Spin-transfer-torque, a transfer of angular momentum between the electron spin and the local magnetic moments, is a promising and key mechanism to control ferromagnetic materials in modern spintronic devices . However, much less attention has been paid to the same effect in antiferromagnets. For the sake of investigating how the spin current interacts with the magnetic moments in antiferromagnets, we perform spin-torque ferromagnetic resonance measurements on Co20Fe60B20 4nm/Ir25Mn75 tIrMn nm/Pt 4 nm multilayers under a spin Hall effect of Pt. The effective magnetic damping in Co20Fe60B20 is modified by the spin current injected from the Pt layer via the IrMn layer. The results indicate that the spin current interacts with IrMn magnetic moments and exerts the anti-damping torque on the magnetic moments of Co20Fe60B20 through the IrMn. It is also found that the reduction of the exchange bias in the IrMn/Pt interface degrades the anti-damping torque exerted on the Co20Fe60B20 layer, suggesting the transmission of the spin torque becomes less efficient as the interface exchange coupling degrades. Our work infers that the magnetic moments in IrMn can be manipulated by spin torque similarly to the one in a ferromagnetic layer.
We demonstrate an anomalous spin-orbit torque induced by the broken magnetic symmetry in the antiferromagnet IrMn. We study the magnetic structure of three phases of IrMn thin films using neutron diffraction technique. The magnetic mirror symmetry M' is broken laterally in both L10-IrMn and L12-IrMn3 but not γ-IrMn3. We observe an out-of-plane damping-like spin-orbit torque in both L10-IrMn/permalloy and L12-IrMn3/permalloy bilayers but not in γ-IrMn3/permalloy. This is consistent with both the symmetry analysis on the effects of a broken M' on spin-orbit torque and the theoretical predictions…
We employ the recently discovered antiferromagnetic tunneling anisotropic magnetoresistance to study the behavior of antiferromagnetically ordered moments in IrMn exchange coupled to NiFe. Experiments performed by common laboratory tools for magnetization and electrical transport measurements allow us to directly link the broadening of the NiFe hysteresis loop and its shift (exchange bias) to the rotation and pinning of antiferromagnetic moments in IrMn. At higher temperatures, the broadened loops show zero shift which correlates with the observation of fully rotating antiferromagnetic moments…
We report measurement of a spin pumping effect due to fluctuations of the magnetic order of IrMn thin films. A precessing NiFe ferromagnet injected spins into IrMn spin sinks, and enhanced damping was observed around the IrMn magnetic phase transition. Our data was compared to a recently developed theory and converted into interfacial spin mixing conductance enhancements. By spotting the spin pumping peak, the thickness dependence of the IrMn critical temperature could be determined and the characteristic length for the spin-spin interactions was deduced.
We study the perpendicular magnetic anisotropy (PMA) of the CoFeB/MgO bilayers in contact with W, Ta, IrMn and Ti which has been suggested as the spin-orbit-torque-related underlayers. The saturation magnetization of the CoFeB depends on the underlayer materials due to formation of a dead-layer, affecting PMA strength of each film. The x-ray magnetic circular dichroism measurement reveals that the interfacial intermixing suppresses only the perpendicular orbital moment of Fe, while the intermixing simultaneously suppresses both the perpendicular and in-plane orbital moments of Co.
Antiferromagnetic spintronics is an emerging field; antiferromagnets can improve the functionalities of ferromagnets with higher response times, and having the information shielded against external magnetic field. Moreover, a large list of aniferromagnetic semiconductors and metals with Néel temperatures above room temperature exists. In the present manuscript, we persevere in the quest for the limits of how large can anisotropic magnetoresistance be in antiferromagnetic materials with very large spin-orbit coupling. We selected IrMn as a prime example of first-class moment (Mn) and spin-orbit…
We investigated a possible correlation between spin Hall angles and exchange bias in Ni$_{81}$Fe$_{19}$/IrMn samples by performing spin torque ferromagnetic resonance measurements. This correlation is probed by patterning of Ni$_{81}$Fe$_{19}$/IrMn bilayers in different relative orientations with respect to the exchange bias direction. The measured voltage spectra allow a quantitative determination of spin Hall angles, which are independent of the orientation around 2.8\pm0.3%.
We investigate the impact of pinned antiferromagnetic order on the decoherence of spin current in polycrystalline IrMn. In NiFe/Cu/IrMn/CoFe multilayers, we coherently pump an electronic spin current from NiFe into IrMn, whose antiferromagnetic order is globally pinned by static exchange-bias coupling with CoFe. We observe no anisotropic spin decoherence with respect to the orientation of the pinned antiferromagnetic order. We also observe no difference in spin decoherence for samples with and without pinned antiferromagnetic order. Moreover, although there is a pronounced resonance linewidth …
We present a comprehensive study of the exchange bias effect in a model system. Through numerical analysis of the exchange bias and coercive fields as a function of the antiferromagnetic layer thickness we deduce the absolute value of the averaged anisotropy constant of the antiferromagnet. We show that the anisotropy of IrMn exhibits a finite size effect as a function of thickness. The interfacial spin disorder involved in the data analysis is further supported by the observation of the dual behavior of the interfacial uncompensated spins. Utilizing soft x-ray resonant magnetic reflectometry …
We provide a complete and systematic first-principles study of the thermodynamic stability, structural parameters, and magnetic properties of the T1 non-collinear antiferromagnetic L1$_2$-IrMn$_3$ surface and L1$_2$-IrMn$_3$/Fe heterostructure. Furthermore, we investigate the exchange-bias effect in the heterostructure and describe its previously unknown complex magnetic coupling at the interface. We consider four atomic configurations and four magnetic arrangements, finding two stable terminations for the surface and for the heterostructure, which are in good agreement with experimental HR-TE…
We report an investigation of temperature and IrMn layered thickness dependence of anomalous-Hall resistance (AHR), anisotropic magnetoresistance (AMR), and magnetization on Pt/Ir20Mn80/Y3Fe5O12 (Pt/IrMn/YIG) heterostructures. The magnitude of AHR is dramatically enhanced compared with Pt/YIG bilayers. The enhancement is much more profound at higher temperatures and peaks at the IrMn thickness of 3 nm. The observed spin-Hall magnetoresistance (SMR) in the temperature range of 10-300 K indicates that the spin current generated in the Pt layer can penetrate the entire thickness of the IrMn layer…
Antiferromagnetic spintronic devices have the potential to outperform conventional ferromagnetic devices due to their ultrafast dynamics and high data density. A challenge in designing these devices is the control and detection of the orientation of the anti-ferromagnet. One of the most promising ways to achieve this is through the exchange bias effect. This is of particular importance in large scale multigranular devices. Due to the large system sizes, previously, only micromagnetic simulations have been possible, these have an assumed distribution of antiferromagnetic anisotropy directions. …
Manipulation of antiferromagnetic (AFM) spins by electrical means is on great demand to develop the AFM spintronics with low power consumption. In spite of the electrical modulation of insulated AFMs through coupling between their intrinsic ferroelectricity and antiferromagnetism, direct electrical control of AFM metals remains challenging due to the screening effect by the surface charge, and the manipulation is confined to a limited depth of atomic dimensions, which is insufficient to form a stable AFM exchange spring. In the present letter we primarily report a reversible electrical control…
Using an atomistic spin model, we have simulated spin wave injection and propagation into antiferromagnetic IrMn from an exchange coupled CoFe layer. The spectral characteristics of the exited spin waves have a complex beating behavior arising from the non-collinear nature of the antiferromagnetic order. We find that the frequency response of the system depends strongly on the strength and frequency of oscillating field excitations. We also find that the strength of excited spin waves strongly decays away from the interfacial layer with a frequency dependent attenuation. Our findings suggest t…
There has been considerable interest in spin-orbit torques for the purpose of manipulating the magnetization of ferromagnetic (FM) films or nano-elements for spintronic technologies. Spin-orbit torques are derived from spin currents created from charge currents in materials with significant spin-orbit coupling that diffuse into an adjacent FM material. There have been intensive efforts to search for candidate materials that exhibit large spin Hall angles, i.e. efficient charge to spin current conversion. Here we report, using spin torque ferromagnetic resonance, the observation of a giant spin…
Non-collinear antiferromagnets can have additional spin Hall effects due to the net chirality of their magnetic spin structure, which provides for more complex spin-transport phenomena compared to ordinary non-magnetic materials. Here we investigated how ferromagnetic resonance of permalloy ($Ni_{80}Fe_{20}$) is modulated by spin Hall effects in adjacent epitaxial $IrMn_{3}$ films. We observe a large dc modulation of the ferromagnetic resonance linewidth for currents applied along the [001] $IrMn_{3}$ direction. This very strong angular dependence of spin-orbit torques from dc currents through…
Anti-ferromagnetic materials have the possibility to offer ultra fast, high data density spintronic devices. A significant challenge is the reliable detection of the state of the antiferromagnet, which can be achieved using exchange bias. Here we develop an atomistic spin model of the athermal training effect, a well known phenomenon in exchange biased systems where the bias is significantly reduced after the first hysteresis cycle. We find that the setting process in granular thin films relies on the presence of interfacial mixing between the ferromagnetic and antiferromagnetic layers. We sys…
We demonstrate that an antiferromagnet can be employed for a highly efficient electrical manipulation of a ferromagnet. In our study we use an electrical detection technique of the ferromagnetic resonance driven by an in-plane ac-current in a NiFe/IrMn bilayer. At room temperature, we observe antidamping-like spin torque acting on the NiFe ferromagnet, generated by the in-plane current driven through the IrMn antiferromagnet. A large enhancement of the torque, characterized by an effective spin-Hall angle exceeding most heavy transition metals, correlates with the presence of the exchange-bias…
The exchange interaction determines the ferromagnetic (FM) or antiferromagnetic (AFM) ordering of atomic spins. When ferromagnets and antiferromagnets are coupled together, they often exhibit the exchange bias effect, a unidirectional interface exchange field causing a shift of the magnetic hysteresis loop. The effective magnitude of this interface exchange field is at most a few percent of the bulk exchange, arising from pinned interfacial spins in the antiferromagnet. The pinned spins are known to comprise a small fraction of the total number of interface spins, yet their exact nature and ph…
Tunneling junctions containing no ferromagnetic elements have been fabricated and we show that distinct resistance states can be set by field cooling the devices from above the Néel along different orientations. Variations of the resistance up to 10% are found upon field cooling in applied fields of 2T, in-plane or out of plane. Below TN, we found that the metastable states are insensitive to magnetic fields thus constituting a memory element robust against external magnetic fields. Our work provides the demonstration of an electrically readable magnetic memory device, which contains no ferrom…
We theoretically investigate dynamics of antiferromagnetic domain walls driven by spin-orbit torques in antiferromagnet/heavy metal bilayers. We show that spin-orbit torques drive antiferromagnetic domain walls much faster than ferromagnetic domain walls. As the domain wall velocity approaches the maximum spin-wave group velocity, the domain wall undergoes Lorentz contraction and emits spin-waves in the terahertz frequency range. The interplay between spin orbit torques and the relativistic dynamics of antiferromagnetic domain walls leads to the efficient manipulation of antiferromagnetic spin…
In this work, IrMn$_{3}$/insulating-Y$_{3}$Fe$_{5}$O$_{12}$ exchange-biased bilayers are studied. The behavior of the net magnetic moment $Δm_{AFM}$ in the antiferromagnet is directly probed by anomalous and planar Hall effects, and anisotropic magnetoresistance. The $Δm_{AFM}$ is proved to come from the interfacial uncompensated magnetic moment. We demonstrate that the exchange bias and rotational hysteresis are induced by the irreversible switching of the $Δm_{AFM}$. In the training effect, the $Δm_{AFM}$ changes continuously. This work highlights the fundamental role of the $Δm_{AFM}$ in th…
Spin torque is a promising tool for driving magnetization dynamics for novel computing technologies. These torques can be easily produced by spin-orbit effects, but for most conventional spin source materials, a high degree of crystal symmetry limits the geometry of the spin torques produced. Magnetic ordering is one way to reduce the symmetry of a material and allow exotic torques, and antiferromagnets are particularly promising because they are robust against external fields. We present spin torque ferromagnetic resonance measurements and second harmonic Hall measurements characterizing the …
Noncollinear antiferromagnets have promising potential to replace ferromagnets in the field of spintronics as high-density devices with ultrafast operation. To take full advantage of noncollinear antiferromagnets in spintronics applications, it is important to achieve efficient manipulation of noncollinear antiferromagnetic spin. Here, using the anomalous Hall effect as an electrical signal of the triangular magnetic configuration, spin-orbit torque switching with no external magnetic field is demonstrated in noncollinear antiferromagnetic antiperovskite manganese nitride Mn$_3$GaN at room tem…
We investigated spin injection by spin pumping from a spin-injector(NiFe) into a spin-sink to detect spin fluctuations in the spin-sink. By scanning the ordering-temperature of several magnetic transitions, we found that enhanced spin pumping due to spin fluctuations applies with several ordering states: ferromagnetic(Tb) and antiferromagnetic(NiO, NiFeOx, BiFeO3, exchange-biased and unbiased IrMn). Results also represent systematic experimental investigation supporting that the effect is independent of the metallic and insulating nature of the spin-sink, and is observed whether the spin curre…
As electrical control of Néel order opens the door to reliable antiferromagnetic spintronic devices, understanding the microscopic mechanisms of antiferromagnetic switching is crucial. Spatially-resolved studies are necessary to distinguish multiple nonuniform switching mechanisms; however, progress has been hindered by the lack of tabletop techniques to image the Néel order. We demonstrate spin Seebeck microscopy as a sensitive, table-top method for imaging antiferromagnetic order in thin films, and apply this technique to study spin-torque switching in NiO/Pt and Pt/NiO/Pt heterostructures. …
Electrical switching and readout of antiferromagnets allows to exploit the unique properties of antiferromagnetic materials in nanoscopic electronic devices. Here we report experiments on the spin-orbit torque induced electrical switching of a polycrystalline, metallic antiferromagnet with low anisotropy and high Néel temperature. We demonstrate the switching in a Ta / MnN / Pt trilayer system, deposited by (reactive) magnetron sputtering. The dependence of switching amplitude, efficiency, and relaxation are studied with respect to the MnN film thickness, sample temperature, and current densit…
Spin current generated by spin Hall effect in the heavy metal would diffuse up and down to adjacent ferromagnetic layers and exert torque on their magnetization, called spin-orbit torque. Antiferromagnetically coupled trilayers, namely the so-called synthetic antiferromagnets (SAF), are usually employed to serve as the pinned layer of spintronic devices based on spin valves and magnetic tunnel junctions to reduce the stray field and/or increase the pinning field. Here we investigate the spin-orbit torque in MgO/CoFeB/Ta/CoFeB/MgO perpendicularly magnetized multilayer with interlayer antiferrom…
For a long time, there have been no efficient ways of controlling antiferromagnets. Quite a strong magnetic field was required to manipulate the magnetic moments because of a high molecular field and a small magnetic susceptibility. It was also difficult to detect the orientation of the magnetic moments since the net magnetic moment is effectively zero. For these reasons, research on antiferromagnets has not been progressed as drastically as that on ferromagnets which are the main materials in modern spintronic devices. Here we show that the magnetic moments in NiO, a typical natural antiferro…
The spin-splitter effect is theoretically predicted to generate an unconventional spin current with $\mathit{x}$- and $\mathit{z}$- spin polarization via the spin-split band in antiferromagnets. The generated torque, namely spin-splitter torque, is effective for the manipulation of magnetization in an adjacent magnetic layer without an external magnetic field for spintronic devices such as MRAM. Here, we study the generation of torque in collinear antiferromagnetic RuO$_2$ with (100), (101), and (001) crystal planes. Next we find all $\mathit{x}$-, $\mathit{y}$-, and $\mathit{z}$-polarized spi…
We achieve current-induced switching in collinear insulating antiferromagnetic CoO/Pt, with fourfold in-plane magnetic anisotropy. This is measured electrically by spin Hall magnetoresistance and confirmed by the magnetic field-induced spin-flop transition of the CoO layer. By applying current pulses and magnetic fields, we quantify the efficiency of the acting current-induced torques and estimate a current-field equivalence ratio of $4x10^{-11} T A^{-1} m^2$. The Néel vector final state ($n \perp j$) is in line with a thermomagnetoelastic switching mechanism for a negative magnetoelastic cons…
We show how compensated antiferromagnetic spins can be implemented in the micromagnetic simulation program MuMax3. We demonstrate that we can model spin flop coupling as a uniaxial anisotropy for small canting angles and how we can take into account the exact energy terms for strong coupling between a ferromagnet and compensated antiferromagnet. We also investigate the training effect in biaxial antiferromagnets and reproduce the training effect in a polycrystalline IrMn/CoFe bilayer.
We study the anomalous Hall-like effect (AHLE) and the effective anisotropic magnetoresistance (EAMR) in antiferromagnetic γ -IrMn3/Y3Fe5O12(YIG) and Pt/YIG heterostructures. For γ -IrMn3/YIG, the EAMR and the AHLE resistivity change sign with temperature due to the competition between the spin Hall magnetoresistance (SMR) and the magnetic proximity effect (MPE) induced by the interfacial antiferromagnetic uncompensated magnetic moment. In contrast, for Pt/YIG the AHLE resistivity changes sign with temperature whereas no sign change is observed in the EAMR. This is because the MPE and the SMR …
Antiferromagnets are magnetically ordered materials which exhibit no net moment and thus are insensitive to magnetic fields. Antiferromagnetic spintronics aims to take advantage of this insensitivity for enhanced stability, while at the same time active manipulation up to the natural THz dynamic speeds of antiferromagnets is possible, thus combining exceptional storage density and ultra-fast switching. However, the active manipulation and read-out of the Néel vector (staggered moment) orientation is challenging. Recent predictions have opened up a path based on a new spin-orbit torque, which c…
Crossover between intrinsic and temperature-assisted regimes in spin-orbit torque switching of antiferromagnetic order
The University of Tokyo（东京大学）；The Johns Hopkins University（约翰斯·霍普金斯大学）；Japan Science and Technology Agency（日本科学技术振兴机构）；Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；Osaka University（大阪大学）；RIKEN（理化学研究所）；Trans-scale Quantum Science Institute, The University of Tokyo（东京大学跨尺度量子科学研究所）；Canadian Institute for Advanced Research（加拿大高级研究院）
With ultra-fast writing capacity and high reliability, the spin-orbit torque is regarded as a promising alternative to fabricate next-generation magnetic random access memory. However, the three-terminal setup can be challenging when scaling down the cell size. In particular, the thermal stability is an important issue. Here we demonstrate that the current-pulse-induced perpendicular exchange bias can significantly relieve the concern of thermal stability. The switching of the exchange bias direction is induced by the spin-orbit torque when passing current pulses through the Pt/Co system with …
We numerically compute current-induced spin-transfer torques for antiferromagnetic domain walls, based on a linear response theory in a tight-binding model. We find that, unlike for ferromagnetic domain wall motion, the contribution of adiabatic spin torque to antiferromagnetic domain wall motion is negligible, consistent with previous theories. As a result, the non-adiabatic spin-transfer torque is a main driving torque for antiferromagnetic domain wall motion. Moreover, the non-adiabatic spin-transfer torque for narrower antiferromagnetic domain walls increases more rapidly than that for fer…
Damping in magnetization dynamics characterizes the dissipation of magnetic energy and is essential for improving the performance of spintronics-based devices. While the damping of ferromagnets has been well studied and can be artificially controlled in practice, the damping parameters of antiferromagnetic materials are nevertheless little known for their physical mechanisms or numerical values. Here we calculate the damping parameters in antiferromagnetic dynamics using the generalized scattering theory of magnetization dissipation combined with the first-principles transport computation. For…
Magnetic random-access memory (MRAM) driven by spin-transfer torque (STT) is a major contender for future memory applications. The energy dissipation involved in writing remains problematic, even with the advent of more efficient perpendicular magnetic anisotropy (PMA) devices. A promising alternative switching mechanism employs spin-orbit torques and the spin-Hall effect (SHE) in particular, but additional symmetry breaking is required to achieve deterministic switching in PMA devices. Currently used methods rely on in-plane magnetic fields or anisotropy gradients, which are not suitable for …
Terahertz spin-orbit torque as a drive of spin dynamics in the insulating antiferromagnet Cr$_{2}$O$_{3}$
Ioffe Institute, Russian Academy of Sciences（俄罗斯科学院 IOFFE 研究所）；Institute of Molecule and Crystal Physics, Ufa Federal Research Center, Russian Academy of Sciences（俄罗斯科学院乌法联邦研究中心分子与晶体物理研究所）；Ufa University of Science and Technology（乌法科学技术大学）；Institute for Molecules and Materials, Radboud University（拉德堡德大学分子与材料研究所）；New Spintronic Technologies LLC（新自旋技术有限责任公司）；Prokhorov General Physics Institute, Russian Academy of Sciences（俄罗斯科学院普罗霍罗夫普通物理研究所）；Amirkhanov Institute of Physics, Dagestan Federal Research Center, Russian Academy of Sciences（俄罗斯科学院达吉斯坦联邦研究中心阿米汉诺夫物理研究所）
We consider a synthetic antiferromagnetic spin-torque oscillator with anisotropic interlayer exchange coupling. This system exhibits exceptional points in its linearized dynamics. We find the non-linear dynamics and the dynamical phase diagram of the system both analytically and numerically. Moreover, we show that, near one of the exceptional points, the power of the oscillator depends extremely sensitively on the injected spin current. Our findings may be useful for designing sensitive magnetometers and for other applications of spin-torque oscillators.
We explore the possibility of exciting spin waves in insulating antiferromagnetic films by injecting spin current at the surface. We analyze both magnetically compensated and uncompensated interfaces. We find that the spin current induced spin-transfer torque can excite spin waves in insulating antiferromagnetic materials and that the chirality of the excited spin wave is determined by the polarization of the injected spin current. Furthermore, the presence of magnetic surface anisotropy can greatly increase the accessibility of these excitations.
In-situ, device level measurement of thermal mag-noise spectral linewidths in 60nm diameter CPP-GMR spin-valve stacks of IrMn/ref/Cu/free, with reference and free layer of similar CoFe/CoFeGe alloy, are used to simultaneously determine the intrinsic Gilbert damping for both magnetic layers. It is shown that careful alignment at a "magic-angle" between free and reference layer static equilibrium magnetization can allow direct measurement of the broadband intrinsic thermal spectra in the virtual absence of spin-torque effects which otherwise grossly distort the spectral line shapes and require l…
Using a three-dimensional focused-ion beam lithography process we have fabricated nanopillar devices which show spin transfer torque switching at zero external magnetic fields. Under a small in-plane external bias field, a field-dependent peak in the differential resistance versus current is observed similar to that reported in asymmetrical nanopillar devices. This is interpreted as evidence for the low-field excitation of spin waves which in our case is attributed to a spin-scattering asymmetry enhanced by the IrMn exchange bias layer coupled to a relatively thin CoFe fixed layer.
We use \textit{ab-initio} calculations to investigate spin-orbit torques (SOTs) in FeRh(001) deposited on W(100). Since FeRh undergoes a ferromagnetic-antiferromagnetic phase transition close to room temperature, we consider both phases of FeRh. In the antiferromagnetic case we find that the effective magnetic field of the even torque is staggered and therefore ideal to induce magnetization dynamics or to switch the antiferromagnet (AFM). At the antiferromagnetic resonance the inverse SOT induces a current density, which can be determined from the SOT. In the ferromagnetic case our calculation…
In spintronics applications, ultrafast spin dynamics have to be controlled at femtosecond (fs) timescales via fs-laser radiation. At such ultrafast timescales, the effect of the Gilbert damping factor α on ultrafast demagnetization time should be considered. In previous explorations for the relationship between these two parameters, it was found that the theoretical calculations based on the local spin-flip scattering model do not agree with the experimental results. Here, we find that in Fe81Ga19(FeGa)/Ir20Mn80(IrMn) bilayers, the unconventional IrMn thickness dependence of α results from the…
We prepare the high quality epitaxial MgO(001)[100]/Pt(001)[100]/NiO(001)[100]/FeNi/SiO2 films to investigate the spin transport in the NiO antiferromagnetic insulator. The ferromagnetic resonance measurements of the FeNi under a spin current injection from the Pt by the spin Hall effect revealed the change of the ferromagnetic resonance linewidth depending on the amount of the spin current injection. The results can be interpreted that there is an angular momentum transfer through the NiO. A high efficient angular momentum transfer we observed in the epitaxial NiO can be attributed to the wel…
There is little doubt that the magnetization dynamics of ferromagnetic systems is governed by the Landau-Lifshitz-Gilbert equation or its generalization with various spin torques. In contrast, there are several sets of dynamic equations for two-sublattice antiferromagnets (AFMs) in literature that have different forms of dissipative torques and no proper dynamic equations for multi-sublattice AFMs and ferrimagnets in general. Here we introduce the general Rayleigh dissipation functional into the Lagrange equation and derive the proper form of the dissipative torques in the phenomenological equ…
