Nonlocal Anomalous Hall Effect.

Ferromagnet (FM)\ifmmode\text{\textcelsius}\else\textcelsius\fi{}heavy-metal (HM) nanostructures can be used for magnetic state readout in the proposed magnetoelectric spin-orbit logic by locally injecting a spin-polarized current and measuring the spin-to-charge conversion via the spin Hall effect. However, this local configuration is prone to spurious signals. Here, we address spurious Hall effects that can contaminate the spin Hall signal in these FM/HM T-shaped nanostructures. The most pronounced Hall effects in our ${\mathrm{Co}}_{50}{\mathrm{Fe}}_{50}/\mathrm{Pt}$ nanostructures are the planar Hall effect and the anomalous Hall effect generated in the ferromagnetic nanowire. We find that the planar Hall effect, induced by misalignment between magnetization and current direction in the ferromagnetic wire, is manifested as a shift in the measured baseline resistance, but does not alter the spin Hall signal. In contrast, the anomalous Hall effect, arising from the charge-current distribution within the FM, adds to the spin Hall signal. However, the effect can be made insignificant by minimizing the shunting effect via proper design of the device. We conclude that local spin injection in FM/HM nanostructures is a suitable tool for measuring spin Hall signals and, therefore, a valid method for magnetic state readout in prospective spin-based logic.

The spin Hall effect (SHE) is closely related to the spin-orbit coupling. Therefore, heavy metals are usually considered as materials of choice to obtain large spin Hall angles. Among these materials, some 5d transition metals such as Pt, Ta or W offer the largest spin-charge interconversion efficiency for spin current detections through the inverse spin Hall effect or to obtain efficient current induced magnetization switching in SOT-MRAM devices [1].Despite a weaker spin orbit coupling, spin Hall angles comparable to that of the best 5d materials has also been obtained in 3d ferromagnetic metals such as Co [2] or Ni [3]. More recently nonmagnetic alloys containing Nickel atoms such as NiCu [4] and CoNiB [5] attracted great interest for their large spin Hall angles and their potential for spintronic applications. Using the spin pumping by ferromagnetic resonance technique we could compare the spin to charge current conversion in Ni60Cu40 to the one of Pt. As can be seen in figure 1 a) the normalized spin pumping signal obtained in CFB/NiCu is comparable to that of CFB/Pt. We performed a thickness dependence shown in figure 1 b) to extract the spin Hall angle (θSHE) and spin diffusion length (λs) and obtained results comparable to the one of 5d metals using similar method with λs = 2.4 ± 0.3 nm and θSHE = 4.1 ± 0.5 % [6].Keller and coworkers [4] noticed similarities in the bandstructure of paramagnetic Nickel and Platinum possibly explaining the origin of the effect. The calculated band structure of ferromagnetic Nickel does not resemble that of Platinum and the origin of the large spin Hall effect observed in ferromagnetic Ni [3] could thus be different. The interplay between the spin-orbit coupling and magnetism is known to be at the origin of specific transport properties such as the anomalous magnetoresistance and the anomalous Hall effect. However, the link between the SHE and the magnetic order is complex and still unexplored [7].In order to study the effect of the phase transition we performed spin pumping as a function of temperature in Ni60Cu40 and Ni50Cu50 alloys above and below the Curie temperature. To avoid direct contact between the ferromagnetic spin injector CFB and the magnetic NiCu alloy that would modify CFB resonance properties due to the direct coupling [8] we used a 5 nm thick Cu interlayer. The spin pumping measurements performed in CFB\Cu\ Ni60Cu40 and CFB\Cu\ Ni50Cu50 shown in figure 2 indicate that in NiCu the SHE is insensitive to the magnetic order [6].A large variety of alloys composed of Ni, Co or Fe in the ferromagnetic or paramagnetic phase could be explored as spin current generators and detectors, extending the number of possible light metal systems with a large SHE. In particular, the large spin Hall effect in NiCu shows that it is a promising alternative to obtain large spin orbit torque without heavy metals. **

Spin-orbit coupling gives rise to the spin Hall effect (SHE) in nonmagnetic heavy metals, and to the anomalous Hall effect (AHE) in ferromagnetic metals. This article proposes a mechanism to switch the perpendicular magnetization in a magnetic trilayer, without an assisting magnetic field, by utilizing the SHE or AHE plus out-of-plane spin polarization. The authors derive the analytic expression for the resulting spin torque, and perform macrospin simulation to validate the field-free switching. The proposed switching mechanism combines well-established spin-valve technology with the spin Hall effect, potentially enhancing switching efficiency in practical devices.

In this letter, we study the origin of the anomalous Hall effect (AHE) in ferrimagnetic insulator ${\mathrm{Tm}}_{3}{\mathrm{Fe}}_{5}{\mathrm{O}}_{12}$ (TmIG)/Pt heterostructures. A monotonic decrease of the anomalous Hall resistivity ($\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$) with decreasing temperature is observed for TmIG/Pt, and a sign reversal of $\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$ occurs at around 80 K. With the addition of a Cu interlayer, the $\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$ similarly decreases as a function of temperature, but maintains the same sign across the full temperature range. This indicates that both the magnetic-proximity effect and spin Hall effect in the TmIG/Pt bilayer contribute to the AHE signal with opposing signs. The spin-Hall contribution to the AHE is dominant at room temperature but decreases with decreasing temperature. Meanwhile, the magnetic-proximity contribution to the AHE becomes dominant with decreasing temperatures, leading to a change of sign for $\mathrm{\ensuremath{\Delta}}{\ensuremath{\rho}}_{\mathrm{AHE}}$. We exclude a dominant influence of a ferrimagnetic compensation point in the temperature region by complementary magnetic hysteresis and neutron diffraction measurements. Our work, based on a simple method, sheds light on the origin of the AHE in magnetic insulator heterostructures, where the competition between the magnetic-proximity effect and spin Hall effect governs the sign and amplitude of the AHE.

A normal metallic film sandwiched between two insulators may have strong spin-orbit coupling near the metal-insulator interfaces, even if spin-orbit coupling is negligible in the bulk of the film. In this paper, we study two technologically important and deeply interconnected effects that arise from interfacial spin-orbit coupling in metallic films. The first is the spin Hall effect, whereby a charge current in the plane of the film is partially converted into an orthogonal spin current in the same plane. The second is the Edelstein effect, in which a charge current produces an in-plane, transverse spin polarization. At variance with strictly two-dimensional Rashba systems, we find that the spin Hall conductivity has a finite value even if spin-orbit interaction with impurities is neglected and ``vertex corrections'' are properly taken into account. Even more remarkably, such a finite value becomes ``universal'' in a certain configuration. This is a direct consequence of the spatial dependence of spin-orbit coupling on the third dimension, perpendicular to the plane of the film. The nonvanishing spin Hall conductivity has a profound influence on the Edelstein effect, which we show to consist of two terms, the first with the standard form valid in a strictly two-dimensional Rashba system, and a second arising from the presence of the third dimension. Whereas the standard term is proportional to the momentum relaxation time, the new one scales with the spin relaxation time. Our results, although derived in a specific model, should be valid rather generally, whenever a spatially dependent Rashba spin-orbit coupling is present and the electron motion is not strictly two dimensional.

We have carried out a comprehensive study of the intrinsic anomalous Hall effect and spin Hall effect of several chiral antiferromagnetic compounds, Mn$_3X$ ($X$ = Ge, Sn, Ga, Ir, Rh and Pt) by $ab~initio$ band structure and Berry phase calculations. These studies reveal large and anisotropic values of both the intrinsic anomalous Hall effect and spin Hall effect. The Mn$_3X$ materials exhibit a non-collinear antiferromagnetic order which, to avoid geometrical frustration, forms planes of Mn moments that are arranged in a Kagome-type lattice. With respect to these Kagome planes, we find that both the anomalous Hall conductivity (AHC) and the spin Hall conductivity (SHC) are quite anisotropic for any of these materials. Based on our calculations, we propose how to maximize AHC and SHC for different materials. The band structures and corresponding electron filling, that we show are essential to determine the AHC and SHC, are compared for these different compounds. We point out that Mn$_3$Ga shows a large SHC of about 600 $(\hbar/e)(\Omega\cdot cm)^{-1}$. Our work provides insights into the realization of strong anomalous Hall effects and spin Hall effects in chiral antiferromagetic materials.

We examine a possible spin Hall effect for localized spin systems with no charge degrees of freedom. In this scenario, a longitudinal magnetic field gradient induces a transverse spin current carried by spin wave excitations with an anomalous velocity which is caused by a topological Berry-phase effect associated with spin chirality, in analogy with anomalous Hall effects in itinerant electron systems. Our argument is based on semiclassical equations of motion applicable to general spin systems. Also, a microscopic model of frustrated magnets which exhibits the anomalous spin Hall effect is presented.

We theoretically investigated the anomalous Hall effect (AHE) and spin Hall effect (SHE) transversal to the insulating spacer I, in magnetic tunnel junctions of the form F/I/F where the F's are ferromagnetic layers and I represents a tunnel barrier. We considered the case of purely ballistic (quantum mechanical) transport. These effects arise because of the asymmetric scattering of evanescent wave functions due to the spin-orbit interaction in the tunnel barrier. The AHE and SHE we investigated have a surface nature due to the proximity effect. Their amplitude is of first order in the scattering potential. This contrasts with ferromagnetic metals wherein these effects are of second (side-jump scattering) and third (skew scattering) order in these potentials. The value of the AHE current in the insulating spacer may be much larger than that in metallic ferromagnetic electrodes. For the antiparallel orientation of the magnetizations in the two F electrodes, a spontaneous Hall current exists even at zero applied voltage.

The sign reversal in the anomalous Hall effect (AHE) that occurs for material offers great prospects for AHE-based spintronic devices design. However, the mechanisms are still controversial in ultrathin ferromagnetic/heavy metal thin film systems due to the complicatedly interfacial effects. Here, we investigate the AHE sign reversal in ultrathin ferromagnetic Mn2CoAl/Pd films, a system which has shown unusual AHE, significant spin-orbit coupling, and magnetic texturing. Element-sensitive cross-sectional STEM imaging and the depth-resolved magnetization profile from polarized neutron reflectometry identifies the presence of a second ferromagnetic layer from intermixed Co-Pd. To quantitatively explain the sign reversal of the AHE, we build a model based on two contributions, ferromagnetic Mn2CoAl and the intermixed CoPd layer. We also clarify that contributions to the AHE from magnetic proximity and spin Hall effect are negligible. Our work demonstrates that interfacial alloying can be a critical factor and provides insightful methods to determine the origins of the AHE in ferromagnet/heavy-metal thin film systems.

The direct current electric voltage induced by the Inverse Spin Hall Effect (ISHE) and Anomalous Hall Effect (AHE) was investigated in the TE011 and TE102 cavities. The ISHE and AHE components were distinguishable through the fitting of the voltage spectrum. The unwanted AHE was minimized by placing the DUT (Device Under Test) at the center of both the TE011 and TE102 cavities. The voltage of ISHE in the TE011 cavity was larger than that in the TE102 cavity due to the higher quality factor of the former. Despite optimized centering, AHE voltage from TE011 cavity was also higher. The reason was attributed to the E-field distribution inside the cavity. In the case of the TE011 cavity, the DUT was easily exposed to the E-field in all directions. Therefore, the parasitic AHE voltage in the TE102 cavity was less sensitive than that in the TE011 cavity to decentering problem.

Quantum anomalous Hall state is expected to emerge in Dirac electron systems such as graphene under both sufficiently strong exchange and spin-orbit interactions. In pristine graphene, neither interaction exists; however, both interactions can be acquired by coupling graphene to a magnetic insulator as revealed by the anomalous Hall effect. Here, we show enhanced magnetic proximity coupling by sandwiching graphene between a ferrimagnetic insulator yttrium iron garnet (YIG) and hexagonal-boron nitride (h-BN) which also serves as a top gate dielectric. By sweeping the top-gate voltage, we observe Fermi level-dependent anomalous Hall conductance. As the Dirac point is approached from both electron and hole sides, the anomalous Hall conductance reaches ¼ of the quantum anomalous Hall conductance 2e2/h. The exchange coupling strength is determined to be as high as 27 meV from the transition temperature of the induced magnetic phase. YIG/graphene/h-BN is an excellent heterostructure for demonstrating proximity-induced interactions in two-dimensional electron systems.

Apart from classical and anomalous Hall effects, an appearance of transverse current in chiral systems has been recently discovered named as topological Hall effect (THE) as it was thought to be possible solely for topologically charged spin textures such as magnetic skyrmions. In this talk we focus on asymmetric electron scattering on chiral spin textures and tunnel transport across a semiconductor interface with account for exchange and spin-orbit interactions. We distinguish between chiral Hall effect (CHE) which requires local chirality of an effective magnetic field and the THE which requires also a non-zero topological charge. THE emerges in adiabatic regime of scattering, which can be described via geometrical Berry phase. With semi-classical approach we clarify why the spin Hall current in this case is proportional to the topological charge of a spin texture and does not depend on its size. For non-adiabatic regime we expand the scattering T-matrix in Born series and show that the scattering cross section acquires an asymmetric part, which depends on the local chirality of the spin texture leading to CHE independently of the topological charge. The analysis allowed us to clarify the reason for the spin Hall effect to vanish in this regime while preserving the charge Hall effect. We have also found theoretically the spin swapping effect for the exchange scattering on chiral spin textures. Further, with the T-matrix analysis we explain the asymmetry in the tunneling across a semiconductor interface emerging due to chiral configuration of exchange and spin-orbit fields leading to generation of transverse current.

The current induced spin-orbit-torque (SOT) [1-4] is the key physical parameter describing the generation of magnetization dynamics in bilayers composed by a heavy metal (HM) layer and a ferromagnetic (FM) layer. Such bilayers, especially when associated with a magnetic tunnel junction at the FM layer (then forming the so called 3-terminal device), are considered as promising elements to be applied in novel magnetic memories, microwave generators and even bio-inspired types of computing [1]. Therefore, the robust and accurate determination of the SOT in HM/FM bilayers is of high relevance for optimizing future applications.In HM/FM bilayers the SOT is mainly caused by the spin Hall effect (SHE) of the HM layer, by which a spin current is injected into the FM layer. 3-terminal devices with application as magnetic memories adopt mainly FM films with perpendicular magnetic anisotropy, while for sensor applications also the configuration with in-plane magnetic anisotropy is of interest. The determination and quantification of the SOT can be performed in both cases by different techniques. While in a 3-terminal device the magnetization dynamics of observed via the tunnel magnetoresistance effect, in HM/FM bilayers other methods can be employed, such as the SOT-ferromagnetic resonance (FMR), the harmonic Hall voltage measurement and the spin Hall magnetoresistance (SMR). The SMR is a straightforward method when the FM layer is an insulator, while with metallic FM, in which the testing electric current in not confined in the HM layer only, it is accompanied by additional effects, such as the anisotropic magnetoresistance or the anomalous Nernst effect [5]. Therefore, it is important to analyse the measurement results by accurate models taking into account all aspects. The dominant theoretical approach for SHE-SOT is the drift-diffusion model considering spin accumulation at the interface between the FM and HM by introducing a spin mixing conductance. However, the agreement with experiments may be poor if spin dissipation at the interface (spin memory loss, spin dependent scattering, etc.) is disregarded [2].In this paper we analyze transport measurements on FM/HM bilayers as a function of the HM thickness by employing a non-equilibrium thermodynamic approach to SHE-SOT [6] with the aim to reach a semi-quantitative agreement. The study was performed on HM/FM bilayers with HM=Pt (wedge 4-19nm), Ta (wedge 4-19nm), and FM=Fe60Co20B20 (2nm) prepared at Singulus Technologies AG. All samples were annealed for 2h at 310°C in a magnetic field of 1T in order to develop an in-plane magnetic anisotropy (y axis). For the transport measurements the films were patterned into µm-size Hall bars/crosses (along x axis) by optical lithography and ion beam etching with Ti/Au contact pads. Magneto-optic Kerr effect (MOKE) and SMR measurements were performed following the protocol of [5], while SOT-FMR and harmonic Hall voltage measurements and analysis were based on references [3,4].Fig.1 shows the comparison between MOKE loops and the SMR signal. The loop along the y axis is more rectangular and indicates the magnetic easy anisotropy axis. The SMR signal is much more pronounced with the field along the x axis, while it shows a small peak close to the coercivity when the field is along the y axis. With MOKE the anisotropy was determined and a dependence on HM thickness was observed.Fig.2 presents the summary of the SOT-FMR and harmonic Hall voltage measurements for HM/FM bilayers with different thickness of HM. FMR analysis enabled determination of effective magnetization and magnetization damping (α = 0.015 for Pt and 0.0075 for Ta), while anomalous Hall effect and harmonic Hall measurements resulted in SOT efficiencies dependence vs. HM thickness. While damping-like SOT efficiency for Ta reaches ξDL-Ta = -0.35 and ξDL-Pt = 0.15 for Pt, with high resistivity difference (ρTa > 200 µΩcm and 24 < ρPt < 52 µΩcm) taken into account, the spin Hall conductivity for Pt of σSH-Pt = 2.9 (105 S/m) exceeds the one for Ta σSH-Ta = 1.7 (105 S/m) and is among the highest values reported [8].The HM thickness dependence of the SMR is analyzed by considering both the finite conductance at the interface [7] (an approach valid for small thicknesses) and thermodynamic approach [6] (valid for larger thicknesses). The results are compared with the harmonic Hall voltage measurements the SOT and the SOT efficiency and discussed in relation to the key parameters as the spin Hall angle and the spin diffusion length.Acknowledgements: WS acknowledges National Science Centre Grant No. UMO-2015/17/D/ST3/00500, Poland. Microfabrication was performed at the Academic Centre for Materials and Nanotechnology of AGH. **

The anomalous transverse transport properties, including the anomalous Hall and Nernst effects, are crucial for probing the topological band features in magnetic materials. In this study, we present a comprehensive investigation of the magnetic, electrical, and thermal transport, as well as the electronic band structure, of the noncollinear ferromagnet ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$. Our findings reveal that ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$ exhibits prominent anomalous Hall and Nernst effects. Scaling analysis suggests that the anomalous Hall effect is predominantly influenced by the intrinsic Berry curvature contribution. Additionally, the large anomalous Nernst signal exceeds the magnetization scaling relations observed in conventional ferromagnets, and the calculated anomalous Nernst conductivity exhibits a $T\text{ln}T$ relation. These observations align with the behavior seen in other magnetic topological materials, further suggesting a substantial intrinsic Berry curvature effect. Complemented by detailed theoretical calculations, we clarify the electronic band properties and confirm a substantial net Berry curvature near the Fermi level in this compound. Our work elucidates that the large anomalous Hall and Nernst effects of ${\mathrm{PrMn}}_{2}{\mathrm{Ge}}_{2}$ are dominated by an intrinsic mechanism and presents it as a probable magnetic topological material with a significant Berry phase effect in the momentum space.

We study transport in normal metals in an external magnetic field. This\nsystem exhibits an interplay between a transverse spin imbalance (spin Hall\neffect) caused by the spin-orbit interaction, a Hall effect via the Lorentz\nforce, and spin precession due to the Zeeman effect. Diffusion equations for\nspin and charge flow are derived. The spin and charge accumulations are\ncomputed numerically in experimentally relevant thin film geometries. The\nout-of-plane spin Hall potential is suppressed when the Larmor frequency is\nlarger than the spin-flip scattering rate. The in-plane spin Hall potential\nvanishes at zero magnetic field and attains its maximum at a finite magnetic\nfield before spin precession starts to dominate. Spin-injection via\nferromagnetic contacts creates a transverse charge Hall effect that decays in a\nfinite magnetic field due to spin precession.\n