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1508.00629v2.A_Critical_Analysis_of_the_Feasibility_of_Pure_Strain_Actuated_Giant_Magnetostrictive_Nanoscale_Memories.pdf

1 A Critical Analysis of the Feasibility of Pure Strain -Actuated Giant Magnetostrictive Nano scale Memories P.G. Gowtham1, G.E. Rowlands1, and R.A. Buhrman1 1Cornell University, Ithaca, New York, 14853, USA Abstract Concepts for memories based on the manipulation of giant magnetostrictive nano magnets by stress pulses have garnered recent attention due to their potential for ultra -low energy operation in the high storage density limit . Here we discuss the feasibility of making such memories in light of the fact that the Gilbert damping of such materials is typically quite high. We report the results of numerical simulations for several classes of toggle precessional and non - toggle dissipative magnetoelastic switching modes. M aterial candidates for each o f the several classes are analyzed and f orms for the anisotropy energy density and range s of material parameters appropriate for each material class are employed. Our study indicates that the Gilbert damping as well as the anisotropy and demagnetization energies are all crucial for determining the feasibility of magnetoelastic toggle -mode precessional switching schemes. The role s of thermal stability and thermal fluctuations for stress -pulse switching of gia nt magnetostrictive nanomagnets are also discussed in detail and are shown to be important in the viability, design, and footprint of magnetostrictive switching schemes. 2 I. Introduction In recent years pure electric -field based control of magnetization has become a subject of very active research. It has been demonstrated in a variety of systems ranging from multiferroic single phase materials, gated dilute ferromagnetic semiconductors 1–3, ultra -thin metallic ferromagnet/oxide interfaces 4–10 and piezoelectric /magnetoelastic composites 11–15. Beyond the goal of establishing an understanding of the physics involved in each of these systems, this work has been strongly motivated by the fact that electrical -field based manipulation of magnetization could form the basis for a new generation of ultra -low power, non -volatile memories. Electric - field based magnetic devices are not necessarily limited by Ohmic losses during the write cycle (as can be the case in current based memories such as spin -torque magnetic random access memory (ST -MRAM) ) but rather by the capacitive charging/decharging energies incurred per write cycle. As the capacitance of these devices scale with area the write energies have the potential to be as low as 1 aJ per write cycle or less. One general approach to the electrical control of magnetism utilizes a magnetostrictive magnet/piezoelectric transducer hybrid as the active component of a nanoscale memory element. In this appro ach a mechanical strain is generated by an electric field within the piezoelectric substrate or film and is then transferred to a thin, nanoscale magnetostrictive magnet that is formed on top of the piezoelectric. The physical interaction driving the write cycle of these devices is the magnetoelastic interaction that describes the coupling between strain in a magnetic body and the magnetic anisotropy energy. The strain imposed upon the magnet creates an internal effective magnetic field via the magnetoelast ic interaction that can exert a direct torque on the magnetization. If successfully implemented this torque can switch the magnet from one stable configuration to another, but whether imposed stresses and strains can be used to switch a 3 magnetic element be tween two bi -stable states depend s on the strength of the magnetoelastic coupling (or the magnetostriction). Typical values of the magnetostriction ( = 0.5 -60 ppm) in most ferromagnets yield strain and stress scales that make the process of strain -induced switching inefficient or impossible. However, considerable advances have been made in synthesizing materials both in bulk and in thin film form that have magnetostrictions that are one to two orders of magnitude larger than standard transition metal ferrom agnets. These giant magnetostrictive materials allow the efficient conversion of strains into torque on the magnetization. However it is important to note that a large magnetostrictive (or magnetoelastic) effect tends to also translate into very high magnetic damping by virtue of the strong coupling between magnons and the phonon thermal bath, which has important implicati ons, both positive and negative, for piezoelectric based magnetic devices. In this paper we provide an analysis of the switching modes of several different implementations of piezoelectric/magnetostrictive devices. We discuss how the high damping that is generally associated with giant magnetoelasticity affects the feasibility of different approaches, and we also take other key material properties into consideration, including the saturation magnetization of the magnetostrictive element, and the form and magnitude of its magnetic anisotropy. Th e scope of th is work excludes device concepts and physics circumscribed by magneto -elastic mani pulation of domain walls in magnetic films, wires, and nanoparticle arrays 11,12,16. Instead we focus here on analyzing various magnetoelastic reversal modes, principally within the single domain approximation, but we do extend this work to micromagnetic modeling in cases where it is not clear that the macrospin approximation prov ides a fully successful description of the essential physics. We enumerate potential material s 4 candidates for each of the modes evaluated and discuss the various challenges inherent in constructing reliable memory cells based on each of the reversal modes t hat we consider. II. Toggle -Mode Precessional Switching Stress pulsing of a magnetoelastic element can be used to construct a toggle mode memory. The toggling mechanism between two stable states relies on transient dynamics of the magnetization that are initi ated by an abrupt change in the anisotropy energy that is of fixed and short duration. This change in the anisotropy is created by the stress pulse and under the right conditions can generate precessional dynamics about a new effective field. This effectiv e field can take the magnetization on a path such that when the pulse is turned off the magnetization will relax to the other stable state. This type of switching mode is referred to as toggle switching because the same sign of the stress pulse will take t he magnetization from one state to the other irrespective of the initial state. We can divide the consideration of the toggle switching modes into two cases; one that utilizes a high sM in-plane magnetized element, and the other that employs perpendicular magnetic anisotro py (PMA) materials with a lower sM. We make this distinction largely because of differences in the structure of the torques and stress fields required to induce a switch in these two class es of systems. The switching of in -plane giant magnetostrictive nanomagnets with sizeable out -of-plane demagnetization fields relies on the use of in-plane uniaxial stress -induced effective fields that overcome the in -plane anisotropy (~O( 102 Oe)). The mo ment will experience a torque canting the moment out of plane and causing precession about the large demagnetization field. Thus the precessional time scales for toggling between stable in -plane states will be largely determined by the d emagnetization fiel d (and thus sM ). The dynamics of this mode bears striking resemblance to the dynamics in hard -axis field 5 pulse switching of nanomagnets 17. On the other hand, the dominant energy scale in PMA giant magnetostrictive materials is the perpendicular anisotropy energy. This energy scale can vary substantially (anywhere from uK ~ 105-107 ergs/cm3) depending on the materials utilized and the details of their growth. The anisotropy energy scale in these materials can be tuned into a region where stress -induced anisotropy energies can be comparable to it. A biaxial stress -induced anisotropy energy, i n this geometry, can induce switching by cancelling and/or overcoming the perpendicular anisotropy energy. As we shall see, this fact and the low sM of these systems imply dynamical time scales that are substantially different from the case where in -plane magnetized materials are employed. A. In-Plane Magnetized Magnetostrictive Materials We first treat the macrospin switching dynamics of an in -plane magnetized magnetostrictive nanomagnet with uniaxial anisotropy under a simple rectangular uniaxial stress pulse. Giant magnetostriction in in -plane magnetized systems have been demonstrated for sputtered polycrystalline Tb 0.3Dy0.7Fe2 (Terfenol -D) 18, and more recently in quenched Co xFe1-x thin film systems 19. We assume that the uniaxial anisotropy is defined completely by the shape anisotropy of the elliptical element and that any magneto -crystalline anisotropy in the film is considerably weaker. This is a reasonabl e assumption for the materials considered here in the limit where the grain size is considerably smaller than the nanomagnet’s dimensions. The stress field is applied by voltage pulsing an anisotropic piezoelectric film that is in contact with the nanomagn et. The proper choice of the film orientation of a piezoelectric material such as <110> lead magnesium niobate -lead titanate (PMN -PT) can ensure that an effective uniaxial in -plane strain develops along a particular crystalline axis after poling the piezo in the z -direction. We 6 assume that the nanomagnet major axis lies along such a crystalline direction (the <110> - direction of PMN -PT) so that the shape anisotropy is coincident with the strain axis (see Figure 1 for the relevant geometry) . For the analysis below we use material values appropriate to sputtered, nanocrystalline Tb0.3Dy0.7Fe2 18 ( sM= 600 emu/cm3, s = 670 ppm is the saturation magnetostriction). Nanocrystalline Tb0.3Dy0.7Fe2 films, with a mean crystalline grain diameter graind < 10 nm, can have an extremely high magnetostriction while being relatively magnetica lly soft with coercive fields, cH ~ 50-100 Oe, results which can be achieved by thermal processing during sputter growth at T ~ 375 ºC 20. The nanomagnet dimensions were as sumed to be 80 nm (minor axis) × 135 nm (major axis) × 5 nm (thickness) yielding a shape anisotropy field 4 ( )k y x sH N N M = 323 Oe and 4 ( )demag z y sH N N M = 5.97 kOe. We use demagnetization factors that are correct for an elliptical cylinder 21. The value of the Gilbert damping parameter for the magnetostrictive element is quite important in determining its dynamical behavior during in -plane stress -induced toggle switching. Previous simulation results 22–24 used a value ( 0.1 for Terfenol -D) that, at least arguably, is consid erably lower than is reasonable since that value was extracted from spin pumping in a Ni (2 nm) /Dy(5 nm) bilayer 25. However, that bilayer material is not a good surrogate for a rare - earth transition -metal alloy (especially for 0L rare earth ions). In the latter case the loss contribution from direct magnon to short w avelength phonon conversion is important, as has been directly confirmed by studies of 0L rare earth ion doping into transition metals 26,27. For example in -plane magnetized nanocrystalline 10% Tb -doped Py shows ~ 0.8 when magnetron sputtered at 5 mtorr Ar pressure, even though the magnetostriction is small within this region of Tb doping 27. We contend that a substantial increase in the magnetoelastic interaction in alloys 7 with higher Tb content is likely to make even larger. Magnetization rotation in a highly magnetostrictive magnet will efficiently generate longer wavelength acoustic phonons as well and heat loss will be generated when these phonons thermalize. Unfortunately, measurements of the magnetic damping parameter in polycrystalline Tb0.3Dy0.7Fe2 do not appear to be available in the literature. However, some results on the amo rphous Tb x[FeCo] 1-x system, achieved by using recent ultra -fast demagnetization techniques, have extracted ~ 0.5 for compositions (x ~ 0.3) that have high magnetostriction 28. We can also estimate the scale for the Gilbert damping by using a formalism that takes into account direct magnon to long wavelength phonon conversion via the magnetoelastic interaction and subsequent phonon relaxation to the thermal phonon bath29. The damping can be estimated by the following formula: 2 2236 1 1 22s sT s L s eff ex eff exMc M c M AA (1) Using sM = 600 emu/cm3, the exchange stiffness exA = 0.7x10-6 erg/cm, a mass density ρ = 8.5 g/cm3, Young’s modulus of 65 GPa 30, Poisson ratio 0.3 , and an acoustic damping time = 0.18 ps 29 the result is an estimate of ~1 . Given the uncertainties in the various parameter s determining the Gilbert damping , we examine the magnetization dynamics for values of ranging from 0.3 to 1.0. We simulate the switching dynamics of the magnetic moment of a Terfenol -D nanomagnet at T=300 K using the Landau -Lifshitz -Gilbert form of the equation describing the precession of a magnetic moment m: 8 ( ) ( )eff eff eff Langevinddttdt dt mmm H m H m (2) where eff is the gyromagnetic ratio. As Tb0.3Dy0.7Fe2 is a rare earth – transition metal (RE-TM) ferrimagnet (or more accurately a speromagnet), the gyromagnetic ratio cannot simply be assumed to be the free electron value. Instead we use the value eff = 1.78 107 Hz/Oe as extracted from a spin wave resonance study in the TbFe 2 system 31 which appears appropriate since Dy and Tb are similar in magnetic moment/atom (10 B and 9 B respectively) and g factor ( ~4/3 and ~3/2 respectively). The first term in Equation (2) represents the torque on the magnetization from any applied fields, the effective stress field, and any anisotropy and demagnetization fields that might be present. The third term in the LLG represents the damping torque that acts to relax the magnetization towards the direction of the effective field and hence damp out precessional dynamics. The second term is the Gaussian -distributed Langevin field that takes into account the effect thermal fluctuations on the magnetization dynamics. From the fluctuation -dissipation theorem, 2RMS B Langevin eff skTHM V t where t is the simulation time -step 32. Thermal fluc tuations are also accounted for in our modeling by assuming that the equilibrium azimuthal and polar starting angles ( 0 and 0 /2 respectively) have a random mean fluctuation given by equipartition as 00 2 2RMS BkT EV and 0 24 ( )RMS B z y skT N N M V . A biasH of 100 Oe was 9 used for our simulations which creates two stable energy minima at 0arcsin ~ 18bias kH H and 1162 symmetric about /2 . This non -zero starting angle ensures that 00RMS . This field bias is essential as the initial torque from a stress pulse depends on the initial starting angle. This angular dependence generates much larger thermally -induced fluctu ations in the initial torque than a hard -axis field pulse. The hard axis bias field also reduces the energy barrier between the two stable states. For Hbias = 100 Oe the energy barrier between the two states is Eb = 1.2 eV yielding a room temperature /bBE k T = 49. This ensures the long term thermal stability required for a magnetic memory. To incorporate the effect of a stress pulse in Equation (2) we employ a free energy form for the effective field, ( ) /efftE Hm that expresses the effect of a stress pulse along the x - direction of our in -plane nanomagnet with a uniaxial shape anisotropy in the x -direction. The stress enters the energy as an effective in -plane anisotropy term that adds to the shape anisotropy of the magnet (first term in Equation (3) below). The sign convention here is such that 0 implies a tensile stress on the x -axis while 0 implies a compressive strain. We also include the possibility of a bias field applied along the hard axis in the final term in Equation (3). 22 223( , , ) [2 ( ) ( )]2 2 ( )x y z y x s s x z y s z bias s yE m m m N N M t m N N M m H M m (3) The geometry that we have assumed allows only for fast compressive -stress pulse based toggle mode switching. The application of a DC compressive stress along the x -axis only reduces the magnitude of the anisotropy and changes the position of the equilibriu m magnetic angles 0 10 and 10180 while keeping the potential wells associated with these states symmetric as well. Adiabatically increasing the value of the compressive stress moves the angles toward /2 until 3()2sutK but obviously can never induce a magnetic switch. Thus the magnetoelastic memory in this geometry must make use of the transient behavior of the magnetization under a stress pulse as opposed to re lying on quasistatic changes to the energy landscape. A compressive stress pulse where 3()2sutK creates a sudden change in the effective field. The resultant effective field 32ˆsu eff y bias sKmHM Hy points in the y -direction and causes a torque that brings the magnetization out of plane. At this point the magnetization rotates rapidly about the very large perpendicular demagnetization field ˆ 4demag s z Mm Hz and if the pulse is turned off at the right time will relax down to the opposite state at 1 = 163. Such a switching trajectory for our simulated nanomagnet is shown in the red curve in Figure 2. This mode of switching is set by a minimum characteristic time scale 1~ 7.54sw spsM , but the precession time will in general be longer than sw for moderate stress pulse amplitudes, ( ) 2 / 3us tK , as the magnetization then cants out of plane enough to see only a fraction of the maximum possible demagH . Larger stress pulse amplitudes result in shorter pulse duratio ns being required as the magnetization has a larger initial excursion out of plane. For pulse durations that are longer than required for a rotation (blue and green curves in Figure 2) m will exhibit damped elliptical precession about /2 . If the stress is released during the correct portion of any of these subsequent precessional cycles the magnetization 180 11 should relax down to the 1 state [blue curve in Figure 2], but otherwise it will relax down to the original state [green curve in Figure 2]. The prospect of a practical device working reliably in the long pulse regime appears to be rather poor. The high damping of giant magnetostrictive magnets and the large field scale of the demagnetization field yield very stringent pulse timing requirements and fast damping times for equilibration to /2 . The natural time scale for magnetization damping in the in -plane magnetized thin film case is 1 2d sM , which ranges from 50 ps down to 15 ps for 0.3 1 with sM = 600 emu/cm3. This high damping also results in the influence of thermal noise on the magnetization dynamics being quite strong since LangevinH . Thus large stress levels with extremely short pulse durations are required in order to rotate the magnetization around the /2 minimum within the damping time, and to keep the precession amplitude large enough that the magnetization will deterministically relax to the reversed state. Our simulation results for polycrystalline Tb0.3Dy0.7Fe2 show that a high stress pulse amplitude of 85 MPa with a pulse duration ~ 65 ps is required if 0.5 (Figure 3a). However, the pulse duration window for which the magnetization will deterministically switch is extremely small in this case (<5 ps). This is due to the fact that the precession amplitude about the /2 minimum at this damping gets small enough that thermal fluctuations allow only a very small window for which switching is reliable. For the lowest damping that we consider reasonable to assume, 0.3 , reliable switching is possible between pulse ~ 30-60 ps at 85 MPa . At a larger damping 0.75 we find that the switching is non -deterministic for all pulse widths as the magnetization damps too quickly; instead very high stresses , 200 MPa are required to 1 12 generate deterministic switching of the magnetization with a pulse duration w indow pulse ~ 25- 45 ps ( Figure 3b). Given the high value of the expected damping we have also simulated the magnetization dynamics in the Landau Lifshitz (LL) form: 2(1 ) ( ( ) ( ))LL eff Langevinddttdt dt mmm H H m (4) The LL form and the LLG form are equivalent in low damping limit ( 1 ) but they predict different dynamics at higher damping values. Which of these norm -preserving forms for the dynamics has the right damping form is still a subject of debate 33–37. As one increases α in the LL form the precessional speed is kept the same while the damping is assumed to affect only the rate of decay of the precession amplitude. The damping in the LLG dynamics, on the other hand, is a viscosity term and retards the pre cessional speed. The effect of this retardation can be seen in the LLG dynamics as the precessional cycles move to longer times as a function of increasing damping. Our simulations show that the LL form (for fixed ) predicts highe r precessional speeds than the LLG and hence an even shorter pulse duration window for which switching is deterministic than the LLG, ~12 ps for LL as opposed to ~ 30 ps for LLG ( Figure 3c). The damping clearly plays a crucial role in the stress amplitude scale and pulse duration windows for which deterministic switching is possible, regardless of the form used to describe the dynamics. Even though the magnetostriction of Tb 0.3Dy0.7Fe2 is high and the stress required to entirely overcome the anisotropy energy is only 9.6 MPa, the fast damping time scale and increased thermal noise (set by the large damping and the out -of-plane demagnetization) means 13 that the stress -amplitude that is required to achieve deterministic toggle switching is 10 -20 times larger. In addition, the pulse duration for in -plane toggling must be extremely short, with typical pulse durations of 10 -50 ps with tight time windows of 20 -30 ps within which the acoustic pulse must be turned off. Given ferroelectric switching rise times on the order of ~50 ps extracted from experiment38 and considering the acoustical resonant response of the entire piezoelectric / magnetostrictive nanostructure and acoustic ringing and inertial terms in the lattice dynamics, generation of such large stresses with the strict pulse time requirem ents needed for switching in this mode is likely unfeasible. In addition, the stress scales required to successfully toggle switch the giant magnetostrictive nanomagnet in this geometry are nearly as high or even higher than that for transition metal ferromagnets such as Ni ( ~ 38 ppms with 0.045 ). For example, with a 70 nm × 130 nm elliptical Ni nanomagnet with a thickness of 6 nm and a hard axis bias field of 120 Oe we should obtain switching at stress values = +95 MPa and pulse = 0.75 ns. Therefore the use of giant magnetostrictive nanomagnets with high damping in this toggle mode scheme confers no clear advantage over the use of a more conventional transition metal ferromagnet, and in neither case does this approach appear particularly viable for t echnological implementation. B. Magneto -Elastic Materials with PMA: Toggle Mode Switching Certain amorphous sputtered RE/TM alloy films with perpendicular magnetic anisotropy such as a -TbFe 2 39–42 and a - Tb0.3Dy0.7Fe2 43 have properties that may make these materials feasible for use in stress -pulse toggle switching. In certain composition ranges they exhibit large magnetostriction ( s > 270 ppm for a -TbFe 2, and both s and the effective out of plane 14 anisotropy can be tuned over fairly wide ranges by varying the process gas pressure during sputter deposition, the target atom -substrate incidence angle, and the substrate temperature. We consider the energy of such an out -of-plane magnetostrictive material under the influence of a magnetic field biasH applied in the ˆx direction and a pulsed biaxial stress: 223( , , ) [ 2 ( )]2u x y z s s biaxial z s bias xE m m m K M t m M H m (5) Such a biaxial stress could be applied to the magnet if it is part of a patterned [001] -poled PZT thin film/ferromagnet bilayer. A schematic of this device geometry is depicted in Figure 4.When 0biasH , it is straightforward to see the stress pulse will not result in reliable switching since, when the tensile biaxial stress is large enough, the out of plane anisotropy becomes an easy -plane anisotropy and the equator presents a zero -torque condition on t he magnetization, resulting in a 50%, or random, probability of reversal when the pulse is removed. However, reliable switching is possible for 0biasH since that results in a finite canting of m towards the x -axis. This canting is required for the same reasons a hard -axis bias field was needed for the toggle switching of an in -plane magnetized element as discussed previously. A pulsed biaxial stress field can then in principle lead to deterministic precessional toggle switching between the +z and –z energy minima . This mode of pulsed switching is analogous to voltage pulse switching in the ultra-thin CoFeB|MgO using the voltage -controlled magnetic anisotropy effect.5,8 Previous simulation results have also di scussed this class of macrospin magnetoelast ic switc hing in the context of a Ni|Barium -Titatate multilayer44 and a zero -field, biaxial stress -pulse induced toggle switching scheme taking advantage of micromagnetic inhomogeneities has recently appeared in the literature45. Here we discuss biaxial stress -pulse switching for a broad class of giant 15 magnetostrictive PMA magnets where we argue that the monodomain limit strictly applies throughout the switching process and extend past previous macrospin modeling by systematically think ing about how pulse -timing requirements and critical write stress amplitudes are determined by the damping, the PMA strength, and sM for values reasonable for these materials. For our simulation study of stress -pulse toggle switching of a PMA magnet, we considered a Tb 33Fe67 nanomagnet with an sM = 300 emu/cm3, effK = 4.0×105 ergs/cm3 and s = 270 ppm. To estimate the appropriate value for the damping parameter we noted that ultrafast demagnetization measurements on Tb 18Fe82 have yielded 0.27 . This 18 -82 composition lies in a region where the magnetostriction is moderate ( s ~50 ppm) 43 so we assumed that the damping will be on the same order or higher for a -TbFe 2 due to its high magnetostriction. Therefore we ran simulations for the range of = 0.3 -1. For the gyromagnetic ratio we used eff = 1.78×107 s-1G-1 which is appropriate for a -TbFe 2 31. We assumed an effective exchange constant 611 10effA erg cm 46 implying an exchange length exeff no stress effAlK = 15.8 nm (in the absences of an applied str ess) and 22exeff pulse sAlM = 13.3 nm (assuming that the stress pulse amplitude is just enough to cancel the out of plane anisotropy). A monodomain crossover criterion of cd ~ ~ 56 nm (with the pulse off) and cd ~ 22ex sA M ~ 47 nm (with the pulse on) can be calculated by considering the minimum length -scale associated with supporting thermal λ/2 confined spin wave modes 47. The important point here is that the low sM of these systems ensures that the exchange length is still fairly long even during the switching process, 4ex uA K 16 which suggests that the macrospin approximation should be valid for describing the switching dynamics of this system for reasonably sized nanomagnets. We simulated a circular element with a diameter of 60 nm and a thickness of 10 nm, under an x -axis bias field, biasH = 500 Oe which creates an initial canting angle of 11 degrees from the vertical (z-axis). This starting angle is sufficient to enable deterministic toggle precessional switching between the +z and –z minima via biaxial stress pulsing. The assumed device geometry, anisotropy energy density and bias field corresponded to an energy barrier bE = 4.6 eV for thermally activated reversal, and hence a room temperature thermal stability factor = 185. We show selected results of the macrospin simulations of stress -pulse toggle switching of this modeled TbFe 2 PMA nanomagnet. Typical switching trajectories are shown in Figure 5a. The switching transition can be divided into two stages (see Figure 5b): the precessional stage that occurs when the stress field is applied, during which the dynamics of the magnetization are dominated by precession about the effective field that arises from the sum of the bias field and the easy -plane anisotropy field 3 ( ) 2eff s z stKmM , and the dissipative stage that begins when the pulse is turned off and where the large effK and the large result in a comparatively quick relaxation to the other energy minimum. Thus most of the switching process is spent in the precessional phase and the entire switching process is not much longer than the actual stress pulse duration. For pulse amplitudes a t or not too far above the critical stress for reversal, 2 / 3eff s K the two relevant timescales for the dynamics are set approximately by the precessional period 1/ 100 pssw bias H of the nanomagnet and the damping time 17 ~ 2 /d bias H . Both of these timescales are much longer than the timescales set by precession and damping about the demagnetization field in the in -plane magnetized toggle switching case. The result is that even with quite high damping one can have reliable s witching over much broader pulse width windows, 200 -450 ps . (Figure 6a,b). The relatively large pulse duration windows within which reliable switching is possible (as compared to the in -plane toggle mode) hold for both the LL and LLG damping. However, the diffe rence between the two forms is evident in the PMA case ( Figure 6c). At fixed , the LLG damping predicts a larger pulse duration window than the LL damping. Also the effective viscosity implicit within the LLG equation ensures that the switching time scales are slower than in the LL case as can also be seen in Figure 6c. An additional and important point concerns the factors that determine the critical switching amplitude. In the in -plane toggle mode switching of the previous section, it was found that the in-plane anisotropy field was not the dominant factor in determining the stress scale required to transduce a deterministic toggle switch. Instead, we found that the stress scale was almost exclusively dependent on the need to generate a high enough preces sion amplitude/precession speed during the switching trajectory so as to not be damped out to the temporary equilibrium at /2 (at least within the damping range considered). This means that the critical stress scale to transduce a deterministic switch is essentially determined by the damping. We find that the situation is fundamentally different for the PMA based toggle memories. The critical amplitude c is nearly independent of the damping from a range of 0.3 0.75 up until ~1 where the damping is sufficiently high (i.e. damping times equaling and/or exceeding the p recessional time scale) that at 85 MPa the magnetization traverses too close to the minimum at /2 , 0 . The main reason for this difference between the 18 PMA toggle based memories and the in-plane toggle based memory lies in the role that the application of stress plays in the dynamics. First, in the in -plane case, the initial elliptical amplitude and the initial out of plane excursion of the magnetization is set by the stress pulse magnitu de. Therefore the stress has to be high to generate a large enough amplitude such that the damping does not take the trajectory too close to the minimum at which point Langevin fluctuations become an appreciable part of the total effective field. This is n ot true in the PMA case where the initial precession amplitude about the bias field is large and the effective stress scale for initiating this precession about the bias field is the full cancellation of the perpendicular anisotropy. Since the minimum stre ss-pulse amplitude required to initiate a magnetic reversal in out - of-plane toggle switching scales with effK in the range of damping values considered, lowering the PMA of the nanomagnet is a straightforward way to reduce the stress and write energy requirements for this type of memory cell. Such reductions can be achieved by strain engineering through the choice of substrate, base electrode and transducer layers, by the choice of deposition parameters, and/or by post -growth annealing protocols. For example growing a TbFe 2 film with a strong tensile biaxial strain can substantially lower effK . If the P MA of such a nanomagnet can be reliably r educed to effK = 2105 ergs/cm3 our simulations indicate that this would result in reliable pulse toggle switching at ~ -50 MPa (corresponding to a strain amplitude on the TbFe 2 film of less than 0.1%) with pulse ≈ 400 ps, for 0.3 ≤ ≤ 0.75 and biasH ~ 250 Oe . Electrical actuation of this level of stress/strain in the sub -ns regime, while challenging, may be possible to achieve.48 If we again assume sM =300 emu/cm3, a diameter of 60 nm and a thickness of 10 nm, this low PMA nanomagnet would still have a high thermal stability with 92 . The challenge, 19 of course, is to consistently and uniformly control the residual strain in the magnetostrictive layer. It is important to note that no such tailoring (short of systematically lowering the damping) can exist in the in -plane toggle mode case. III. Two -State Non -Toggle Switching So far we have discussed toggle mode switching where the same polarity strain pulse is applied to reverse the magnetization between two bi -stable states. In this case the strain pulse acts to create a temporary field around which the magnetization precesse s and the pulse is timed so that the energy landscape and magnetization relax the magnetization to the new state with the termination of the pulse. Non -toggle mode magneto -elastic switching differs fundamentally from the precessional dynamics of toggle -mode switching, being an example of dissipative magnetization dynamics where a strain pulse of one sign destabilizes the original state (A) and creates a global energy minimum for the other state (B). The energy landscape and the damping torque completely de termine the trajectory of the magnetization and the magnetization effectively “rolls” down to its new global energy minimum. Reversing the sign of the strain pulse destabilizes state B and makes state A the global energy minimum – thus ensuring a switch ba ck to state A. There are some major advantages to this class of switching for magneto -elastic memories over toggle mode memories. Precise acoustic pulse timing is no longer an issue. The switching time scales, for reasonable stress values, can range from q uasi-static to nanoseconds. In addition, the large damping typical of magnetoelastic materials does not present a challenge for achieving robust switching trajectories in deterministic switching as it does in toggle -mode memories. Below we will discuss det erministic switching for magneto -elastic materials that have two different types of magnetic anisotropy. 20 C. The Case of Cubic Anisotropy We first consider magneto -elastic materials with cubic anisotropy under the influence of a uniaxial stress field pulse. T here are many epitaxial Fe -based magnetostrictive materials that exhibit a dominant cubic anisotropy when magnetron -sputter grown on oriented C u underlayers on Si or on MgO, GaAs , or PMN -PT substrates. For example, Fe 81Ga19 grown on MgO [100] or on GaAs ex hibit a cubic anisotropy 49–51. Given the low cost of these Fe -based materials compared to rare -earth alloys, it is worth investigating whether such films can be used to construct a two state memory. Fe 81Ga19 on MgO exhibits easy axes along <100>. In ad dition, epitaxial Fe 81Ga19 films have been found to have a reasonably high magnetostriction λ100=180 ppm making them suitable for stress induced switching. If we assume that the cubic magnetoelastic thin-film nanomagnet has circular cross section, that the stress field is applied by a transducer along the [100] direction , and that a bias field is applied at 4 degrees, the magnetic free energy is : 2 2 2 2 2 11 2( , ) (1 ) 2 ( ) 3( ) ( )2 2x y x y z z z s z s bias x y s xE m m K m m K m m N N M m MHm m t m (6) Equation (6) shows that, in the absence of a bias field, the anisotropy energy is 4 -fold symmetric in the film -plane. It is rather easy to see that it is im possible to make a two -state non - toggle switching with a simple cubic anisotropy energy and uniaxial stress field along [100]. Figure 7a shows the free energy landscape described by Equation (6) without stress applied. To create a two -state deterministic magnetostrictive device , biasH needs to be strong enough to eradicate the energy minima at and 3 / 2 which strictly requires that 1 0.5 /bias sH K M . 21 Finite temperature considerations can lower this minimum bias field requirement considerably. This is due to the fact that the bias field can make the lifetime to escape the energy minima in th e third quadrant and fourth qua drant small and the energy bar rier to return them from the energy minima in the first quadrant extremely large. We arbitrarily set this requirement for the bias field to correspond to a lifetime of 75 μs. The typical energy barriers to hop from back to the metastable minima in the thi rd and fourth quadrant for device volumes we will consider are on the order of several eV. The requirement for thermal stability of the two minima in the first quadrant , given a diameter d and a thickness filmt for the nanomagnet, sets an upper bound on biasH as we require / 40bbE k T at room temp erature between the two states (see Figure 7c). It is desirable that this upper bound is high enough that there is some degree of tolerance to the value of the bias field at device dimensions that are employed. This sets requirement s on the minimum volume of the cylindical nanomagnet that are dependent on 1K . For a circular element with d = 100 nm, filmt = 12.5 nm and 1K= 1.5 105 ergs/cm3, two - state non -toggle switching with the required thermal stability can only occur for biasH between 50 - 56 Oe. This is too small a range of acceptable bias fields. However , by increasing filmt to 15 nm the bias field range grows to biasH = 50 - 90 Oe wh ich is an acceptable range. For 1K = 2.0×105 erg/cm3 with d= 100 nm and filmt = 12.5 nm , there is an appreciable region of bias field (~65-120 Oe) for which /barrier BE k T > 42. For 1K = 2.5 105 ergs/cm3, the bias range goes from 90 – 190 Oe for the same volume. The main po int here is that, given the scale for the cubic anisotropy in Fe 81Ga19, careful attention must be paid to the actual values of the anisotropy 22 constants, device lateral dimensions, film thickness, and the exchange bias strength in order to ensure device stability in the sub -100 nm diameter regime . We now discuss the dynamics for a simulated case where d = 100 nm, filmt= 12.5 nm, 1K = 2.0×105 ergs/cm3, biasH = 85 Oe, and sM = 1300 emu/cm3. Two stable minima exist at =10o and = 80o. Figure 7b shows the effect of the stress pulse on the energy landscape. When a compressive stress c is applied, the potential minimum at =10o is rendered unstable and the magnetization follows the free energy gradient to = 80o (green curve). Since the stress field is applied along [100] the magnetization first switches to a minima very close to but greater than = 80o and when the stress is released it gently relaxes down to the zero stress minimum at = 80o. In order to switch from = 80o to = 10o we need to reverse the sign of the applied stress field to tensile (red curve). A memory constructed on these principles is thus non -toggle. The magnetization -switching trajectory is simple and follows the dissipative dynamics dictated by the free energy landscape (see Figure 8a). We have assumed a damping of 0.1 for the Fe 81Ga19 system, based on previous measurements52 and as confirmed by our own. Higher damping only ends up speeding up the sw itching and ri ng-down process. Figure 8b shows the simulated stress amplitude and pulse switching probability phas e diagram at room temperature. Ultimately, we must take the macrospin estimates for device parameters as only a roug h guide. The macrospin dynamics approximate the true micromagnetics less and less well as the device diameter gets larger. The mai n reason for this is the large sM of Fe 81Ga19 and the tendency of the magnetization to curl at the sample edges. Accordingly we have performed T = 0 ºK micromagnetic simulations in OOMMF.53 An exchange bias field biasH = 85 Oe was applied 23 at = 45º and we assume 1K = 2.0×105 ergs/cm3, sM = 1300 emu/cm3, and exA = 1.9 × 10-6 erg/cm. Micromagnetics show that the macrospin picture quantitatively captures the switching dynamics, the angular positions of the stables states ( 0~ 10 and 1~ 80 ) and the critical stress amplitude at ( ~ 30 MPa) when the device diameter d < 75 nm. The switching is essentially a rigid in -plane rotation of the magnetization from 0 to 1 . However, we cho se to show the switching for an element with d = 100 nm because it allowed for thermal stability of the devices in a region of thicknes s ( filmt = 12-15 nm) where biasH ~ 50-100 Oe at room temperature could be reasonably expected. The initial average magnetization angle is larger ( 0~ 19 and 1~ 71 ) than would b e predicted by macrospin for a d = 100 nm element. This is due to the magnetization c urling at the devices edges at d = 100 nm (see Figure 8c). Despite the fact that magnetization profile differs from the macrospin picture we find that there is no appreciable difference between the stress scales required for switching , or the basi c switching mechanism. The stress amplitude scale for writing the simulated Fe 81Ga19 element at ~ 30 MPa is not excessively high and there are essentially no demands on the acoustic pulse width requirements. These memories can thus be written at pulse amplitudes of ~ 30 MPa with acoustical pulse widths of ~ 10 ns. These numbers do not represent a major challenge from the acoustical transduction point of view. The drawback s to this scheme are the necessity of growing high quality single crystal thin film s of Fe 81Ga19 on a piezoelectric substrate that can generate large enough strain to switch the magnet (e.g. PMN -PT) and difficulties associated with tailoring the magnetocrystalline anisotropy 1K and ensuring thermal stability at low lateral device dimensions. 24 D. The Case of Uniaxial Anisotropy Lastly we discuss deterministic (non -toggle) switching of an in -plane giant magnetostrictive magnet with uniaxial anisotropy. In -plane magnetized polycrystalline TbDyFe patterned into ellipti cal nanomagnets could serve as a potential candidate material in such a memory scheme. To implement deterministic switching in this geometry a bias field biasH is applied along the hard axis of the nanomagnet. This generates two stable minima at 0 and 0 180 symmetric about the hard axis. The axis of the stress pulse then needs to be non - collinear with respect to the e asy axis in order to break the symmetry of the potential wells and drive the transition to the selected equilibrium position. Figure 9 below shows a schematic of the situation. When a stress pulse is applied in the direction that makes an angle with respect to the easy axis of the nanom agnet, oo0 90 , the free energy within the macrospin approximation becomes: 2 2 2 2 2( , , ) [2 ( ) 2 ( ) 3( ) (cos( ) sin( ) )2x y z y x s x z y s z bias s y s y x sE m m m N N M m N N M m H M m t m mM (7) From Equation (7) it can be seen that a sufficiently strong compressive stress pulse can switch the magnetization between 0 and o 0 180 , but only if 0 is between and . To see why this condition is necessary, we look at the magnetization dynamics in the high stress limit when 0 0 . During such a strong pulse the magnetization will s ee a hard axis appear at and hence will rotate towards the new easy axis at 90 , but when the stress pulse is o90 25 turned off the magnetization will equilibrate back to 0 . This situation is represented by the green trajectory shown in Figure 11a. But when o 090 , a sufficiently strong compressive stress pulse defines a new easy axis close to o90 and when the pulse is turned off the magnetization will relax to 0 180 (blue trajectory in Figure 11a). Similarly the possibility of switching from o180 to with a tensile strain depends on whether o o o90 180 90 . Thus o45 is the optimal situation as then the energy landscape becomes mirror symmetric about the hard axis and the amplitude of the required switching stress (voltage) are equal. This scheme is quite similar to the case of deterministic switching in biaxial anisotropy systems (with the coordinate system rotated by ). We note that a set of papers54–56 have previously proposed this particular case as a candidate for non -toggle magnetoelectric memory and have experimentally demonstrated operation of such a memory in the large feature -size (i.e. extended film ) limit .55 We argue here that in-plane giant magnetostrictive magnets operated in the non -toggle mode could be a good candidate for construct ing memories with low write stress amplitude, and nanosecond -scale write time operation. However , as we will discuss , the prospects of this type of switching mode being suitable for implementation in ultrahigh density memory appear to be rather poor. The m ain reason for this lies in the hard axis bias field requirements for maintaining low write error rates and the effect that such a hard axis bias field will have on the long term thermal stability of the element . At T = 0 ºK the requirement on biasH is only that it be strong enough that 0 > 45º. However, this is no longer sufficient at finite temperature where thermal fluctuations impl y a thermal, Gaussian distribution of the initial orientation of the magnetization o45 26 direction 0 about 0. If a significant componen t of this angular distribution falls below 45 degrees there will be a high write error rate. Thus we must ensure that biasH is high enough that the probability of < 45º is extremely low. We have selected the re quirement that < 45º is a 8 event where is the standard deviation of about 0 and is given by the relation . However, biasH must be low enough to be technologically feasible, but also must not exceed a value that compromises the energy barrier between the two potential minima – thus rendering the nanomagnet thermally unstable . These minimum and maximum requirement s on biasH puts significant constraints on the minimum size of the nanomagnet that can be used in this device approach. It also sets some rather tight requirements on the hard axis bias field, as we shall see. We first disc uss the effects of these requirements in the case of a relatively large magnetostrictive device. We assume the use of a polycrystalline Tb 0.3Dy0.7Fe2 element having sM = 600 emu/cm3 and an elliptical cross section of 400×900 nm2 and a thickness filmt = 12.5 nm. This results in a shape anisotropy field kH ≈ 260 Oe. We find that for an applied hard axis bias field biasH ~ 200 Oe, a field strength that can be reasonably engineered on -chip, the equilibrium angle of the element is 0 ≈ 51º and its root mean square (RMS) angular fluctuation amplitude is RMS ≈ 0.75º. Thus element ’s anisotropy field and the assumed hard axis biasing condition s just satisfy the assumed requirement that 08RMS > 45º (see Figure 10b). The magnetic energy barrier to thermal energy ratio for the element at biasH = 200 Oe is /bBE k T 02 2BkT EV 27 ≈ 350, which easily satisf ies the long-term thermal stability requirement (see Figure 10a), and which also provides some latitude for the use of a slightly higher biasH if desired to further reduce the write error rate . It is straightforward to see from these numbers that if the area of the magnetostrictive element is substantially reduced below 400 ×900 nm2 there must be a corresponding increase in kH and hence in biasH if the write error rate for the device is to remain acceptable. Of course an increase in the thickness of the element can partially reduce the increase in fluctuation amplitude due to the decrease in the magnetic a rea, but the feasible range of thickness variation cannot match the effect of, for example, reducing the cross -sectional area by a factor of 10 to 100, with the latter, arguably, being the minimum required for high density memory applications. While perhaps a strong shape anisotropy and an increased filmt can yield the required kH ≥ 1 kOe, the fact that in this deterministic mode of magnetostrictive switching we must also have biasH ~ kH results in a bias field requirement that is not technologically feasible. We could of course allow the write error rate to be much larger than indicated by an 8 fluctuation probability, but this would only relax the requirement on biasH marginally, which always must be such that 0 > 45o.Thus the deterministic magneto strictive device is not a viable candidate for ultra -high density memory. Instead this approach is only feasible for device s with lateral area ≥ 105 nm2 . While the requir ement of a large footprint is a limitation of the deterministic magneto strictive memory element , this device does have the significant advantage that the stress scale required to switch the memory is quite low. We have simulated T = 300 ºK macrospin switching dynamics for a 400×900 nm2 ellipse with thickness filmt = 12.5 nm with biasH = 200 Oe such that 0 ~ 51º. The Gilbert damping parameter was set to 0.5 and magnetostriction s = 28 670 ppm. The magnetization switches by simple rotation from 0 = 51º to 1129 that is driven by the stress pulse induced change in the energy landscape (see Figure 11a). Phase diagram results are provided in Figure 11b where the switching from 0 = 51º to 1 = 129 º shows a 100% switching probability for stresses as low as = - 5 MPa for pulse widths as short as 1 ns. Since the dimensions of the ellipse are large enough that t he macrospin picture is not strictly valid, we have also conducted T = 0 K micromagnetic simulations of the stress -pulse induced reversal in this geometry. We find that the trajectories are essentially well described by a quasi - coherent rotation with non-uniformities in the magnetization being more pronounced at the ellipse edges (see Figure 11c). The minimum stress pulse amplitude for swi tching is even lower than that predicted by macrospin at = - 3 MPa. This stress scale for switching is substantially lower than any of the switching mode schemes discussed before. Despite the fact that this scheme is not scalable down into the 100 -200 nm size regime, it can be appropriate for larger footprint memori es that can be written at very low write stress pulse amplitudes. IV. CONCLUSION The physical properties of giant magnetostrictive magnets (particularly of the rare -earth based TbFe 2 and Tb 0.3Dy0.7Fe2 alloys) place severe restrictions on the viability of such materials for use in fast, ultra -high density , low energy consumption data storage. We have enumerated the various potential problems that might arise from the characteristically high damping of giant magnetostrictive nanoma gnets in toggle -mode switch ing. We have also discussed the rol e that thermal fluctuation s have on the various switching modes and the challenges involved in 29 maintaining long -time device thermal stability that arise mainly from the necessity of employing hard axis bias fields . It is clear that the task of constructing a reliable memory using pure stress induced reversal of g iant magnetostrictive magnets will be , when pos sible, a question of trade -offs and careful engineering . PMA based giant magnetostrictive nanomagnets can be made extremely small ( d < 50 nm) while still maintaining thermal stability. The small diameter and low cross - sectional area of these PMA giant magnetostrictive devices could , in principle, lead to very low capacitive write energies. The counterpoint is that the stress fields required to switch the device are not necessarily small and the acoustical pulse timing requirements are demanding. However, it might be possible t o tune the magnetostriction s , K , and sM (either by adjustment of the growth conditions of the magnetostrictive magnet or by engineering the RE-TM multilayers appropriately) in order to significantly reduce the pulse amplitudes required f or switching (down into the 20-50 MPa range) and reduce th e required in -plane bias field – without compromising thermal stability of the bit . Such tuning must be carried out carefully. As we have discussed , the Gilbert dampi ng , s , K , and sM can all affect the pure stress -driven switching process and device thermal stability in ways that are certainly interlinked and not necessarily complementary. Two state non-toggle memories such as we described in Section III D could have extremely low stress write amplitudes and non-restrictive pulse requirements . However, the trade -off arises from thermal stability considerations and such a switching scheme is not scala ble down into the 100-200 nm size regime . Despite this limitation there may well be a place for durable memories with very low write stress pulse amplitudes and low write energies that operate reliably in the nanosecond regime . 30 ACKNOWLEDGEMENTS We thank R.B. van Dover, W.E. Bailey, C. Vittoria, J.T. Heron, T. Gosavi, and S. Bhave for fruitful discussions. We also thank D.C. Ralph and T. Moriyama for comments and suggestions on the manuscript. This work was supported by the Office of Naval Research and the Army Research Office. 31 REFERENCES 1 H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature 408, 944 (2000). 2 D. Chiba, M. Sawicki, Y. Nishitani, Y. Nakatani, F. Matsukura, and H. Ohno, Nature 455, 515 (2008). 3 D. Chiba, M. Yamanouchi, F. Matsukura, and H. Ohno, Science 301, 943 (2003). 4 M. Weisheit, S. Fähler, A. Marty, Y. Souche, C. Poinsignon, and D. Givord, Science 315, 349 (2007). 5 T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda, M. Mizuguchi, A.A. Tulapurkar, T. Shinjo, M. Shiraishi, S. Mizukami, Y. Ando, and Y. Suzuki, Nat. Nanotechnol. 4, 158 (2009). 6 M. Endo, S. Kanai, S. Ikeda, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 96, 212503 (2010). 7 W.-G. Wang, M. Li, S. Hageman, and C.L. Chien, Nat. Mater. 11, 64 (2012). 8 Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo, and Y. Suzuki, Nat. Mater. 11, 39 (2012). 9 T. Nozaki, Y. Shiota, S. Miwa, S. Murakami, F. Bonell, S. Ishibashi, H. Kubota, K. Yakushiji, T. Saruya, A. Fukushima, S. Yuasa, T. Shinjo, and Y. Suzuki, Nat. Phys. 8, 492 (2012). 10 P. Khalili Amiri, P. Upadhyaya, J.G. Alzate, and K.L. Wang, J. Appl. Phys. 113, 013912 (2013). 11 T. Br intlinger, S. -H. Lim, K.H. Baloch, P. Alexander, Y. Qi, J. Barry, J. Melngailis, L. Salamanca -Riba, I. Takeuchi, and J. Cumings, Nano Lett. 10, 1219 (2010). 12 D.E. Parkes, S.A. Cavill, A.T. Hindmarch, P. Wadley, F. McGee, C.R. Staddon, K.W. Edmonds, R.P. Campion, B.L. Gallagher, and A.W. Rushforth, Appl. Phys. Lett. 101, 072402 (2012). 13 M. Weiler, A. Brandlmaier, S. Geprägs, M. Althammer, M. Opel, C. Bihler, H. Huebl, M.S. Brandt, R. Gross, and S.T.B. Goennenwein, New J. Phys. 11, 013021 (2009). 14 M. We iler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M.S. Brandt, and S.T.B. Goennenwein, Phys. Rev. Lett. 106, 117601 (2011). 15 M. Weiler, H. Huebl, F.S. Goerg, F.D. Czeschka, R. Gross, and S.T.B. Goennenwein, Phys. Rev. Lett. 108, 176601 (2012). 32 16 V. Novosad, Y. Otani, A. Ohsawa, S.G. Kim, K. Fukamichi, J. Koike, K. Maruyama, O. Kitakami, and Y. Shimada, J. Appl. Phys. 87, 6400 (2000). 17 H. Schumacher, C. Chappert, P. Crozat, R. Sousa, P. Freitas, J. Miltat, J. Fassbender, and B. Hillebrands, Phys. Rev. Lett. 90, 017201 (2003). 18 P.I. Williams, D.G. Lord, and P.J. Grundy, J. Appl. Phys. 75, 5257 (1994). 19 D. Hunter, W. Osborn, K. Wang, N. Kazantseva, J. Hattrick -Simpers, R. Suchoski, R. Takahashi, M.L. Young, A. Mehta, L.A. Bendersky, S.E. Lofland, M. Wuttig, and I. Takeuchi, Nat. Commun. 2, 518 (2011). 20 K. Ried, M. Schnell, F. Schatz, M. Hirscher, B. Lu descher, W. Sigle, and H. Kronmüller, Phys. Status Solidi 167, 195 (1998). 21 M. Beleggia, M. De Graef, Y.T. Millev, D.A. Goode, and G. Rowlands, J. Phys. D. Appl. Phys. 38, 3333 (2005). 22 K. Roy, S. Bandyopadhyay, and J. Atulasimha, Phys. Rev. B 83, 2244 12 (2011). 23 K. Roy, S. Bandyopadhyay, and J. Atulasimha, J. Appl. Phys. 112, 023914 (2012). 24 M. Salehi Fashami, K. Roy, J. Atulasimha, and S. Bandyopadhyay, Nanotechnology 22, 309501 (2011). 25 J. Walowski, M.D. Kaufmann, B. Lenk, C. Hamann, J. McCord, and M. Münzenberg, J. Phys. D. Appl. Phys. 41, 164016 (2008). 26 W. Bailey, P. Kabos, F. Mancoff, and S. Russek, IEEE Trans. Magn. 37, 1749 (2001). 27 S.E. Russek, P. Kabos, R.D. McMichael, C.G. Lee, W.E. Bailey, R. Ewasko, and S.C. Sanders, J. Appl. Phys. 91, 8659 (2002). 28 Y. Ren, Y.L. Zuo, M.S. Si, Z.Z. Zhang, Q.Y. Jin, and S.M. Zhou, IEEE Trans. Magn. 49, 3159 (2013). 29 C. Vittoria, S.D. Yoon, and A. Widom, Phys. Rev. B 81, 014412 (2010). 30 Q. Su, J. Morillo, Y. Wen, and M. Wuttig, J. Appl. Phys. 80, 3604 (1996). 31 S.M. Bhagat and D.K. Paul, AIP Conf. Proc. 29, 176 (1976). 32 W.F. Brown, Phys. Rev. 130, (1963). 33 W.M. Saslow, J. Appl. Phys. 105, 07D315 (2009). 33 34 M. Stiles, W. Saslow, M. Donahue, and A. Zangwill, Phys. Rev. B 75, 214423 (2007). 35 N. Smith, Phys. Rev. B 78, 216401 (2008). 36 S. Iida, J. Phys. Chem. Solids 24, 625 (1963). 37 T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004). 38 J. Li, B. Nagaraj, H. Liang, W. Cao, C.H. Lee, and R. Ramesh, Appl. Phys. Lett. 84, 1174 (2004). 39 R.B. van Dover, M. Hong, E.M. Gyorgy, J.F. Dillon, and S.D. Albiston, J. Appl. Phys. 57, 3897 (1985). 40 F. Hellman, Appl. Phys. Lett. 64, 1947 (1994). 41 F. Hellman, R.B. van Dover, and E.M. Gyorgy, Appl. Phys. Lett. 50, 296 (1987). 42 T. Niihara, S. Takayama, and Y. Sugita, IEEE Trans. Magn. 21, 1638 (1985). 43 P.I. Williams and P.J. Grundy, J. Phys. D Appl. Phys. 27, 897 (1994). 44 M. Ghidini, R. Pellicel li, J.L. Prieto, X. Moya, J. Soussi, J. Briscoe, S. Dunn, and N.D. Mathur, Nat. Commun. 4, 1453 (2013). 45 J.-M. Hu, T. Yang, J. Wang, H. Huang, J. Zhang, L. -Q. Chen, and C. -W. Nan, Nano Lett. 15, 616 (2015). 46 F. Hellman, A.L. Shapiro, E.N. Abarra, R.A. Robinson, R.P. Hjelm, P.A. Seeger, J.J. Rhyne, and J.I. Suzuki, Phys. Rev. B 59, 408 (1999). 47 J.Z. Sun, R.P. Robertazzi, J. Nowak, P.L. Trouilloud, G. Hu, D.W. Abraham, M.C. Gaidis, S.L. Brown, E.J. O’Sullivan, W.J. Gallagher, and D.C. Worledge, Phys. Re v. B 84, 064413 (2011). 48 A. Grigoriev, D.H. Do, D.M. Kim, C.B. Eom, P.G. Evans, B. Adams, and E.M. Dufresne, Appl. Phys. Lett. 89, 1 (2006). 49 A. Butera, J. Gómez, J.L. Weston, and J.A. Barnard, J. Appl. Phys. 98, 033901 (2005). 50 A. Butera, J. Gómez, J.A. Barnard, and J.L. Weston, Phys. B Condens. Matter 384, 262 (2006). 51 A. Butera, J.L. Weston, and J.A. Barnard, J. Magn. Magn. Mater. 284, 17 (2004). 34 52 J. V. Jäger, A. V. Scherbakov, T.L. Linnik, D.R. Yakovlev, M. Wang, P. Wadley, V. Holy, S.A. Cavill, A. V. Akimov, A.W. Rushforth, and M. Bayer, Appl. Phys. Lett. 103, 032409 (2013). 53 M.J. Donahue and D.G. Porter, OOMMF User’s Guide, Version 1.0, Inter agency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (1999). 54 N. Tiercelin, Y. Dusch, V. Preobrazhensky, and P. Pernod, J. Appl. Phys. 109, 07D726 (2011). 55 Y. Dusch, N. Tiercelin, A. Klimov, S. Giordano, V. Preobrazhensky, and P. Pernod, J. Appl. Phys. 113, 17C719 (2013). 56 S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, and V. Preobrazhensky, J. Phys. D. Appl. Phys. 46, 325002 (2013). 35 Figure 1. Magnetoelastic elliptical memory element schematic with associated coordinate system for in - plane stress -pulse induced toggle switching. Here M is the magnetization vector with and being polar and azimuthal angles . For the in -plane t oggle switching case, the initial normalized magnetization 0 0 0ˆˆ cos sinm x y and is in the film plane with 0arcsin[ / ]bias kHH and ˆbias bias H Hy . Figure 2. Toggle switching trajectory for an in -plane magnetized polycrystalline Tb 0.3Dy 0.7Fe2 element with LLG = 0.3, = -120 MPa, and pulse = 50 ps (red) and 125 ps (blue) and 160 ps (green). 36 Figure 3. a) Effect of the Gilbert damping on pulse switching probability statistics for = -85 MPa. b) Effect of increasing stress pulse amplitude for high damping LLG = 0.75. Very high stress pulses ( >200 MPa) are required to allow precession to be fast enough to cause a switch before dynamics are damped out. c) Comparison of switching statistics for the LL and LLG dynamics at = -200 MPa, = 0.75. The LL dynamics exhibits faster precession than the LLG for a given torque implying shorter windows of reliability and requirements for faster pulses. Figure 4. Schematic of TbFe 2 magnetic element under biaxial stress generated by a PZT layer. Here the initial normalized magnetization 0 0 0ˆˆ cos sinm z x is predominantly out of the film plane with a cant 0arcsin[ / ]bias kHH in the x -direction provided by ˆbias bias H Hx . 37 Figure 5. a) Switching trajectories for a TbFe 2 nanomagnet under a pulsed biaxial stress = -85 MPa, pulse = 400 ps ( green ) and = -120 MPa and pulse = 300 ps (blue ) b) Switching trajectory time trace for {m x,my,mz} for = -85 MPa . The pulse is initiated at t = 500 ps. The blue region denotes when precession about biasH dominates (i.e. while the pulse is on) and the red when the dissipative dynamics rapidly damp the system down to the other equilibrium point. 38 Figure 6. a) Dependence of the simulated pulse switching probability on for = -85 MPa . b) Dependence of pulse switching probability on stress amplitude. Stress -induced switching is possible even for = 1.0. c) Comparison of pulse switching probability for LL and LLG dynamics for = -85 MPa and = 0.75. Here the difference between the LL and LLG dynamics has a significant effect on the width of the pulse window where reliable switching is predicted by the simulations ( LL = 200 ps and LLG =320 ps.) Figure 7. a) Energy (normalized to 1K ) landscape as a function of angle for various values of exchange bias energy. b) = 80º ( = 10 º) is the only stab le equilibrium for compressive ( tensi le) stress. Dissipative dynamics and the free energy landscape then dictate the non -toggle switching dynamics. c) Shows the energy barrier dependence on the [110] bias field for a d = 100 nm, filmt = 12.5 nm circular element with (curve 1) 1K = 2.5x105 ergs/cm3, (curve 2) 1K = 2.0×105 ergs/cm3, and ( curve 4) 1K =1.5×105 ergs/cm3. Curve 3 shows the energy barrier dependence for 1K=1.5x105 ergs/cm3 and d = 100 nm & filmt = 15 nm . 39 Figure 8. a) Magnetoelastic switching trajectory for Fe 81Ga19 with = -45 MPa and pulse = 3 ns. The main part of the switching occurs within 200 ps. The magnetization relaxes to the equilibrium defined when the pulse is on and then relaxes to the final equilibrium when the pulse is turned off. b) Switchin g probability phase diagram for Fe 81Ga19 with biaxial anisotropy at T = 300 ºK. c) T = 0 ºK OOMMF simulations showing the equilibrium m icromagnetic configuration for 1K = 2×105 ergs/cm3 and sM = 1300 emu/cm3. Subsequent shots show the rotational switching mode for a 45 MPa uniaxial compressive stress along [100]. Color scale is blue -white -red indicating the local projection 1xm (blue), 0xm (white), 1xm (red). 40 Figure 9. Schematic of magnetostrictive device geometry that utilizes uniaxial anisotropy to achieve deterministic switching. Polycrystalline Tb 0.3Dy 0.7Fe2 on PMN -PT with 1 axis oriented at angle with respect to the easy axis. In this geometry, M lies in the x -y plane (film -plane) with the normalized ˆˆ cos sinm x y . 41 Figure 10. a) In-plane shape anisotropy field ( kH ) and hard axis bias field ( biasH ) for a 400×900 nm2 ellipse as a function of film thickness required to ensure 0 = 51º . Thermal stability parameter plotted versus film thickness with kH , biasH such that 0 = 51º . b) Eight times the RMS angle fluctuation about three different average 0 > 45º versus film thickness for a 400×900 nm2 ellipse at T = 300 ºK. 42 Figure 11. a) Magnetization trajectories for = 45º, = -5 MPa , pulse = 3 ns, with ~ 200 Oe yielding 0 = 51º ( red) and = 45º, = -20 MPa with biasH = 120 Oe yielding 0 = 28º ( green). b) T = 300 ºK stress pulse (compressive) switching prob ability phase diagram for a 400×90 0 nm2 ellipse with filmt = 12.5 nm , = 45º, 0 = 51º c) Micromagneti c switching trajectory of a 400×90 0 nm2 ellipse under a DC compressive stress of -3 MPa transduced along 45 degrees. Color scale is blue -white -red indicating the local projection 1xm (blue), 0xm (white), 1xm (red). biasH

1610.04598v2.Nambu_mechanics_for_stochastic_magnetization_dynamics.pdf

arXiv:1610.04598v2 [cond-mat.mes-hall] 19 Jan 2017Nambu mechanics for stochastic magnetization dynamics Pascal Thibaudeaua,∗, Thomas Nusslea,b, Stam Nicolisb aCEA DAM/Le Ripault, BP 16, F-37260, Monts, FRANCE bCNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350), F´ ed´ eration de Recherche ”Denis Poisson” (FR2964), D´ epartement de Physi que, Universit´ e de Tours, Parc de Grandmont, F-37200, Tours, FRANCE Abstract The Landau-Lifshitz-Gilbert (LLG) equation describes the dynamic s of a damped magnetization vector that can be understood as a generalization o f Larmor spin precession. The LLG equation cannot be deduced from the Hamilton ian frame- work, by introducing a coupling to a usual bath, but requires the int roduction of additional constraints. It is shown that these constraints can be formulated ele- gantly and consistently in the framework of dissipative Nambu mecha nics. This has many consequences for both the variational principle and for t opological as- pects of hidden symmetries that control conserved quantities. W e particularly study how the damping terms of dissipative Nambu mechanics affect t he con- sistent interaction of magnetic systems with stochastic reservoir s and derive a master equation for the magnetization. The proposals are suppor ted by numer- ical studies using symplectic integrators that preserve the topolo gical structure of Nambu equations. These results are compared to computations performed by direct sampling of the stochastic equations and by using closure a ssumptions for the moment equations, deduced from the master equation. Keywords: Magnetization dynamics, Fokker-Planck equation, magnetic ordering ∗Corresponding author Email addresses: pascal.thibaudeau@cea.fr (Pascal Thibaudeau), thomas.nussle@cea.fr (Thomas Nussle), stam.nicolis@lmpt.univ-tours.fr (Stam Nicolis) Preprint submitted to Elsevier September 18, 20181. Introduction In micromagnetism, the transverse Landau-Lifshitz-Gilbert (LLG ) equation (1 +α2)∂si ∂t=ǫijkωj(s)sk+α(ωi(s)sjsj−ωj(s)sjsi) (1) describes the dynamics of a magnetization vector s≡M/MswithMsthe sat- uration magnetization. This equation can be seen as a generalization of Larmor spin precession, for a collection of elementary classical magnets ev olving in an effective pulsation ω=−1 ¯hδH δs=γBand within a magnetic medium, charac- terized by a damping constant αand a gyromagnetic ratio γ[1].His here identified as a scalar functional of the magnetization vector and ca n be consis- tently generalized to include spatial derivatives of the magnetizatio n vector [2] as well. Spin-transfer torques, that are, nowadays, of particula r practical rele- vance [3, 4] can be, also, taken into account in this formalism. In the following, we shall work in units where ¯ h= 1, to simplify notation. It is well known that this equation cannot be derived from a Hamiltonia n variational principle, with the damping effects described by coupling t he magne- tization to a bath, by deforming the Poisson bracket of Hamiltonian m echanics, even though the Landau–Lifshitz equation itself is Hamiltonian. The r eason is that the damping cannot be described by a “scalar” potential, but b y a “vector” potential. This has been made manifest [5] first by an analysis of the quantum ve rsion of the Landau-Lifshitz equation for damped spin motion including arb itrary spin length, magnetic anisotropy and many interacting quantum spin s. In par- ticular, this analysis has revealed that the damped spin equation of m otion is an example of metriplectic dynamical system [6], an approach which t ries to unite symplectic, nondissipative and metric, dissipative dynamics into one com- mon mathematical framework. This dissipative system has been see n afterwards nothing but a natural combination of semimetric dynamics for the dis sipative part and Poisson dynamics for the conservative ones [7]. As a conse quence, this provided a canonical description for any constrained dissipative sy stems through 2an extension of the concept of Dirac brackets developed originally f or conserva- tive constrained Hamiltonian dynamics. Then, this has culminated rec ently by observing the underlying geometrical nature of these brackets a s certain n-ary generalizations of Lie algebras, commonly encountered in conserva tive Hamilto- nian dynamics [8]. However, despite the evident progresses obtaine d, no clear direction emerges for the case of dissipative n-ary generalizations, and even no variational principle have been formulated, to date, that incorp orates such properties. What we shall show in this paper is that it is, however, possible to de- scribe the Landau–Lifshitz–Gilbert equation by using the variationa l principle of Nambu mechanics and to describe the damping effects as the resu lt of in- troducing dissipation by suitably deforming the Nambu–instead of th e Poisson– bracket. In this way we shall find, as a bonus, that it is possible to de duce the relation between longitudinal and transverse damping of the ma gnetization, when writing the appropriate master equation for the probability de nsity. To achieve this in a Hamiltonian formalism requires additional assumptions , whose provenance can, thus, be understood as the result of the prope rties of Nambu mechanics. We focus here on the essential points; a fuller account will be pro- vided in future work. Neglecting damping effects, if one sets H1≡ −ω·sandH2≡s·s/2, eq.(1) can be recast in the form ∂si ∂t={si,H1,H2}, (2) where for any functions A,B,Cofs, {A,B,C} ≡ǫijk∂A ∂si∂B ∂sj∂C ∂sk(3) is the Nambu-Poisson (NP) bracket, or Nambu bracket, or Nambu t riple bracket, a skew-symmetric object, obeying both the Leibniz rule and the Fun damental Identity [9, 10]. One can see immediately that both H1andH2are constants of motion, because of the anti-symmetric property of the bracket. This provides the generalization of Hamiltonian mechanics to phase spaces of arbitrar y dimension; 3in particular it does not need to be even. This is a way of taking into acc ount constraints and provides a natural framework for describing the magnetization dynamics, since the magnetization vector has, in general, three co mponents. The constraints–and the symmetries–can be made manifest, by no ting that it is possible to express vectors and vector fields in, at least, two wa ys, that can be understood as special cases of Hodge decomposition. For the three–dimensional case that is of interest here, this mean s that a vector field V(s) can be expressed in the “Helmholtz representation” [11] in the following way Vi≡ǫijk∂Ak ∂sj+∂Φ ∂si(4) whereAis a vector potential and Φ a scalar potential. On the other hand, this same vector field V(s) can be decomposed according to the “Monge representation” [12] Vi≡∂C1 ∂si+C2∂C3 ∂si(5) which defines the “Clebsch-Monge potentials”, Ci. If one identifies as the Clebsch–Monge potentials, C2≡H1,C3≡H2and C1≡D, Vi=∂D ∂si+H1∂H2 ∂si, (6) and the vector field V(s)≡˙s, then one immediately finds that eq. (2) takes the form ∂s ∂t={s,H1,H2}+∇sD (7) that identifies the contribution of the dissipation in this context, as the expected generalization from usual Hamiltonian mechanics. In the absence of the Gilbert term, dissipation is absent. More generally, the evolution equation for any function, F(s) can be written as [13] ∂F ∂t={F,H1,H2}+∂D ∂si∂F ∂si(8) for a dissipation function D(s). 4The equivalence between the Helmholtz and the Monge representat ion im- plies the existence of freedom of redefinition for the potentials, CiandDand Aiand Φ. This freedom expresses the symmetry under symplectic tra nsforma- tions, that can be interpreted as diffeomorphism transformations , that leave the volume invariant. These have consequences for the equations of m otion. For instance, the dissipation described by the Gilbert term in the Lan dau– Lifshitz–Gilbert equation (1) ∂D ∂si≡α(˜ωi(s)sjsj−˜ωj(s)sjsi) (9) cannot be derived from a scalar potential, since the RHS of this expr ession is not curl–free, so the function Don the LHS is not single valued; but it does conserve the norm of the magnetization, i.e. H2. Because of the Gilbert expression, bothωandηare rescaled such as ˜ω≡ω/(1 +α2) andη→η/(1 +α2). So there are two questions: (a) Whether it can lead to stochastic e ffects, that can be described in terms of deterministic chaos and/or (b) Whethe r its effects can be described by a bath of “vector potential” excitations. The fi rst case was described, in outline in ref. [14], where the role of an external to rque was shown to be instrumental; the second will be discussed in detail in the following sections. While, in both cases, a stochastic description, in terms of a probability density on the space of states is the main tool, it is much easier to pre sent for the case of a bath, than for the case of deterministic chaos, which is much more subtle. Therefore, we shall now couple our magnetic moment to a bath of flu ctuating degrees of freedom, that will be described by a stochastic proces s. 2. Nambu dynamics in a macroscopic bath To this end, one couples linearly the deterministic system such as (8) , to a stochastic process, i.e. a noise vector, random in time, labelled ηi(t), whose law of probability is given. This leads to a system of stochastic differen tial equations, that can be written in the Langevin form ∂si ∂t={si,H1,H2}+∂D ∂si+eij(s)ηj(t) (10) 5whereeij(s) can be interpreted as the vielbein on the manifold, defined by the dynamical variables, s. It should be noted that it is the vector nature of the dynamical variables that implies that the vielbein, must, also, carry in dices. We may note that the additional noise term can be used to “renorma lize” the precession frequency and, thus, mix, non-trivially, with the Gilb ert term. This means that, in the presence of either, the other cannot be ex cluded. When this vielbein is the identity matrix, eij(s) =δij, the stochastic cou- pling to the noise is additive, whereas it is multiplicative otherwise. In th at case, if the norm of the spin vector has to remain constant in time, t hen the gradient of H2must be orthogonal to the gradient of Dandeij(s)si= 0∀j. However, it is important to realize that, while the Gilbert dissipation te rm is not a gradient, the noise term, described by the vielbein is not so co nstrained. For additive noise, indeed, it is a gradient, while for the case of multiplic ative noise studied by Brown and successors there can be an interesting interference between the two terms, that is worth studying in more detail, within N ambu mechanics, to understand, better, what are the coordinate art ifacts and what are the intrinsic features thereof. Because {s(t)}, defined by the eq.(10), becomes a stochastic process, we can define an instantaneous conditional probability distribution Pη(s,t), that depends, on the noise configuration and, also, on the magnetizatio ns0at the initial time and which satisfies a continuity equation in configuration sp ace ∂Pη(s,t) ∂t+∂( ˙siPη(s,t))) ∂si= 0. (11) An equation for /an}b∇acketle{tPη/an}b∇acket∇i}htcan be formed, which becomes an average over all the possible realizations of the noise, namely ∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂t+∂/an}b∇acketle{t˙siPη/an}b∇acket∇i}ht ∂si= 0, (12) once the distribution law of {η(t)}is provided. It is important to stress here that this implies that the backreaction of the spin degrees of freed om on the bath can be neglected–which is by no means obvious. One way to chec k this is by showing that no “runaway solutions” appear. This, however, do es not ex- 6haust all possibilities, that can be found by working with the Langevin equation directly. For non–trivial vielbeine, however, this is quite involved, so it is useful to have an approximate solution in hand. To be specific, we consider a noise, described by the Ornstein-Uhlen beck process [15] of intensity ∆ and autocorrelation time τ, /an}b∇acketle{tηi(t)/an}b∇acket∇i}ht= 0 /an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht=∆ τδije−|t−t′| τ where the higher point correlation functions are deduced from Wick ’s theorem and which can be shown to become a white noise process, when τ→0. We assume that the solution to eq.(12) converges, in the sense of ave rage over-the- noise, to an equilibrium distribution, that is normalizable and, whose co rrelation functions, also, exist. While this is, of course, not at all obvious to p rove, evi- dence can be found by numerical studies, using stochastic integra tion methods that preserve the symplectic structure of the Landau–Lifshitz e quation, even under perturbations (cf. [16] for earlier work). 2.1. Additive noise Walton [17] was one of the first to consider the introduction of an ad ditive noise into an LLG equation and remarked that it may lead to a Fokker- Planck equation, without entering into details. To see this more thoroughly and to illustrate our strategy, we consider the case of additive noise, i.e. w heneij= δijin our framework. By including eq.(10) in (12) and in the limit of white noise, expressions like /an}b∇acketle{tηiPη/an}b∇acket∇i}htmust be defined and can be evaluated by either an expansion of the Shapiro-Loginov formulae of differentiation [18] an d taking the limit ofτ→0, or, directly, by applying the Furutsu-Novikov-Donsker theore m [19, 20, 21]. This leads to /an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂si. (13) 7where ˜∆≡∆/(1 +α2). Using the dampened current vector Ji≡ {si,H1,H2}+ ∂D ∂si, the (averaged) probability density /an}b∇acketle{tPη/an}b∇acket∇i}htsatisfies the following equation ∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂t+∂ ∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂2/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂si∂si= 0 (14) where˜˜∆≡∆/(1 +α2)2and which is of the Fokker-Planck form [22]. This last partial differential equation can be solved directly by several nume rical methods, including a finite-element computer code or can lead to ordinary differ ential equations for the moments of s. For example, for the average of the magnetization, one obtains th e evolution equation d/an}b∇acketle{tsi/an}b∇acket∇i}ht dt=−/integraldisplay dssi∂/an}b∇acketle{tPη(s,t)/an}b∇acket∇i}ht ∂t=/an}b∇acketle{tJi/an}b∇acket∇i}ht. (15) For the case of Landau-Lifshitz-Gilbert in a uniform precession field B, we obtain the following equations, for the first and second moments, d dt/an}b∇acketle{tsi/an}b∇acket∇i}ht=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α[˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht] (16) d dt/an}b∇acketle{tsisj/an}b∇acket∇i}ht= ˜ωl(ǫilk/an}b∇acketle{tsksj/an}b∇acket∇i}ht+ǫjlk/an}b∇acketle{tsksi/an}b∇acket∇i}ht) +α[˜ωi/an}b∇acketle{tslslsj/an}b∇acket∇i}ht + ˜ωj/an}b∇acketle{tslslsi/an}b∇acket∇i}ht−2˜ωl/an}b∇acketle{tslsisj/an}b∇acket∇i}ht] + 2˜˜∆δij (17) where ˜ω≡γB/(1 +α2). In order to close consistently these equations, one can truncate the hierarchy of moments; either on the second /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 or third cumulants /an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e. /an}b∇acketle{tsisj/an}b∇acket∇i}ht=/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht, (18) /an}b∇acketle{tsisjsk/an}b∇acket∇i}ht=/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht+/an}b∇acketle{tsisk/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht+/an}b∇acketle{tsjsk/an}b∇acket∇i}ht/an}b∇acketle{tsi/an}b∇acket∇i}ht −2/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht. (19) Because the closure of the hierarchy is related to an expansion in po wers of ∆, for practical purposes, the validity of eqs.(16,17) is limited to low v alues of the coupling to the bath (that describes the fluctuations). For example, if one sets /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, eq.(16) produces an average spin motion independent of value that ∆ may take. This is in contradiction with the numerical expe riments 8performed by the stochastic integration and noise average of eq.( 10) quoted in reference [23] and by experiments. This means that it is mandatory to keep at least eqs.(16) and (17) together in the numerical evaluation of t he thermal behavior of the dynamics of the average thermal magnetization /an}b∇acketle{ts/an}b∇acket∇i}ht. This was previously observed [24, 25] and circumvented by alternate secon d-order closure relationships, but is not supported by direct numerical experiment s. This can be illustrated by the following figure (1). For this given set of Figure 1: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, connected to an additive noise. The upper graphs (a) plot som e of the first–order moments of the averaged magnetization vector over 102realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text). Parameters of the simulations : {∆ = 0.13 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{ts1(0)s1(0)/an}bracketri}ht= 1. parameters, the agreement between the stochastic average an d the effective model is fairly decent. As expected, for a single noise realization, th e norm of the spin vector in an additive stochastic noise cannot be conserv ed during the dynamics, but, by the average-over-the-noise accumulation process, this is 9observed for very low values of ∆ and very short times. However, t his agreement with the effective equations is lost, when the temperature increase s, because of the perturbative nature of the equations (16-17). Agreement c an, however, be restored by imposing this constraint in the effective equations, for a given order in perturbation of ∆, by appropriate modifications of the hierarchic al closing relationships /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Bij(∆) or/an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Cijk(∆). It is of some interest to study the effects of the choice of initial con ditions. In particular, how the relaxation to equilibrium is affected by choosing a c omponent of the initial magnetization along the precession axis in the effective m odel, e.g. s(0) = (1/√ 2,0,−1/√ 2) and by taking all the initial correlations, /an}b∇acketle{tsi(0)sj(0)/an}b∇acket∇i}ht= 1 20−1 2 0 0 0 −1 201 2 (20) The results are shown in figure (2). Both in figures (1) and (2), it is observed that the average norm of the spin vector increases over time. This can be understood with the above arguments. In general, according to eq.(10) and because Jis a transverse vector, (1 +α2)sidsi dt=eij(s)siηj(t). (21) This equation describes how the LHS depends on the noise realization ; so the average over the noise can be found by computing the averages of the RHS. The simplest case is that of the additive vielbein, eij(s) =δij. Assuming that the average-over-the noise procedure and the time derivative commu te, we have d dt/angbracketleftbig s2/angbracketrightbig =2/an}b∇acketle{tsiηi/an}b∇acket∇i}ht 1 +α2. (22) For any Gaussian stochastic process, the Furutsu-Novikov-Don sker theorem states that /an}b∇acketle{tsi(t)ηi(t)/an}b∇acket∇i}ht=/integraldisplay+∞ −∞dt′/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht/angbracketleftbiggδsi(t) δηj(t′)/angbracketrightbigg . (23) In the most general situation, the functional derivativesδsi(t) δηj(t′)can be calculated [26], and eq.(23) admits simplifications in the white noise limit. In this limit, 10-2-1012 0 1 2 3 4 5 t (ns)-2-1012 sxsy sz(a) (b) Figure 2: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con- nected to an additive noise. The upper graphs (a) plot some of the first–order moments of the averaged magnetization vector over 103realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text). Parameters of the simulations : {∆ = 0.0655 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆ t= 10−4ns,s(0) =/an}bracketle{ts(0)/an}bracketri}ht= (1/√ 2,0,−1/√ 2),/an}bracketle{tsisj/an}bracketri}ht(0) = 0 except for (11)=1/2, (13)=(31)=-1/2, (33)=1/2 }. 11the integration is straightforward and we have /angbracketleftbig s2(t)/angbracketrightbig =s2(0) + 6˜˜∆t, (24) which is a conventional diffusion regime. It is also worth noticing that w hen computing the trace of (17), the only term which remains is indeed d dt/an}b∇acketle{tsisi/an}b∇acket∇i}ht= 6˜˜∆ (25) which allows our effective model to reproduce exactly the diffusion re gime. Fig- ure (3) compares the time evolution of the average of the square n orm spin vector. Numerical stochastic integration of eq.(10) is tested by in creasing the 0 1 2 3 4 5 t (ns)11,522,53 <|s|2>mean over 103 runs mean over 104 runs diffusion regime Figure 3: Mean square norm of the spin in the additive white no ise case for the following conditions: integration step of 10−4ns; ∆ = 0 .0655 rad.GHz; s(0) = (0 ,1,0);α= 0.1; ω= (0,0,18) rad.GHz compared to the expected diffusion regime (see te xt). size of the noise sampling and reveals a convergence to the predicte d linear diffusion regime. 122.2. Multiplicative noise Brown [27] was one of the first to propose a non–trivial vielbein, tha t takes the form eij(s) =ǫijksk/(1 +α2) for the LLG equation. We notice, first of all, that it is present, even if α= 0, i.e. in the absence of the Gilbert term. Also, that, since the determinant of this matrix [ e] is zero, this vielbein is not invertible. Because of its natural transverse character, this vie lbein preserves the norm of the spin for any realization of the noise, once a dissipation fu nctionD is chosen, that has this property. In the white-noise limit, the aver age over-the- noise continuity equation (12) cannot be transformed strictly to a Fokker-Planck form. This time /an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂ ∂sj(eji/an}b∇acketle{tPη/an}b∇acket∇i}ht), (26) which is a generalization of the additive situation shown in eq.(13). The conti- nuity equation thus becomes ∂/an}b∇acketle{tPη/an}b∇acket∇i}ht ∂t+∂ ∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂ ∂si/parenleftbigg eij∂ ∂sk(ekj/an}b∇acketle{tPη/an}b∇acket∇i}ht)/parenrightbigg = 0. (27) What deserves closer attention is, whether, in fact, this equation is invariant under diffeomeorphisms of the manifold [28] defined by the vielbein, o r whether it breaks it to a subgroup thereof. This will be presented in future w ork. In the context of magnetic thermal fluctuations, this continuity equatio n was encoun- tered several times in the literature [22, 29], but obtaining it from fir st principles is more cumbersome than our latter derivation, a remark already qu oted [18]. Moreover, our derivation presents the advantage of being easily g eneralizable to non-Markovian noise distributions [23, 30, 31], by simply keeping th e partial derivative equation on the noise with the continuity equation, and so lving them together. Consequently, the evolution equation for the average magnetizat ion is now supplemented by a term provided by a non constant vielbein and one h as d/an}b∇acketle{tsi/an}b∇acket∇i}ht dt=/an}b∇acketle{tJi/an}b∇acket∇i}ht+˜˜∆/angbracketleftbigg∂eil ∂skekl/angbracketrightbigg . (28) With the vielbein proposed by Brown and assuming a constant extern al field, 13one gets d/an}b∇acketle{tsi/an}b∇acket∇i}ht dt=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α(˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht) −2∆ (1 +α2)2/an}b∇acketle{tsi/an}b∇acket∇i}ht. (29) This equation highlights both a transverse part, coming from the av erage over the probability current Jand a longitudinal part, coming from the average over the extra vielbein term. By imposing, further, the second-or der cumulant approximation /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e. “small” fluctuations to keep the distribution of sgaussian, a single equation can be obtained, in which a longitudinal rela xation timeτL≡(1 +α2)2/2∆ may be identified. This is illustrated by the content of figure (4). In that case, the ap proxima- Figure 4: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con- nected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments of the averaged magnetization vector over 102realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param- eters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{tsx(0)sx(0)/an}bracketri}ht= 1. 14tion/an}b∇acketle{t/an}b∇acketle{tsisjsj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 has been retained in order to keep two sets of equations, three for the average magnetization components and nine on the averag e second-order moments, that have been solved simultaneously using an eight-orde r Runge- Kutta algorithm with variable time-steps. This is the same numerical im ple- mentation that has been followed for the studies of the additive nois e, solving eqs.(16) and (17) simultaneously. We have observed numerically tha t, as ex- pected, the average second-order moments are symmetrical by an exchange of their component indices, both for the multiplicative and the additive n oise. In- terestingly, by keeping identical the number of random events tak en to evaluate the average of the stochastic magnetization dynamics between th e additive and multiplicative noise, we observe a greater variance in the multiplicative case. As we have done in the additive noise case, we will also investigate briefl y the behavior of this equation under different initial conditions, and in par ticular with a non vanishing component along the z-axis. This is illustrated by the c ontent of figure (5). It is observed that for both figures (4) and (5), th e average spin converges to the same final equilibrium state, which depends ultimat ely on the value of the noise amplitude, as shown by equation (27). 3. Discussion Magnetic systems describe vector degrees of freedom, whose Ha miltonian dynamics implies constraints. These constraints can be naturally ta ken into account within Nambu mechanics, that generalizes Hamiltonian mecha nics to phase spaces of odd number of dimensions. In this framework, diss ipation can be described by gradients that are not single–valued and thus do no t define scalar baths, but vector baths, that, when coupled to external torques, can lead to chaotic dynamics. The vector baths can, also, describe non-tr ivial geometries and, in that case, as we have shown by direct numerical study, the stochastic description leads to a coupling between longitudinal and transverse relaxation. This can be, intuitively, understood within Nambu mechanics, in the fo llowing way: 15-1-0.500.51 0 1 2 3 4 5 t (ns)-1-0.500.51 sxsy sz(a) (b) Figure 5: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con- nected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments of the averaged magnetization vector over 104realizations of the noise, when the lower graphs (b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param- eters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep ∆t= 10−4ns}. Initial conditions: s(0) =/parenleftbig 1/√ 2,0,1/√ 2/parenrightbig ,/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 except for /an}bracketle{ts1(0)s1(0)/an}bracketri}ht=/an}bracketle{ts1(0)s3(0)/an}bracketri}ht=/an}bracketle{ts3(0)s3(0)/an}bracketri}ht= 1/2. 16The dynamics consists in rendering one of the Hamiltonians, H1≡ω·s, stochastic, since ωbecomes a stochastic process, as it is sensitive to the noise terms–whether these are described by Gilbert dissipation or couplin g to an external bath. Through the Nambu equations, this dependence is “transferred” toH2≡ ||s||2/2. This is one way of realizing the insights the Nambu approach provides. In practice, we may summarize our numerical results as follows: When the amplitude of the noise is small, in the context of Langevin- dynamics formalism for linear systems and for the numerical modeling ofsmall thermal fluctuations in micromagnetic systems, as for a linearized s tochastic LLG equation, the rigorous method of Lyberatos, Berkov and Cha ntrell might be thought to apply [32] and be expected to be equivalent to the app roach presented here. Because this method expresses the approach t o equilibrium of every moment, separately, however, it is restricted to the limit of s mall fluctua- tions around an equilibrium state and, as expected, cannot captur e the transient regime of average magnetization dynamics, even for low temperatu re. This is a useful check. We have also investigated the behaviour of this system under differe nt sets of initial conditions as it is well-known and has been thoroughly studied in [ 1] that in the multiplicative noise case (where the norm is constant) this syst em can show strong sensitivity to initial conditions and it is possible, using ste reographic coordinates to represent the dynamics of this system in 2D. In our additive noise case however, as the norm of the spin is not conserved, it is not eas y to get long run behavior of our system and in particular equilibrium solutions. Mor eover as we no longer have only two independent components of spin, it is not p ossible to obtain a 2D representation of our system and makes it more comp licated to study maps displaying limit cycles, attractors and so on. Thus under standing the dynamics under different initial conditions would require somethin g more and, as it is beyond the scope of this work, will be done elsewhere. Therefore, we have focused on studying the effects of the prese nce of an initial longitudinal component and of additional, diagonal, correlation s. No 17differences have been observed so far. Another issue, that deserves further study, is how the probabilit y density of the initial conditions is affected by the stochastic evolution. In th e present study we have taken the initial probability density to be a δ−function; so it will be of interest to study the evolution of other initial distributions in d etail, in particular, whether the averaging procedures commute–or not. In general, we expect that they won’t. This will be reported in future work. Finally, our study can be readily generalized since any vielbein can be ex - pressed in terms of a diagonal, symmetrical and anti-symmetrical m atrices, whose elements are functions of the dynamical variable s. Because ˙sis a pseu- dovector (and we do not consider that this additional property is a cquired by the noise vector), this suggests that the anti-symmetric part of the vielbein should be the “dominant” one. Interestingly, by numerical investigations , it appears that there are no effects, that might depend on the choice of the n oise connection for the stochastic vortex dynamics in two-dimensional easy-plane ferromagnets [33], even if it is known that for Hamiltonian dynamics, multiplicative and a d- ditive noises usually modify the dynamics quite differently, a point that also deserves further study. References [1] Giorgio Bertotti, Isaak D. Mayergoyz, and Claudio Serpico. Nonlinear Magnetization Dynamics in Nanosystems . Elsevier, April 2009. Google- Books-ID: QH4ShV3mKmkC. [2] Amikam Aharoni. Introduction to the Theory of Ferromagnetism . Claren- don Press, 2000. [3] Jacques Miltat, Gon¸ calo Albuquerque, Andr´ e Thiaville, and Caro le Vouille. Spin transfer into an inhomogeneous magnetization distribution. Journal of Applied Physics , 89(11):6982, 2001. [4] Dmitry V. Berkov and Jacques Miltat. Spin-torque driven magnet ization 18dynamics: Micromagnetic modeling. Journal of Magnetism and Magnetic Materials , 320(7):1238–1259, April 2008. [5] Janusz A. Holyst and Lukasz A. Turski. Dissipative dynamics of qu antum spin systems. Physical Review A , 45(9):6180–6184, May 1992. [6] /suppress Lukasz A. Turski. Dissipative quantum mechanics. Metriplectic d ynamics in action. In Zygmunt Petru, Jerzy Przystawa, and Krzysztof Ra pcewicz, editors,From Quantum Mechanics to Technology , number 477 in Lecture Notes in Physics, pages 347–357. Springer Berlin Heidelberg, 1996. [7] Sonnet Q. H. Nguyen and /suppress Lukasz A. Turski. On the Dirac approa ch to constrained dissipative dynamics. Journal of Physics A: Mathematical and General , 34(43):9281–9302, November 2001. [8] Josi A. de Azc´ arraga and Josi M. Izquierdo. N-ary algebras: A review with applications. Journal of Physics A: Mathematical and Theoretical , 43(29):293001, July 2010. [9] Yoichiro Nambu. Generalized Hamiltonian Dynamics. Physical Review D , 7(8):2405–2412, April 1973. [10] Ra´ ul Ib´ a˜ nez, Manuel de Le´ on, Juan C. Marrero, and Dav id Martı´ n de Diego. Dynamics of generalized Poisson and Nambu–Poisson bracket s. Journal of Mathematical Physics , 38(5):2332, 1997. [11] Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and Symmetry , volume 17 of Texts in Applied Mathematics . Springer New York, New York, NY, 1999. [12] Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry . John Wiley & Sons, August 2014. [13] Minos Axenides and Emmanuel Floratos. Strange attractors in dissipative Nambu mechanics: Classical and quantum aspects. Journal of High Energy Physics , 2010(4), April 2010. 19[14] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Quantum Mag- nets and Matrix Lorenz Systems. Journal of Physics: Conference Series , 574(1):012146, 2015. [15] George Eugene Uhlenbeck and Leonard S. Ornstein. On the The ory of the Brownian Motion. Physical Review , 36(5):823–841, September 1930. [16] Pascal Thibaudeau and David Beaujouan. Thermostatting the atomic spin dynamics from controlled demons. Physica A: Statistical Mechanics and its Applications , 391(5):1963–1971, March 2012. [17] Derek Walton. Rate of transition for single domain particles. Journal of Magnetism and Magnetic Materials , 62(2-3):392–396, December 1986. [18] V. E. Shapiro and V. M. Loginov. “Formulae of differentiation” an d their use for solving stochastic equations. Physica A: Statistical Mechanics and its Applications , 91(3-4):563–574, May 1978. [19] Koichi Furutsu. On the statistical theory of electromagnetic waves in a fluctuating medium (I). Journal of Research of the National Bureau of Standards , 67D:303–323, May 1963. [20] Evgenii A. Novikov. Functionals and the Random-force Method in Tur- bulence Theory. Soviet Physics Journal of Experimental and Theoretical Physics , 20(5):1290–1294, May 1964. [21] Valery I. Klyatskin. Stochastic Equations through the Eye of the Physicist: Basic Concepts, Exact Results and Asymptotic Approximatio ns. Elsevier, Amsterdam, 1 edition, 2005. OCLC: 255242261. [22] Hannes Risken. The Fokker-Planck Equation , volume 18 of Springer Series in Synergetics . Springer-Verlag, Berlin, Heidelberg, 1989. [23] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Closing th e hier- archy for non-Markovian magnetization dynamics. Physica B: Condensed Matter , 486:57–59, April 2016. 20[24] Dmitry A. Garanin. Fokker-Planck and Landau-Lifshitz-Bloch e quations for classical ferromagnets. Physical Review B , 55(5):3050–3057, February 1997. [25] Pui-Wai Ma and Sergei L. Dudarev. Langevin spin dynamics. Physical Review B , 83:134418, April 2011. [26] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. A functio nal calculus for the magnetization dynamics. arXiv:1606.02137 [cond-mat, physics:nlin, physics:physics] , June 2016. [27] William Fuller Brown. Thermal Fluctuations of a Single-Domain Partic le. Physical Review , 130(5):1677–1686, June 1963. [28] Jean Zinn-Justin. QuantumField Theory and Critical Phenomena . Number 113 in International series of monographs on physics. Clarendon P ress, Oxford, 4. ed., reprinted edition, 2011. OCLC: 767915024. [29] Jos´ e Luis Garc´ ıa-Palacios and Francisco J. L´ azaro. Langev in-dynamics study of the dynamical properties of small magnetic particles. Physical Review B , 58(22):14937–14958, December 1998. [30] Pascal Thibaudeau, Julien Tranchida, and Stam Nicolis. Non-Mar kovian Magnetization Dynamics for Uniaxial Nanomagnets. IEEE Transactions on Magnetics , 52(7):1–4, July 2016. [31] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Colored- noise magnetization dynamics: From weakly to strongly correlated noise. IEEE Transactions on Magnetics , 52(7):1300504, 2016. [32] Andreas Lyberatos, Dmitry V. Berkov, and Roy W. Chantrell. A method for the numerical simulation of the thermal magnetization fluctuat ions in micromagnetics. Journal of Physics: Condensed Matter , 5(47):8911–8920, November 1993. 21[33] Till Kamppeter, Franz G. Mertens, Esteban Moro, Angel S´ an chez, and A. R. Bishop. Stochastic vortex dynamics in two-dimensional easy- plane ferromagnets: Multiplicative versus additive noise. Physical Review B , 59(17):11349–11357, May 1999. 22

1708.02008v2.Chiral_damping__chiral_gyromagnetism_and_current_induced_torques_in_textured_one_dimensional_Rashba_ferromagnets.pdf

arXiv:1708.02008v2 [cond-mat.mes-hall] 31 Aug 2017Chiral damping, chiral gyromagnetism and current-induced torques in textured one-dimensional Rashba ferromagnets Frank Freimuth,∗Stefan Bl¨ ugel, and Yuriy Mokrousov Peter Gr¨ unberg Institut and Institute for Advanced Simula tion, Forschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y (Dated: May 17, 2018) We investigate Gilbert damping, spectroscopic gyromagnet ic ratio and current-induced torques in the one-dimensional Rashba model with an additional nonc ollinear magnetic exchange field. We find that the Gilbert damping differs between left-handed and right-handed N´ eel-type magnetic domain walls due to the combination of spatial inversion asy mmetry and spin-orbit interaction (SOI), consistent with recent experimental observations o f chiral damping. Additionally, we find that also the spectroscopic gfactor differs between left-handed and right-handed N´ eel- type domain walls, which we call chiral gyromagnetism. We also investig ate the gyromagnetic ratio in the Rashba model with collinear magnetization, where we find that scatt ering corrections to the gfactor vanish for zero SOI, become important for finite spin-orbit couplin g, and tend to stabilize the gyromagnetic ratio close to its nonrelativistic value. I. INTRODUCTION In magnetic bilayer systems with structural inversion asymmetry the energies of left-handed and right-handed N´ eel-type domain walls differ due to the Dzyaloshinskii- Moriya interaction (DMI) [1–4]. DMI is a chiral interac- tion, i.e., it distinguishes between left-handed and right- handed spin-spirals. Not only the energy is sensitive to the chirality of spin-spirals. Recently, it has been re- ported that the orbital magnetic moments differ as well between left-handed and right-handed cycloidal spin spi- rals in magnetic bilayers [5, 6]. Moreover, the experi- mental observation of asymmetry in the velocity of do- main walls driven by magnetic fields suggests that also the Gilbert damping is sensitive to chirality [7, 8]. In this work we show that additionally the spectro- scopic gyromagnetic ratio γis sensitive to the chirality of spin-spirals. The spectroscopic gyromagnetic ratio γ can be defined by the equation dm dt=γT, (1) whereTis the torque that acts on the magnetic moment mand dm/dtis the resulting rate of change. γenters the Landau-Lifshitz-Gilbert equation (LLG): dˆM dt=γˆM×Heff+αGˆM×dˆM dt,(2) whereˆMis a normalized vector that points in the direc- tionofthemagnetizationandthetensor αGdescribesthe Gilbert damping. The chiralityofthe gyromagneticratio provides another mechanism for asymmetries in domain- wall motion between left-handed and right-handed do- main walls. Not only the damping and the gyromagnetic ratio exhibit chiral corrections in inversion asymmetric sys- tems but also the current-induced torques. Amongthese torques that act on domain-walls are the adia- batic and nonadiabatic spin-transfer torques [9–12] and the spin-orbit torque [13–16]. Based on phenomenologi- cal grounds additional types of torques have been sug- gested [17]. Since this large number of contributions are difficult to disentangle experimentally, current-driven domain-wall motion in inversion asymmetric systems is not yet fully understood. The two-dimensionalRashbamodel with an additional exchange splitting has been used to study spintronics effects associated with the interfaces in magnetic bi- layer systems [18–22]. Recently, interest in the role of DMI in one-dimensional magnetic chains has been trig- gered [23, 24]. For example, the magnetic moments in bi-atomic Fe chains on the Ir surface order in a 120◦ spin-spiral state due to DMI [25]. Apart from DMI, also other chiral effects, such as chiral damping and chiral gyromagnetism, are expected to be important in one- dimensional magnetic chains on heavy metal substrates. The one-dimensional Rashba model [26, 27] with an ad- ditional exchange splitting can be used to simulate spin- orbit driven effects in one-dimensional magnetic wires on substrates [28–30]. While the generalized Bloch theo- rem[31]usuallycannotbeusedtotreatspin-spiralswhen SOI is included in the calculation, the one-dimensional Rashba model has the advantage that it can be solved with the help of the generalized Bloch theorem, or with a gauge-field approach [32], when the spin-spiral is of N´ eel- type. WhenthegeneralizedBlochtheoremcannotbeem- ployed one needs to resort to a supercell approach [33], use open boundary conditions [34, 35], or apply pertur- bation theory [6, 9, 36–39] in order to study spintronics effects in noncollinear magnets with SOI. In the case of the one-dimensional Rashba model the DMI and the ex- changeparameterswerecalculatedbothdirectlybasedon agauge-fieldapproachandfromperturbationtheory[38]. The results from the two approaches were found to be in perfect agreement. Thus, the one-dimensional Rashba2 model provides also an excellent opportunity to verify expressions obtained from perturbation theory by com- parisonto the resultsfromthe generalizedBlochtheorem or from the gauge-field approach. In this work we study chiral gyromagnetism and chi- ral damping in the one-dimensional Rashba model with an additional noncollinear magnetic exchange field. The one-dimensional Rashba model is very well suited to study these SOI-driven chiral spintronics effects, because it can be solved in a very transparent way without the need for a supercell approach, open boundary conditions or perturbation theory. We describe scattering effects by the Gaussian scalar disorder model. To investigate the role of disorder for the gyromagnetic ratio in general, we studyγalso in the two-dimensional Rashba model with collinear magnetization. Additionally, we compute the current-induced torques in the one-dimensional Rashba model. This paper is structured as follows: In section IIA we introduce the one-dimensional Rashba model. In sec- tion IIB we discuss the formalism for the calculation of the Gilbert damping and of the gyromagnetic ratio. In section IIC we present the formalism used to calcu- late the current-induced torques. In sections IIIA, IIIB, and IIIC we discuss the gyromagnetic ratio, the Gilbert damping, and the current-induced torques in the one- dimensionalRashbamodel, respectively. Thispaperends with a summary in section IV. II. FORMALISM A. One-dimensional Rashba model The two-dimensional Rashba model is given by the Hamiltonian [19] H=−/planckover2pi12 2me∂2 ∂x2−/planckover2pi12 2me∂2 ∂y2+ +iαRσy∂ ∂x−iαRσx∂ ∂y+∆V 2σ·ˆM(r),(3) where the first line describes the kinetic energy, the first twotermsin thesecondline describethe RashbaSOI and the last term in the second line describes the exchange splitting. ˆM(r) is the magnetization direction, which may depend on the position r= (x,y), andσis the vector of Pauli spin matrices. By removing the terms with the y-derivatives from Eq. (3), i.e., −/planckover2pi12 2me∂2 ∂y2and −iαRσx∂ ∂y, one obtains a one-dimensional variant of the Rashba model with the Hamiltonian [38] H=−/planckover2pi12 2me∂2 ∂x2+iαRσy∂ ∂x+∆V 2σ·ˆM(x).(4) Eq. (4) is invariant under the simultaneous rotation ofσand of the magnetization ˆMaround the yaxis.Therefore, if ˆM(x) describes a flat cycloidal spin-spiral propagating into the xdirection, as given by ˆM(x) = sin(qx) 0 cos(qx) , (5) we can use the unitary transformation U(x) =/parenleftBigg cos(qx 2)−sin(qx 2) sin(qx 2) cos(qx 2)/parenrightBigg (6) in order to transform Eq. (4) into a position-independent effective Hamiltonian [38]: H=1 2m/parenleftbig px+eAeff x/parenrightbig2−m(αR)2 2/planckover2pi12+∆V 2σz,(7) wherepx=−i/planckover2pi1∂/∂xis thexcomponent of the momen- tum operator and Aeff x=−m e/planckover2pi1/parenleftbigg αR+/planckover2pi12 2mq/parenrightbigg σy (8) is thex-component of the effective magnetic vector po- tential. Eq. (8) shows that the noncollinearity described byqacts like an effective SOI in the special case of the one-dimensional Rashba model. This suggests to intro- duce the concept of effective SOI strength αR eff=αR+/planckover2pi12 2mq. (9) Based on this concept of the effective SOI strength one can obtain the q-dependence of the one-dimensional Rashba model from its αR-dependence at q= 0. That a noncollinear magnetic texture provides a nonrelativistic effective SOI has been found also in the context of the intrinsic contribution to the nonadiabatic torque in the absence of relativistic SOI, which can be interpreted as a spin-orbit torque arising from this effective SOI [40]. While the Hamiltonian in Eq. (4) depends on position xthrough the position-dependence of the magnetization ˆM(x) in Eq. (5), the effective Hamiltonian in Eq. (7) is not dependent on xand therefore easy to diagonalize. B. Gilbert damping and gyromagnetic ratio In collinear magnets damping and gyromagnetic ratio can be extracted from the tensor [16] Λij=−1 Vlim ω→0ImGR Ti,Tj(/planckover2pi1ω) /planckover2pi1ω, (10) whereVis the volume of the unit cell and GR Ti,Tj(/planckover2pi1ω) =−i∞/integraldisplay 0dteiωt/angbracketleft[Ti(t),Tj(0)]−/angbracketright(11)3 is the retarded torque-torque correlation function. Tiis thei-th component of the torque operator [16]. The dc- limitω→0 in Eq. (10) is only justified when the fre- quency of the magnetization dynamics, e.g., the ferro- magnetic resonance frequency, is smaller than the relax- ationrateoftheelectronicstates. In thin magneticlayers and monoatomicchains on substratesthis is typically the case due to the strong interfacial disorder. However, in very pure crystalline samples at low temperatures the relaxation rate may be smaller than the ferromagnetic resonance frequency and one needs to assume ω >0 in Eq. (10) [41, 42]. The tensor Λdepends on the mag- netization direction ˆMand we decompose it into the tensorS, which is even under magnetization reversal (S(ˆM) =S(−ˆM)), and the tensor A, which is odd un- der magnetization reversal ( A(ˆM) =−A(−ˆM)), such thatΛ=S+A, where Sij(ˆM) =1 2/bracketleftBig Λij(ˆM)+Λij(−ˆM)/bracketrightBig (12) and Aij(ˆM) =1 2/bracketleftBig Λij(ˆM)−Λij(−ˆM)/bracketrightBig .(13) One can show that Sis symmetric, i.e., Sij(ˆM) = Sji(ˆM), while Ais antisymmetric, i.e., Aij(ˆM) = −Aji(ˆM). The Gilbert damping may be extracted from the sym- metric component Sas follows [16]: αG ij=|γ|Sij Mµ0, (14) whereMis the magnetization. The gyromagnetic ratio γis obtained from Λ according to the equation [16] 1 γ=1 2µ0M/summationdisplay ijkǫijkΛijˆMk=1 2µ0M/summationdisplay ijkǫijkAijˆMk. (15) It is convenient to discuss the gyromagnetic ratio in terms of the dimensionless g-factor, which is related to γthrough γ=gµ0µB//planckover2pi1. Consequently, the g-factor is given by 1 g=µB 2/planckover2pi1M/summationdisplay ijkǫijkΛijˆMk=µB 2/planckover2pi1M/summationdisplay ijkǫijkAijˆMk.(16) Due to the presence of the Levi-Civita tensor ǫijkin Eq. (15) and in Eq. (16) the gyromagnetic ratio and the g-factoraredetermined solelyby the antisymmetriccom- ponentAofΛ. Various different conventions are used in the literature concerning the sign of the g-factor [43]. Here, we define the sign of the g-factor such that γ >0 forg >0 and γ <0 forg <0. According to Eq. (1) the rate of change ofthemagneticmomentisthereforeparalleltothetorqueforpositive gandantiparalleltothetorquefornegative g. While we are interested in this work in the spectroscopic g-factor, and hence in the relation between the rate of change of the magnetic moment and the torque, Ref. [43] discusses the relation between the magnetic moment m andtheangularmomentum Lthatgeneratesit, i.e., m= γstaticL. Since differentiation with respect to time and use ofT= dL/dtleads to Eq. (1) our definition of the signs ofgandγagrees essentially with the one suggested in Ref. [43], which proposes to use a positive gwhen the magnetic moment is parallel to the angular momentum generatingitandanegative gwhenthemagneticmoment is antiparallel to the angular momentum generating it. Combining Eq. (14) and Eq. (15) we can express the Gilbert damping in terms of AandSas follows: αG xx=Sxx |Axy|. (17) IntheindependentparticleapproximationEq.(10)can be written as Λij= ΛI(a) ij+ΛI(b) ij+ΛII ij, where ΛI(a) ij=1 h/integraldisplayddk (2π)dTr/angbracketleftbig TiGR k(EF)TjGA k(EF)/angbracketrightbig ΛI(b) ij=−1 h/integraldisplayddk (2π)dReTr/angbracketleftbig TiGR k(EF)TjGR k(EF)/angbracketrightbig ΛII ij=1 h/integraldisplayddk (2π)d/integraldisplayEF −∞dEReTr/angbracketleftbigg TiGR k(E)TjdGR k(E) dE − TidGR k(E) dETjGR k(E)/angbracketrightbigg .(18) Here,dis the dimension ( d= 1 ord= 2 ord= 3),GR k(E) is the retarded Green’s function and GA k(E) = [GR k(E)]†. EFis the Fermi energy. ΛI(b) ijis symmetric under the interchange of the indices iandjwhile ΛII ijis antisym- metric. The term ΛI(a) ijcontains both symmetric and antisymmetric components. Since the Gilbert damping tensor is symmetric, both ΛI(b) ijand ΛI(a) ijcontribute to it. Since the gyromagnetic tensor is antisymmetric, both ΛII ijand ΛI(a) ijcontribute to it. In order to account for disorder we use the Gaus- sian scalardisordermodel, wherethe scatteringpotential V(r) satisfies /angbracketleftV(r)/angbracketright= 0 and /angbracketleftV(r)V(r′)/angbracketright=Uδ(r−r′). This model is frequently used to calculate transport properties in disordered multiband model systems [44], but it has also been combined with ab-initio electronic structure calculations to study the anomalous Hall ef- fect [45, 46] and the anomalous Nernst effect [47] in tran- sition metals and their alloys. In the clean limit, i.e., in the limit U→0, the an- tisymmetric contribution to Eq. (18) can be written as4 Aij=Aint ij+Ascatt ij, where the intrinsic part is given by Aint ij=/planckover2pi1/integraldisplayddk (2π)d/summationdisplay n,m[fkn−fkm]ImTi knmTj kmn (Ekn−Ekm)2 = 2/planckover2pi1/integraldisplayddk (2π)d/summationdisplay n/summationdisplay ll′fknIm/bracketleftbigg∂/angbracketleftukn| ∂ˆMl∂|ukn/angbracketright ∂ˆMl′/bracketrightbigg × ×/summationdisplay mm′ǫilmǫjl′m′ˆMmˆMm′. (19) The second line in Eq. (19) expresses Aint ijin terms of the Berry curvature in magnetization space [48]. The scattering contribution is given by Ascatt ij=/planckover2pi1/summationdisplay nm/integraldisplayddk (2π)dδ(EF−Ekn)Im/braceleftBigg −/bracketleftbigg Mi knmγkmn γknnTj knn−Mj knmγkmn γknnTi knn/bracketrightbigg +/bracketleftBig Mi kmn˜Tj knm−Mj kmn˜Ti knm/bracketrightBig −/bracketleftbigg Mi knmγkmn γknn˜Tj knn−Mj knmγkmn γknn˜Ti knn/bracketrightbigg +/bracketleftBigg ˜Ti knnγknm γknn˜Tj kmn Ekn−Ekm−˜Tj knnγknm γknn˜Ti kmn Ekn−Ekm/bracketrightBigg +1 2/bracketleftbigg ˜Ti knm1 Ekn−Ekm˜Tj kmn−˜Tj knm1 Ekn−Ekm˜Ti kmn/bracketrightbigg +/bracketleftBig Tj knnγknm γknn1 Ekn−Ekm˜Ti kmn −Ti knnγknm γknn1 Ekn−Ekm˜Tj kmn/bracketrightBig/bracerightBigg . (20) Here,Ti knm=/angbracketleftukn|Ti|ukm/angbracketrightare the matrix elements of the torque operator. ˜Ti knmdenotes the vertex corrections of the torque, which solve the equation ˜Ti knm=/summationdisplay p/integraldisplaydnk′ (2π)n−1δ(EF−Ek′p) 2γk′pp× ×/angbracketleftukn|uk′p/angbracketright/bracketleftBig ˜Ti k′pp+Ti k′pp/bracketrightBig /angbracketleftuk′p|ukm/angbracketright.(21) The matrix γknmis given by γknm=−π/summationdisplay p/integraldisplayddk′ (2π)dδ(EF−Ek′p)/angbracketleftukn|uk′p/angbracketright/angbracketleftuk′p|ukm/angbracketright (22) and the Berry connection in magnetization space is de- fined as iMj knm=iTj knm Ekm−Ekn. (23) The scattering contribution Eq. (20) formally resembles the side-jump contribution to the AHE [44] as obtainedfrom the scalar disorder model: It can be obtained by replacing the velocity operators in Ref. [44] by torque operators. We find thatin collinearmagnetswithoutSOI this scattering contribution vanishes. The gyromagnetic ratio is then given purely by the intrinsic contribution Eq. (19). This is an interesting difference to the AHE: Without SOI all contributions to the AHE are zero in collinear magnets, while both the intrinsic and the side- jump contributions are generally nonzero in the presence of SOI. In the absence of SOI Eq. (19) can be expressed in terms of the magnetization [48]: Aint ij=−/planckover2pi1 2µB/summationdisplay kǫijkMk. (24) Inserting Eq. (24) into Eq. (16) yields g=−2, i.e., the expected nonrelativistic value of the g-factor. Theg-factor in the presence of SOI is usually assumed to be given by [49] g=−2Mspin+Morb Mspin=−2M Mspin,(25) whereMorbis the orbital magnetization, Mspinis the spin magnetization and M=Morb+Mspinis the total magnetization. The g-factor obtained from Eq. (25) is usually in good agreementwith experimental results [50]. When SOI is absent, the orbital magnetization is zero, Morb= 0, and consequently Eq. (25) yields g=−2 in that case. Eq. (16) can be rewritten as 1 g=Mspin MµB 2/planckover2pi1Mspin/summationdisplay ijkǫijkAijˆMk=Mspin M1 g1,(26) with 1 g1=µB 2/planckover2pi1Mspin/summationdisplay ijkǫijkAijˆMk. (27) From the comparison of Eq. (26) with Eq. (25) it follows that Eq. (25) holds exactly if g1=−2 is satisfied. How- ever, Eq. (27) usually yields g1=−2 only in collinear magnets when SOI is absent, otherwise g1/negationslash=−2. In the one-dimensionalRashbamodel the orbitalmagnetization is zero,Morb= 0, and consequently 1 g=µB 2/planckover2pi1Mspin/summationdisplay ijkǫijkAijˆMk. (28) The symmetric contribution can be written as Sij= Sint ij+SRR−vert ij+SRA−vert ij, where Sint ij=1 h/integraldisplayddk (2π)dTr/braceleftbig TiGR k(EF)Tj/bracketleftbig GA k(EF)−GR k(EF)/bracketrightbig/bracerightbig (29)5 and SRR−vert ij=−1 h/integraldisplayddk (2π)dTr/braceleftBig ˜TRR iGR k(EF)TjGR k(EF)/bracerightBig (30) and SAR−vert ij=1 h/integraldisplayddk (2π)dTr/braceleftBig ˜TAR iGR k(EF)TjGA k(EF)/bracerightBig , (31) whereGR k(EF) =/planckover2pi1[EF−Hk−ΣR k(EF)]−1is the retarded Green’s function, GA k(EF) =/bracketleftbig GR k(EF)/bracketrightbig†is the advanced Green’s function and ΣR(EF) =U /planckover2pi1/integraldisplayddk (2π)dGR k(EF) (32) is the retarded self-energy. The vertex corrections are determined by the equations ˜TAR=T+U /planckover2pi12/integraldisplayddk (2π)dGA k(EF)˜TAR kGR k(EF) (33) and ˜TRR=T+U /planckover2pi12/integraldisplayddk (2π)dGR k(EF)˜TRR kGR k(EF).(34) In contrast to the antisymmetric tensor A, which be- comes independent of the scattering strength Ufor suf- ficiently small U, i.e., in the clean limit, the symmetric tensorSdepends strongly on Uin metallic systems in the clean limit. Sint ijandSscatt ijdepend therefore on U through the self-energy and through the vertex correc- tions. In the case of the one-dimensional Rashba model, the equations Eq. (19) and Eq. (20) for the antisymmet- ric tensor Aand the equations Eq. (29), Eq. (30) and Eq. (31) for the symmetric tensor Scan be used both for the collinear magnetic state as well as for the spin- spiral of Eq. (5). To obtain the g-factor for the collinear magnetic state, we plug the eigenstates and eigenvalues of Eq. (4) (with ˆM=ˆez) into Eq. (19) and into Eq. (20). In the case of the spin-spiral of Eq. (5) we use instead the eigenstates and eigenvalues of Eq. (7). Similarly, to ob- tain the Gilbert damping in the collinear magnetic state, we evaluate Eq. (29), Eq. (30) and Eq. (31) based on the Hamiltonian in Eq. (4) and for the spin-spiral we use instead the effective Hamiltonian in Eq. (7). C. Current-induced torques The current-induced torque on the magnetization can be expressed in terms of the torkance tensor tijas [15] Ti=/summationdisplay jtijEj, (35)whereEjis thej-th component of the applied elec- tric field and Tiis thei-th component of the torque per volume [51]. tijis the sum of three terms, tij= tI(a) ij+tI(b) ij+tII ij, where [15] tI(a) ij=e h/integraldisplayddk (2π)dTr/angbracketleftbig TiGR k(EF)vjGA k(EF)/angbracketrightbig tI(b) ij=−e h/integraldisplayddk (2π)dReTr/angbracketleftbig TiGR k(EF)vjGR k(EF)/angbracketrightbig tII ij=e h/integraldisplayddk (2π)d/integraldisplayEF −∞dEReTr/angbracketleftbigg TiGR k(E)vjdGR k(E) dE − TidGR k(E) dEvjGR k(E)/angbracketrightbigg .(36) We decompose the torkance into two parts that are, respectively, even and odd with respect to magnetiza- tion reversal, i.e., te ij(ˆM) = [tij(ˆM) +tij(−ˆM)]/2 and to ij(ˆM) = [tij(ˆM)−tij(−ˆM)]/2. In the clean limit, i.e., for U→0, the even torkance can be written as te ij=te,int ij+te,scatt ij, where [15] te,int ij= 2e/planckover2pi1/integraldisplayddk (2π)d/summationdisplay n/negationslash=mfknImTi knmvj kmn (Ekn−Ekm)2(37) is the intrinsic contribution and te,scatt ij=e/planckover2pi1/summationdisplay nm/integraldisplayddk (2π)dδ(EF−Ekn)Im/braceleftBigg /bracketleftBig −Mi knmγkmn γknnvj knn+Aj knmγkmn γknnTi knn/bracketrightBig +/bracketleftBig Mi kmn˜vj knm−Aj kmn˜Ti knm/bracketrightBig −/bracketleftBig Mi knmγkmn γknn˜vj knn−Aj knmγkmn γknn˜Ti knn/bracketrightBig +/bracketleftBig ˜vj kmnγknm γknn˜Ti nn Ekn−Ekm−˜Ti kmnγknm γknn˜vj knn Ekn−Ekm/bracketrightBig +1 2/bracketleftBig ˜vj knm1 Ekn−Ekm˜Ti kmn−˜Ti knm1 Ekn−Ekm˜vj kmn/bracketrightBig +/bracketleftBig vj knnγknm γknn1 Ekn−Ekm˜Ti kmn −Ti knnγknm γknn1 Ekn−Ekm˜vj kmn/bracketrightBig/bracerightBigg . (38) is the scattering contribution. Here, iAj knm=ivj knm Ekm−Ekn=i /planckover2pi1/angbracketleftukn|∂ ∂kj|ukm/angbracketright(39) is the Berry connection in kspace and the vertex correc- tions of the velocity operator solve the equation ˜vi knm=/summationdisplay p/integraldisplaydnk′ (2π)n−1δ(EF−Ek′p) 2γk′pp× ×/angbracketleftukn|uk′p/angbracketright/bracketleftbig ˜vi k′pp+vi k′pp/bracketrightbig /angbracketleftuk′p|ukm/angbracketright.(40)6 The odd contribution can be written as to ij=to,int ij+ tRR−vert ij+tAR−vert ij, where to,int ij=e h/integraldisplayddk (2π)dTr/braceleftbig TiGR k(EF)vj/bracketleftbig GA k(EF)−GR k(EF)/bracketrightbig/bracerightbig (41) and tRR−vert ij=−e h/integraldisplayddk (2π)dTr/braceleftBig ˜TRR iGR k(EF)vjGR k(EF)/bracerightBig (42) and tAR−vert ij=e h/integraldisplayddk (2π)dTr/braceleftBig ˜TAR iGR k(EF)vjGA k(EF)/bracerightBig .(43) The vertex corrections ˜TAR iand˜TRR iof the torque op- erator are given in Eq. (33) and in Eq. (34), respectively. While the even torkance, Eq. (37) and Eq. (38), be- comes independent of the scattering strength Uin the clean limit, i.e., for U→0, the odd torkance to ijdepends strongly on Uin metallic systems in the clean limit [15]. In the case of the one-dimensional Rashba model, the equations Eq. (37) and Eq. (38) for the even torkance te ijand the equations Eq. (41), Eq. (42) and Eq. (43) for the odd torkance to ijcan be used both for the collinear magnetic state as well as for the spin-spiral of Eq. (5). To obtain the even torkance for the collinear magnetic state, we plug the eigenstates and eigenvalues of Eq. (4) (withˆM=ˆez) into Eq. (37) and into Eq. (38). In the case of the spin-spiral of Eq. (5) we use instead the eigen- states and eigenvalues of Eq. (7). Similarly, to obtain the odd torkance in the collinear magnetic state, we evaluate Eq. (41), Eq. (42) and Eq. (43) based on the Hamilto- nian in Eq. (4) and for the spin-spiral we use instead the effective Hamiltonian in Eq. (7). III. RESULTS A. Gyromagnetic ratio We first discuss the g-factor in the collinear case, i.e., whenˆM(r) =ˆez. Inthis casetheenergybandsaregiven by E=/planckover2pi12k2 x 2m±/radicalbigg 1 4(∆V)2+(αRkx)2.(44) When ∆ V/negationslash= 0 orαR/negationslash= 0 the energy Ecan become negative. The band structure of the one-dimensional Rashba model is shown in Fig. 1 for the model param- etersαR=2eV˚A and ∆ V= 0.5eV. For this choice of parameters the energy minima are not located at kx= 0 but instead at kmin x=±/radicalBig (αR)4m2−1 4/planckover2pi14(∆V)2 /planckover2pi12αR,(45)-0.4 -0.2 0 0.2 0.4 k-Point kx [Å-1]00.511.5Band energy [eV] FIG. 1: Band structure of theone-dimensional Rashbamodel. and the corresponding minimum of the energy is given by Emin=−m(αR)4+1 4/planckover2pi14 m(∆V)2 2/planckover2pi12(αR)2. (46) The inverse g-factor is shown as a function of the SOI strength αRin Fig. 2 for the exchange splitting ∆ V= 1eV and Fermi energy EF= 1.36eV. At αR= 0 the scattering contribution is zero, i.e., the g-factor is de- termined completely by the intrinsic Berry curvature ex- pression, Eq. (24). Thus, at αR= 0 it assumes the value 1/g=−0.5, which is the expected nonrelativistic value (see the discussion below Eq. (24)). With increasing SOI strength αRthe intrinsic contribution to 1 /gis more and more suppressed. However, the scattering contribution compensates this decrease such that the total 1 /gis close to its nonrelativistic value of −0.5. The neglect of the scattering corrections at large values of αRwould lead in this case to a strong underestimation of the magnitude of 1/g, i.e., a strong overestimation of the magnitude of g. However, at smaller values of the Fermi energy, the gfactor can deviate substantially from its nonrelativis- tic value of −2. To show this we plot in Fig. 3 the in- verseg-factor as a function of the Fermi energy when the exchange splitting and the SOI strength are set to ∆V= 1eV and αR=2eV˚A, respectively. As discussed in Eq. (44) the minimal Fermi energyis negativ in this case. The intrinsic contribution to 1 /gdeclines with increas- ing Fermi energy. At large values of the Fermi energy this decline is compensated by the increase of the vertex corrections and the total value of 1 /gis close to −0.5. Previous theoretical works on the g-factor have not considered the scattering contribution [52]. It is there- fore important to find out whether the compensation of the decrease of the intrinsic contribution by the in-7 00.511.52 SOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/gscattering intrinsic total FIG. 2: Inverse g-factor vs. SOI strength αRin the one- dimensional Rashba model. 0 1 2 3 4 5 6 Fermi energy [eV]-0.6-0.4-0.201/gscattering intrinsic total FIG. 3: Inverse g-factor vs. Fermi energy in the one- dimensional Rashba model. crease of the extrinsic contribution as discussed in Fig. 2 and Fig. 3 is peculiar to the one-dimensional Rashba model or whether it can be found in more general cases. For this reason we evaluate g1for the two-dimensional Rashba model. In Fig. 4 we show the inverse g1-factor in the two-dimensional Rashba model as a function of SOI strength αRfor the exchange splitting ∆ V= 1eV and the Fermi energy EF= 1.36eV. Indeed for αR< 0.5eV˚A the scattering corrections tend to stabilize g1at its non-relativistic value. However, in contrast to the one-dimensional case (Fig. 2), where gdoes not deviate much from its nonrelativistic value up to αR= 2eV˚A, g1starts to be affected by SOI at smaller values of αR in the two-dimensional case. According to Eq. (26) the fullgfactor is given by g=g1(1+Morb/Mspin). There- fore, when the scattering corrections stabilize g1at its00.511.52 SOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/g1 scattering intrinsic total FIG. 4: Inverse g1-factor vs. SOI strength αRin the two- dimensional Rashba model. nonrelativistic value the Eq. (25) is satisfied. In the two- dimensional Rashba model Morb= 0 when both bands are occupied. For the Fermi energy EF= 1.36eV both bands are occupied and therefore g=g1for the range of parameters used in Fig. 4. The inverse g1of the two-dimensional Rashba model is shown in Fig. 5 as a function of Fermi energy for the parameters ∆ V= 1eV and αR= 2eV˚A. The scattering correction is as large as the intrinsic contribution when EF>1eV. While the scattering correction is therefore important, it is not sufficiently large to bring g1close to its nonrelativistic value in the energy range shown in the figure, which is a major difference to the one-dimensional case illustrated in Fig. 3. According to Eq. (26) the g factor is related to g1byg=g1M/Mspin. Therefore, we show in Fig. 6 the ratio M/Mspinas a function of Fermi energy. AthighFermienergy(whenbothbandsareoccu- pied) the orbital magnetization is zeroand M/Mspin= 1. At low Fermi energy the sign of the orbital magnetiza- tionis oppositeto the signofthe spin magnetizationsuch that the magnitude of Mis smaller than the magnitude ofMspinresulting in the ratio M/Mspin<1. Next, we discuss the g-factor of the one-dimensional Rashba model in the noncollinear case. In Fig. 7 we plot the inverse g-factor and its decomposition into the intrinsic and scattering contributions as a function of the spin-spiral wave vector q, where exchange splitting, SOI strength and Fermi energy are set to ∆ V= 1eV, αR= 2eV˚A andEF= 1.36eV, respectively. Since the curves are not symmetric around q= 0, the g- factor at wave number qdiffers from the one at −q, i.e., thegyromagnetism in the Rashba model is chiral . At q=−2meαR//planckover2pi12theg-factorassumesthevalueof g=−2 and the scattering corrections are zero. Moreover, the curves are symmetric around q=−2meαR//planckover2pi12. These8 0 2 4 6 Fermi energy [eV]-0.5-0.4-0.3-0.2-0.101/g1 scattering intrinsic total FIG. 5: Inverse g1-factor 1 /g1vs. Fermi energy in the two- dimensional Rashba model. -2 0 2 4 6 Fermi energy [eV]00.511.52M/Mspin FIG. 6: Ratio of total magnetization and spin magnetization , M/Mspin, vs. Fermi energy in the two-dimensional Rashba model. observationscan be explained by the concept of the effec- tive SOI introduced in Eq. (9): At q=−2meαR//planckover2pi12the effective SOI is zero and consequently the noncollinear magnet behaves like a collinear magnet without SOI at this value of q. As we have discussed above in Fig. 2, the g-factor of collinear magnets is g=−2 when SOI is ab- sent, which explains why it is also g=−2 in noncollinear magnets with q=−2meαR//planckover2pi12. If only the intrinsic con- tribution is considered and the scattering corrections are neglected, 1 /gvaries much stronger around the point of zero effective SOI q=−2meαR//planckover2pi12, i.e., the scattering corrections stabilize gat its nonrelativistic value close to the point of zero effective SOI.-2 -1 0 1 Wave vector q [Å-1]-0.8-0.6-0.4-0.201/g scattering intrinsic total FIG. 7: Inverse g-factor 1 /gvs. wave number qin the one- dimensional Rashba model. 0 1 2 3 4 Scattering strength U [(eV)2Å]-0.4-0.200.20.4Gilbert Damping αG xx RR-Vertex AR-Vertex intrinsic total FIG. 8: Gilbert damping αG xxvs. scattering strength Uin the one-dimensional Rashba model without SOI. In this case the vertex corrections and the intrinsic contribution sum up to zero. B. Damping We first discuss the Gilbert damping in the collinear case, i.e., we set ˆM(r) =ˆezin Eq. (4). The xxcom- ponent of the Gilbert damping is shown in Fig. 8 as a function of scattering strength Ufor the following model parameters: exchange splitting ∆ V=1eV, Fermi energyEF= 2.72eV and SOI strength αR= 0. All three contributions are individually non-zero, but the contribution from the RR-vertex correction (Eq. (30)) is muchsmallerthanthe onefromthe AR-vertexcorrection (Eq. (31)) and much smaller than the intrinsic contribu- tion (Eq. (29)). However, in this case the total damping is zero, because a non-zero damping in periodic crystals with collinear magnetization is only possible when SOI is present [53].9 1 2 3 4 Scattering strength U [(eV)2Å]050100150200250300Gilbert Damping αG xx RR-Vertex AR-Vertex intrinsic total FIG. 9: Gilbert damping αG xxvs. scattering strength Uin the one-dimensional Rashba model with SOI. In Fig. 9 we show the xxcomponent of the Gilbert damping αG xxas a function of scattering strength Ufor the model parameters ∆ V= 1eV, EF= 2.72eV and αR= 2eV˚A. ThedominantcontributionistheAR-vertex correction. The damping as obtained based on Eq. (10) diverges like 1 /Uin the limit U→0, i.e., proportional to the relaxation time τ[53]. However, once the relax- ation time τis larger than the inverse frequency of the magnetization dynamics the dc-limit ω→0 in Eq. (10) is not appropriate and ω >0 needs to be used. It has beenshownthattheGilbertdampingisnotinfinite inthe ballistic limit τ→ ∞whenω >0 [41, 42]. In the one- dimensional Rashba model the effective magnetic field exerted by SOI on the electron spins points in ydirec- tion. Since a magnetic field along ydirection cannot lead toatorquein ydirectionthe yycomponentoftheGilbert damping αG yyis zero (not shown in the Figure). Next, we discuss the Gilbert damping in the non- collinear case. In Fig. 10 we plot the xxcomponent of the Gilbert damping as a function of spin spiral wave number qfor the model parameters ∆ V= 1eV, EF= 1.36eV,αR= 2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The curves are symmetric around q=−2meαR//planckover2pi12, because the damping is determined by the effective SOI defined in Eq. (9). At q=−2meαR//planckover2pi12 the effective SOI is zero and therefore the total damp- ing is zero as well. The damping at wave number qdif- fers from the one at wave number −q, i.e.,the damp- ing is chiral in the Rashba model . Around the point q=−2meαR//planckover2pi12the damping is described by aquadratic parabola at first. In the regions -2 ˚A−1< q <-1.2˚A−1 and 0.2˚A−1< q <1˚A−1this trend is interrupted by a W- shape behaviour. In the quadratic parabola region the lowest energy band crosses the Fermi energy twice. As shown in Fig. 1 the lowest band has a local maximum at-2-1.5-1-0.500.51 Wave vector q [Å-1]05101520Gilbert damping αxxG RR-Vertex AR-Vertex intrinsic total FIG. 10: Gilbert damping αG xxvs. spin spiral wave number q in the one-dimensional Rashba model. q= 0. In the W-shape region this local maximum shifts upwards, approaches the Fermi level and finally passes it such that the lowest energy band crosses the Fermi level four times. This transition in the band structure leads to oscillations in the density of states, which results in the W-shape behaviour of the Gilbert damping. Since the damping is determined by the effective SOI, we can use Fig. 10 to draw conclusions about the damp- ing in the noncollinear case with αR= 0: We only need to shift all curves in Fig. 10 to the right such that they are symmetric around q= 0 and shift the Fermi energy. Thus, for αR= 0 the Gilbert damping does not vanish ifq/negationslash= 0. Since for αR= 0 angular momentum transfer from the electronic system to the lattice is not possible, the damping is purely nonlocal in this case, i.e., angular momentum is interchanged between electrons at differ- ent positions. This means that for a volume in which the magnetization of the spin-spiral in Eq. (5) performs exactly one revolution between one end of the volume and the other end the total angular momentum change associated with the damping is zero, because the angu- lar momentum is simply redistributed within this volume and there is no net change of the angular momentum. A substantial contribution of nonlocal damping has also been predicted for domain walls in permalloy [35]. In Fig. 11 we plot the yycomponent of the Gilbert damping as a function of spin spiral wave number qfor the model parameters ∆ V= 1eV,EF= 1.36eV,αR= 2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The totaldampingiszerointhiscase. Thiscanbeunderstood from the symmetry properties of the one-dimensional Rashba Hamiltonian, Eq. (4): Since this Hamiltonian is invariant when both σandˆMare rotated around the yaxis, the damping coefficient αG yydoes not depend on the position within the cycloidal spin spiral of Eq. (5).10 -3 -2 -1 0 1 2 Wave vector q [Å-1]-0.4-0.200.20.4Gilbert Damping αG yyRR-Vertex AR-Vertex intrinsic total FIG. 11: Gilbert damping αG yyvs. spin spiral wave number q in the one-dimensional Rashba model. Therefore, nonlocal damping is not possible in this case andαG yyhas to be zero when αR= 0. It remains to be shown that αG yy= 0 also for αR/negationslash= 0. However, this fol- lows directly from the observation that the damping is determined by the effective SOI, Eq. (9), meaning that any case with q/negationslash= 0 and αR/negationslash= 0 can always be mapped onto a case with q/negationslash= 0 and αR= 0. As an alternative argumentation we can also invoke the finding discussed abovethat αG yy= 0 in the collinearcase. Since the damp- ing is determined by the effective SOI, it follows that αG yy= 0 also in the noncollinear case. C. Current-induced torques We first discuss the yxcomponent of the torkance. In Fig. 12 we show the torkance tyxas a function of the Fermi energy EFfor the model parameters ∆ V= 1eV andαR= 2eV˚A when the magnetization is collinear and points in zdirection. We specify the torkance in units of the positive elementary charge e, which is a convenient choice for the one-dimensional Rashba model. When the torkance is multiplied with the electric field, we ob- tain the torque per length (see Eq. (35) and Ref. [51]). Since the effective magnetic field from SOI points in ydirection, it cannot give rise to a torque in ydirec- tion and consequently the total tyxis zero. Interest- ingly, the intrinsic and scattering contributions are indi- vidually nonzero. The intrinsic contribution is nonzero, because the electric field accelerates the electrons such that/planckover2pi1˙kx=−eEx. Therefore, the effective magnetic fieldBSOI y=αRkx/µBchanges as well, i.e., ˙BSOI y= αR˙kx/µB=−αRExe/(/planckover2pi1µB). Consequently, the electron spin is no longer aligned with the total effective magnetic field (the effective magnetic field resulting from both SOI-1 0 1 2 3 4 5 6 Fermi energy [eV]-0.2-0.100.10.2Torkance tyx [e]scattering intrinsic total FIG. 12: Torkance tyxvs. Fermi energy EFin the one- dimensional Rashba model. and from the exchange splitting ∆ V), when an electric field is applied. While the total effective magnetic field lies in the yzplane, the electron spin acquires an xcom- ponent, because it precesses around the total effective magnetic field, with which it is not aligned due to the applied electric field [54]. The xcomponent of the spin density results in a torque in ydirection, which is the reason why the intrinsic contribution to tyxis nonzero. The scattering contribution to tyxcancels the intrinsic contribution such that the total tyxis zero and angular momentum conservation is satisfied. Using the concept of effective SOI, Eq. (9), we con- clude that tyxis also zero for the noncollinear spin-spiral described by Eq. (5). Thus, both the ycomponent of the spin-orbit torque and the nonadiabatic torque are zero for the one-dimensional Rashba model. To show that tyx= 0 is a peculiarity of the one- dimensional Rashba model, we plot in Fig. 13 the torkance tyxin the two-dimensional Rashba model. The intrinsic and scattering contributions depend linearly on αRfor small values of αR, but the slopes are opposite such that the total tyxis zero for sufficiently small αR. However, for largervalues of αRthe intrinsic and scatter- ing contributions do not cancel each other and therefore the total tyxbecomes nonzero, in contrast to the one- dimensional Rashba model, where tyx= 0 even for large αR. Several previous works determined the part of tyx that is proportionalto αRin the two-dimensionalRashba model and found it to be zero [21, 22] for scalar disor- der, which is consistent with our finding that the linear slopes of the intrinsic and scattering contributions to tyx are opposite for small αR. Next, we discuss the xxcomponent of the torkance in the collinear case ( ˆM=ˆez). In Fig. 14 we plot the torkance txxvs. scattering strength Uin the one-11 00.511.52 SOI strength αR [eVÅ]-0.00500.0050.01Torkance tyx [e/Å] scattering intrinsic total FIG. 13: Nonadiabatic torkance tyxvs. SOI parameter αRin the two-dimensional Rashba model. dimensional Rashba model for the parameters ∆ V= 1eV,EF= 2.72eV and αR= 2eV˚A. The dominant con- tribution is the AR-type vertex correction (see Eq. (43)). txxdiverges like 1 /Uin the limit U→0 as expected for the odd torque in metallic systems [15]. In Fig. 15 and Fig. 16 we plot txxas a function of spin-spiral wave number qfor the model parameters ∆V= 1eV,EF= 2.72eV and U= 0.18(eV)2˚A. In Fig. 15 the case with αR= 2eV˚A is shown, while Fig. 16 illus- trates the case with αR= 0. In the case αR= 0 the torkance txxdescribes the spin-transfer torque (STT). In the case αR/negationslash= 0 the torkance txxis the sum of contribu- tions from STT and spin-orbit torque (SOT). The curves withαR= 0 andαR/negationslash= 0 are essentially related by a shift of ∆q=−2meαR//planckover2pi12, which can be understood based on the concept of the effective SOI, Eq. (9). Thus, in the special case of the one-dimensional Rashba model STT and SOT are strongly related. IV. SUMMARY We study chiral damping, chiral gyromagnetism and current-induced torques in the one-dimensional Rashba model with an additional N´ eel-type noncollinear mag- netic exchange field. In order to describe scattering ef- fects we use a Gaussian scalar disorder model. Scat- tering contributions are generally important in the one- dimensional Rashba model with the exception of the gy- romagnetic ratio in the collinear case with zero SOI, where the scattering correctionsvanish in the clean limit. In the one-dimensional Rashba model SOI and non- collinearity can be combined into an effective SOI. Us- ing the concept of effective SOI, results for the mag- netically collinear one-dimensional Rashba model can be used to predict the behaviour in the noncollinear case.1 2 3 4 Scattering strength U [(eV)2Å]-6-4-20Torkance txx [e] RR-Vertex AR-Vertex intrinsic total FIG. 14: Torkance txxvs. scattering strength Uin the one- dimensional Rashba model. -2 -1 0 1 Wave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex AR-Vertex intrinsic total FIG. 15: Torkance txxvs. wave vector qin the one- dimensional Rashba model with SOI. In the noncollinear Rashba model the Gilbert damp- ing is nonlocal and does not vanish for zero SOI. The scattering corrections tend to stabilize the gyromagnetic ratio in the one-dimensional Rashba model at its non- relativistic value. Both the Gilbert damping and the gyromagnetic ratio are chiral for nonzero SOI strength. The antidamping-like spin-orbit torque and the nonadi- abatic torque for N´ eel-type spin-spirals are zero in the one-dimensional Rashba model, while the antidamping- like spin-orbit torque is nonzero in the two-dimensional Rashba model for sufficiently large SOI-strength. ∗Corresp. author: f.freimuth@fz-juelich.de12 -1-0.500.51 Wave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex AR-Vertex intrinsic total FIG. 16: Torkance txxvs. wave vector qin the one- dimensional Rashba model without SOI. [1] T. Moriya, Phys. Rev. 120, 91 (1960). [2] I. Dzyaloshinsky, Journal of Physics and Chemistry of Solids4, 241 (1958). [3] M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Phys. Rev. B 78, 140403 (2008). [4] P. Ferriani, K. von Bergmann, E. Y. Vedmedenko, S. Heinze, M. Bode, M. Heide, G. Bihlmayer, S. Bl¨ ugel, and R. Wiesendanger, Phys. Rev. Lett. 101, 027201 (2008). [5] K. Yamamoto, A.-M. Pradipto, K. Nawa, T. Akiyama, T. Ito, T. Ono, and K. Nakamura, AIP Advances 7, 056302 (2017). [6] F. R. Lux, F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, ArXiv e-prints (2017), 1706.06068. [7] E. Ju´ e, C. K. Safeer, M. Drouard, A. Lopez, P. Balint, L. Buda-Prejbeanu, O. Boulle, S. Auffret, A. Schuhl, A. Manchon, et al., Nature materials 15, 272 (2015). [8] C. A. Akosa, I. M. Miron, G. Gaudin, and A. Manchon, Phys. Rev. B 93, 214429 (2016). [9] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W. Bauer, Phys. Rev. B 74, 144405 (2006). [10] K. Gilmore, I. Garate, A. H. MacDonald, and M. D. Stiles, Phys. Rev. B 84, 224412 (2011). [11] I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDon- ald, Phys. Rev. B 79, 104416 (2009). [12] K.-W.Kim, K.-J.Lee, H.-W.Lee, andM. D.Stiles, Phys. Rev. B92, 224426 (2015). [13] A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane, J. Grollier, and A. Fert, Phys. Rev. B 87, 020402 (2013). [14] E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406 (2012). [15] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B 90, 174423 (2014). [16] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B 92, 064415 (2015). [17] K. M. D. Hals and A. Brataas, Phys. Rev. B 88, 085423 (2013). [18] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008).[19] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, Nature materials 14, 871 (2015). [20] D. A. Pesin and A. H. MacDonald, Phys. Rev. B 86, 014416 (2012). [21] A. Qaiumzadeh, R. A. Duine, and M. Titov, Phys. Rev. B92, 014402 (2015). [22] I. A. Ado, O. A. Tretiakov, and M. Titov, Phys. Rev. B 95, 094401 (2017). [23] V. Kashid, T. Schena, B. Zimmermann, Y. Mokrousov, S. Bl¨ ugel, V. Shah, and H. G. Salunke, Phys. Rev. B 90, 054412 (2014). [24] B. Schweflinghaus, B. Zimmermann, M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Phys. Rev. B 94, 024403 (2016). [25] M. Menzel, Y. Mokrousov, R. Wieser, J. E. Bickel, E. Vedmedenko, S. Bl¨ ugel, S. Heinze, K. von Bergmann, A. Kubetzka, and R. Wiesendanger, Phys. Rev. Lett. 108, 197204 (2012). [26] I. Barke, F. Zheng, T. K. R¨ ugheimer, and F. J. Himpsel, Phys. Rev. Lett. 97, 226405 (2006). [27] T. Okuda, K. Miyamaoto, Y. Takeichi, H. Miyahara, M. Ogawa, A.Harasawa, A.Kimura, I.Matsuda, A.Kak- izaki, T. Shishidou, et al., Phys. Rev. B 82, 161410 (2010). [28] A. B. Shick, F. Maca, and P. M. Oppeneer, Journal of Magnetism and Magnetic Materials 290, 257 (2005). [29] M. Komelj, D. Steiauf, and M. F¨ ahnle, Phys. Rev. B 73, 134428 (2006). [30] P. Gambardella, A. Dallmeyer, K. Maiti, M. C. Malagoli, S. Rusponi, P. Ohresser, W. Eberhardt, C. Carbone, and K. Kern, Phys. Rev. Lett. 93, 077203 (2004). [31] L. M. Sandratskii, physica status solidi (b) 136, 167 (1986). [32] T. Fujita, M. B. A. Jalil, S. G. Tan, and S. Murakami, Journal of applied physics 110, 121301 (2011). [33] H. Yang, A. Thiaville, S. Rohart, A. Fert, and M. Chshiev, Phys. Rev. Lett. 115, 267210 (2015). [34] Z. Yuan and P. J. Kelly, Phys. Rev. B 93, 224415 (2016). [35] Z. Yuan, K. M. D. Hals, Y. Liu, A. A. Starikov, A. Brataas, and P. J. Kelly, Phys. Rev. Lett. 113, 266603 (2014). [36] F. Freimuth, R. Bamler, Y. Mokrousov, and A. Rosch, Phys. Rev. B 88, 214409 (2013). [37] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Journal of physics: Condensed matter 26, 104202 (2014). [38] F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B 95, 184428 (2017). [39] H. Kohno, Y. Hiraoka, M. Hatami, and G. E. W. Bauer, Phys. Rev. B 94, 104417 (2016). [40] K.-W. Kim, K.-J.Lee, H.-W.Lee, andM.D. Stiles, Phys. Rev. B92, 224426 (2015). [41] D. M. Edwards, Journal of Physics: Condensed Matter 28, 086004 (2016). [42] A. T. Costa and R. B. Muniz, Phys. Rev. B 92, 014419 (2015). [43] J. M. Brown, R. J. Buenker, A. Carrington, C. D. Lauro, R. N. Dixon, R. W. Field, J. T. Hougen, W. H¨ uttner, K. Kuchitsu, M. Mehring, et al., Molecular Physics 98, 1597 (2000). [44] A. A. Kovalev, J. Sinova, and Y. Tserkovnyak, Phys. Rev. Lett. 105, 036601 (2010). [45] J. Weischenberg, F. Freimuth, J. Sinova, S. Bl¨ ugel, an d Y. Mokrousov, Phys. Rev. Lett. 107, 106601 (2011). [46] P. Czaja, F. Freimuth, J. Weischenberg, S. Bl¨ ugel, and13 Y. Mokrousov, Phys. Rev. B 89, 014411 (2014). [47] J. Weischenberg, F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B 87, 060406 (2013). [48] Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404 (2002). [49] C. Kittel, Phys. Rev. 76, 743 (1949). [50] M. A. W. Schoen, J. Lucassen, H. T. Nembach, T. J. Silva, B. Koopmans, C. H. Back, and J. M. Shaw, Phys. Rev. B95, 134410 (2017). [51] In Ref. [15] and Ref. [16] we consider the torque per unit cell and use therefore charge times length as unit of torkance. In this work we consider instead the torque per volume. Consequently, the unit of torkance used inthis work is charge times length per volume. In the one- dimensional Rashba model the volume is given by the length and consequently the unit of torkance is charge. [52] S. Lounis, M. dos Santos Dias, and B. Schweflinghaus, Phys. Rev. B 91, 104420 (2015). [53] I. Garate and A. MacDonald, Phys. Rev. B 79, 064404 (2009). [54] H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. D. Skinner, J. Wunderlich, V. Nov´ ak, R. P. Campion, B. L. Gallagher, E. K. Vehstedt, et al., Nature nanotechnology 9, 211 (2014).

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