Observation of Floquet-Bloch States on the Surface of a Topological Insulator

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

In the present study, we employ the angle-resolved photoemission spectroscopy (ARPES) technique to study the electronic structure of topological states of matter. In particular, the so-called topological crystalline insulators (TCIs) Pb1-xSnxSe and Pb1-xSnxTe, and the Mn-doped Z2 topological insulators (TIs) Bi2Te3 and Bi2Se3. The Z2 class of strong topological insulators is protected by time-reversal symmetry and is characterized by an odd number of metallic Dirac type surface states in the surface Brillouin zone. The topological crystalline insulators on the other hand are protected by the individual crystal symmetries and exhibit an even number of Dirac cones. The topological properties of the lead tin chalcogenides topological crystalline insulators can be tuned by temperature and composition. Here, we demonstrate that Bi-doping of the Pb1-xSnxSe(111) epilayers induces a quantum phase transition from a topological crystalline insulator to a Z2 topological insulator. This occurs because Bi-doping lifts the fourfold valley degeneracy in the bulk. As a consequence a gap appears at ⌈¯, while the three Dirac cones at the M points of the surface Brillouin zone remain intact. We interpret this new phase transition is caused by lattice distortion. Our findings extend the topological phase diagram enormously and make strong topological insulators switchable by distortions or electric field. In contrast, the bulk Bi doping of epitaxial Pb1-xSnxTe(111) films induces a giant Rashba splitting at the surface that can be tuned by the doping level. Tight binding calculations identify their origin as Fermi level pinning by trap states at the surface. Magnetically doped topological insulators enable the quantum anomalous Hall effect (QAHE) which provide quantized edge states for lossless charge transport applications. The edge states are hosted by a magnetic energy gap at the Dirac point which has not been experimentally observed to date. Our low temperature ARPES studies unambiguously reveal the magnetic gap of Mn-doped Bi2Te3. Our analysis shows a five times larger gap size below the Tc than theoretically predicted. We assign this enhancement to a remarkable structure modification induced by Mn doping. Instead of a disordered impurity system, a self-organized alternating sequence of MnBi2Te4 septuple and Bi2Te3quintuple layers is formed. This enhances the wave-function overlap and gives rise to a large magnetic gap. Mn-doped Bi2Se3 forms similar heterostructure, but only a nonmagnetic gap is observed in this system. This correlates with the difference in magnetic anisotropy due to the much larger spin-orbit interaction in Bi2Te3 compared to Bi2Se3. These findings provide crucial insights for pushing lossless transport in topological insulators towards room-temperature applications.

The unique electronic properties of the surface electrons in a topological insulator [1] are protected by time-reversal symmetry. Circularly polarized light naturally breaks time-reversal symmetry, which may lead to an exotic surface quantum Hall state [2]. Using time- and angle-resolved photoemission spectroscopy [3] [4], we show that an intense ultrashort mid-infrared pulse with energy below the bulk band gap hybridizes with the surface Dirac fermions of a topological insulator to form Floquet-Bloch bands when the pump pulse is present [Fig. 1] [5]. These photon dressed surface bands exhibit polarization-dependent band gaps at avoided crossings [Fig. 2A and B]. Circularly polarized photons induce an additional gap at the Dirac point [Fig. 2C], which is a signature of broken time-reversal symmetry on the surface. The size of the gaps are consistent with the calculation based on Floquet-Bloch theory [6]. This results in a Chern insulator as originally proposed by Haldane [7]. These observations establish the Floquet-Bloch bands in solids and pave the way for optical manipulation of topological quantum states of matter.

A key feature of topological insulators is the robustness of the electron energy spectrum. At a surface of a topological insulator, the Dirac point is protected by the characteristic symmetry of the system. The breaking of the symmetry opens a gap in the energy spectrum. Therefore, topological insulators are very sensitive to magnetic fields, which can open a gap in the electronic spectrum. Concerning "internal" magnetic effects, for example, the situation with doped magnetic impurities, is not trivial. A single magnetic impurity is not enough to open the band gap, while in the case of a ferromagnetic chain of deposited magnetic impurities the Dirac point is lifted. However, a much more interesting case is when localized magnetic impurities form a chiral spin order. Our first principle density functional theory calculations have shown that this is the case for Fe deposited on the surface of a Bi2Se3 topological insulator. But not only magnetic impurities can form a chiral helicoidal spin texture. An alternative way is to use chiral multiferroics (prototype material is LiCu2O2) that induce a proximity effect. The theoretical approach we present here is valid for both cases. We observed that opposite to a ferromagnetically ordered case, a chiral spin order does not destroy the Dirac point. We also observed that the energy gap appears at the edges of the new Brillouin zone. Another interesting result concerns the spin dynamics. We derived an equation for the spin density dynamics with a spin current and relaxation terms. We have shown that the motion of the conductance electron generates a magnetic torque and exerts a certain force on the helicoidal texture.

We study the properties of a family of anti-pervoskite materials, which are topological crystalline insulators with an insulating bulk but a conducting surface. Using ab-initio DFT calculations, we investigate the bulk and surface topology and show that these materials exhibit type-I as well as type-II Dirac surface states protected by reflection symmetry. While type-I Dirac states give rise to closed circular Fermi surfaces, type-II Dirac surface states are characterized by open electron and hole pockets that touch each other. We find that the type-II Dirac states exhibit characteristic van-Hove singularities in their dispersion, which can serve as an experimental fingerprint. In addition, we study the response of the surface states to magnetic fields.

Condensed matter physics is a vibrant branch of physics, which addresses a very broad spectrum of issues related to electronic, magnetic, thermal, structural and optical proper- ties of condensed phases of matter. The interplay between structural and magnetic phases, interactions between different components of a material, spin-orbit coupling (SOC) and other effects, make condensed matter physics a rich and colorful field. In this thesis I will focus on topological materials, excitonic insulators and atomically thin films, which are being explored intensely both theoretically and experimentally. Specifically, topological materials, including topological (crystalline) insulators and topological Weyl semimetals are covered in Chapters 2 to 4. Chapter 5 is mainly concerned about the excitonic in- sulator (EI) phase in 'slow graphene'. The electronic structure of atomically thin MoS2 films will be discussed in Chapter 6. 3D Topological insulators (TIs), known as quantum spin Hall insulators in 2D, are in- sulating in the bulk while conducting on the surface in sharp contrast with conventional insulators. In Chapter 2, by using first-principles and tight-binding model calculations, we identify a 2-ML (monolayer) Bi(110) thin film as a candidate quantum-spin-Hall in- sulator. In the absence of spin-orbit coupling, 2-ML Bi(110) thin films have two types of Dirac cones in the Brillouin zone (BZ). The Dirac cones, carrying non-zero winding numbers, serve as the starting or ending points of the edge bands in the ribbon spectrum. After the inclusion of the SOC, all Dirac nodes are gapped out. Correspondingly, a Dirac cone formed by the gapless edge states was found at the boundary of the ribbon reflecting the topologically nontrivial nature of the system. In Chapter 3, we present our work on the topological Weyl semimetal phase in the transi- tion metal monopnictide TaAs. A topological Weyl semimetal is a new kind of topological phase, which shows exotic properties in the bulk and on the surface. In the bulk, the valence and conduction bands touch each other at discrete K points, termed as Weyl points or Weyl nodes. On the surface, the Weyl nodes can induce Fermi arcs which are non-closed surface states connecting Weyl nodes with opposite chiralities. TaAs breaks the inversion symmetry and therefore it can be a possible system to host the topologi- cal Weyl semimetal phase. Our first-principles and tight-binding model calculations find that there are 24 Weyl points in the Brillouin zone. The Fermi arcs on (001) surface are identified by surface-state calculations and observed in angle-resolved photoemission experiments (ARPES). Topological crystalline insulators (TCIs) are new kind of topological insulators, which are protected by the crystal symmetries instead of time-reversal symmetry. In Chapter 4, we consider a rotational symmetry protected topological crystalline phase in TaAs2 family of materials. The TaAs2 class crystalizes in a monoclinic structure with space group No. 12. By checking the parity eigenvalues of the occupied bands at time reversal invariant momenta (TRIM), the topological invariants (symmetry indicators) of TaAs2 are found to be (Z2Z2Z2; Z4) = (111; 2), suggesting that TaAs2 can host two Dirac cones on the (010) surface protected by C2 rotational symmetry. Our surface state calculation identified two clear Dirac cones on the (010) surface with the bulk band gap as large as 300 meV. An excitonic insulator instability can arise in narrow gap semiconductors and semimetals when the binding energy of an electron-hole pair exceeds the band gap. Although many experiments have shown some signs of an excitonic insulator state in various systems, conclusive experimental evidence still remains elusive. In Chapter 5 we discuss the ex- citonic instability in 'slow graphene'. We approach the EI transition from two different directions. First, we apply a commonly used mean-field approach that can give us the overall phase diagram of the EI transition. In another approach, we solve the Bethe- Salpeter equation (BSE) and follow the evolution of the lowest excitons. Since graphene is gapless, the presence of an excitonic state at negative energy signals an instability of the assumed ground state. By studying properties of the exciton, we can infer properties of the resulting EI phase, finding overall consistency between the two approaches. Finally, in Chapter 6, we consider the electronic structure of few-layer MoS2. Transition metal dichalcogenides are a family of layered materials, which exhibit metal to semicon- ductor transitions and many other interesting properties as a function of the number of layers. For example, bulk MoS2 is a semiconductor with indirect gap, while monolayer MoS2 has a direct gap at the K-point. I will investigate the evolution of the electronic structure and related properties of MoS2 films as the number of layers is increased within the first-principles density functional theory (DFT) framework. Wannier-function based tight-binding models will be used to gain insight into the first-principles results. I will summarize my thesis in Chapter 7.

Topological insulator surfaces support metallic surface states with closed Fermi contours, encircling an odd number of Dirac points. Experimental studies have so far concentrated on surfaces with only one Dirac point, but three Dirac points can be expected for certain surface orientations of several topological insulator materials. Here we experimentally realize the Bi1−xSbx(110) surface for which an electronic structure with three Dirac points has been predicted (Teo et al 2008 Phys. Rev. B 78 045426), in contrast to the closed-packed (111) surface of the same material that supports only one Dirac point. We study the electronic structure of Bi1−xSbx(110) with angle-resolved photoemission and tight-binding calculations. We observe several metallic surface states, confirming not only the expectation that a topological insulator should be enclosed by metallic surfaces on all faces, but also the prediction of the surface state topology. Tight-binding calculations of the electronic structure are found to reproduce the expected topology of the surface states but they show one Dirac point that is not observed in the experiment, in the mirror line of the surface Brillouin zone. As in the case of Bi1−xSbx(111), this can be ascribed to an incorrect value of the mirror Chern number in the tight-binding parameters employed for the calculation. The quantitative agreement of the tight-binding calculation and the experiment is poorer than in the case of the (111) surface, something that is ascribed to the existence of dangling bonds on the (110) surface.

We propose to Floquet-engineer Dirac cones at the surface of a three-dimensional topological insulator. We show that a large tunability of the Fermi velocity can be achieved as a function of the polarization, direction and amplitude of the driving field. Using this external control, the Dirac cones in the quasienergy spectrum may become elliptic or massive, in accordance to experimental evidences. These results help us to understand the interplay of surface states and external ac driving fields in topological insulators. In our work we use the full Hamiltonian for the three-dimensional system instead of effective surface Hamiltonians, which are usually considered in the literature. Our findings show that the Dirac cones in the quasienergy spectrum remain robust even in the presence of bulk states and, therefore, they validate the usage of effective surface Hamiltonians to explore the properties of Floquet-driven topological boundaries. Furthermore, our model allows us to introduce new out-of-plane field configurations, which cannot be accounted for by effective surface Hamiltonians.

The recent observation of zero-biased photocurrent on the surface of topological insulators allows to spin-orbit-coupled two-dimensional Dirac cones as ideal platforms for the manipulation of the tilt of Dirac cone. We show that the in-plane effective magnetic field B̃ implements a moving frame transformation on the topological insulators' helical surface states. As a result, photo-excited electrons on the surface undergo a Galilean boost proportional to the effective in-plane magnetic field B̃. The boost velocity is transversely proportional to B̃. This explains why the experimentally observed photocurrent depends linearly on B̃. Our theory, while consistent with the observation that at leading order the effect does not depend on the polarization of the incident radiation, at next leading order in B̃ predicts a polarization dependence in both parallel and transverse directions to the polarization. We also predict two induced Fermi-surface effects that can serve as further confirmation of our moving frame theory. Based on the estimated value ζ≈0.34 of the tilt parameter for a magnetic field of B̃∼3T, our geometric picture qualifies the surface Dirac cone of magnetic topological insulators as an accessible platform for the synthesis and experimental investigation of strong analog gravitational phenomena. Published by the American Physical Society 2024

Parity-time symmetry plays an essential role for the formation of Dirac states in Dirac semimetals. So far, all of the experimentally identified topologically nontrivial Dirac semimetals (DSMs) possess both parity and time reversal symmetry. The realization of magnetic topological DSMs remains a major issue in topological material research. Here, combining angle-resolved photoemission spectroscopy with density functional theory calculations, it is ascertained that band inversion induces a topologically nontrivial ground state in EuCd2 As2 . As a result, ideal magnetic Dirac fermions with simplest double cone structure near the Fermi level emerge in the antiferromagnetic (AFM) phase. The magnetic order breaks time reversal symmetry, but preserves inversion symmetry. The double degeneracy of the Dirac bands is protected by a combination of inversion, time-reversal, and an additional translation operation. Moreover, the calculations show that a deviation of the magnetic moments from the c-axis leads to the breaking of C3 rotation symmetry, and thus, a small bandgap opens at the Dirac point in the bulk. In this case, the system hosts a novel state containing three different types of topological insulator: axion insulator, AFM topological crystalline insulator (TCI), and higher order topological insulator. The results provide an enlarged platform for the quest of topological Dirac fermions in a magnetic system.

The electronic structure of magnetically doped topological insulator Bi1.09Gd0.06Sb0.85Te3 is studied in the vicinity of the Dirac point at various temperatures (above and below the Néel temperature, 1–35 K) and synchrotron radiation polarizations using angle-resolved photoelectron spectroscopy. It is shown that the energy gap exists in photoemission spectra at the Dirac point, which remains open above the long-range magnetic ordering temperature TN. Measurements of magnetic properties by the superconducting magnetometry method show antiferromagnetic ordering with the paramagnetic transition temperature of 8.3 K. The studies of the temperature dependence of the Dirac cone state intensity by photoelectron spectroscopy confirm the existence of the magnetic transition and show the possibility of its indication directly from photoemission spectra. A more detailed analysis of the splitting between states of upper and lower Dirac cones (i.e., the energy gap) at the Dirac point in the photoelectron spectra shows the dependence of the gap at the Dirac point on the synchrotron radiation polarization type (28–30 meV for p-polarization and 22–25 meV for circularly polarized radiation of opposite chirality). The gap opening mechanism at the Dirac point above TN due to “coupling” of Dirac fermions with opposite momenta and spin orientations due to their interaction with the spin texture formed immediately during photoemission in the region of the photoemission hole at the magnetic impurity atom (Gd). It is shown that the gap at the Dirac point, measured above TN, is dynamic and is formed immediately during photoemission. In this case, the gap nature remains magnetic (even in the absence of long-range magnetic ordering) and is caused by properties of magnetic topological insulator, which does control the gap invariability when passing through TN. The dynamic nature of the generated gap is confirmed by its dependence on synchrotron radiation polarization.

The low-energy theory of the surface of the topological crystalline insulator (TCI) is characterized by four Dirac cones anisotropic into the $x$ and $y$ directions. Recent experiments have shown that the band gap can be introduced in these Dirac cones by crystal distortion by applying strain to the crystal structure. The TCI surface provides us with a new way to valleytronics when gaps are given to Dirac cones. Indeed the system has the Chern number and three valley-Chern numbers. We investigate the optical absorption on the TCI surface. It shows a strong elliptic dichroism though the four Dirac cones have the same chiralities. Namely, it is found that the absorptions of the right- and left-polarized light are different, depending on the sign of mass and the location of the Dirac cones, owing to the anisotropy of the Dirac cone. By measuring this elliptic dichroism it is possible to determine the anisotropy of a Dirac cone experimentally.

We study the effects of strong electron-electron interactions on the surface of cubic topological Kondo insulators (such as samarium hexaboride, SmB$_6$). Cubic topological Kondo insulators generally support three copies of massless Dirac nodes on the surface, but only two of them are energetically degenerate and exhibit an energy offset relative to the third one. With a tunable chemical potential, when the surface states host electron and hole pockets of comparable size, strong interactions may drive this system into rotational symmetry breaking nematic and translational symmetric breaking excitonic spin- or charge-density-wave phases, depending on the relative chirality of the Dirac cones. Taking a realistic surface band structure into account we analyze the associated Ginzburg-Landau theory and compute the mean field phase diagram for interacting surface states. Beyond mean field theory, this system can be described by a two-component isotropic Ashkin-Teller model at finite temperature, and we outline the phase diagram of this model. Our theory provides a possible explanation of recent measurements which detect a two-fold symmetric magnetoresistance and an upturn in surface resistivity with tunable gate voltage in SmB$_6$. Our discussion can also be germane to other cubic topological insulators, such as ytterbium hexaboride (YbB$_6$), plutonium hexaboride (PuB$_6$).

We present a series of models of three-dimensional rotation-symmetric fragile topological insulators in class AI (time-reversal symmetric and spin-orbit-free systems), which have gapless surface states protected by time-reversal ($T$) and $n$-fold rotation ($C_n$) symmetries ($n=2,4,6$). Our models are generalizations of Fu's model of a spinless topological crystalline insulator, in which orbital degrees of freedom play the role of pseudo-spins. We consider minimal surface Hamiltonian with $C_n$ symmetry in class AI and discuss possible symmetry-protected gapless surface states, i.e., a quadratic band touching and multiple Dirac cones with linear dispersion. We characterize topological structure of bulk wave functions in terms of two kinds of topological invariants obtained from Wilson loops: $\mathbb{Z}_2$ invariants protected by $C_n$ ($n=4,6$) and time-reversal symmetries, and $C_2T$-symmetry-protected $\mathbb{Z}$ invariants (the Euler class) when the number of occupied bands is two. Accordingly, our models realize two kinds of fragile topological insulators. One is a fragile $\mathbb{Z}$ topological insulator whose only nontrivial topological index is the Euler class that specifies the number of surface Dirac cones. The other is a fragile $\mathbb{Z}_2$ topological insulator having gapless surface states with either a quadratic band touching or four (six) Dirac cones, which are protected by time-reversal and $C_4$ ($C_6$) symmetries. Finally, we discuss the instability of gapless surface states against the addition of $s$-orbital bands and demonstrate that surface states are gapped out through hybridization with surface-localized $s$-orbital bands.

Topological insulators (TIs) host robust edge or surface states protected by time-reversal symmetry (TRS), which makes them prime candidates for applications in spintronic devices. A promising avenue of research for the development of functional TI devices has involved doping of three-dimensional (3D) TI thin film and bulk materials with magnetic elements. This approach aims to break the TRS and open a surface band gap near the Dirac point. Utilizing this gapped surface state allows for a wide range of novel physical effects to be observed, paving a way for applications in spintronics and quantum computation. This review focuses on the research of 3D TIs doped with manganese (Mn). We summarize major progress in the study of Mn doped chalcogenide TIs, including Bi2Se3, Bi2Te3, and Bi2(Te,Se)3. The transport properties, in particular the anomalous Hall effect, of the Mn-doped Bi2Se3 are discussed in detail. Finally, we conclude with future prospects and challenges in further studies of Mn doped TIs.