The critical current of the spin transfer torque-driven magnetization dynamics was studied by taking into account both spin pumping and the finite penetration depth of the transverse spin current. We successfully reproduced the recent experimental results obtained by Chen et al. [Phys. Rev. B {\bf 74}, 144408 (2006)] and found that the critical current remains finite even in the zero-thickness limit of the free layer. We showed that the remaining value of the critical current is determined mainly by spin pumping.
We present a calculation of current-induced torques in metallic magnetic multilayers derived from the spin-dependent transmission and reflection properties of the magnetic layers. A scattering formalism is employed to calculate the torques in a magnetic-nonmagnetic-magnetic trilayer, for currents perpendicular to the layers, in both the ballistic and diffusive regimes.
We report on an {\em ab-initio} study of giant magnetoresistance (GMR) and current-induced-torques (CITs) in Cr/Au multilayers that is based on non-equilibrium Green's functions and spin density functional theory. We find substantial GMR due primarily to a spin-dependent resonance centered at the Cr/Au interface and predict that the CITs are strong enough to switch the antiferromagnetic order parameter at current-densities $\sim 100$ times smaller than typical ferromagnetic metal circuit switching densities.
We develop a simple theoretical framework for transport in magnetic multilayers, based on Landauer-Buttiker scattering formalism and Random Matrix Theory. A simple transformation allows one to go from the scattering point of view to theories expressed in terms of local currents and electrochemical potential. In particular, our theory can be mapped onto the well established classical Valet Fert theory for collinear systems. For non collinear systems, in the absence of spin-flip scattering, our theory can be mapped onto the generalized circuit theory. We apply our theory to the angular dependanc…
We consider a quasi one-dimensional configuration consisting of two small pieces of ferromagnetic material separated by a metallic one and contacted by two metallic leads. A spin-polarized current is injected from one lead. Our goal is to investigate the correlation induced between the magnetizations of the two ferromagnets by spin-transfer torque. This torque results from the interaction between the magnetizations and the spin polarization of the current. We discuss the dynamics of a single ferromagnet, the extension to the case of two ferromagnets, and give some estimates for the parameters …
We study the influence of the spin torque, which depends on the space and time derivative of magnetization, on magnetization reversal time in a ferromagnetic fine particle. The spin torque operates to dissipate the angular momentum of the magnetization precession, and the torque increases in a spin vortex structure. We calculate the magnetization reversal time under a DC magnetic field using the Landau-Lifshitz-Gilbert equation containing a spin torque term. We found that the spin torque changes the magnetization switching speed significantly during the reversal process by maintaining a spin v…
It is well known that skyrmions can be driven using spin-orbit torques due to the spin-Hall effect. Here we show an additional contribution in multilayered stacks arises from vertical spin currents due to inter-layer diffusion of a spin accumulation generated at a skyrmion. This additional interfacial spin torque is similar in form to the in-plane spin transfer torque, but is significantly enhanced in ultra-thin films and acts in the opposite direction to the electron flow. The combination of this diffusive spin torque and the spin-orbit torque results in skyrmion motion which helps to explain…
The spin-torque ferromagnetic resonance (ST-FMR) in a bilayer system consisting of a magnetic insulator such as Y3Fe5O12 and a normal metal with spin-orbit interaction such as Pt is addressed theoretically. We model the ST-FMR for all magnetization directions and in the presence of field-like spin-orbit torques based on the drift-diffusion spin model and quantum mechanical boundary conditions. ST-FMR experiments may expose crucial information about the spin-orbit coupling between currents and magnetization in the bilayers.
We study the role of thermal fluctuations on the spin dynamics of a thin permalloy film with a focus on the behavior of spin torque and find that the thermally assisted spin torque results in new aspects of the magnetization dynamics. In particular, we uncover the formation of a finite, spin torque-induced, in-plane magnetization component. The orientation of the in-plane magnetization vector depends on the temperature and the spin-torque coupling. We investigate and illustrate that the variation of the temperature leads to a thermally-induced rotation of the in-plane magnetization.
To address thermal stability issues for spintronic devices with a reduced size, we investigate spin-orbit torque in Co/Pt multilayers with strong perpendicular magnetic anisotropy. Note that the spin-orbit torque arises from the global imbalance of the spin currents from the top and bottom interfaces for each Co layer. By inserting Ta or Cu layers to strengthen the top-down asymmetry, the spin-orbit torque efficiency can be greatly modified without compromised perpendicular magnetic anisotropy. Above all, the efficiency builds up as the number of layers increases, realizing robust thermal stab…
Current induced spin-orbit torques have been studied in ferromagnetic nanowires made of 20 nm thick Co/Pd multilayers with perpendicular magnetic anisotropy. Using Hall voltage and lock-in measurements, it is found that upon injection of an electric current both in-plane (Slonczewski-like) and perpendicular (field-like) torques build up in the nanowire. The torque efficiencies are found to be as large as 1.17 kOe and 5 kOe at 108 A/cm2 for the in-plane and perpendicular components, respectively, which is surprisingly comparable to previous studies in ultrathin (~ 1 nm) magnetic bilayers. We sh…
In systems with small spin-orbit coupling, current-induced torques on the magnetization require inhomogeneous magnetization textures. For large spin-orbit coupling, such torques exist even without gradients in the magnetization direction. Here, we consider current-induced torques in ferromagnetic metals with both Rashba spin-orbit coupling and inhomogeneous magnetization. We first phenomenologically construct all torques that are allowed by the symmetries of the system, to first order in magnetization-direction gradients and electric field. Second, we use a Boltzmann approach to calculate the …
Oscillation frequency of spin torque oscillator with a perpendicularly magnetized free layer and an in-plane magnetized pinned layer is theoretically investigated by taking into account the field-like torque. It is shown that the field-like torque plays an important role in finding the balance between the energy supplied by the spin torque and the dissipation due to the damping, which results in a steady precession. The validity of the developed theory is confirmed by performing numerical simulations based on the Landau-Lifshitz-Gilbert equation.
We investigate the spin-transfer torque in a magnetic multilayer structure by means of a spin-diffusion model. The torque in the considered system, consisting of two magnetic layers separated by a conducting layer, is caused by a perpendicular-to-plane current. We compute the strength of the field-like and the damping-like torque for different material parameters and geometries. Our studies suggest that the field-like torque highly depends on the exchange coupling strength of the itinerant electrons with the magnetization both in the pinned and the free layer. While a low coupling leads to ver…
It is known that the transfer of spin angular momenta between current carriers and local moments occurs near the interface of magnetic layers when their moments are non-collinear. However, to determine the magnitude of the transfer, one should calculate the spin transport properties far beyond the interface regions. Based on the spin diffusion equation, we present a self-consistent approach to evaluate the spin torque for a number of layered structures. One of the salient features is that the longitudinal and transverse components of spin accumulations are inter-twined from one layer to the ne…
The optimum condition of the magnetization alignment to maximize the spin torque diode voltage is derived by solving the Landau-Lifshitz-Gilbert equation. We show that the optimized diode voltage can be one order of magnitude larger than that of the conventional alignment where the easy axes of the free and the pinned layers are parallel. These analytical predictions are confirmed by numerical simulations.
Spin torque majority gates are modeled and several regimes of magnetization switching (some leading to failure) are discovered. The switching speed and noise margins are determined for STMGs and an adder based on it. With switching time of 3ns at current of 80uA, the adder computational throughput is comparable to that of a CMOS adder.
Spin torque from spin current applied to a nanoscale region of a ferromagnet can act as negative magnetic damping and thereby excite self-oscillations of its magnetization. In contrast, spin torque uniformly applied to the magnetization of an extended ferromagnetic film does not generate self-oscillatory magnetic dynamics but leads to reduction of the saturation magnetization. Here we report studies of the effect of spin torque on a system of intermediate dimensionality - a ferromagnetic nanowire. We observe coherent self-oscillations of magnetization in a ferromagnetic nanowire serving as the…
We study the magnetization dynamics in thin ferromagnetic films and small ferromagnetic particles in contact with paramagnetic conductors. A moving magnetization vector causes \textquotedblleft pumping\textquotedblright of spins into adjacent nonmagnetic layers. This spin transfer affects the magnetization dynamics similar to the Landau-Lifshitz-Gilbert phenomenology. The additional Gilbert damping is significant for small ferromagnets, when the nonmagnetic layers efficiently relax the injected spins, but the effect is reduced when a spin accumulation build-up in the normal metal opposes the s…
Extensive efforts have been devoted to the study of spin-orbit torque in ferromagnetic metal/heavy metal bilayers and exploitation of it for magnetization switching using an in-plane current. As the spin-orbit torque is inversely proportional to the thickness of the ferromagnetic layer, sizable effect has only been realized in bilayers with an ultrathin ferromagnetic layer. Here we demonstrate that, by stacking ultrathin Pt and FeMn alternately, both ferromagnetic properties and current induced spin-orbit torque can be achieved in FeMn/Pt multilayers without any constraint on its total thickne…
We apply a recently developed quantum theory of spin transfer torque to study the effect of the quantum noise in spin transfer process on the magnetization switching in spin-torque-driven devices. The quantum noise induces considerable fluctuation of the switching time at zero temperature. By including the thermal noise, the temperature dependence of the expectation value and standard deviation of the switching time are obtained in the Monte Carlo simulations, and the results are fitted to an effective first passage model. We expect that the predictions here are observable in the single-shot m…
We demonstrate magnetization switching in out-of-plane magnetized Ta\CoFeB\MgO nanowires by current pulse injection along the nanowires, both with and without a constant and uniform magnetic field collinear to the current direction. We deduce that an effective torque arising from spin-orbit effects in the multilayer drives the switching mechanism. While the generation of a component of the magnetization along the current direction is crucial for the switching to occur, we observe that even without a longitudinal field thermally generated magnetization fluctuations can lead to switching. Analys…
We study current-induced switching in magnetic tunnel junctions (MTJs) in the presence of a field-like spin-transfer torque and titled pinned-layer magnetization in the high current limit at finite temperature. We consider both the Slonczewski and field-like torques with coefficients $a_J$ and $b_J $, respectively. At finite temperatures, $σ= b_J/a_J = \pm1$ leads to a smaller mean switching time compared that with $σ= 0$. The reduction of switching time in the presence of the field-like term is due to the alignment effect (for $σ> 0$) and the initial torque effect.
We demonstrate efficient current induced spin orbit torque switching in perpendicularly magnetized epitaxial MgO(001)//Pd/Co2FeAl/MgO heterostructures grown by magnetron sputtering. The advantage of such heterostructures for spin orbit torque MRAM devices is that they allow a unique combination of critical ingredients: i) low resistivity required for reduced electric energy consumption during writing, ii) high TMR ratio required for efficient reading and iii) strong perpendicular magnetic anisotropy for increased thermal stability.
In this article we analyze by modeling two possible mechanisms for magnetization switching using spin orbit torques, which have been reported to cause field-free deterministic switching in experiments. Here we compare the field-free magnetization switching due to a tilt of the anisotropy direction against the use of an antiferromagnetic bias field. Simple results obtained analytically show that a bias field not only causes the magnetization reversal but also reduces the corresponding energy barrier. The critical current required for magnetization switching is analyzed on the basis of a macrosp…
A common ploy to reduce the switching current and energy dissipation in spin-transfer-torque driven magnetization switching of shape-anisotropic single-domain nanomagnets is to employ magnets with low saturation magnetization $M_s$ and high shape-anisotropy. The high shape-anisotropy compensates for low $M_s$ to keep the static switching error rate constant. However, this ploy increases the switching delay, its variance in the presence of thermal noise, and the dynamic switching error rate. Using the stochastic Landau-Lifshitz-Gilbert equation with a random torque emulating thermal noise, we s…
An experimental study of current-induced magnetization reversal of the Ru/Co/Ru and Ru/Co/Ru/W structures was carried out. In the considered structures, due to the small value of the coercive force comparable in magnitude to the Oersted field and the SOT effect field, magnetization reversal is carried out by moving a domain wall parallel to the direction of current injection. For such a case, a new method for estimating the effective field of SOT based on the analysis of the domain wall position taking into account the distribution of the Oersted field was proposed. This method allowed determi…
Current-induced dynamics of twisted domain walls and skyrmions in ferromagnetic perpendicularly magnetized multilayers is studied through three-dimensional micromagnetic simulations and analytical modeling. It is shown that such systems generally exhibit a Walker breakdown-like phenomenon in the presence of current-induced damping-like spin-orbit torque. Above a critical current threshold, corresponding to typical velocities of the order tens of m/s, domain walls in some layers start to precess with frequencies in the gigahertz regime, which leads to oscillatory motion and a significant drop i…
The dependence of the critical current of spin transfer torque-driven magnetization dynamics on the free-layer thickness was studied by taking into account both the finite penetration depth of the transverse spin current and spin pumping. We showed that the critical current remains finite in the zero-thickness limit of the free layer for both parallel and anti-parallel alignments. We also showed that the remaining value of the critical current of parallel to anti-parallel switching is larger than that of anti-parallel to parallel switching.
Strong magnetic field pulses associated with a relativistic electron bunch can imprint switching patterns in magnetic thin films that have uniaxial in-plane anisotropy. In experiments with Fe and FeCo alloy films the pattern shape reveals an additional torque acting on magnetization during the short (in the 100fs time scale) magnetic field pulse. The magnitude of the torque is as high as 15% of the torque from the magnetic field. The torque symmetry is that of a uniaxial anisotropy along the direction of the eddy current screening the magnetic field. Spin-orbit interaction acting on the conduc…
We predict the existence of a new metastable magnetization state in a single-domain nanomagnet with uniaxial shape anisotropy. It emerges when a spin-polarized current, delivering a spin-transfer-torque, is injected into the nanomagnet. It can trap the magnetization vector and prevent spin-transfer-torque from switching the magnetization from one stable state along the easy axis to the other. Above a certain threshold current, the metastable state no longer appears. This has important technological consequences for spin-transfer-torque based magnetic memory and logic systems.
To study the effect of spin-orbit coupling (SOC) on spin-transfer torque in magnetic materials, we have implemented two theoretical formalisms that can accommodate SOC. Using the "charge-pumping" formalism, we find two contributions to the out-of-plane spin-transfer torque parameter $β$ in ballistic Ni domain walls (DWs). For short DWs, the nonadiabatic reflection of conduction electrons caused by the rapid spatial variation of the exchange potential results in an out-of-plane torque that increases rapidly with decreasing DW length. For long DWs, the Fermi level conduction channel anisotropy t…
We present a comparative study of high frequency dynamics and low frequency noise in elliptical magnetic tunnel junctions with lateral dimensions under 100 nm presenting current-switching phenomena. The analysis of the high frequency oscillation modes with respect to the current reveals the onset of a steady-state precession regime for negative bias currents above $J=10^7 A/cm^2$, when the magnetic field is applied along the easy axis of magnetization. By the study of low frequency noise for the same samples, we demonstrate the direct link between changes in the oscillation modes with the appl…
We study the effects of the coupling between magnetization dynamics and the electronic degrees of freedom in a heterostructure of a metallic nanomagnet with dynamic magnetization coupled with a superconductor containing a steady spin-splitting field. We predict how this system exhibits a non-linear spin torque, which can be driven either with a temperature difference or a voltage across the interface. We generalize this notion to arbitrary magnetization precession by deriving a Keldysh action for the interface, describing the coupled charge, heat and spin transport in the presence of a precess…
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 le…
In this paper we present an overview of recent progress made in the understanding of the spin-torque induced magnetization dynamics in nanodevices using mesoscopic micromagnetic simulations. We first specify how a spin-torque term may be added to the usual Landau-Lifshitz-Gilbert equation of magnetization motion and detail its physical meaning. After a brief description of spin-torque driven dynamics in the macrospin approximation, we discuss the validity of this approximation for various experimentally relevant geometries. Next, we perform a detailed comparison between accurate experimental d…
The critical current needed to destabilize the magnetization of a perpendicular ferromagnet via the spin Hall effect is studied. Both the dampinglike and fieldlike torques associated with the spin current generated by the spin Hall effect is included in the Landau-Lifshitz-Gilbert equation to model the system. In the absence of the fieldlike torque, the critical current is independent of the damping constant and is much larger than that of conventional spin torque switching of collinear magnetic systems, as in magnetic tunnel junctions. With the fieldlike torque included, we find that the crit…
We present a time-resolved study of the magnetization dynamics in a microstructured Cr$|$Heusler$|$Pt waveguide driven by the Spin-Hall-Effect and the Spin-Transfer-Torque effect via short current pulses. In particular, we focus on the determination of the threshold current at which the spin-wave damping is compensated. We have developed a novel method based on the temporal evolution of the magnon density at the beginning of an applied current pulse at which the magnon density deviates from the thermal level. Since this method does not depend on the signal-to-noise ratio, it allows for a robus…
Magnetic multilayers of (Ir/Co/Pt)$_6$ with interfacial Dzyaloshinskii-Moriya interaction (IDMI) were deposited by magnetron sputtering with Co thickness $d=1.8$ nm. Exploiting magneto-optical Kerr effect in longitudinal mode microscopy, magnetic force microscopy, and vibrating sample magnetometry, the magnetic field-driven evolution of domain structures and magnetization hysteresis loops have been studied. The existence of weak stripe domains structure was deduced -- tens micrometers size domains with in-plane "core" magnetization modulated by hundred of nanometers domains with out-of-plane m…
We consider current-induced spin-transfer torque on an antiferromagnet in a dual spin-valve setup. It is demonstrated that a net magnetization may be induced in the AFM by partially or completely aligning the sublattice magnetizations via a current-induced spin-transfer torque. This effect occurs for current densities ranging below 10$^6$ A/cm$^2$. The direction of the induced magnetization in the AFM is shown to be efficiently controlled by means of the magnetic configuration of the spin-valve setup, with the anti-parallell configuration yielding the largest spin-transfer torque. Interestingl…
The exchange interaction between electrons and magnetic spins is considerably enhanced near interfaces, in magnetic multilayers. As a result, a dc current can be used to generate spin oscillations. We review theory and experimental evidence. The s-d exchange interaction causes a rapid precession of itinerant conduction-electron spins s around the localized spins S of magnetic electrons. Because of the precession, the time-averaged interaction torque between s and S vanishes. An interface between a magnetic layer and a spacer causes a local coherence between the precession phases of differnt el…
The FMR linewidth and its anisotropy in F$_1$/f/F$_2$/AF multilayers, where spacer f has a low Curie point compared to the strongly ferromagnetic F$_1$ and F$_2$, is investigated. The role of the interlayer exchange coupling in magnetization relaxation is determined experimentally by varying the thickness of the spacer. It is shown that stronger interlayer coupling via thinner spacers enhances the microwave energy exchange between the outer ferromagnetic layers, with the magnetization of F$_2$ exchange-dragged by the resonance precession in F$_1$. A weaker mirror effect is also observed: the m…
We show that a slowly decaying current pulse can lead to nearly deterministic precessional switching in the presence of noise. We consider a biaxial macrospin, with an easy axis in the plane and a hard axis out-of-the plane, typical of thin film nanomagnets patterned into asymmetric shapes. Out-of-plane precessional magnetization orbits are excited with a current pulse with a component of spin polarization normal to the film plane. By numerically integrating the stochastic Landau-Lifshitz-Gilbert-Slonczewski equation we show that thermal noise leads to strong dephasing of the magnetization orb…
We address the nature of spin-orbit torques at the magnetic surfaces of topological insulators, using both Kubo formula and Keldysh formalism. Through the analysis of the derived spin-charge equations, we find that the so-called Dirac torques possess a different symmetry compared to their Rashba counterpart, as well as a high anisotropy as a function of the magnetization direction. In particular, the damping torque vanishes when the magnetization lies in the plane of the topological insulator surface. This peculiarity has important consequences in terms of magnetization dynamics and switching,…
Current-induced magnetization switching by spin-orbit torque (SOT) holds considerable promise for next generation ultralow-power memory and logic applications. In most cases, generation of spin-orbit torques has relied on an external injection of out-of-plane spin currents into the magnetic layer, while an external magnetic field along the electric current direction is generally required for realizing deterministic switching by SOT. Here, we report deterministic current-induced SOT full magnetization switching by lateral spin-orbit torque in zero external magnetic field. The Pt/Co/Pt magnetic …
For graphene nanoribbons with Rashba spin-orbit coupling, the peculiar magnetic response due to the presence of a magnetization and geometric confinement are analyzed within a tight-binding model. We observe a sizable transverse susceptibility that can be considered as a gate voltage-induced magnetoelectric torque without the need of a bias voltage, with different directions for zigzag and armchair ribbons. The local torque generates non-collinear spin polarization between the two edges and/or along the ribbon, and the net torque averages to zero if the magnetization is homogeneous. Neverthele…
Spin-orbit torque (SOT) magnetization switching of ferromagnets with large perpendicular magnetic anisotropy has a great potential for the next-generation non-volatile magnetoresistive random-access memory (MRAM). It requires a high-performance pure spin current source with a large spin Hall angle and high electrical conductivity, which can be fabricated by a mass production technique. In this work, we demonstrate ultrahigh efficient and robust SOT magnetization switching in all-sputtered BiSb topological insulator - perpendicularly magnetized Co/Pt multilayers. Despite fabricated by the indus…
We study deterministic magnetic reversal of a perpendicularly magnetized Co layer in a Co/MgO/Ta nano-square driven by spin Hall torque from an in-plane current flowing in an underlying Pt layer. The rate-limiting step of the switching process is domain-wall (DW) depinning by spin Hall torque via a thermally-assisted mechanism that eventually produces full reversal by domain expansion. An in-plane applied magnetic field collinear with the current is required, with the necessary field scale set by the need to overcome DW chirality imposed by the Dzyaloshinskii-Moriya interaction. Once Joule hea…
Spin transport in magnetic tunnel junctions comprising a single magnetic layer in the presence of interfacial spin-orbit interaction (SOI) is investigated theoretically. Due to the presence of interfacial SOI, a current-driven spin torque can be generated at the second order in SOI, even in the absence of an external spin polarizer. This torque possesses two components, in-plane and perpendicular to the plane of rotation, that can induce either current-driven {\em magnetization switching} from in-plane to out-of-plane configuration or {\em magnetization precessions}, similarly to Spin Transfer…
We investigate theoretically the spin torque and magnetization dynamic in a thin ferromagnetic (FM) layer with spatially varying magnetization. The FM layer is deposited on the surface of a topological insulator (TI). In the limit of the adiabatic relaxation of electron spin along the magnetization, the interaction between the exchange interaction and the Rashba-like surface texture of a TI yields a topological gauge field. Under the gauge field and an applied current, spin torque is induced according to the direction of the current. We derived the corresponding effective anisotropy field and …
We have studied the spin orbit torque (SOT) in Pt/Co/Ir multilayers with 3 repeats of the unit structure. As the system exhibits oscillatory interlayer exchange coupling (IEC) with varying Ir layer thickness, we compare the SOT of films when the Co layers are coupled ferromagnetically and antiferromagnetically. SOT is evaluated using current induced shift of the anomalous Hall resistance hysteresis loops. A relatively thick Pt layer, serving as a seed layer to the multilayer, is used to generate spin current via the spin Hall effect. In the absence of antiferromagnetic coupling, the SOT is con…
Pure spin currents transport angular momentum without an associated charge flow. This unique property makes them attractive for spintronics applications, such as torque induced magnetization control in nanodevices that can be used for sensing, data storage, interconnects and logics. Up to now, however, most spin transfer torque studies focused on metallic ferromagnets, while magnetic insulators were largely ignored, in spite of superior magnetic quality factors. Here, we report the observation of spin torque-induced magnetization dynamics in a magnetic insulator. Our experiments show that in u…
Current-driven magnetization dynamics in ferromagnetic metals are studied in a self-consistent adiabatic local-density approximation in the presence of spin-conserving and spin-dephasing impurity scattering. Based on a quantum kinetic equation, we derive Gilbert damping and spin-transfer torques entering the Landau-Lifshitz equation to linear order in frequency and wave vector. Gilbert damping and a current-driven dissipative torque scale identically and compete, with the result that a steady current-driven domain-wall motion is insensitive to spin dephasing in the limit of weak ferromagnetism…
Using the Kubo linear-response formalism we derive expressions to calculate the heat current generated by magnetization dynamics in magnets with broken inversion symmetry and spin-orbit interaction (SOI). The effect of producing heat currents by magnetization dynamics constitutes the Onsager reciprocal of the thermal spin-orbit torque (TSOT), i.e., the generation of torques on the magnetization due to temperature gradients. We find that the energy current driven by magnetization dynamics contains a contribution from the Dzyaloshinskii-Moriya interaction (DMI), which needs to be subtracted from…
