willgenerate an energy gap for the electronic structure andthe Dirac nodes are protected by extra physical symme-tries apart from the lattice translational symmetry.The search for 3D Dirac semimetal with simple elec-tronic structure continues after the discovery of the 2DDirac semimetals which can exist with or without spin-orbit coupling. The most direct generalization of 2Dto 3D is the Weyl point of two bands, such as H = v F ( p x σ x + p y σ y + p z σ z ). The three Pauli matrices are allnow used up in above equation and there is no local massterm. The Weyl semimetal phase is robust against per-turbation and the robustness results from a topological(spin-orbit and symmetry together) consideration, whichmeans that there cannot be only one Weyl point on theFermi surface as the total Chern number must be zero perBrillouin zone (BZ). In materials with both time-reversal(T) and space inversion (I) symmetries, Weyl points mustcome together in pairs, degenerate in energy to form 3DDirac points. Due to the inevitable degeneracy in T andI symmetric systems, one can only see 3D Dirac statesrather than Weyl states. In order to achieve the Weylstates, one has to break either T or I symmetry.Three-dimensional (3D) Dirac fermion metals, some-times noted as the bulk Dirac semimetal phases, areof great interest if the material possesses 3D isotropicor anisotropic relativistic dispersion in the presence ofstrong spin-orbit coupling. It has been theoretically pre-dicted that a topological (spin-orbit) 3D spin-orbit Diracsemimetal can be viewed as a composite of two sets ofWeyl fermions where broken time-reversal or space inver- sion symmetry can lead to a surface Fermi-arc semimetalphase or a topological insulator [54]. In the absenceof spin-orbit coupling, topological phases cannot be de-rived from a 3D Dirac semimetal. Thus the parent bulkDirac semimetal phase with strong spin-orbit couplingis of great interest. Moreover, it is theoretically pre-dicted that Weyl semimetal (WS) phase can be existedin HgCr Se [163], Pyrochlore Iridates [51, 164] and β -cristobalite BiO [165]. Despite their predicted exis-tence of bulk Dirac semimetal phase [53, 54, 119] and WSphase [51, 163–165], experimental studies have been lack-ing since it has been difficult to realize these phases in realmaterials, especially in stoichiometric single crystallinenon-metastable systems with high mobility. It has alsobeen noted that the bulk Dirac semimetal state can beachieved at the critical point of a topological phase tran-sition [34, 35, 66] between a normal insulator and a topo-logical insulator (Fig. 13), which requires fine-tuning ofthe chemical doping composition thus by effectively vary-ing the spin-orbit coupling strength. This approach alsointroduces chemical disorder into the system. In stoichio-metric bulk materials, the known 3D Dirac fermions inbismuth are in fact of massive variety since there clearlyexists a band gap in the bulk Dirac spectrum [159]. Onthe other hand, the bulk Dirac fermions in the Bi − x Sb x system coexist with additional Fermi surfaces [13].Recently, several theoretical studies have predicted theexistence of the topological Dirac semimetal (TDS) [52–54]. In a 3D TDS, the 3D Dirac band touchings arise fromthe protection of certain space group crystalline sym-metries, and are therefore proposed to be more robustto disorders or chemical alloying [34, 52–54, 66]. More-over, the topological Dirac semimetal differs from othertypes of 3D Dirac semimetals because it possesses strongspin-orbit coupling that leads to an inverted bulk bandstructure, making it possible to realize 3D Dirac multi-plet states and host nontrivial topological order as wellas novel spin-momentum locked Fermi arc surface states[53, 54]. In Ref. [40], we report experimental discoveryof the gapless TDS phase in high-mobility stoichiometricmaterial Cd As . Similar experimental results are alsoreported in Ref. [41]. Furthermore, experimental realiza-tion of the 3D Dirac phase in a metastable low mobilitycompound, Na Bi has also been reported [42, 43] .Fig. 13 a,b show the crystal structure of Cd As , whichhas a tetragonal unit cell with a = 12 .
67 ˚ A and c =25 .
48 ˚ A for Z = 32 with symmetry of space group I cd.In this structure, arsenic ions are approximately cubicclose-packed and Cd ions are tetrahedrally coordinated,which can be described in parallel to a fluorite structureof systematic Cd/As vacancies. There are four layersper unit and the missing Cd-As tetrahedra are arrangedwithout the central symmetry as shown with the (001)projection view in Fig. 13 b , with the two vacant sitesbeing at diagonally opposite corners of a cube face [166].In order to resolve a low-lying small dispersion featurenear the Fermi level, we perform high-resolution ARPESdispersion measurements in the close vicinity of the Fermi5 FIG. 10.
Topological phase transition through a 3D Dirac Semimetal (3D analog of graphene) phase inBiTl(S − δ Se δ ) . (A) High resolution ARPES dispersion mappings along a pair of time-reversal invariant points or Kramers’points (¯Γ and ¯ M ). (B) ARPES mapped native Fermi surfaces for varying chemical compositions. (C)
Left- and right-most:Energy-distribution curves for δ = 0 . (D) Compositional evolution of band structure measured over a wide energy and momentum range. At the critical pointa 3D Dirac Semimetal (3D analog of graphene) is realized [Adapted from S.-Y. Xu et al. , Science
560 (2011). [34]], Alsosee, M. Neupane et. al., Nature Commun. (2014) for the critical point data on 3D Dirac Semimetal [40]] level as shown in Fig. 13 c . Remarkably, a linearly disper-sive upper Dirac cone is observed at the surface BZ center¯Γ point, whose Dirac node is found to locate at a bindingenergy of E B ≃ . c and Fig. 14 b ). From the observedsteep Dirac dispersion (Fig. 13 c ), we obtain a surpris-ingly high Fermi velocity of about 9.8 eV · ˚ A ( ≃ . × ms − ). This is more than 10-fold larger than the the- oretical prediction of 0.15 eV · ˚ A at the correspondinglocation of the chemical potential [54]. Compared to themuch-studied 2D Dirac systems, the Fermi velocity ofthe 3D Dirac fermions in Cd As is thus about 3 timeshigher than that of in the topological surface states (TSS)of Bi Se [29], 1 . [45, 136]. The observed largeFermi velocity of the 3D Dirac band provides clues tounderstand Cd As ’s unusually high mobility reportedin previous transport experiments [168, 169].6 FIG. 11.
Bulk band inversion and spin texture inversion in BiTl(S − δ Se δ ) . (A) High resolution ARPES measuredBi Te Fermi surface. Direction and relative position of the spin-polarization measurement cuts A and B are indicated. (B)
Out-of-plane spin-polarization profile of Bi Te for cuts A and B. (C) Fitted values of direction of the 3D spin vectors obtainedfrom the Bi Te spin-resolved data. A 3D modulated spin texture is revealed from the data. (D) Experimental geometryemployed to obtain the spin-polarization components. (E)
ARPES measured BiTl(S Se ) dispersion along ¯Γ − ¯ M momentumspace cut, indicating the energy positions of cuts C, D and E. The binding energies for the cuts are: E B (Cuts C,D)=0.01eV, E B (Cuts E)=0.50eV. (F) Measured out-of-plane spin-polarization profile of cuts C and E on BiTl(S Se ) . (G) A map of themomentum space spin-resolved cuts C, D and E across the Fermi surfaces of BiTl(S Se ) . The hexagonal Fermi surface islocated 0.40eV above the Dirac node, whereas the circular Fermi surface is located 0.10eV below the Dirac node. (H) SurfaceFermi surface topology evolution of BiTl(S Se ) across the Dirac node. The corresponding binding energies of constant energycontours are indicated. Observed spin textures are schematically drawn at various binding energies. [Adapted from S.-Y. Xu et al. , Science
560 (2011). [34]]
In theory, there are two 3D Dirac nodes that are ex-pected at two special k points along the Γ − Z momentumspace cut-direction (Figs. 13 d,e ). At the (001) surface,these two k points along the Γ − Z axis project on to the¯Γ point of the (001) surface BZ (Fig. 13 d ). Therefore,at the (001) surface, theory predicts one 3D Dirac coneat the BZ center ¯Γ point (Fig. 14 a ). These results arein qualitative agreement with our data, which supportsour experimental observation of the 3D TDS phase inCd As . We also study the ARPES measured constantenergy contour maps (Figs. 14 c,d ). At the Fermi level,the constant energy contour consists of a single pocketcentered at the ¯Γ point. With increasing binding energy,the size of the pocket decreases and eventually shrinks toa point (the 3D Dirac point) near E B ≃ . As is better un-derstood from ARPES data if we compare our resultswith that of the prototype TI, Bi Se . In Bi Se as shownin Fig. 15 b , the bulk conduction and valence bands arefully separated (gapped), and a linearly dispersive topo-logical surface state is observed that connect across thebulk band-gap. In the case of Cd As (Fig. 15 a ), theredoes not exist a full bulk energy gap. On the otherhand, the bulk conduction and valence bands “touch” (and only “touch”) at one specific location in the mo-mentum space, which is the 3D band-touching node, thusrealizing a 3D TDS. For comparison, we further showthat a similar TDS state is also realized by tuning thechemical composition δ (effectively the spin-orbit cou-pling strength) to the critical point of a topological phasetransition between a normal insulator and a topologicalinsulator. Fig. 15 c,d present the surface electronic struc-ture of two other TDS phases in the BiTl(S − δ Se δ ) and(Bi − δ In δ ) Se systems. In both systems, it has beenshown that tuning the chemical composition δ can drivethe system from a normal insulator state to a topologicalinsulator state [34, 35, 66]. The critical compositions forthe two topological phase transitions are approximatelynear δ = 0 . δ = 0 .
04, respectively. Fig. 15 c,d show the ARPES measured surface electronic structureof the critical compositions for both BiTl(S − δ Se δ ) and(Bi − δ In δ ) Se systems, which are expected to exhibitthe TDS phase. Indeed, the bulk critical compositionswhere bulk and surface Dirac bands collapse also showDirac cones with intensities filled inside the cones, whichis qualitatively similar to the case in Cd As .Based on the ARPES data in Fig. 14 c,d , the Fermi ve-locity is estimated to be ∼ · ˚A and ∼ · ˚A for the3D Dirac fermions in BiTl(S − δ Se δ ) and (Bi − δ In δ ) Se respectively, which is much lower than that of what weobserve in Cd As , thus likely limiting the carrier mo-7 FIG. 12.
The 3D Dirac semimetal phase realized byfine-tuning to the critical point of a topological phasetransition. a,
Schematic view of topological phase tran-sition. The critical point ( δ c ) is marked by an arrow anda 3D Dirac cone is presented in the upper inset. The cal-culated electronic bulk bands structure [energy (eV) versusmomentum (˚A − )] at critical point is shown in the lower in-set. b, Crystal structure of TlBi(S/Se) with repeating Tl-Se-Bi-Se layers. c, ARPES dispersion map of TlBi(S − δ Se δ ) ( δ = 0 . d, Crystal structure of (Bi/In) Se with repeatingBi/In-Se layers. e, ARPES dispersion map of (Bi − δ In δ ) Se ( δ = 0 . et. al., Nature Com-mun. (2014) [40]. bility. The mobility is also limited by the disorder dueto strong chemical alloying. More importantly, the finecontrol of doping/alloying δ value and keeping the com-position exactly at the bulk critical composition is dif-ficult to achieve [34], especially while considering thechemical inhomogeneity introduced by the dopants. Wehave experimentally identified the crystalline-symmetry-protected 3D spin-orbit TDS phase in a stoichiometricsystem Cd As . Our experimental identification of theDirac-like bulk topological semimetal phase in high mo-bility Cd As opens the door for exploring higher dimen-sional spin-orbit Dirac physics in a real material TOPOLOGICAL CRYSTALLINE INSULATORS
In this section, we review the research on the topolog-ical crystalline insulator (TCI). In particular, we focuson the mirror symmetry protected TCI phase and its ex-perimental discovery in the Pb − x Sn x Te(Se) system. Wereview the methodology developed to uniquely determinethe topological number (the mirror Chern number n M ) inPb − x Sn x Te by measuring its surface state spin textureincluding the chirality (or handness) using spin-resolvedARPES.The 3D Z (Kane-Mele) topological insulator repre-sents the first example in nature of a topologically or-dered electronic phase existing in bulk solids [1]. In a 3DZ (Kane-Mele) TI, it is the protection of time-reversalsymmetry that gives rise to a nontrivial Z topological in-variant. With the explosion of research interest on 3D Z TI materials, a new research topic that focuses on search-ing for new topologically nontrivial phases protected byother discrete symmetries emerged. In 2011, a new topo-logical phase of matter, which is now usually referred asthe topological crystalline insulator (TCI), was theoret-ically proposed by Fu [48]. In a TCI, space group sym-metries of the crystalline system replace the role of time-reversal symmetry in an otherwise Z TI. Therefore, theTCI phase is topologically distinct from the much-studiedZ TI, and it is believed to host many exotic topologi-cal quantum properties, such as higher order (non-linear)surface band crossings, topological state without spin-orbit coupling, and crystalline symmetry protected topo-logical superconductivity or Chern currents [48]. So far,possible TCI phases have been theoretically discussed forsystems possessing four-fold ( C ) or six-fold ( C ) rota-tional symmetry as well as the mirror symmetry [48, 49].And the mirror symmetry case gained particular interestssince a real material prediction, namely the Pb − x Sn x Tesystem, was made by Hsieh et al via first-principles bandstructure calculations [49].Pb − x Sn x Te is a pseudobinary semiconducting systemwidely used for infrared optoelectronic and thermoelec-tric devices. It is known that the band-gap at the fourL points in the bulk Brillouin zone (BZ) closes itself andre-opens upon increasing x in the Pb − x Sn x Te system[170] (Fig. 16). The fact that band inversion occurs at8
FIG. 13.
Topological Dirac semimetal phase in Cd As . a, Cd As crystalizes in a tetragonal body center structurewith space group of I cd, which has 32 number of formula units in the unit cell. The tetragonal structure has lattice constantof a = 12 .
670 ˚A, b = 12 .
670 ˚A, and c = 25 .
480 ˚A. FIG. 13. b, The basic structure unit is a 4 corner-sharingCdAs -trigonal pyramid. c, ARPES E B − k x cut of Cd As near the Fermi level at around surface BZ center ¯Γ point. d, Cartoon view of dispersion of 3D Dirac semimetal. e, Schematic view of the Fermi surface above the Dirac point(left panel), at the Dirac point (middle panel) and below theDirac point (right panel). [Figures are adapted from M. Ne-upane et. al., Nature Commun. (2014) [40]]. even number of points per bulk BZ excludes the pos-sibility of the Z -type (Kane-Mele) topological insulatorphase in the Pb − x Sn x Te system under ambient pressure[1]. However, Hsieh et al noticed that any two of thefour L points along with the Γ point form a momentum-space mirror plane, making it possible to realize a noveltopological phase related to the crystalline mirror sym-metry in Pb − x Sn x Te [49]. Detailed theoretical analysisin Ref. [49] showed that the Pb − x Sn x Te system can the-oretically host a unique mirror symmetry protected TCIphase with a nontrivial topological invariant that is themirror Chern number n M , whereas the Z invariant ν equals to 0 for Pb − x Sn x Te showing the predicted TCIphase’s irrelevance to time-reversal symmetry. Therefore,the experimental identification of the mirror symmetryprotected TCI phase in Pb − x Sn x Te requires to not onlyobserve surface states within the bulk energy gap butalso find a way that can uniquely measure its topologicalnumber n M .It turns out that studying the surface state spin po-larization and its momentum-space texture chirality (orhandness) serves as keys to probing the role of topology inthe predicted TCI phase in Pb − x Sn x Te. This is becauseof the distinct property of its topological number, namelythe mirror Chern number n M . Unlike the Z invariant ν that can only be 0 or 1, the mirror Chern number n M cantake any integer value. While the absolute value of n M isdetermined by the number of surface states that dispersealong each momentum-space mirror direction, the signof n M is uniquely fixed by the chirality of the surfacestate spin texture [28, 49, 128]. Furthermore, becauseof the predicted four band inversions in Pb − x Sn x Te, itis in principle more favorable to study the inverted end-compound SnTe because it has the largest inverted band-gap. However, it has been known that SnTe is heavily p − type, due to the fact that Sn vacancies are thermody-namically stable [171], which makes the chemical poten-tial cut deeply inside the bulk valence bands [172] (seeFig. 16). Therefore, one needs to work with the system inthe Pb-rich (yet still inverted) regime, in order to accessand study the predicted surface states via photoemissionexperiments.Following the theoretical prediction in Ref.[49], ARPES experiments have been performed inPb − x Sn x Te, Pb − x Sn x Se and SnTe [36–38], and theexistence of Dirac surface states inside the bulk energygap has been observed in both Pb-rich Pb . Sn . Te[37] and Pb . Sn . Se [36] systems. More importantly, the helical spin texture and its chirality (or handness)have been systematically mapped out by spin-resolvedARPES experiments in Pb . Sn . Te.Utilizing spin-resolved angle-resolved photoemissionspectroscopy, Xu et al for the first time experimen-tally determined the topological mirror Chern numberof n M = − − x Sn x Te(Se), which experimentallyrevealed its topological mirror nontriviality of the TCIphase in the Pb − x Sn x Te(Se) system. The experimentaldata were reported in Ref. [37] and are summarized inFigs. 16-18. As shown in Fig. 17, two distinct surfacestates that cross the Fermi level are observed on the op-posite sides of each ¯ X point along the ¯Γ − ¯ X − ¯Γ directionat the (001) surface of Pb . Sn . Se. Therefore, in totalfour surface states are observed within one surface BZ,consistent with the predicted four bulk band inversions.All surface states are observed to locate along the mo-mentum space mirror line direction ¯Γ − ¯ X − ¯Γ, whichreflects the mirror symmetry protection to the observedtopological surface states. Each ¯Γ − ¯ X − ¯Γ mirror line pos-sesses two surface states, from which the absolute valueof the topological mirror Chern number of | n M | = 2 is de-termined. It is also interesting to notice that since thereare two surface Dirac cones that are located very close tonear each ¯ X point, they inevitably touch and hybridizewith each other, giving rise to a topological change in theband contours, also known as a Lifshitz transition in theelectronic structure. Such Fermi surface Lifshitz transi-tion is clearly observed in Fig. 17 d . At the energy wherethe Lifshitz transition happens, a saddle point type of vanHove singularity (VHS) is expected leading to the diver-gence of the density of states at the energy. Such VHSis also observed in our ARPES data shown in Fig. 17 f .Observation of saddle point singularity on the surface ofPb . Sn . Se paves the way for realizing correlated phys-ical phenomena in topological Dirac surface states.To uniquely determine the topological mirror Chernnumber n M , spin-resolved ARPES measurements wereperformed on the TCI surface states in Pb . Sn . Te asshown in Fig. 18. Fig. 18 h shows that four distinct spinpolarizations with the configuration of ↓ , ↑ , ↓ , ↑ are ob-served along the mirror line ¯Γ − ¯ X − ¯Γ. These mea-surements clearly identify the one to one helical spin-momentum locking in the surface states. More impor-tantly, the right-handed chirality of the spin texture isexperimentally measured for the lower-Dirac-cone states,which therefore determines the topological mirror Chernnumber of n M = −
2. These systematic measurementsreported in Ref. [37] for the first time conclusively iden-tified a novel mirror symmetry protected TCI phase bymeasuring its topological mirror number using a spin-sensitive probe.We present a comparison of the Pb . Sn . Te and asingle Dirac cone Z topological insulator (TI) systemGeBi Te [73, 84]. As shown in Fig. 18 a-d , for the Z TI system GeBi Te , a single surface Dirac cone is ob-served enclosing the time-reversal invariant Kramers’ mo-menta ¯Γ in both ARPES and calculation results, demon-0 FIG. 14.
In-plane dispersion in Cd As . a, Left: First principles calculation of the bulk electronic structure along the( π, π, . πc ∗ ) − (0 , , . πc ∗ ) direction ( c ∗ = c/a ). Right: Projected bulk band structure on to the (001) surface, where theshaded area shows the projection of the bulk bands. b, ARPES measured dispersion map of Cd As , measured with photonenergy of 22 eV and temperature of 15 K along the ( − π, − π ) − (0 , − ( π, π ) momentum space cut direction. c, ARPES constantenergy contour maps using photon energy of 22 eV on Cd As . d, ARPES constant energy contour maps using photon energyof 102 eV on Cd As . [This figure is adapted from M. Neupane et al., Nature Commun. (2014) [40]] strating its Z topological insulator state and the time-reversal symmetry protection of its single Dirac cone sur-face states. On the other hand, for the Pb . Sn . Te sam-ples (Fig. 18 e-h ), none of the surface states is observedto enclose any of the time-reversal invariant momentum,suggesting their irrelevance to the time-reversal symme-try related protection. With future ultra-high-resolutionexperimental studies to prove the strict gapless nature ofthe Pb . Sn . Te surface states and therefore their pre-dicted topological protection by the crystalline mirrorsymmetries, it is then possible to realize magnetic yettopologically protected surface states in the Pb − x Sn x Tesystem due to its irrelevance to the time-reversal symme-try related protection, which is fundamentally not possi-ble in the Z topological insulator systems.The experimental discovery of mirror-protected TCIphase in the Pb − x Sn x Te(Se) systems have attractedmuch interest in condensed matter physics and openedthe door for many further theoretical and experimentalstudies on this novel TCI phase [39, 90, 99, 173–190].These following works include scanning tunneling spec- troscopies [39, 173–175], thermal and electrical transport[176], further systematic studies on the surface spin andorbital textures [99, 177–179], as well as theoretical andexperimental efforts in realizing mirror symmetry pro-tected topological superconductivity [185–188] or mag-netic Chern current on the TCI surfaces [189, 190].
MAGNETIC AND SUPERCONDUCTING DOPEDTOPOLOGICAL INSULATORS
In this section, we review the photoemission studieson magnetic or superconducting topological insulators.The goal of the ARPES and spin-resolved ARPES stud-ies on magnetic topological insulators is to resolve themagnetic gap opened at the surface Dirac point as well asthe magnetically-driven spin texture near the gap edge.The magnetic gap and its spin texture are the keys torealizing the proposed novel effects based on a mag-netic topological insulator, including quantum anoma-lous Hall effect [70] and topological magneto-electrical1
FIG. 15.
Surface electronic structure of 2D and 3D Dirac fermions. a,
ARPES measured surface electronic struc-ture dispersion map of Cd As and its corresponding momentum distribution curves (MDCs). b, ARPES measured surfacedispersion map of the prototype TI Bi Se and its corresponding momentum distribution curves. Both spectra are measuredwith photon energy of 22 eV and at a sample temperature of 15 K. The black arrows show the ARPES intensity peaks in theMDC plots. c and d ARPES spectra of two Bi-based 3D Dirac semimetals, which are realized by fine tuning the chemicalcomposition to the critical point of a topological phase transition between a normal insulator and a TI: c, TlBi(S − δ Se δ ) ( δ = 0 .
5) (Xu et al. [34]), and (Bi − δ In δ ) Se ( δ = 0 .
04) (Brahlek et al. [66]) d, . Spectrum in panel c is measured with photonenergy of 16 eV and spectrum in panel d is measured with photon energy of 41 eV. For the 2D topological surface Dirac conein Bi Se , a distinct in-plane ( E B − k x ) dispersion is observed in ARPES, whereas for the 3D bulk Dirac cones in Cd As ,TlBi(S . Se . ) , and (Bi . In . ) Se , a Dirac-cone-like intensity continuum is also observed. [This figure is adapted from M.Neupane et al., Nature Commun. (2014) [40]] effect [69, 191, 192]. On the other hand, the goal ofARPES studies on superconducting topological insula-tors is to resolve the superconducting gap in the topo-logical surface states, which is predicted as a promisingplatform in realizing Majorana fermion modes [68, 193].We first focus on the research on magnetic topologi-cal insulators. Since the discovery of three dimensionaltopological insulators [1], topological order proximity toferromagnetism or superconductivity has been consideredas one of the core interest of the field [61, 62, 64, 65, 69,70, 191, 192, 194–196]. Such interest is strongly moti-vated by the proposed time-reversal (TR) breaking topo-logical physics such as quantized anomalous chiral Hallcurrent, spin current, axion electrodynamics, and inversespin-galvanic effect [69, 70, 191, 192, 194], all of whichcritically rely on finding a way to break TR symmetry toopen up a magnetic gap at the Dirac point on the surface and to further utilize the unique TR broken spin texturefor applications.Experimentally, a number of photoemission experi-ments have been performed in magnetically doped topo-logical insulators, in order to observe the energy gap atthe Dirac point opened by the breaking time-reversalsymmetry via magnetic doping. Although gap-like fea-ture at the Dirac point has been reported and inter-preted as the magnetic gap [35, 62], a number of otherfactors, such as spatial fluctuation of momentum andenergy near the Dirac point [97] and surface chemicalmodifications [62, 197], contribute to the observed gap[62, 78, 81, 97, 197, 198]. The photoemission probe pre-viously used to address the gap cannot distinguish orisolate these factors that respect TR symmetry from theTR breaking effect as highlighted in recent STM works[97]. In fact, photoemission Dirac point spectral sup-2 FIG. 16.
Band inversion transition and double Dirac surface states in Pb − x Sn x Te. a,
The lattice of Pb − x Sn x Tesystem is based on the “sodium chloride” crystal structure. The Pb-rich side of the Pb − x Sn x Te possesses the ideal “sodiumchloride” crystal structure without rhombohedral distortion. b, The first Brillouin zone (BZ) of Pb − x Sn x Te lattice. Themirror planes are shown using green and light-brown colors. These mirror planes project onto the (001) crystal surface as the¯X − ¯Γ − ¯X mirror lines. c, ARPES measured core level spectra (incident photon energy 75 eV) of two representative compositions,namely Pb . Sn . Te and Pb . Sn . Te. The photoemission (spin-orbit coupled) core levels of tellurium 4d, tin 4d, and lead5d orbitals are observed. d, The bulk band-gap of Pb − x Sn x Te alloy system undergoes a band inversion upon changing thePb/Sn ratio. A TCI phase with metallic surface states is theoretically predicted when the band-gap is inverted (toward SnTe)[49]. The Pb-rich inverted regime lies on the inverted compositional range yet still with Pb% > Sn% ( x inversion < x < / e,f, First-principles based calculation of band dispersion (e) and iso-energetic contour with energy set 0.02eV below the Dirac nodeenergy (f) of the inverted end compound SnTe as a qualitative reference for the ARPES experiments. The surface states areshown by the red lines whereas the bulk band projections are represented by the green shaded area in e . Adapted from S.-Y.Xu et al. , Nature Commun. , 1192 (2012) [37]. pression including a gap is also observed even on stoi-chiometric TI crystals without magnetic dopants or fer-romagnetism [81]. This is because surface can acquirenontrivial energy gaps due to ad-atom hybridization, sur-face top layer relaxation, Coulomb interaction from de-posited atoms, and other forms of surface chemistry suchas in situ oxidation [62, 78, 81, 97, 197, 198]. Under suchconditions, it was not possible to isolate TR breaking ef-fect from the rest of the extrinsic surface gap phenomena[62, 78, 81, 97, 197, 198]. Therefore, the establishment ofTR breaking effect fundamentally requires measurementsof electronic groundstate with a spin -sensitive probe.In Ref. [63], the authors utilized spin-resolved angle-resolved photoemission spectroscopy to measure the mo-mentum space spin configurations in systematically mag-netically doped, non-magnetically doped, and ultra-thin quantum coherent topological insulator films, in order tounderstand the nature of electronic groundstates undertwo extreme limits vital for magnetic topological devices.These measurements allow to make definitive conclusionsregarding magnetism on topological surfaces, and makeit possible to quantitatively isolate the TR breaking ef-fect in generating the surface electronic gap from manyother physical or chemical changes also leading to gap-like behavior [35, 78, 97, 198] often observed on the sur-faces. Spin reorientation measurements and the system-atic methodology demonstrated here can be utilized toprobe quantum magnetism on the surfaces of other ma-terials as well. Furthermore, following this spin-resolvedARPES work [63], surface magnetism mediated by thesurface Dirac fermions were again confirmed by trans-port experiments [64]. And very recently, the long-sought3 FIG. 17.
Observation of the topological crystalline surface states and saddle point singularity. a,
ARPES dispersionmaps upon in situ
Sn deposition on the Pb . Sn . Se surface. The dosage (time) for Sn deposition is noted. A different batchof sample, which is p − type with the chemical potential below the Dirac points, is used for the Sn deposition data shown in thispanel. b and c, Schematics of surface band dispersion of the TCI phase along the mirror line ¯Γ − ¯X − ¯Γ and the ¯M − ¯X − ¯Mmomentum space cut-directions. Five important features of the surface states, including Dirac point of the upper part of theDirac cones (UDP), van Hove singularity of the upper Dirac cones (VH1), two Dirac points along the ¯Γ − ¯X − ¯Γ mirror line(DP), van Hove singularity of the lower part of the Dirac cones (VH2) and Dirac point of the lower part of the Dirac cones(LDP) are marked. c, Calculated density of state (DOS) for the surface states and the bulk bands using the k · p model. FIG. 17. (Previous page.) d, Experimental observation of the Lifshitz transition - the binding energies are noted on theconstant energy contours. e, ARPES measured dispersion plots along ¯Γ − ¯X − ¯Γ and ¯M − ¯X − ¯M. f, Momentum ( k x and k y ) integrated ARPES intensity as a function of binding energy (left). 2 nd derivative of the ARPES intensity with respect tobinding energy is presented to further highlight the features. The upper Dirac point (UDP), upper van Hove singularity (VH1),Dirac point (DP), lower van Hove singularity (VH2) and lower Dirac point (LDP) are marked. Adapted from M. Neupane etal. , Preprint at http://arXiv:1403.1560 (2014) [90].FIG. 18. The topological distinction between Z (Kane-Mele) topological insulator and topological crystallineinsulator phases. (a-d) ARPES, spin-resolved ARPES and calculation results of the surface states of a Z topological insulatorGeBi Te [73], an analog to Bi Se [29]. (a) ARPES measured Fermi surface with the chemical potential tuned near the surfaceDirac point. (b) First-principles calculated iso-energetic contour of the surface states near the Dirac point. The solid blueline shows the momentum-space cut used for spin-resolved measurements. Right: A stack of ARPES iso-energetic contoursnear the ¯Γ point of the surface BZ. (d) Measured spin polarization of Bi Se , in which a helical spin texture is revealed. (e-h)ARPES and spin-resolved ARPES measurements on the Pb . Sn . Te ( x = 0 .
4) samples and band calculation results on theend compound SnTe [49]. (e) ARPES measured Fermi surface map of Pb . Sn . Te. (f) First-principles calculated iso-energeticcontour of SnTe surface states near the Dirac point. The solid blue line shows the momentum-space cut near the surface BZedge center ¯ X point, which is used for spin-resolved measurements shown in panel (h). (g) A stack of ARPES iso-energeticcontours near the ¯ X point of the surface BZ, revealing the double Dirac cone contours near each ¯ X point on the surface ofPb . Sn . Te. (h) Measured spin polarization of Pb . Sn . Te near the native Fermi energy along the momentum space cutdefined in panel (f), in which two spin helical Dirac cones are observed near an ¯ X point. [Adapted from S.-Y. Xu et al. , NatureCommun. , 1192 (2012). [37]] quantum anomalous Hall currents have been observed inmagnetically-doped topological insulator thin films [65].Fig. 19 presents the key spin-resolved measure-ments on magnetically-doped topological insulator thinfilms, which reveals the exotic time-reversal breaking(hedgehog-like [63]) spin texture near the edge of themagnetic gap. Fig. 19 a shows a hysteretic measure-ment using x-ray circular dichroism in the out-of-planedirection, which suggests a ferromagnetically orderedgroundstate mediated by the surface Dirac fermions [64].Fig. 19 b shows the out-of-plane spin polarization ( P z )measurements of the electronic states in the vicinityof the Dirac point gap of a Mn(2.5%)-Bi Se sample. The surface electrons at the time-reversal invariant ¯Γpoint (red curve in Fig. 19 b ) are clearly observed tobe spin polarized in the out-of-plane direction. The op-posite sign of P z for the upper and lower Dirac bandshows that the Dirac point spin degeneracy is indeedlifted up (E( k // = 0 , ↑ ) =E( k // = 0 , ↓ )), which mani-festly breaks the time-reversal symmetry on the surface ofour Mn(2.5%)-Bi Se samples. Systematic spin-resolvedmeasurements as a function of binding energy and mo-mentum reveal a Hedgehog-like spin texture (inset ofFig. 19 b ). As demonstrated recently [34], the quan-tum Berry’s phase (BP) defined on the spin texture ofthe surface state Fermi surface bears a direct correspon-5 FIG. 19.
Hedgehog spin texture and Berry’s phase tuning in a magnetic topological insulator. (a) Magnetizationmeasurements using magnetic circular dichroism shows out-of-plane ferromagnetic character of the Mn-Bi Se MBE film surfacethrough the observed hysteretic response. The inset shows the ARPES observed gap at the Dirac point in the Mn(2.5%)-Bi Se film sample. (b) Spin-integrated and spin-resolved measurements on a representative piece of Mn(2.5%)-Bi Se film sampleusing 9 eV photons. Left: Spin-integrated ARPES dispersion map. The blue arrows represent the spin texture configurationin close vicinity of the gap revealed by our spin-resolved measurements. Right, Measured out-of-plane spin polarization as afunction of binding energy at different momentum values. The momentum value of each spin polarization curve is noted on thetop. The polar angles ( θ ) of the spin polarization vectors obtained from these measurements are also noted. The 90 ◦ polar angleobserved at ¯Γ point suggests that the spin vector at ¯Γ is along the vertical direction. The spin behavior at ¯Γ and its surroundingmomentum space reveals a hedgehog-like spin configuration for each Dirac band separated by the gap. Inset shows a schematicof the revealed hedgehog-like spin texture. (c) Measured surface state dispersion upon in situ NO surface adsorption on theMn-Bi Se surface. The NO dosage in the unit of Langmuir (1L = 1 × − torr · sec) and the tunable Berry’s phase (BP)associated with the topological surface state are noted on the top-left and top-right corners of the panels, respectively. The redarrows depict the time-reversal breaking out-of-plane spin texture at the gap edge based on the experimental data. (d) Thetime-reversal breaking spin texture features a singular hedgehog-like configuration when the chemical potential is tuned to liewithin the magnetic gap, corresponding to the experimental condition presented in the last panel in panel (c). [Adapted fromS.-Y. Xu et al. , Nature Physics , 616 (2012). [63]]. dence to the bulk topological invariant realized in thebulk electronic band structure via electronic band inver-sion [31, 34]. We experimentally show that a BP tun-ability can be realized on our magnetic films which isimportant to prepare the sample condition to the ax-ion electrodynamics limit. On the Mn-Bi Se film, spinconfiguration pattern can be understood as a competi-tion between the out-of-plane TR breaking componentand the in-plane helical component of spin. The in-planespin that can be thought of winding around the Fermisurface in a helical pattern contributes to a nonzero BP[31], whereas the out-of-plane TR breaking spin directionis constant as one loops around the Fermi surface hence does not contribute to the Berry’s phase (BP). Such ex-otic spin groundstate in a magnetic topological insulatorenables a tunable Berry’s phase on the magnetized topo-logical surface [63], as experimentally demonstrated byour chemical gating via NO surface adsorption methodshown in Figs. 19 c,d .The interplay between the topological order and super-conductivity may lead to many proposals of novel quan-tum phenomena such as time-reversal invariant topolog-ical superconductors [68, 193, 199], Majorana fermions[68, 193, 199], and fault-tolerant quantum computation[193, 199]. Currently, researchers have been focused ontwo approaches to introduce superconductivity into the a6 FIG. 20.
Superconducting doped topological insulator. (a) Topologically protected surface states cross the Fermi levelbefore merging with the bulk valence and conduction bands in a lightly doped topological insulator. (b) If the superconductingwavefunction has even parity, the surface states will be gapped by the proximity effect, and vortices on the crystal surfacewill host braidable Majorana fermions. (c) If superconducting parity is odd, the material will be a so-called topologicalsuperconductor, and new states will appear below T c to span the bulk superconducting gap. (d) Majorana fermion surfacevortices are found at the end of bulk vortex lines and could be manipulated for quantum computation if superconducting pairingis even. [Adapted from L. Wray et al ., Nature Phys. , 855 (2010). [57]] TI. The first approach is to bulk dope a TI material in or-der to make it a bulk superconductor. The most notableexample is the bulk superconductivity with T c ∼ . . Bi Se .The second approach is to utilize the superconductingproximity effect by interfacing a TI with a superconduc-tor. We review the ARPES studies for both approachesas follows.The bulk superconductivity reported in copper-dopedbismuth selenide Cu x Bi Se [56] has attracted much in-terests [56, 57, 200–202]. A major contribution made byARPES measurements [57] is that ARPES shows that thetopological surface states remain well defined and non-degenerate with bulk electronic states at the Fermi levelof optimally doped superconducting Cu . Bi Se . Thisobservation is important for the following reasons: Sincethe bulk is superconducting, then it is possible to use thenatural proximity effect between the bulk and surface toinduce superconductivity on the surface of Cu . Bi Se .The superconductivity in these spin-helical Dirac sur-face states can realize a 2D topological superconductor.And Majorana fermion bound states may exist in themagnetic vortices at the surface schematically shown in Fig.20. However, this exciting scenario is only possibleif the surface states are non-degenerate with the bulkbands at the Fermi level. Because otherwise the surfaceand bulk superconductivity are strongly coupled and theMajorana fermion bound state trapped in a surface vor-tex can leak to the bulk, causing decoherence and annihi-lation of the Majorana fermion. Therefore, the ARPESobservation of non-degenerate nature of the surface statesat the Fermi level of optimally doped superconductingCu . Bi Se reported in [57] serves as the key for re-alizing topological superconductivity on the surface ofsuperconducting Cu . Bi Se .The superconductivity physics in Cu x Bi Se can beeven richer. In Ref. [199], the authors proposed thetheoretical possibility that the bulk superconductivity inCu x Bi Se may also be topologically nontrivial. If theintra- and inter-orbital hopping parameters lie in an ap-propriate regime, theory in Ref. [199] shows that theCu x Bi Se system is a bulk odd-parity topological su-perconductor, and one would expect helical Majoranasurface states. However, the nature of the bulk super-conductivity is still controversial. Although zero-biaspeak in a point contact experiment has been reported7and interpreted as the signature for the bulk topologicalsuperconductivity, high-resolution ARPES [201] in factdid not resolve any observable superconducting gap nei-ther in the bulk bands nor in the surface states, and STMmeasurements [202] suggest the pairing Cu x Bi Se seemsto be conventional (topologically trivial). Furthermore,recent theoretical and experimental studies suggest thatthe nature of the zero-bias peak can be very complex[203–206], which therefore cannot serve as conclusive sig-nature for the topological superconductivity or the Ma-jorana fermions.As for the superconducting proximity effect approach,there have been many transport and STM studies onthis topic [58, 207–219]. However, due to the lack ofmomentum and spin-resolution of transport and STM,these experiments cannot show that the topological sur-face states are indeed superconducting since topologicalsurface states, bulk bands, and potentially trivial surfacestates or impurity states all contribute to the transportor STM signals. ARPES studies on TI/superconductorproximity effect samples are on the other hand very lim-ited and under debate [59, 60]. Although extensive exper-imental efforts are underway, critical signatures regarding the observation of unambiguous Majorana mode are stilllacking. Without the demonstration of helical Cooperpairing in the topological (spin only) Dirac surface states(which can presently be done only via ARPES thanks toits momentum and spin -resolution and buried interfacesensitivity), critical evidence for time-reversal invarianttopological superconductivity (TRI-TSC) is still lacking. Acknowledgement
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