Helical Surface States 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

Helical Topological Superconducting Pairing at Finite Excitation Energies.

We propose helical topological superconductivity away from the Fermi surface in three-dimensional time-reversal-symmetric odd-parity multiband superconductors. In these systems, pairing between electrons originating from different bands is responsible for the corresponding topological phase transition. Consequently, a pair of helical topological Dirac surface states emerges at finite excitation energies. These helical Dirac surface states are tunable in energy by chemical potential and strength of band splitting. They are protected by time-reversal symmetry combined with crystalline twofold rotation symmetry. We suggest concrete materials in which this phenomenon could be observed.

Fully gapped pairing state in spin-triplet superconductor UTe2.

The recently discovered superconductor UTe2 is a promising candidate for spin-triplet superconductors, but the symmetry of the superconducting order parameter remains highly controversial. Here, we determine the superconducting gap structure by the thermal conductivity of ultraclean UTe2 single crystals. We find that the a-axis thermal conductivity divided by temperature κ/T in zero-temperature limit is vanishingly small for both magnetic field H‖a and H‖c axes up to H/Hc2 ∼ 0.2, demonstrating the absence of nodes around the a axis contrary to the previous belief. The present results, combined with the reduction of nuclear magnetic resonance Knight shift, indicate that the superconducting order parameter belongs to the isotropic Au representation with a fully gapped pairing state, analogous to the B phase of superfluid 3He. These findings reveal that UTe2 is likely to be a long-sought three-dimensional strong topological superconductor, hosting helical Majorana surface states on any crystal plane.

Electric field-dependent photoexcited spin-polarized electron transport in Bi2Se3 topological insulator

Three-dimensional (3D) topological insulators (TIs) exhibit spin-polarized surface states in which the spin of electrons is locked to their momentum. The helical surface states can be explored from circularly polarized light-induced spin photocurrent, a phenomenon called circular photogalvanic effect (CPGE). In this work, we fabricate a TI transistor with the Bi2Se3 channel layer synthesized using vapor deposition. The photocurrent response of Bi2Se3 TI is characterized under horizontal and longitudinal electric fields. CPGE and linear photogalvanic effect (LPGE) contribute to the surface state photocurrent at room temperature. The longitudinal electric field provides kinetic energy to the electrons so that the transition of electrons to higher energy bands in momentum space occurs. Under a photon excitation with the energy far above the TI bandgap, we observed a photocurrent difference between left and right circularly polarized light excitation. The photocurrent variation with gate voltage (longitudinal field) is also investigated. Adjusting the Fermi level with the gate bias leads to changes in the population of bulk state carriers and spin electrons in surface states. By shifting the gate bias toward negative, the CPGE current increases because of the reduced scattering with bulk carriers. Our work reveals that longitudinal and horizontal electric fields can manipulate the helical spin-polarized photocurrent of Bi2Se3. From the asymmetry of circularly polarized photoconductive differential current (CPDC) under the horizontal field, we found evidence of spin-polarized electron transition to other conduction band valleys at room temperature under a high-energy photon excitation.

Chiral and helical states in selective-area epitaxial heterostructure

The quasi-1D chiral edge states in a quantum anomalous Hall insulator are dissipationless, while the 2D helical surface states in a topological insulator are insensitive to spin-independent scatterings due to the topological protection. Both serve as essential ingredients for topological electronics. Here, we integrate these states into a single device using selective area epitaxy based on the molecular beam epitaxy technique. The chiral edge state comes from the quantum anomalous Hall insulator Cr:(Bi,Sb)2Te3, while the helical surface state comes from the intrinsic topological insulator (Bi,Sb)2Te3 which only interfaces with a partial edge of the former, forming a selective-area heterostructure. At the heterointerface, the chiral state in Cr:(Bi,Sb)2Te3 is allowed to be scattered into (Bi,Sb)2Te3 so that the incoming current will be redistributed according to the coordination between the chirality and helicity. Our device enables the collaboration between chiral and helical states for low-dissipative transport with tunable current dimension.

Parallel and anti-parallel helical surface states for topological semimetals

Weyl points, carrying a Z-type monopole charge C, have bulk-surface correspondence (BSC) associated with helical surface states (HSSs). When |C| >1, multi-HSSs can appear in a parallel manner. However, when a pair of Weyl points carrying C=pm 1 meet, a Dirac point carrying C = 0 can be obtained and the BSC vanishes. Nonetheless, a recent study in Zhang et al. (Phys Rev Res 4:033170, 2022) shows that a new BSC can survive for Dirac points when the system has time-reversal ({T})-glide ({G}) symmetry ({tilde{Theta }}=TG), i.e., anti-parallel double/quad-HSSs associated with a new Z_{2}-type monopole charge {Q} appear. In this paper, we systematically review and discuss both the parallel and anti-parallel multi-HSSs for Weyl and Dirac points, carrying two different kinds of monopole charges. Two material examples are offered to understand the whole configuration of multi-HSSs. One carries the Z-type monopole charge C, showing both local and global topology for three kinds of Weyl points, and it leads to parallel multi-HSSs. The other carries the Z_{2}-type monopole charge {Q}, only showing the global topology for {tilde{Theta }}-invariant Dirac points, and it is accompanied by anti-parallel multi-HSSs.

Magnetic skyrmion crystal at a topological insulator surface

We consider a magnetic skyrmion crystal formed at the surface of a topological insulator. Incorporating the exchange interaction between the helical Dirac surface states and the periodic N\'eel or Bloch skyrmion texture, we obtain the resulting electronic band structure and discuss the constraints that symmetries impose on the energies and Berry curvature. We find substantive qualitative differences between the N\'eel and Bloch cases, with the latter generically permitting a multiband low energy tight-binding representation whose parameters are tightly constrained by symmetries. We explicitly compute the associated Wannier orbitals, which resemble the ringlike chiral bound states of helical Dirac fermions coupled to a single skyrmion in a ferromagnetic background. We construct a two-band tight-binding model with real nearest-neighbor hoppings which captures the salient topological features of the low-energy bands. Our results are relevant to magnetic topological insulators (TIs), as well as to TI-magnetic thin film heterostructures, in which skyrmion crystals may be stabilized.

Symmetry-enforced topological band crossings in orthorhombic crystals: Classification and materials discovery

We identify all symmetry-enforced band crossings in nonmagnetic orthorhombic crystals with and without spin-orbit coupling and discuss their topological properties. We find that orthorhombic crystals can host a large number of different band degeneracies, including movable Weyl and Dirac points with hourglass dispersions, fourfold double Weyl points, Weyl and Dirac nodal lines, almost movable nodal lines, nodal chains, and topological nodal planes. Interestingly, spin-orbit coupled materials in the space groups 18, 36, 44, 45, and 46 can have band pairs with only two Weyl points in the entire Brillouin zone. This results in a simpler connectivity of the Fermi arcs and more pronounced topological responses than in materials with four or more Weyl points. In addition, we show that the symmetries of the space groups 56, 61, and 62 enforce nontrivial weak $\mathbb{Z}_2$ topology in materials with strong spin-orbit coupling, leading to helical surface states. With these classification results in hand, we perform extensive database searches for orthorhombic materials crystallizing in the relevant space groups. We find that Sr$_2$Bi$_3$ and Ir$_2$Si have bands crossing the Fermi energy with a symmetry-enforced nontrivial $\mathbb{Z}_2$ invariant, CuIrB possesses nodal chains near the Fermi energy, Pd$_7$Se$_4$ and Ag$_2$Se exhibit fourfold double Weyl points, the latter one even in the absence of spin-orbit coupling, whereas the fourfold degeneracies in AuTlSb are made up from intersecting nodal lines. For each of these examples we compute the ab-initio band structures, discuss their topologies, and for some cases also calculate the surface states.

Surface step states and Majorana end states in profiled topological-insulator thin films

The protected helical surface states in thin films of topological insulators (TI) are subject to inter-surface hybridisation. This leads to gap opening and spin texture changes as witnessed in photoemission and quasiparticle interference investigations. Theoretical studies show that universally the hybridisation energy exhibits exponential decay as well as sign oscillations as a function of film thickness, depending on the effective band parameters of the material. When a step is introduced in the TI film e.g. by profiling the substrate such that the hybridisation has different signs on both sides of the step, 1D bound states appear within the hybridisation gap which decay exponentially with distance from the step. The step bound states have linear dispersion and inherit the helical spin locking from the surface states and are therefore non-degenerate. When the substrate becomes an s-wave superconductor Majorana zero modes located at the step ends are created inside the superconducting gap. The proposed scenario involves just a suitably stepped interface of superconductor and TI and therefore may be a most simple device being able to host Majorana zero modes.

Dual Topological Features of Weyl Semimetallic Phases in Tetradymite BiSbTe3 Supported by the National Natural Science Foundation of China (Grant Nos. 11604032, 11674040, and 51672270), and the Fundamental Research Funds for the Central Universities (Grant No. 106112016CDJZR308808).

Based on first-principles calculations and symmetry arguments, we reveal that the non-centrosymmetric ternary tetradymite BiSbTe3 possesses exotic dual topological features of Weyl semimetallic phases with Z 2 index (1:000). The results show that the helical Dirac-type surface states protected by the time-reversal symmetry are present in the vicinity of the Brillouin zone center, which is consistent with the experimental report. Furthermore, we show that four pairs of Weyl points reside exactly at the Fermi level, which are guaranteed to be located on high-symmetry planes due to mirror symmetries. The helical surface states and the projected Weyl nodes are well separated in the momentum space, facilitating their observations in experiments. This work not only uncovers a unique quantum phenomenon with dual topological features in the tetradymite family but also paves a fascinating avenue for exploring the coexistence of multi-topological states with wide applications.

Majorana bound states in vortex lattices on iron-based superconductors

Majorana quasi-particles may arise as zero-energy bound states in vortices on the surface of a topological insulator that is proximitized by a conventional superconductor. Such a system finds its natural realization in the iron-based superconductor FeTe0.55Se0.45 that combines bulk s-wave pairing with spin helical Dirac surface states, and which thus comprises the ingredients for Majorana modes in absence of an additional proximitizing superconductor. In this work, we investigate the emergence of Majorana vortex modes and lattices in such materials depending on parameters like the magnetic field strength and vortex lattice disorder. A simple 2D square lattice model here allows us to capture the basic physics of the underlying materials system. To address the problem of disordered vortex lattice, which occurs in real systems, we adopt the technique of the singular gauge transformation which we modify such that it can be used in a system with periodic boundary conditions. This approach allows us to go to larger vortex lattices than otherwise accessible, and is successful in replicating several experimental observations of Majorana vortex bound states in the FeTe0.55Se0.45 platform. Finally it can be related to a simple disordered Majorana lattice model that should be useful for further investigations on the role of interactions, and toward topological quantum computation.

High-harmonic generation from topological surface states

Three-dimensional topological insulators are a phase of matter that hosts unique spin-polarized gapless surface states that are protected by time-reversal symmetry. They exhibit unconventional charge and spin transport properties1,2. Intense laser fields can drive ballistic charge dynamics in Dirac bands3,4 or they can coherently steer spin5 and valley pseudospin6. Similarly, high-harmonic generation (HHG) in solids provides insights into the dynamics of the electrons in topological insulators7–13. Despite several theoretical attempts to identify a topological signature in the high-harmonic spectrum14–16, a unique fingerprint has yet to be found experimentally. Here, we observe HHG that arises from topological surface states in the intrinsic topological insulator BiSbTeSe2. The components of the even-order harmonics that are polarized along the pump polarization stem from the spin current in helical surface states, whereas the perpendicular components originate from the out-of-plane spin polarization related to the hexagonal wrapping effect17. The dependence of HHG on surface doping in ambient air also suggests the presence of a Rashba-split two-dimensional electron gas, whose strength can be enhanced by an increase in the intensity of the mid-infrared pump. High-harmonic generation up to the seventh harmonic is observed from the intrinsic three-dimensional topological insulator BiSbTeSe2. The parallel components of the even-order harmonics arise directly from the topological surface states.

Terahertz photoresistivity of a high-mobility 3D topological insulator based on a strained HgTe film

We report on a detailed study of terahertz (THz) photoresistivity in a strained HgTe three-dimensional topological insulator (3D TI) for all Fermi level positions: inside the conduction and valence bands and in the bulk gap. In the presence of a magnetic field, we detect a resonance corresponding to the cyclotron resonance (CR) in the top surface Dirac fermions (DFs) and examine the nontrivial dependence of the surface state cyclotron mass on the Fermi level position. We also detect additional resonant features at moderate electron densities and demonstrate that they are caused by the mixing of surface DFs and bulk electrons. At high electron densities, we observe THz radiation-induced 1/B-periodic low-field magneto-oscillations coupled to harmonics of the CR and demonstrate that they have a common origin with microwave-induced resistance oscillations previously observed in high mobility GaAs-based heterostructures. This observation attests the superior quality of a 2D electron system formed by helical surface states in strained HgTe films.

Scattering symmetry-breaking induced spin photocurrent from out-of-plane spin texture in a 3D topological insulator

We theoretically study helicity-dependent photocurrent in a three-dimensional topological insulator Bi2Te3 under elastic scattering of different symmetries. By exploring spin-selective optical transitions and symmetry-breaking scattering, we are able to address the out-of-plane spin texture of the topological helical surface states and to generate directional, spin-polarization tunable photocurrent that is otherwise forbidden for the original C3v symmetry of the surface. This can be achieved regardless of the Fermi level, even under the condition when the topological states are inaccessible in dark. This work paves the way to robustly explore the out-of-plane spin texture for harvesting opto-spintronic functionalities of topological insulators.

Reducing the Impact of Bulk Doping on Transport Properties of Bi‐Based 3D Topological Insulators

The observation of helical surface states in Bi‐based 3D topological insulators (TIs) has been a challenge since their theoretical prediction. The main issue arises when the Fermi level shifts deep into the bulk conduction band due to unintentional doping. This results in the metallic conduction of the bulk which dominates the transport measurements and hinders the probing of the surface states in these experiments. Here, various strategies are investigated to reduce the residual doping in Bi‐based TIs. Flakes of Bi2Se3 and Bi1.5Sb0.5Te1.7Se1.3 are grown by physical vapor deposition and their structural and electronic properties are compared with mechanically exfoliated flakes. Using Raman spectroscopy, the role of the substrate in this process is explored, and the optimal conditions for the fabrication of high‐quality crystals are presented. Despite this improvement, it is shown that the vapor phase‐deposited flakes still suffer from structural disorder which leads to the residual n‐type doping of the bulk. Using magneto‐transport measurements, we find that exfoliated flakes show better electrical properties and are thus more promising for the probing of surface states.

Correction to Large Tunable Spin-to-Charge Conversion Induced by Hybrid Rashba and Dirac Surface States in Topological Insulator Heterostructures.

Topological insulators (TIs) have emerged as some of the most efficient spin-to-charge convertors because of their correlated spin-momentum locking at helical Dirac surface states. While endeavors ...

Time-reversal symmetry breaking in topological superconductor Sr0.1Bi2Se3

The single helical Fermi surface on the surface state of three-dimensional topological insulator ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ is constrained by the time-reversal invariant bulk topology to possess a spin-singlet superconducting pairing symmetry. In fact, the Cu-doped and pressure-tuned superconducting ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ show no evidence of the time-reversal symmetry (TRS) breaking. We report on the detection of the TRS breaking in the topological superconductor ${\mathrm{Sr}}_{0.1}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, probed by zero-field $\ensuremath{\mu}\mathrm{SR}$ measurements. The TRS breaking provides strong evidence for the existence of a spin-triplet pairing state. The existence of TRS breaking is also verified by longitudinal-field $\ensuremath{\mu}\mathrm{SR}$ measurements, which negates the possibility of magnetic impurities as the source of TRS breaking. The temperature-dependent superfluid density deduced from transverse-field $\ensuremath{\mu}\mathrm{SR}$ measurements yields nodeless superconductivity with low superconducting carrier density and penetration depth $\ensuremath{\lambda}=1622(134)\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$. From the microscopic theory of unconventional pairing, we find that such a fully gapped spin-triplet pairing channel is promoted by the complex interplay between the structural hexagonal warping and higher order Dresselhaus spin-orbit-coupling terms. Based on Ginzburg-Landau analysis, we delineate the mixing of singlet- to triplet-pairing symmetry as the chemical potential is tuned far above from the Dirac cone. Our observation of such spontaneous TRS breaking chiral superconductivity on a helical surface state, protected by the TRS invariant bulk topology, can open avenues for interesting research and applications.

Interlayer RKKY coupling in bulk Rashba semiconductors under topological phase transition

The bulk Rashba semiconductors BiTeX (X=I, Cl and Br) with intrinsically enhanced Rashba spin-orbit coupling provide a new platform for investigation of spintronic and magnetic phenomena in materials. We theoretically investigate the interlayer exchange interaction between two ferromagnets deposited on opposite surfaces of a bulk Rashba semiconductor BiTeI in its trivial and topological insulator phases. In the trivial phase BiTeI, we find that for ferromagnets with a magnetization orthogonal to the interface, the exchange coupling is reminiscent of that of a conventional three-dimensional metal. Remarkably, ferromagnets with a magnetization parallel to the interface display a magnetic exchange qualitatively different from that of conventional three-dimensional metal due to the spin-orbit coupling. In this case, the interlayer exchange interaction acquires two periods of oscillations and decays as the inverse of the thickness of the BiTeI layer. For topological BiTeI, the magnetic exchange interaction becomes mediated only by the helical surface states and acts between the one-dimensional spin chains at the edges of the sample. The surface state-mediated interlayer exchange interaction allows for the coupling of ferromagnets with non-collinear magnetization and displays a decay power different from that of trivial BiTeI, allowing the detection of the topological phase transition in this material. Our work provides insights into the magnetic properties of these newly discovered materials and their possible functionalization.

Electrical detection of the surface spin polarization of the candidate topological Kondo insulator SmB6

The Kondo insulator compound $\mathrm{Sm}{\mathrm{B}}_{6}$ has emerged as a strong candidate for the realization of a topologically nontrivial state in a strongly correlated system, a topological Kondo insulator, which can be a novel platform for investigating the interplay between nontrivial topology and emergent correlation-driven phenomena in solid-state systems. Electronic transport measurements on this material, however, so far showed only the robust surface-dominated charge conduction at low temperatures, lacking evidence of its connection to the topological nature by showing, for example, spin polarization due to spin-momentum locking. Here, we find evidence for surface-state spin polarization by electrical detection of a current-induced spin chemical-potential difference on the surface of a $\mathrm{Sm}{\mathrm{B}}_{6}$ single crystal. We clearly observe that a surface-dominated spin voltage, which is proportional to the projection of the spin polarization onto the contact magnetization, is determined by the direction and magnitude of the charge current and is strongly temperature dependent due to the crossover from surface to bulk conduction. We estimate the lower bound of the surface-state net spin polarization as 25% based on the quantum transport model, providing direct evidence that $\mathrm{Sm}{\mathrm{B}}_{6}$ supports metallic spin helical surface states.

Dephasing effects in topological insulators

Topological insulators, a class of typical topological materials in both two dimensions and three dimensions, are insulating in bulk and metallic at surface. The spin-momentum locked surface states and peculiar transport properties exhibit promising potential applications on quantum devices, which generate extensive interest in the last decade. Dephasing is the process of the loss of phase coherence, which inevitably exists in a realistic sample. In this review, we focus on recent progress in dephasing effects on the topological insulators. In general, there are two types of dephasing processes: normal dephasing and spin dephasing. In two-dimensional topological insulators, the phenomenologically numerical investigation shows that the longitudinal resistance plateaus is robust against normal dephasing but fragile with spin dephasing. Several microscopic mechanisms of spin dephasing are then discussed. In three-dimensional topological insulators, the helical surface states exhibit a helical spin texture due to the spin-momentum locking mechanism. Thus, normal dephasing has close connection to spin dephasing in this case, and gives rise to anomalous “gap-like” feature. Dephasing effects on properties of helical surface states are investigated.