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Abstract 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.

Magnetic domain walls (DWs) are the finite boundaries that separate the regions of uniform magnetization in a magnetic material. They constitute a key research topic in condensed matter physics due to their intriguing physics and relevance in technological applications. A multitude of spintronic concepts for memory, logic, and data processing applications have been proposed, relying on the precise control of DWs via magnetic fields and electric currents. Intensive research into DWs has also spurred interest into chiral magnetic interactions, topology, and relativistic physics. In this article, we will first review the rapid evolution of magnetic DW research and, in particular, the current-driven DW motion enabled by the improved understanding of DW dynamics and the development of suitable ferrimagnetic thin films. We will then provide an outlook on future directions in DW dynamics research exploiting ferrimagnetic garnets as a tunable material platform.

Abstract Spin–orbit torques (SOTs) have emerged as an efficient means to electrically control the magnetization in ferromagnetic heterostructures. Lately, 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 is, thus, vital for the fundamental understanding of charge–spin interrelations and designing novel devices. However, the accurate determination of SOTs in MIs is limited due to low electrical signals and dominant spurious thermoelectric effects caused by Joule heating. Here, a simple method based on harmonic Hall voltage detection and macrospin simulations is reported that accurately quantifies 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 1.7 × 1011 A m−2, respectively. It is further revealed 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 methodology can be beneficial for the fundamental understanding of SOTs in MI‐based heterostructures and device designs where accurate SOTs knowledge is necessary.

Magnetic domain walls (DWs) are the finite boundaries that separate the regions of uniform magnetization in a magnetic material. They constitute a key research topic in condensed matter physics due to their intriguing physics and relevance in technological applications. A multitude of spintronic concepts for memory, logic, and data processing applications have been proposed, relying on the precise control of DWs via magnetic fields and electric currents. Intensive research into DWs has also spurred interest into chiral magnetic interactions, topology, and relativistic physics. In this article, we will first review the rapid evolution of magnetic DW research and, in particular, the current-driven DW motion enabled by the improved understanding of DW dynamics and the development of suitable ferrimagnetic thin films. We will then provide an outlook on future directions in DW dynamics research exploiting ferrimagnetic garnets as a tunable material platform. [ABSTRACT FROM AUTHOR]

Current‐induced spin‐orbit torques (SOTs) have emerged as a powerful tool to control magnetic elements and non‐uniform magnetic textures such as domain walls and skyrmions. SOT‐induced switching of perpendicular magnetization generally requires an external field to break the rotational symmetry of the spin‐orbit effective fields responsible for the deterministic reversal. The proposed mechanisms to eliminate this requirement often rely on complex multilayer structures that necessitate laborious optimization in the material and spin transport properties, making them less attractive for applications. Herein, current‐induced, external field‐free switching of an epitaxial MgO/Pt/Co trilayer with an extremely large perpendicular anisotropy in excess of 3 Tesla is reported. It is found that switching occurs due to the interplay of strong SOTs, local anisotropy fluctuations, and the Dzyaloshinkii‐Moriya interaction inherent to this epitaxial system. Given that these layers constitute the base stack of a magnetic tunnel junction, this switching mechanism offers the most technologically viable path toward devices such as field‐free SOT‐based magnetic random‐access memories. [ABSTRACT FROM AUTHOR]

Spin–orbit torques (SOTs) have emerged as an efficient means to electrically control the magnetization in ferromagnetic heterostructures. Lately, 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 is, thus, vital for the fundamental understanding of charge–spin interrelations and designing novel devices. However, the accurate determination of SOTs in MIs is limited due to low electrical signals and dominant spurious thermoelectric effects caused by Joule heating. Here, a simple method based on harmonic Hall voltage detection and macrospin simulations is reported that accurately quantifies 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 1.7 × 1011 A m−2, respectively. It is further revealed 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 methodology can be beneficial for the fundamental understanding of SOTs in MI‐based heterostructures and device designs where accurate SOTs knowledge is necessary. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We report on the combination of current-induced spin–orbit torques and giant magnetoresistance in a single device to achieve all-electrical write and readout 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 a moderate effort. [ABSTRACT FROM AUTHOR]

We study the role of Pt crystal orientation in spin-orbit torques in Co/Pt bilayers by means of the harmonic Hall effect and current-induced switching measurements. Perpendicularly magnetized Co/Pt bilayers were fabricated with the Pt layer exhibiting either a polycrystalline grain structure or an epitaxial (111)-oriented film on MgO substrates by magnetron sputtering. We find that the damping-like spin-orbit torque is 1.3 times smaller in the epitaxial Co/Pt(111) bilayers compared to the polycrystalline films, whereas the field-like spin-orbit torque values are of comparable magnitude. Current-induced magnetization switching measurements show good agreement with the results of harmonic measurements of damping-like torque, i.e., the critical switching current is about 30% higher in epitaxial Co/Pt(111). These results highlight the importance of crystal orientation effects on spin-orbit torques in nominally identical bilayer structures. [ABSTRACT FROM AUTHOR]

This research was conducted for the purpose of determining the effects of bat guano on plant nutrient contents in the soil and in lettuce (Lactuca sativa L.) during the spring of both 2014 and 2015 with three replications according to randomized blocks experiment design. Bat guano was applied to the soil in two different forms; According to the results organic matter, P, Cu and Mn amounts in the soil were found to be statistically significant and increased compared to the control. Likewise, in the application of bat guano dust, organic matter, P and Mn amounts in the soil were found to be statistically significant and increased compared to the control. In lettuce, N, K, Mg, Fe, Cu, Zn and Mn amounts in the application of bat guano powder and K, Ca, Mg, Fe, Zn and Mn amounts in the application of bat guano dust were found to be statistically significant. [ABSTRACT FROM AUTHOR]

We report fast and efficient current-induced switching of a perpendicular anisotropy magnetic insulator thulium iron garnet by using spin-orbit torques (SOT) from the Pt overlayer. We first show that, with quasi-DC (10 ms) current pulses, SOT-induced switching can be achieved with an external field as low as 2 Oe, making TmIG an outstanding candidate to realize efficient switching in heterostructures that produce moderate stray fields without requiring an external field. We then demonstrate deterministic switching with fast current pulses (≤20 ns) with an amplitude of ~1012 A/m2, similar to all-metallic structures. We reveal that, in the presence of an initially nucleated domain, the critical switching current is reduced by up to a factor of five with respect to the fully saturated initial state, implying efficient current-driven domain wall motion in this system. Based on measurements with 2 ns-long pulses, we estimate the domain wall velocity of the order of ~400 m/s per j=1012 A/m2. [ABSTRACT FROM AUTHOR]

The article focuses on a memory device concept based on a unidirectional spin Hall magnetoresistance (USMR), which can store multiple bits of information in single ferromagnetic heterostructure. It notes that USMR can give rise to four different 2nd harmonic resistance levels corresponding to four magnetization states. It proposes the possibility for building an all-electrical lateral two-terminal multi-bit-per-cell memory device.

Magnetoresistive effects are usually invariant on inversion of the magnetization direction. In non-centrosymmetric 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. [ABSTRACT FROM AUTHOR]

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(6 nm)/Co(2.5 nm) and negligible in Pt(6 nm)/Co(2.5 nm) 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 fieldlike SOTs are about an order of magnitude smaller. [ABSTRACT FROM AUTHOR]

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 Ic as a function of pulse length, amplitude, and external field. Our data evidence two distinct regimes: a short-time intrinsic regime, where Ic scales linearly with the inverse of the pulse length, and a long-time thermally assisted regime, where Ic varies weakly. Both regimes are consistent with magnetization reversal proceeding by nucleation and fast propagation of domains. We find that Ic 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. [ABSTRACT FROM AUTHOR]

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 °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 (fieldlike) and longitudinal (antidampinglike) 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 fieldlike and antidamping torques. Measurements of the switching probability using sub-μs 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 and 270 °C. [ABSTRACT FROM AUTHOR]

We report on the current-induced magnetization switching of a three-terminal perpendicular magnetic tunnel junction by spin-orbit torque and its 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 5×1011 A/m2 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. [ABSTRACT FROM AUTHOR]

Recent demonstrations of magnetization switching induced by in-plane current injection in heavy metal/ferromagnetic heterostructures have drawn increasing attention to spin torques based on orbital-to-spin momentum transfer. The symmetry, magnitude and origin of spin-orbit torques (SOTs), however, remain a matter of debate. Here we report on the three-dimensional vector measurement of SOTs in AlOx/Co/Pt and MgO/CoFeB/Ta trilayers using harmonic analysis of the anomalous and planar Hall effects. We provide a general scheme to measure the amplitude and direction of SOTs as a function of the magnetization direction. Based on space and time inversion symmetry arguments, we demonstrate that heavy metal/ferromagnetic layers allow for two different SOTs having odd and even behaviour with respect to magnetization reversal. Such torques include strongly anisotropic field-like and spin transfer-like components, which depend on the type of heavy metal layer and annealing treatment. These results call for SOT models that go beyond the spin Hall and Rashba effects investigated thus far. [ABSTRACT FROM AUTHOR]

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 monotonically 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. [ABSTRACT FROM AUTHOR]

We demonstrate a conceptually new mechanism to generate an in-plane spin current with out-of-plane polarization in a nonmagnetic metal, detected by nonlocal thermoelectric voltage measurement. We generate out-of-plane (∇TOP) and in-plane (∇TIP) temperature gradients, simultaneously, acting on a magnetic insulator-Pt bilayer. When the magnetization has a component oriented perpendicular to the plane, ∇TOP drives a spin current into Pt with out-of-plane polarization due to the spin Seebeck effect. ∇TIP then drags the resulting spin-polarized electrons in Pt parallel to the plane against the gradient direction. This finally produces an inverse spin Hall effect voltage in Pt, transverse to ∇TIP and proportional to the out-of-plane component of the magnetization. This simple method enables the detection of the perpendicular magnetization component in a magnetic insulator in a nonlocal geometry. [ABSTRACT FROM AUTHOR]

Recent studies evidenced 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. [ABSTRACT FROM AUTHOR]

We characterize the spin Hall magnetoresistance (SMR), spin Seebeck effect (SSE), and dampinglike spin-orbit torque (SOT) in thulium iron garnet/platinum bilayers with perpendicular magnetic anisotropy by using harmonic Hall effect measurements. By consecutive annealing steps followed by measurements on a single device, we reveal that the spin-dependent effects gradually decrease in amplitude as the annealing temperature increases. We attribute this behavior primarily to the changes in the spin-mixing conductance, which sensitively depends on the interface quality. However, further analysis demonstrates that although the SSE scales closely with the SMR, the dampinglike SOT shows a significantly different trend upon annealing, contrary to theoretical expectations. By comparing the dampinglike SOT with the field-induced Hall effect, we found evidence that scattering from Fe impurities in the Pt at the interface might be responsible for the distinct annealing temperature dependence of the dampinglike SOT. [ABSTRACT FROM AUTHOR]

Current-induced spin-orbit torques are one of the most effective ways to manipulate the magnetization in spintronic devices, and hold promise for fast switching applications in non-volatile memory and logic units. Here, we report the direct observation of spin-orbit-torque-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 subnanosecond 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 the damping-like and field-like spin-orbit torques and the Dzyaloshinskii-Moriya interaction, and show that reproducible switching events can be obtained for over 10 12 reversal cycles.

The spin Hall effect in heavy metals converts charge current into pure spin current, which can be injected into an adjacent ferromagnet to exert a torque. This spin-orbit torque (SOT) has been widely used to manipulate the magnetization in metallic ferromagnets. In the case of magnetic insulators (MIs), although charge currents cannot flow, spin currents can propagate, but current-induced control of the magnetization in a MI has so far remained elusive. Here we demonstrate spin-current-induced switching of a perpendicularly magnetized thulium iron garnet film driven by charge current in a Pt overlayer. We estimate a relatively large spin-mixing conductance and damping-like SOT through spin Hall magnetoresistance and harmonic Hall measurements, respectively, indicating considerable spin transparency at the Pt/MI interface. We show that spin currents injected across this interface lead to deterministic magnetization reversal at low current densities, paving the road towards ultralow-dissipation spintronic devices based on MIs.