Topological insulators are a new state of quantum matter with a bulk gap and odd number of relativistic Dirac fermions on the surface. The bulk of such materials is insulating but the surface can conduct electric current with well-defined spin texture. These materials, with spin-orbit interaction playing an important role, are classified differently from the traditional Landau theory of phase transitions in matter, rather they are classified by quantities in the electrons wave-function, the so-called Chern number. In addition, the relativistic energy-momentum relationship of electrons in these materials provides a great opportunity to study the physics of relativity in a condensed matter system with the velocity of massless particles about 200 times slower than the light speed in vacuum.

Unlike other materials where the fragile surface states can be easily altered by details in the surface geometry and chemistry, topological insulators are predicted to have unusually robust surface states due to the protection of time-reversal symmetry. These unique states are protected against all time-reversal-invariant perturbations, such as scattering by non-magnetic impurities, crystalline defects, and distortion of the surface itself, and can lead to striking quantum phenomena such as quantum spin Hall and quantum anomalous Hall effects, an image magnetic monopole induced by an electric charge, and Majorana fermions (whose anti-particle is itself) induced by proximity effect from a superconductor.

Extracting the electronic and structural properties of topological insulators is essential for both the understanding of the underlying physics and potential applications. As a direct method to study the electron band structures of solids, ARPES can yield rich information of the electronic bands of topological insulators and even demonstrate control over the electronic surface states of the topological insulators in the time domain. For 3D topological insulators, ARPES has emerged as the leading tool to elucidate the topological nature of these materials.

### Electronic structure of the 3D topologic Insulator Bi_{2}Te_{3}

Bi_{2}Te_{3} has been proposed to be the simplest 3D topological insulators whose surface states consist of a single Dirac cone at the Γ point. Our ARPES results on Bi_{2}Te_{3} demonstrate that the surface state consists of a single non-degenerate Dirac cone by scanning over the Brillouin zone. Furthermore, we explicitly showed the 100 meV energy gap for the bulk state by tuning the Fermi Level. Our results prove that Bi_{2}Te_{3} is among the possible candidates for high-temperature spintronics applications.

### Probing the unoccupied states in topological insulator Bi_{2}Se_{3}

In addition to Bi_{2}Te_{3}, Bi_{2}Se_{3} is also predicted to be a nearly idealized single Dirac cone topological insulator. Bi_{2}Se_{3} offers the potential for topologically protected behavior in ordinary crystals at room temperature and zero magnetic field. It has a large band gap of 0.3 eV at 3600 K, which is promising for spintronics applications. For topological insulators, understanding the interplay of spin-polarized surface electrons with non-spin-polarized bulk electrons is critical for potential spintronics applications. Therefore, our work reveals the coupling between bulk and surface electrons of the topological insulator Bi_{2}Se_{3} by probing the dynamics of optically excited electrons directly in the material's electronic band structure using time-resolved ARPES(trARPES).

_{2}Se

_{3}depicts how the excited electrons fill unoccupied bands, but are prevented from relaxing immediately back to equilibrium due to the bulk band gap. These hot electrons accumulate at the bulk conduction band edge and act as a reservoir, which continues to populate the spin-polarized surface states for >10 ps. This finding may pave the way for development of ultrafast optical switches of spin-polarized conduction channels. We have also applied optically excited the topological insulator Bi

_{2}Se

_{3}to the unoccupied state to obtain a complete picture of the electronic structure from the Fermi level up to the vacuum level. We found that the unoccupied states host a second, topologically protected Dirac surface state which can be resonantly excited by 1.5 eV photons.

In addition, the knowledge of the unoccupied band structure was instrumental in our investigations of the photocurrents generated via circularly polarized optical excitations in Bi_{2}Se_{3}. These photocurrents can be measured in conventional transport configurations and are carried by topological surface states. The light helicity controls the photocurrent direction, making such systems interesting for electronic applications. The spectroscopic signature of photocurrents in trARPES is an asymmetric electron population in momentum space. We found that in Bi_{2}Se_{3} only resonant optical transitions lead to such momentum-asymmetric electron populations and therefore contribute to photocurrent generation. We also observed that different bands contribute to the net current in opposite directions. Our work provides a microscopic understanding of how to control photocurrents in topological materials and sets the stage for further studies leading toward spintronic applications.

### Spin-resolved imaging of the surface state of topological insulator Bi_{2}Se_{3}

Spin-orbital texture in topological surfaces states is a special character in topological insulators where the spin direction is locked to the wave vector and winds by 2 around the Fermi surface. Probing the spin texture in TI is thus crucial for understanding topological insulators. Our newly-developed spin-ARPES technique has the ability to probe the spin texture in the electronic band structures. As shown in the figure is our measured spin-polarized surface states of Bi_{2}Se_{3}. Red and blue colors indicate opposite in-plane spin polarization. The spin-momentum locking feature of the topological surface state is clearly shown, where opposite momenta possess opposite spin polarizations.

### Massive Dirac fermion on the surface of the magnetically doped topological insulator Bi_{2}Se_{3}

Topological insulators accommodate a conducting, linearly dispersed Dirac surface state in addition to the bulk energy gap. This state is predicted to become massive if time reversal symmetry (TRS) is broken, and to become insulating if the Fermi energy is positioned inside both the surface and bulk gaps. Furthermore, if Fermi energy (EF) can be tuned into this surface-state gap, an insulating massive Dirac fermion state is formed; this state may support many striking topological phenomena, such as the image magnetic monopole induced by a point charge, the half quantum Hall effect on the surface, and a topological contribution to the Faraday and Kerr effects.

We introduced long-range magnetic order to the 3D topological insulator Bi_{2}Se_{3} to break the time-reversal symmetry by adding magnetic dopants and further position the Fermi energy inside the gaps by simultaneous magnetic and charge doping. Using ARPES, we studied electronic structures of both undoped and doped Bi_{2}Se_{3}. For undoped Bi_{2}Se_{3}, EF was found to always reside above the Drac fermion state. By introducing Mn dopants to Bi_{2}Se_{3}, we not only lifted the degeneracy at the Dirac point with an approximately 7meV large TRS gap formed, but also tuned the EF into this gap, removing the excess n-type carriers while maintaining the magnetic doping effect. In other words, Mn dopants not only introduce magnetic moments into the system, but also naturally p-dope the samples. The resulting insulating massive Dirac fermion state we discovered paves the way for studying a range of topological phenomena relevant to both condensed matter and particle physics.

### Discovery of the topological semimetal Na_{3}Bi

In contrast to 2D Dirac fermions in graphene or on the surface of 3D topological insulators, Topological Dirac Semimetals (TDS) possess 3D Dirac fermions in the bulk and can be viewed as a 3D counterpart of graphene. The bulk conduction and valence bands touch only at Dirac points and disperse linearly along momentum directions, forming 3D Dirac fermions in the bulk. TDS state is the neighbor to various quantum states ranging from regular band insulators to topological superconductors, hence is useful for studying topological phase transitions.

We have performed angle-resolved photoemission spectroscopy (ARPES) measurements to investigate the electronic structures of Na_{3}Bi single crystals and found that it is a 3D counterpart of graphene. Our ARPES data clearly observed the Dirac cone in Na_{3}Bi and showed that the Dirac cone is 3D/bulk in nature. We also found that the bulk Dirac cone persists despite such surface deterioration, supporting the notion that the Dirac fermion observed is protected by the bulk crystal symmetry. The discovery of the Topological Dirac Semimetal Na_{3}Bi opens the door to exploring other 3D TDSs and is potentially useful for spintronics applications given its unique properties.

### Quantum Spin Hall Insulator WTe_{2}

A quantum spin Hall insulator (QSH), or a 2-dimentional topological insulator results from topologically protected dissipationless edge states with an insulating bulk that bridge the energy gap opened by band inversion and strong spin-orbit coupling. A QSH state features quantized Hall conductance in the absence of a magnetic field and is thus promising for spintronics applications.

In our study, we demonstrated successful growth of monolayer 1T’ phase of WTe_{2} using molecular beam epitaxy (MBE) on a bilayer graphene substrate. More information on Thin Films and 2D materials can be found in this section. Using ARPES and STS (Scanning tunneling spectroscopy), we established evidence that monolayer 1T’-WTe_{2} is a new class of QSH insulator, with nontrivial band inversion, the opening of a 55meV bulk band gap, and a conducting edge state in contrast to the insulating bulk. Such findings provide a platform for studying QSH insulator in 2D transition metal dichalcogenides and for developing novel device applications.