Skyrmions with the topological number $Q$ equal an integer larger than 1 are called high-topological-number skyrmions or high-$Q$ skyrmions. In this work, we theoretically study the topological Hall effect in square-lattice high-$Q$ skyrmion crystals (SkX) with $Q=2$ and $Q=3$. As a result of the emergent magnetic field, Landau-level-like electronic band structure gives rise to quantized Hall conductivity when the Fermi energy is within the gaps between adjacent single band or multiple bands intertwined. We found that different from conventional ($Q=1$) SkX the Hall quantization number increases by $1/Q$ in average when the elevating Fermi energy crosses each band. We attribute the result to the fact that the Berry phase ${\cal{C}}$ is measured in the momentum space and the topological number of a single skyrmion $Q$ is measured in the real space. The reciprocality does not affect the conventional SkX because $Q=1=1/Q$.
We numerically examine thermal effects on the skyrmion Hall angle for driven skyrmions interacting with quenched disorder. We identify a creep regime in which motion occurs via intermittent jumps between pinned and flowing states. Here the skyrmion Hall angle is zero since the skyrmions have time to relax into equilibrium positions in the pinning sites, eliminating the side-jump motion induced by the Magnus force. At higher drives we find a crossover to a viscous flow regime where the skyrmion Hall angle is finite and increases with increasing drive or temperature. Our results are in agreement…
Improvements in computing performance have significantly slowed down over the past few years owing to the intrinsic limitations of computing hardware. However, the demand for data computing has increased exponentially. To solve this problem, tremendous attention has been focused on the continuous scaling of Moore's Law as well as the advanced non-von Neumann computing architecture. A rich variety of unconventional computing paradigms has been raised with the rapid development of nanoscale devices. Magnetic skyrmions, spin swirling quasiparticles, have been endowed with great expectations for u…
Exploring a classical solution of the non-linear sigma model for a quantum Hall ferromagnet, a skyrmion-magnon effective hamiltonian is obtained via the collective coordinates method. Using the Feynman-Vernon functional integral formalism for this model we find the temperature dependent transport coefficients which characterize a single skyrmion dynamics.
A new framework for quantum Hall skyrmions in O(4) nonlinear sigma model is studied here. The size and energy of the skyrmions are determined incorporating the quartic stability term in the Lagrangian. Moreover, the introduction of a $θ$-term determines the spin and statistics of these skyrmions.
It is widely believed that the skyrmion Hall effect, often disruptive for device applications, vanishes for overall topologically trivial structures such as (synthetic) antiferromagnetic skyrmions and skyrmioniums due to a compensation of Magnus forces. In this manuscript, however, we report that in contrast to the case of spin-transfer torque driven skyrmion motion, this notion is generally false for spin-orbit torque driven objects. We show that the skyrmion Hall angle is directly related to their helicity and imposes an unexpected roadblock for developing faster and lower input racetrack me…
We numerically study the interacting quantum Hall skyrmion system based on the Chern-Simons action. By noticing that the action is invariant under global spin rotations in the spin space with respect to the magnetic field direction, we obtain the low-energy effective action for a many skyrmion system. Performing extensive molecular dynamics simulations, we establish the thermodynamic phase diagram for a many skyrmion system.
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…
Electron spin-momentum locking together with background magnetic textures can significantly alter the electron transport properties. We theoretically investigate the electron transport at the interface between a topological insulator and a magnetic insulator with magnetic skyrmions on the top. In contrast to the conventional topological Hall effect in normal metals, the skyrmions yield an additional contribution to the anomalous Hall conductivity even in the absence of in-plane magnetic texture, arising from the phase factor characteristic to Dirac electrons acquired at skyrmion boundary.
We have discussed here the Magnus force acting on the vortices and skyrmions in the quantum Hall systems. We have found that it is generated by the chirality of the system which is associated with the Berry phase and is same for both the cases.
Magnetic skyrmions are whirl-like nano-objects with topological protection. When driven by direct currents (DC), skyrmions move but experience a transverse deflection. This so-called skyrmion Hall effect is often regarded a drawback for memory applications. Herein, we show that this unique effect can also be favorable for spintronic applications: We show that in a racetrack with a broken inversion symmetry, the skyrmion Hall effect allows to translate an alternating current (AC) into a directed motion along the track, like in a ratchet. We analyze several modes of the ratchet mechanism and sho…
Magnetic skyrmions are stable nanosized spin structures that can be displaced at low electrical current densities. Because of these properties, they have been proposed as building blocks of future electronic devices with unprecedentedly high information density and low energy consumption. The electrical detection of skyrmions via the Topological Hall Effect (THE), has so far been demonstrated only at cryogenic temperatures. Here, we report the observation of a skyrmion Topological Hall Effect near room temperature (276 K) in a mesoscopic lamella of FeGe. This region unambiguously coincides wit…
Magnetic skyrmions can be driven by an applied spin-polarized electron current which exerts a spin-transfer torque on the localized spins constituting the skyrmion. However, the longitudinal dynamics is plagued by the skyrmion Hall effect which causes the skyrmions to acquire a transverse velocity component. We show how to use spin-orbit interaction to control the skyrmion Hall angle and how the interplay of spin-transfer and spin-orbit torques can lead to a complete suppression of the transverse motion. Since the spin-orbit torques can be controlled all-electronically by a gate voltage, the s…
Bilayer quantum Hall states at $ν=1$ have been demonstrated to possess a distinguished state with interlayer phase coherence. The state has both excitations of Skyrmion with spin and pseudoSkyrmion with pseudospin. We show that Skyrmion $\leftrightarrow$ pseudoSkyrmion transition arises in the state by changing imbalance between electron densities in both layers; PseudoSkyrmion is realized at balance point, while Skyrmion is realized at large imbalance. The transition can be seen by observing the dependence of activation energies on magnetic field parallel to the layers.
We calculate in an analyical fashion the energies and net spins of skyrmions in fractional quantum Hall systems, based on the suggestion that skyrmion states are spontaneously $L_Z$ and $S_Z$ symmetry-breaking states. The quasihole-skyrmion state with a charge $-e/3$ around $ν$ = 1/3, where the ground state is known as a spin-polarized ferromagnetic state, is found to exist even in high magnetic fields up to about 7 T for GaAs samples.
The well-known Hall effect describes the transverse deflection of charged particles (electrons or holes) in an electric-current carrying conductor under the influence of perpendicular magnetic fields, as a result of the Lorentz force. Similarly, it is intriguing to examine if quasi-particles without an electric charge, but with a topological charge, show related transverse motion. Chiral magnetic skyrmions with a well-defined spin topology resulting in a unit topological charge serve as good candidates to test this hypothesis. In spite of the recent progress made on investigating magnetic skyr…
Magnetic skyrmions are topologically non-trivial nanoscale objects. Their topology, which originates in their chiral domain wall winding, governs their unique response to a motion-inducing force. When subjected to an electrical current, the chiral winding of the spin texture leads to a deflection of the skyrmion trajectory, characterized by an angle with respect to the applied force direction. This skyrmion Hall angle was believed to be skyrmion diameter-dependent. In contrast, our experimental study finds that within the plastic flow regime the skyrmion Hall angle is diameter-independent. At …
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…
We consider topological Hall effect (THE) in thin ferromagnetic films due to electron scattering on magnetic skyrmions in the presence of the relatively strong external magnetic field. We account for the effect of the magnetic field on a skyrmion structure and describe the hallmarks of THE differentiating it from ordinary and anomalous Hall effects. We have found that, although in typical ferromagnets the variation of magnetic field in the range 1-5 T substantially affects the skyrmion size, THE changes rather weakly remaining quite robust. Therefore, the magnitude of THE is primarily determin…
The Quantum Hall ground state away from $ν= 1$ can be described by a collection of interacting skyrmions. We show within the context of a nonlinear sigma model, that the classical ground state away from $ν= 1$ is a skyrmion crystal with a generalized Néel order. We show that as a function of filling $ν$, the skyrmion crystal undergoes a triangle to square to triangle transition at zero temperature. We argue that this structural transition, driven by a change in the shape of the individual skyrmions, is stable to thermal and quantum fluctuations and may be probed experimentally.
We propose an improved composite-boson theory of quantum Hall ferromagnets, where the field operator describes solely the physical degrees of freedom representing the deviation from the ground state. In this scheme skyrmions appear merely as generic excitations confined in the lowest Landau level. We evaluate the excitation energy of one skyrmion. Our theoretical estimation accounts for the activation-energy data due to Schmeller et al. remarkably well.
To enable functional skyrmion based spintronic devices, the controllable generation and manipulation of skyrmions is essential. While the generation of skyrmions by using a magnetic geometrical constriction has already been demonstrated, this approach is difficult to combine with a subsequent controlled manipulation of skyrmions. The high efficiency of skyrmion generation from magnetic constrictions limits the useful current density, resulting in stochastic skyrmion motion, which may obscure topological phenomena such as the skyrmion Hall effect. In order to address this issue, we designed a n…
We revisit a theory of skyrmion transport in ferromagnets. On a basis of an effective U(1) gauge theory for spin-chirality fluctuations in double-exchange ferromagnets, we derive an expression for the velocity of a skyrmion core driven by the dc electric field. We find that mutual feedback effects between conduction electrons and localized spins give rise to Chern-Simons terms, suggesting a dissipationless mechanism for the skyrmion Hall current. A conventional description of the current-induced skyrmion motion, appearing through the spin transfer torque and scattering events, is reproduced in…
We have investigated quantum Hall skyrmions at filling factor ν=1 carrying more than one unit of topological, and hence electric, charge. Using a combination of analytic and numerical methods we find the counterintuitive result that when the Zeeman energy is tuned to values much smaller than the interaction energy (g μ_B B/(e^2/ε\ell) < 9*10^{-5}),the creation energy of a charge two skyrmion becomes less than twice the creation energy of a charge one skyrmion, i.e. skyrmions bind in pairs. The doubly charged skyrmions are stable to further accretion of charge and exhibit a 10% larger spin per …
We have investigated skyrmions in N-component quantum Hall systems. We find that SU(N) skyrmions are the lowest energy charged excitations for filling factors ν= 1,2,...,N-1 for small enough symmetry breaking terms. N>2 skyrmions can be realized in Si QH systems based on the (110) or (111) interfaces of Si, or perhaps in Si (100) systems, where the spin and valley isospin together provide an SU(4)-symmetry, or in multilayer QH systems. We also present Hartree-Fock results for a phenomenological easy-axis SU(2)-breaking model appropriate to valley degeneracy.
Recently, an O(3) type of effective action was proposed for the quantum Hall ferromagnet, which accounts for bag formation observed in microscopic Hartree- Fock calculations. We apply this action in the soliton sector and compare with Hartree-Fock results. We find good agreement over the whole parameter range where skyrmions exist. The standard minimal O(3) model cannot explain the bag and has further shortcomings connected with that fact.
We discuss the properties of Skyrmions in the Fractional Quantum Hall effect (FQHE). We begin with a brief description of the Chern-Simons-Landau-Ginzburg description of the FQHE, which provides the framework in which to understand a new derivation of the properties of FQHE Skyrmions (S. Baez, A.P. Balachandran, A. Stern and A. Travesset cond-mat/9712151) from anomaly and edge considerations.
Recent experiments have shown that magnetic skyrmion bubbles can be generated and injected at room temperature in thin films [W. Jiang et al, Science vol. 349, 283 (2015)]. Here, we demonstrate, using micromagnetic modeling, that such skyrmions can be generated by an inhomogeneous spin Hall torque in the presence of Dzyaloshinskii-Moriya interactions (DMIs). In the experimental Ta/Co$_{20}$Fe$_{60}$B$_{20}$ thin films, the DMI is rather small; nevertheless, the skyrmion bubbles are stable, or at least metastable on observational time scales.
We report on the observation of a new spin mode in a quantum Hall system in the vicinity of odd electron filling factors under experimental conditions excluding the possibility of Skyrmion excitations. The new mode having presumably zero energy at odd filling factors emerges at small deviations from odd filling factors and couples to the spin-exciton. The existence of an extra spin mode assumes a nontrivial magnetic order at partial fillings of Landau levels surrounding quantum Hall ferromagnets other then the Skyrmion crystal.
We explore different skyrmion types in the lowest Landau level of graphene at a filling factor $ν=\pm 1$. In addition to the formation of spin and valley pseudospin skyrmions, we show that another type of spin-valley entangled skyrmions can be stabilized in graphene due to an approximate SU(4) spin-valley symmetry that is affected by sublattice symmetry-breaking terms. These skyrmions have a clear signature in spin-resolved density measurements on the lattice scale, and we discuss the expected patterns for the different skyrmion types.
We demonstrate that the Magnus force acting on magnetic skyrmions can be efficiently tuned via modulation of the spin-orbit interaction strength. We show that the skyrmion Hall effect, which is a direct consequence of the non-vanishing Magnus force on the magnetic structure can be suppressed in certain limits. Our calculations show that the emergent magnetic fields in the presence of spin-orbit coupling (SOC) renormalize the Lorentz force on itinerant electrons and thus influence the topological transport. In particular, we show that for a Néel-type skyrmion and Bloch-type antiskyrmion, the sk…
Numerical results are presented for the spin excitations of a two-dimensional electron gas confined to a quantum well of width w. Spin waves and charged skyrmion excitations are studied for filling factors nu=1, 3, and 1/3. Phase diagrams for the occurrence of skyrmions of different size as a function of w and the Zeeman energy are calculated. For nu=3, skyrmions occur only if w is larger than about twice the magnetic length. A general necessary condition on the interaction pseudopotential for the occurrence of stable skyrmion states is proposed.
Skyrmions are topologically protected spin textures, characterized by a topological winding number N , that occur spontaneously in some magnetic materials. Recent experiments have demonstrated the capability to grow graphene on top Fe/Ir, a system that exhibits a two dimensional Skyrmion lattice. Here we show that a weak exchange coupling between the Dirac electrons in graphene and a two dimensional Skyrmion lattice with $N = \pm 1$ drives graphene into a quantum anomalous Hall phase, with a band-gap in bulk, a Chern number ${\cal C}=2N$ and chiral edge states with perfect quantization of cond…
We present a Hartree-Fock study that incorporates the effects of Landau level mixing and screening due to filled levels into the computation of energies and states of quasiparticles in quantum Hall ferromagnets. We use it to construct a phase diagram for skyrmion stability as a function of magnetic field and Zeeman coupling strengths. We find that Landau level mixing tends to favor spin-polarized quasiparticles, while finite thickness corrections favor skyrmions. Our studies show that skyrmion stability in high Landau levels is very sensitive to the way in which electron-electron interactions …
Skyrmions can be driven by spin-orbit torques as a result of the spin Hall effect. Here we model an additional contribution in ultra-thin multilayers, arising from the spin accumulation at heavy metal / ferromagnetic interfaces and observe the effects on a large range of skyrmion diameters. The combination of the interfacial spin-transfer torque and the spin-orbit torque results in skyrmion motion which helps to explain the observation of small skyrmion Hall angles for skyrmion diameters less than 100 nm. We show that this additional term has a significant effect on the skyrmion dynamics and l…
Spin-Hall-Like Magnon Transport in a Synthetic Antiferromagnetic Skyrmion Lattice
Dongguan Institute of Materials Science and Technology, Chinese Academy of Sciences（中国科学院东莞材料基因高等理工研究院）；Songshan Lake Materials Laboratory（松山湖材料实验室）；Shenzhen Technology University（深圳技术大学）；Hunan University（湖南大学）；Central South University（中南大学）
We report on a study of the classical field theory description of charged skyrmions in quantum Hall ferromagnets. The appropriate field theory is a non-linear $σ$ model generalized to include Coulomb and Zeeman interaction terms. We have tested the range of validity of the classical field theory by comparing with microscopic descriptions of the single skyrmion state based on the Hartree-Fock approximation, exact diagonalization calculations, and many-body trial wavefunctions. We find that the field theory description is accurate for skyrmions with moderate spin quantum numbers ($\gtrsim 10$) a…
Using a particle-based model we examine the depinning motion of solitons in skyrmion chains in quasi-one dimensional (1D) and two-dimensional (2D) systems containing embedded 1D interfaces. The solitons take the form of a particle or hole in a commensurate chain of skyrmions. Under an applied drive, just above a critical depinning threshold the soliton moves with a skyrmion Hall angle of zero. For higher drives, the entire chain depins, and in a 2D system we observe that both the solitons and chain move at zero skyrmion Hall angle and then transition to a finite skyrmion Hall angle as the driv…
The understanding of the dynamical properties of skyrmion is a fundamental aspect for the realization of a competitive skyrmion based technology beyond CMOS. Most of the theoretical approaches are based on the approximation of a rigid skyrmion. However, thermal fluctuations can drive a continuous change of the skyrmion size via the excitation of thermal modes. Here, by taking advantage of the Hilbert-Huang transform, we demonstrate that at least two thermal modes can be excited which are non-stationary in time. In addition, one limit of the rigid skyrmion approximation is that this hypothesis …
Skyrmions, novel topologically stable spin vortices, hold promise for next-generation magnetic storage due to their nanoscale domains to enable high information storage density and their low threshold for current-driven motion to enable ultralow energy consumption. One-dimensional (1D) nanowires are ideal hosts for skyrmions since they not only serve as a natural platform for magnetic racetrack memory devices but also can potentially stabilize skyrmions. Here we use the topological Hall effect (THE) to study the phase stability and current-driven dynamics of the skyrmions in MnSi nanowires. Th…
A novel type of charged excitation, known as a Skyrmion, has recently been discovered in quantum Hall systems with filling factor near ν= 1. A Skyrmion -- which can be thought of as a topological twist in the spin density of the electron gas -- has the same charge as an electron, but a much larger spin. In this review we present a detailed theoretical investigation of the optical properties of Skyrmions. Our results provide means for the optical detection of Skyrmions using photoluminescence (PL) spectroscopy. We first consider the optical properties of Skyrmions in disordered systems. A calcu…
A field theory of quantum Hall effects is constructed based on the \CB picture. It is tightly related with the microscopic wave-function theory. The characteristic feature is that the field operator describes solely the physical degrees of freedom representing the deviation from the Laughlin state. It presents a powerful tool to analyze excited states within the \LLL. It is shown that all excitations are nonlocal topological solitons in the spinless quantum Hall system. On the other hand, in the presence of the spin degree of freedom it is shown that a quantum coherence develops spontaneously,…
An improved composite-boson theory of quantum Hall ferromagnets is proposed. It is tightly related with the microscopic wave-function theory. The characteristic feature is that the field operator describes solely the physical degrees of freedom representing the deviation from the ground state. It presents a powerful tool to analyze excited states within the \LLLd. Excitations include a Goldstone mode and nonlocal topological solitons. Solitons are vortices and Skyrmions carrying the U(1) and SU(2) topological charges, respectively. Their classical configurations are derived from their microsco…
We have discussed here the polarization of quantum Hall states in the framework of the hierarchical analysis of IQHE and FQHE in terms of Berry phase. It is observed that we have fully polarized states for the filling factor $ν=1$ as well as $ν=\frac{1}{2m+1}$, $m$ being an integer. However, for $ν=p$ as well as $ν=\frac{p}{q}$, with $p>1$ and odd, $q$ odd we have partially polarized states and for $ν=\frac{p}{q}$, $p$ even, $q$ odd we have unpolarized states. It has been found that skyrmion excitations exist only for fully polarized states and for partially polarized and unpolarized states sk…
We study the anomalous Hall effect (AHE) for the double exchange model with the exchange coupling $|J_H|$ being smaller than the bandwidth $|t|$ for the purpose of clarifying the following unresolved and confusing issues: (i) the effect of the underlying lattice structure, (ii) the relation between AHE and the skyrmion number, (iii) the duality between real and momentum spaces, and (iv) the role of the disorder scatterings; which is more essential, $σ_H$ (Hall conductivity) or $ρ_H$ (Hall resistivity)? Starting from a generic expression for $σ_H$, we resolve all these issues and classify the r…
The quantum spin Hall effect is conventionally thought to require a strong spin-orbit coupling, producing an effective spin-dependent magnetic field. However, spin currents can also be present without transport of spins, for example, in spin-waves or skyrmions. In this paper, we show that topological skyrmionic spin textures can be used to realize a quantum spin Hall effect. From basic arguments relating to the single-valuedness of the wave function, we deduce that loop integrals of the derivative of the Hamiltonian must have a spectrum that is integer multiples of $ 2 π$. By relating this to …
The interplay of topological characteristics in real space and reciprocal space can lead to the emergence of unconventional topological phases. In this Letter, we implement a novel mechanism for generating higher-Chern flat bands on the basis of twisted bilayer graphene (TBG) coupled to topological magnetic structures in the form of the skyrmion lattice. In particular, we discover a scenario for generating $|C|=2$ dispersionless electronic bands when the skyrmion periodicity and the moiré periodicity are matched. Following the Wilczek argument, the statistics of the charge-carrying excitations…
Graphene exhibits quantum Hall ferromagnetism in which an approximate SU(4) symmetry involving spin and valley degrees of freedom is spontaneously broken. We construct a set of integer and fractional quantum Hall states that break the SU(4) spin/valley symmetry, and study their neutral and charged excitations. Several properties of these ferromagnets can be evaluated analytically in the SU(4) symmetric limit, including the full collective mode spectrum at integer fillings. By constructing explicit wave functions we show that the lowest energy skyrmion states carry charge $\pm 1$ for {\em any} …
We investigate the classical ground state configurations of a collection of skyrmions in the limit of vanishing Lánde $g$ factor. We show that in this limit, the skyrmions which are large and overlapping, behave qualitatively differently from small skyrmions. We investigate the system in various regimes of spin stiffness. We find that in all regimes, at $g=0$, the ground state configuration is a rectangular skyrmion lattice with Néel orientational ordering. The charge distribution has a higher symmetry of a face-centered rectangular lattice and the $z$-component of the spin polarization is zer…
We demonstrate that the nontrivial magnetic texture of antiferromagnetic skyrmions (AFM-Sks) promotes a non-vanishing topological spin Hall effect (TSHE) on the flowing electrons. This results in a substantial enhancement of the non-adiabatic torque and hence improves the skyrmion mobility. This non-adiabatic torque increases when decreasing the skyrmion size, and therefore scaling down results in a much higher torque efficiency. In clean AFM-Sks, we find a significant boost of the TSHE close to van Hove singularity. Interestingly, this effect is enhanced away from the band gap in the presence…
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…
Motivated by the recent discoveries of magnets harboring short-pitch skyrmion lattices, we investigate the skyrmion-size dependence of the topological Hall effect. By means of large-scale real-space calculations, we find that the Hall conductivity takes its extreme value in the crossover region where both the real-space and momentum-space Berry curvature play a crucial role. We also investigate how the optimum skyrmion size ($λ_{\rm sk}^*$) depends on the lifetime of itinerant electrons ($τ$) and coupling constant between electrons and localized spins ($J$). For the former, we show that $λ_{\r…
We report on a study of the charged-skyrmion or spin-texture excitations which occur in quantum Hall ferromagnets near odd Landau level filling factors. Particle-hole symmetry is used to relate the spin-quantum numbers of charged particle and hole excitations and neutral particle-hole pair excitations. Hartree-Fock theory is used to provide quantitative estimates of the energies of these excitations and their dependence on Zeeman coupling strength, Landau level quantum numbers, and the thicknesses of the two-dimensional electron layers. For the case of $ν$ near three we suggest the possibility…
