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 magnetic field, a manifestation of the skyrmion Berry-phase. The proposal that THE can be used to detect skyrmions needs to be tested quantitatively. For that it is imperative to develop comprehensive understanding of skyrmions and other chiral textures, and their electrical fingerprint. Here, using Hall transport and magnetic imaging, we track the evolution of magnetic textures and their THE signature in a technologically viable multilayer film as a function of temperature ($T$) and out-of-plane applied magnetic field ($H$). We show that topological Hall resistivity ($ρ_\mathrm{TH}$) scales with the density of isolated skyrmions ($n_\mathrm{sk}$) over a wide range of $T$, confirming the impact of the skyrmion Berry-phase on electronic transport. We find that at higher $n_\mathrm{sk}$ skyrmions cluster into worms which carry considerable topological charge, unlike topologically-trivial spin spirals. While we establish a qualitative agreement between $ρ_\mathrm{TH}(H,T)$ and areal density of topological charge $n_\mathrm{T}(H,T)$, our detailed quantitative analysis shows a much larger $ρ_\mathrm{TH}$ than the prevailing theory predicts for observed $n_\mathrm{T}$.
Skyrmions are nanoscale spin configurations with topological properties that hold great promise for spintronic devices. Here, we establish their Néel texture, helicity, and size in Ir/Fe/Co/Pt multilayer films by constructing a multipole expansion to model their stray field signatures and applying it to magnetic force microscopy (MFM) images. Furthermore, the demonstrated sensitivity to inhomogeneity in skyrmion properties, coupled with a unique capability to estimate the pinning force governing dynamics, portends broad applicability in the burgeoning field of topological spin textures.
Magnetic skyrmions are topologically wound nanoscale textures of spins whose ambient stability and electrical manipulation in multilayer films have led to an explosion of research activities. While past efforts focused predominantly on isolated skyrmions, recently ensembles of chiral spin textures, consisting of skyrmions and magnetic stripes, were shown to possess rich interactions with potential for device applications. However, several fundamental aspects of chiral spin texture phenomenology remain to be elucidated, including their domain wall structure, thermodynamic stability, and morphol…
We show that scanning electron microscopy with polarization analysis (SEMPA) that is sensitive to both in-plane magnetization components can be used to image the out-of-plane magnetized multi-domain state in multilayered chiral spin textures. By depositing a thin layer of Fe on top of the multilayer we image the underlying out-of-plane domain state through the mapping of its stray fields in the Fe. We also demonstrate that SEMPA can be used to image the domain wall chirality in these systems after milling away the capping layer and imaging the topmost magnetic layer directly.
Twisted graphene multilayers provide tunable platforms to engineer flat bands and exploit the associated strongly correlated physics. The two-dimensional nature of these systems makes them suitable for encapsulation by materials that break specific symmetries. In this context, recently discovered two-dimensional helimagnets, such as the multiferroic monolayer NiI$_2$, are specially appealing for breaking time-reversal and inversion symmetries due to their nontrivial spin textures. Here we show that this spin texture can be imprinted on the electronic structure of twisted bilayer graphene by pr…
We discuss the quantum Hall effect of bilayer graphene with finite gate voltage where the Fermi energy exceeds the interlayer hopping energy. We calculated magnetic susceptibility, diagonal and off-diagonal conductivities in finite-magnetic-field formalism, and observed crossover of integer quantum Hall effect from two independent monolayer type system to strongly coupled bilayer systems by changing the ratio of interlayer hopping energy and the gate voltage. We also discuss the case of multilayer systems with Bernal stacking
Three-Dimensional Non-Foliated Fractional Quantum Hall Phases with Irrational Anyons in Twisted van der Waals Multilayers
Pohang University of Science and Technology（浦项科技大学）；Daegu Gyeongbuk Institute of Science and Technology (DGIST)（大邱庆北科学技术院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Institute for Basic Science（基础科学研究院）
We experimentally image the magnetic precession phase of patterned spin Hall multilayer samples to study the rf driving field vector using time-resolved anomalous Nernst effect (TRANE) microscopy. Our ferromagnetic resonance (FMR) measurements quantify the phase and amplitude for both the magnetic precession and the electric current, which allows us to establish the total driving field orientation and the strength of spin Hall effect. In a channel of uniform width, we observe spatial variation of the FMR phase laterally across the channel. We interpret our findings in the context of electrical…
A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the way ideal ground state wave functions go to zero as various clusters of electrons are brought together. In this paper we generalize this approach to classify multilayer FQH states. Such a classification leads to the construction of a class of non-Abelian multilayer FQH states that are closely related to $\hat{g}_k$ parafermion conformal field theories, where $\hat{g}_k$ is an affine simple Lie algebra. We discuss the possibility of some of the simplest of these non-A…
We evidenced unconventionnal Anomalous Hall Effects (AHE) in 3d/5d (Co0.2nm/Ni0.6nm)N multilayers grown on a thin Pt layer or thin Au:W alloy. The inversion observed on AHE originates from the opposite sign of the spin-orbit coupling of Pt compared to Ni. Via advanced simulations methods for the description of the spin-current profiles based on the spin-dependent Boltzmann formalism, we extracted the spin Hall angle (SHA) of Pt and (Co/Ni) as well as the relevant transport parameters. The extracted SHA for Pt, +20%, is opposite to the one of (Co/Ni), giving rise to an effective AHE inversion f…
This is a Reply to a Comment by Tanaka and Machida. We provide some details of the derivation of the effective field theory for integer quantum Hall transitions using the non-Abelian chiral anomaly.
We report on a theoretical study of one-dimensional (1D) states localized at few-layer graphene system ribbon edges, and at interfaces between few-layer graphene systems with different valley Hall conductivities. These 1D states are topologically protected when valley mixing is neglected. We address the influence on their properties of stacking arrangement, interface structure, and external electric field perpendicular to the layers. We find that 1D states are generally absent at multilayer ribbon armchair direction edges, but present irrespective of crystallographic orientation at any interna…
We experimentally investigate the current-induced magnetization reversal in Pt/[Co/Ni]$_3$/Al multilayers combining the anomalous Hall effect and magneto-optical Kerr effect techniques in crossbar geometry. The magnetization reversal occurs through nucleation and propagation of a domain of opposite polarity for a current density of the order of 0.3 TA/m$^2$. In these experiments we demonstrate a full control of each stage: i)the Ørsted field controls the domain nucleation and ii) domain-wall propagation occurs by spin torque from the Pt spin Hall effect. This scenario requires an in-plane magn…
We theoretically study the correlated insulator states, quantum anomalous Hall (QAH) states, and field-induced topological transitions between different correlated states in twisted multilayer graphene systems. Taking twisted bilayer-monolayer graphene and twisted double-bilayer graphene as examples, we show that both systems stay in spin polarized, $C_{3z}$-broken insulator states with zero Chern number at 1/2 filling of the flat bands under finite displacement fields. In some cases these spin polarized, nematic insulator states are in the quantum valley Hall phase by virtue of the nontrivial…
We propose a mechanism for the quenching of the Shubnikov de Haas oscillations and the quantum Hall effect observed in epitaxial graphene. Experimental data show that the scattering time of the conduction electron is magnetic field dependent and of the order of the cyclotron orbit period, \textit{i.e.} can be much smaller than the zero field scattering time. Our scenario involves the extraordinary graphene $n=0$ Landau level of the uncharged layers that produces a high density of states at the Fermi level. We find that the coupling between this $n=0$ Landau level and the conducting states of t…
We study charged spin textures (CST) over the Moore-Read quantum Hall state at filling factor 5/2. We develop an algebraic framework and show that the pairing condition that is inherent in the Moore-Read state naturally leads to a class of charged spin textures, labeled by winding numbers [w_I, w_II]. The fundamental CST, with labels [1,0] and electric charge e/4, is identified with the polar core vortex known in the spin-1 BEC literature. The spin texture carried by the fusion product of fundamental CSTs is correlated with the fusion channel of the underlying non-Abelian quasiholes.
Three-dimensional (3D) topological spin textures emerge as promising quasi-particles for encoding information in future spintronic devices. The third dimension provides more malleability regarding their magnetic properties as well as more flexibility for potential applications. However, the stabilization and characterization of such quasi-particles in easily implementable systems remain a work in progress. Here we observe a new type of 3D magnetic textures that we called skyrmionic cocoons that sits in the interior of magnetic thin films multilayers and possesses a characteristic ellipsoidal s…
The Landau level spectra and the quantum Hall effect of ABA-stacked multilayer graphenes are studied in the effective mass approximation. The low-energy effective mass Hamiltonian may be partially diagonalized into an approximate block-diagonal form, with each diagonal block contributing parabolic bands except, in a multilayer with an odd number of layers, for an additional block describing Dirac-like bands with a linear dispersion. We fully include the band parameters and, taking into account the symmetry of the lattice, we analyze their affect on the block-diagonal Hamiltonian. Next-nearest …
Domain walls, topological defects that define the frontier between regions of different stacking in multilayer graphene, have proved to host exciting physics. The ability of tuning these topological defects in-situ in an electronic transport experiment brings a wealth of possibilities in terms of fundamental understanding of domain walls as well as for electronic applications. Here, we demonstrate through a MEMS (micro-electromechanical system) actuator and magnetoresistance measurements the effect of domain walls in multilayer graphene quantum Hall effect. Reversible and controlled uniaxial s…
The quantum anomalous Hall (QAH) effect has been recently discovered in experiment using thin-film topological insulator with ferromagnetic ordering and strong spin-orbit coupling. Here we investigate the spin degree of freedom of a QAH insulator and uncover a fundamental phenomenon that the edge states exhibit topologically stable spin texture in the boundary when a chiral-like symmetry is present. This result shows that edge states are chiral in both the orbital and spin degrees of freedom, and the chiral edge spin texture corresponds to the bulk topological states of the QAH insulator. We a…
Investigation of the spin Hall effect in gold has triggered increasing interest over the past decade, since gold combines the properties of a large bulk spin diffusion length and strong interfacial spin-orbit coupling. However, discrepancies between the values of the spin Hall angle of gold reported in the literature have brought into question the microscopic origin of the spin Hall effect in Au. Here, we investigate the thickness dependence of the spin-charge conversion efficiency in single Au films and ultrathin Au/Si multilayers by non-local transport and spin-torque ferromagnetic resonance…
We study the effect that the electron-electron interaction has on the properties of a multilayer electron system. We consider the case corresponding to filling factor unity in each layer. We find that as a function of the sample parameters the system has ferromagnetic, canted antiferromagnetic or paramagnetic interlayer spin correlations. These three ground states are QHE phases, because of the existence of a finite activation energy. In the ferromagnetic phase the gap is due to the intrawell exchange energy, whereas in the paramagnetic phase the gap appears due to the spatial modulation of th…
We report a comprehensive study of the maximization of the spin Hall ratio (θSH) in Pt thin films by the insertion of sub-monolayer layers of Ti to decrease carrier lifetime while minimizing the concurrent reduction in the spin Hall conductivity. We establish that the intrinsic spin Hall conductivity of Pt, while robust against the strain and the moderate interruption of crystal order caused by these insertions, begins to decrease rapidly at high resistivity level because of the shortening carrier lifetime. The unavoidable trade-off between the intrinsic spin Hall conductivity and carrier life…
Dirac points in two-dimensional electronic structures are a source for topological electronic states due to the $\pm π$ Berry phase that they sustain. Here we show that two rutile multilayers (namely (WO$_2$)$_2$/(ZrO$_2$)$_n$ and (PtO$_2$)$_2$/(ZrO$_2$)$_n$, where an active bilayer is sandwiched by a thick enough (n=6 is sufficient) band insulating substrate, show semi-metallic Dirac dispersions with a total of four Dirac cones along the $Γ-M$ direction. These become gapped upon the introduction of spin-orbit coupling, giving rise to an insulating ground state comprising four edge states. We …
In ultra-thin film of topological insulator, the hybridization between the top and bottom surfaces opens an energy gap and forms two degenerate quantum anomalous Hall states, which give rise to a quantum spin Hall state. In this paper, we demonstrate that a Weyl semimetal can be realized in an ultra-thin film of topological insulator heterostructure in a similar way to that of the surface state of a strong three-dimensional (3D) topological insulator studied by Burkov and Balents . We find that the system realizes both 3D quantum anomalous Hall phase and 3D quantum spin Hall phase, and the Wey…
Impacts of domain textures on low-lying neutral excitations in the bulk of fractional quantum Hall effect (FQHE) systems are probed by resonant inelastic light scattering. We demonstrate that large domains of quantum fluids support long-wavelength neutral collective excitations with well-defined wave vector (momentum) dispersion that could be interpreted by theories for uniform phases. Access to dispersive low-lying neutral collective modes in large domains of FQHE fluids such as long wavelength magnetorotons at filling factor v=1/3 offer significant experimental access to strong electron corr…
Competing interactions produce finite-size textures in myriad condensed matter systems, typically forming elongated stripe or round bubble domains. Transitions between stripe and bubble phases, driven by field or temperature, are expected to be reversible in nature. Here we report on the distinct character of the analogous transition for nanoscale spin textures in chiral Co/Pt-based multilayer films, known to host Néel skyrmions, using microscopy, magnetometry, and micromagnetic simulations. Upon increasing field, individual stripes fission into multiple skyrmions, and this transition exhibits…
We report an unconventional quantum spin Hall phase in the monolayer T$_\text{d}$-WTe$_2$, which exhibits hitherto unknown features in other topological materials. The low-symmetry of the structure induces a canted spin texture in the $yz$ plane, which dictates the spin polarization of topologically protected boundary states. Additionally, the spin Hall conductivity gets quantized ($2e^2/h$) with a spin quantization axis parallel to the canting direction. These findings are based on large-scale quantum simulations of the spin Hall conductivity tensor and nonlocal resistances in multi-probe geo…
A Zener diode is a paradigmatic device in semiconductor-based electronics that consists of a pn junction where an external electric field induces a switching behavior in the current-voltage characteristics. We study Zener tunneling in HgTe quantum wells and graphene multilayers. We find that the tunneling transition probability depends asymmetrically on the parallel momentum of the carriers to the barrier. In HgTe quantum wells the asymmetry is the opposite for each spin, whereas for graphene multilayers it is the opposite for each valley degree of freedom. In both cases, a spin/valley current…
The so called quantum spin Hall phase is a topologically non trivial insulating phase that is predicted to appear in graphene and graphene-like systems. In this work we address the question of whether this topological property persists in multilayered systems. We consider two situations: purely multilayer graphene and heterostructures where graphene is encapsulated by trivial insulators with a strong spin-orbit coupling. We use a four orbital tight-binding model that includes the full atomic spin-orbit coupling and we calculate the $Z_{2}$ topological invariant of the bulk states as well as th…
The anomalous Hall effect is caused by magnetic textures such as skyrmions. We derive an analytical formula of the Hall conductivity on the surface of a topological insulator up to third order in magnetization, $\boldsymbol{M}(\boldsymbol{x})$, based on a perturbative approach. We identify the magnetic textures that contribute to the Hall conductivity up to third order in magnetization and second order in spatial differentiation. We treat magnetization as a perturbation to calculate the Hall conductivity for each magnetic texture based on the linear response theory. Furthermore, we estimate th…
Symmetry breaking together with strong spin-orbit interaction give rise to many exciting phenomena within condensed matter physics. A recent example is the existence of chiral spin textures, which are observed in magnetic systems lacking inversion symmetry. These chiral spin textures, including domain walls and magnetic skyrmions, are both fundamentally interesting and technologically promising. For example, they can be driven very efficiently by electrical currents, and exhibit many new physical properties determined by their real-space topological characteristics. Depending on the details of…
We proposed that the simultaneous presence of both Rashba and band inversion can lead to a Rashba-like spin-splitting formed by two bands with the same in-plane helical spin texture. Because of this unconventional spin texture, the backscattering is forbidden in edge and bulk conductivity channels. We propose a new non-centrosymmetric honeycomb-lattice quantum spin Hall (QSH) insulator family formed by the IV, V, and VII elements with this property. The system formed by Bi, Pb and I atoms is mechanically stable and has both a large Rashba spin-splitting of 60 meV and a large nontrivial band ga…
Two-dimensional magnetic insulators can be promising hosts for topological magnons. In this study, we show that ABC-stacked honeycomb lattice multilayers with alternating Dzyaloshinskii-Moriya interaction (DMI) reveal a rich topological magnon phase diagram. Based on our bandstructure and Berry curvature calculations, we demonstrate jumps in the thermal Hall behavior that corroborate with topological phase transitions triggered by adjusting the DMI and interlayer coupling. We connect the phase diagram of generic multilayers to a bilayer and a trilayer system. We find an even-odd effect amongst…
We carry out a throughout study of the topological Hall and topological spin Hall effects in disordered skyrmionic systems: the dimensionless (spin) Hall angles are evaluated across the energy band structure in the multiprobe Landauer-Büttiker formalism and their link to the effective magnetic field emerging from the real space topology of the spin texture is highlighted. We discuss these results for an optimal skyrmion size and for various sizes of the sample and found that the adiabatic approximation still holds for large skyrmions as well as for few atomic size-nanoskyrmions. Finally, we te…
Current induced spin-orbit torques have been studied in ferromagnetic nanowires made of 20 nm thick Co/Pd multilayers with perpendicular magnetic anisotropy. Using Hall voltage and lock-in measurements, it is found that upon injection of an electric current both in-plane (Slonczewski-like) and perpendicular (field-like) torques build up in the nanowire. The torque efficiencies are found to be as large as 1.17 kOe and 5 kOe at 108 A/cm2 for the in-plane and perpendicular components, respectively, which is surprisingly comparable to previous studies in ultrathin (~ 1 nm) magnetic bilayers. We sh…
Magnetic skyrmions are nanoscale spin textures touted as next-generation computing elements. When subjected to lateral currents, skyrmions move at considerable speeds. Their topological charge results in an additional transverse deflection known as the skyrmion Hall effect (SkHE). While promising, their dynamic phenomenology with current, skyrmion size, geometric effects and disorder remain to be established. Here we report on the ensemble dynamics of individual skyrmions forming dense arrays in Pt/Co/MgO wires by examining over 20,000 instances of motion across currents and fields. The skyrmi…
We extend our ensemble density functional approach to quantum Hall systems to include non-collinear spins to study charge-spin textures in inhomogeneous quantum Hall systems. We have studied the edge reconstruction in quantum dots at unit bulk filling factor and at 1/3 bulk filling factor as a function of the stiffness of an external confining potential. For soft enough edges, these systems reconstruct to a state in which the electron spins rotate gently as the edge is approached, with a non-trivial spin-charge texture at the edge of the system.
The current proposals for producing non-Abelian anyons and Majorana particles, which are neither fermions nor bosons, are primarily based on the realization of topological superconductivity in two dimensions. We show theoretically that the unique Landau level structure of bilayer graphene provides a new possible avenue for achieving such exotic particles. Specifically, we demonstrate the feasibility of a "parton" fractional quantum Hall (FQH) state, which supports non-Abelian particles without the usual topological superconductivity. Furthermore, we advance this state as the fundamental explan…
We investigate electron transport through an antidot embedded in a narrow strip of two-dimensional topological insulator. We focus on the most generic and experimentally relevant case with broken axial spin symmetry. Spin-non-conservation allows additional scattering processes which change the transport properties profoundly. We start from an analytical model for noninteracting transport, which we also compare with a numerical tight-binding simulation. We then extend this model by including Coulomb repulsion on the antidot, and we study the transport in the Coulomb-blockade limit. We investiga…
We show theoretically that both intrinsic spin Hall effect (SHE) and orbital Hall effect (OHE) can arise in centrosymmetric systems through momentum-space orbital texture, which is ubiquitous even in centrosymmetric systems unlike spin texture. OHE occurs even without spin-orbit coupling (SOC) and is converted into SHE through SOC. The resulting spin Hall conductivity is large (comparable to that of Pt) but depends on the SOC strength in a nonmonotonic way. This mechanism is stable against orbital quenching. This work suggests a path for an ongoing search for materials with stronger SHE. It al…
The orbital structure of the quantum Hall ferromagnetic states in the zero-energy Landau level in chiral multilayer graphene (AB, ABC, ABCA, etc.\ stackings) is determined by the exchange interaction with all levels, including deep-lying states in the Dirac sea. This exchange field favors orbitally coherent states with a U(1) orbital symmetry if the filling factor $ν$ is not a multiple of the number of layers. If electrons fill the orbital sector of a fixed spin/valley component to one-half, e.g., at $ν=\pm3,\pm1$ in the bilayer and at $ν=\pm2,\pm6$ in the ABCA four-layer, there is a transitio…
Magnetic excitations in ferromagnetic systems with a noncollinear ground state magnetization experience a fictitious magnetic field due to the equilibrium magnetic texture. Here, we investigate how such fictitious fields lead to thermal Hall effects in two-dimensional insulating magnets in which the magnetic texture is caused by spin-orbit interaction. Besides the well-known geometric texture contribution to the fictitious magnetic field in such systems, there exists also an equally important contribution due to the original spin-orbit term in the free energy. We consider the different possibl…
Tellurium is a gyrotropic, p-type Weyl semiconductor with remarkable electronic, optical, and transport properties. It has been argued that some of these properties might stem from Weyl nodes at crossing points in the band structure, and their nontrivial topological textures. However, Weyl nodes in time-reversal invariant semiconductors are split up in energy, rather than in momentum, and located deep below (far above) the top (bottom) of the valence (conduction) band, challenging such an interpretation. Here, instead, we use a 4-band kp Hamiltonian for $p-$type tellurium to show how the k-dep…
Realizing effective manipulation and explicit identification of topological spin textures are two crucial ingredients to make them as information carrier in spintronic devices with high storage density, high data handling speed and low energy consumption. Electric-field manipulation of magnetism has been achieved as a dissipationless method compared with traditional regulations. However, the magnetization is normally insensitive to the electric field since it does not break time-reversal symmetry directly, and distribution of topological magnetic quasiparticles is difficult to maintain due to …
