Surface Dirac Cone Research Articles

Moving frame theory of zero-bias photocurrent on the surface of topological insulators

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

Control of magnetism on the topological SnTe(001) surface by doping, strain, and gap opening

The spin–spin exchange interaction is an important observable that helps in understanding the magnetism of materials. Topological materials are of particular interest due to their strong spin–orbit coupling, which makes them promising for applications in valleytronics and spintronics. In this study, we have theoretically demonstrated the ability to control the switching of ferromagnetic (FM) and antiferromagnetic (AFM) couplings between two magnetic impurities on the (001) surface of the topological crystalline insulator SnTe. By considering experimental parameters such as doping, strain, and gap opening at surface Dirac cones, we provide a generic way to build the FM-AFM and clockwise–counterclockwise phase diagrams of SnTe(001), which are still missing in the literature, especially for the spintronics community.

Crystalline finite-size topology

Topological phases stabilized by crystalline point group symmetry protection are a large class of symmetry-protected topological phases subjected to considerable experimental scrutiny. Here, we show that the canonical three-dimensional (3D) crystalline topological insulator protected by time-reversal symmetry T and fourfold rotation symmetry C4 individually or the product symmetry C4T, generically realizes finite-size crystalline topological phases in thin film geometry [a quasi-(3−1)-dimensional, or q(3−1)D, geometry]: response signatures of the 3D bulk topology coexist with topologically protected, quasi-(3−2)D, and quasi-(3−3)D boundary modes within the energy gap resulting from strong hybridization of the Dirac cone surface states of the underlying 3D crystalline topological phase. Importantly, we find qualitative distinctions between these gapless boundary modes and those of strictly 2D crystalline topological states with the same symmetry protection and develop a low-energy, analytical theory of the finite-size topological magnetoelectric response. Published by the American Physical Society 2024

Nontrivial Topological Phases in Ternary Borides M2XB2 (M=W, Mo; X=Co, Ni)

The nontrivial band topologies protected by certain symmetries have attracted significant interest in condensed matter physics. The discoveries of nontrivial topological phases in real materials provide a series of archetype materials to further explore the topological physics. Ternary borides M2XB2 (M = W, Mo; X = Co, Ni) have been widely investigated as the wear-resistant and high-hardness materials. Based on first-principles calculations, we find the nontrivial topological properties in these materials. Taking W2NiB2 as an example, this material shows the nodal line semimetal state in the absence of spin-orbit coupling. Two types of nodal lines appear near the Fermi level simultaneously. One is protected by the combined space-inversion and time-reversal symmetry, and the other is by the mirror symmetry. Part of these two-type nodal lines form nodal chains. When spin-orbit coupling is included, these nodal lines are fully gapped and the system becomes a strong topological insulator with nontrivial Z 2 index (1;000). Our calculations demonstrate that a nontrivial spin-momentum locked surface Dirac cone appears on the (1¯10) surface. We also find that other isostructural ternary borides Mo2NiB2, Mo2CoB2, and W2CoB2 possess similar topological band structures. Therefore, our work not only enriches the understanding of band topology for ternary borides, but also lays the foundation for the further study of topological phases manipulation and potential spintronic applications in realistic materials.

Anomalous linear and quadratic nodeless surface Dirac cones in three-dimensional Dirac semimetals

Anomalous linear and quadratic nodeless surface Dirac cones in three-dimensional Dirac semimetals

Aberrant photoelectric effect in the topological insulator/n-GaN heterojunction (Bi2Te3/n-GaN) under unpolarized illumination.

A topological insulator has a unique graphene-like Dirac cone conducting surface state, which is excellent for broadband absorption and photodetector applications. Experimental investigations on the Bi2Te3/n-GaN heterojunction exhibited an aberrant photoelectric effect under the influence of unpolarized light. Transport measurements of the Bi2Te3/n-GaN heterojunction revealed a negative photoconductance, with a sudden increase in resistance. This was consistent with the applied range of wavelength and power used for incident light while it was contrary to the usual gap-state transition model, which states that a negative conductance is due to the trapping of charge carriers. The observed aberrant photoelectric effect seen in Bi2Te3/n-GaN heterojunction devices was due to the polycrystalline nature of the Bi2Te3 topological insulator film, where the incident photon-induced bandgap in the Dirac cone surface state resulted in a negative photoelectric effect. This phenomenon opens the possibility for applications in highly sensitive photodetectors and non-volatile memories, along with employing the bandgap-opening concept in retinomorphic devices.

First Principles Studies of Topological Insulating Behavior in Lanthanum‐Monopnictides and their Heterostructures

AbstractThe topological insulating behavior of LaX (X = Bi, Sb, As, and P) and their heterostructure combinations are investigated. The system's bulk‐band‐inversion is analyzed, and the ‐topological indexes are calculated using LDA and mBJLDA exchange functionals. Strong topological insulating behavior in LaBi is confirmed from bulk band inversion. Linearly dispersive Dirac‐cone in the (001)‐surface band structure is observed. The other LaX binaries also exhibit bulk‐band‐inversion under LDA+SOC settings but disappear after including the mBJ potential. The robustness of the surface Dirac cone is tested by estimating the surface band structure under uniaxial strain. The result shows that Dirac‐cone in the (001)‐surface is unaltered only for the LaBi system and destroyed in all other three binaries, concluding the topologically trivial nature of LaP, LaSb, and LaAs. The investigation is further extended to study the multilayer LaX1/LaX2 systems composed of an alternate stacking of two different LaX binaries. The findings suggest that the multilayer structures exhibit topological insulating behavior only when LaBi is present in the system. Bulk‐band‐inversion and surface Dirac‐cone structures in LaBi/LaAs, LaBi/LaSb, and LaBi/LaP multilayers are observed. Furthermore, the surface Dirac cones remain unchanged under the influence of TR‐preserving perturbation, confirming these systems possess strong topological insulating character.

ZrOsSi: a Z2 topological metal with a superconducting ground state

The silicide superconductors (Ta, Nb, Zr)OsSi are among the best candidate materials for investigating the interplay of topological order and superconductivity. Here, we investigate in detail the normal-state topological properties of (Ta, Nb, Zr)OsSi, focusing on ZrOsSi, by employing a combination of 29Si nuclear magnetic resonance (NMR) measurements and first-principles band-structure calculations. We show that, while (Ta, Nb)OsSi behave as almost ideal metals, characterized by weak electronic correlations and a relatively low density of states, the replacement of Ta (or Nb) with Zr expands the crystal lattice and shifts ZrOsSi towards an insulator. Our ab initio calculations indicate that ZrOsSi is a Z2 topological metal with clear surface Dirac cones and properties similar to a doped strong topological insulator.

Hall effect of ferro/antiferromagnetic wallpaper fermions

Nonsymmorphic crystals can host characteristic double surface Dirac cones with fourfold degeneracy on the Dirac points, called wallpaper fermion, protected by wallpaper group symmetry. We clarify the charge and spin Hall effect of wallpaper fermions in the presence of the (anti)ferromagnetism. Based on a four-sublattice model, we construct the effective Hamiltonian of wallpaper fermions coupled with the ferromagnetic or antiferromagnetic moment. Both ferromagnetic and antiferromagnetic moments induce an energy gap for the wallpaper fermions, leading to quantized (spin) Hall conductivity. The ferromagnetic wallpaper fermion induces the Hall conductivity quantized into ${e}^{2}/h$, which is twice that for a single Dirac fermion on the surface of topological insulators. On the other hand, the spin Hall conductivity decays and reaches to be a finite value as the antiferromagnetic coupling increases. We also show that the results above are valid for a general model of wallpaper fermions from symmetry consideration.

Engineering axion insulator and other topological phases in superlattices without inversion symmetry

We study theoretically the interplay between magnetism and topology in three-dimensional HgTe/MnTe superlattices stacked along the (001) axis. Our results show the evolution of the magnetic topological phases with respect to the magnetic configurations. An axion insulator phase is observed for the antiferromagnetic order with the out-of-plane N\'eel vector direction below a critical thickness of MnTe, which is the ground state amongst all magnetic configurations. Defining $T$ as the time-reversal symmetry, this axion insulator phase is protected by a magnetic two-fold rotational symmetry $C_2{\cdot}T$. The axion insulator phase evolves into a trivial insulator as we increase the thickness of the magnetic layers. By switching the N\'eel vector direction into the $ab$ plane, the system realizes different antiferromagnetic topological insulators depending on the thickness of MnTe. These phases feature gapless surface Dirac cones shifted away from high-symmetry points on surfaces perpendicular to the N\'eel vector direction of the magnetic layers. In the presence of ferromagnetism, the system realizes a magnetic Weyl semimetal and a ferromagnetic semimetal for out-of-plane and in-plane magnetization directions, respectively. We observe large anomalous Hall conductivity in the presence of ferromagnetism in the three-dimensional superlattice.

Perfect one-dimensional interface states in a twisted stack of three-dimensional topological insulators

We theoretically study the electronic structure of interface states in twisted stacks of three-dimensional topological insulators. When the center of the surface Dirac cone is located at a midpoint of a side of the Brillouin zone boundary, we find that an array of nearly independent one-dimensional channels is formed by the interface hybridization of the surface states, even when the moir\'e pattern itself is isotropic. The two counterpropagating channels have opposite spin polarization, and they are robust against scattering by spin-independent impurities. The coupling between the parallel channels can be tuned by the twist angle. The unique one-dimensional states can be understood as effective Landau levels where the twist angle works as a fictitious magnetic field.

Non-trivial band topology in Bi doped Lanthanum monopnictides (LaX; X = As and Sb)

In this study, we report first-principles investigation of non-trivial topological characteristics in Bi doped Lanthanum monopnictides, particularly LaAs and LaSb. For this, we varied the concentration of Bi atoms in 1 × 1 × 2 supercell of LaAs and LaSb unit cells along [001] direction, and phonon-dispersion spectra of all these doped arrangements are observed to be dynamically stable.HSE06 band structure calculations with spin–orbit coupling reveal that the non-trivial topological characteristics starts appearing in 50% concentration of Bi atoms in the given supercells of LaAs and LaSb. The non-trivial topological characteristics in the proposed doped arrangements are verified by bulk band inversion, unit Z2 topological invariant, and a surface Dirac cone. Our findings provide promising candidates of non-trivial topological family, which may carry huge potential for extremely large magnetoresistance and spintronics applications.

Observation of anisotropic Dirac cones in the topological material Ti2Te2P

Anisotropic bulk Dirac (or Weyl) cones in three-dimensional systems have recently gained intense research interest as they are examples of materials with tilted Dirac (or Weyl) cones indicating the violation of Lorentz invariance. In contrast, the studies on anisotropic surface Dirac cones in topological materials which contribute to anisotropic carrier mobility have been limited. By employing angle-resolved photoemission spectroscopy and first-principles calculations, we reveal the anisotropic surface Dirac dispersion in a tetradymite material ${\mathrm{Ti}}_{2}{\mathrm{Te}}_{2}\mathrm{P}$ on the (001) plane of the Brillouin zone. We observe quasielliptical Fermi pockets at the $\overline{M}$ point of the Brillouin zone forming the anisotropic surface Dirac cones. Our calculations of the ${\mathbb{Z}}_{2}$ indices confirm that the system is topologically nontrivial with multiple topological phases in the same material. In addition, the observed nodal-line-like feature formed by bulk bands makes this system topologically rich.

Nanometric Moiré Stripes on the Surface of Bi2Se3 Topological Insulator.

Mismatch between adjacent atomic layers in low-dimensional materials, generating moiré patterns, has recently emerged as a suitable method to tune electronic properties by inducing strong electron correlations and generating novel phenomena. Beyond graphene, van der Waals structures such as three-dimensional (3D) topological insulators (TIs) appear as ideal candidates for the study of these phenomena due to the weak coupling between layers. Here we discover and investigate the origin of 1D moiré stripes on the surface of Bi2Se3 TI thin films and nanobelts. Scanning tunneling microscopy and high-resolution transmission electron microscopy reveal a unidirectional strained top layer, in the range 14–25%, with respect to the relaxed bulk structure, which cannot be ascribed to the mismatch with the substrate lattice but rather to strain induced by a specific growth mechanism. The 1D stripes are characterized by a spatial modulation of the local density of states, which is strongly enhanced compared to the bulk system. Density functional theory calculations confirm the experimental findings, showing that the TI surface Dirac cone is preserved in the 1D moiré stripes, as expected from the topology, though with a heavily renormalized Fermi velocity that also changes between the top and valley of the stripes. The strongly enhanced density of surface states in the TI 1D moiré superstructure can be instrumental in promoting strong correlations in the topological surface states, which can be responsible for surface magnetism and topological superconductivity.

Magnetic moiré surface states and flat Chern bands in topological insulators

We propose the magnetic moir\'e surface states of topological insulators as a platform for time-reversal symmetry breaking flat Chern bands. We construct a generic continuum model of Dirac electrons with a single Dirac cone moving in a time-reversal symmetry breaking periodic potential containing minimal parameters. The Zeeman-type moir\'e potentials generically gap out the moir\'e surface Dirac cones and give rise to isolated flat Chern minibands with Chern number $\ifmmode\pm\else\textpm\fi{}1$. Interestingly, the bandwidth of such isolated flat minibands is insensitive to the twisting angle, which originates from the completely flat band of the $p$ orbitals on the honeycomb lattice. This result provides a promising platform for realizing time-reversal breaking correlated topological phases and circumvents two main experimental difficulties in correlated states with twisted graphene: that the bandwidth varies rapidly with angle as well as the moir\'e disorder. Furthermore, in a ${C}_{6}$ periodic potential, with the interplay of the scalar potential and the Zeeman moir\'e potential, the $\mathrm{\ensuremath{\Gamma}}$-valley moir\'e surface electrons simulate two-dimensional honeycomb lattice physics, leading to emergent moir\'e surface Dirac cones as well as the Haldane model.

Higher-order topological semimetals and nodal superconductors with an order-two crystalline symmetry

Using a systematic relation between topological gapless phases in three dimensions and topological gapped phases in two dimensions, we identify four types of higher-order topological semimetals or nodal superconductors (HOTS), hosting (i) flat zero-energy "Fermi arcs" at crystal hinges, (ii) flat zero-energy hinge arcs coexisting with surface Dirac cones, (iii) chiral or helical hinge modes, or (iv) flat zero-energy hinge arcs connecting nodes only at finite momentum. Bulk-boundary correspondence relates the hinge states to the bulk topology protecting the nodal point or loop. We classify all HOTS for all tenfold-way classes with an order-two crystalline (anti-)symmetry, such as mirror, twofold rotation, or inversion.

Yu-Shiba-Rusinov States in a Superconductor with Topological Z_{2} Bands.

A Yu-Shiba-Rusinov (YSR) state is a localized in-gap state induced by a magnetic impurity in a superconductor. Recent experiments used an STM tip to manipulate the exchange coupling between an Fe adatom and the FeTe_{0.55}Se_{0.45} superconductor possessing a Z_{2} nontrivial band structure with topological surface states. As the tip moves close to the single Fe adatom, the energy of the in-gap state modulates and exhibits a zero-energy crossing followed by an unusual return to zero energy, which cannot be understood by coupling the magnetic impurity to the superconducting topological surface Dirac cone. Here, we numerically and analytically study the YSR states in superconductors with nontrivial Z_{2} bands and show the emergence of the two zero-energy crossings as a function of the exchange coupling between the magnetic impurity and the bulk states. We analyze the role of the topological surface states and compare in-gap states to systems with trivial Z_{2} bands. The spin polarization of the YSR states is further studied for future experimental measurement.

Surface Bogoliubov-Dirac cones and helical Majorana hinge modes in superconducting Dirac semimetals

In the presence of certain symmetries, three-dimensional Dirac semimetals can harbor not only surface Fermi arcs, but also surface Dirac cones. Motivated by the experimental observation of rotation-symmetry-protected Dirac semimetal states in iron-based superconductors, we investigate the potential intrinsic topological phases in a $C_{4z}$-rotational invariant superconducting Dirac semimetal with $s_{\pm}$-wave pairing. When the normal state harbors only surface Fermi arcs on the side surfaces, we find that an interesting gapped superconducting state with a quartet of Bogoliubov-Dirac cones on each side surface can be realized, even though the first-order topology of its bulk is trivial. When the normal state simultaneously harbors surface Fermi arcs and surface Dirac cones, we find that a second-order time-reversal invariant topological superconductor with helical Majorana hinge states can be realized. The criteria for these two distinct topological phases have a simple geometric interpretation in terms of three characteristic surfaces in momentum space. By reducing the bulk material to a thin film normal to the axis of rotation symmetry, we further find that a two-dimensional first-order time-reversal invariant topological superconductor can be realized if the inversion symmetry is broken by applying a gate voltage. Our work reveals that diverse topological superconducting phases and types of Majorana modes can be realized in superconducting Dirac semimetals.

Fragile topological insulators protected by rotation symmetry without spin-orbit coupling

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.

Multiple magnetism-controlled topological states in EuAgAs

The interplay between magnetism and band topology is a focus of current research on magnetic topological systems. Based on first-principle calculations and symmetry analysis, we reveal multiple intriguing topological states can be realized in a single system EuAgAs, controlled by the magnetic ordering. The material is Dirac semimetal in the paramagnetic state, with a pair of accidental Dirac points. Under different magnetic configurations, the Dirac points can evolve into magnetic triply-degenerate points, magnetic linear and double Weyl points, or being gapped out and making the system a topological mirror semimetal characterized by mirror Chern numbers. The change in bulk topology is also manifested in the surface states, including the surface Fermi arcs and surface Dirac cones. In addition, the antiferromagnetic states also feature a nontrivial Z4 index, implying a higher order topology. These results deepen our understanding of magnetic topological states and provide new perspectives for spintronic applications.