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The coset construction is a tool for systematically building low energy effective actions for Nambu-Goldstone modes. This technique is typically used to compute time-ordered correlators appropriate for $S$-matrix computations for systems in their ground state. In this paper, we extend this technique to the Schwinger-Keldysh formalism, which enables one to calculate a wider variety of correlators and applies also to systems in a mixed state. We focus our attention on internal symmetries and demonstrate that, after identifying the appropriate symmetry breaking pattern, Schwinger-Keldysh effective actions for Nambu-Goldstone modes can be constructed using the standard rules of the coset construction. Particular emphasis is placed on the thermal state and ensuring that correlators satisfy the KMS relation. We also discuss explicitly the power counting scheme underlying our effective actions. We comment on the similarities and differences between our approach and others that have previously appeared in the literature. In particular, our prescription does not require the introduction of additional ``diffusive'' symmetries and retains the full non-linear structure generated by the coset construction. We conclude with a series of explicit examples, including a computation of the finite-temperature two-point functions of conserved spin currents in non-relativistic paramagnets, antiferromagnets, and ferromagnets. Along the way, we also clarify the discrete symmetries that set antiferromagnets apart from ferromagnets, and point out that the dynamical KMS symmetry must be implemented in different ways in these two systems., Comment: 38 pages, 1 figure, 1 table; v2: matches published version

We report the structural, magnetic, and interfacial spin transport properties of epitaxial terbium iron garnet (TbIG) ultrathin films deposited by magnetron sputtering. High crystallinity was achieved by growing the films on gadolinium gallium garnet (GGG) substrates either at high temperature, or at room temperature followed by thermal annealing, above 750 {\deg}C in both cases. The films display large perpendicular magnetic anisotropy (PMA) induced by compressive strain, and tunable structural and magnetic properties through growth conditions or the substrate lattice parameter choice. The ferrimagnetic compensation temperature (TM) of selected TbIG films was measured through temperature-dependent anomalous Hall effect (AHE) in Pt/TbIG heterostructures. In the studied films, TM was found to be between 190-225 K, i.e., approximately 25-60 K lower than the bulk value, which is attributed to the combined action of Tb deficiency and oxygen vacancies in the garnet lattice evidenced by x-ray photoelectron spectroscopy measurements. Sputtered TbIG ultrathin films with large PMA and highly tunable properties reported here can provide a suitable material platform for a wide range of spintronic experiments and device applications.

Spin-orbit torques (SOTs) have emerged as an efficient means to electrically control the magnetization in ferromagnetic heterostructures. Lately, an increasing attention has been devoted to SOTs in heavy metal (HM)/magnetic insulator (MI) bilayers owing to their tunable magnetic properties and insulating nature. Quantitative characterization of SOTs in HM/MI heterostructures are, thus, vital for fundamental understanding of charge-spin interrelations and designing novel devices. However, the accurate determination of SOTs in MIs have been limited so far due to small electrical signal outputs and dominant spurious thermoelectric effects caused by Joule heating. Here, we report a simple methodology based on harmonic Hall voltage detection and macrospin simulations to accurately quantify the damping-like and field-like SOTs, and thermoelectric contributions separately in MI-based systems. Experiments on the archetypical Bi-doped YIG/Pt heterostructure using the developed method yield precise values for the field-like and damping-like SOTs, reaching -0.14 and -0.15 mT per 1.7x$10^{ 11}$ A/$m^2$, respectively. We further reveal that current-induced Joule heating changes the spin transparency at the interface, reducing the spin Hall magnetoresistance and damping-like SOT, simultaneously. These results and the devised method can be beneficial for fundamental understanding of SOTs in MI-based heterostructures and designing new devices where accurate knowledge of SOTs is necessary.

We measured the spin-orbit torques (SOTs), current-induced switching, and domain wall (DW) motion in synthetic ferrimagnets consisting of Co/Tb layers with differing stacking order grown on a Pt underlayer. We find that the SOTs, magnetic anisotropy, compensation temperature and SOT-induced switching are highly sensitive to the stacking order of Co and Tb and to the element in contact with Pt. Our study further shows that Tb is an efficient SOT generator when in contact with Co, such that its position in the stack can be adjusted to generate torques additive to those generated by Pt. With optimal stacking and layer thickness, the dampinglike SOT efficiency reaches up to 0.3, which is more than twice that expected from the Pt/Co bilayer. Moreover, the magnetization can be easily switched by the injection of pulses with current density of about 0.5-2*107A/cm2 despite the extremely high perpendicular magnetic anisotropy barrier (up to 7.8 T). Efficient switching is due to the combination of large SOTs and low saturation magnetization owing to the ferrimagnetic character of the multilayers. We observed current-driven DW motion in the absence of any external field, which is indicative of homochiral N\'eel-type DWs stabilized by the interfacial Dzyaloshinkii-Moriya interaction. These results show that the stacking order in transition metal/rare-earth synthetic ferrimagnets plays a major role in determining the magnetotransport properties relevant for spintronic applications.

We report on the occurrence of strong interlayer Dzyaloshinskii-Moriya interaction (DMI) between an in-plane magnetized Co layer and a perpendicularly magnetized TbFe layer through a Pt spacer. The DMI causes a chiral coupling that favors one-handed orthogonal magnetic configurations of Co and TbFe, which we reveal through Hall effect and magnetoresistance measurements. The DMI coupling mediated by Pt causes effective magnetic fields on either layer of up to 10-15 mT, which decrease monotonously with increasing Pt thickness. Ru, Ta, and Ti spacers mediate a significantly smaller coupling compared to Pt, highlighting the essential role of Pt in inducing the interlayer DMI. These results are relevant to understand and maximize the interlayer coupling induced by the DMI as well as to design spintronic devices with chiral spin textures.

Topological insulators have emerged as an important material class for efficient spin-charge interconversion. Most topological insulators considered to date are binary or ternary compounds, with the exception of $\alpha$-Sn. Here we report a comprehensive characterization of the growth, magnetotransport properties, and current-induced spin-orbit torques of $\alpha$-Sn and $\beta$-Sn-based ferromagnetic heterostructures. We show that $\alpha$-Sn grown with a Bi surfactant on CdTe(001) promotes large spin-orbit torques in a ferromagnetic FeCo layer at room temperature, comparable to Pt, whereas $\alpha$-Sn grown without Bi surfactant and the non-topological phase, $\beta$-Sn, induce lower torques. The dampinglike and fieldlike spin-orbit torque efficiency in $\alpha$-Sn with Bi are 0.12 and 0.18, respectively. Further, we show that $\alpha$-Sn grown with and without Bi presents a spin Hall-like magnetoresistance comparable to that found in heavy metal/ferromagnet bilayers. Our work demonstrates direct and efficient charge-to-spin conversion in $\alpha$-Sn ferromagnetic heterostructures, showing that $\alpha$-Sn is a promising material for current-induced magnetization control in spintronic devices.

The research into insulating ferrimagnetic garnets has gained enormous momentum in the past decade. This is partly due to the improvement in the techniques to grow high-quality ultrathin films with desirable properties and the advances in understanding the spin transport within the ferrimagnetic garnets and through their interfaces with conducting materials. In recent years, we have seen remarkable progress in controlling the magnetization state of ferrimagnetic garnets by electrical means in suitable heterostructures and device architectures. These advances have readily placed ferrimagnetic garnets in a favorable position for the future development of insulating spintronic concepts. The purpose of this article is to review recent experimental results of the current-induced magnetization control and associated phenomena in ferrimagnetic garnets, as well as to discuss future directions in this rapidly evolving area of spintronics., Comment: Short Review, 31 pages, 10 figures

We report on the combination of current-induced spin-orbit torques and giant magnetoresistance in a single device to achieve all-electrical write and read out of the magnetization. The device consists of perpendicularly magnetized TbCo and Co layers separated by a Pt or Cu spacer. Current injection through such layers exerts spin-orbit torques and switches the magnetization of the Co layer while the TbCo magnetization remains fixed. Subsequent current injection of lower amplitude senses the relative orientation of the magnetization of the Co and TbCo layers, which results in two distinct resistance levels for parallel and antiparallel alignment due to the current-in-plane giant magnetoresistance effect. We further show that the giant magnetoresistance of devices including a single TbCo/spacer/Co trilayer can be improved from 0.02% to 6% by using a Cu spacer instead of Pt. This type of devices offers an alternative route to a two terminal spintronic memory that can be fabricated with moderate effort., Comment: 18 pages, 5 figures

We show that the coupling between two ferromagnetic layers separated by a nonmagnetic spacer can be used to control the depinning of domain walls and induce unidirectional domain wall propagation. We investigated CoFeB/Ti/CoFeB trilayers where the easy axis of the magnetization of the top CoFeB layer is out-of-plane and that of the bottom layer is in-plane. Using Magneto-optic Kerr effect microscopy, we find that the depinning of a domain wall in the perpendicularly magnetized CoFeB layer is influenced by the orientation of the magnetization of the in-plane layer, which gives rise to a field-driven asymmetric domain expansion. This effect occurs due to the magnetic coupling between the internal magnetization of the domain wall and the magnetization of the in-plane CoFeB layer, which breaks the symmetry of up-down and down-up homochiral N\'eel domain walls in the perpendicular CoFeB layer. Micromagnetic simulations support these findings by showing that the interlayer coupling either opposes or favors the Dzyaloshinskii-Moriya interaction in the domain wall, thereby generating an imbalance in the depinning fields. This effect also allows for artificially controlling the chirality and dynamics of domain walls in magnetic layers lacking a strong Dzyaloshinskii-Moriya interaction., Comment: 21 pages, 6 figures

Oxidation strongly influences the properties of magnetic layers employed in spintronic devices. We study the effect of oxidation on the structural, magnetic, and electrical properties as well as current-induced spin-orbit torques (SOTs) in Pt/Co/AlOx, Pt/CoOx/Co/AlOx, and PtOx/Co/AlOx layers. We show how the saturation magnetization, perpendicular magnetic anisotropy, anomalous Hall resistance, and SOT are systematically affected by the degree of oxidation of both the Pt/Co and Co/Al interfaces. Oxidation of the Co/Al interface results in a 21% and 42% variation of the dampinglike and fieldlike SOT efficiencies, which peak at 0.14 and 0.07, respectively. The insertion of a paramagnetic CoOx layer between Pt and Co maintains a very strong perpendicular magnetic anisotropy and improves the dampinglike and fieldlike SOT efficiencies, up to 0.26 and 0.20, respectively. In contrast with recent reports, we do not find that the oxidation of Pt leads to a significant enhancement of the torques. Rather, we find that oxygen migrates from Pt to the Co and Al layers, leading to a time-dependent oxidation profile and an effective spin Hall conductivity that decreases with increasing oxygen concentration. Finally, we study current-induced switching in Pt/Co/AlOx with different degrees of oxidation and find a linear relationship between the critical switching current and the effective magnetic anisotropy controlled by the oxidation of Al. These results highlight the importance of interfaces and oxidation effects on the SOT and magnetotransport properties of heavy metal/ferromagnet/oxide trilayers and provide information on how to improve the SOT efficiency and magnetization-switching characteristics of these systems.

We studied the effect of inserting 0.5 nm-thick spacer layers (Ti, V, Cr, Mo, W) at the Pt/Co interface on the spin-orbit torques, Hall effect, magnetoresistance, saturation magnetization, and magnetic anisotropy. We find that the damping-like spin-orbit torque decreases substantially for all samples with a spacer layer compared to the reference Pt/Co bilayer, consistently with the opposite sign of the atomic spin-orbit coupling constant of the spacer elements relative to Pt. The reduction of the damping-like torque is monotonic with atomic number for the isoelectronic 3d, 4d, and 5d elements, with the exception of V that has a stronger effect than Cr. The field-like spin-orbit torque almost vanishes for all spacer layers irrespective of their composition, suggesting that this torque predominantly originates at the Pt/Co interface. The anomalous Hall effect, magnetoresistance, and saturation magnetization are also all reduced substantially, whereas the sheet resistance is increased in the presence of the spacer layer. Finally, we evidence a correlation between the amplitude of the spin-orbit torques, the spin Hall-like magnetoresistance, and the perpendicular magnetic anisotropy. These results highlight the significant influence of ultrathin spacer layers on the magnetotransport properties of heavy metal/ferromagnetic systems., Comment: 29 pages, 5 figures

Recent studies evidence the emergence of asymmetric electron transport in layered conductors owing to the interplay between electrical conductivity, magnetization, and the spin Hall or Rashba- Edelstein effects. Here, we investigate the unidirectional magnetoresistance (UMR) caused by the current-induced spin accumulation in Co/Pt and CoCr/Pt bilayers. We identify three competing mechanisms underpinning the resistance asymmetry, namely interface and bulk spin-dependent electron scattering and electron-magnon scattering. Our measurements provide a consistent description of the current, magnetic field, and temperature dependence of the UMR and show that both positive and negative UMR can be obtained by tuning the interface and bulk spin-dependent scattering terms relative to the magnon population.

Current-induced spin-orbit torques (SOTs) represent one of the most effective ways to manipulate the magnetization in spintronic devices. The orthogonal torque-magnetization geometry, the strong damping, and the large domain wall velocities inherent to materials with strong spin-orbit coupling make SOTs especially appealing for fast switching applications in nonvolatile memory and logic units. So far, however, the timescale and evolution of the magnetization during the switching process have remained undetected. Here, we report the direct observation of SOT-driven magnetization dynamics in Pt/Co/AlO$_x$ dots during current pulse injection. Time-resolved x-ray images with 25 nm spatial and 100 ps temporal resolution reveal that switching is achieved within the duration of a sub-ns current pulse by the fast nucleation of an inverted domain at the edge of the dot and propagation of a tilted domain wall across the dot. The nucleation point is deterministic and alternates between the four dot quadrants depending on the sign of the magnetization, current, and external field. Our measurements reveal how the magnetic symmetry is broken by the concerted action of both damping-like and field-like SOT and show that reproducible switching events can be obtained for over $10^{12}$ reversal cycles.

Magnetic heterostructures that combine large spin-orbit torque efficiency, perpendicular magnetic anisotropy, and low resistivity are key to develop electrically-controlled memory and logic devices. Here we report on vector measurements of the current-induced spin orbit torques and magnetization switching in perpendicularly magnetized Pd/Co/AlO$_x$ layers as a function of Pd thickness. We find sizeable damping-like (DL) and field-like (FL) torques, of the order of 1~mT per $10^7$~A/cm$^2$, which have different thickness and magnetization angle dependence. The analysis of the DL torque efficiency per unit current density and electric field using drift-diffusion theory leads to an effective spin Hall angle and spin diffusion length of Pd larger than 0.03 and 7~nm, respectively. The FL SOT includes a significant interface contribution, is larger than estimated using drift-diffusion parameters, and is further strongly enhanced upon rotation of the magnetization from the out-of-plane to the in-plane direction. Finally, taking advantage of the large spin-orbit torques in this system, we demonstrate bipolar magnetization switching of Pd/Co/AlO$_x$ layers with similar current density as used for Pt/Co layers with comparable perpendicular magnetic anisotropy.

We studied the magnetoresistance of normal metal (NM)/ferromagnet (FM) bilayers in the linear and nonlinear (current-dependent) regimes and compared it with the amplitude of the spin-orbit torques and thermally induced electric fields. Our experiments reveal that the magnetoresistance of the heavy NM/Co bilayers (NM = Ta, W, Pt) is phenomenologically similar to the spin Hall magnetoresistance (SMR) of YIG/Pt, but has a much larger anisotropy, of the order of 0.5%, which increases with the atomic number of the NM. This SMR-like behavior is absent in light NM/Co bilayers (NM = Ti, Cu), which present the standard AMR expected of polycrystalline FM layers. In the Ta, W, Pt/Co bilayers we find an additional magnetoresistance, directly proportional to the current and to the transverse component of the magnetization. This so-called unidirectional SMR, of the order of 0.005%, is largest in W and correlates with the amplitude of the antidamping spin-orbit torque. The unidirectional SMR is below the accuracy of our measurements in YIG/Pt., Comment: 14 pages, 5 figures

Magnetoresistive effects are usually invariant upon inversion of the magnetization direction. In noncentrosymmetric conductors, however, nonlinear resistive terms can give rise to a current dependence that is quadratic in the applied voltage and linear in the magnetization. Here we demonstrate that such conditions are realized in simple bilayer metal films where the spin-orbit interaction and spin-dependent scattering couple the current-induced spin accumulation to the electrical conductivity. We show that the longitudinal resistance of Ta|Co and Pt|Co bilayers changes when reversing the polarity of the current or the sign of the magnetization. This unidirectional magnetoresistance scales linearly with current density and has opposite sign in Ta and Pt, which we associate with the modification of the interface scattering potential induced by the spin Hall effect in these materials. Our results suggest a route to control the resistance and detect magnetization switching in spintronic devices using a two-terminal geometry, which applies also to heterostructures including topological insulators.

We present harmonic transverse voltage measurements of current-induced thermoelectric and spin-orbit torque (SOT) effects in ferromagnet/normal metal bilayers, in which thermal gradients produced by Joule heating and SOT coexist and give rise to ac transverse signals with comparable symmetry and magnitude. Based on the symmetry and field-dependence of the transverse resistance, we develop a consistent method to separate thermoelectric and SOT measurements. By addressing first ferromagnet/light metal bilayers with negligible spin-orbit coupling, we show that in-plane current injection induces a vertical thermal gradient whose sign and magnitude are determined by the resistivity difference and stacking order of the magnetic and nonmagnetic layers. We then study ferromagnet/heavy metal bilayers with strong spin-orbit coupling, showing that second harmonic thermoelectric contributions to the transverse voltage may lead to a significant overestimation of the antidamping SOT. We find that thermoelectric effects are very strong in Ta(6nm)/Co(2.5nm) and negligible in Pt(6nm)/Co(2.5nm) bilayers. After including these effects in the analysis of the transverse voltage, we find that the antidamping SOTs in these bilayers, after normalization to the magnetization volume, are comparable to those found in thinner Co layers with perpendicular magnetization, whereas the field-like SOTs are about an order of magnitude smaller., Comment: 26 pages, 6 figures, 1 table

We present a comprehensive study of the current-induced spin-orbit torques in perpendicularly magnetized Ta/CoFeB/MgO layers. The samples were annealed in steps up to 300 degrees C and characterized using x-ray absorption spectroscopy, transmission electron microscopy, resistivity, and Hall effect measurements. By performing adiabatic harmonic Hall voltage measurements, we show that the transverse (field-like) and longitudinal (antidamping-like) spin-orbit torques are composed of constant and magnetization-dependent contributions, both of which vary strongly with annealing. Such variations correlate with changes of the saturation magnetization and magnetic anisotropy and are assigned to chemical and structural modifications of the layers. The relative variation of the constant and anisotropic torque terms as a function of annealing temperature is opposite for the field-like and antidamping torques. Measurements of the switching probability using sub-{\mu}s current pulses show that the critical current increases with the magnetic anisotropy of the layers, whereas the switching efficiency, measured as the ratio of magnetic anisotropy energy and pulse energy, decreases. The optimal annealing temperature to achieve maximum magnetic anisotropy, saturation magnetization, and switching efficiency is determined to be between 240 degrees and 270 degrees C.

We report on the current-induced magnetization switching of a three-terminal perpendicular magnetic tunnel junction by spin-orbit torque and the read-out using the tunnelling magnetoresistance (TMR) effect. The device is composed of a perpendicular Ta/FeCoB/MgO/FeCoB stack on top of a Ta current line. The magnetization of the bottom FeCoB layer can be switched reproducibly by the injection of current pulses with density $5\times10^{11}$ A/m$^2$ in the Ta layer in the presence of an in-plane bias magnetic field, leading to the full-scale change of the TMR signal. Our work demonstrates the proof of concept of a perpendicular spin-orbit torque magnetic memory cell., Comment: 17 pages, 5 figures

Spin-orbit torques induced by spin Hall and interfacial effects in heavy metal/ferromagnetic bilayers allow for a switching geometry based on in-plane current injection. Using this geometry, we demonstrate deterministic magnetization reversal by current pulses ranging from 180~ps to ms in Pt/Co/AlOx dots with lateral dimensions of 90~nm. We characterize the switching probability and critical current $I_c$ as function of pulse length, amplitude, and external field. Our data evidence two distinct regimes: a short-time intrinsic regime, where $I_c$ scales linearly with the inverse of the pulse length, and a long-time thermally assisted regime where $I_c$ varies weakly. Both regimes are consistent with magnetization reversal proceeding by nucleation and fast propagation of domains. We find that $I_c$ is a factor 3-4 smaller compared to a single domain model and that the incubation time is negligibly small, which is a hallmark feature of spin-orbit torques.

Current-induced spin torques are of great interest to manipulate the orientation of nanomagnets without applying external magnetic fields. They find direct application in non-volatile data storage and logic devices, and provide insight into fundamental processes related to the interdependence between charge and spin transport. Recent demonstrations of magnetization switching induced by in-plane current injection in ferromagnetic heterostructures have drawn attention to a class of spin torques based on orbital-to-spin momentum transfer, which is alternative to pure spin transfer torque (STT) between noncollinear magnetic layers and amenable to more diversified device functions. Due to the limited number of studies, however, there is still no consensus on the symmetry, magnitude, and origin of spin-orbit torques (SOTs). Here we report on the quantitative vector measurement of SOTs in Pt/Co/AlO trilayers using harmonic analysis of the anomalous and planar Hall effects as a function of the applied current and magnetization direction. We provide an all-purpose scheme to measure the amplitude and direction of SOTs for any arbitrary orientation of the magnetization, including corrections due to the interplay of Hall and thermoelectric effects. Based on general space and time inversion symmetry arguments, we show that asymmetric heterostructures allow for two different SOTs having odd and even behavior with respect to magnetization reversal. Our results reveal a scenario that goes beyond established models of the Rashba and spin Hall contributions to SOTs. The even SOT is STT-like but stronger than expected from the spin Hall effect in Pt. The odd SOT is composed of a constant field-like term and an additional component, which is strongly anisotropic and does not correspond to a simple Rashba field., Comment: Supplementary Informations follows Paper in the .pdf file