# Topological Materials

**Topological insulators** are a newly discovered state of quantum matter. They feature a bulk gap and an odd number of relativistic Dirac fermions on their surfaces. While their bulk is insulating, the surfaces can conduct electric current with a well-defined spin texture. Spin-orbit interaction plays an important role. As a result, these materials are classified differently from the traditional Landau theory of phase transitions in matter. Instead, they are categorized by a topological quantity in the electron's wave-function, the *Chern number*. In addition, the *linear energy-momentum relationship* (*E* = *pv*) of electrons in these materials also appears in relativity (*E* = *pc*). As the velocity (*v*) of massless particles here are about 200 times slower than the speed of light (*c*) in vacuum, they are a great way to study the physics of relativity manifest in a condensed matter system.

Unlike non-topological ('trivial') materials, where fragile surface states can be easily altered by imperfections in surface geometry and chemistry, topological insulators are predicted to have unusually robust surface states protected by **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. This can produce striking quantum phenomena such as quantum spin Hall (QSH) and quantum anomalous Hall (QAH) effects, image magnetic monopoles induced by electric charges, and Majorana fermions (which are their own anti-particle — expected for bosons but not fermions) induced by proximity to a superconductor.

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

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

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

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

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

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

##### Quantum Spin Hall Insulator WTe_{2}

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

Bi_{2}Te_{3} has been proposed to be the simplest 3D topological insulator, with a surface state consisting of a **single Dirac cone** at the Γ point (* k* = 0 in the Brillouin zone). By scanning across the Brillouin zone, our ARPES results on Bi

_{2}Te

_{3}demonstrate that the surface state consists of a single non-degenerate Dirac cone. Furthermore, we explicitly showed the existence of a 100 meV energy gap for the bulk state by tuning the Fermi level. Our results demonstrate that Bi

_{2}Te

_{3}is a possible candidate for high-temperature spintronic applications.

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

Just like Bi_{2}Te_{3}, Bi_{2}Se_{3} is also predicted to be a near-ideal topological insulator with a single Dirac cone. 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 360 K, which is promising for spintronic applications. For topological insulators, understanding the interplay of **spin-polarized surface electrons** with non-spin-polarized bulk electrons is critical for these potential applications. 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 electronic band structure using time-resolved ARPES (trARPES).

This femtosecond 'movie' on Bi_{2}Se_{3} depicts how excited electrons fill unoccupied bands, and are subsequently 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 more than 10 picoseconds. This finding may pave the way for development of ultrafast optical switches of spin-polarized conduction channels. By resonantly exciting electrons in Bi_{2}Se_{3} to unoccupied states, we 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, our knowledge of the unoccupied band structure was instrumental in our investigation of **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. 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 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 surface states is a special characteristic of topological insulators in which spin direction is locked to the wave vector and winds twice around the Fermi surface. Probing spin texture in topological insulators is thus crucial for understanding them. Our newly-developed spin-ARPES technique has the ability to probe spin texture in electronic band structure. The figure above shows spin-polarized surface states we have measured in 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 — that is, they form a gap and lose their linear dispersion — if time reversal symmetry is broken. It is also insulating if the Fermi energy (*E*_{F}) is positioned inside both the surface and bulk gaps. This is shown in the diagram above. We can force an insulating massive **Dirac fermion state** to form by tuning *E*_{F} into the surface-state gap. This state may then support many striking topological phenomena, such as an 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}, breaking time-reversal symmetry by adding magnetic dopants. We positioned the *E*_{F} inside the gaps by simultaneous magnetic and charge doping. Using ARPES, we studied the electronic structures of undoped and doped Bi_{2}Se_{3}. For undoped Bi_{2}Se_{3}, *E*_{F} was found to always reside above the Dirac fermion state. Introducing Mn dopants to Bi_{2}Se_{3} lifted the degeneracy at the Dirac point, forming an approximately 7 meV time-reversal symmetry-broken gap. It also tuned the *E*_{F} into this gap, removing 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 the *surfaces* 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. The TDS state is proximate to various quantum states, ranging from regular band insulators to topological superconductors, and is thus 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 validated its description as a 3D counterpart of graphene. Our data clearly observed the Dirac cone in Na_{3}Bi and showed that it is 3D (bulk) in nature. We also found that the bulk Dirac cone persists despite 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 spintronic applications.

## Quantum Spin Hall Insulator WTe_{2}

A **quantum spin Hall** insulator (QSH), or a two-dimensional topological insulator, possesses an insulating bulk, and topologically protected dissipationless edge states 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 spintronic applications.

In our study, we demonstrated successful growth of monolayer 1T’ phase of WTe_{2} on a bilayer graphene substrate using molecular beam epitaxy (MBE). Using ARPES and STS (scanning tunneling spectroscopy), we established that monolayer 1T’-WTe_{2} is a new class of QSH insulator, with nontrivial band inversion, opening of a 55 meV bulk band gap, and a conducting edge state coexistent with an insulating bulk. This finding provides a platform for studying QSH insulators in 2D transition metal dichalcogenides, and for developing novel device applications. More information on thin films and 2D materials may be found here.