Studies on the electronic structures of three-dimensional topological insulators by angle resolved photoemission spectroscopy

Abstract

Three-dimensional (3D) topological insulators represent a new state of quantum matter with a bulk gap and odd number of relativistic Dirac fermions on the surface. The unusual surface states of topological insulators rise from the nontrivial topology of their electronic structures as a result of strong spin–orbital coupling. In this review, we will briefly introduce the concept of topological insulators and the experimental method that can directly probe their unique electronic structure: angle resolved photoemission spectroscopy (ARPES). A few examples are then presented to demonstrate the unique band structures of different families of topological insulators and the unusual properties of the topological surface states. Finally, we will briefly discuss the future development of topological quantum materials.

Graphical abstract

Keywords

topological insulator (TI) / photoemission / ARPES

1 Introduction to topological insulators

In condensed matter physics, one of the central topics is to study why a handful of elements on earth can make up of thousands and thousands of substances with vastly different properties. To work on such a large quantity of materials, classification of them is thus of great importance. Until recently, different states of matter are usually classified by the spontaneous breaking of symmetries [1]. As examples, a crystal breaks translation symmetry; a magnet breaks rotation symmetry; a superconductor breaks the more subtle gauge symmetry.

The deficiency of this classification was first realized in 1980, when the quantum Hall (QH) state, a new state of matter that does not have spontaneous broken symmetry was discovered [2]. In this state, the bulk of the two-dimensional (2D) material is insulating, while the dissipationless chiral electric current can flow only along the sample edge, giving rise to a quantized Hall effect. The topological nature of the QH state soon came to light [3, 4] and a topological invariant, the first Chern number defined by the global properties of all occupied Bloch states of the materials can be used to characterize the state, making the QH state topologically distinct from all previously known states of matter.

Recently, a new class of topological state, Z₂ topological insulators (TIs), have emerged. TIs represent a novel quantum matter with a bulk gap and odd number of relativistic Dirac fermions on the surface [5–7]. Similar to the QH state, this state can also be characterized by a topological invariant, Z₂ number [6–8]; but different from the QH state where the Chern number characterizing the topological order can have any integer values, the Z₂ number of TIs can possess only two values: 0 or 1. The value 0 indicates that the material’s electronic structure is similar to a “normal” insulator, which is topologically equivalent to vacuum; and the value 1 indicates that the material is topologically non-trivial, forming a new state topologically distinct from all other known states of matter, including the QH state.

The prediction of this new state of quantum matter immediately inspired intensive studies in the field of condensed matter physics [5–28] including both theoretical and experimental efforts.

The 2D TI state, or the quantum spin Hall (QSH) state, was first investigated based on graphene [9]. However, it was soon realized that the spin-orbital coupling in graphene was too small to make the experimental observations feasible [29, 30]. Before long, a more realistic material system was proposed in the HgTe quantum well [10]. Shortly after the theoretical prediction [10], the QSH state was observed by electric transport experiments in the HgTe quantum well [11]. In the thick HgTe quantum well where the QSH state exist, there is an insulating gap in the bulk and gapless states at the edge where electrons with opposite spins counter-propagate [Fig. 1(a)], forming a single massless Dirac fermion at the edge [Fig. 1(b)]. The crossing of their dispersion branches at a time reversal invariant point (Dirac point) is protected by the time reversal symmetry (TRS) [6–8].

Soon the TI state was generalized to three-dimension (3D) [7, 18, 21, 22]. In a 3D TI, there exists an insulating bulk energy gap, but electrons can flow on its 2D surface [Fig. 1(c)], with their spin polarization locked to the momentum [Fig. 1(c)], forming a cone-shape dispersion with helical spin structure [Dirac cone, see Fig. 1(d)]. The spin degenerate points of these Dirac cones (Dirac points) reside at the time reversal invariant points in k-space [7, 18, 21, 22]. Up to date, several families of 3D TIs including tens of materials have been theoretically predicted, such as the Bi_1−xSb_x alloy (x=0.07\sim 0.22) [6]; {\mathrm{V}}_2\text{–}{\mathrm{VI}}_3 binary compounds (Bi₂Te₃, Bi₂Se₃ and Sb₂Te₃) [13], \mathrm{I}\mathrm{I}\mathrm{I}\text{–}\mathrm{V}\text{–}{\mathrm{VI}}_2 ternary compounds ({\mathrm{TlBiSe}}_2) [27, 31] and many Heusler alloys [32, 33]. Experimental efforts soon followed and proved many of the predictions.

Unlike in the 2D QSH state, where the edge current flows strictly dissipationless and make the electric transport experiments most effective in determining its topological nature [11], the transport measurements can hardly determine the 3D TI materials. It is not only because the contribution from the surface state is smaller in the conductivity and it is hard to distinguish the bulk and surface contributions; but also because the 2D surface state of 3D TIs does not lead to complete suppression of electric resistance as in the 2D QSH states due to small angle scatterings [7]. Thus an experimental method is needed which can measure the electronic structure and directly separate the bulk and surface contributions.

With these requirements, angle-resolved photoemission spectroscopy (ARPES) stands as a natural choice due to its capability of directly observing electronic band structure in solids. Because of the suitable mean free path of photoelectrons (1\sim 10 nm), the ARPES signal comes primarily from the top most unit-cells of the materials (Fig. 2), thus making the surface state contribution prominent in the total spectra. In addition, by performing photon-energy dependent measurements (Fig. 3), ARPES can easily separate the bulk and surface band structures because of their different k_z dispersions. These advantages, together with its in plane k-space resolution, overcome the disadvantages encountered in the transport measurements as discussed above, making ARPES the most important method for studying the unique electronic structure of 3D TIs [12, 14–16, 25]. Furthermore, the ability of coupling the spin-detector to ARPES spectrometer makes it suitable to study the non-trivial helical spin-structure of TIs.

In the following sections, we will first briefly review this powerful experimental tool, and then present a few examples of the application of ARPES on 3D TIs.

2 Angle resolved photoemission spectroscopy

2.1 Basic concept

Angle resolved photoemission spectroscopy is a direct experimental technique to observe the distribution of electrons in solid, or more precisely, the density of single particle electronic excitations in the reciprocal (momentum, or k-) space. With the high angular and energy resolutions now achievable, this technique can measure the electronic structure with great precision, and thus represents the best technique choice to visualize the energy and momentum phase space of electrons.

The discovery of photoemission process can be traced back to 1839 when Alexandre Edmond Becquerel observed the photoelectric effect via an electrode in a conductive solution exposed to light [35]. In 1887, Hertz further observed the photoelectric effect along with the production and reception of electromagnetic (EM) waves [36]. But the underlined principle was not well understood until 1905 when Albert Einstein introduced the concept of light quanta – Photon [37]. A schematic of photoemission experiment is shown in Fig. 4, where we send in energetic photons to excite the electrons out of the sample, then record its emission directions and kinetic energy with an electron analyzer.

The energetics of the photoemission process is shown in Fig. 5. In a crystal, electrons form energy bands (usually called core or valence bands depending on their energy levels), and can have non-uniform distribution of electron density of states. To study these states by photoemission process, we need impinge photons (with quantized energy E=h\nu) upon the sample surface to excite electrons out before they can be detected by an electron analyzer. Due to the existence of a potential barrier \Phi between the Fermi-energy E_{\mathrm{F}} and vacuum level E_{\mathrm{vac}}, the condition h\nu >\Phi must be satisfied. By measuring the photoelectron kinetic energy E_{\mathrm{k}}, we can extract the electrons’ original binding energy E_{\mathrm{b}}. From Fig. 5, we see the following relationship immediately as a result of energy conservation:

\begin{array}{}\text{(1)}& E_{\mathrm{b}}=h\nu -\Phi -E_{\mathrm{k}}\end{array}

With additional information on the photoelectrons’ emission angle with respect to the sample surface, we can extract the electron momentum, both parallel (k_{//}) and perpendicular (k_{\perp }) to the sample surface. Due to the existence of work function \Phi, an electric field exists along the sample surface normal, thus only k_{//} is conserved during the photoemission process, i.e.,

\begin{array}{}\text{(2)}& k_{//\mathrm{photoelectron}}=k_{//\mathrm{sampleelectron}}\end{array}

Equations (1) and (2) show that ARPES is an ideal method for studying materials with 2D electronic structure; if it is apply to 3D materials, the work function \Phi and the details of the surface field have to be understood before we reconstruct the sample’s 3D electronic structure from ARPES data.

2.2 General principle

What we have discussed above is the energetics of the photoemission process; below we will give a brief description of the photocurrent we collect in the photoemission experiment [38].

The photoemitted electric current comes from the excitation of electrons from the initial state |{\psi }_i⟩ to the final state |{\psi }_f⟩ by the introduced photon field (with vector potential A) associated with the interaction Hamiltonian H_{\mathrm{int}}=P\cdot A, thereby the transition probability P can be computed by Fermi’s Golden Rule or the first Born approximation. In the diploe approximation [76], we get

\begin{array}{}\text{(3)}& P\propto {\displaystyle \frac{2\pi }{\mathrm{\hslash }}}{|⟨{\psi }_f|H_{\mathrm{int}}|{\psi }_i⟩|}^2\delta (E_f-E_i-\mathrm{\hslash }\omega )\end{array}

To study the transition matrix element above, we have to make some assumptions about the wavefunction. The simplest approximation is to use the one-electron wavefunction for the initial and final states. For the final state, we have an additional free electron with the energy E_k, accordingly for a system with N electrons, we can write the initial states as:

\begin{array}{}\text{(4)}& |{\psi }_i⟩=|{\varphi }_{i,k}⟩|{\psi }_{i,R(N-1)}^k⟩\end{array}

where |{\varphi }_{i,k}⟩ is the orbital from which the photoemitted free electron is excited, and |{\psi }_{i,R(N-1)}^k⟩ represents the remaining electrons.

In the same manner, the final state can be written as a product of the photoemitted electron and that of the remaining N-1 electrons:

\begin{array}{}\text{(5)}& |{\psi }_f⟩=|{\varphi }_{f,E_k}⟩|{\psi }_{f,R(N-1)}^k⟩\end{array}

Therefore, the transition matrix element becomes

\begin{array}{}\text{(6)}& \begin{array}{cc}⟨{\psi }_f|H_{\mathrm{int}}|{\psi }_i⟩\hfill & =⟨{\psi }_{f,R(N-1)}^k⟨{\varphi }_{f,E_{\mathrm{k}}}|H_{\mathrm{int}}|{\varphi }_{i,k}⟩{\psi }_{i,R(N-1)}^k⟩\hfill \\ \hfill & =⟨{\varphi }_{f,E_{\mathrm{k}}}|H_{\mathrm{int}}|{\varphi }_{i,k}⟩⟨{\psi }_{f,R(N-1)}^k|{\psi }_{i,R(N-1)}^k⟩\hfill \end{array}\end{array}

which is a product consisting of a one-electron matrix element and an (N-1) electron overlap integral. If, for simplicity, we assume that the remaining (N-1) orbitals are the same for the initial and final state (the frozen-orbital or sudden approximation), i.e., |{\psi }_{i,R(N-1)}^k⟩=|{\psi }_{f,R(N-1)}^k⟩, Then ⟨{\psi }_{f,R(N-1)}^k|{\psi }_{i,R(N-1)}^k⟩=1 and the transition matrix element ⟨{\psi }_f|H_{\mathrm{int}}|{\psi }_i⟩=⟨{\varphi }_{f,E_k}|H_{\mathrm{int}}|{\varphi }_{i,k}⟩, which is just the one-electron matrix element. Under this assumption, the PES experiment measures the negative Hartree–Fock orbital energy, and the experimental spectra will be given by the delta functions [Fig. 6(b)] that trace the real band dispersion [Fig. 6(a)].

As the TIs currently under investigations are mostly weakly interacting systems, the single particle picture discussed above can be applied well. The ARPES results then on the whole represent the real band structure of these TI materials.

In other materials where the electron–electron interaction is strong, however, the situation is more complicated. After the photoemission, the remaining (N-1) electron system is not necessarily left in its ground state, thus the system will readjust its charge distribution and minimize its energy. Let us assume the final (N-1) electrons has S excited states, |{\psi }_{f,S(N-1)}^k⟩, with energy E_{S(N-1)}, the probability that the system is remained in the excited S is {|⟨{\psi }_{f,S(N-1)}^k|{\psi }_{i,R(N-1)}^k⟩|}^2, thus the total photocurrent measured by the experiment is the sum over the contributions of all the possible final states:

\begin{array}{}\text{(7)}& \begin{array}{ccc}I\hfill & \propto \hfill & {\displaystyle \sum _{f,i,k}}{|⟨{\varphi }_{f,E_{\mathrm{k}}}|H_{\mathrm{int}}|{\varphi }_{i,k}⟩|}^2\hfill \\ \hfill & \hfill & \cdot {\displaystyle \sum _S}|⟨{\psi }_{f,S(N-1)}^k|{\psi }_{i,R(N-1)}^k⟩{|}^2\hfill \\ \hfill & \hfill & \cdot \delta (E_{f,E_{\mathrm{k}}}+E_{S(N-1)}-E_{0(N)}-\mathrm{\hslash }\omega )\hfill \\ \hfill & \triangleq \hfill & {\displaystyle \sum _{f,i,k}}{|⟨{\varphi }_{f,E_{\mathrm{k}}}|H_{\mathrm{int}}|{\varphi }_{i,k}⟩|}^{2}A(k,E)\hfill \end{array}\end{array}

where

\begin{array}{}\text{(8)}& \begin{array}{cc}A(k,E)=\hfill & {\displaystyle \sum _S}{|⟨{\psi }_{f,S(N-1)}^k|{\psi }_{i,R(N-1)}^k⟩|}^2\hfill \\ \hfill & \cdot \delta (E_{f,E_{\mathrm{k}}}+E_{S(N-1)}-E_{0(N)}-\mathrm{\hslash }\omega )\hfill \end{array}\end{array}

is the spectral function for the wave number k and energy E which describes the probability of removing () or adding (E>E_{\mathrm{F}}) an electron with the energy E and wave vector k from or to the interacting N-electron system. In this case, the PES spectra will become more complicated and the δ function like PES peaks will be accompanied with the satellite peaks [Fig. 6(c)].

By definition, the spectra function A(k,E) can be related to the single-particle Green’s function by

\begin{array}{}\text{(9)}& A(k,E)={\displaystyle \frac{1}{\pi }}|\mathrm{Im}(G(k,E))|\end{array}

For a non interacting system with the one-electron energy E_k^0, the Green’s function is the following:

\begin{array}{}\text{(10)}& G_0(k,E)=\underset{ϵ\to 0}{lim}{\displaystyle \frac{1}{E-E_k^0-\mathrm{i}ϵ}}\end{array}

thus

\begin{array}{}\text{(11)}& A_0(k,E)=\delta (E-E_k^0)\end{array}

which again tells us that the spectral function is a δ fucntion at E=E_k^0, agreeing with our previous discussion on the non-interacting systems.

In an interacting system, however, the electron energy gets renormalized by the self energy \mathrm{\Sigma }(k,E), which makes the Green’s function becomes

\begin{array}{}\text{(12)}& G(k,E)={\displaystyle \frac{1}{E-E_k^0-\mathrm{\Sigma }(k,E)}}\end{array}

and

\begin{array}{}\text{(13)}& A(k,E)={\displaystyle \frac{1}{\pi }}\left|{\displaystyle \frac{\mathrm{Im}[\mathrm{\Sigma }(k,E)]}{{\{E-E_k^0-\mathrm{Re}[\mathrm{\Sigma }(k,E)]\}}^2+{\{\mathrm{Im}[\mathrm{\Sigma }(k,E)]\}}^2}}\right|\end{array}

Therefore, for the study of future TI materials with strong correlation, such as the Ir-related compounds recently proposed [39, 40], the more sophisticated treatment on the ARPES spectra as above should be applied before the real band structure can be deduced from the experimental measurements.

2.3 Experimental setup for angle resolved photoemission spectroscopy

The light sources for ARPES vary a lot in forms and in the experimental side, a basic photoemission ARPES station usually comprises the following four elements: i) Vacuum system, ii) Photon source, iii) Electron analyzer, iv) Sample manipulating/cooling system. Figure 7 shows a typical ARPES spectrometer. There can also be other accessaries attached to the chamber for sample surface preparation, but the four parts mentioned above are the necessary ones for all ARPES spectrometer. In the following, we will briefly introduce the function of each part.

(i) Vacuum system

The vacuum system is the housing of the whole spectrometer, which usually consists of a vacuum chamber and the supporting pump system. Reducing the residue gas molecules in the chamber is necessary as they not only contaminate the sample surface, but also scatter the photoelectrons and prevent them from reaching the detector.

Depending on the materials studied, the vacuum requirement for photoemission measurements can be very different, ranging from {10}^{-7} Torr for regular gas phase to {10}^{-11} Torr for solid state samples so as to allow enough data collection time before the sample surface is contaminated by adsorpted residue gas molecules.

(ii) Light sources

A light source is required for any ARPES spectrometer. It produces photons for exciting the photoemission process. For gas phase photoemission experiments, the energy of the photons must exceed the lowest binding energy; while for the solid state ARPES measurements, it must be larger than the working function of the materials. The light sources for ARPES vary much in forms and operating principles. Commonly used sources include X-ray tube [41], gas discharge lamp [42], synchrotron radiation [43] and laser [44].

Due to its brightness and the capability of adjusting photon energy in a broad range (from a few eVs to hundreds of eVs), synchrotron-radiation became the most popular light source. A synchrotron has a storage ring for electrons and several bending magnets or undulators. Since the charged particle emit electromagnetic radiation when accelerated [45], people purposely bend the trajectory (by magnet or undulator) of electrons in the storage ring to get the electromagnetic radiation (A more detailed description of synchrotron can be found in Ref. [46].).

(iii) Electron analyzer

After the electrons in the sample are photoemitted, an electron analyzer collects them and measures their energy and emission direction for extracting the electron momentum. At present, 2D electrostatic spherical deflection analyzer (SDA) is the most commonly used analyzer in most ARPES stations, which can acquire two-dimensional information in a single measurement.

The schematics of a typical SDA analyzer is shown in Fig. 8, which consists of two concentric hemispheres. For transmission of electrons with initial energy E₀ along the central path with R_0=(R_{\mathrm{in}}+R_{\mathrm{out}})/2, the potential of the outer hemisphere has to be V_{\mathrm{out}}=E_0[3-2/(R_0/R_{\mathrm{out}})]/e, and that of the inner hemisphere V_{\mathrm{in}}=E_0[3-2/(R_0/R_{\mathrm{in}})]/e. For electrons with energy slightly higher/lower than E₀, the trajectory will be closer to the outer/innner hemisphere, respectively. Thus the electrons of different energy will disperse along the longitudinal direction of the detector [Fig. 8(a)]. Almost all SDA analyzers use an additional pre-retardation stage prior to the energy analysis, enabling the analyzer to measure a larger energy range and improve the resolution for high energy electrons (Likewise, after the pre-retardation, a lower pass energy mode can be used.).

In addition to the energy detection, modern SDA analyzer can also collect electrons’ angle information by focusing electrons from different emission angle to different spacial points. As can be seen in Fig. 8(a), where the front retardation lens can preserve the angular information of the incoming electrons, which will arrive at different lateral positions of the detector. As the angle and energy dispersion of electrons on the detector are perpendicular, a single measurement of the SDA analyzer can acquire a 2D dispersion of the band structure.

Figure 8(b) shows a real SDA analyzer (Scienta R4000) in which the hemispherical lens cover and the pre-lens in front can be clearly seen. This is the analyzer we use the most for the ARPES study that we will present in the latter part of this review.

(iv) Sample manipulating/colling system

Consider that a typical SDA electron analyzer only gets a 2D dispersion cut during each measurement, to acquire the 3D electronic structure, we have to sample the data along the 2nd direction perpendicular to the lateral direction of the 2D SDA detector (the \Phi angle, see right panel of Fig. 7). It is obvious that if the sampling step of the \Phi angle becomes smaller, we can get finer momentum information along this direction – at the cost of longer data acquisition time. The right panel of Fig. 7 shows a typical set of ARPES data from HERS that directly visualize the electron structures in the sample.

To realize the sampling in \Phi direction, one has two choices: rotating the sample or rotating the analyzer. Rotating the sample seems a much easier way (as it is much smaller than the analyzer) and it is in fact more commonly adopted. However, it has some disadvantages: First, as the relative orientation of the sample and light beam is changed during this rotation, the matrix elements [Eq. (6)] also change, making it harder to interpret the data. Second, for samples with multiple domains or rough surface, the light beam can be blocked by extruded parts on the surface when the sample rotates. Third, due to the inevitable error of the concentricity of the manipulator, for small samples and multi-domain samples, rotation of the sample often causes the photon beam to miss the sample or to shine at different domains.

To overcome these disadvantages, sometimes we need to adopt the rotating analyzer scheme, even though it is technically harder due to the large size of the analyzer and the difficulty to maintain the ultra-high vacuum during the rotation. Nevertheless, we managed to address these technical issues and realize it in HERS endstation.

Finally, as the high resolution ARPES measurements have to be performed at low temperature to reduce the thermal broadening effect (\sim k_{\mathrm{B}}T) [47], a sample cooling system (that can reach the temperature region where k_{\mathrm{B}}T is comparable to the analyzer energy resolution) is also needed. Obviously, in addition to reducing the temperature broadening, the cooling system will also be helpful in the study of low temperature electronic structures.

3 Electronic structure of 3D topological insulators (TIs)

After the prediction of 3D TI materials, ARPES was soon applied to these candidates and yielded fruitful results [5–28]. Many of the materials have been confirmed to be 3D TIs with their unique electronic structure with an odd number of Dirac fermions on the surface with helical spin structure protected by the time reversal symmetry. Furthermore, the rich information acquired by ARPES has helped the understanding of other experimental techniques such as electric transport, STM. In the following, a few examples are presented to demonstrate the rich unusual properties of 3D TIs.

3.1 Bi_xSb_1-x alloy

Bi_1-xSb_x alloy was first proposed to be a 3D TI within the doping range in the pioneering work on the band theory of 3D TIs [6], as is illustrated in Fig. 9 that summarizes the evolution of the Bi_1-xSb_x band structure.

Pure Bismuth is a semimetal with strong spin-orbital coupling, and the overlap between the bands around L points and that at T point leads to the simultaneous existence of electron and hole pockets at E_{\mathrm{F}} (left most axis of Fig. 9). The substitution of Bismuth by Antimony results in the downshift of the T and L_{\mathrm{s}} bands and the upshift of the L_{\mathrm{a}} band, thus the direct bulk gap at the L points diminishes with the Antimony concentration and eventually vanishes at (x\approx 0.04). Further Antimony doping makes the gap turn negative, but the bulk semimetalicity remains since the T band is still above the L bands. When x\gtrsim 0.07, the T band falls below the L_{\mathrm{s}} band, turning the system to a semiconductor with an inverted band gap between L_{\mathrm{a}} and L_{\mathrm{s}} bands. The semiconductor state persists till x\approx 0.22, when another band around H point crosses the L_{\mathrm{s}} band, converting the system back to a semimetal.

Due to the inverted band structure and the opening of the bulk gap, Bi_1-xSb_x was predicted [6] to be a (strong) 3D TI in the semiconductor region () with an odd number of the E_F crossings of its surface state between the two time invariant point Γ and M.

The ARPES experiments were soon carried out [12] (Fig. 10), and the surface state bands of Bi_0.9Sb_0.1 was shown to cross E_{\mathrm{F}} five times [The E_{\mathrm{F}} crossing points 4 and 5 are seen in the plot of the second derivative of the ARPES intensity spectra, Fig. 10(b).]; the subsequent spin resolved measurements [48, 76] also indicated the non-degenerate spin polarization of the surface state (at least at the E_{\mathrm{F}} corssings 1–3), and the bulk gap of the studied sample was estimated to be \sim 50 meV at the M point of the surface Brillouin zone (BZ).

Although the Bi_1-xSb_x alloy shows an odd number of Fermi-crossings, its complicated band structure [five Fermi-crossings in total, with that of 4, 5 very close to each other thus making the spin signature vanish in the measurement shown in Fig. 10(d)] and the small energy gap (\sim 50 meV) made it highly desirable to search for simpler 3D TI materials with the signature of Dirac fermions explicit (ideally with a single Dirac fermion on the surface – similar to the simplicity of Hydrogen atom in atomic physics) and a larger bulk gap. Materials with these nice characters will facilitate not only the realization of many non-trival topological phenomena, but also their potential applications.

Under this circumstance, a family of single dirac cone TIs with a large bulk gap in V₂–VI₃ materials were soon put on the table.

3.2 Single Dirac cone TIs from V₂–VI₃ compounds

After theoretical analysis and ab initio calculations, three layered materials Bi₂Te₃, Bi₂Se₃ and Sb₂Te₃ from the V₂–VI₃ family were proposed to be 3D TIs [13]. Better than the Bi_1-xSb_x alloy discussed above, these new family of 3D TIs are stoichiometric semiconductors with a single in-gap surface Dirac fermion located at the \Gamma point of the surface BZ. The bulk energy gap of this family of materials is also much larger (up to \sim300 meV in Bi₂Se₃) than the energy scale of room temperature (\sim 30 meV) and that found in the Bi_1-xSb_x alloy (\sim 50 meV).

The underlined physics of these materials can be explained by a simple effective model of four bands [13]. In short, the strong spin–orbit interaction in these materials causes the inversion of the Bismuth/Antimony and Selenium/Tellurium P_z orbitals, resulting in the bulk gap and a single in-gap Dirac fermion centered at the \Gamma point. The simplicity of a single surface Dirac cone makes this family of materials ideal candidates for realizing the many proposed topological phenomena, such as the magneto–electric effect [7]; and the large bulk gap (compared to the energy scale of room temperature) makes them possible candidates for high temperature spintronics applications.

The non-trivial band structures of these materials were soon confirmed by ARPES studies. The characteristic Dirac fermions at the BZ center and the large bulk gap were observed in all the three compounds [14, 16, 49]. With such properties and application potentials, this family of materials are now under intensive investigations. In the following, we will use two examples (Bi₂Te₃ and Bi₂Se₃) to demonstrate their unique band structures and the non-trivial surface state properties.

The band structure of Bi₂Te₃ is shown in Fig. 11. Besides the bulk conduction and valance bands, the single Dirac cone is clearly seen [panel (a) and (c)] at the center of the BZ, whose surface nature is confirmed by the energy dependent ARPES study [panel (d)]. The indirect bulk gap measured is \sim 165 meV, and the velocity of the Dirac fermion is \sim 4\times {10}^5m/s, about half of that of graphene [16].

Due to the vacancies and defects in the crystal, as grown Bi₂Te₃ samples usually have excess n-type carriers in bulk [16], as is indicated by the small BCB pocket in the ARPES measurement (Fig. 11).

To realize the topological insulating phase (tune E_{\mathrm{F}} into the gap), one can either reduce these imperfections by tuning the crystal growth condition, or by doping holes into the material to compensate for these excess n-type carriers. So far both methods have been successfully implemented and Fig. 12 and Fig. 13 [16] show two such examples, respectively. For the bulk doping method, introduction of more hole concentrations can further drive the material into p-type [Fig. 13(d)], making many applications feasible such as topological p–n junction. We note that unlike the surface state of regular materials that is extremely sensitive to impurities, the topological surface state in Bi₂Te₃ persists in all doping regions (Fig. 13) as long as the dopants are non-magnetic, due to the protection of time reversal symmetry (as will be discussed later).

The spin polarization of Bi₂Te₃ surface state was also investigated with the spin-resolved ARPES (Fig. 14) [15], which has confirmed its unique helical structure.

Despite the overall agreement with theoretical predictions, the experiments revealed a surprise. From Fig. 11 and Fig. 13, we see that the FS of the surface Dirac fermion shows an hexagram shape in conflict with the circular shape predicted by simple k\cdot Hamiltonian. This unconventional warping of the surface state FS [Fig. 11(b)] provides an intrinsic test for the topological theory. To look into this discrepancy, an additional third order term was added into the surface Hamiltonian (Ref. [19] and others) to explain the hexagonal warping effect.

The validity of the revised model can be further checked by the observable consequences it predicts, such as the surface quasi-particle interference and the out of plane surface spin polarization. Both effects were soon observed by scanning tunneling microscopy (STS) [50] and the spin-ARPES measurement [51, 52].

The band structure of another compound in the family, Bi₂Se₃, is shown in Fig. 15 [14, 16]. Similar to Bi₂Te₃, there is a surface Dirac cone centered at the \Gamma point. The bulk gap (\sim 200 meV) of Bi₂Se₃ is slightly larger than that of Bi₂Te₃; and unlike Bi₂Te₃ where the Dirac point is below the valence band top, the Dirac point of Bi₂Se₃ is “exposed” and resides at the top of the bulk valence band [16]. This exposure of the Dirac point makes it the easiest to observe the unique phenomena happening at the topological transport point when E_{\mathrm{F}} coincides with the E_{\mathrm{Dirac}\mathrm{point}}, at which k_f=0 and there is no interference from the bulk carriers. Similar to Bi₂Te₃, by surface [Fig. 15(b)] or bulk doping [Fig. 15(c)], E_{\mathrm{F}} can be tuned to make the bulk crystal either p- or n-types, or at the topological transport point.

Due to the finite extension of the surface state [54], when the thickness of the Bi₂Se₃ crystal becomes comparable to the surface state wave function, the surface state from the top and bottom surfaces can overlap, leading to an opening of a gap at the Dirac point due to their interaction. The effect is shown in the ARPES, resulting from MBE grown Bi₂Se₃ thin films (Fig. 16) where the thickness of the samples can be precisely controlled.

3.3 Effects of non-magnetic and magnetic impurities

Comparing to the surface state of regular materials (e.g. crystals of noble metal and some insulators such as Si), which is usually very fragile and sensitive to external perturbations such as impurities, the Dirac surface state of TIs are unusually robust. This robustness originates from the topology of its bulk electronic structure and is thus protected by TRS. As a consequence, in the presence of TRS, the massless surface Dirac fermions of TIs will persist against any impurities or crystal defects as long as the topology of the bulk band structure of the crystal does not change.

To test this ususual property, one can purposely introduce non-magnetic impurities into the bulk or onto the surface to verify the robustness of the massless surface Dirac cone. The existence of a continuous Dirac point that connects the upper and lower surface Dirac cones [Fig. 17(b)] should not be broken even if the system is severely perturbed by non-magnetic dopants [Fig. 17(a)]. As we will see below, the ARPES experiments carried out on both the bulk and surface doped Bi₂Se₃ have confirmed this TRS protection.

A bulk example is shown in Figs. 17(c) and (d) on non-magnetically doped Bi₂Se₃, where the band structures of an intrinsic and a Thulium doped samples are shown, respectively (for more examples, see Ref. [25]). In both cases, the continuity at the Dirac point is indicated by the strong spectral intensity (left sub-panels) and the single peak structure of the energy distribution curve (EDC) at the Dirac point (right sub-panels). In Fig. 17(d), the charge doping effect of Thulium is clearly seen from the dramatic shift of E_{\mathrm{F}} into the bulk gap. Nonetheless, the topology of the surface Dirac cone remains the same with a continuous Dirac point.

Besides the bulk, the perturbation can also be applied directly onto the TI surface. As an example, in Fig. 18, we demonstrate a dramatic treatment of the the Bi₂Se₃ surface by photo-assisted oxygen deposition. The accumulation of the Oxygen on the sample surface is clearly reflected by the valence spectra evolution in Fig. 18(a). The surface state and bulk valence band spectra from five different Oxygen dosage are shown in Fig. 18(b). We can see that with the increase of the Oxygen doping, the valence band dispersion of Bi₂Se₃ gradually deteriorates (bottom row), while the surface Dirac state persists (top row) despite the strong hole-doping effect of Oxygen, indicating its robustness (Note that the broadening of the surface state dispersion is due to the scattering of photoelectron from the increasing surface impurities such as adsorbed Oxygen and oxides.).

However, if we introduce magnetic impurities into the system, the TRS can be broken, thus the protection of the Dirac point is lifted [Fig. 19(a)], resulting in a gap that separates the upper and lower branches of the Dirac cone that makes the Dirac fermion massive [Fig. 19(b)]. This effect is illustrated in the band structure of an Fe-doped sample [Fig. 19(c)]. Unlike the non-magnetically doped cases [Fig. 17(c), (d)], the SSB dispersion at the Dirac point is broken, as indicated by the suppressed intensity regions in the spectral density plots (left sub-panels) and the twin-peak structure around the Dirac point in the EDC plots (right sub-panels).

Given the SSB gap formation at the Dirac point with broken TRS, we can further realize the insulating massive Dirac fermion state, a state capable of hosting many novel topological phenomena [7, 55–57]. To do so, we need to tune the E_{\mathrm{F}} into this gap, which can be achieved by Manganese doping, as Manganese dopants not only introduce magnetic moments into the system, but also naturally p-dope the samples to lower the E_{\mathrm{F}} (as Manganese has one less valance electron than Bi). The measurements on an optimally Mn-doped sample clearly show that the E_{\mathrm{F}} resides right inside the SSB gap [Fig. 19(d), left panel] of at least 7 meV [Fig. 19(d) right panel], indicating the full realization of the insulating massive Dirac fermion state in this material.

To maintain this insulating massive Dirac fermion state at higher temperatures, a further increase of the Dirac gap is required (while keeping E_{\mathrm{F}} inside it). However, because of the hole–doping effect of Manganese dopants, one cannot simply increase the Manganese concentration in (Bi_1-δMn{}_{\delta }{}^{}){}_2Se₃ to acquire a larger Dirac gap, as the system will become p-type before the gap magnitude increases appreciably [25]. Instead, one can combine the magnetic and charge doping together to achieve this. For example, it is possible to introduce much Fe dopants into Bi₂Se₃ to increase the gap size without substantially altering the E_{\mathrm{F}} position relative to the undoped Bi₂Se₃; then we can move E_{\mathrm{F}} into the gap by introducing additional p-type dopants such as Magnesium [25]. We can achieve a larger gap while preserving the insulating nature of the state.

Besides the example above, the unique robustness of the surface state of TIs and the effect of broken TRS by another study can be seen in Ref. [58].

3.4 Search for other TI materials

Despite the exciting progress made in the last couple of years, the field of TIs is still in its early development. Many of the amazing topological effects and their potential applications are still limited by the non-ideal properties of current TI materials.

For example, to date, the 2D quantum spin Hall effect was discovered only in the HgTe/CdTe quantum well system, and the small gap (\sim20 meV) of HgTe/CdTe quantum well prevents it from being used for high (room) temperature applications. In the 3D case, while the Bi₂Te₃/Bi₂Se₃ family of materials have the bulk gap up to 200 meV, the layered nature and the van der Waals force bonding between quintuple layers makes it difficult to shape into the desired forms for transport measurement and applications; and foreign atoms can be easily adsorbed on the surface (see Fig. 20 for the aging curve of a pristine Bi₂Se₃ in the UHV environments) and even get into the interstitial space between quintuple layers [59], introducing undesired impurities and charge dopings [16, 25, 60]. These problems thus provide the motivation for the search for more TI materials with better chemical, physical and electronic properties.

In Fig. 21, we give an example to illustrate the recent discovery of a new TI family in III–V–VI₂ compounds [28]. From panels (a)–(d), the characteristic single Dirac cone at the \Gamma point immediately discoveres the topological nontrivial nature of this compound, and a bulk gap of \sim 200 meV in TlBiSe₂ is shown in Fig. 21(b), similar to that of Bi₂Se₃. However, compared to the Bi₂Se₃ family of compounds, this new family has much better physical properties due to the covalence bonding between atomic layers in TlBiSe₂, which is much stronger [27, 31] than the van de Waals force that bonds quintuple layer units of Bi₂Te₃ or Bi₂Se₃ [13]. This makes this material much more favorable for other measurements and applications.

Besides the single surface Dirac fermion, materials in the III–V–VI₂ family can also possess other interesting properties. One such example is shown in Fig. 22, where the band structure of another compound, TlBiTe₂, is summarized. Besides having the Dirac surface state, TlBiTe₂ was reported to show a bulk superconducting transition at T=0.14 K when p-doped to the carrier density 6\times {10}^{20}/{\mathrm{cm}}^3 [61]. At this doping level, the E_{\mathrm{F}} of the system is \sim 150 meV below the BCB bottom and resides in a region [see region III in Fig. 22(c)] where the FS geometry is characterized by a ringlike SSB FS and six surrounding p-type bulk pockets [Fig. 22(d)].

As the six leaf-like bulk pockets are the only bulk electronic features at this doping [see Fig. 23(a), (b)], it naturally indicates that they are the origin of the the bulk superconductivity of p-type TlBiTe₂. Furthermore, in the superconducting state, the surface state can also become superconducting due to the proximity effect. For such a superconductor, it has been proposed [23] that each vortex line has two Majorana zero modes related by the time reversal symmetry, making it a candidate for the long sought topological superconductors (which will be discussed below) and for the topological quantum computation [62].

Interestingly, the ternary III–V–VI₂ family compounds are also known as good high-temperature thermoelectric materials [63–68], and a long standing puzzle since 1960s has been the unusually low thermoelectric figure of merit (ZT) of TlBiTe₂ (ZT\sim 0.15) due to its much smaller thermoelectric power S (\sim 70 µV/K) [63, 65] compared to that of other compounds (ZT\sim 0.9,S\sim 200 µV/K) [66, 67] in the family. By observing a negative band gap in the band structure [Fig. 23(c)], we demonstrate that TlBiTe₂ is a semi-metal, in contrast to a narrow-gap semiconductor as conventionally believed [64, 68], and naturally explains its mysteriously small thermoelectric power [28]. The opposite contributions of the simultaneous existing electron and hole pockets greatly reduce the total thermoelectric power S, which leads to further suppress of its thermoelectric figure of merit as ZT=S^2\sigma /(\kappa T).

Besides the III–V–VI₂ compounds shown above, there are also other TIs proposed, such as the many half heusler materials [32, 33]. In addition, surprises keep coming from the exiting family of materials such as the dramatic reduction of residual bulk carrier density in the variation of V₂–VI₃ TI materials [69, 70] and the Dirac surface state arising from the bulk HgTe (Fig. 24 [71]). Furthermore, besides the TIs in weakly interacting systems, TIs from strongly correlated materials are also actively investigated [39, 40]. The study of TIs in these materials will not only extend our understanding on topological properties of future TI materials, but also help us improve the materials’ properties of future TIs (e.g., more insulating and larger gap) for practical applications.

3.5 Other topological states and perspectives

Following the discovery in 2D and 3D materials, TIs have grown as one of the most intensively studied fields in condensed matter physics. The swift development of TIs also inspires the study of other topological states, such as quantum anomalous Hall insulators [72] and topological superconductors [19].

In a quantum anomalous Hall insulator, spontaneous magnetic moments and spin–orbit coupling combine to build a topologically nontrivial electronic structure, leading to the quantized Hall effect without an external magnetic field. The quantum anomalous Hall effect is the last quantized effect in the Hall effect family besides the Hall effect and the spin Hall effect. Given the large separation between the conducting channels, the back scattering of the electron can be suppressed even in the presence of magnetic impurities, signaling promising low power applications.

A topological superconductor has a pairing gap in the bulk and topologically protected surface state consisting of Majorana fermions [23]. Unlike Dirac fermions in TIs that can have the form of particles or holes, Majorana fermions are their own antiparticles [23, 26, 73–75]. The simplest 3D topological superconductor consists of a single Majorana cone on the surface, containing half the degree of freedom of the Dirac surface state for a single cone 3D TI. This fractionalization of the degree of freedom introduces quantum non-locality and is essential for the topological quantum computing based on Majorana fermions [62].

The study of these new quantum materials will not only be of great scientific interests, but also provide new opportunities for novel applications. We believe that the ARPES technique will continue to be an important tool in the research on novel topological quantum materials, and greatly contribute to the development in this new frontier.

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