First direct observation of Spin-textures in Topological Insulators : Spin-resolved ARPES as a probe of topological quantum spin Hall effect and Berry's phase
D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, F. Meier, J. Osterwalder, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava, M. Z. Hasan
备注：
15 pages, 3 figures, first Submitted to SCIENCE on July-22, 2008
A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry based classification of condensed matter. Exotic spin transport phenomena such as the dissipationless quantum spin Hall effect have been speculated to originate from a novel topological order whose identification requires a spin sensitive measurement, which does not exist to this date in any system (neither in Hg(Cd)Te quantum wells nor in the topological insulator BiSb). Using Mott polarimetry, we probe the spin degrees of freedom of these quantum spin Hall states and demonstrate that topological quantum numbers are uniquely determined from spin texture imaging measurements. Applying this method to the Bi{1-x}Sb{x} series, we identify the origin of its novel order and unusual chiral properties. These results taken together constitute the first observation of surface electrons collectively carrying a geometrical quantum (Berry's) phase and definite chirality (mirror Chern number, n_M =-1), which are the key electronic properties for realizing topological computing bits with intrinsic spin Hall-like topological phenomena. Our spin-resolved results not only provides the first clear proof of a topological insulating state in nature but also demonstrate the utility of spin-resolved ARPES technique in measuring the quantum spin Hall phases of matter.
The energy-momentum relationship of electrons on the surface of an ideal "Hydrogen-Atom" Topological Insulator forms a cone - a Dirac cone, which, when warped and distorted (no longer described by the Dirac equation), can lead to unusual phenomena such as enhanced electronic interference around defects (observed in STM) and a magnetically ordered broken symmetry surface with lost topological protection. A detailed spin-texture and hexagonal warping maps on Bi2Te3 are presented in this context as a Viewpoint on L.Fu, Phys. Rev. Lett. \textbf{103}, 266801 (2009).
Quantum spin Hall insulator is characterized by the helical edge states, with the spin polarization of electron being locked to its direction of motion. Although the edge-state conduction has been observed, unambiguous evidence of the helical spin texture is still lacking. Here, we investigate the coherent edge-state transport in an interference loop pinched by two point contacts. Due to the helical character, the forward inter-edge scattering enforces a $π$ spin rotation. Two successive processes can only produce a nontrivial $2π$ or trivial $0$ spin rotation, which can be controlled by the R…
Antiferromagnets (AFMs), in contrast to ferromagnets, show a nontrivial magnetic structure with zero net magnetization. However, they share a number of spintronic effects with ferromagnets, including spin-pumping and spin transfer torques. Both phenomena stem from the coupled dynamics of free carriers and localized magnetic moments. In the present paper I study the adiabatic dynamics of a spin-polarized electrons in a metallic AFM exhibiting a noncollinear 120$^\circ$ magnetic structure. I show that the slowly varying AFM spin texture produces a non-Abelian gauge potential related to the time/…
Nonequilibrium quantum dynamics of many-body systems is the frontier of condensed matter physics; recent advances in various time-resolved spectroscopic techniques continue to reveal rich phenomena. Angle-resolved photoemission spectroscopy (ARPES) as one powerful technique can resolve electronic energy, momentum, and spin along the time axis after excitation. However, dynamics of spin textures in momentum space remains mostly unexplored. Here we demonstrate theoretically that the photoexcited surface state of genuine or magnetically doped topological insulators shows novel topological spin te…
We study Bi2Se3 by polarization-dependent angle-resolved photoemission spectroscopy (ARPES) and density-functional theory slab calculations. We find that the surface state Dirac fermions are characterized by a layer-dependent entangled spin-orbital texture, which becomes apparent through quantum interference effects. This explains the discrepancy between the spin polarization from spin-resovled ARPES - ranging from 20 to 85% - and the 100% value assumed in phenomenological models. It also suggests a way to probe the intrinsic spin texture of topological insulators, and to continuously manipula…
Understanding and control of spin degrees of freedom on the surfaces of topological materials are key to future applications as well as for realizing novel physics such as the axion electrodynamics associated with time-reversal (TR) symmetry breaking on the surface. We experimentally demonstrate magnetically induced spin reorientation phenomena simultaneous with a Dirac-metal to gapped-insulator transition on the surfaces of manganese-doped Bi2Se3 thin films. The resulting electronic groundstate exhibits unique hedgehog-like spin textures at low energies, which directly demonstrate the mechani…
The high index (441) surface of bismuth has been studied using Scanning Tunnelling Microscopy (STM), Angle Resolved Photoemission Spectroscopy (APRES) and spin-resolved ARPES. The surface is strongly corrugated, exposing a regular array of (110)-like terraces. Two surface localised states are observed, both of which are linearly dispersing in one in-plane direction ($k_x$), and dispersionless in the orthogonal in-plane direction ($k_y$), and both of which have a Dirac-like crossing at $k_x$=0. Spin ARPES reveals a strong in-plane polarisation, consistent with Rashba-like spin-orbit coupling. O…
We theoretically study the Berry curvature of the magnon induced by the hybridization with the acoustic phonons via the spin-orbit and dipolar interactions. We first discuss the magnon-phonon hybridization via the dipolar interaction, and show that the dispersions have gapless points in momentum space, some of which form a loop. Next, when both spin-orbit and dipolar interactions are considered, we show anisotropic texture of the Berry curvature and its divergence with and without gap-closing. Realistic evaluation of the consequent anomalous velocity is given for yttrium iron garnet.
Topologically nontrivial two-dimensional materials hold great promise for next-generation optoelectronic applications. However, measuring the Hall or spin-Hall response is often a challenge and practically limited to the ground state. An experimental technique for tracing the topological character in a differential fashion would provide useful insights. In this work, we show that circular dichroism angle-resolved photoelectron spectroscopy (ARPES) provides a powerful tool which can resolve the topological and quantum-geometrical character in momentum space. In particular, we investigate how to…
We propose to use quantized Berry phases as local order parameters of gapped quantum liquids, which are invariant under some anti-unitary operation. After presenting a general prescription, the scheme is applied for Heisenberg models with frustrations and two dimensional extended Su-Schrieffer-Heeger models associated with a random dimer covering. In each phases, the quantized Berry phases, as 0 or $π$, describe the ground state texture pattern of local singlet bonds and dimer bonds. Also possible applications to large classes of correlated electron systems are discussed.
In two-dimensional time-reversal symmetric topological insulators described by Dirac models, the ${\mathbb Z}_{2}$ topological invariant can be described by the spin Chern number. We present a linear response theory for the spin Berry curvature that integrates to the spin Chern number, and introduce its spectral function that can be measured at finite temperature by momentum- and spin-resolved circular dichroism, which may be achieved by pump-probe type of experiments using spin- and time-resolved ARPES. As a result, the sign of the Pfaffian of the ${\mathbb Z}_{2}$ invariant can be directly m…
The unambiguous detection of the band topology for topological nodal systems remains an urgent problem in this field. Usually in experiments this relies on the detection for the topological edge modes, which may requires the high demands for the preparation and purifications of the materials and may give ambiguities. In this work, we propose a new approach towards the unambiguous detection of the bulk band crossing nodes of topological nodal systems without evolving the edge mode. The spirit of our proposal is to couple our targeted Hamiltonian to an weak external field $\mathbf {A}$ with a sp…
The spin orientation of electronic wavefunctions in crystals is an internal degree of freedom, typically insensitive to electrical knobs. We argue from a general symmetry analysis and a $\vec k \cdot \vec p$ perspective, that monolayer 1T'-WTe$_2$ possesses a gate-activated canted spin texture that produces an electrically tunable bulk band quantum geometry. In particular, we find that due to its out-of-plane asymmetry, an applied out-of-plane electric field breaks inversion symmetry to induce both in-plane and out-of-plane electric dipoles. These in-turn generate spin-orbit coupling to lift t…
In a magnetic texture, the spin of a conduction electron is forced to be aligned to the localized moment. As a result, the topology of the magnetic texture affects the electron dynamics in nontrivial ways. A representative example is the topological Hall effect in noncoplanar spin textures with finite spin chirality. While propagating in the noncoplanar spin texture, electrons acquire Berry phase, and their motion is deflected as if they were in a magnetic field. Here, we report a distinct Berry phase effect in a coplanar helimagnet: the spin moment of the conduction electron is polarized unde…
Relative hybridization textures as local coordinates for band geometry and topology
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Condensed Matter Physics Data Center of Chinese Academy of Sciences（中国科学院凝聚态物理数据中心）
We study the role of different orientations of an applied magnetic field as well as the interplay of structural asymmetries on the characteristics of eigenstates in a quantum ring system. We use a nearly analytical model description of the quantum ring, which allows for a thorough study of elliptical deformations and their influence on the spin content and Berry phase of different quantum states. The diamagnetic shift and Zeeman interaction compete with the Rashba spin-orbit interaction, induced by confinement asymmetries and external electric fields, to change spin textures of the different s…
Using a semiclassical Green's function formalism, we discover the emergence of chiral and topological orbital magnetism in two-dimensional chiral spin textures by explicitly finding the corrections to the orbital magnetization, proportional to the powers of the gradients of the texture. We show that in the absence of spin-orbit coupling, the resulting orbital moment can be understood as the electronic response to the emergent magnetic field associated with the real-space Berry curvature. By referring to the Rashba model, we demonstrate that by tuning the parameters of surface systems the engin…
We study the Andreev reflection of a hybrid mesoscopic ring in the presence of a crown-like magnetic texture. By calculating the linear-response conductance as a function of the Zeeman splitting and the magnetic flux through the ring, we are able to identify signatures of the Berry phase acquired by the electrons during transport. This is proposed as a novel detection scheme of the spin-related Berry phase, having the advantage of a larger signal contrast and robustness against ensemble averaging.
We provide a unified semiclassical theory for the conserved current of nonconserved quantities, and manifest it in two physical contexts: the spin current of Bloch electrons and the charge current of mean-field Bogoliubov quasiparticles. Several longstanding problems that limit the playground of the conserved spin current of electrons are solved. We reveal that the hitherto overlooked torque quadrupole density and Berry phase correction to the torque dipole density are essential to assure a circulating spin current with vanishing net flow at equilibrium. The band geometric origin of bulk spin …
We present an open-source software package WannierTools, a tool for investigation of novel topological materials. This code works in the tight-binding framework, which can be generated by another software package Wannier90. It can help to classify the topological phase of given materials by calculating the Wilson loop and can get the surface state spectrum which is detected by angle-resolved photoemission (ARPES) and in scanning tunneling microscopy (STM) experiments. It also identifies positions of Weyl/Dirac points and nodal line structures, calculates the Berry phase around a closed momentu…
A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry based classification of condensed matter. Exotic spin transport phenomena such as the dissipationless quantum spin Hall effect have been speculated to originate from a novel topological order whose identification requires a spin sensitive measurement. Using Spin-resolved-ARPES, we probe the spin degrees of freedom and demonstrate that topological quantum numbers are uniquely determined from spin-texture Berry Phase imaging measurements. Applyi…
Hall experiments in chiral magnets are often analyzed as the sum of an anomalous Hall effect, dominated by momentum-space Berry curvature, and a topological Hall effect, arising from the real-space Berry curvature in the presence of skyrmions, in addition to the ordinary Hall resistivity. This raises the questions of how one can incorporate, on an equal footing, the effects of the anomalous velocity and the real space winding of the magnetization, and when such a decomposition of the resistivity is justified. We provide definitive answers to these questions by including the effects of all phas…
Motivated by recent progress in 2D spintronics, we present Eu deposited on a 1H-WSe$_2$ as a promising platform for engineering spin-orbit effects and Berry curvature. By first-principles calculations based on density functional theory, we show that Eu/WSe$_2$ exhibits intriguing properties such as high magnetic anisotropy, valley-dependent polarization of spin and orbital angular momenta, and their Rashba textures. These originate from magnetic and spin-orbit proximity effects at the interface and the interplay between localized $4f$ magnetic moments of Eu and mobile charge carriers of WSe$_2…
We reveal the meron and antimeron spin textures in momentum space in a photonic crystal slab. These spin textures in momentum space have not been previously noted either in electronic or photonic systems. Breaking the inversion symmetry of a honeycomb photonic crystal gaps out the Dirac cones at the corners of Brillouin zone. The spin textures of photonic bands near the gaps exhibit a meron or antimeron. Unlike the electronic systems, the spin texture of the photonic modes manifests directly in the polarization of the leakage radiation, as the Dirac points can be above the light line. The spin…
Chiral quasiparticles in Bernal-stacked bilayer graphene have valley-contrasting Berry phases of 2π. This nontrival topological structure, associated with the pseudospin winding along a closed Fermi surface, is responsible for various novel electronic properties, such as anti-Klein tunneling, unconventional quantum Hall effect, and valley Hall effect1-6. Here we show that the quantum interference due to intervalley scattering induced by atomic defects/impurities provides further insights into the topological nature of the bilayer graphene. The scattered chiral quasiparticles between distinct v…
Effective dynamics of conduction electrons in antiferromagnetic (AFM) materials with slowly varying spin texture is developed via non-Abelian gauge theory. Quite different from the ferromagnetic (FM) case, the spin of a conduction electron does not follow the background texture even in the adiabatic limit due to the accumulation of a SU(2) non-Abelian Berry phase. Correspondingly, it is found that the orbital dynamics becomes spin-dependent and is affected by two emergent gauge fields. While one of them is the non-Abelian generalization of what has been discovered in FM systems, the other lead…
Starting from the density-matrix equation of motion, we derive a semiclassical kinetic equation for a general two-band electronic Hamiltonian, systematically including quantum-mechanical corrections up to second order in space-time gradients. We find, in addition to band-projected corrections to the single-particle equation of motion due to phase-space Berry curvature, interband terms that we attribute to the nonorthorgonality of the projected Hilbert spaces. As examples, we apply our kinetic equation to electronic systems in the presence of spatially inhomogeneous and dynamical spin textures …
We uncover a novel two-particle Berry phase mechanism to realize exotic corner modes in second-order topological insulators (TIs) and topological superconductors (TSCs) with time-reversal symmetry. We show that the nontrivial pseudospin textures of edge states in two different types of two-dimensional TIs give rise to the novel two-particle geometric phases in the particle-hole and particle-particle channels, respectively, for which the edge mass domain walls or intrinsic $π$-junctions emerge across corners when an external magnetic field or $s$-wave superconductivity is considered, hosting Di…
We develop the hydrodynamical theory of collinear spin currents coupled to magnetization dynamics in metallic ferromagnets. The collective spin density couples to the spin current through a U(1) Berry-phase gauge field determined by the local texture and dynamics of the magnetization. We determine phenomenologically the dissipative corrections to the equation of motion for the electronic current, which consist of a dissipative spin-motive force generated by magnetization dynamics and a magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque on the magne…
We study the thermal transport in two-dimensional systems with a nontrivial Berry curvature texture. The physical realizations are many: for a sake of definiteness we consider undoped graphene gapped by the presence of an aligned hexagonal-Boron-Nitride substrate. The same phenomenology applies, i.e., to surface states of 3D topological insulators in the presence of a uniform magnetization. We find that chiral valley-polarized second-sound collective modes propagate along the edges of the system. The localization length of the edge modes has topological origin stemming from the anomalous veloc…
We study Berry phase effects on conductance properties of diffusive mesoscopic conductors, which are caused by an electron spin moving through an orientationally inhomogeneous magnetic field. Extending previous work, we start with an exact, i.e. not assuming adiabaticity, calculation of the universal conductance fluctuations in a diffusive ring within the weak localization regime, based on a differential equation which we derive for the diffuson in the presence of Zeeman coupling to a magnetic field texture. We calculate the field strength required for adiabaticity and show that this strength …
The protected surface states of topological insulators (TIs) form gapless Dirac cones corresponding non-degenerate eigenstates with helical spin polarisation. The presence of a warping term deforms the isotropic cone of the most simple model into snowflake Fermi surfaces as in Bi2Se3 and Bi2Te3. Their features have been identified in STM quasiparticle interference (QPI) experiments on isolated surfaces. Here we investigate the QPI spectrum for the TI thin-film geometry with finite tunnelling between the surface states. This leads to a dramatic change of spectrum due to gapping and a change in …
The topological structure of the wavefunctions of particles in periodic potentials is characterized by the Berry curvature $Ω_{kn}$ whose integral on the Brillouin zone is a topological invariant known as the Chern number. The bulk-boundary correspondence states that these numbers define the number of edge or surface topologically protected states. It is then of primary interest to find experimental techniques able to measure the Berry curvature. However, up to now, there are no spectroscopic experiments that proved to be capable to obtain information on $Ω_{kn}$ to distinguish different topol…
We study charge and energy transport in a quasi-1D magnetic wire in the presence of magnetic textures. The energy flows can be expressed in a fashion similar to charge currents, leading to new energy-current induced spin torques. In analogy to charge currents, we can identify two reciprocal effects: spin-transfer torque on the magnetic order parameter induced by energy current and the Berry-phase gauge field induced energy flow. In addition, we phenomenologically introduce new $β-$like viscous coupling between magnetic dynamics and energy current into the LLG equation, which originates from sp…
Multiple-$Q$ spin textures, such as magnetic bubble and skyrmion lattices, can be driven into motion by external stimuli. The motion of spin textures affects the electronic states. Here we show that to describe correctly the electronic dynamics, the momentum space needs to be transcended to higher dimensions by including the ancillary dimensions associated with phason modes of the translational motion of the spin textures. The electronic states have non-trivial topology characterized by the first and second Chern numbers in the high dimensional hybrid momentum space. This gives rise to an anom…
Angle-resolved photoemission spectroscopy (ARPES) is the most powerful technique to investigate the electronic band structure of crystalline solids. To completely characterize the electronic structure of topological materials, one needs to go beyond band structure mapping and access information about the momentum-resolved Bloch wavefunction, namely orbitals, Berry curvature, and topological invariants. However, because phase information is lost in the process of measuring photoemission intensities, retrieving the complex-valued Bloch wavefunction from photoemission data has yet remained elusiv…
We predict that a single oscillation of a strong optical pulse can significantly populate the surface conduction band of a three-dimensional topological insulator, Bi2Se3. Both linearly- and circularly-polarized pulses generate chiral textures of interference fringes of population in the surface Brillouin zone. These fringes constitute a self-referenced electron hologram carrying information on the topology of the surface Bloch bands, in particular, on the effect of the warping term of the low-energy Hamiltonian. These electron-interference phenomena are in a sharp contrast to graphene where t…
Multiple types of spin textures and robust valley physics in MP$_2$X$_6$
School of Physics and Technology, Nanjing Normal University（南京师范大学物理与天文学院）；Center for Quantum Transport and Thermal Energy Science, Nanjing Normal University（南京师范大学量子输运与热能源科学中心）
Electron motion in crystals is governed by the coupling between crystal momentum and internal degrees of freedom such as spin implicit in the band structure. The description of this coupling in terms of a momentum-dependent effective field and the resultant Berry phase has greatly advanced our understanding of diverse phenomena including various Hall effects and has lead to the discovery of new states of matter exemplified by topological insulators. While experimental studies on topological systems have focused on the gapless states that emerge at the surfaces or edges, the underlying nontrivi…
The spatial texture of internal degree of freedom of electrons has profound effects on the properties of materials. Such texture in real space can manifest as an emergent magnetic field (or Berry curvature), which is expected to induce interesting valley/spin-related transport phenomena. Moiré pattern, which emerges as a spatial variation at the interface of 2D atomic crystals, provides a natural platform for investigating such real space Berry curvature effects. Here we study moiré structures formed in homobilayer transition metal dichalcogenides (TMDs) due to twisting, various uniform strain…
The quantum anomalous Hall (QAH) effect is a quintessential consequence of non-zero Berry curvature in momentum-space. The QAH insulator harbors dissipation-free chiral edge states in the absence of an external magnetic field. On the other hand, the topological Hall (TH) effect, a transport hallmark of the chiral spin textures, is a consequence of real-space Berry curvature. While both the QAH and TH effects have been reported separately, their coexistence, a manifestation of entangled chiral edge states and chiral spin textures, has not been reported. Here, by inserting a TI layer between two…
Skyrmions are topologically protected, two-dimensional, localized hedgehogs and whorls of spin. Originally invented as a concept in field theory for nuclear interactions, skyrmions are central to a wide range of phenomena in condensed matter. Their realization at room temperature (RT) in magnetic multilayers has generated considerable interest, fueled by technological prospects and the access granted to fundamental questions. The interaction of skyrmions with charge carriers gives rise to exotic electrodynamics, such as the topological Hall effect (THE), the Hall response to an emergent magnet…
Electrons moving through a noncoplanar magnetic texture acquire a Berry phase, which can be described as an effective magnetic field. This effect is known as the topological Hall effect and has been observed in topological spin textures. Motivated by recent experimental realizations, here we study the Hall effect in a nontopological magnetic texture known as a target skyrmion. We start from a simplified semiclassical picture and show that the Hall signal is a nonmonotonic function of both the electronic energy and target skyrmion radius. That observation carries over to the fully quantum mecha…
This work brings forward an alternative experimental approach to infer the topological character of phase transitions in insulators. This method relies on subjecting the target system to a set of external fields, each of which consists of two parts, i.e., a weak spatiotemporally slowly-varying component on top of a constant offset. The fields are chosen in such a way, so that they respectively induce slow variations in the wave vector describing the bulk band structure, as well as a parameter which allows tuning the bulk gap. Such a process maps the Berry singularities of the base space to a s…
Pseudospin Hall Transport Induced by Berry Curvature
Tsinghua University（清华大学）；The University of Hong Kong（香港大学）；Suzhou Laboratory（苏州实验室）；Materials Innovation Institute for Life Sciences and Energy (MILES), HKUSIRI（香港大学科创研究院生命与能源材料创新研究院（MILES））；Quantum Science Center of Guangdong-Hong Kong-Macao Great Bay Area（粤港澳大湾区量子科学中心）
