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1 Electron -Phonon Scattering governs both Ultrafast and Precessional Magnetization Dynamics in Co -Fe Alloys Ramya Mohan1, Victor H. Ortiz2, Luat Vuong2, Sinisa Coh2, Richard B. Wilson1,2* 1Materials Science & Engineering Program, University of California, Riverside 2Department of Mechanical Engineering, University of California, Riverside * Corresponding Author : rwilson@ucr.edu Abstract Recent investigations have advanced the understanding of how structure -property relationships in ferromagnetic metal alloys affect the magnetization dynamics on nanosecond time -scales . A similar understanding for magnetization dynamics on femto - to pico -second time -scales does not yet exist. To address this, we perform time -resolved magneto optic Kerr effect (TRMOKE) measurements of magnetization dynamics in Co -Fe alloys on femto - to nano-second regimes . We show that Co -Fe compositions that exhibit low Gilbert damping parameters also feature prolonged ultrafast demagnetization upon photoexcitation. We analyze our experimental TR - MOKE data with the three-temperature -model (3TM) and the Landau -Lifshitz -Gilbert equation . These analyses reveal a strong compositional dependence of the dynamics across all time -scales on the strength of electron -phonon interactions. Our findings are beneficial to the spintronics and magnonics community , and wil l aid in the quest for energy -efficient magnetic storage applications. Introduction Laser excitation of a magnetic metal causes energy to cascade from photoexcited electrons into spin and vibrational degrees of freedom1–3. In ferromagnetic 3d transiti on metals such as Fe, Co, and Ni, the rapid increase in thermal energy stored by spin degrees of freedom causes femtosecond quenching of the magnetization2,3, followed by a partial recover over the next few picoseconds . Subsequently, on nanosecond time-scales , a temperature induced change in equilibrium properties causes oscillatory precessions of the magnetic moment . Both ultrafast and precessional magnetization dynamics involve energy exchange between magnetic and vibrational degrees of freedom . The energy exchange is mediated by quasi -particle interactions . The strength of quasi -particle interactions in a ferromagnet depends on e lectronic band structure4,5. In 3d ferromagnetic alloys, the electronic energy bands near the Fermi -level vary strong ly with composition6. Several recent investigations of nanosecond precessional dynamics in ferromagnetic alloy s have explored the relationship between electronic band structure, quasi -particle interactions , and magnetic damping6–8. Schoen et al. report a n intrinsic 2 damping parameter less than 10-3 for Co 0.25Fe0.756, which is unusually low for a metal . They conclude that the low damping in Co0.25Fe0.75 is a result of a minimization in the density of states at the Fermi -level, which decreases the rate of electron -phonon scattering . Researchers have not yet reached a unified understanding of how quasi -particle interactions govern the magnetization dynamics in the femtosecond regime2,9–15. Some studies have hypothesized that spin-flips caused by electron -phonon interactions are key drivers of femtose cond magnetization dynamics9,11. Other experimental and theoretical studies have explored the importance of electron -magnon interactions12–15. Encouraged by the recent advances in the materials science of nanosecond precessional dynamics6–8, we study the compositional dependence of ultrafast magnetization dynamics in Co -Fe alloys. Our study’s goal is to understand the relationship between electronic band structure , quasi -particle interactions, and femto -magnetism properties of ferromagnetic metal alloys. We perform time -resolved magneto optic Kerr effect (TR-MOKE) measurements to characterize the magnetization dynamics of thin CoxFe1-x alloy films (capped and seeded with Ta/Cu layers on a sapphire substrate) on femto - to nanosecond time-scale s. See Methods for details on sample geometry. We observe that the ultrafast magnetization dynamics are a strong function of Co - concentration , see Figure . 1a. The ultrafast dynamics of Co xFe1-x differ most significantly from those of Co and Fe at a composition of x = 0.25. We also analyze the time -resolved macroscopic precessional dynamics and report the effective damping parameter of our samples , see Figure 2a. After linewidth analyses, f or CoxFe1-x, we observe that the Gilbert damping parameter varies from 3.6 ×10−3 to 5.6 ×10−3 for compositions between x = 0 and 1, with a minimum value of 1.5 ×10−3 at x = 0.25 , in good agreement with previously reported results , see Figure 3b . To determine the strength and composition dependence of electron -magnon and electron -phonon quasi -particle interactions , we analyze our ultrafast magnetization dynamics data with a three- temperature -model (3TM)2,16. Our results reveal a strong composition al dependence of the electron -phonon energy transfer coefficient, 𝑔𝑒𝑝, suggesting that the variation in the ultrafast dynamics in Co xFe1-x alloys occurs primarily due to electron -phonon scattering. We draw this conclusion because t he value of 𝑔𝑒𝑝 depends on the rate of phonon emission by hot electrons 17. Electron -phonon scattering is also predicted to govern the dampin g of nanosecond precessional dynamics 6,18,19. Therefore, o ur results demonstrate that the same microscopic electron -phonon interactions responsible for Gilbert damping also play a dominant role in femto -magnetism properties of ferromagnetic alloys. Results Ultrafast Magnetization Dynamics We plot the normalized u ltrafast magnetization dynamics response , ∆M(t), for Co, Fe, and Co0.25Fe0.75 as a function of time delay in Figure . 1a. Data for the rest of the Co -Fe compositions are plotted in Supplementary Figure 1. All our measurements were performed with an incident 3 laser fluence less than ~15 J/m2. This is a sufficiently small fluence for the dynamics in our experiments to follow a linear regime. In other words, decreasing the incident f luence by a factor of two decreases the optical signal by a factor of two, but does not change the time-dependence of the signal . We use a polar TR -MOKE configuration t o measure the ultrafast magnetization dynamics at femtosecond time delays. A schemati c of our experimental setup is shown in Supplementary Figure 2a. We apply a n external 2.2 Tesla (T) field perpendicular to the plane of the sample using an electromagnet (GMW 3480). This external field is strong enough to effectively overcome the in-plane shape anisotropy of the Co -Fe alloys and saturate the moment in the out - of-plane direction. Since the equilibrium orientation of the moment is in the out -of-plane direction , both, before and after laser irradiation, this geometry allows us to quan tify the femtosecond demagnetization response of the Co -Fe alloys , without the presence of macroscopic precessional dynamics , see schematic in Fig ure 1b. Upon excitation with the pump pulse, the magnetic moment decreases on a sub -picosecond time- scale due to the flow of energy from electrons to magnons2,3,16,20,21. Then, on picosecond time- scale s, the magnetization partially recovers as energy is transferr ed to the lattice and temperature gradients across the film thickness relax. After a few picoseconds, the magnetic film reaches a new equilibrium at an elevated temperature. Ultrafast dynamics with sub -picosecond demagnetization followed by picosecond re-magnetization are commonly categorized as “type I” dynamics , and are characteristic of 3d ferromagnetic metals such as Fe, Co, and Ni9. To elucidate how the de - and re -magnetization dynamics change with composition, we define two data descriptors : τD and R. We define the demagnetization time , τD, as the delay time where d∆M(t)/dt reaches its maximum value. We define R as the ratio of the maximum of 𝛥𝑀(𝑡) to 𝛥𝑀(𝑡≈10ps). We plot τD and R as a function of composition in Figure 3a. τD varies weakly with composition and has a minimum value of 40 fs at x = 0.25. In contrast , we observe that R varies strongly with composition and is a maximum of 4 at x = 0.25. Nanosecond Precessional Dynamics We show measurements of the macroscopic precessional dynamics of Fe, Co, and Co 0.25Fe0.75 in Figure 2a. Data for the other Co -Fe compositions are plotted in Supplementary Figure 3. We use a polar TR -MOKE experimental setup, with an obliquely angled external magnetic field, to measure the macroscopic precessional dynamics of our samples. A schematic of our experimental setup is shown in Supplementary Figure 2b. Tilting the electromagnet to an angle of 11° , with respect to the plane of the sample, allows us to apply a canted external magnetic field so that the magnetic moment has an out -of-plane component. The equilibrium orientation of the moment depends on the balancing between the applied external field and the thin -film shape anisotropy field. The shape anisotropy field in the z -direction is proportional to the out-of-plane component of the magnetic moment. Upon heating, the total magnetic moment decreases . This decrease results in an ultrafast change to the out-of-plane anisotropy field and equilibrium 4 orientation . As a result, t he magnetic moment will precess to a new equilibrium orientation , see schematic in Figure 2b. Our polar TR -MOKE setup detects changes in the out -of-plane moment , so we can sensitively measure the frequency and amplitude of the precessional dynamics. We collect between 6 and 12 TR-MOKE scans of precessional dynamics for each sample . Each of these scans is co llected with a different applied external magnetic field , ranging from 0. 2 T to 2.2 T. The TR -MOKE signals include precessional dynamics in addition with a background related to temperature -induced demagnetization. To analyze the precessional dynamics, we subtract the background with a biexponential decay function . We fit the resulting dataset with a damped harmonic function, V(t)=Asin(ωt+∅)exp (−t/τ). Our fits yield unique values of A (amplitude), ∅ (the initial phase of the oscillation), T (period), and τ (the exponential decay time of the precession). Using these values, we determine the effective dimensionless damping parameter , αeff = ω.τ-1. The resonance frequency is a function of applied external magnetic field and magnetic moment, 𝜔=γ √Heff(Heff+μ0Ms). Here, ɣ is the gyromagnetic ratio, μ0 is the vacuum permeability, Heff is the out -of-plane component of the external magnetic field as measured by a Hall probe , and Ms is the saturation magnetization of the sample . We derive the magnetic moment of the sample by treating Ms as a fit parameter . We also perform VSM measurements of the moment of some of the samples and find that the magnetic moment obtained is in good agreement with the value that we derive by fitting our precessional dynamics data . See Supplementary Figure 4 for more details . The effective damping parameter α eff that we deduce from our precessional dynamics measurements includes effects from damping and inhomogeneous broadening. The effect of inhomogeneous broadening is independent of the applied field at high frequencies22. To obtain the Gilbert damping parameter intrinsic to the sample geometry (not intrinsic to the material) , we plot the effective linewidth, αeff∙f, as a function of frequency, and linearly fit to the equation , αeff∙f=α∙f+∆H, where ∆H is the inhomogeneous broadening component and α is the Gilbert damping parameter . Further details can be found in Supplementary Figure 5. In contra st to prior investigations that performed FMR measurements in the frequency range from 16 -18 GHz8 and 40 GHz6, our TR -MOKE experimental setup allows us to study dynamics at frequencies as large as 90 GHz. At such high frequency, we can be confident that our measured Gilbert damping parameter is dominate d by the intrinsic linewidth over inhomogeneous broadening effects. The Gilbert damping parameter we observe of α = 1.5 ×10−3 for Co 0.25Fe0.75 is amongst the lowest ever reported for a ferromagnetic metal. Schoen et al. report α=2.1 ×10−3 for Co0.25Fe0.75. After accounting for radiative and spin -pumping contributions, they estimate an intrinsic damping parameter for Co0.25Fe0.75 to be αint=5 ×10−4 . Lee et al. 8 performed FMR measurements of Co0.25Fe0.75 epitaxial films and report α=1.4 ×10−3. Wei et al. report α=5 1.5 ×10−3 for Fe 0.75Al0.25 films 7. We note that our measured damping parameter likely includes significant contributions from spin -pumping into the adjoining Ta /Cu layers, but we did not experimentally examine the effect s of spin -pumping in our samples. Analysis and Discussion The c omparison of 𝑅 and 𝛼 in Figure 3a and Figure 3b reveals that the two quantities depend on composition in a similar manner. R is at a maximum and 𝛼 is at a minimum at x = 0.25 . Fe and CoxFe1-x alloys with x ≥ 0.5 have small R and high 𝛼. Alternatively, C oxFe1-x alloys with 0.1< x < 0.5 have both high 𝑅 and low 𝛼. To confirm this correlation , we performed a hierarchical cluster analysis of the raw data at both femtosecond and nanosecond time-scale s. The clustering algorithm divides the Co -Fe alloys into groups based on similarit ies in the dynamics data . The clustering results as a function of composition are nearly identical when based on the femto - /pico -second time -scale data vs. the nanosecond time -scale data. We include further details on the clustering analysis in Supplementary Note 1 and Supplementary Figure 6 . We now explain the correlation between ultrafast and precessional dynamics by considering how electronic scattering processes depend on composition. Similar to prior studies of damping in Co-Fe alloys6,7,23, our results for 𝛼 vs. x are in good agreement with the “breathing Fermi surface ” model for damping24. In this model , spin -orbit coupling causes the Fermi -level to shift with the precessi ons of the magnetic moment25. A shift in the equilibrium Fermi -level leads to a nonequilibrium electron population . As the Fermi -level repopulates, i ntra-band electron -phonon scattering transfers energy to the lattice . The “breathing Fermi surface” model predicts that the damping parameter is directly proportional to 𝐷(𝜀𝑓), because more electronic states near 𝜀𝑓 leads to higher rates of electron -phonon scattering . We observe that the 𝛼 value for Co0.25Fe0.75 is ~2.5x lower th an 𝛼 for Fe. Density functional theory predicts a ~2x difference in 𝐷(𝜀𝑓) for Co0.25Fe0.75 vs. Fe, see Supplementary Note 2 or Ref.6. Therefore, like prior studies of Co -Fe alloys6,7,23, we conclude that intra -band electron -phono n scattering governs precessional damping. To better understand how composition affects electron -magnon and electron -phonon energy transfer mechanisms , we analyze our 𝛥𝑀(𝑡) data with a phenomenological three temperature model (3TM) , see Figure 4. The 3TM describes how heat flows between electrons, phonons, and magnons after laser excitation of the Co-Fe sample . (See Methods for additional details. ) The 3TM predicts that τD depends on two groupings of model parameters: 𝜏𝑒𝑚≈𝐶𝑚/𝑔𝑒𝑚 and 𝜏𝑒𝑝≈ 𝐶𝑒/𝑔𝑒𝑝. Here 𝐶𝑚 and 𝐶𝑒 are the magnon and electron heat-capacity per unit volume, and 𝑔𝑒𝑚 and 𝑔𝑒𝑝 are the energy transfer coefficients from electrons to magnons an d phonons, respectively. We estimate v alues for 𝐶𝑒 vs. composition using the Sommerfeld model together with the electronic density of states vs. composition reported in Ref.6. The 3TM also predicts that the parameter R is determined by the following grouping of parameters: 𝑅= 𝐶𝑝𝑔𝑒𝑚/𝐶𝑚𝑔𝑒𝑝 16, where 𝐶𝑝 is the phonon heat -capacity per unit volume . We assume that the value of 𝐶𝑝 is 3.75 6 MJ m-3 K-1 for Co, Fe and Co -Fe alloys. With these estimates for 𝐶𝑒 and 𝐶𝑝, and other relevant model parameters, summarized in Supplementary Table 1, we can deduce unique values for 𝐶𝑚/𝑔𝑒𝑚 and 𝐶𝑝/𝑔𝑒𝑝 as a function of composition from our TR-MOKE data, see Figure 4b. Based on our 3TM analysis, we conclude that the strong composition dependence of R is due to the composition dependence of 𝑔𝑒𝑝. Boltzmann rate -equation modelling of the nonequilibrium electron dynamics after photoexcitation predicts that the electron -phonon energy -trans fer coefficient is 𝑔𝑒𝑝=[𝜋ℏ𝑘𝐵𝐷(𝜀𝐹)]𝜆⟨𝜔2⟩ 5. Here, 𝜆⟨𝜔2⟩ is the second frequency moment of the Eliashberg function and is a measure of the strength of electron -phonon interactions . Most of the composition al dependence we observe in 𝑔𝑒𝑝 is explained by the composition al dependence of 𝐷(𝜀𝑓). To show this, we include a prediction for 𝑔𝑒𝑝 in Figure 4b. Our prediction uses the 𝐷(𝜀𝑓) vs. x reported in6 and treats 𝜆⟨𝜔2⟩ as a composition independent fit parameter . We find 𝜆⟨𝜔2⟩=260 meV2 provides an excellent fit to our data . The best-fit value for 𝜆⟨𝜔2⟩ is in good agreement with 𝜆⟨𝜔2⟩≈𝜆𝑅Θ𝐷22⁄=280 meV2. Here, 𝜆𝑅 is derived from electrical resistivity data for Fe 26, and Θ𝐷=470𝐾 is the Debye temperature of Fe. Before beginning our experimental study, we hypothesized that the energy transfer coefficient between electrons and magnons, emg , would be correlated with the phase -space for electron - magnon scattering . We expected the phase -space for electron -magnon scattering to be a strong function of band -structure near the Fermi -level 12–15. We also expected the phase -space to be minimized at a composition of x = 0.25, because of the minimum in the density of states at the fermi -level. To explore how the phase -space for electron -magnon scattering depends on composition, we performed density functional theory calculations for the electronic band structure with x = 0 and x = 0.25, see Supplementa ry Note 2. Our DFT calculations suggest that the phase -space for electron -magnon scattering is an order of magnitude higher for x = 0 vs. 0.25. However, we do not see evidence that this large theoretical difference in electron -magnon scattering phase -space affects ultrafast dynamics . The time -scale for magnons to heat up after photoexcitation, /em m emCg , decreases monotonically with increasing x, and does display structure near x ~ 0.25. Several theoretical models predict a strong correlation between τ D and αint. For example, Koopmans et al. predicts τ D will be inversely proportional to α by assuming that the dissipative processes responsible for damping also drive ultrafast demagnetization 27. Alternatively, Fähnle et al. predict s that τD should be proportional to αint 28. In our experiments on Co -Fe thin films, w e observe only a weak correlation between τD and αint. While α int varies with composition by a factor of three , τD for 8 of the 9 compositions we study fall within 20% of 75 fs. The τD value we obtained for Fe (= 76 fs) agrees well with experimental results reported in 9,12,29. 7 Conclusions We have measured the magnetization dynamics of Co xFe1-x thin-films , and we observe that both ultrafast and precessional dynamics of Co 0.25Fe0.75 differ significantly from Co and Fe . When the moment of Co0.25Fe0.75 is driven away from its equilibrium orientation , the time -scale for the moment to return to equilibrium is 3 -4x as long as for Fe or Co. Similarly, when spins of Co0.25Fe0.75 are driven into a nonequilibrium state by ultrafast laser heating, the time -scale for thermalization with the lattice is 2 -3x as long as for Fe or Co. Through 3TM analyses, we demonstrate that this occurs primarily due to the effect of the electronic band -structure on electron -phonon interactions , consistent with the “breathing Fermi surface” theory . Our findings are of fundamental importance to the field of ul trafast magnetism, which seeks to control magnetic order on femto - to picosecond time-scale s. Such control requires a thorough understanding of how and why energy is exchanged between electronic, spin, and vibrational degrees of freedom. Prior studies have shown that 𝑔𝑒𝑝 is correlated with a wide range of physical properties, e.g the superconducting transition temperature30, electrical resistivity 26, photoelectron emission31, and the laser fluence required for ablation32. To our knowledge, o ur study provides the first demonstration that 𝑔𝑒𝑝 in ferromagnetic metals is also correlated to the Gilbert damping parameter 𝛼. Our findings also have implications for the ongoing search for magnetic materials with ultrafast magnetic switching functionality. Atomistic spin dynamics simulations predict that the energy required for ultrafast electrical or optical switching of rare -earth ferromagnetic alloys, e.g. GdFeCo, is governed by the electron -phonon energy transfer coefficient33. To date, most studies aimed at exploring the materials science of ultrafast switching have used alloy composition as a way to control magnetic properties 34–37. Our work suggests an alternative strategy for reducing the energy requirements for ultrafast magnetic switching. The alloy composition should be chosen to minimize the electronic density of states at the Fermi -level. Such metals will have lower electron -phonon energy trans fer coefficients, and therefore more energy efficient ultrafast switching 33. Finally , our findings offer a new route for discovering ferro magnetic materials with ultra -low damping as a result of low 𝑔𝑒𝑝. Current methods for identifying low damping materials involve labor -intensive ferromagnetic resonance measurements of one alloy composition at a time. Alternatively, high-throughput localized measurements of ultrafast demagnetization dynamics of samples produced using combinatorial techniques38 would allow promising alloy compounds with weak electron -phonon interactions to be rapidly identified 39–41. 8 Materials and Methods Sample Preparation We sputter deposit the Co -Fe samples onto sapphire substrates with a direct current (DC) magnetron sputtering system (Orion, AJA International). The base pressure prior to deposition is less than 3.5 × 10-7 torr. We sputter with an Ar gon pressure of ~3.5 × 10-3 torr. The geometry of the samples is sapphire/Ta(2nm)/Cu(3nm)/Co xFe1-x(15nm)/Cu(3nm)/Ta(1nm). The Co xFe1-x layer is deposited by co -sputtering two 4N purity Co and Fe targets at different powers. We chose this film geometry to mimic the samples in Ref.6 which demonstrated low damping at x = 0.25. To ensure an accurate thickness of each layer in our samples, we calibrate the deposition rates of each metal by sputtering individual Co, Fe, Ta, and Cu films onto SiO 2/Si substrates and/or BK -7 glass substrates. We use picosecond acoustics42 and time-domain thermo -reflectance (TDTR) measurements43,44 to determine the thicknesses of these individual films. We validate the composition of the Co -Fe alloy layer by perf orming Energy Dispersive X -Ray Spectroscopy (EDS) analyses with a scanning electron microscope ( FEI Nova Nano SEM 450) at an operating voltage of 15 kV and working distance of 14 mm. We analyze the EDS data using Aztec Synergy software ( Oxford Instruments ). Time -Resolved MOKE Experimental Setup We use a pump/probe laser system to perform TR -MOKE measurements of the magne tization dynamics. The pulsed laser is a Ti:sapphire oscillator with an 80 MHz repetition rate. The laser beam is split into a pump and probe beam, that are modulated to frequencies of 10.7 MHz and 200 Hz , respectively. A time -delayed pump beam irradiates the sample surface and heats the metal film. The ultrafast heating causes a change in the magnetic moment. We measure the time - evolution of the magnetic moment by monitoring the polarization of the probe beam reflected of f the sample surface. The reflected probe beam’s polarization state is affected by the out -of-plane magnetic moment of the sample due to the polar Kerr effect. Additional details about the MOKE experiment set -up are in Ref.45. The t ime-resolution of our experiment is controlled by the convolution of the intensity vs. time of the pump and probe pulses. The wavelength of our pump and probe beams is tunable. Employing a red (900 nm ) pump and blue (450 nm ) probe yields higher time-resolution capabilities , allowing us to accurately measure the ultrafast magnetization at fe mtosecond time delays . We measure the full-width -at-half-maximum ( FWHM ) of the convolution of the pump and probe pulses by performing an inverse Faraday effect (IFE) measurement on Pt . We obtain a FWHM value of 390 fs for the convoluted pulses , and a pulse duration of 2 10 fs for the 900 nm pump/450 nm probe beam setup . For further details on our IFE measurements and pulse duration calculations, please refer to Supplementary Figure 8. 9 To investigate the precessional dynami cs on longer time -scales, we use a pump and probe wavelength of 783 nm. The pulse duration for this setup is 610 fs due to pulse broadening from a two-tint setup we use to prevent pump light from reaching the balanced detector45,46. Three Temperature Modeling To determine the electron, phonon, and magnon energy transfer coefficients, we use t he phenomenological three -temperature model (3TM), given by the following set of equations : 𝐶𝑒𝑑𝑇𝑒 𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+ 𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑒𝑑2𝑇𝑒 𝑑𝑧2+𝑆(𝑧,𝑡) (1) 𝐶𝑝𝑑𝑇𝑝 𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+Ʌ𝑝𝑑2𝑇𝑝 𝑑𝑧2 (2) 𝐶𝑚𝑑𝑇𝑚 𝑑𝑡=𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑚𝑑2𝑇𝑚 𝑑𝑧2 (3) 𝑆(𝑧,𝑡)= 𝑆0𝑃(𝑡)𝐴(𝑧) (4) Equations 1 – 3 describe the temperature evolution of electrons (e), phonons (p) and magnons (m), as a function of time delay (t). C, T, and Ʌ are the heat capacity per unit volume, temperature, and thermal conductivity, respectively. We use the density of states (DOS) at the Fermi level as a function of Co -concentration6 to calculate the electronic heat capacity (C e) using the Sommerfeld model . We assume that the phonon -magnon energy transfer is negligible compared to electron -magnon coupling, and thus, neglect 𝑔𝑝𝑚. We calculate the laser energy absorption by electrons (S), as a function of depth (z) and time delay (t), as described in Equation 4. The terms P(t) and A(z) denote the time -dependent laser pulse intensity and the optical absorption profile as a function of stack thickness. We calculat e A(z) us ing the refractive indices of each metal constituent of the stack47–49. The material parameters that are used to numerically solve equations 1 – 4 are listed in Supplementary Table 1. 10 Figures: Figure 1. Ultrafast magnetization dynamics of Co, Fe, and Co 0.25Fe0.75 thin films (a) Polar TR - MOKE data showing ultrafast demagnetization behavior at short delay times. (b) Schematic illustration of the three phases of an ultrafast magnetization dynamics experiment. Stage I: A large external magnetic field oriented normal to the plane of t he sample leads to an equilibrium moment , 𝑀⃗⃗ in the out -of-plane direction. Stage II: Upon heating with a pump beam, ultrafast demagnetization ( 𝑀′⃗⃗⃗⃗ ) occurs within ~100s of fs. Energy from hot electrons is transferred to the magnons, increasing the amplitude of precession. Stage III: Over the next few picoseconds, energy is transferred from magnons and electrons to the lattice. Additionally, spatial t emperature gradients relax. As a result, magnons cool, i.e. the average precessional amplitude of individual spins decreases. As a result, the magnetization partially recovers to 𝑀′′⃗⃗⃗⃗⃗⃗ . The time -scale for the partial recovery in stage III depends strongly o n the composition. 11 Figure 2 . Precessional dynamics in Co, Fe, and Co 0.25Fe0.75 thin films (a) Polar TR -MOKE data on sub -nanosecond time-scale s. (b) Illustration of the three stages for precessional dynamics after laser excitation . Stage I: Prior to laser excitation, the presence of a canted external magnetic field, 𝐻𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ , oriented at an angle θ. This results in the orientation of the out -of-plane moments, 𝑀⃗⃗ 𝑧. Stage II: Laser -induced photoexcitation leads to the disorder of the magnetic moment, causing a decay in the net magnetization , denoted by 𝑀′⃗⃗⃗⃗ . The net torque imba lance causes macroscopic precessions of the magnons, towards equilibrium, 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ , over several ~100s of picoseconds . Stage III : Eventually, after ~1 ns, the magnetic moment re -equilibrates to 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ . The lifetime of the magnetic precessions depends o n the effective damping parameter, α eff. The time -scale for the precessional dynamics to cease ( in stage III) depends strongly on composition, and is a maximum for x = 0.25. 12 Figure 3. Compositional dependence of descriptors for the ultrafast dynamics data . (a) R describes the maximum change in the magnetic moment, i.e. how far from equilibrium spin - degrees of freedom are driven after ultrafast excitation. τD describes the lag between zero delay time and demagnetization, as a function of Co -concentration. (b) α denotes the Gilbert damping parameter, as a function of Co concentration. Data obtained from our TR -MOKE experiments described in this study (plotted in orange), agree reasonably with data from Ref. [6] (plotted in green). Co 0.25Fe0.75 features the largest deviation in R and α, when compared to its constituent elements Co and Fe. 13 Figure 4. Analyses of Ultrafast Demagnetization Results using the Three Temperature Model (3TM) in Co -Fe alloys . (a) Polar TR -MOKE dataset of the Co 0.25Fe0.75 composition (black circles) with best -fit results of the 3TM. The 3TM describes the temperature excursions of the electrons (blue curve), magnons (red curve) and phonons (green curve) after laser excitation. (b) We treat 𝑔𝑒𝑝 and 𝑔𝑒𝑚 as fit parameters when solving the 3TM. Using literature values of C p and C m (further details available in Supplementary Table 1), we calculate and plot the electron -phonon (τ ep) and electron -magnon (τ em) relaxation times, as a function of Co -concentration. The red -line is a best - fit value for the electron -phonon relaxation time as a function of composition, with the assumption of a composition -independent value for the electron -phonon coupling parameter λ . 14 References: 1. Kirilyuk, A., Kimel, A. V & Rasing, T. Ultrafast optical manipulation of magnetic order. Rev. Mod. Phys. 82, 2731 (2010). 2. Beaurepaire, E., Merle, J. C., Daunois, A. & Bigot, J. Y. Ultrafast spin dynamics in ferromagnetic nickel. Phys. Rev. Lett. 76, 4250 –4253 (1996). 3. Hellman, F. et al. Interface -Induced Phenomena in Magnetism. Rev. Mod. Phys. 89, 025006 (2017). 4. McMillan , W. L. Transition Temperature of Strong -Coupled Superconductors. Phys. 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High sensitivity pump -probe measurements of magnetic, thermal, and acoustic phenomena with a spectrally tunable oscillator. Rev. Sci. Instrum. 91, (2020). 46. Kang, K., Koh, Y. K., Chiritescu, C., Zheng, X. & Cahill, D. G. Two -tint pump -probe measurements using a femtosecond laser oscillator and sharp -edged optical filters. Rev. Sci. Instrum. 79, (2008). 47. Johnson, P. B. & Christy, R. W. Optical constants of transition metals. Phys. Rev. B 9, 5056 –5070 (1974). 48. P. B. Johnson and R. W. Chri sty. Optical Constant of the Nobel Metals. Phys. Rev. B 6, 17 4370 –4379 (1972). 49. Ordal, M. A., Bell, R. J., Alexander, R. W., Newquist, L. A. & Querry, M. R. Optical properties of Al, Fe, Ti, Ta, W, and Mo at submillimeter wavelengths. Appl. Opt. 27, 1203 (1988). Acknowledgements The work by R. M., V. H. O, and R. B. W. was primarily supported by the U.S. Army Research Laboratory and the U.S. Army Research Office under contract/grant number W911NF -18-1- 0364 and W911NF -20-1-0274. R. M. and R. B. W. also acknowledge support by NSF (C BET – 1847632). The work by L. V. and S. C. was supported by the U.S. Army Research Laboratory and U.S. Army Research Office under contract/grant number W911NF -20-1-0274. Energy Dispersive X -Ray Spectroscopy (EDS) analyses were performed at the Central Fac ility for Advanced Microscopy and Microanalysis (CFAMM) at UC Riverside. Author Contributions R. M. and R. B. W. designed the experiments. R. M. prepared all the samples and characterized them , and performed TR-MOKE experiments . V. H. O performed VSM measurements. L. V. performed hierarchical clustering analyses. S. C. performed DFT calculations. R. M. and R . B. W. analyzed the data and wrote the manuscript, with discussions and contributions from L. V. and S. C . Additional Information: Supplementary information is provided with this manuscript. Competing Interests: The authors declare no competing interest. Data Availability: The data that supports the findings of this paper are available from the corresponding author upon reasonable request. Correspondence: Correspondence and request for additional information must be addressed to rwilson@ucr.edu

2021-07-24

Recent investigations have advanced the understanding of how structure-property relationships in ferromagnetic metal alloys affect the magnetization dynamics on nanosecond time-scales. A similar understanding for magnetization dynamics on femto- to pico-second time-scales does not yet exist. To address this, we perform time-resolved magneto optic Kerr effect (TRMOKE) measurements of magnetization dynamics in Co-Fe alloys on femto- to nano-second regimes. We show that Co-Fe compositions that exhibit low Gilbert damping parameters also feature prolonged ultrafast demagnetization upon photoexcitation. We analyze our experimental TR-MOKE data with the three-temperature-model (3TM) and the Landau-Lifshitz-Gilbert equation. These analyses reveal a strong compositional dependence of the dynamics across all time-scales on the strength of electron-phonon interactions. Our findings are beneficial to the spintronics and magnonics community, and will aid in the quest for energy-efficient magnetic storage applications.

Electron-Phonon Scattering governs both Ultrafast and Precessional Magnetization Dynamics in Co-Fe Alloys

2107.11699v1

arXiv:2206.10948v1 [math.AP] 22 Jun 2022HOMOGENIZATION OF THE LANDAU-LIFSHITZ-GILBERT EQUATION WITH NATURAL BOUNDARY CONDITION JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN Abstract The full Landau-Lifshitz-Gilbert equation with periodic material coe fficients and natural boundary condition is employed to model the magnetization dynamics in composite ferromagnets. In this work, we establish the converge nce between the homogenized solution and the original solution via a Lax equivalence th eorem kind of argument. There are a few technical difficulties, including: 1) it is p roven the classic choice of corrector to homogenization cannot provide the c onvergence re- sult in the H1norm; 2) a boundary layer is induced due to the natural boundary condition; 3) the presence of stray field give rise to a multiscale pote ntial problem. To keep the convergence rates near the boundary, we introduce the Neumann cor- rector with a high-order modification. Estimates on singular integra l for disturbed functions and boundary layer are deduced, to conduct consisten cy analysis of stray field. Furthermore, inspired by length conservation of magnetizat ion, we choose proper correctors in specific geometric space. These, together with a uniform W1,6 estimate on original solution, provide the convergence rates in the H1sense. 1.Introduction The intrinsic magnetic order of a rigid single-crystal ferr omagnet over a region Ω ⊂Rn,n= 1,2,3 is described by the magnetization Msatisfying M=Ms(T)m,a.e. in Ω , where the saturation magnetization Msdepends on the material and the temperature T. Below Curie temperature, Msis modeled as a constant. A stable structure of a ferromagnet is mathematically chara cterized as the local minimizers of the Landau-Lifshitz energy functio nal [7] GL[m] :=/integraldisplay Ωa(x)|∇m|2dx+/integraldisplay ΩK(x)(m·u)2(m)dx −µ0/integraldisplay Ωhd[Msm]·Msmdx−/integraldisplay Ωha·Msmdx =:E(m)+A(m)+W(m)+Z(m). Date: June 23, 2022. 2010Mathematics Subject Classification. 35B27; 65M15; 82D40. Key words and phrases. Homogenization; Landau-Lifshitz-Gilbert equation; Boun dary layer; Magnetization dynamics; Micromagnetics. 12 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN E(m) is the exchange energy, which penalizes the spatial variat ion ofm. The matrix a= (aij)1≤i,j≤3is symmetric, uniformly coercive and bounded, i.e., (1) n/summationdisplay i,j=1aij(x)ηiηj≥amin|η|2for anyx∈Rn,η∈Rn, n/summationdisplay i,j=1aij(x)ηiξj≤amax|η||ξ|for anyx∈Rn,η, ξ∈Rn. Intheanisotropyenergy A(m),uistheeasy-axisdirectionwhichdependson thecrystallographicstructureofthematerial. Theanisot ropyenergydensity is assumed to be a non-negatively even and globally Lipschit z continuous function that vanishes only on a finite set of unit vectors (th e easy axis). The third term W(m) is the magnetostatic self-energy due to the dipolar magnetic field, also known as the stray field hd[m]. For an open bounded domain Ω with a Lipschitz boundary, the magnetization m∈Lp(Ω,R3) generates a stray field satisfying (2) hd[m] =∇Um, where the potential Umsolves (3) ∆ Um=−div(mXΩ),inD′(R3) withmXΩthe extension of mtoR3that vanishes outside Ω. The exis- tence and uniqueness of Umfollows from the Lax-Milgram Theorem and Um satisfies the estimate [ 10] (4) /ba∇dblhd[m]/ba∇dblLp(Ω)≤ /ba∇dblm/ba∇dblLp(Ω)1< p <∞. The last term Z(m) is the Zeeman energy that models the interaction be- tweenmand the externally applied magnetic field ha. For a composite ferromagnet with periodic micorstructures , the material constants are modeled with periodic material coefficients wi th period ε, i.e., aε=a(x/ε),Kε=K(x/ε),Mε=Ms(x/ε), with functions a,K,Ms periodic over Y= [0,1]n. The associated energy reads as (5)Gε L[m] :=/integraldisplay Ωaε(x)|∇m|2dx+/integraldisplay ΩKε(x)(m·u)2dx −µ0/integraldisplay Ωhd[Mεm]·Mεmdx−/integraldisplay Ωha·Mεmdx. It is proved in [ 2] thatGε L[m] is equi-mild coercive in the metric space (H1(Ω,S2),dL2(Ω,S2)) and Γ-converges to the functional Ghomdefined as Ghom[m] =/integraldisplay Ωa0|∇m|2dx+/integraldisplay ΩK0(m·u)2dx−µ0(M0)2/integraldisplay Ωhd[m]·mdx −µ0/integraldisplay Ω×Y/vextendsingle/vextendsinglem·Hd[Ms(y)](y)/vextendsingle/vextendsingle2dxdy−M0/integraldisplay Ωha·mdx, (6)HOMOGENIZATION OF THE LLG EQUATION 3 wherea0is the homogenized tensor a0 ij=/integraldisplay Y/parenleftBigg aij+n/summationdisplay k=1aik∂χj ∂yk/parenrightBigg dy, the constants M0andK0are calculated by M0=/integraldisplay YMs(y)dy, K0=/integraldisplay YK(y)dy, and the symmetric matrix-valued function Hd[Ms(y)](y) =∇yU(y) with potential function given by (7)/integraldisplay YMs(y)∇yϕ(y)dy=−/integraldisplay Y∇yU(y)·∇yϕ(y)dy, U(y) isY-periodic ,/integraldisplay YU(y)dy= 0, for any periodic function ϕ∈H1 per(Y). In the current work, we are interested in the convergence of t he dynamic problem driven by the Landau-Lifshitz energy ( 5) to the dynamics problem driven by the homogenized energy ( 6) asεgoes to 0. It is well known that the time evolution of the magnetization over Ω T= Ω×[0,T] follows the Landau-Lifshitz-Gilbert (LLG) equation [ 7,6] (8) ∂tmε−αmε×∂tmε=−(1+α2)mε×Hε e(mε) a.e. in Ω T, ν·aε∇mε= 0,a.e. on∂Ω×[0,T], mε(0,x) =mε init(x),|mε init(x)|= 1 a.e. in Ω , whereα >0isthedampingconstant, andtheeffective field Hε e(mε) =−δGε L δmε associated to the Landau-Lifshitz energy ( 5) is given by (9)Hε e(mε) = div(aε∇mε)−Kε(mε·u)u+µ0Mεhd[Mεmε]+Mεha. Meanwhile, the LLG equation associated to the homogenized e nergy (6) reads as (10) ∂tm0−αm0×∂tm0=−(1+α2)m0×H0 e(m0) ν·a0∇m0= 0,a.e. on∂Ω×[0,T] m0(0,x) =m0 init(x),|minit(x)|= 1 a.e. in Ω with homogenized effective field H0 e(m0) =−δGhom δm0calculated by (11)H0 e(m0) =div/parenleftbig a0∇m0/parenrightbig −K0(m0·u)u +µ0(M0)2hd[m0]+µ0H0 d·m0+M0ha, where the matrix H0 d=/integraltext YMs(y)Hd[Ms(y)](y)dy. Works related tothehomogenization oftheLLGequationinth eliterature include [ 11,5,1,8,9,4]. As for the convergence rate, most relevantly, the LLG equation ( 8) with only the exchange term and with the periodic4 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN boundarycondition is studied in [ 8]. Convergence rates between mεandm0 in time interval [0 ,εσT] are obtained under the assumption (12) /ba∇dbl∇mε/ba∇dblL∞(Ω)≤C,for anyt∈[0,εσT], whereCis a constant independent of εandσ∈[0,2). As a special case, whenσ= 0 in assumption ( 12), i.e.,/ba∇dbl∇mε/ba∇dblL∞(Ω)is uniformly boundedover a time interval independent of ε, it is proven that /ba∇dblmε−m0/ba∇dblL∞(0,T;L2(Ω))= O(ε)while/ba∇dblmε−m0/ba∇dblL∞(0,T;H1(Ω))isonlyuniformlyboundedwithoutstrong convergence rate. In this work, we consider the full LLG model ( 8) equipped with the Neu- mann boundary condition, which is the original model derive d by Landau and Lifshitz [ 7]. We prove the convergence rates between mεandm0in the L∞(0,T;H1(Ω)) sense without the strong assumption ( 12). It is worth men- tioning that, the trick to improve the convergence result in toH1sense is to find proper correctors m1,m2, such that they satisfy geometric properties (13) m0·m1= 0,andm0·m2=−|m1|2, which are motivated by the length-preserving property of ma gnetization and asymptotic expansion. A familiar definition of classic fi rst-order ho- mogenization corrector m1in (32) would naturally satisfies first property in (13); see [8]. In this article, the suitable corrector m2in (13) is obtained. By the usage of these properties, we are able to derive the estim ate of consis- tency error, which is induced by an equivalent form of LLG equ ation, given in (22), and a sharper estimate than [ 8] inL∞(0,T;H1(Ω)) sense is finally obtained. Instead of the assumption ( 12), we prove a weak result that /ba∇dbl∇mε/ba∇dblL6(Ω) is uniformly bounded over a time interval independent of ε. Such a uniform estimate is nontrivial for the LLG equation, since the stand ard energy es- timate usually transforms the degenerate (damping) term in to the diffusion term and thus the upper bound becomes ε-dependent. To overcome this difficulty, we introduce the interpolation inequality when n≤3 (14)/ba∇dbldiv(aε∇m)/ba∇dbl3 L3(Ω)≤C+C/ba∇dbldiv(aε∇m)/ba∇dbl6 L2(Ω) +C/ba∇dblm×∇{div(aε∇m)}/ba∇dbl2 L2(Ω), for theS2-value function msatisfying homogeneous Neumann boundary condition. This inequality can help us derive a structure-p reserving energy estimate, in which the degenerate term is kept in the energy. The full LLG model ( 8) we considered contains the stray field, where an independent homogenization problem of potential function in the distribu- tion sense arises, and this complicated the problem when we a rrive at the consistency analysis. By using results in [ 10] and Green’s representation formula, the stray field is rewritten as the derivatives of Ne wtonian poten- tial. Then we are able to obtain the consistency error by deri ving detailed estimate of singular integral for disturbed function and bo undary layer.HOMOGENIZATION OF THE LLG EQUATION 5 The effect of boundary layer exists when we apply classic homog eniza- tion corrector to the Neumann boundary problem, which would cause the approximation deterioration on the boundary. To avoid this , a Neumann corrector is introduced, which is usually used in elliptic h omogenization problems (see [ 12] for example). In this article, we provide a strategy to apply the Neumann corrector to evolutionary LLG equations, by finding a proper higher-order modification. For a big picture, let us w rite ahead the linear parabolic equation of error ∂teε−Lεeε+fε=0, whose detailed derivation can be found in ( 26). Following the notation of eε b=eε−ωbwith boundary corrector ωb, one can find by above equa- tion that an L∞(0,T;H1(Ω)) norm of eε brelies on the boundary data and inhomogeneous term induced by ωb, which read as (15) /ba∇dblν·aε∇{eε+ωb}/ba∇dblB−1/2,2(∂Ω)and/ba∇dblLεωb/ba∇dblL2(Ω). In this end, we divide the corrector ωbinto two parts as ωb=ωN−ωM, such that they can control two terms in ( 15) respectively. Here ωNis the Neumann corrector used in elliptic problems (see [ 12]), andωMis a modi- fication to be determined. We point out the modification ωMis necessary since calculation implies some bad terms in LεωNdo not converge in L2 sense. Therefore we construct following elliptic problem t o determine ωM: div(aε∇ωM) =/parenleftBig Bad Terms in LεωN/parenrightBig with proper Neumann boundary condition. Such a solution ωMcan be proved to have better estimates than ωN, by the observation that all “Bad Terms in LεωN” can be written in the divergence form. At this point, ωM can be viewed as a high-order modification. This paper is organized as follows. In the next Section, we in troduce the main result of our article and outline the main steps of the pr oof. In Sec- tion3, multiscale expansions are used to derive the second-order corrector m2. In Section 4, we deduce that the consistency error fεonly relies on the consistency error of the stray field, which can be estimated b y calculation of singular integral for disturbed function and boundary la yer. In Section 5, we introduce the boundary corrector ωb, and derive several relevant es- timates of it. Section 6contains the stability analysis in L2andH1sense respectively. And we finally give a uniform regularity analy sis ofmε, by deriving a structure-preserving energy estimate in Sectio n7. 2.Main result To proceed, we make the following assumption Assumption 1. 1.Smoothness We assume Y-periodic functions a(y) = (aij(y))1≤i,j≤3, K(y),Ms(y), and the time-independent external field ha(x), alone with6 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN boundary ∂Ω, are sufficiently smooth. These together with the definition in (32),(46),(66),(76)leads to the smoothness of m1,m2andωb. 2.Initial data Assumem0 init(x)andmε init(x)are smooth enough and satisfy the Neumann compatibility condition: (16) ν·aε∇mε init(x) =ν·a0∇m0 init(x) = 0,x∈∂Ω. Furthermore, we might as well set them satisfying periodica lly disturbed el- liptic problem: (17) div(aε∇mε init(x)) = div( a0∇m0 init(x)),x∈Ω. (16)-(17)implym0 init(x)is the homogenization of mε init(x). Note that the as- sumption (17)is necessary not only for the convergence analysis in Theorem 4, but also for the uniform estimate of mεin Theorem 8. Now let us state our main result: Theorem 1. Letmε∈L∞(0,T;H2(Ω)),m0∈L∞(0,T;H6(Ω))be the unique solutions of (8)and(10), respectively. Under Assumption 1, there exists some T∗∈(0,T]independent of ε, such that for any t∈(0,T∗)and forn= 2,3, it holds (18)/ba∇dblmε(t)−m0(t)/ba∇dblL2(Ω)≤β(ε),/ba∇dblmε(t)−m0(t)/ba∇dblH1(Ω)≤Cε1/2, where (19) β(ε) =/braceleftBigg Cε[ln(ε−1+1)]2,whenn= 2, Cε5/6,whenn= 3. In the absence of the stray field, i.e., µ0= 0, then it holds for any t∈(0,T∗) and forn= 1,2,3 (20)/ba∇dblmε(t)−m0(t)/ba∇dblL2(Ω)≤Cε[ln(ε−1+1)]2, /ba∇dblmε(t)−m0(t)−(Φ−x)∇m0(t)/ba∇dblH1(Ω)≤Cε[ln(ε−1+1)]2, wherexis spatial variable, Φ= (Φi)1≤i≤nis the corrector defined in (67). Constant Cdepends on the initial data mε initandm0 init, but is independent ofε. Remark 2.1. Comparing (18)and(20), one can see that in the L2norm, the stray field makes little influence when n= 2, but causes 1/6-order loss of rate when n= 3. In the H1norm, however, the stray field leads to 1/2- order loss of rate in both cases. Such a deterioration of conv ergence rate is induced since the zero-extension has been applied for stray field(3), which introduces a boundary layer. Remark 2.2. The logarithmic growth [ln(ε−1+1)]2in(20)is caused by the Neumann corrector (Φ−x)∇m0. For problems (8)and(10)with periodic boundary condition over a cube, by replacing the Neumann cor rector in (20)HOMOGENIZATION OF THE LLG EQUATION 7 with the classical two-scale corrector, a similar argument in the current work leads to (21) /ba∇dblmε−m0−χ∇m0/ba∇dblH1(Ω)≤Cε, whereχ= (χi)1≤i≤nis defined in (33). Note that (21)is consistent with the L2result in [8]. However, only the uniform boundedness in H1has been shown in [8], while our results (20)and (21)imply that it maintains the same convergence rate in L2andH1norm, by choosing the correctors satisfying specific geometric pr operty(13). 2.1.Some notations and Lax equivalence type theorem. Recall that a classical solution to ( 8) also satisfies an equivalent form of equation, reads (22)LLLG(mε) :=∂tmε−αHε e(mε)+mε×Hε e(mε)−αgε l[mε]mε= 0, where the gε l[·] is the energy density calculated by (23)gε l[mε] =aε|∇mε|2+Kε(mε·u)u−hd[Mεmε]·Mεmε−ha·Mεmε. For convenience, we also define a bilinear operator deduced f rom (23), which reads Bε[m,n] =aε∇m·∇n+Kε(m·u)(n·u)−µ0hd[Mεm]·Mεn. Now let us set up the equation of error, in terms of Lax equival ence theorem kind of argument. Define the approximate solution (24) /tildewidemε(x) =m0(x)+εm1(x,x ε)+ε2m2(x,x ε), wherem0is the homogenized solution to ( 10),m1is the first-order corrector definedin( 32), andm2isthesecond-ordercorrectordeterminedbyTheorem 2. Then replacing mεby/tildewidemεin (22) provides the notation of consistence errorfε: (25) LLLG(/tildewidemε) =fε. Together ( 22) and (25), we can obtain the equation of error eε=mε−/tildewidemε, denoted by (26) ∂teε−Lεeε+fε=0, whereLεis second-order linear elliptic operator depending on mεand/tildewidemε, that can be characterized as (27) Lε(eε) =α/tildewideHε e(eε)−D1(eε)−D2(eε). Here/tildewideHε e(mε) is the linear part of Hε e(mε), i.e., /tildewideHε e(mε) :=Hε e(mε)−Mεha, procession term D1is calculated by (28)D1(eε) =mε×Hε e(mε)−/tildewidemε×Hε e(/tildewidemε) =mε×/tildewideHε e(eε)+eε×Hε e(/tildewidemε),8 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN and the degeneracy term D2reads as D2(eε) =−αgε l[mε]mε+αgε l[/tildewidemε]/tildewidemε =−α/parenleftbig Bε[eε,mε]+Bε[/tildewidemε,eε]+Mε(ha·eε)/parenrightbig mε−αgε l[/tildewidemε]eε. Moreover, we define a correctional error eε bas (29) eε b=eε−ωb, whereωbis the boundary corrector satisfying ωb=ωN−ωM, forωNthe Neumann corrector given in ( 66), andωMthe modification determined in (76). Then equation ( 26) leads to (30) ∂teε b−Lεeε b+/parenleftbig ∂tωb−Lεωb+fε/parenrightbig = 0. By the Lax equivalence theorem kind of argument, the estimat e of error eε bfollows from consistency analysis of ( 25), energy estimate of boundary corrector, and stability analysis of ( 30). 2.2.Proof of Theorem 1. Proof.Following the above notations, for the consistency error fε, Theorem 3says that it can be divided as fε=f0+/tildewidef, satisfying /ba∇dbl/tildewidef(t)/ba∇dblL2(Ω)≤Cε, and /ba∇dblf0(t)/ba∇dblL2(Ω)= 0,whenµ0= 0, /ba∇dblf0(t)/ba∇dblLr(Ω)≤Crµ0/parenleftbig ε1/r+εln(ε−1+1)/parenrightbig ,whenµ0>0,n/ne}ationslash= 1, where constants CrandCare independent of ε, for any t∈(0,T), and 1≤r <+∞. Consideringtheboundarycorrectortermsin( 30), byTheorem 5there exists C=C(/ba∇dbl∇mε/ba∇dblL2(Ω)) such that /ba∇dbl∂tωb(t)/ba∇dblL2(Ω)≤Cεln(ε−1+1), /ba∇dblLεωb(t)/ba∇dblL2(Ω)≤Cε[ln(ε−1+1)]2+C/ba∇dbleε b(t)/ba∇dblH1(Ω), for anyt∈(0,T). As for initial-boundary data of eε b, using Theorem 4we write with C=C(/ba∇dbl∇mε/ba∇dblL2(Ω)), /ba∇dbleε b(x,0)/ba∇dblH1(Ω)+/ba∇dbl∂ ∂νεeε b/ba∇dblW1,∞(0,T;B−1/2,2(∂Ω))≤Cεln(ε−1+1). Now let us turn to stability analysis of ( 30). For the L∞(0,T;L2(Ω)) norm, let σ= 1 when n= 1,2, andσ= 6/5 whenn= 3, we can apply Theorem 6to derive for n= 1,2,3 (31)/ba∇dbleε b/ba∇dbl2 L∞(0,T;L2(Ω))+/ba∇dbl∇eε b/ba∇dbl2 L2(0,T;L2(Ω)) ≤Cδ/parenleftBig /ba∇dbleε b(x,0)/ba∇dbl2 L2(Ω)+/ba∇dbl∂ ∂νεeε b/ba∇dbl2 L2(0,T;B−1/2,2(∂Ω))+/ba∇dbl/tildewidef/ba∇dbl2 L2(0,T;L2(Ω)) +/ba∇dbl∂tωb/ba∇dbl2 L2(0,T;L2(Ω))+γ(ε)/ba∇dblf0/ba∇dbl2 L2(0,T;Lσ(Ω))/parenrightBig +δ/ba∇dblLεωb/ba∇dbl2 L2(0,T;L2(Ω))+ε2/ba∇dblAεeε b/ba∇dbl2 L2(0,T;L2(Ω)).HOMOGENIZATION OF THE LLG EQUATION 9 with /braceleftBigg γ(ε) = 1, whenn= 1,3, γ(ε) = [ln(ε−1+1)]2,whenn= 2. Constant Cδ=Cδ(/ba∇dbl∇mε/ba∇dblL4(Ω)). Now taking δsmall enough in ( 31), and using the fact /ba∇dblAεeε b/ba∇dblL2(0,T;L2(Ω))≤Cln(ε−1+1) withC=C/parenleftbig /ba∇dblAεmε/ba∇dblL2(Ω)/parenrightbig from Theorem 5, we finally obtain /ba∇dbleε b/ba∇dblL∞(0,T;L2(Ω))≤/braceleftBiggβ(ε), whenµ0>0,n= 2,3, Cε[ln(ε−1+1)]2,whenµ0= 0,n= 1,2,3, whereβ(ε) satisfies ( 19). Using the fact mε−m0=eε b+εm1+ε2m2+ωb, along with the estimates of εm1,ε2m2,ωbin Lemma 4-5, we obtain the L2 estimates in Theorem 1. As for the stability of ( 30) inL∞(0,T;H1(Ω)) norm, we can apply The- orem7to obtain for n= 1,2,3 /ba∇dbl∇eε b/ba∇dbl2 L∞(0,T;L2(Ω))≤C/parenleftBig /ba∇dbleε b(x,0)/ba∇dbl2 H1(Ω)+/ba∇dbl∂ ∂νεeε b/ba∇dbl2 H1(0,T;B−1/2,2(∂Ω)) +/ba∇dblLεωε b/ba∇dbl2 L2(0,T;L2(Ω))+/ba∇dblfε/ba∇dbl2 L2(0,T;L2(Ω))+/ba∇dbl∂tωb/ba∇dbl2 L2(0,T;L2(Ω))/parenrightBig , where constant C=C(/ba∇dblAεmε/ba∇dblL2(Ω),/ba∇dbl∇mε/ba∇dblL4(Ω)). Together with above results, and estimate for /ba∇dbl∇eε b/ba∇dbl2 L2(0,T;L2(Ω))in (31), we arrive at /ba∇dbl∇eε b(t)/ba∇dbl2 L∞(0,T;L2(Ω))≤/braceleftBigg Cε1/2, whenµ0>0,n= 2,3, Cε[ln(ε−1+1)]2,whenµ0= 0,n= 1,2,3, by the representation of eε bin (80), together with estimate of m2andωM in Lemma 5, it leads to the H1estimates in Theorem 1. Notice that all the constants in our estimate depend on the va lue of /ba∇dblAεmε(t)/ba∇dblL2(Ω)and/ba∇dbl∇mε(t)/ba∇dblL4(Ω), which from Theorem 8are uniformly bounded with respect to εandtfor anyt∈(0,T∗), with some T∗∈(0,T]. This completes the proof. /square 3.The Asymptotic Expansion In this section, we derive the second-order corrector using the formal asymptotic expansion. First, let us define the first-order co rrectorm1by (32) m1(x,y) =n/summationdisplay j=1χj(y)∂ ∂xjm0(x), whereχj,j= 1,...,nare auxiliary functions satisfying cell problem (33) div/parenleftbig a(y)∇χj(y)/parenrightbig =−n/summationdisplay i=1∂ ∂yiaij(y), χjY-periodic ,10 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN such that the first geometric property in ( 13) holds. As for the second-order corrector m2, we assume it as a two-scale function satisfying /braceleftBigg m2(x,y) is defined for x∈Ω andy∈Y, m2(·,y) isY-periodic . For notational convenience, given a two-scale function in t he form of m(x,x ε), we denote the fast variable y=x εand have the following chain rule (34) ∇m(x,x ε) = [(∇x+ε−1∇y)m](x,y). Moreover, denoting Aε= div(aε∇), one can rewrite Aεm(x,x ε) = [(ε−2A0+ε−1A1+A2)m](x,y), where (35) A0= divy/parenleftbig a(y)∇y/parenrightbig , A1= divx/parenleftbig a(y)∇y/parenrightbig +divy/parenleftbig A(y)∇x/parenrightbig , A2= divx/parenleftbig a(y)∇x/parenrightbig . The procedure to determine m2is standard. With the notation in ( 24), assumemεcan be written in form of (36) mε(x) =/tildewidemε(x)+o(ε2). One can derive m2by substituting ( 36) into (8) and comparing like terms of ε. However, it is a bit fussy in the presence of stray field. Let u s outline the main steps here. Revisiting the stray field hd[Mεmε(x)] =∇Uεin (2)-(3), one finds that the potential function Uε=Uε[Mεmε(x)] satisfies (37) ∆Uε=−div(Ms(x ε)mεXΩ). Substituting Uε= Σ2 j=0εjUj(x,x ε)+o(ε2) and (36) into (37) and combining like terms of εleads to (38) divy(∇yU0(x,y)) = 0, divy(∇yU1(x,y)) =−divy(Ms(y)m0(x)XΩ(x)), divx(∇xU0(x,y))+2div y(∇xU1(x,y))+div y(∇yU2(x,y)) =−Ms(y)divxm0(x)XΩ(x)−divy(Ms(y)m1(x,y)XΩ(x)). The first equation in ( 38) implies that U0(x,y) =U0(x) since the Lax- Milgram Theorem ensures the uniqueness and existence of sol ution (up to a constant). Integrating the third equation in ( 38) with respect to yyields ∆U0(x) =−div(M0m0XΩ). An application of ( 2)-(3) implies that U0is actually the potential function ofhd[Mhm0], i.e., (39) ∇U0(x) =hd[Mhm0] =Mhhd[m0].HOMOGENIZATION OF THE LLG EQUATION 11 With notation given in ( 7), one can deduce from the second equation in ( 38) thatm0(x)XΩ(x)U(y) =U1(x,y) up to a constant in the H1(Y) space. Hence it follows that by ( 7) (40) ∇yU1(x,y) =XΩ(x)m0(x)·Hd[Ms(y)](y). Substituting ( 39) and (40) into the expansion of Uε, one can deduce that, forx∈Ω, (41)hd[Mεmε] =∇Uε=hd[Mhm0]+m0(x)·Hd[Ms(y)](x ε)+O(ε). Substituting ( 36), (32), (35), (41) into (8) and collecting terms of O(ε0), we obtain the following equations (42)/braceleftBigg ∂tm0−αm0×∂tm0=−(1+α2)m0×{A0m2+Ha e}, m2Y-periodic in y, where (43)Ha e=A1m1+A2m0−Kε(m0·u)u +µ0Mshd[Mhm0]+µ0Msm0·Hd[Ms(y)]+Msha. Substituting ( 10) into (42) leads to (44)/braceleftBigg m0×A0m2=m0×/braceleftbig H0 e(m0)−Ha e/bracerightbig , m2Y-periodic in y. (44) is the degenerate system that determines m2in terms of m0. 3.1.Second-order corrector. Thewell-posednessof ( 44)isnontrivial due tothedegeneracy. InthefollowingTheorem, bysearchingas uitablesolution satisfying ( 13), we give the existence result, andderive an explicit expre ssion ofm2in terms of m0and some auxiliary functions. Theorem 2. Givenm0∈L∞/parenleftbig [0,T];H2(Ω)/parenrightbig the homogenization solution andm1calculated in (32), define V=/braceleftBig m∈H2(Y)∩H1 per(Y) :m·m0=−1 2|m1|2a.e. inΩ×Y/bracerightBig , then(44)admits a unique solution m2(x,y)∈ V/Tm0(S2), with notation Tm0(S2)denoting the tangent space of m0. Proof.Assume m2(x,y)∈ V, i.e.,m2·m0=−1 2|m1|2. Applying A0to both sides of it yields (45) m0·A0m2=−a(y)∇ym1·∇ym1−m1·A0m1. Taking the cross-product with m0to (44) and substituting ( 45) lead to (46)A0m2=−{Ha e−H0 e(m0)}+/braceleftbig m0·/parenleftbig Ha e−H0 e(m0)/parenrightbig/bracerightbig m0 −(m1·A0m1+a(y)∇ym1·∇ym1)m0.12 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN Now using the fact (47)/integraldisplay YH0 e(m0)−Ha edy= 0, together with ( 45), one can check equation ( 46) satisfies the compatibility condition for Y-periodic function m2iny. Thus by the application of Lax- Milgram Theorem andsmoothnessassumption, ( 46) admitsauniqueregular solution up to a function independent of y, denoted by m2(x,y)+/tildewidem2(x). Moreover, one can determine /tildewidem2(x) by m0·(m2+/tildewidem2) =−1 2|m1|2, such that m2+/tildewidem2∈ V, and therefore is also a solution to ( 44) by taking the above transformation inversely. /square Remark 3.1. One can check that equation (46)has a solution (48) m2=n/summationdisplay i,j=1θij∂2m0 ∂xi∂xj+n/summationdisplay i,j=1(θij+1 2χiχj)/parenleftbigg∂m0 ∂xi·∂m0 ∂xj/parenrightbigg m0+Tlow−(m0·Tlow)m0 with low-order terms Tlowcalculated by Tlow=−κ(m0·u)u+µ0ρhd[Mhm0]+µ0m0·Λ+Msha, whereθijandκ,ρ,Λare given by (49) A0θij=a0 ij−/parenleftbig aij+n/summationdisplay k=1aik∂χj ∂yk/parenrightbig −n/summationdisplay k=1∂(aikχj) ∂yk, A0ρ=Ms(y)−M0,A0κ=K(y)−K0, A0Λ=Ms(y)Hd[Ms(y)](y)−H0 d, θij, κ, ρ,Λ,areY-periodic . Moreover, one can find m2defined above satisfies geometric property (13), therefore is also the solution to equation (44). In the following, we may assume second-order correct m2takes the form in (48). 4.Consistency Estimate In this section, we aim to estimate the consistence error fεdefined in (25). Following the notation in ( 34)-(35), by the definition of /tildewidemε, (25) can be written in terms of fε=ε−2f−2+ε−1f−1+f0+εf1+ε2f2. It is easy to check that f−2=f−1=0by the definition of m0,m1in Section3. Along the same line, by the H¨ older’s inequality, one has /ba∇dblf1/ba∇dblL2(Ω)+/ba∇dblf2/ba∇dblL2(Ω)≤C,HOMOGENIZATION OF THE LLG EQUATION 13 whereCdepends on the L2(Ω) andL∞(Ω) norms of mi(x,x ε),∇xmi(x,x ε), ∇ymi(x,x ε),i= 0,1,2, and thus is bounded from above by /ba∇dbl∇m0/ba∇dblH4(Ω) with the help of smoothness assumption and Sobolev inequali ty. It remains to estimate f0, let us prove that f0only depends on the con- sistence error of stray field, by the help of geometric proper ty (13). Denote the consistence error of stray field by (50)/tildewideh=µMεhd[(Mε−Mh)m0]−µMεHd[Ms(y)](x ε)·m0 withHdgiven in ( 7). After some algebraic calculations and the usage of (42) and (43), one has (51)f0=∂tm0−α/braceleftBig A0m2+Ha e+/tildewideh/bracerightBig +m0×/braceleftBig A0m2+Ha e+/tildewideh/bracerightBig −αgε l[m0]m0−(aε∇ym1·∇ym1)m0−2(aε∇ym1·∇m0)m0. Notice that the classical solution m0to (10) also satisfies the equivalent form ∂tm0−αH0 e(m0)+m0×H0 e(m0)−αg0 l[m0]m0= 0, (52) where g0 l[m] :=a0|∇m|2+K0(m·u)u−µ0(M0)2hd[m]·m −µ0m·H0 d·m−ha·M0m. Substituting ( 52) into (51) and using ( 46) lead to f0=−α/tildewideh+m0×/tildewideh−α/braceleftbig m0·/parenleftbig Ha e−H0 e(m0)/parenrightbig/bracerightbig m0 +α(m1·A0m1−2aε∇ym1·∇m0)m0+αg0 l[m0]m0−αgε l[m0]m0. (53) Note that A2m0=Aεm0−ε−1A1m0, one can deduce Ha e=Hε e(m0)+A1m1+ε−1A0m1−/tildewideh. Substituting it into ( 53), and using the fact m0·Hε e(m0) =−gε l[m0],m0·H0 e(m0) =−g0 l[m0], one has (54)f0=−α/tildewideh+m0×/tildewideh−α/braceleftbig m0·/parenleftbig A1m1+ε−1A0m1−/tildewideh/parenrightbig/bracerightbig m0 +α(m1·A0m1−2aε∇ym1·∇m0)m0. ApplyA0andA1to both sides of m0·m1= 0 respectively, and substitute resulting equations into ( 54). After simplification, we finally obtain (55) f0=−α/tildewideh+m0×/tildewideh+α/parenleftBig m0·/tildewideh/parenrightBig m0. (55) implies that the convergence of fεdepends on the convergence of stray field error /tildewideh. In fact, we have14 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN Lemma 1. For any 1≤r <∞, andn= 2,3, it holds that (56)/vextenddouble/vextenddoubleMεhd[(Mε−Mh)m0]−MεHd[Ms(y)](x ε)·m0/vextenddouble/vextenddouble Lr(Ω) ≤Cε1/r+Cεln(ε−1+1), whereCdepends on /ba∇dbl∇m0/ba∇dblW1,∞(Ω),/ba∇dbl∇Ms(y)/ba∇dblH1(Y)and is independent of ε. Proof of Lemma 1will be given in Section 4.2. Lemma 1directly leads to the consistency error: Theorem 3. (Consistency) Given fεdefined in (25), it can be divided as fε=f0+/tildewidef, satisfying /ba∇dbl/tildewidef/ba∇dblL2(Ω)≤Cε, and (57)/ba∇dblf0/ba∇dblL2(Ω)= 0,whenµ0= 0, /ba∇dblf0/ba∇dblLr(Ω)≤Crµ0/parenleftbig ε1/r+εln(ε−1+1)/parenrightbig ,whenµ0>0,n/ne}ationslash= 1, for any1≤r <+∞. Here constant CandCrdepend on /ba∇dbl∇m0/ba∇dblH4(Ω), /ba∇dbl∇Ms(y)/ba∇dblH1(Y), and are independent of ε. 4.1.Estimate of some singular integral. The strategy to prove Lemma 1is to rewrite the stray field into derivatives of Newtonian po tential, thus the consistency estimate turns into the estimate of singula r integrals. The following Lemmas introduce the estimate of singular integr al in terms of distribution function and boundary layer. We will use the cu t-off function ηεwithin the interior of area away from boundary: (58) 0≤ηε≤1,|∇ηε| ≤Cε−1, ηε(x) = 1 if dist( x,∂Ω)≥2 3ε, ηε(x) = 0 if dist( x,∂Ω)≤1 3ε. where dist( x,∂Ω) denotes the distance between xand∂Ω, and cut-off func- tionφεin a small ball: (59) 0≤φε≤1,|∇φε| ≤Cε−1, φε(x) = 1 if |x| ≤1 3ε, φε(x) = 0 if |x| ≥2 3ε. Denote the boundary layer Ωεas Ωε={x∈Ω,dist(x,∂Ω)≤ε}. Lemma 2. Assume that scalar functions f(y)∈C1(Rn)isY-periodic, g(x)∈C1(¯Ω), define for x∈Ω u(x) =/integraldisplay Ω/vextendsingle/vextendsinglef(x ε)−f(z ε)/vextendsingle/vextendsingle |x−z|ndz, v(x) =/integraldisplay Ωε|g(x)−g(z)| |x−z|ndz, thenu(x)∈L∞(Ω)logarithmically grows with respect to ε, satisfying /ba∇dblu/ba∇dblL∞(Ω)≤Cln(ε−1+1)/ba∇dblf(y)/ba∇dblL∞(Y)+C/ba∇dbl∇f(y)/ba∇dblL∞(Y),HOMOGENIZATION OF THE LLG EQUATION 15 andv(x)∈Lr(Ω)decreases at speed of O(ε1/r)for any1≤r <∞, satisfying /ba∇dblv/ba∇dblLr(Ω)≤Cε1/rln(ε−1+1)/parenleftbig /ba∇dblg(x)/ba∇dblL∞(Ω)+ε/ba∇dbl∇g(x)/ba∇dblL∞(Ω)/parenrightbig . Constant Cis independent of ε. Proof.Splitting the integral in uinto/integraltext Ω−B(x,ε)+/integraltext B(x,ε), one can estimate it by |u(x)| ≤C/integraldisplay Ω−B(x,ε)/ba∇dblf(y)/ba∇dblL∞(Y) |x−z|ndz+Cε−1/integraldisplay B(x,ε)/ba∇dbl∇f(y)/ba∇dblL∞(Y) |x−z|n−1dz, therefore the estimate of uin Lemma follows by simple integral. As for the estimate of v, by application of cut-off function φε=φε(x−z), one has |v(x)|=/integraldisplay Ωεφε|g(x)−g(z)| |x−z|ndz+/integraldisplay Ωε(1−φε)|g(x)−g(z)| |x−z|ndz ≤C/ba∇dbl∇g/ba∇dblL∞(Ω)/integraldisplay Ωεφε |x−z|n−1dz+C/ba∇dblg/ba∇dblL∞(Ω)/integraldisplay Ωε1−φε |x−z|ndz =R1+R2. ForR1, one can write by Fubini’s Theorem /ba∇dblR1/ba∇dblr Lr(Ω)≤C/ba∇dbl∇g/ba∇dblr L∞(Ω)/integraldisplay Ω/parenleftBig/integraldisplay Ωεφε |x−z|n−1dz/parenrightBigr dx ≤C/ba∇dbl∇g/ba∇dblr L∞(Ω)sup x∈Ω/parenleftBig/integraldisplay Ωεφε |x−z|n−1dz/parenrightBigr−1 ×sup z∈Ωε/integraldisplay Ωφε |x−z|n−1dx/integraldisplay Ωε1dz ≤C/ba∇dbl∇g/ba∇dblr L∞(Ω)·Cεr−1·Cε·Cε. As forR2, applying the same argument leads to /ba∇dblR2/ba∇dblr Lr(Ω)≤C/ba∇dblg/ba∇dblr L∞(Ω)sup x∈Ω/parenleftBig/integraldisplay Ωε1−φε |x−z|ndz/parenrightBigr−1 ×sup z∈Ωε/integraldisplay Ω1−φε |x−z|ndx/integraldisplay Ωε1dz ≤C/ba∇dblg/ba∇dblr L∞(Ω)·C[ln(ε−1+1)]r−1·Cln(ε−1+1)·Cε. /square Lemma 3. Assume that a scalar function fε(x)∈L∞(Ω)satisfiesfε(x) = 0whenx∈Ω−Ωε, which means fεis nonzero only in boundary layer. Let w(x)be the Newtonian potential of fεinΩ, i.e., w(x) =/integraldisplay ΩΦ(x−z)fε(z)dz,x∈Ω, whereΦis the fundamental solution of Laplace’s equation. Then w(x)∈ W2,p(Ω)satisfies for any 1≤p <+∞ /ba∇dbl∇2w/ba∇dblLp(Ω)≤C/parenleftbig ε1/p+εln(ε−1+1)/parenrightbig/parenleftbig /ba∇dblfε(x)/ba∇dblL∞(Ω)+ε/ba∇dbl∇fε(x)/ba∇dblL∞(Ω)/parenrightbig .16 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN Constant Cis independent of ε. Proof.The case of 1 < p <+∞follows directly by the property of Newto- nian potential: /ba∇dbl∇2w/ba∇dblLp(Ω)≤C/ba∇dblfε/ba∇dblLp(Ω)≤C/ba∇dblfε/ba∇dblLp(Ωε), and the fact (60) /ba∇dblfε/ba∇dblLp(Ωε)≤ |Ωε|1/p/ba∇dblfε/ba∇dblL∞(Ω). Now let us consider the case of p= 1 and write ∂2w ∂xi∂xj=/integraldisplay Ω∂2 ∂xi∂xj/braceleftbig Φ(x−z)/bracerightbig ·/braceleftbig fε(z)−fε(x)/bracerightbig dz +fε(x)/integraldisplay ∂Ωνi·∂ ∂xj/braceleftbig Φ(x−z)/bracerightbig dz =:S1+S2. ForS1, one can apply Lemma 2to derive /ba∇dblS1/ba∇dblL1(Ω)≤Cεln(ε−1+1)/parenleftbig /ba∇dblfε(x)/ba∇dblL∞(Ω)+ε/ba∇dbl∇fε(x)/ba∇dblL∞(Ω)/parenrightbig . As forS2, we can split the integral into/integraltext ∂Ω−B(x,ε)+/integraltext ∂Ω∩B(x,ε), and write /ba∇dblS2/ba∇dblL1(Ω)≤sup x∈Ω/integraldisplay ∂Ω−B(x,ε)νi·∂ ∂xj/braceleftbig Φ(x−z)/bracerightbig dz×/integraldisplay Ωfε(x)dx + sup z∈∂Ω/integraldisplay Ω∩B(z,ε)νi·∂ ∂xj/braceleftbig Φ(x−z)/bracerightbig ·fε(x)dx×/integraldisplay ∂Ω1dz ≤Cln(ε−1+1)×ε/ba∇dblfε/ba∇dblL∞(Ω)+Cε/ba∇dblfε/ba∇dblL∞(Ω), here in the second line we have used the Fubini’s theorem. Thu s the Lemma is proved. /square 4.2.Consistency error of stray field. Now we are ready to prove the consistency error of stray field /tildewidehin Lemma 1. The idea is to use result in [10] and Green’s representation formula, to rewrite /tildewidehinto singular integral that of the types estimated in above Lemmas. Proof.(Proof of Lemma 1) Recall from ( 2) the stray field in LLG equation can be calculated by (61) hd[(Mε−Mh)m0] =∇U, whereU=U[(Mε−Mh)m0] satisfies ∆U=−div[(Mε−Mh)m0XΩ] inD′(Rn). Denotes the ith component of m0bym0,i. Using the fact |m0|= 1, one can write [10] U(x) =−n/summationdisplay i=1/integraldisplay Ω∂ ∂xiΦ(x−z)(Ms(z ε)−Mh)m0,i(z)dz.HOMOGENIZATION OF THE LLG EQUATION 17 Substituting above representation of U(x) into (61) and making the use of cut-off function ηεdefined in ( 58), one can derive (62)hd[(Mε−Mh)m0] =−∇/parenleftBign/summationdisplay i=1/integraldisplay Ω∂ ∂xiΦ(x−z)ηε(z)(M(z ε)−Mh)m0,i(z)dz/parenrightBig −∇/parenleftBign/summationdisplay i=1/integraldisplay Ωε∂ ∂xiΦ(x−z)(1−ηε(z))(M(z ε)−Mh)m0,i(z)dz/parenrightBig =:Pε+/tildewidePε, where/tildewidePεis the derivative of Newtonian potential in boundary layer t hat can be estimated by Lemma 3. Define/tildewideU(y) as the solution of (63) ∆/tildewideU(y) =−(Ms(y)−Mh), U(y) isY-periodic in y, then one can write from ( 7) and (63) that (64)Hd[Ms(y)](x ε) =ε2∇2/tildewideU(x ε) =ε2∇2/braceleftbig ηε(x)/tildewideU(x ε)/bracerightbig +ε2∇2/braceleftbig (1−ηε(x))/tildewideU(x ε)/bracerightbig . Note that by Green’s representation formula, ε2ηε(x)/tildewideU(x ε) =−/integraldisplay ΩΦ(x−z)∆/parenleftbig ε2ηε(z)/tildewideU(z ε)/parenrightbig dz. Substituting the above formula into ( 64) and using the fact of /tildewideU −∆/parenleftbig ε2ηε(z)/tildewideU(z ε)/parenrightbig =ηε(M(z ε)−Mh)−/braceleftbig ε2∆ηε(z)·/tildewideU(z ε)+2ε2∇ηε(z)·∇/tildewideU(z ε)/bracerightbig , we finally obtain m0·Hd[Ms(y)](x ε) =m0·∇2/integraldisplay ΩΦ(x−z)ηε(z)(M(z ε)−Mh)dz+m0·/braceleftBig ε2∇2/braceleftbig (1−ηε(x))/tildewideU(x ε)/bracerightbig +∇2/integraldisplay ΩΦ(x−z)/braceleftbig ε2∆ηε(z)·/tildewideU(z ε)+2ε2∇ηε(z)·∇/tildewideU(z ε)/bracerightbig dz/bracerightBig =:Qε+/tildewideQε, where the boundary layer term /tildewideQεcan be estimated by Lemma 3and (60). Now in order to estimate the left-hand side of ( 56) in the Lemma, it only remains to consider the term Pε−Qε. Notice that one can write (65)Pε−Qε=n/summationdisplay i=1/integraldisplay Ω∂ ∂xi/braceleftbig ∇xΦ(x−z)/bracerightbig ηε(z) ×(M(z ε)−Mh)/parenleftbig m0,i(x)−m0,i(z)/parenrightbig dz.18 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN With the notation ( 63), one has (M(z ε)−Mh) =∇z·/braceleftbig ∇z(ε2/tildewideU(z ε))−∇x(ε2/tildewideU(x ε))/bracerightbig . After substituting it into ( 65) and applying integration by parts, the leading integrals are estimated directly by application of Lemma 2. /square 5.Boundary Corrector 5.1.Neumann corrector. Let usgive thedefinition of Neumanncorrector ωNas (66) ωN=n/summationdisplay i=1/parenleftbig Φi−xi−εχε i/parenrightbig∂m0 ∂xi with the notation χε i(x) =χi(x ε),xiis theith component of spatial variable, and (Φ i)1≤i≤nis given by (67) div(aε∇Φi) = div(a0∇xi) in Ω, ∂ ∂νεΦi=∂ ∂νhxion∂Ω. Here we denote∂ ∂νε=ν·aε∇,∂ ∂νh=ν·a0∇. Thusxiis the homogenized solution of Φ ifrom (67). Since Φ iis unique up to a constant, one may assume Φ i(˜x)−˜x= 0 for some ˜x∈Ω. We introduce that Φ i−xi−εχε ihas following property. Lemma 4. ForΦigiven in (67), under the smoothness assumption on A(y) and∂Ω, it holds that (see [12]) (68) /ba∇dbl∇Φi−∇xi−ε∇χε i/ba∇dblL∞(Ω)≤C,/ba∇dbl∇2Φi/ba∇dblL∞(Ω)≤C, and (69) /ba∇dblΦi−xi/ba∇dblL∞(Ω)≤Cεln(ε−1+1), whereCis independent of ε. Proof.In fact, one has the estimate /vextendsingle/vextendsingle∇Φi−∇xi−ε∇χε i/vextendsingle/vextendsingle≤Cmax{1,ε[dist(x,∂Ω)]−1} from Lemma 7 .4.5 in [12]. This, together with the fact Φ i(˜x)−˜x= 0 , yields (69) by following integrals: |Φi(x)−xi|=/vextendsingle/vextendsingle/vextendsingle/integraldisplay1 0d ds/braceleftBig Φi/parenleftbig ˜x+s(x−˜x)/parenrightbig −/parenleftbig ˜xi+s(xi−˜xi)/parenrightbig/bracerightBig ds/vextendsingle/vextendsingle/vextendsingle ≤C/integraldisplay1 0max{1,ε(1−s)−1}ds≤Cεln(ε−1+1), for anyx∈Ω.HOMOGENIZATION OF THE LLG EQUATION 19 As for the second inequality in ( 68), we prove by making use of the Neu- mann function for operator Aεfrom [12] Section 7.4, denoted by Nε(x,z), and write from ( 67) that (70) Φi(x) =−n/summationdisplay k=1/integraldisplay ∂Ωνk·a0 kiNε(x,z)dz+1 |∂Ω|/integraldisplay ∂ΩΦi(z)dz. Let us denote the projection of∂ ∂xjalong∂ ∂νεbyPxj, and define P⊥ xj= ∂ ∂νε−Pxj, one can write for z∈∂Ω ∂ ∂zjNε(x,z) =/parenleftbig Pzj+P⊥ zj/parenrightbig Nε(x,z) =P⊥ zjNε(x,z). Now applying∂2 ∂xl∂xjto both sides of ( 70), using above formula and integra- tion by parts on ∂Ω lead to (71)∂2 ∂xl∂xjΦi(x) =−n/summationdisplay k=1/integraldisplay ∂ΩP⊥ zlP⊥ zjνk(z)·a0 kiNε(x,z)dz, here we have used the fact that P⊥ zlis a tangential derivative on the bound- ary, and Nε(x,z) =Nε(z,x) by the symmetry of Aε. (71) implies the second inequality in ( 68) by the smoothness assumption of boundary. /square 5.2.A high-order modification. As noted in Section 1, we use ωNto control the Neumann boundary data, and use a modification fun ctionωM to control the inhomogeneous term that induced by ωN, written in ( 15) separately. In order to explain the construction of the modi fication function, we point out that there are some bad terms appear whenwe calcu lateLεωN, which have no convergence in L2norm. Denote the bad terms by T1 badand T2 bad, then they can be written as (72)T1 bad=2n/summationdisplay i,j,k=1∂ ∂xk/braceleftBig aε ki/parenleftbig Φj−xj−εχε j/parenrightbig ·∂2m0 ∂xi∂xj/bracerightBig −n/summationdisplay i,j,k=1/braceleftBig∂ ∂xkaε ik·/parenleftbig Φj−xj−εχε j/parenrightbig/bracerightBig∂2m0 ∂xi∂xj, and T2 bad=αn/summationdisplay i,j=1/parenleftbig aε ij∂ωN ∂xi·∂{2/tildewidemε+ωN} ∂xj/parenrightbig/parenleftbig/tildewidemε+ωN/parenrightbig . Notice that these terms cannot converge for the existence of∂ωN ∂xiand∂aε ik ∂xk. Now let us rewrite T1 badandT2 badinto divergence form up to a small term. ForT1 bad, notice that/summationtextn k=1∂aε ik ∂xk=Aε(εχε i) from (33), substitute it into the20 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN second term on the right-hand side of ( 72), it leads to (73)T1 bad=n/summationdisplay k,l=1∂ ∂xk/parenleftbig aε klG1 l(x)/parenrightbig +n/summationdisplay i,j=1εχε i·Aε/braceleftBig/parenleftbig Φj−xj−εχε j/parenrightbig ·∂2m0 ∂xi∂xj/bracerightBig , whereG1 l(x) in the divergence term reads G1 l(x) =2n/summationdisplay j=1/parenleftbig Φj−xj−εχε j/parenrightbig∂2m0 ∂xl∂xj+n/summationdisplay i,j=1/braceleftBig∂ ∂xl(εχε i) ·/parenleftbig Φj−xj−εχε j/parenrightbig −εχε i·∂ ∂xl/parenleftbig Φj−xj−εχε j/parenrightbig/bracerightBig∂2m0 ∂xi∂xj. As forT2 bad, a direct calculation implies it can be rewritten as T2 bad=n/summationdisplay k,l=1∂ ∂xk/parenleftbig aε klG2 l(x)/parenrightbig −αn/summationdisplay i,j=1aε ij/parenleftbig ωN·∂{2/tildewidemε+ωN} ∂xj/parenrightbig/parenleftbig /tildewidemε+ωN/parenrightbig −αn/summationdisplay i,j=1/parenleftbig ωN·Aε{2/tildewidemε+ωN}/parenrightbig/parenleftbig /tildewidemε+ωN/parenrightbig , (74) whereG2 l(x) in the divergence term can be calculated by G2 l(x) =α/parenleftbig ωN·∂{2/tildewidemε+ωN} ∂xl/parenrightbig/parenleftbig /tildewidemε+ωN/parenrightbig . Moreover, one can apply Lemma 4to deduce from ( 73) and (74) that for i= 1,2 (75)/vextenddouble/vextenddoubleTi bad−n/summationdisplay k,l=1∂ ∂xk/parenleftbig aε klGi l(x)/parenrightbig/vextenddouble/vextenddouble L2(Ω)≤Cε[ln(ε−1+1)]2, here we have use the fact Aε(Φi−xi−εχε i) = 0. Constant Cdepends on /ba∇dbl∇m0/ba∇dblW2,∞(Ω),/ba∇dblAε/tildewidemε/ba∇dblL∞(Ω), but is independent of ε. Now we define the modification function ωM=ω1 M+ω2 M, whereωi M, i= 1,2 satisfies (76) Aεωi M=n/summationdisplay k,l=1∂ ∂xk/parenleftbig aε klGi l(x)/parenrightbig in Ω, ∂ ∂νεωi M=n/summationdisplay k,l=1νk·aε klGi l(x) on∂Ω, hereνkis thek-th component of vector ν. By the Lax-Milgram theorem, onecan obtain theexistence anduniquenessof ωi M,i= 1,2 upto aconstant. Let/integraltext ∂Ωωi Mdx= 0, then the correctors yield the following estimate.HOMOGENIZATION OF THE LLG EQUATION 21 Lemma 5. Forωi M,i= 1,2defined in (76), under smooth assumption of m0and∂Ω, it holds that for n≤3 (77) /ba∇dblωi M/ba∇dblL∞(Ω)≤Cε,/ba∇dbl∇ωi M/ba∇dblL∞(Ω)≤Cεln(ε−1+1), whereCdepends on /ba∇dbl∇m0/ba∇dblW3,∞(Ω)and is independent of ε. Proof.Here we use the Neumann function Nε(x,z) for operator Aε, see [12] Section 7.4. ( 76) implies for i= 1,2 ωi M=n/summationdisplay k,l=1/integraldisplay Ωaε kl∂ ∂zk/braceleftbig Nε(x,z)/bracerightbig Gi l(z)dz. Using the fact ∇zNε(x,z)≤C|x−z|1−n, see [12] p.159, we can derive /ba∇dblωi M/ba∇dblL∞(Ω)≤C/ba∇dblGi l/ba∇dblL∞(Ω)≤Cεln(ε−1+1). As for the second inequality in ( 77), it follows from [ 12], Lemma 7.4.7: /ba∇dbl∇ωi M/ba∇dblL∞(Ω)≤Cln(ε−1+1)/ba∇dblGi l/ba∇dblL∞(Ω)+Cε/ba∇dbl∇Gi l/ba∇dblL∞(Ω) with the estimate /ba∇dbl∇Gi l/ba∇dblL∞(Ω)≤C, from Lemma 4. Here constant Cdepends on /ba∇dbl∇m0/ba∇dblW3,∞(Ω),/ba∇dbl∇(Φj−xj− εχε j)/ba∇dblL∞(Ω), but is independent of εby Lemma 4. /square 5.3.Estimates of initial-boundary data. Theorem 4. Foreε bgiven in (29), withωb=ωN−ωMgiven in (66), under the smooth assumption, it holds that initial data of eε bsatisfies (78) /ba∇dbleε b(x,0)/ba∇dblH1(Ω)≤Cεln(ε−1+1), whereCdepends on /ba∇dbl∇2m0 init/ba∇dblH1(Ω)and is independent of ε. And for the boundary data, it holds that /ba∇dbl∂ ∂νεeε b/ba∇dblW1,∞(0,T;B−1/2,2(∂Ω))≤Cεln(ε−1+1), (79) whereCdepends on /ba∇dbl∇2m0/ba∇dblW1,∞(0,T;B−1/2,2(∂Ω))and is independent of ε. Proof.We rewrite eε bfrom its definition as (80) eε b=mε−m0−n/summationdisplay i=1(Φi−xi)∂m0 ∂xi−ε2m2+ωM. First, let us prove ( 78). By the initial condition of mεandm0, along with the smoothness condition, one can check (81)eε b(x,0) =mε init−m0 init−n/summationdisplay i=1(Φi−xi)∂m0 init ∂xi−ε2m2,init+ωM,init with notation m2,init,ωM,initdefined the same as m2,ωMexcept we replace m0bym0 init. From the assumption ( 16)-(17),m0 initis the homogenized22 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN solution of mε init, by classical homogenization theorem of elliptic problems in [12], one has /ba∇dblmε init−m0 init−(Φ−x)∇m0 init/ba∇dblH1(Ω)≤Cεln(ε−1+1). Also note that by definition of m2and Lemma 5, one has /ba∇dblε2m2,init/ba∇dblH1(Ω)+/ba∇dblωM,init/ba∇dblH1(Ω)≤Cεln(ε−1+1)/ba∇dbl∇2m0 init/ba∇dblH1(Ω). Therefore the inequality ( 78) follows from ( 81) and above estimates. Notice that by the boundary condition of m0in (10) and Φ kin (67), we have n/summationdisplay k=1∂ ∂νε(Φk−xk)·∂m0 ∂xk=−∂ ∂νεm0,x∈∂Ω, therefore applying∂ ∂νεto both sides of ( 80) leads to (82) ∂ ∂νεeε b=−n/summationdisplay k=1(Φk−xk)·∂ ∂νε(∂m0 ∂xk)−ε2∂ ∂νεm2+∂ ∂νεωM,x∈∂Ω. Under the smoothness assumption of m0andaε, we can also derive the smoothness of ( Φ−x),m2andωMover¯Ω. Thus by Lemma 4and Lemma 5, one can directly obtain from ( 82) /ba∇dbl∂ ∂νεeε b/ba∇dblB−1/2,2(∂Ω)≤C/ba∇dblΦ−x/ba∇dblL∞(Ω)+Cε2/ba∇dbl∇m2/ba∇dblL∞(Ω)+C/ba∇dbl∇ωM/ba∇dblL∞(Ω) ≤Cεln(ε−1+1), whereCdepends on /ba∇dbl∇2m0/ba∇dblB−1/2,2(∂Ω)and/ba∇dbl∇2(∂tm0)/ba∇dblB−1/2,2(∂Ω). The same argument for /ba∇dbl∂ ∂νε(∂teε b)/ba∇dblB−1/2,2(∂Ω)leads to ( 79). /square 5.4.Estimates of inhomogeneous terms. Fromtheabovedefinitionand property, we get the main result of this section. Theorem 5. Foreε bgiven in (29), withωb=ωN−ωMgiven in (66)and Lεdefined in (27), under the smooth assumption, it holds that /ba∇dbl∂tωb/ba∇dblL2(Ω)≤Cεln(ε−1+1), (83) /ba∇dblLεωb/ba∇dblL2(Ω)≤Cε[ln(ε−1+1)]2+C/ba∇dblmε−/tildewidemε−ωb/ba∇dblH1(Ω), (84) whereCdepends on /ba∇dblmε/ba∇dblH1(Ω),/ba∇dbl∇2m0/ba∇dblW2,∞(Ω),/ba∇dbl∇(∂tm0)/ba∇dblW1,∞(Ω)and is independent of ε. Moreover, one has the estimate (85) /ba∇dblAεeε b/ba∇dblL2(Ω)≤Cln(ε−1+1), whereCdepends on /ba∇dblAεmε/ba∇dblL2(Ω),/ba∇dbl∇2m0/ba∇dblW2,∞(Ω),/ba∇dbl∇(∂tm0)/ba∇dblW1,∞(Ω)and is independent of ε.HOMOGENIZATION OF THE LLG EQUATION 23 Proof.Inordertoestimateleft-handsideof ( 84), wesplititas /ba∇dblLεωb/ba∇dblL2(Ω)= /ba∇dblLεωN−LεωM/ba∇dblL2(Ω)≤R1+R2+R3, with R1=/vextenddouble/vextenddoubleLεωN−/braceleftbig AεωN−mε×AεωN−D2(ωN)/bracerightbig/vextenddouble/vextenddouble L2(Ω), R2=/vextenddouble/vextenddouble/braceleftbig Aεω1 M−mε×Aεω1 M−Aεω2 M/bracerightbig −LεωM/vextenddouble/vextenddouble L2(Ω), R3=/vextenddouble/vextenddoubleAε(ωN−ω1 M)−mε×Aε(ωN−ω1 M)−/parenleftbig D2(ωN)−Aεω2 M/parenrightbig/vextenddouble/vextenddouble L2(Ω). One can check by definition of LεthatR1does not have derivative of ωN, andR2onlycontains first-orderderivativeof ωM, thustheycanbeestimated by Lemma 4and Lemma 5as R1+R2≤Cεln(ε−1+1), whereCdepends on /ba∇dblmε/ba∇dblH1(Ω),/ba∇dbl∇2m0/ba∇dblW2,∞(Ω). As forR3, in the view of (75), it can be bounded from above by /ba∇dblAεωN−T1 bad/ba∇dblL2(Ω)+/ba∇dblmε×(AεωN−T1 bad)/ba∇dblL2(Ω)+/ba∇dblD2(ωN)−T2 bad/ba∇dblL2(Ω). In above terms, the first term can be estimated by applying Aε(Φi−xi− εχε i) = 0 to derive that /ba∇dblAεωN− T1 bad/ba∇dblL2(Ω)≤Cεln(ε−1+ 1), with C independent of ε. The same result holds for the second term. Now let us estimate the last term. We assert that (86)/ba∇dblD2(ωN)−T2 bad/ba∇dblL2(Ω)≤Cεln(ε−1+1)+C/ba∇dblmε−/tildewidemε−ωb/ba∇dblH1(Ω), whereCdepends on /ba∇dbl∇2m0/ba∇dblW1,∞(Ω)and/ba∇dblmε/ba∇dblH1(Ω). In fact, we denote the terms in D2(ωN) that contain derivatives of ωNby/tildewideD2(ωN), then it reads /tildewideD2(ωN) =αn/summationdisplay i,j=1/parenleftbig aε ij∂ωN ∂xi·∂mε ∂xj+aε ij∂/tildewidemε ∂xi·∂ωN ∂xj/parenrightbig mε, and one can check the remaining terms satisfy /ba∇dblD2(ωN)−/tildewideD2(ωN)/ba∇dblL2(Ω)≤C(1+/ba∇dbl∇m0/ba∇dbl2 L∞(Ω))/ba∇dblωN/ba∇dblL2(Ω)≤Cεln(ε−1+1). Substituting mε= (mε−/tildewidemε−ωN)+(/tildewidemε+ωN) into/tildewideD2(ωN), one can write /tildewideD2(ωN) =T2 bad+αn/summationdisplay i,j=1/parenleftbig aε ij∂ωN ∂xi·∂{mε−/tildewidemε−ωN} ∂xj/parenrightbig mε +αn/summationdisplay i,j=1/parenleftbig aε ij∂ωN ∂xi·∂{/tildewidemε+ωN} ∂xj/parenrightbig/parenleftbig mε−/tildewidemε−ωN/parenrightbig . Hence it follows that /ba∇dbl/tildewideD2(ωN)−T2 bad/ba∇dblL2(Ω)≤C(1+/ba∇dbl∇m0/ba∇dbl2 L∞(Ω))/ba∇dblmε−/tildewidemε−ωN/ba∇dblH1(Ω). The assertion is proved.24 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN As for (83), one can deduce from Lemma 4and the proof of Lemma 5to obtain /ba∇dbl∂tωN/ba∇dblL∞(Ω)≤Cεln(ε−1+1), /ba∇dbl∂tωi M/ba∇dblL∞(Ω)≤C/ba∇dbl∂tGi l/ba∇dblL∞(Ω)≤Cεln(ε−1+1), whereCdepends on /ba∇dbl∇(∂tm0)/ba∇dblW1,∞(Ω). In order to prove ( 85), we use Lemma4, Lemma 5and definition of /tildewidemε, to deduce the estimates /ba∇dblAε/tildewidemε/ba∇dblL2(Ω)+/ba∇dblAεωb/ba∇dblL2(Ω)≤Cln(ε−1+1), then (85) follows with some constant Cdepending on /ba∇dblAεmε/ba∇dblL2(Ω). There- fore Theorem is proved. /square 6.Stability Analysis In this section, we will discuss the stability of following i nitial-boundary problem, which is motivated by equation ( 26): (87) ∂te−Lε(e) =Fin Ω, ν·aε∇e=gon∂Ω, e(0,x) =hin Ω. The following two inequalities will be used. The first inequa lity is motivated byW1,pestimate for oscillatory elliptic problem. Lemma 6. Assumeu∈H2(Ω),ν·aε∇u=gon∂Ω, withg∈B−1/2,2(∂Ω), then it holds that for n≤3, /ba∇dbl∇u/ba∇dblL6(Ω)≤C/ba∇dblAεu/ba∇dblL2(Ω)+C/ba∇dblg/ba∇dblB−1/2,2(∂Ω), moreover, if g= 0, then one has for n≤3 /ba∇dbl∇u/ba∇dblL6(Ω)≤C/ba∇dblAεu/ba∇dblL2(Ω). Constant Cis independent of ε. Proof.We refer that Lemma 6is a direct corollary of Theorem 6.3.2 in [ 12]. One can find the proof in [ 12] [Pages 144-152]. /square We also introduce Sobolev inequality with small coefficient w henn= 2. Lemma 7. For any function f∈H2(Ω), one has when n= 2 /ba∇dblf/ba∇dblL∞(Ω)≤Cln(ε−1+1)/ba∇dblf/ba∇dblH1(Ω)+ε/ba∇dblAεf/ba∇dblL2(Ω), where constant Cis independent of ε. Proof.Using the Neumann function, see [ 12] Section 7.4, one has f=−/integraldisplay Ω∇zNε(x,z)·aε∇f(z)dz+1 |∂Ω|/integraldisplay ∂Ωfdz=:P1+P2.HOMOGENIZATION OF THE LLG EQUATION 25 Applying cut-off function φε=φε(x−z), the first term yields by integration by parts P1=−/integraldisplay Ω(1−φε)∇zNε(x,z)·aε∇f(z)dz+/integraldisplay ΩφεNε(x,z)·Aεf(z)dz +/integraldisplay Ω∇zφε(x−z)·Nε(x,z)·aε∇f(z)dz ≤Cε/ba∇dblAεf/ba∇dblL2(Ω)+Cln(ε−1+1)/ba∇dbl∇f/ba∇dblL2(Ω), here in the last line we have used the fact ∇zNε(x,z)≤C|x−y|−1and Nε(x,z)≤C{1+ln[|x−z|−1]}forn= 2, see [ 12] page 159. As for P2, one has by trace inequality that P2≤C/ba∇dblf/ba∇dblH1(Ω). The Lemma is proved. /square Now let us give the stability of system ( 87) in terms of h,g,Fin L∞(0,T;L2(Ω)) and L∞(0,T;H1(Ω)) norm, respectively. 6.1.Stability in L∞(0,T;L2(Ω)). Theorem 6. Lete∈L∞(0,T;H2(Ω))be a strong solution to (87). Assume h∈L2(Ω),g∈L∞(0,T;B−1/2,2(∂Ω)), andF=F1+F2satisfies (88) F1∈L2(0,T;Lσ(Ω)),F2∈L2(0,T;L2(Ω)) withσ= 1whenn= 1,2, andσ= 6/5whenn= 3, then it holds that, for any0≤t≤T (89)/ba∇dble/ba∇dbl2 L∞(0,T;L2(Ω))+/ba∇dbl∇e/ba∇dbl2 L2(0,T;L2(Ω)) ≤Cδ/parenleftBig /ba∇dblh/ba∇dbl2 L2(Ω)+/ba∇dblg/ba∇dbl2 L2(0,T;B−1/2,2(∂Ω))+γ(ε)/ba∇dblF1/ba∇dbl2 L2(0,T;Lσ(Ω))/parenrightBig +δ/ba∇dblF2/ba∇dbl2 L2(0,T;L2(Ω))+ε2/ba∇dblAεe/ba∇dbl2 L2(0,T;L2(Ω)), for any small δ >0, where /braceleftBigg γ(ε) = 1, whenn= 1,3, γ(ε) = [ln(ε−1+1)]2,whenn= 2. Cδis a constant depending on /ba∇dbl∇mε/ba∇dblL4(Ω),/ba∇dbl∇/tildewidemε/ba∇dblL4(Ω), but is independent oftandε. Proof.The inner product between ( 87) andeinL2(Ω) leads to (90)1 2d dt/integraldisplay Ω|e|2dx−α/integraldisplay Ω/tildewideHε e(e)·edx =−/integraldisplay ΩD1(e)·edx−/integraldisplay ΩD2(e)·edx−/integraldisplay ΩF·edx. Now let us give the estimates to ( 90) term by term. Integration by parts for the second term on the left-hand side yields −/integraldisplay Ω/tildewideHε e(e)·edx≥n/summationdisplay i,j=1/integraldisplay Ωaε ij∂e ∂xi·∂e ∂xjdx−/integraldisplay ∂Ωg·edx−C/integraldisplay Ω|e|2dx26 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN with the boundary term satisfying /integraldisplay ∂Ωg·edx≤ /ba∇dblg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblB1/2,2(∂Ω)≤C/ba∇dblg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblH1(Ω). By integration by parts and the same argument for the boundar y term, the first term on the right-hand side can be estimated as −/integraldisplay ΩD1(e)·edx≤C/integraldisplay Ω|e|2dx+δC/integraldisplay Ω|∇e|2dx−C/ba∇dblg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblH1(Ω). For the second term on the right-hand side of ( 90), using the estimates /integraldisplay Ω/parenleftbig Bε[e,mε]/parenrightbig mε·edx≤C/ba∇dbl∇mε/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dbl∇e/ba∇dblL2(Ω) +C/ba∇dblmε/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dble/ba∇dblL2(Ω),/integraldisplay Ωgε l[/tildewidemε]e·edx≤C/ba∇dbl∇/tildewidemε/ba∇dbl2 L4(Ω)/ba∇dble/ba∇dbl2 L4(Ω)+C/ba∇dbl/tildewidemε/ba∇dbl2 L4(Ω)/ba∇dble/ba∇dbl2 L4(Ω), and the same argument can be applied to the other terms, we fina lly obtain by Sobolev inequality −/integraldisplay ΩD2(e)·edx≤C+C/integraldisplay Ω|e|2dx+δC/integraldisplay Ω|∇e|2dx, whereC=C0/parenleftbig 1+/ba∇dbl∇mε/ba∇dbl2 L4(Ω)+/ba∇dbl∇/tildewidemε/ba∇dbl2 L4(Ω)/parenrightbig . For the last term in ( 90), by the assumption ( 88), we apply Sobolev inequality for n= 1,3, and apply Lemma7forn= 2, it follows that −/integraldisplay ΩF1·edx≤ C/ba∇dblF1/ba∇dbl2 L1(Ω)+δ/ba∇dble/ba∇dbl2 H1(Ω), n = 1 C[ln(ε−1+1)]2/ba∇dblF1/ba∇dbl2 L1(Ω) +δ/ba∇dble/ba∇dbl2 H1(Ω)+ε2/ba∇dblAεe/ba∇dbl2 L2,n= 2 C/ba∇dblF1/ba∇dbl2 L6/5(Ω)+δ/ba∇dble/ba∇dbl2 H1(Ω), n = 3 −/integraldisplay ΩF2·edx≤δ∗/ba∇dblF2/ba∇dbl2 L2(Ω)+C/ba∇dble/ba∇dblL2(Ω), with any small δ,δ∗>0. Substituting above estimates, one can derive from (90) that 1 2d dt/integraldisplay Ω|e|2dx+(αamin−2δ)/integraldisplay Ω|∇e|2dx≤C/integraldisplay Ω|e|2dx+C/ba∇dblg/ba∇dbl2 B−1/2,2(∂Ω) +C[ln(ε−1+1)]2/ba∇dblF1/ba∇dbl2 Lσ(Ω)+δ∗/ba∇dblF2/ba∇dbl2 L2(Ω)+ε2/ba∇dblAεe/ba∇dbl2 L2. Then (89) follows directly by taking δsmall enough, and the application of Gr¨ onwall’s inequality. /squareHOMOGENIZATION OF THE LLG EQUATION 27 6.2.Stability in L∞(0,T;H1(Ω)). Theorem 7. Lete∈L∞(0,T;H2(Ω))be a strong solution to (87). Assume h∈H1(Ω),g∈H1(0,T;B−1/2,2(∂Ω)), andF∈L2(0,T;L2(Ω)), it holds (91) /ba∇dbl∇e/ba∇dbl2 L∞(0,T;L2(Ω))≤C/parenleftbig /ba∇dblh/ba∇dbl2 H1(Ω)+/ba∇dblg/ba∇dbl2 H1(0,T;B−1/2,2(∂Ω))+/ba∇dblF/ba∇dbl2 L2(0,T;L2(Ω))/parenrightbig , whereCdepends on /ba∇dbl∇mε/ba∇dblL4(Ω),/ba∇dbl∇/tildewidemε/ba∇dblL4(Ω)and/ba∇dblHε e(/tildewidemε)/ba∇dblL4(Ω), but is independent of tandε. Proof.The inner product between ( 87) and/tildewideHε e(e) inL2(Ω) leads to (92)−/integraldisplay Ω∂te·/tildewideHε e(e)dx+α/integraldisplay Ω/tildewideHε e(e)·/tildewideHε e(e)dx =/integraldisplay ΩD1(e)·/tildewideHε e(e)dx+/integraldisplay ΩD2(e)·/tildewideHε e(e)dx+/integraldisplay ΩF·/tildewideHε e(e)dx. In the following we give the estimates to ( 92) term by term. Note that integration by parts yields −/integraldisplay Ω∂te·/tildewideHε e(e)dx=d dtGε L[e]−/integraldisplay ∂Ω∂te·gdx, =d dtGε L[e]−∂t/integraldisplay ∂Ωe·gdx−C/ba∇dbl∂tg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblH1(Ω). Using the fact mε×/tildewideHε e(e)·/tildewideHε e(e) = 0, the first term on the right-hand side of (92) can be estimate by Sobolev inequality as /integraldisplay ΩD1(e)·/tildewideHε e(e)dx≤C/ba∇dblHε e(/tildewidemε)/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dbl/tildewideHε e(e)/ba∇dblL2(Ω) ≤C/ba∇dble/ba∇dbl2 H1(Ω)+δC/ba∇dbl/tildewideHε e(e)/ba∇dbl2 L2(Ω), whereC=C0/parenleftbig 1+/ba∇dblHε e(/tildewidemε)/ba∇dbl2 L4(Ω)/parenrightbig . For the second term on the right-hand side of (92), note that we have the estimate /integraldisplay Ω/parenleftbig Bε[e,mε]/parenrightbig mε·/tildewideHε e(e)dx≤C/ba∇dbl∇mε/ba∇dblL4(Ω)/ba∇dbl∇e/ba∇dblL4(Ω)/ba∇dbl/tildewideHε e(e)/ba∇dblL2(Ω) +C/ba∇dblmε/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dbl/tildewideHε e(e)/ba∇dblL2(Ω), in which one can deduce /ba∇dbl∇e/ba∇dblL4(Ω)≤/ba∇dbl∇e/ba∇dbl1/3 L2(Ω)/ba∇dbl∇e/ba∇dbl2/3 L6(Ω) ≤C/ba∇dbl∇e/ba∇dbl1/3 L2(Ω)(1+/ba∇dbl/tildewideHε e(e)/ba∇dblL2(Ω)+/ba∇dblg/ba∇dblB−1/2,2(∂Ω))2/3, using interpolation inequality and Lemma 6. The other terms can be esti- mated in the same fashion. After the application of Young’s i nequality, one finally obtains /integraldisplay ΩD2(e)·/tildewideHε e(e)dx≤C/ba∇dble/ba∇dbl2 H1(Ω)+δC/ba∇dbl/tildewideHε e(e)/ba∇dbl2 L2(Ω)+C/ba∇dblg/ba∇dbl2 B−1/2,2(∂Ω),28 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN whereC=C0(1+/ba∇dbl∇mε/ba∇dbl2 L4(Ω)+/ba∇dbl∇/tildewidemε/ba∇dbl2 L4(Ω)). Substitutingaboveestimates into (92), we arrive at d dtGε L[e]+(α−Cδ)/integraldisplay Ω|/tildewideHε e(e)|2dx−∂t/integraldisplay ∂Ωe·gdx ≤C/parenleftbig /ba∇dble/ba∇dbl2 H1(Ω)+/ba∇dblF/ba∇dbl2 L2(Ω)+/ba∇dblg/ba∇dblB−1/2,2(∂Ω)+/ba∇dbl∂tg/ba∇dblB−1/2,2(∂Ω)/parenrightbig . Integrating the above inequality over [0 ,t] with 0 < t < T and using the facts/integraldisplay ∂Ωe·gdx≤δ/ba∇dble/ba∇dbl2 H1(Ω)+C/ba∇dblg/ba∇dbl2 B−1/2,2(∂Ω), Gε L[e]≥amin 2/ba∇dbl∇e/ba∇dbl2 L2(Ω)−C, one can finally derive (amin 2−δ)/ba∇dbl∇e(t)/ba∇dbl2 L2(Ω)+(α−Cδ)/integraldisplayt 0/ba∇dbl/tildewideHε e(e)/ba∇dbl2 L2(Ω)dτ ≤C/integraldisplayt 0/parenleftbig /ba∇dble/ba∇dbl2 L2(Ω)+/ba∇dbl∇e/ba∇dbl2 L2(Ω)+/ba∇dblF/ba∇dbl2 L2(Ω)+/ba∇dbl∂tg/ba∇dblB−1/2,2(∂Ω)/parenrightbig dτ+J(h), whereJ(h) yields J(h) =Gε L[h]−/integraldisplay ∂Ωh·∂ ∂νεhdx≤C/ba∇dblh/ba∇dbl2 H1(Ω)+/ba∇dbl∂ ∂νεh/ba∇dbl2 B−1/2,2(∂Ω). (91) is then derived after taking δsmall enough and the application of Gr¨ onwall’s inequality. /square 7.Regularity In the estimate of boundary corrector and stability analysi s by Theorem 5, Theorem 6, Theorem 7, the constant we deduced rely on the value of /ba∇dblAεmε/ba∇dblL2(Ω)and/ba∇dbl∇mε/ba∇dblL6(Ω). In this section, we introduce the uniform regularity on mε, over a time interval independent of ε. For this purpose, we intend to derive a structure-preserving energy inequali ty, in which the degenerate term are kept in the energy. First, let us introduce an interpolation inequality of the e ffective field Hε e(mε) for some S2-valued function mε, which is the generalization of ( 14). The following estimates will be used: a−1 max/ba∇dblmε·Aεmε/ba∇dbl3 L3(Ω)≤ /ba∇dbl∇mε/ba∇dbl6 L6(Ω)≤a−1 min/ba∇dblAεmε/ba∇dbl3 L3(Ω), (93) /ba∇dblAεmε/ba∇dblLp(Ω)−Cp≤ /ba∇dblHε e(mε)/ba∇dblLp(Ω)≤ /ba∇dblAεmε/ba∇dblLp(Ω)+Cp, (94) with 1< p <+∞, here the first line follows from the fact −aε|∇mε|2= mε·Aεmεby|mε|= 1 and assumption of aεin (1), and in second line the estimate ( 4) is used. We also introduce a orthogonal decomposition to an y vectoraas (95) a= (mε·a)mε−mε×(mε×a).HOMOGENIZATION OF THE LLG EQUATION 29 Lemma 8. Givenmε∈H3(Ω)that satisfies |mε|= 1and Neumann bound- ary condition ν·aε∇mε= 0, then it holds for n≤3and any 0< δ <1 (96)/ba∇dblHε e(mε)/ba∇dbl3 L3(Ω)≤Cδ+Cδ/ba∇dblHε e(mε)/ba∇dbl6 L2(Ω)+δ/ba∇dblmε×∇Hε e(mε)/ba∇dbl2 L2(Ω), whereCδis a constant depending on δbut independent of ε. Proof.Applying decomposition ( 95) by taking a=Hε e(mε), one can write (97)/ba∇dblHε e(mε)/ba∇dbl3 L3(Ω)≤/integraldisplay Ω|mε·Hε e(mε)|3dx +/integraldisplay Ω|mε×Hε e(mε)|3dx=:I1+I2. Now let us estimate the right-hand side of ( 97) separately. For I1, we apply (93) and Remark 6to derive I1≤C+C/ba∇dbl∇mε/ba∇dbl6 L6(Ω)≤C+C/ba∇dblHε e(mε)/ba∇dbl6 L2(Ω). As forI2, we have by Sobolev inequality for n≤3 I2≤C+C/ba∇dblmε×Hε e(mε)/ba∇dbl6 L2(Ω)+δ∗/ba∇dblmε×Hε e(mε)/ba∇dbl2 H1(Ω), here in the last term, we can apply ( 93)-(94) to derive: δ∗/ba∇dbl∇mε×Hε e(mε)/ba∇dbl2 L2(Ω)≤δ∗/ba∇dbl∇mε/ba∇dbl6 L6(Ω)+δ∗/ba∇dblHε e(mε)/ba∇dbl3 L3(Ω) ≤C+Cδ∗/ba∇dblHε e(mε)/ba∇dbl3 L3(Ω). Now let us turn back to ( 97), we finally obtain (1−Cδ∗)/ba∇dblHε e(mε)/ba∇dbl3 L3(Ω)≤C+C/ba∇dblHε e(mε)/ba∇dbl6 L2(Ω)+δ∗/ba∇dblmε×∇Hε e(mε)/ba∇dbl2 L2(Ω). Letδ∗<1 2C, one can derive ( 96) withδ=δ∗/(1−Cδ∗)<1. /square Now let us recall some energy property of LLG equation, and gi ve the uniform regularity result. Using the formula of vector oute r production (98) a×(b×c) = (a·c)b−(a·b)c, one can rewrite LLG equation ( 22) into a degenerate form (99) ∂tmε+αmε×/parenleftbig mε×Hε e(mε)/parenrightbig +mε×Hε e(mε) = 0. Multiplying ( 99) byHε e(mε) and integrating over (0 ,t), we derive the energy dissipation identity (100) Gε L[mε(t)]+α/integraldisplayt 0/ba∇dblmε×Hε e(mε)/ba∇dbl2 L2(Ω)dτ=Gε L[mε(0)], together ( 99) and (100) leads to the integrable of kinetic energy α 1+α2/integraldisplayt 0/ba∇dbl∂tmε/ba∇dbl2 L2(Ω)dτ≤ Gε L[mε(0)]. (101) Theenergyidentity ( 100)impliestheuniformregularityof /ba∇dblmε×Aεmε/ba∇dbl2 L2(Ω), however, this is not enough to obtain the regularity of /ba∇dblAεmε/ba∇dbl2 L2(Ω)due to the degeneracy. In this end we introduce that:30 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN Theorem 8. Letmε∈L2([0,T];H3(Ω))be a solution to (8). Assume n≤ 3, then there exists T∗∈(0,T]independent of ε, such that for 0≤t≤T∗, /ba∇dblAεmε(t)/ba∇dbl2 L2(Ω)+/integraldisplayt 0/vextenddouble/vextenddoublemε×∇Hε e(mε)(τ)/vextenddouble/vextenddouble2 L2(Ω)dτ≤C, and therefore, by the Sobolev-type inequality in Remark 6, /ba∇dbl∇mε(·,t)/ba∇dbl2 L6(Ω)≤C, whereCis a constant independent of εandt. Proof.Applying ∇to (99) and multiplying by aε∇Hε e(mε) lead to (102)−/integraldisplay Ω∇(∂tmε)·aε∇Hε e(mε)dx =α/integraldisplay Ω∇/parenleftbig mε×(mε×Hε e(mε))/parenrightbig ·aε∇Hε e(mε)dx +n/summationdisplay i,j=1/integraldisplay Ω∂ ∂ximε×Hε e(mε)·aε ij∂ ∂xjHε e(mε)dx=:J1+J2. Denote Γε(mε) =Hε e(mε)−Aεmε. After integration by parts, the left-hand side of (102) becomes −/integraldisplay Ω∇(∂tmε)·aε∇Hε e(mε)dx=/integraldisplay ΩAε(∂tmε)·/parenleftbig Aεmε+Γε(mε)/parenrightbig dx, where the right-hand side can be rewritten as 1 2d dt/integraldisplay Ω|Aεmε|2dx+d dt/integraldisplay ΩAεmε·Γε(mε)dx−/integraldisplay ΩAε(mε)·Γε(∂tmε)dx. Now let us consider the right-hand side of ( 102). ForJ1, one can derive by swapping the order of mixed product J1=−αn/summationdisplay i,j=1/integraldisplay Ω/parenleftbig mε×∂ ∂xiHε e(mε)/parenrightbig ·aε ij/parenleftbig mε×∂ ∂xjHε e(mε)/parenrightbig dx+F1, herethefirstterm onright-handsideissign-preservedduet otheuniformco- erciveness of aεin (1). As for J2, we apply ( 95) by taking a=aε ij∂jHε e(mε), it leads to (103)J2=n/summationdisplay i,j=1/integraldisplay Ωmε×/parenleftbig∂ ∂ximε×Hε e(mε)/parenrightbig ·/parenleftbig mε×aε ij∂ ∂xjHε e(mε)/parenrightbig dx −n/summationdisplay i,j=1/integraldisplay Ω/parenleftbig mε×Hε e(mε)·∂ ∂ximε/parenrightbig mε·aε ij∂ ∂xjHε e(mε)dx.HOMOGENIZATION OF THE LLG EQUATION 31 Usingpropertyof vector outer production( 98) forfirstterm, andintegration by parts for the second term, ( 103) becomes J2=2n/summationdisplay i,j=1/integraldisplay Ω/parenleftbig mε·Hε e(mε)/parenrightbig/parenleftbig mε×aε ij∂ ∂xjHε e(mε)·∂ ∂ximε/parenrightbig dx+F2 ≤C/ba∇dbl∇mε/ba∇dbl6 L6(Ω)+C/ba∇dblHε e(mε)/ba∇dbl3 L3(Ω)+δ/ba∇dblmε×∇Hε e(mε)/ba∇dbl2 L2(Ω)+F2. Here low-order terms Fi,i= 1,2 satisfies by ( 94) and H¨ older’s inequality Fi≤C+C/ba∇dbl∇mε/ba∇dbl6 L6(Ω)+C/ba∇dblHε e(mε)/ba∇dbl3 L3(Ω). Substituting above estimates into ( 102), applying estimate ( 93) and Lemma 8, we finally arrive at (104)1 2d dt/ba∇dblAεmε/ba∇dbl2 L2(Ω)+(αamin−Cδ)/ba∇dblmε×∇Hε e(mε)/ba∇dbl2 L2(Ω) ≤C+C/ba∇dblAεmε/ba∇dbl6 L2(Ω)+C/ba∇dbl∂tmε/ba∇dbl2 L2(Ω)−d dt/integraldisplay ΩAεmε·Γε(mε)dx, Integrating ( 104) over [0 ,t], using the integrability of kinetic energy ( 101) and the following inequality /integraldisplay ΩAεmε·Γε(mε)dx≤C/ba∇dblΓε(mε)/ba∇dbl2 L2(Ω)+1 4/ba∇dblAεmε/ba∇dbl2 L2(Ω), one has for any t∈(0,T] (105)1 4/ba∇dblAεmε(t)/ba∇dbl2 L2(Ω)≤C+C/integraldisplayt 0/ba∇dblAεmε(τ)/ba∇dbl6 L2(Ω)dτ, whereCdependson /ba∇dblAεmε init/ba∇dblL2(Ω),Gε L[mε init] thus is independent of εandt by assumption ( 16)-(17) and Lemma 6. Denote the right-hand side of ( 105) byF(t) and write d dtF(t)≤CF3(t). By the Cauchy-Lipshitz-Picard Theorem [ 3] and comparison principle, there existsT∗∈(0,T] independent of ε, such that F(t) is uniformly bounded on [0,T∗], thus/ba∇dblAεmε(t)/ba∇dbl2 L2(Ω)is uniformly bounded by ( 105). The Lemma is proved. /square Acknowledgments J. Chen was supported by National Natural Science Foundatio n of China via grant 11971021. J.-G. Liu was supported by Natural Scien ce Foun- dation via grant DMS-2106988. Z. Sun was supported by the Pos tgradu- ate Research & Practice Innovation Program of Jiangsu Provi nce via grant KYCX21 2934.32 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN References 1. Francois Alouges, AnneDeBouard, Benoˆ ıt Merlet, andL´ e aNicolas, Stochastic homog- enization of the Landau-Lifshitz-Gilbert equation , Stochastics and Partial Differential Equations: Analysis and Computations 9(2021), 789–818. 2. Francois Alouges and Giovanni Di Fratta, Homogenization of composite ferromagnetic materials , Proceedings of the Royal Society A: Mathematical, Physica l and Engineer- ing Sciences 471(2015), 20150365. 3. Haim Brezis and Haim Br´ ezis, Functional analysis, sobolev spaces and partial differ- ential equations , vol. 2, Springer, 2011. 4. Jingrun Chen, Rui Du, Zetao Ma, Zhiwei Sun, and Zhang Lei, On the multiscale Landau-Lifshitz-Gilbert equation: Two-scale convergenc e and stability analysis , Mul- tiscale Modeling & Simulation (2022), in press. 5. Catherine Choquet, Mohammed Moumni, and Mouhcine Tiliou a,Homogenization of the Landau-Lifshitz-Gilbert equation in a contrasted comp osite medium , Discrete & Continuous Dynamical Systems-S 11(2018), 35. 6. T. L.Gilbert, A Lagrangian formulation of gyromagnetic equation of the ma gnetization field, Physical Review D 100(1955), 1243–1255. 7. Lev Davidovich Landau and E Lifshitz, On the theory of the dispersion of mag- netic permeability in ferromagnetic bodies , Physikalische Zeitschrift der Sowjetunion 8(1935), 153–169. 8. Lena Leitenmaier and Olof Runborg, On homogenization of the Landau-Lifshitz equa- tion with rapidly oscillating material coefficient , Communications in Mathematical Sciences 20(2022), 653–694. 9. ,Upscaling errors in heterogeneous multiscale methods for t he Landau-Lifshitz equation, Multiscale Modeling & Simulation 20(2022), 1–35. 10. Dirk Praetorius, Analysis of the operator delta div arising in magnetic model s, Zeitschrift Fur Analysis Und Ihre Anwendungen 23(2004), 589–605. 11. K´ evin Santugini-Repiquet, Homogenization of the demagnetization field operator in periodically perforated domains , Journal of Mathematical Analysis and Applications 334(2007), 502–516. 12. Zhongwei Shen, Periodic homogenization of elliptic systems , Springer, 2018. School of MathematicalSciences, Universityof Science and Technology of China, Hefei, Anhui 230026, China; Suzhou Institute for Adv anced Research, Universityof Science and Technology of China, Suzhou, Jian gsu 215123, China Email address :jingrunchen@ustc.edu.cn Department of Mathematics and Department of Physics, Duke U niversity, Box 90320, Durham NC 27708, USA Email address :jliu@phy.duke.edu School of Mathematical Sciences, Soochow University, Suzh ou, Jiangsu 215006, China Email address :20194007008@stu.suda.edu.cn

2022-06-22

The full Landau-Lifshitz-Gilbert equation with periodic material coefficients and natural boundary condition is employed to model the magnetization dynamics in composite ferromagnets. In this work, we establish the convergence between the homogenized solution and the original solution via a Lax equivalence theorem kind of argument. There are a few technical difficulties, including: 1) it is proven the classic choice of corrector to homogenization cannot provide the convergence result in the $H^1$ norm; 2) a boundary layer is induced due to the natural boundary condition; 3) the presence of stray field give rise to a multiscale potential problem. To keep the convergence rates near the boundary, we introduce the Neumann corrector with a high-order modification. Estimates on singular integral for disturbed functions and boundary layer are deduced, to conduct consistency analysis of stray field. Furthermore, inspired by length conservation of magnetization, we choose proper correctors in specific geometric space. These, together with a uniform $W^{1,6}$ estimate on original solution, provide the convergence rates in the $H^1$ sense.

Homogenization of the Landau-Lifshitz-Gilbert equation with natural boundary condition

2206.10948v1

arXiv:1008.0674v1 [cond-mat.mtrl-sci] 3 Aug 2010Determinationofthe spin-flip timeinferromagnetic SrRuO 3from time-resolved Kerr measurements C.L.S.Kantner,1,2M.C.Langner,1,2W.Siemons,3J.L.Blok,4G.Koster,4A.J.H.M.Rijnders,4R.Ramesh,1,3andJ.Orenstein1,2 1Department of Physics, University of California, Berkeley , CA 94720 2Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, CA 94720 3Department of Materials Science and Engineering, Universi ty of California, Berkeley, CA 94720 4MESA+Institute for Nanotechnology, University of Twente, 7500 A E Enschede, The Netherlands (Dated: December 6, 2018) Wereport time-resolvedKerr effectmeasurements of magnet izationdynamics inferromagnetic SrRuO 3. We observe that the demagnetization timeslows substantially at temperatures within15K of theCurie temperature, whichis∼150K. We analyze the data witha phenomenological model that relates the demagnetization timeto the spinfliptime. Inagreement withour observations the mod el yields a demagnetization timethat is inversely proportional toT-T c. Wealsomake adirectcomparisonofthespinfliprateandtheG ilbertdampingcoefficient showing thattheir ratioveryclose tok BTc,indicating a common originfor these phenomena. I: Introduction Thereisincreasinginterestincontrollingmagnetisminfe r- romagnets. Of particular interest are the related question s of howquicklyandbywhatmechanismthemagnetizationcanbe changed by external perturbations. In addition to advancin g our basic understanding of magnetism, exploring the speed withwhichthemagneticstatecanbechangediscrucialtoap- plications such as ultrafast laser-writing techniques. De spite its relevance, the time scale and mechanisms underlying de- magnetizationarenotwell understoodata microscopicleve l. BeforeBeaurepaireetal.’spioneeringworkonlaser-excit ed Ni in 1996,it was thoughtthat spins wouldtake nanoseconds torotate,withdemagnetizationresultingfromtheweakint er- actionofspinswiththelattice. TheexperimentsonNishowe d that this was not the case and that demagnetizationcould oc- cur on time scales significantly less than 1 ps1. Since then demagnetization is usually attributed to Elliott-Yafet me cha- nism, in which the rate of electron spin flips is proportional to the momentum scattering rate. Recently Koopmans et al. have demonstrated that electron-phononor electron-impur ity scattering can be responsible for the wide range of demag- netization time scales observed in different materials2. Also recentlyit hasbeenproposedthat electron-electronscatt ering should be included as well as a source of Elliott-Yafet spin flipping,andconsequently,demagnetization3. AlthoughRef.3 specifically refers to interband scattering at high energie s, it is plausible that intrabandelectron scattering can lead to spin memorylossaswell. Time-resolved magneto-optical Kerr effect (TRMOKE) measurementshavebeendemonstratedtobeausefulprobeof ultrafast laser-induceddemagnetization1. In this paper we re- portTRMOKEmeasurementsonthinfilmsofSRO/STO(111) between 5 and 165K. Below about 80 K we observe damped ferromagneticresonance (FMR), from which we determine a Gilbert damping parameter consistent with earlier measure - ments on SrTiO 3with (001) orientation6. As the the Curie temperature ( ∼150K) is approached the demagnetization time slows significantly, as has been observed in other mag- netic systems4. The slowing dynamics have been attributed to critical slowing down, due to the similarities between th e temperature dependencies of the demagnetization time andthe relaxation time5. In this paper we develop an analytical expression relating the demagnetization time to the spin-fl ip timenearthe Curietemperature. Thisprovidesa newmethod of measuring the spin-flip time, which is essential to under- standingthedynamicsoflaser-induceddemagnetization. II: SampleGrowthandCharacterization SRO thin films were grown via pulsed laser deposition at 700◦C in 0.3 mbar of oxygenand argon(1:1) on TiO 2termi- nated STO(111)7. A pressed pellet of SRO was used for the targetmaterial and the energyon the targetwas kept constan t at 2.1 J/cm2. High-pressure reflection high-energy electron diffraction (RHEED) was used to monitor the growth speed and crystallinity of the SRO film in situ. RHEED patterns andatomicforcemicroscopyimagingconfirmedthepresence of smooth surfaces consisting of atomically flat terraces se p- arated by a single unit cell step (2.2 ˚Ain the [111] direction). X-ray diffraction indicated fully epitaxial films and x-ray re- flectometry was used to verify film thickness. Bulk magneti- zationmeasurementsusingaSQUIDmagnetometerindicated a Curie temperature,T c, of∼155K.Electrical transportmea- surementswere performedin the Vander Pauwconfiguration andshowtheresidualresistanceratiotobeabout10forthes e films. III: ExperimentalMethods IntheTRMOKEtechniqueamagneticsampleisexcitedby theabsorptionofapumpbeam,resultinginachangeofpolar- izationangle, ∆ΘK(t),ofatimedelayedprobebeam. Theul- trashortpulsesfroma Ti:Sapphlaser are used to achievesub - picosecondtime resolution. Near normalincidence,as in th is experiment, ∆ΘKis proportional to the ˆzcomponent of the perturbedmagnetization, ∆Mz.∆ΘKis measured via a bal- anceddetectionscheme. Foradditionalsensitivity,thede riva- tiveof∆ΘKt)withrespecttotimeismeasuredbylockinginto thefrequencyofasmallamplitude( ∼500fs)fastscanningde- lay line in the probe beam path as time is stepped throughon anotherdelayline. IV.1: ExperimentalResults: Low Temperature Fig. 1 shows the time derivative of ∆ΘKfor an 18.5nm SRO/STO(111)sample forthe 16psfollowingexcitationbya pump beam, for temperatures between 5 and 85K. Clear fer- romagnetic resonance (FMR) oscillations are present, gene r-2 FIG.1. DerivativeofthechangeinKerrrotationasafunctio noftime delay followingpulsed photoexcitation, for 5 <T<85 K ated bya suddenshift in easy axisdirectionuponthermal ex- citation by a pump beam6. This motion is described by the Landau-Lifshitz-Gilbertequationwith thefrequencyofos cil- lation proportional to the strength of the magnetocrystall ine anisotropy field, and the damping described by dimension- less phenomenological parameter, α. The motion appears as a decaying oscillation to TRMOKE. The orientation of the anisotropyfield, closer to in-planewith the sample surface in SRO/STO(111) than in SRO/STO(001), makes these oscilla- tions more prominent when observed with the polar Kerr ge- ometrycomparedtopreviousmeasurements. Attempting to model the time derivative of ∆ΘKwith a dampedcosine revealsthat it cannot be fit by such a function for t<2ps. The feature at short times in Fig. 1 contains higherfrequencycomponents,whereastheoscillationswhi ch become clear after 2 ps are at a single frequency. A com- parison of the amplitude of the first peak (at t ∼.5 ps) with the amplitude of the subsequent oscillations (defined as the difference between d ∆ΘK/dt at the peak at ∼3.5 ps and the dip at∼5.5 ps), is shown as a functionof temperaturein Fig. 2. The constant offset between the two amplitudes indicates that d∆ΘK/dt is comprised of a superposition of a tempera- tureindependent,short-livedcomponentwiththelongerli ved dampedoscillations. Fitting the oscillatory portion of the signal to a damped cosine, the temperature dependencies of the amplitude, fre - quency,anddampingparameterarefound,asshowninFig. 3. ComparingtheseparametersforSRO/STO(111)topreviously published work on SRO/STO(001), the frequencyis found to be somewhat smaller and to change more with temperature. Of particular interest is α, which is also smaller in this ori- entation of SRO, consistent with the more pronounced FMR oscillations. Strikingly, in both orientations there is a d ip in αaround45K,whichisrelativelystrongerinSRO/STO(111). This further strengthens the link between αand the anoma- lous hall conductivity, speculated in that paper, through n ear degeneraciesin thebandstructure6. IV.2: ExperimentalResults: HighTemeperatureFIG. 2. Comparing amplitudes of the short time feature and th e fer- romagnetic resonance oscillations By taking the time derivative of ∆ΘK, the FMR oscilla- tionscanbe followeduntilthey disappearat elevatedtempe r- atures,atwhichpointitbecomessimplertolookat ∆ΘKthan its time derivative. Fig. 4 shows ∆ΘKas a function of time for the first 38 ps after excitation by the pump laser, for tem- peratures between 120K and 165K. A property of a second order phase transitions is that the derivative of the order p a- rameter divergesnear the transition temperature. The peak in magnitudeof ∆ΘKin figure 4, shown in figure5, can be un- derstood as the result of the derivative of magnetization wi th respect to temperature becoming steeper near the Curie tem- perature. A strongtemperaturedependenceofthedemagneti - zationtime, τM, is seen,with τMsignificantlyenhancednear 150K,consistentwithpreviousreportsonSRO4,6. ∆ΘK(t) in Fig. 4, normalized by the largest value of ∆ΘK(t) in the first 38 ps, can be fit with the following func- tion: fort <0∆ΘK(t) ∆Θmax(t)= 0 fort >0∆ΘK(t) ∆Θmax(t)=C−Ae−t/τM(1) where the decay time is τM. The resulting τMis plotted as a functionof temperaturein Fig. 6. Notably, τMincreases by a factor of 10 from 135K to 150K. Taking the fit value of Tc= 148.8K, as will be discussed later, τMis plotted log- log as function of reduced temperature, tR= (Tc−T)/Tc. Theresult looksapproximatelylinear,indicatinga powerl aw dependenceof τMonthereducedtemperature. V: Discussion ofResults: Efforts to explain demagnetization have been largely phe- nomenological thus far, understandably, given the dauntin g challenge of a full microscopic model. Beaurepaire et al. in - troduced the three temperature model (3TM) to describe de- magnetization resulting from the interactions of the elect ron, phonon,andspinbaths1. In3TMthedynamicsaredetermined3 FIG. 3. Temperature dependence of (a) Amplitude of oscillat ions, (b) FMRfrequency, and, (c)damping parameter FIG.4. ChangeinKerrrotationasafunctionoftimedelayfol lowing pulsed photoexcitation, for 120 <T<165 KFIG. 5. Magnitude of change in Kerr rotation at 38ps as a funct ion of temperature FIG.6. Demagnetization timeat hightemperature by the specific heats of each bath as well as the coupling constants between them. Demagnetization can generally be described with the appropriate choice of coupling constant s, providing a guide into the microscopic mechanism. Koop- mans et al. also offer a phenomenological description of de- magnetization considering three baths, but one that follow s spin in additionto heat8. Spinis treated asa two state system with energylevels separatedby an exchangegapand Fermi’s goldenrule is usedto relate demagnetizationto electronsc at- teringwhichflipsaspin. Equationsforcouplingconstantsa re derived based on parameters such as the density of states of electrons, phonons, and spins, the electron-phononscatte ring rate,andthe probabilityofspinflipat a scatteringevent. In the following we attempt to understand the behavior of the demagnetization time near T cwith an approach based on the two spin state model. A general relationship between the laser-induced τMand the spin flip time, τsf, can be derived near the transition temperature based on the concept of de- tailed balance9. In equilibrium,the ratio of the probabilityof4 FIG. 7. Log-log plot of demagnetization time as a function of re- duced temperature aspinflippingfrommajoritytominoritytothereverseofthi s process is the Boltzmann factor, e−∆ex/kT, where∆exis the exchange energy gap. The time derivative of the number of majorityandminorityelectronscanthenbewritten: ˙Nmaj=−˙Nmin=Nmin τsf−Nmaj τsfe−∆ex/kBT(2) Whenthesampleisthermallyexcitedbyapumpbeam,the electron temperature is increased by δTe. The rate of change ofspinsisthenalteredinthefollowingway: ˙Nmaj=−˙Nmin=Nmin τsf−Nmaj τsfe−∆ex/kB(T+δTe)(3) The demagnetization time is related to the total change in spin,∆S,frominitialtofinaltemperature,where,setting /planckover2pi1=1, Sis definedby: S= 1/2(Nmaj−Nmin)/Ntotal (4) Assumingthat ∆S,asa functionoftime,canbewritten: ∆S(t) = [S(Tf)−S(Ti)](1−e−t/τM)(5) thedemagnetizationtimecanbewrittenas: τM=∆S ˙S(0)(6) where˙S(0)is the initial change in the time derivative of the spin. The total change in spin can be calculated by taking the derivative of Swith respect to T, and multiplying by ∆Teq, theincreaseintemperatureonceelectrons,phonons,andsp inshavecomeintothermalequilibriumwitheachother. S(T)and ∆Scanbe written: S(T) =−1 2tanh/parenleftbigg∆ 2kT/parenrightbigg (7) and: ∆S=dS dT/vextendsingle/vextendsingle/vextendsingle T=T0∆Teq=−∆ex 4kBT2 0/bracketleftbigg T0∆′ ex ∆ex−1/bracketrightbigg ∆Teq (8) where we have relied on the fact that near the transition tem- perature, ∆ex≪kBTand made the approximation that δTe≪Tforlowlaserpower. Inthelastequation T0∆′ ex ∆ex≫1 nearTc,so onlythefirst termwill beconsidered. Thequantity ˙S(0),where˙S= 1/2(˙Nmaj−˙Nmin)/Ntotal, can be found by taking the derivative of ˙S(0)with respect to Te, since immediately after excitation the electron tempera- turehasincreased,butthespin temperature, T,hasnot. ˙S(0) =d˙S dTe/vextendsingle/vextendsingle/vextendsingle T=T0∆Teq=Nmaj N0τsf∆ex kBT∆Teq(9) Near the Curie temperature Nmaj∼Nmin∼1 2Ntotal. Usingthisapproximationsandequation(6), wefind: τM=/parenleftbigg∆′ ex ∆ex/parenrightbiggTcτsf 2(10) where∆′ existhederivativeof ∆exwithrespecttotemperature and∆ex∼(Tc−T)β, whereβis the critical exponent of the order parameter. Taking the derivative, we find ∆′ ex∼ −β(Tc−T)β−1,andthuscanwrite τM=βτsf 2/parenleftbiggTc Tc−T/parenrightbigg (11) Therefore τMis predicted to scale as 1/(Tc−T)near the transition temperature. A fit of T c∼148.8K is found for the datainFig 6. Note that detailed balance suggests that the demagnetiza- tion time scales as 1/tRnear the transition temperature re- gardlessoftheunderlyingmechanismofthedemagnetizatio n. Additionally,the critical exponentfoundis independento fβ. It should also be noted that the current situation, where the sample has been excited by a laser, is distinct from critical behavior as typically considered. In general, divergent ti me scales are linked to divergent length scales, but here excit a- tions of various length scales are not being excited. Instea d thelengthscale is alwayseffectivelyinfinite, havingbeen de- terminedby the laser spot size. τsfis plotted as a functionof temperature for the mean field value of β= 1/2, which has beenshowntobesuitableforSRO10,inFig. 8. τsfisrevealed to be approximately 200 fs and nearly constant as a function oftemperature.5 FIG.8. Spinfliptime at hightemperature Previous reports of conductivity in SRO give a scattering time of∼20 fs near the transition temperature11. A compar- ison of the spin flip time with the scattering time implies a probabilityof0.1thatascatteringeventsresultsinaspin flip. Thoughelectron-phononinteractionsare the most commonly considered source of demagnetization, as mentioned previ- ously, Eliot Yafet-like electron-electron coulomb scatte ring canalsoresultindemagnetization3. Thisisespeciallytruefor materials with strong spin orbit coupling, such as SRO. Ad- ditionally in SRO the interaction with the crystal field mean s that total spin is not conserved[Goodenough], so every elec - troninteractioncanperturbthespinstate. Having found a relationship between the demagnetization time and the spin flip time we would like to explore the rela- tionship between these parameters and the damping param- eter,α. Intuitively, the damping parameter should be pro- portional to the spin flip scattering rate, or inversely prop or- tionaltothe spinflip scatteringtime: α∼1/τsf. Elliot-Yafet type scattering dissipates energy from motion described by the LLG equation by disrupting the coherent, collective pre - cession of spins. Spins that have had their angular momen- tum changed through electron collisions must be pulled back into the precession through the exchange interaction, repr e-senting a transfer of energy away from the precessional mo- tion. These collision-mediatedspin-orbitcouplingeffec ts are thought to be the primary source of Gilbert-type damping in ferromagnets12. Again, this should be particularly true in a ferromagnetwith strongspinorbitcoupling. Combining the spin flip time and the damping parameter with Planck’s constant reveals an energy scale, E, given by theconditionthat: 1 α∼E /planckover2pi1τsf (12) Noting that the valuesfor αandτsffoundin figures 3 and 7, respectively, are approximately constant as a function o f temperature, this energy scale for SRO is ∼7 meV. The fun- damental energy scales applicable to the magnetic system in SRO aretheFermienergy,the exchangeenergy,andthecriti- caltemperature,thelasttwoofwhichareinterdependent. T he Fermi energy is orders of magnitude larger than 7 meV, but the energy associated with the critical temperature, kBTc∼ 13 meV, is of the same order. This suggests an underlying connectionbetween the critical temperature(and thus the e x- changeenergy),Gilbertdamping,andspinflip scattering. A relationshipsimilar to equation(12) hasbeenfoundpre- viously between τM(rather than τsf) andαby Koopmanset al. at lowtemperature: τM=1 4/planckover2pi1 kBTc1 α(13) Applying this equation to SRO at 5K yields τm∼30fs, which is unphysical since it is below the total scattering ra te of∼100fsatlowtemperature11. Whetherthefundamentalre- lationshipisbetweentransitiontemperatureandthedemag ne- tizationtimeorthespin-flipscatteringtimeremainsaques tion fora microscopicmodeltoresolve. ACKNOWLEDGMENTS This research is supported by the US Department of En- ergy, Office of Science under contract number DE-AC02- 05CH1123. 1E.Beaurepaire etal.,Phys. Rev. Lett. 76, 4250 (1996). 2B.Koopmans et al.,Nat.Mater. 9,259 (2009). 3M. Krauss et al.,Phys.Rev. B 80, 180407 (2009). 4T. Ogasawara et al.,Phys.Rev. Lett. 94, 087202 (2005). 5T. Kiseet al.,Phys.Rev. Lett. 85, 1986 (2000). 6M. Langner etal.,Phys. Rev. Lett. 102, 177601 (2009). 7G.Koster etal.,Appl. Phys.Lett. 73, 2920 (1998). 8B.Koopmans et al.,Phys.Rev. Lett. 95, 267207 (2005). 9N.Metropolis et al.,J.Chem. Phys. 21, 1087 (1953). 10D.Kim et al.,Phys.Rev. B 67, 100406 (2003).11J.S.Dodge et al.,Phys.Rev. Lett. 85, 4932 (2000). 12B. Heinrich, in Ultrathin Magnetic Structures III (Springer- Verlag,Berlin,Germany, 2005). 13J.Fabian, and S.Das Sarma, Phys.Rev. Lett. 81, 5624 (1998). 14P.Monod, andF.Beuneu, Phys.Rev. B 19, 911 (1979). 15X.Wanget al.,Phys.Rev. B. 74, 195118 (2006). 16D.L.Mills and S.M.Rezende, in Spin dynamics in confined mag- netic structures II , edited by B. Hillebrands and K. Ounadjela (Springer-Verlag,Berlin,Germany, 2003). 17M. Pickelet al.,Phys.Rev. Lett. 101, 066402 (2008).

2010-08-03

We report time-resolved Kerr effect measurements of magnetization dynamics in ferromagnetic SrRuO3. We observe that the demagnetization time slows substantially at temperatures within 15K of the Curie temperature, which is ~ 150K. We analyze the data with a phenomenological model that relates the demagnetization time to the spin flip time. In agreement with our observations the model yields a demagnetization time that is inversely proportional to T-Tc. We also make a direct comparison of the spin flip rate and the Gilbert damping coefficient showing that their ratio very close to kBTc, indicating a common origin for these phenomena.

Determination of the spin-flip time in ferromagnetic SrRuO3 from time-resolved Kerr measurements

1008.0674v1

1 Enhancing Perpendicular Magnetic Anisotropy in Garnet Ferrimagnet by Interfacing with Few -Layer WTe 2 Guanzhong Wu1*, Dongying Wang1, Nishchhal Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, Guixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun Ning Lau1, Fengyuan Yang1, Mohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA 2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air Force Base, Dayton, OH, 45433, USA 3. Research Center for Functional Materials, National Institute for Materials Science, 1 -1 Namiki, Tsukuba 305 -0044, Japan 4. International Center for Materials Nanoarchitectonics, Nationa l Institute for Materials Science, 1-1 Namiki, Tsukuba 305 -0044, Japan *wu.2314@osu.edu 2 Abstract: Engineering magnetic anisotropy in a ferro - or ferrimagnetic (FM) thin film is crucial in spintronic device. One way to modify the magnetic anisotropy is through the surface of the FM thin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced by interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y 3Fe5O12 (YIG), and the low -symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, WTe 2. At the same time, we also observed an enhancement in Gilbert damping in the WTe 2 covered YIG area. Both the magnitude of interface -induced PMA and the Gilb ert damping enhancement have no observable WTe 2 thickness dependence down to single quadruple -layer, indicating that the interfacial interaction plays a critical role. The ability of WTe 2 to enhance the PMA in FM thin film, combined with its previously rep orted capability to generate out -of-plane damping like spin torque, makes it desirable for magnetic memory applications. Key words: perpendicular magnetic anisotropy, magnetic resonance force microscope, transition metal dichalcogenides, ferrimagnetic i nsulator 3 Perpendicular magnetic anisotropy (PMA) in a ferromagnetic thin film is of great interest in spintronics research and application s. Ferromagnetic nano -element s with PMA overcome their shape anisotropy , greatly ease the memory cell size reduction and improves memory retention . These exceptional properties, improving the performance of magnetic devices , make PMA highly desirable for magnetic memory application s. PMA becomes even more important in the recent development of solid state magnetic random -access memory (MRAM) since it allows MRAM to have lower switching current and faster switching speed compare d to in-plane magnetized materials 1, 2. Magnetic storage devices generally rely on metallic magnetic material s due to their robust electrical response . Interfacial magnetic anisotropy plays a critical role in generat ing PMA in metallic ferromagnet s. When interfacing with a nonmagnetic material (NM), electron orbital angular momentum of the magnetic ions at the ferromagnet surface will be modified, in some cases enabling strong covalent bonding, resulting in distinct magnetic properties compare d to the single layer 3-6. However, spintronics devices made of metallic magnetic materials are inherently energy consumptive due to resistive losses. Recently, complex oxide ferro - or ferrimagnet insulator s (FMI) have attracted substantial interest due to their ability to transport spin excitation s with low dissipation 7. Inducing PMA in FMIs naturally becomes an important topic both for scientific and technologic al reasons . Several successful route s to achiev ing PMA in FMIs ha ve been reported using bulk intrinsic anisotropy 8 or lattice strain 9-12. But in most experiments, the sign of the resulting interfacial anisotropy in FMI/NM heterostructures is such as to enhance the easy-plane anisotropy 13-15. Only one recent experiment has shown the possibility of generating interfacial PMA, and this was attributed to topological surface states 16. Nevertheless, t hese results demonstrate the possibility of controlling magnetic anisotropy through interfac ial interaction s in 4 FMI/NM heterostructures . Here, we report a study on YIG/WTe 2/hBN heterostructures, which shows that when interfacing with a low symmetry nonmagnetic van der Waals material , WTe 2, an additional interfac e-induced PMA (iPMA) term emerges in the magnetic anisotropy of the YIG thin film . The absence of topological surface states at room temperature in WTe 2 17, 18 forces us to seek an explanation for our observation of enhanced PMA that is distinct from that proposed for top ological insulator/YIG bilayers 16. We therefore turn to an analysis of the broken symmetries in WTe 2. We point out that low symmetry WTe 2 has recently shown the capability of generating both in -plane and out -of-plane spin polarization in charge -spin conversion experiments 19-22. It also enables field-free switching of PMA magnet ic material , which ease s the application of PMA material s in MRAM application s 23-25. Ferrimagnetic insulator YIG is of significant research interest in spintronics due to its exceptionally low Gilbert damping 26, which describes the relaxation rate of magnetization precession . And 1T’-WTe 2 is a semi -metallic transition metal dichalcogenide (TMD) layered material with strong SOC 27, 28. The crystal structure of 1T’-WTe 2 lacks twofold rotational symmetry about the c -axis (Fig. 1a). The only symmetry in the WTe 2 crystal lattice ab plane is the mirror symmetry about the bc plane 29. This u nique symmetry breaking allows out-of-plane damping -like torque to be generated 30, 31, enabling efficient switching of the out-of-plane magnetization of the adjacent magnetic material 24. A 20nm thick YIG thin film used in our experiment is epitaxially grown on (111) -oriented Gd3Ga5O12 (GGG) substrate by off -axis sputtering 32. WTe 2 flakes are then mechanically exfoliated from a flux-grown crystal, and dry transferred on to the clean top YIG surface without touching any other substances. This whole process is carried out in an Ar -filled glove box with <0.1 ppm of H 2O and 5 O2 to protect the flakes from degradation and ensure the clean liness of the YIG/WTe 2 interface. We employ hexagonal boron nitride ( hBN ) encapsulation to protect the WTe 2 flakes from oxidation after being removed from the glove box. We make two samples and focus on the data taken from sample 1 in the main text. The raw data taken from sample 2 can be found in Supporting Information Fig. S 2. Fig. 1 Crystal structure of WTe 2 and sample schematic . a) Crystal lattice structure of WTe 2 viewed from the top along the c-axis and looking from the side along the a-axis. The black dashed box in the side view indicates a monolayer of WTe 2. b) Schematic of the ferromagnetic resonance force microscope. RF excitation is generated by a stripline underneath the sample , where the hBN encapsulation is not shown . The region of localized mode is shown as a yellow dot adjacent to the WTe 2 flake, and the probe magnetic moment is shown as a yellow arrow on the particle. The cantilever oscillation is detected by a fiber laser interferometer. Figure 2a shows an optical image of the sample 1. Due to the small lateral size of the exfoliate d WTe 2 and hBN flake s having length scale s of 10 μm, we use a home -built ferromagnetic resonance force microscope (FMRFM) to measure the local ferromagnetic resonance (FMR) signal. FMRFM is a sensitive technique to detect the local magnetic properties with high spatial and spectral resolution 33. In our FMRFM, the external magnetic field 𝐻⃗⃗ ext is aligned perpendicular to the sample plane. The cantilever tip holds a high coercivity SmCo 5 magnetic particle , whose moment is magnetized in the 6 direction opposite to 𝐻⃗⃗ ext to create a magnetic field well . The field well supports a set of localized standing spin wave modes (LMs). During the measurement, we excite spin precession uniformly by a stripline underneath the sample at a fixed RF frequency (2 GHz) and sweep the magnetic field. The resonance of each LM generate s a stray field, whic h can then be detected by the SmCo 5 magnetic particle attached on the cantilever through their magnetic dipole -dipole interaction (Fig. 1b). During the measurement, we keep the probe -to-sample separation around 4 μm. The operation of FMRFM is described in detail in Ref s. 34-36. For reference, w e separate a region of YIG that does not contain WTe 2/hBN heterostructures and measur e its Gilbert damping using broadband FMR. To eliminate two - magnon scattering, w e perform broadband FMR in the out -of-plane field geometry. The FMR linewidth as a function of frequency measured on bare YIG (sample 1) shows a linear dependence (Fig. 2b), from which we can extract the Gilbert damping of bare YIG 𝛼YIG=1.05×10−3. We also confirm that the WTe 2 used in the experiment is indeed the 1T’ phase through polarized Raman measurements. The polarization angle dependence of the Raman peak at 212 cm-1 (spectrum is shown in Fig. S4) exhibits minimum intensity when the excitation laser polarization is along the crystallographic a axis of WTe 2 37 as shown by the polar plot in Fig. 2c and Raman intensity plot in Fig. 2d . We find t he position of the YIG/WTe 2/hBN heterostructure with the assist ance of magnetic alignment markers (Fig. 2a) . Figure 2e shows t wo raw FMRFM scans taken in the region of YIG/hBN and YIG/WTe 2/hBN , indicated by the blue and the red dot in Fig. 2a, respectively , which reveals the change in FMRFM spectra at two different location s. Here we focus on the 𝑛=1 LM because it has the mode radius of around 1 μm and gives the highest spatial resolution. Higher order modes have increasing mode radius and therefore, detect less local ized magnetic properties. This is the reason why 7 the quasi -uniform mode at ~ 3325 Oe does not show obvious change in resonance field or signal amplitude. We further take a line scan across the edge of WTe 2 flake (Fig. 2 f) to resolve the spatial evolution of FMRFM spectra . The line scan in Fig. 2f (along the dashed line shown in Fig. 2a) shows three main features : first, the magnitude of the LM resonance signal is reduced in the YIG/WTe 2/hBN region compare d to the YIG/hBN region; second, the LM resonance field for all LMs is decreased by ~40 Oe in the YIG/WTe 2/hBN region; third , the LMs show complex splitting and crossing when the probe is close to the boundary (−5 μm<𝑋<10 μm). Fig. 2 FMRFM and Raman measurement data . a) An optical micrograph of the YIG/WTe 2/hBN heterostructure under study . WTe 2 crystal a and b axis are labeled. b) Broadband FMR measurement of the frequency -dependent linewidth of the YIG thin film. The measurement is done on the same piece of YIG used to make sample shown in Fig. 1b. c) Polar plot of the 21 2 cm-1 peak Raman intensity. Angle denote s the relative angle between the measurement laser polarization and the WTe 2 a axis. d) 2D intensity plot showing Raman peak intensities versus polarization angle . e) FMRFM spectra, one over the YIG/hBN region (blue line) and the second over the YIG/WTe 2/hBN region (red line); these locations are indicated by the blue and red dot s in Fig. 2a respectively . f) Color plot of field -dependence 8 FMRFM scans as a function of position along the trace indicated by the black dashed line in Fig. 2a. A constant background is subtracted to show only the signal from the several LM resonance s. In the following, we will explain the origin of the three observed effects using spin pumping and magnetic anisotropy. The first effect , i.e. signal reduction in the YIG/WTe 2/hBN area relative to the YIG/hBN area, is the result of enhanced relaxation due to spin pumping from YIG to WTe 2 38. The 𝑛= 1 LM resonance signal amplitude ∆𝐴 is inversely proportional to the square of Gilbert damping , 𝛼2. We determine the Gilbert damping constant 𝛼 for YIG/WTe 2/hBN using 𝛼YIG/WTe2/hBN= 𝛼YIG/hBN×√∆𝐴YIG/hBN∆𝐴YIG/WTe2/hBN ⁄ (see Ref. 39), where 𝛼YIG/hBN is assumed to be the same as 𝛼YIG=1.05×10−3 due to the low SOC and insulating character of hBN . The second effect is the decrease of 𝑛=1 LM resonance field 𝐻r,1 by ~40 Oe . And the third effect is splitting and crossing of complex modes in the region −5 μm<𝑋<10 μm. The second and the third effects are due to an abrupt change of uniaxial anisotropy across the boundary separating the YIG/WTe 2/hBN and YIG/hBN regions 15. Here, the uniaxial anisotropy refers to the magnetic free energy depends on the angle between magnetization and sample normal ℱu=−𝐾u𝒎z2, where 𝒎z is the component of magnetization unit vector in the direction normal to sample plane and 𝐾u is the uniaxial anisotropy constant specific to sample and depends on the total interaction in the sample . When 𝐾u is positive, ℱu is called to be of PMA type, on the other hand, if 𝐾u is negative, ℱu is called to be of easy -plane type. This uniaxial anisotropy will lead to an effective uniaxial magnetic field 𝑯u=−𝜕ℱu𝜕𝑴⁄ , where 𝑴 is the magnetization . And therefore, a change in 𝐾u can modify the resonance field in a FMR measurement . In FMRFM spatial mapping, a n abrupt change in 𝐾u spatially could disturb the LM and lead to mode splitting and crossing as described in Ref. 15. Moreover, i n striking contrast to the previously studied 9 YIG/Au interface 15, which result s in a 32 Oe increase of 𝐻r,1 due to the enhanced easy -plane anisotropy, the observed decrease of 𝐻r,1 indicates that the WTe 2 overlayer induces an iPMA in YIG. We note that the magnitude of the shift in 𝐻r,1 is comparable to the easy -plane anisotropy induced by a heavy metal 15, 40 or the iPMA generated by topological surface state 16 on garnet ferrimagnetic material . In order to probe the global effect of a WTe 2 overlayer on YIG, we spatial ly map 𝐻r,1 using the 𝑛=1 LM. Figure 3a present s an optical image of WTe 2 flakes on a Si/SiO 2 (285nm ) substrate , where different c olors of WTe 2 flakes indicat e different WTe 2 thickness es. Figure 3b and 3 c show spatial maps of magnetic properties in the region enclosed by the black dashed rectangle in Fig. 3a . We acquire the maps using the procedure described in Ref. 39, i.e., simultaneously measuring spatial variation of the magnetic anisotropy and Gilbert damping using the 𝑛=1 LM resonance field 𝐻r,1 and signal amplitude ∆𝐴. The entire WTe 2-covered area show s uniformly lower ed 𝐻r,1 and increased Gilbert damping relative to the area without WTe 2. In Fig. 3c, despite the not great signal to noise ratio in damping imaging, the re is a clear Gilbert damping enhancement in WTe 2-covered area . The averaged Gilbert damping of YIG in WTe 2-covered area is 𝛼̅YIG/WTe2/hBN≈1.30×10−3, about 24% higher than 𝛼YIG. We note that due to the slight relative tilting of the scan plane and the sample plane, there is a color shift in Fig. 3b that might conceal the contrast difference in different WTe 2 thickness region . Therefore, to study the WTe 2 thickness dependence, we will show fine line scans across edge s of flakes having different WTe 2 thickness es. 10 Fig. 3 Two -dimensional FMRFM scan resolving the spatial variation of magnetic anisotropy and Gilbert damping. a ) Optical micrograph showing the color contrast of different thickness WTe 2 flake s (ranging from 4.7 nm to 44.8 nm) on Si/SiO 2(300 nm). Black dashed box outlines the FMRFM scanned area for 2D mapping. b) 2D map of the 𝑛=1 LM resonance field . The d ashed line s labeled 1 -4 correspond to the four line-scans shown in Fig. S1a-S1d. c) 2D mapping of the Gilbert damping extracted from the 𝑛=1 LM resonance peak amplitude. 11 Next, we want to understand what gives rise to the PMA in WTe 2/YIG. We rule out the effect induced by a modification of the gyromagnetic ratio by showing the resonance field shift across the WTe 2 edge does not depend on RF excitation frequency (See Fig. S3). We also exclude a strain induced effect given the absence of an epitaxial relation and the weakness of the van der Waals interaction between YIG and WTe 2. We further note that we can ignore the role of topological surface states 16 in our analysis; they are not relevant for our room temperature experiment since WTe 2 is a topological Weyl semimetal only below 100 K 17, 18. We show how an analysis based on symmetry and the nature of the interfacial SOC , generalizing the theory in Ref. 41, gives insight into the PMA observed in our experiment. This will also help us understand why the easy-axis anisotropy we observe in WTe 2/YIG is so different from the results of Ref. 13, 15 on YIG interfaces with a dozen different metallic and semiconducting materials, all of which exhibit interface -induced easy-plane anisotropy, as is predicted by theory 41. YIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via antiferromagnetic (AFM) superexchange interactions. We focus on how interfacial SOC impacts AFM superexchange in YIG and show that it leads to a very specific form of the mag netic anisotropy that is governed by the direction of the effective B-field (see Supporting Information for a details). Before turning to WTe 2/YIG, it is useful to first consider the simpler case when the only broken symmetry is the mirror plane defined b y the interface. The abrupt change in lattice potential then results in an effective electric field that points normal to the interface, which in turn leads to an effective magnetic field in the rest frame of the electron that couples to its spin. Since t he E-field points normal to 12 the interfacial plane in which the electron moves, the resulting B-field arising from SOC lies within the interfacial plane. As we show in the SI, this leads to a SOC -induced correction to AFM superexchange that necessarily lead s to an easy-plane anisotropy. In the case of WTe 2/YIG, however, when there are additional broken symmetries. Not only does the interface break inversion symmetry , but the crystal structure of WTe 2 itself breaks in -plane inversion symmetry . The electric field is now no longer normal to the interface, and the effective B-field arising from SOC necessarily has an out -of-plane component, as shown in Fig S5b in SI. Thus, we see why the lower symmetry of WTe 2/YIG can naturally result in an easy-axis or perpendicular magnetic anisotropy (PMA); see Supporting Information for details. We note that the lack of two -fold rotational symmetry in the ab plane in WTe 2 that plays a critical role in our understanding of PMA in WTe 2/YIG, has also been pointed out be crucial for the out - of-plane damping -like torque in WTe 2/Permalloy30. We note, however, that the out -of-plane damping - like torque necessarily involves current flow in WTe 2, while the PMA is an equilibrium property of the system independent of current flow. We further demonstrate the interfacial origin of the observed effect by studying the influence of WTe 2 thickness. We show four line-scans , labeled in Fig. 3b, across the edges of WTe 2 with different thickness es, ranging from 4. 7 nm to 44.8 nm . From these four line -scans, we extract the 𝑛=1 LM resonance field 𝐻r,1 and the 𝑛=1 LM resonance signal amplitude ∆𝐴. Figures S1a-d in the Supporting Information show the evolution of 𝐻r,1 and ∆𝐴 along the traces labeled correspondingly . The thickness of WTe 2 at each measurement location is later measured using atomic force microscop y. From these 13 line-scans , we choose the region s where the probe is far away from the edge of WTe 2 so that the magnetic propert ies are uniform, to obtain spatial average s of 𝐻r,1 and ∆𝐴, which are denoted 𝐻̅r,1,YIG/hBN and ∆𝐴̅̅̅̅YIG/hBN in the YIG/hBN region , and 𝐻̅r,1,YIG/WTe2/hBN and ∆𝐴̅̅̅̅YIG/WTe2/hBN in the YIG/WTe 2/hBN region , respectively . We further extract the 𝑛=1 LM resonance field difference between two regions using ∆𝐻r,1=𝐻̅r,1,YIG/hBN−𝐻̅r,1,YIG/WTe2/hBN, as well as the Gilbert damping difference using ∆𝛼=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ −1) as a function of the WTe 2 thickness . We note that the hBN overlayer does not change the Gilbert damping in YIG . The summarized results containing the data from both sample 1 and sample 2 are shown in Fig s. 4a and 4 b. Raw data from sample 2 can be found in Supporting Information Fig. S 2. The thinnest WTe 2 acquired in the experiment is 3.2nm from sample 2, which is approximately the thickness of a quadruple -layer WTe 2. Figures 4a and 4 b indicate that both ∆𝐻r,1 and ∆𝛼 have almost no WTe 2 thickness dependence . There is a small sample -to-sample variation possibly due to different YIG/WTe 2 interfacial quality . The change of 𝑛=1 LM resonance field, ∆𝐻r,1, is as large as ~38 Oe even when the WTe 2 thickness approaches the quadruple -layer thickness . This indicates that the modification of magnetic anisotropy is due to the YIG/WTe 2 interfac ial interaction , with no bulk contribution. For the increase of Gilbert damping ∆𝛼, no obvious thickness dependence is observed when comparing the data from the same sample. In sample 2, the Gilbert damping enhancement due to the quadruple -layer WTe 2 has almost the same value as the 50 nm thick WTe 2 flake, indicating that no thickness dependence of spin pumping can be resolved from our measurement. There are two possible interpretations of these results . First, if the 14 spin current injected into WTe 2 is mainly relaxed due to spin relaxation in the bulk, then the experimental result is a demonstration of ultra -short spin diffusion length along the c axis38, smaller or comparable to the thinnest WTe 2 flake (3.2 nm), employed in this experiment . It is much smaller than the 8nm spin diffusio n length in the in-plane direction measured using inverse spin Hall effect 22. Note that due to the chang e in mo bility and the metal -insulator transition in few layer WTe 2 when its thickness reduces 42, the spin diffusion length approximated here could be inaccurate . Alternatively , it is possible that the spin relaxation is primarily due to the interfacial SOC induced by inversion symmetry breaking at the interface and in the WTe 2 crystal lattice. In this case, the Gilbert damping enhancement will have no WTe 2 thickness dependence. 15 Fig. 4 WTe 2 thickness dependence of resonance field and damping enhancement . a) 𝐻r,1 in the YIG/hBN and YIG/WTe 2/hBN regions are averaged respectively to get 𝐻̅r,1,YIG and 𝐻̅r,1,YIG/WTe2, and ∆𝐻r,1=𝐻̅r,1,YIG−𝐻̅r,1,YIG/WTe2. b) ∆𝛼 as a function of WTe 2 thickness , and ∆𝛼=𝛼YIG/WTe2−𝛼YIG where 𝛼YIG is the Gilbert damping of bare YIG measured using broadband FMR for each sample , and 𝛼YIG/WTe2=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ ). In conclusion, we have shown that the YIG/WTe 2 interface plays a critical role in both interfacial magnetic anisotropy and spin relaxation , making WTe 2 a promising material in magnetic memory 16 application s. Combining the iPMA created by WTe 2 with the out-of-plane spin orbit torque generated by flowing a charge current along the a axis of WTe 2, one can possibly achieve field -free switching of a PMA magnetic cell for magnetic memory application s. It will improv e the scalability , reduc e the power consumption and increas e operation speed of magnetic solid -state devices . Our result reveals new possibilities in selecting materials and designing spintronic devices. For example, one can consider other materials with low lattice symmetry and strong SOC to induce larger PMA type interfacial ani sotropy in FMIs. To achieve a fully PMA material, one could utilize thinner FMIs to magnify the role of iPMA. Moreover, interfacial SOC also plays an important role in generat ing topologically protected magnetic textures in the FMIs 43. These findings will motivate further research to reveal the fundamental physics arising at the interface between FMIs and nonmagnetic materials. Data availability: The data generated by the present study are available from the corresponding author on request. Supporting Information: A description of raw data on WTe 2 thickness dependence, a FMRFM measurement on a second sample, a FMRFM measurement at different RF frequency, a description of polarized Raman measurement result, and a detailed illustration of impact of broken mirror reflection symmetries on the magnetic anisotropy. Ackno wledgements: 17 This work was primarily supported by the Center for Emergent Materials: an NSF MRSEC under award number DMR -2011876 (GW, NV , YC, SG, FY , MR and PCH) . KW and TT acknowledge support from the Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) and JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233). DW, GC, CNL, and MB are supported by NSF under award DMR -2004801. We gr atefully acknowledge N. Trivedi for insightful discussions. Fabrication and some characterization were performed in the Ohio State University NanoSystems Laboratory. 18 Methods: Sample fabrication Our YIG/WTe 2/hBN heterostructure was prepared by means of dry transfer and stacking 44. hBN crystals were mechanically exfoliated under ambient conditions onto SiO 2/Si substrates (285 nm thick SiO 2). 20- 40 nm thick hBN flakes were identified under an optical microscope and used for the capping lay er for the stack. The hBN was picked up using a polymer -based dry transfer technique and then moved into an Ar-filled glove box with oxygen and water level below 0.1 ppm. Flux -grown WTe 2 crystals 45 were exfoliated inside the glove box and flakes with different thicknesses were optically identified and quickly picked up with the capping hBN layer then transferred to the YIG substrate. Finally, we removed the fully encapsulated sample from the glove b ox and performed the e -beam lithography and metallization (Ni/Au) step for alignment in our ferromagnetic resonance force microscope (FMRFM). Polarized Raman measurement Polarized Raman spectra from the WTe 2 sample were collected using 633 nm excitation w avelength in an inVia Renishaw Raman microscope. The sample was loaded onto the microscope stage and positioned in such a way that the long edge of the flake was aligned parallel to the laser polarization ( θ = 0°). In this configuration, the incident illu mination is polarized vertically coming out of the laser and is aligned with the long axis of the WTe 2 flake. The polarization of the incident laser was rotated from 0 to 360° by 10° increments using a polarization rotator, while an analyzer was set to onl y allow vertically polarized light to enter the spectrometer. Raman spectra were collected at each polarization for 3 acquisitions with a 20 s time per acquisition. The laser power was set to 0.5 mW at the sample to avoid any damage by heating. Followin g spectral collection, the (baseline corrected) integrated intensities under each peak were calculated to make the contour plots and polar plots in Fig. 2c and 2d. FMRFM measurement and signal fitting Our FMRFM perform s local ly measures FMR at room temper ature in vacuum. The cantilever has natural frequency of ~18 KHz, spring constant of 0.2 N/m and Q factor of ~20000, resulting in force detection sensitivity of 10-15 N/Hz1/2. The SmCo 5 magnetic particle attached on the cantilever has a magnetic moment of ~4 nemu. When a LM is on resonance, the local reduction of magnetization in out - of-plane direction will generate a stray field, which will couple the altered magnetization to the magnet ic tip thus changing the cantilever oscillation amplitude and frequency. 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Anisotropic magnetotransport and exotic longitudinal linear magnetoresistance in WTe2 crystals. Phys. Rev. B 2015, 92, (4), 041104. 23 Enhancing Perpendicular Magnetic Anisotropy in Garnet Ferrimagnet by Interfacing with Few-Layer WTe 2 Guanzhong Wu1*, Dongying Wang1, Nish chhal Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, Guixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun Ning Lau1, Fengyuan Yang1, Mohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA 2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air Force Base, Dayton, OH, 45433, USA 3. Research Center for Functional Materials, National Institute for Materials Science, 1 -1 Namiki, Tsukuba 305 -0044, Japan 4. International Center for Materials Nanoarchitectonics , National Institute for Materials Science, 1-1 Namiki, Tsukuba 305 -0044, Japan *wu.2314@osu.edu 24 FMRFM line-scan across edge of WTe 2 with different thickness Fig. S1 a-d, FMRFM line-scans along the traces 1~4 indicated in Fig. 3b respectively. The gray shaded area in four figures are outlining the location of WTe2 flake . 𝐻r,1 and ∆𝐴 at each position are derived by fitting the 𝑛=1 LM to a Lorentzian line shape. The thickness of WTe2 flake at each location are measured by atomic force microscope. 25 FMRFM measurement on Sample 2 Fig. S 2 FMRFM measurement on sample 2. a, The optical picture of the YIG/WTe 2/hBN heterostructure. b, 2D mapping of the 𝑛=1 LM resonance field in the black dash line circled area. c, 2D mapping of the Gilbert damping extracted from 𝑛=1 LM resonance peak amplitude in the black dash line circled area. d, FMRFM line scan along the trace indicated by the solid black line in Fig. S 2a. A constant background is subtracted to show only the signal from the LMs resonance. e, Fine scan zoomed in o n the quadruple layer WTe 2 stripe area 26 FMRFMR measurement at 4 GHz Fig. S 3 FMRFM measurement across WTe 2 edge at 4 GHz. FMRFM line scan is measured at 4 GHz across the WTe 2 flake edge. The shift of the resonance field 𝐻r,1 is 36 Oe, similar to the 𝐻r,1 shift measured at 2 GHz. This result excludes the possibility that the resonance field shift arises from modification of the gyromagnetic ratio. 27 Polarized Raman measurement Fig. S 4 Polarized Raman measurement. As shown by the red curve, the Raman spectrum taken on GGG/YIG/WTe 2/hBN heterostructure contains more peaks than WTe 2. The Raman spectrum taken in the GGG/YIG/hBN area identifies the peaks arising from the substrate GGG/YIG or top hBN encapsulation layer. By subtracting the Raman spectrum in the GGG/YIG/hBN area, the Raman spectra from WTe 2 layer are extracted and plotted in Fig. 2d. The black dash line are the markers indicating the Raman peaks of WTe 2 28 Impact of broken mirror reflection symmetr ies on the magnetic anisotropy We describe theo retical constraints on the interface -induced magnetic anisotropy in the WTe 2/YIG bilayer. We first show that symmetry arguments alone do not provide strong constraints on the anisotropy tensor, given that we are dealing with an interface between two crystalline materials at an arbitrary orientation with respect to each other . We then present qualitative arguments, based on the interfacia l spin - orbit coupling, that give insight into the magnetic anisotropy in WTe 2/YIG. This helps us understand why the easy-axis anisotropy that we observe in WTe 2/YIG differ s from the results of Lee et al. [1] on YIG interfaces with a dozen different metallic and semiconducting materials , all of which exhibit interface - induced easy-plane anisotropy as predicted by theory [2]. On general grounds, the anisotropy (free) energy can be written as ℱ𝑎𝑛𝑖𝑠= ∑ 𝐾𝑎𝑏𝑎,𝑏 𝑚𝑎 𝑚𝑏, (S1) where a and b take on values x,y,z. We focus here on the leading term, quadratic in the magnetization, and ignore higher order anisotropy terms like (mx4+my4+mz4) or (mx2my2+my2mz2+mz 2mx2). The form of 𝐾𝑎𝑏= 𝐾𝑏𝑎 is constrained by symmetry. Let us consider three cases , going from the most symmetric to the least . Case I: The only broken symmetry is interfacial inversion (z → - z), which is relevant for the experiments of Ref. [1]. The magnetization is an axial vector (or pseudovector) that transforms under rotation s like a vector but is unchanged under inversion . Thus (𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧)→ (𝑚𝑥 ,− 𝑚𝑦 ,−𝑚𝑧) under reflection in a mirror plane with normal 𝑥̂. Using reflection symmetry in mirror planes normal to 𝑥̂ and to 29 𝑦̂ , we can see that all off -diagonal components of 𝐾𝑎𝑏 vanish. Further, f our-fold rotational symmetry about the 𝑧̂ axis shows that 𝐾𝑥𝑥= 𝐾𝑦𝑦. Using mx2+my2+mz2=1, we write 𝐾𝑥𝑥(m𝐱2+m𝐲2) in terms of m𝑧2, and d efining 𝐾𝑢= (𝐾𝑥𝑥− 𝐾𝑧𝑧), we obtain ℱ𝑎𝑛𝑖𝑠= − 𝐾𝑢 m𝑧2. (S2) This symmetry analysis only constrains the form of the anisotropy energy, but not the sign of 𝐾𝑢. We will give below a simple microscopic argument [2] that shows that 𝐾𝑢<0 (easy plane) for Case I. Case II: In addition to broken interfacial inversion (z → - z), let u s also break reflection symmetry in the plane normal to 𝑥̂. This would be the case if the crystalline axes of WTe 2 were aligned with YIG. This also breaks four-fold rotational symmetry about 𝑧̂, so that 𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. However, we can still use reflection symmetry in the plane normal to 𝑦̂ to conclude that 𝐾𝑥𝑦= 𝐾𝑦𝑧=0. Thus we find that K = (𝐾𝑥𝑥0 𝐾𝑥𝑧 0 𝐾𝑦𝑦0 𝐾𝑥𝑧0 𝐾𝑧𝑧) (S3) Case III: When the crystalline axes of WTe 2 are not aligned with YIG, which is the experimentally relevant case, all mirror reflection and rotation symmetries are broken. Then there are no symmetr y constrain ts on 𝐾𝑎𝑏 and all six components of this symmetric tensor are in general non -zero. Let us now see how, despite the lack of general symmetry -based constrai nts, we can still get some qualitative insight about the form of the anisotropy from simple microscopic considerations informed by symmetry. YIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via 30 antiferromagnetic (AFM) superexch ange interactions. We thus focus on how interfacial spin -orbit coupling (SOC) impacts AFM superexchange. The broken symmetry at the interface leads to an electric field ℇ=−𝛁𝑉(𝒓), whose direction will be discussed in detail below for three cases. This in turn produces a magnetic field in the rest frame of the electron which underlies SOC. As the electron moves along 𝐫̂ij from site i to j, it experiences an SOC field in the direction 𝐝̂ij which is determined by ℇ ×𝐫̂ij . The SOC Hamiltonian is thus given by −𝑖𝜆∑𝑐𝐢𝛼†(𝐝̂ij∙𝝈𝛼𝛽)𝑐𝐣𝛃 𝛼𝛽 . Including the effect of this term in addition to the usual hopping t and Hubbard U in the standard strong coupling expansion calculation leads to the Hamiltonian ℋex=J∑𝐒i∙𝐒j <𝐢,𝐣>+D∑𝐝̂ij∙𝐒i×𝐒j <𝐢,𝐣> +K0∑(𝐝̂ij∙𝐒i)(𝐝̂ij∙𝐒j). <𝐢,𝐣> (S4) Here the spin 𝐒i at site i is coupled to its neighbors via the AFM superexchange 𝐽 ~𝑡2 𝑈 and the Dzyaloshinskii -Moriya interaction (DMI) 𝐷 ~𝑡𝜆 𝑈. The K0 term will be the focus of our attention belo w as it leads to magnetic anisotropy. We note that the general form of ℋex is in fact substantially independent [2] of the microscopic mechanism and very similar results are obtained not only for superexchange but also for Zener double exchange and RKKY interactions. Case I: Let us again return to the simplest case with broken interfacial inversion (z → - z). This leads to an electric field ℇ=−𝛁𝑉(𝒓) along ẑ , the normal to the interface. The SOC magnetic field direction is then given by 𝐝̂ij= ẑ ×𝐫̂ij ; see Figure S4(a). This is the well -known Rashba SOC at interfaces . We note in passing that 𝐝̂ij is antis ymmetric under the interchange of i and j, and thus leads to a DMI term where 𝐒i×𝐒j is also antisymmetric. 31 Fig. S 5 Symmetry based selection of magneto -crystalline anisotropy. Interfacial SOC originates from an effective Electric field ℇ=−𝛁𝑉(𝒓) whose direction is determined by the broken mirror planes in the system. This electric field leads to spin-orbit coupling (SOC), with the 𝐝̂ij= ℇ̂ ×𝐫̂ij, the direction of the SOC magnetic field. Note that the direction of the electron hop 𝐫̂ij lies in the xy plane of the interface. As shown in the text 𝐝̂ij controls the interface -induced magnetic anisotropy. (a) When only surface inversion is broken, 𝐝𝐢𝐣 is constrained to lie in the int erface and interfacial SOC leads to easy -plane anisotropy. (b) If there are other broken mirror planes, the 𝐝𝐢𝐣 must lie outside the interfacial plane. This can lead to a perpendicular magnetic anisotropy in systems like YIG/WTe 2 bilayers. We see that in Case I, 𝐝̂ij lies in the plane of the interface, and the third term in eq. (S 4) then takes the form K0∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦) 𝐢 for a square lattice . To make the connection with magnetic anisotropy, we look at a continuum approximation with a s lowly varying magnetization 𝐦(𝐫). We make a Taylor expan sion of 𝐒r in terms of its value at 𝒓, denoted by 𝐦(𝐫), and its spatial derivatives . The exchange and DMI terms involve gradients of 𝐦(𝐫), but we focus here on local terms that do not involve derivatives to 32 understand the magnetic anisotropy . The leading term is + K0(m𝐱2+m𝐲2) which can be rewritten as – K0 m𝑧2 using the fact that mx2+my2+mz2=1 at each 𝒓. Thus, we may identify K0 with the anisotropy K𝑢 defined in eq. (S2). The microscopic analysis leads to the result K0= − 𝜆2 𝑈 < 0 and this explains the easy-plane anisotropy arising Rashba SOC at the interface . The easy-plane nature of the anisotropy is in fact a general feature of various microscopic models as emphasized in Ref. [2]. We note however that these author s use d the opposite sign convention for anisotropies from the one we use here . The easy plane vs. easy -axis character is , of course, independent of sign conventions. The FMR experiments of Ref. [1] have seen the interface -induced easy-plane anisotropy predicted by the theory in a YIG interfaces with several metallic and semiconducting materials . The key difference between the YIG/WTe 2 bilayer studied here and systems studied earlier [1] is that WTe 2 has a broken mirror plane (the ac plane ) as shown in Fig. 1(a) of the paper . We now look at the effect of this lower symmetry on the microscopic analysis. Case II: Let us break reflection symmetry in the plane normal to 𝑥̂ in addition to broken interfacial inversion. We choose x̂ parallel to the b axis, ŷ parallel to a, and ẑ parallel to c. Reflection symmetry in the ŷ mirror plane constrains the electric field ℇ =−𝛁𝑉(𝒓) to lie in the xz plane, at an angle 𝜃 from the z-axis as shown in Fig. S 5(b). Thus 𝐝ij=(sin𝜃𝑥̂+cos𝜃𝑧̂)×𝐫̂ij (S5) 33 where 𝐫̂ij is a vector in the interface (xy plane ) and 0≤𝜃≤𝜋. Using eq. (S5), we may rewrite the last term in the Hamiltonian (S4) as K0 sin2𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑧) 𝐢+K0 cos2𝜃∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦) 𝐢 −K0sin𝜃cos𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑥+S𝐢𝑥𝑆𝐢+𝑦𝑧) 𝐢 As before, we make a continuum approximation with a smoothly varying 𝐦(𝐫) and focus only on the local terms, without gradients, to obtain the magnetic anisotropy . We find that the leading order contribution to anisotropy is −K0cos2𝜃m𝑧2+K0sin2𝜃 mzmx. This analysis correctly captures the non - zero K𝑥𝑧 expected on general grounds; see eq. (S3). We did not include here , for simplicity, the effects of broken four -fold rotation that would have led to 𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. Case III: When we lose all mirror symmetries, the case relevant to the YIG/WTe 2 experiment, the electric field ℇ =−𝛁𝑉(𝒓) will point in a general direction specified by 0≤𝜃≤𝜋 and 0≤𝜑≤2𝜋, and there will be no symmetry constraints on the anisotropy tensor 𝐾𝑎𝑏. Let us conclude by highlighting the key qualitative difference between Case I on the one hand and Cases II and III on the other . In Case I, the only broken symmetry is interfaci al inversion (z → - z). Then symmetry constrains the 𝐝̂ij, the direction of the SOC B-field, to lie in the plane of the interface and this leads to easy -plane anisotropy as described above. In Cases II and III, there are other additional broken mirror planes, and this leads to the 𝐝̂ij vector being pulled out of the plane of the interface. This immediately leads to the possibility of an easy -axis like character to the anisotropy, although in the general case one has a non -trivial anisotropy tensor 𝐾𝑎𝑏. 34 Reference [1] Lee, A. J.; Ahmed, A. S.; McCullian, B. A.; Guo, S. D.; Zhu, M. L.; Yu, S. S.; Woodward, P. M.; Hwang, J.; Hammel, P. C.; Yang, F. Y . Interfacial Rashba -Effect-Induced Anisotropy in Nonmagnetic - Material -Ferrimagnetic -Insulator Bilayers. Phys. Rev. Lett. 2020, 124, (25), 257202. [2] Banerjee, S.; Rowland, J.; Erten, O.; Randeria, M. Enhanced Stability of Skyrmions in Two - Dimensional Chiral Magnets with Rashba Spin -Orbit Coupling. Physical Review X 2014, 4, (3), 031045.

2022-02-06

Engineering magnetic anisotropy in a ferro- or ferrimagnetic (FM) thin film is crucial in spintronic device. One way to modify the magnetic anisotropy is through the surface of the FM thin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced by interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y3Fe5O12 (YIG), and the low-symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, WTe2. At the same time, we also observed an enhancement in Gilbert damping in the WTe2 covered YIG area. Both the magnitude of interface-induced PMA and the Gilbert damping enhancement have no observable WTe2 thickness dependence down to single quadruple-layer, indicating that the interfacial interaction plays a critical role. The ability of WTe2 to enhance the PMA in FM thin film, combined with its previously reported capability to generate out-of-plane damping like spin torque, makes it desirable for magnetic memory applications.

Enhancing Perpendicular Magnetic Anisotropy in Garnet Ferrimagnet by Interfacing with Few-Layer WTe2

2202.02834v1

arXiv:1906.08987v1 [math.AP] 21 Jun 2019UNIQUE DETERMINATION OF THE DAMPING COEFFICIENT IN THE WAVE EQUATION USING POINT SOURCE AND RECEIVER DATA MANMOHAN VASHISTH Abstract. Inthisarticle, weconsidertheinverseproblemsofdetermi ningthedampingcoefficientappearing in the wave equation. We prove the unique determination of th e coefficient from the data coming from a single coincident source-receiver pair. Since our problem is under-determined, so some extra assumption on the coefficient is required to prove the uniqueness. Keywords : Inverse problems, wave equation, point source-receiver, d amping coefficient Mathematics subject classification 2010: 35L05, 35L10, 35R30, 74J25 1.Introduction We consider the following initial value problem (IVP), (/square−q(x)∂t)u(x,t) =δ(x,t) (x,t)∈R3×R u(x,t)|t<0= 0 x∈R3(1) where/square:=∂2 t−∆xdenotes the wave operator and the coefficient q∈C∞(R3) is known as damping coefficient. In this paper, we study the problem of determinat ion of coefficient qappearing in (1) from the knowledge of solution measured at a single point for a cer tain period of time. We are interested in the uniqueness of determination of coefficients qfrom the knowledge of u(0,t) fort∈[0,T] withT >0 in Equation (1). The problem studied here is motivated by geoph ysics, where geophysicists wish to determine the properties of earth structure by sending the waves from t he surface of the earth and measuring the correspondingscattered responses(see [2, 24] andreferen ces therein). Sincethecoefficient tobedetermined here depends on three variables while the given data depends on one variable as far as the parameter count is concerned, the problem studied here is under-determined . Thus some extra assumptions on coefficient q are required in order to make the inverse problem solvable. W e prove the uniqueness result for the radial coefficient. There are several results related to the inverse problems fo r the wave equation with point source. We list them here. Romanov in [18] considered the problem for determ ining the damping and potential coefficient in the wave equation with point source and proved unique dete rmination of these coefficients by measuring the solution on a set containing infinite points. In [12] the p roblem of determining the radial potential from the knowledge of solution measured on a unit sphere for s ome time interval is studied. Rakesh and Sacks in [16] established the uniqueness for angular contro lled potential in the wave equation from the knowledge of solution and its radial derivative measured on a unit sphere. In the above mentioned works the measurement set is an infinite set. Next we mention the wor k where uniqueness is established from the measurement of solution at a single point. Determinatio n of the potential from the data coming from a single coincident source-receiver pair is considered in [ 15] and the uniqueness result is established for the potentials which are either radial with respect a point differ ent from source location or the potentials which are comparable. Recently author in [25] extended the result of [15] to a separated point source and receiver data. To the best of our understanding, very few results exis t in the literature involving the recovery of the damping coefficient from point source and receiver data. O ur result, Theorem 1.1, is work in this direction. In the 1-dimensional inverse problems context, several results exist involving the uniqueness of 12 VASHISTH recovery of the coefficient which depends on the space variabl e corresponding to the first order derivative; see [9, 10, 11, 13, 19, 22]. We refer to [1, 3, 8, 14, 17] and refe rences therein for more works related to the point source inverse problems for the wave equation. We now state the main results of this article. Theorem 1.1. Suppose qi(x)∈C∞(R3),i= 1,2withqi(x) =Ai(|x|)for some C∞function Aion[0,∞). Letuibe the solution of the IVP (/square−qi(x)∂t)ui(x,t) =δ(x,t) (x,t)∈R3×R ui(x,t)|t<0= 0 x∈R3.(2) Ifu1(0,t) =u2(0,t)for allt∈[0,T]for some T >0, thenq1(x) =q2(x)for allxwith|x| ≤T/2, provided q1(0) =q2(0). The proof of the above theorem is based on an integral identit y derived using the solution to an adjoint problem as used in [21] and [23]. This idea was used in [4, 17, 2 5] as well. The article is organized as follows. In Section 2, we state th e existence and uniqueness results for the solution of Equation (1), the proof of which is given in [5, 8, 20]. Section 3contains the proof of Theorem 1.1. 2.Preliminaries Proposition 2.1. [5, pp.139,140] Suppose q∈C∞(R3)andu(x,t)satisfies the following initial value problem Pu(x,t) := (/square−q(x)∂t)u(x,t) =δ(x,t),(x,t)∈R3×R u(x,t)|t<0= 0, x ∈R3(3) thenu(x,t)is given by u(x,t) =R(x,t)δ(t−|x|) 4π|x|+v(x,t) (4) wherev(x,t) = 0fort <|x|and in the region t >|x|,v(x,t)is aC∞solution of the characteristic boundary value problem (Goursat Problem) Pv(x,t) = 0, for t > |x| v(x,|x|) =−R(x,|x|) 8π1/integraldisplay 0PR(sx,s|x|) R(sx,s|x|)ds,∀x∈R3(5) andR(x,t)is given by [5, pp. 134] R(x,t) = exp −1 21/integraldisplay 0q(sx)tds . (6) 3.Proof of Theorem 1.1 In this section, we prove Theorem 1.1. We will first prove an in tegral identity which will be used to prove our main result. Lemma3.1.Letui(x,t)fori= 1,2be the solution to Equation (2). Then the following integral identity holds for all σ≥0/integraldisplay R3/integraldisplay Rq(x)∂tu2(x,t)u1(x,2σ−t)dtdx=u(0,2σ) (7) whereq(x) :=q1(x)−q2(x)andu(x,t) = (u1−u2)(x,t).AN INVERSE PROBLEM WITH UNDER-DETERMINED DATA 3 Proof.Here we have usatisfies the following IVP /squareu(x,t)−q1(x)∂tu(x,t) =q(x)∂tu2(x,t) (x,t)∈R3×R u(x,t)|t<0= 0 x∈R3.(8) Multiplying Equation (8) by u1(x,2σ−t) and integrating over R3×R, we have /integraldisplay R3/integraldisplay Rq(x)∂tu2(x,t)u1(x,2σ−t)dtdx=/integraldisplay R3/integraldisplay R(/squareu(x,t)−q1(x)∂tu(x,t))u1(x,2σ−t)dtdx =/integraldisplay R3/integraldisplay Ru(x,t)(/squareu(x,t)−q1(x)∂tu1(x,2σ−t))dxdt where in the last step above we have used integration by parts and the properties of vin Proposition 2.1. Thus finally using the fact that u1is solution to (2), we get /integraldisplay R3/integraldisplay Rq(x)∂tu2(x,t)u1(x,2σ−t)dtdx=u(0,2σ); for all σ≥0. This completes the proof of the lemma. /square Using Lemma 3.1and the fact that u(0,t) = 0 for all t∈[0,T], we see that /integraldisplay R3/integraldisplay Rq(x)∂tu2(x,t)u1(x,2σ−t)dtdx= 0; for all σ∈[0,T/2]. Now using Equation (4), we get /integraldisplay R3/integraldisplay Rq(x)∂t/parenleftBigg R2(x,t)δ(t−|x|) 4π|x|+v2(x,t)/parenrightBigg/parenleftBigg R1(x,2σ−t)δ(2σ−t−|x|) 4π|x|/parenrightBigg dtdx +/integraldisplay R3/integraldisplay Rq(x)v1(x,2σ−t)∂t/parenleftBigg R2(x,t)δ(t−|x|) 4π|x|+v2(x,t)/parenrightBigg dtdx= 0.4 VASHISTH This gives /integraldisplay R3/integraldisplay Rq(x)∂tR2(x,t)R1(x,2σ−t)δ(t−|x|)δ(2σ−t−|x|) 16π2|x|2dtdx /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright I1 +/integraldisplay R3/integraldisplay Rq(x)R2(x,t)R1(x,2σ−t)∂tδ(t−|x|)δ(2σ−t−|x|) 16π2|x|2dtdx /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright I2 +/integraldisplay R3/integraldisplay Rq(x)∂t/parenleftBigR2(x,t)δ(t−|x|) 4π|x|/parenrightBig v1(x,2σ−t)dtdx /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright I3 +/integraldisplay R3/integraldisplay Rq(x)∂tv2(x,t)R1(x,2σ−t)δ(2σ−t−|x|) 4π|x|dtdx /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright I4 +/integraldisplay R3/integraldisplay Rq(x)∂tv2(x,t)v1(x,2σ−t)dtdx /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright I5= 0; for all σ∈[0,T/2].(9) In a compact form, this can be written as I1+I2+I3+I4+I5= 0. (10) Next we simplify each Ijwithj= 1,2,.....,5. We will use the fact that vi(x,t) = 0 for t <|x|. We have I1=/integraldisplay R3/integraldisplay Rq(x)∂tR2(x,t)R1(x,2σ−t)δ(t−|x|)δ(2σ−t−|x|) 16π2|x|2dtdx =/integraldisplay |x|=σq(x)∂tR2(x,|x|)R1(x,|x|) 16π2|x|2dSx =−/integraldisplay |x|=σq(x)R1(x,|x|)R2(x,|x|) 32π2|x|2/parenleftBig1/integraldisplay 0q2(sx)ds/parenrightBig dSx. Next we simplify the integral I2. We use the following formula [7, Page 231, Eq.(10)] /integraldisplay δ′(r−|x|)ϕdx=−1 |x|2/integraldisplay |x|=r∂ ∂r/parenleftbig ϕr2/parenrightbig dSx. (11) Note that from this formula, by a change of variable, we have /integraldisplay δ′(2r−2|x|)ϕdx=−1 2|x|2/integraldisplay |x|=r∂ ∂r/parenleftbig ϕr2/parenrightbig dSx. (12)AN INVERSE PROBLEM WITH UNDER-DETERMINED DATA 5 Now I2=/integraldisplay R3/integraldisplay Rq(x)R2(x,t)R1(x,2σ−t)∂tδ(t−|x|)δ(2σ−t−|x|) 16π2|x|2dtdx =/integraldisplay R3/integraldisplay Rq(x)R2(x,t)R1(x,2σ−t)δ′(t−|x|)δ(2σ−t−|x|) 16π2|x|2dtdx =/integraldisplay R3q(x)R2(x,2σ−|x|)R1(x,|x|)δ′(2σ−2|x|) 16π2|x|2dx =−1 32π2σ2/integraldisplay |x|=σ∂ ∂r{q(x)R1(x,|x|)R2(x,2σ−|x|)}dSx. In the last step above, we used Equation (12). Next we have I3=/integraldisplay R3/integraldisplay Rq(x)∂t/parenleftBigR2(x,t)δ(t−|x|) 4π|x|/parenrightBig v1(x,2σ−t)dxdt. We can view the derivative above as a limit of the difference quo tients in the distribution topolgy [6, pp.48]. Combining this with the fact that v1isC2in{(x,t) :|x| ≤t}, we get, I3=−/integraldisplay R3/integraldisplay Rq(x)R2(x,t)δ(t−|x|) 4π|x|∂t/parenleftBig v1(x,2σ−t)/parenrightBig dxdt =/integraldisplay R3q(x)R2(x,|x|)∂tv1(x,2σ−|x|) 4π|x|dx. Again using the fact that v1(x,t) = 0 for t <|x|, we get, I3=/integraldisplay |x|≤σq(x)R2(x,|x|)∂tv1(x,2σ−|x|) 4π|x|dx. Next we simplify I4. Similiar to I3, we have I4=/integraldisplay R3/integraldisplay Rq(x)∂tv2(x,t)R1(x,2σ−t)δ(2σ−t−|x|) 4π|x|dtdx =/integraldisplay |x|≤σq(x)R1(x,|x|)∂tv2(x,2σ−|x|) 4π|x|dx. Finally, we have I5=/integraldisplay R3/integraldisplay Rq(x)∂tv2(x,t)v1(x,2σ−t)dtdx =/integraldisplay |x|≤σ2σ−|x|/integraldisplay |x|q(x)∂tv2(x,t)v1(x,2σ−t)dtdx.6 VASHISTH Now, we use the fact that qiis a radial function, that is, qi(x) =Ai(|x|). Then note that Ri(x,|x|) = exp −|x| 21/integraldisplay 0qi(sx)ds = exp −|x| 21/integraldisplay 0Ai(s|x|)ds is also radial. For simplicity, we denote R(x,|x|) byR(|x|). With this, we have I1=−A(σ)R1(σ)R2(σ) 8π1/integraldisplay 0A2(sσ)ds. Next we consider I2. First let us consider the derivative: Dr:=∂ ∂r(A(r)R1(x,r)R2(x,2σ−r)). After a routine calculation, we get, Dr=A′(r)R1(x,r)R2(x,r)−1 2A(r)2R1(x,r)R2(x,2σ−r) −σA(r)R1(x,r)R2(x,2σ−r)1/integraldisplay 0A′ 2(rs)sds =A′(r)R1(x,r)R2(x,r)−1 2A(r)2R1(x,r)R2(x,2σ−r) −A(r)R1(x,r)R2(x,2σ−r) σ r A2(r)−1/integraldisplay 0A2(rs)ds . On|x|=σ, we have Dr||x|=σ=R1(σ)R2(σ) A′(σ)−1 2A(σ)2−A(σ)A2(σ)+A(σ)1/integraldisplay 0A2(sσ)ds =R1(σ)R2(σ) A′(σ)−1 2A(σ)(A1+A2)(σ)+A(σ)1/integraldisplay 0A2(sσ)ds . Hence I2=−1 8π R1(σ)R2(σ) A′(σ)−1 2A(σ)(A1+A2)(σ)+A(σ)1/integraldisplay 0A2(sσ)ds . Let us denote ˜A(σ) =A(σ)R1(σ)R2(σ). Then I2=−1 8πd dσ˜A(σ)−1 8π˜A(σ)1/integraldisplay 0A2(sσ)ds. Therefore I1+I2=−1 8π 2˜A(σ)1/integraldisplay 0A2(sσ)ds+d dσ˜A(σ) .AN INVERSE PROBLEM WITH UNDER-DETERMINED DATA 7 Considering the following integrating factor for I1+I2 exp 2σ/integraldisplay 01/integraldisplay 0A2(ts)dtds , we have I1+I2=−1 8πexp −2σ/integraldisplay 01/integraldisplay 0A2(ts)dtds d dσ exp 2σ/integraldisplay 01/integraldisplay 0A2(ts)dtds ˜A(σ) . Now from Equation (10), we have 1 8πd dσ ˜A(σ)exp 2σ/integraldisplay 01/integraldisplay 0A2(st)dsdt = exp 2σ/integraldisplay 01/integraldisplay 0A2(st)dsdt /bracketleftBigg/integraldisplay |x|≤σq(x)R2(x,|x|)∂t{R1v1}(x,2σ−|x|) 4π|x|dx +/integraldisplay |x|≤σq(x)R1(x,|x|)∂tv2(x,2σ−|x|) 4π|x|dx +/integraldisplay |x|≤σ2σ−|x|/integraldisplay |x|q(x)∂tv2(x,t)v1(x,2σ−t)dtdx/bracketrightBigg for allσ∈[0,T/2].(13) Integrating on both sides with respect to σunder the assumption that ˜A(0) = 0, we get exp ˜σ/integraldisplay 01/integraldisplay 02A2(st)dsdt ˜A(˜σ) =˜σ/integraldisplay 0exp σ/integraldisplay 01/integraldisplay 02A2(st)dsdt /braceleftBigg/integraldisplay |x|≤σq(x)R2(x,|x|)∂tv1(x,2σ−|x|) 4π|x|dx +/integraldisplay |x|≤σq(x)R1(x,|x|)∂tv2(x,2σ−|x|) 4π|x|dx +/integraldisplay |x|≤σ2σ−|x|/integraldisplay |x|q(x)∂tv2(x,t)v1(x,2σ−t)dtdx/bracerightBigg dσ,for all ˜σ∈[0,T/2]. Now using the fact that R′ isare continuous, non-zero functions, and v′ isare continuous, we have the following inequality: |˜A(˜σ)| ≤C˜σ/integraldisplay 0|˜A(r)|drfor all ˜σ∈[0,T/2]. Now by Gronwall’s inequality, we have ˜A(σ) = 0 for all ˜ σ∈[0,T/2], which gives us q1(x) =q2(x) for all x∈R3such that |x| ≤T/2. This completes the proof.8 VASHISTH Acknowledgement The author would like to thank Dr. Venky Krishnan for useful d iscussions. He is supported by NSAF grant (No. U1530401). References [1] T. Aktosun, A. Machuca and P. Sacks; Determining the shap e of a human vocal tract from pressure measurements at the lips, Inverse Problems, vol. 33, (2017), 115002, 33 pages. [2] K.P. Bube and R. Burridge; The one-dimensional inverse p roblem of reflection seismology. SIAM Rev. 25 (1983), no. 4, 497–559. [3] R. Burridge; The Gel’fand-Levitan, the Marchenko, and t he Gopinath-Sondhi integral equations of inverse scatteri ng theory, regarded in the context of inverse impulse-respons e problems. Wave Motion 2 (1980), no. 4, 305–323. [4] E. Bl˚ asten; Well-posedness of the Goursat problem and s tability for point source inverse backscattering, 2017 Inv erse Problems 33 125003. [5] F.G. Friedlander; The wave equation on a curved space-ti me. Cambridge University Press, Cambridge-New York- Melbourne, 1975. Cambridge Monographs on Mathematical Phy sics, No. 2. [6] F.G. Friedlander and M. Joshi; The Theory of Distributio ns, 2nd Edition, Cambridge University Press, 1998. [7] I.M. Gel’fand and G. E. Shilov; Generalized functions. V ol. 1. Properties and operations. Translated from the 1958 R ussian original by Eugene Saletan. [8] M. M. Lavrent’ev, V. G. Romanov and S. P. Shishat ·ski˜i; Ill-posed problems of mathematical physics and analysis . Translated from the Russian by J. R. Schulenberger. [9] W. Ning and M. Yamamoto; The GelfandLevitan theory for on e-dimensional hyperbolic systems with impulsive inputs, Inverse Problems 24 (2008) 025004 (19pp). [10] Rakesh; An inverse impedance transmission problem for the wave equation. Comm. Partial Differential Equations 18 (1993), no. 3-4, 583–600. [11] Rakesh and P. Sacks; Impedance inversion from transmis sion data for the wave equation. Wave Motion 24 (1996), no. 3, 263–274. [12] Rakesh; Inversion of spherically symmetric potential s from boundary data for the wave equation. Inverse Problems 14 (1998), no. 4, 999–1007. [13] Rakesh; Characterization of transmission data for Web ster’s horn equation. Inverse Problems 16 (2000), no. 2, L9– L24. [14] Rakesh; An inverse problem for a layered medium with a po int source. Problems 19 (2003), no. 3, 497–506. [15] Rakesh; Inverse problems for the wave equation with a si ngle coincident source-receiver pair. Inverse Problems 24 (2008), no. 1, 015012, 16 pp. [16] Rakesh and P. Sacks; Uniqueness for a hyperbolic invers e problem with angular control on the coefficients. J. Inverse Ill-Posed Probl. 19 (2011), no. 1, 107–126. [17] Rakesh and G. Uhlmann; The point source inverse back-sc attering problem. Analysis, complex geometry, and mathema t- ical physics: in honor of Duong H. Phong, 279–289. [18] V.G. Romanov; On the problem of determining the coefficie nts in the lowest order terms of a hyperbolic equation. (Russian. Russian summary) Sibirsk. Mat. Zh. 33 (1992), no. 3, 156–160, 220; translation in Siberian Math. J. 33 (1992), no. 3, 497–500. [19] V. G. Romanov and D. I. Glushkova; The problem of determi ning two coefficients of a hyperbolic equation. (Russian) Dokl. Akad. Nauk 390 (2003), no. 4, 452–456. [20] V.G.Romanov; Integralgeometryandinverseproblemsf or hyperbolicequations, volume26.SpringerScienceandBu siness Media, 2013. [21] F. Santosa and W.W. Symes; High-frequency perturbatio nal analysis of the surface point-source response of a layer ed fluid. J. Comput. Phys. 74 (1988), no. 2, 318–381. [22] M.M. Sondhi; A survey of the vocal tract inverse problem : theory, computations and experiments. Inverse problems o f acoustic and elastic waves (Ithaca, N.Y., 1984), 1–19, SIAM , Philadelphia, PA, 1984. [23] P.D. Stefanov; A uniqueness result for the inverse back -scattering problem. Inverse Problems 6 (1990), no. 6, 1055 –1064. [24] W. W. Symes; The seismic reflection inverse problem. Inv erse Problems 25 (2009), no. 12, 123008, 39 pp. [25] M. Vashisth; An inverse problems for the wave equation w ith source and receiver at distinct points, Journal of Inver se and Ill-posed Problems, http://doi.org/10.1515/jiip-20 18-0004. Beijing Computational Science Research Center, Beijing 10 0193, China. E-mail: manmohanvashisth@gmail.com

2019-06-21

In this article, we consider the inverse problems of determining the damping coefficient appearing in the wave equation. We prove the unique determination of the coefficient from the data coming from a single coincident source-receiver pair. Since our problem is under-determined, so some extra assumption on the coefficient is required to prove the uniqueness.

Unique determination of the damping coefficient in the wave equation using point source and receiver data

1906.08987v1

arXiv:2305.10111v1 [cond-mat.mes-hall] 17 May 2023Journal of the Physical Society of Japan FULL PAPERS Material Parameters for Faster Ballistic Switching of an In -plane Magnetized Nanomagnet Toshiki Yamaji1*and Hiroshi Imamura1 † 1National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan High-speed magnetization switching of a nanomagnet is nece ssary for faster information processing. The ballistic switching by a pulsed magnetic filed is a promising candidate for the high-speed switching. It is known that the switch- ing speed of the ballistic switching can be increased by incr easing the magnitude of the pulsed magnetic field. However it is difficult to generate a strong and short magnetic field pulse in a sm all device. Here we explore another direction to achieve the high-speed ballistic switching by designing material parameters such as anisotropy constant, saturati on magnetization, and the Gilbert damping constant. We perfor m the macrospin simulations for the ballistic switching of in-plane magnetized nano magnets with varying material par ameters. The results are analyzed based on the switching dynamics on the energy density contour. We show that the puls e width required for the ballistic switching can be re- duced by increasing the magnetic anisotropy constant or by d ecreasing the saturation magnetization. We also show that there exists an optimal value of the Gilbert damping constan t that minimizes the pulse width required for the ballistic switching. 1. Introduction In modern information technologies huge amount of data are represented as the direction of the magnetization in a sm all magnet such as magnetic grains in magnetic tapes or hard disk drives. To write information on the conventional mag- netic recording media an external magnetic field is applied i n the opposite direction of the magnetization to switch the di - rection of the magnetization. During the switching the mag- netization undergoes multiple precessions around the loca l ef- fective field consisting of the external field, anisotropy fie ld, and demagnetizing field. The typical switching time or write time is of the order of nanoseconds. To meet the growing demand for fast information process- ing it is important to develop a faster switching scheme. The ballistic switching is a promising candidate for high-spee d switching, and much e ffort has been devoted to developing the ballistic switching both theoretically1–8)and experimen- tally.9–16)In ballistic switching a pulsed magnetic field is ap- plied perpendicular to the easy axis to induce the large-ang le precession around the external magnetic field axis. The dura - tion of the pulse is set to a half of the precession period. Aft er the pulse the magnetization relaxes to the equilibrium dire c- tion opposite to the initial direction. The switching speed of the ballistic switching can be increased by increasing the m ag- nitude of the pulsed field. However, it is di fficult to generate a strong and short field pulse in a small device. It is desired to find a way to speed up the ballistic switching without increas - ing magnetic field. The magnetization dynamics of the ballistic switching is determined by the torques due to the external magnetic field, the uniaxial anisotropy field, the demagnetizing field, and t he Gilbert damping. The torques other than the external mag- netic field torque are determined by the material parameters such as the anisotropy constant, the saturation magnetizat ion, and the Gilbert damping constant. There is room to speed up *toshiki-yamaji@aist.go.jp †h-imamura@aist.go.jpthe ballistic switching by designing the appropriate mater ial parameters. In the early 2000s the several groups each independently reported the optical microscope measurements of the ballis - tic switching by picosecond pulse magnetic field.9–13)Then the mechanism of a ballistic switching was analyzed in terms of the nonlinear dynamics concepts such as a fixed point, at- tractors, and saddle point.2, 3, 6)Especially the minimal field required for a ballistic switching was investigated by comp ar- ing the so-called Stoner-Wohlfarth (SW) type.2, 3)The damp- ing constant dependence of the minimal switching field was also studied.2)The characteristics of the parameters of a pulse magnetic field, i.e., magnitude, direction, and rise /fall time on the mechanism of a ballistic switching had been also studied by the simulations and experiments.6, 7, 14, 15) As described above, in 2000s and 2010s a ballistic switch- ing technique had received much attention for the fast magne - tization reversal with ringing suppression by fine-tuning t he magnetic pulse parameters. Due to the recent advance of an ultra-fast measurement17)the studies of a ballistic switching have attracted much attention again. Last year the in-plane magnetization switching dynamics as functions of the pulse magnetic field duration and amplitude was calculated and analyzed by using the conventional Landau-Lifshitz-Gilbe rt (LLG) equation and its inertial form, the so-called iLLG equation.16)The solutions of both equations were compared in terms of the switching characteristics, speed and energy density analysis. Both equations return qualitatively sim ilar switching dynamics. However the extensive material param- eter dependences of a ballistic switching region have not yet been sufficiently explored. Therefore it is worth clearing the extensive material parameter dependences of the ballis tic switching of an in-plane magnetized nanomagnet. In this paper, we study the ballistic switching of an in- plane magnetized nanomagnet with systematically varying the material parameters by using the macrospin simulations . The results show that the pulse width required for the bal- listic switching can be reduced by increasing the magnetic 1J. Phys. Soc. Jpn. FULL PAPERS Hp mz yx(a) (c) my at t = 10 ns (b) (d) 0 200 400-1 01 t [ps]my tp [ps]0 1 2 3 4 5tSW [ps] 110 10 210 3 tl tutSW 0 1 -1 0 1 2 3 4 502.55.010.0 7.5 tp [ps]Hp [T] Fig. 1. (a) Schematic illustration of the in-plane magnetized nano magnet. The pulse field, Hp, is applied along the x-direction. The initial direction of the magnetization is in the positive y-direction. (b) Gray scale map of myat t=10 ns as a function of the pulse field width, tp, and Hp. The black and white regions represent the success and failure of switchin g. The parameters areµ0Ms=0.92 T,µ0HK=0.1 T, andα=0.023. (c) Typical example of the time evolution of mywhen the magnetization switches ( Hp=5 T and tp =0.4 ps). The switching time, tSW, is defined as the time when mychanges the sign. (d) tpdependence of tSWalong the dashed horizontal line at Hp=5 T shown in Fig. 1(b). tlandtuare 3.15 ps and 3.93 ps, respectively. tSWat tl≤tp≤tuis 1.7 ps. anisotropy constant or by decreasing the saturation magnet i- zation. There exists an optimal value of the Gilbert damping constant that minimizes the pulse width required for ballis - tic switching. The simulation results are intuitively expl ained by analyzing the switching trajectory on the energy density contour. 2. Model and Method In this section we show the theoretical model, the numer- ical simulation method, and the analysis using the trajecto ry in the limit ofα→0. The macrospin model of the in-plane magnetized noanomagnet and the equations we solve to simu- late the magnetization dynamics are given in Sec. 2.1. In Sec . 2.2 we show that the switching conditions can be analyzed by using the trajectory on the energy density contour in the lim it ofα→0 if theα≪1. 2.1 Macrospin Model Simulation Figure 1(a) shows the schematic illustration of the in- plane magnetized nanomagnet. The pulsed magnetic field, Hp, is applied along the x-direction. The unit vector m= (mx,my,mz) indicates the direction of the magnetization. The size of the nanomagnet is assumed to be so small that the dy- namics of mcan be described by the macrospin LLG equation dm dt=−γm×/parenleftBigg Heff−α γdm dt/parenrightBigg , (1) where tis time,γis the gyromagnetic ratio, αis the Gilbert damping constant. The e ffective field, Heff=Hp+HK+Hd, comprises the pulse field, Hp, the anisotropy field, HK, andthe demagnetizing field, Hd. The anisotropy field and the de- magnetizing field are defined as HK=/bracketleftbig2K/(µ0Ms)/bracketrightbigmyey, (2) and Hd=µ0Msmzez, (3) respectively, where Kis the uniaxial anisotropy constant, µ0 is the magnetic permeability of vacuum, Msis the saturation magnetization, and ejis the unit vector along the j-axis ( j= x,y,z). The switching dynamics are calculated by numerically solving the LLG equation. The initial ( t=0) direction is set asmy=1. The rectangular shaped pulse magnetic field with duration of tpis applied at t=0. The time evolution of magne- tization dynamics are calculated for 10 ns. Success or failu re of switching is determined by whether my<−0.5 att=10 ns. Figure 1(b) shows the gray scale plot of myatt=10 ns on the tp-Hpplane. Following Ref. 16 the parameters are as- sumed to beµ0Ms=0.92 T, K=2.3 kJ/m3, i.e.µ0HK= 0.1 T, andα=0.023. The black and white regions represent the success and failure of switching, respectively. The wid e black region at upper right of Fig. 1(b) represents the balli stic switching region (BSR). A typical example of the time evolu- tion of mywhen the magnetization switches is shown in Fig. 1(c). The switching time, tSW, is defined as the time when my changes the sign. Figure 1(d) shows the tpdependence of tSW along the horizontal line shown in Fig. 1(b), i.e. at Hp=5 T. The BSR indicated by shade appears between tl=3.15 ps and tu=3.93 ps, where tSW=1.7 ps independent of tp. The lower and upper boundary of the BSR are represented by tlandtu, respectively. We investigate the material parameter dependence of tlandtuwith keeping Hp=5 T. 2.2 Analysis of the Switching Conditions for α≪1 If the Gilbert damping constant is much smaller than unity the approximate value of tlandtucan be obtained without performing macrospin simulations. In the limit of α→0, the trajectory is represented by the energy contour because the en- ergy is conserved during the motion of m. The energy density, E, of the nanomagnet is defined as18) E=1 2µ0M2 scos2θ+K(1−sin2θsin2φ), (4) whereθandφare the polar and azimuthal angles of the mag- netization, respectively. The color plot of the energy dens ity contour is shown in Fig. 2. The separatrix representing the energy contour with E=Kis indicated by the white curve, which is expresses as 1 2µ0M2 scos2θ−Ksin2θsin2φ=0. (5) The green dot indicates the initial direction of matt=0. The black curve represents the trajectory of munder the pulse field ofHpin the limit ofα→0. Under the pulse field the energy density is given by E=1 2µ0M2 scos2θ+K(1−sin2θsin2φ) −µ0MsHpsinθcosφ. (6) 2J. Phys. Soc. Jpn. FULL PAPERS 01 5 4 3 26E/K tltu θ φ Fig. 2. (Color online) Color plot of the energy density contour give n by Eq. (4).θandφare the polar and azimuthal angles of the magnetization, re- spectively. The material parameters, MsandKare same as in Fig. 1. The separatrix given by Eq. (5) is indicated by the white curve. T he initial direc- tion of mis indicated by the green dot at ( θ,φ)=(π/2,π/2). The black curve represents the trajectory of the magnetization under the fie ld of Hp=5 T in the limit ofα→0, which is given by Eq. (7). The yellow stars indicate the intersection points of the separatrix and the trajectory, w hich correspond to tp =tlandtu. If the pulse is turned o ffattl≤t≤tu, the magnetization switches ballistically. The yellow triangle indicates the turning p oint of the trajectory of the magnetization near mz=1, at whichφ=0. Since the energy density of the initial direction, θ=φ=π/2, isE=0, the trajectory under the pulse field is expressed as 1 2µ0M2 scos2θ+K(1−sin2θsin2φ) −µ0MsHpsinθcosφ=0. (7) The yellow stars indicate the points where the trajectory crosses the separatrix surrounding the equilibrium point a t φ=−π/2. The upper and lower points indicates the direc- tion of mat the end of the pulse with tp=tuandtl, re- spectively. The corresponding angles ( θl,φl) and (θu,φu) can be obtained by solving Eqs. (5) and (7) simultaneously. If tl≤tp≤tu, the magnetization relaxes to the equilibrium di- rection at (θ,φ)=(π/2,−π/2) after the pulse to complete the switching. We can obtain the approximate expressions of tl andtuas follows. Assuming that the pulse field is much larger than the other fields, the angular velocity of the precession ,ω, is approximated as γHp/(1+α2), and tlandtuare analytically obtained as tl=π−2θturn ω−1 2∆θ ω, (8) and tu=π−2θturn ω+1 2∆θ ω, (9) where∆θ=θu−θl, andθturnis the polar angle at the turning point (φ=0) indicated by the yellow triangle.3. Results and Discussion In this section we discuss the dependence of the BSR on the material parameters by analyzing the numerical simula- tion results and Eqs. (8) and (9). The results for the variati on of the magnetic anisotropy constant, K, saturation magnetiza- tion, Ms, and the Gilbert damping constant, α, will be given in Secs. 3.1, 3.2, and 3.3, respectively. 3.1 Anisotropy Constant Dependence of the BSR Figure 3(a) shows the anisotropy constant, K, dependence of the BSR. The parameters are Hp=5 T,µ0Ms=0.92 T, and α=0.023. The simulation results of tlandtuare indicated by the orange and blue dots, respectively. The analytical ap - proximations of tlandtuobtained by solving Eqs. (5),(7),(8), and (9) are represented by the orange and blue curves, respec - tively. The simulation and analytical results agree well wi th each other because the Gilbert damping constant is as small a s 0.023. As shown in Fig. 3(a), tlis a monotonically decreasing function of Kwhile tuis a monotonically increasing function ofK. As a result the width of the BSR, tu-tl, is a monoton- ically increasing function of Kas shown in the inset of Fig. 3(a). In the left panel of Fig. 3(b) the separatrix and the trajecto ry withα=0 for K=2.3 kJ/m3are shown by the blue and black curves, respectively. The same plot for K=9.3 kJ/m3 is shown in the right panel. As shown in these panels, the increase of Kdoes not change the trajectory much. However, the increase of Kchanges the separatrix significantly through the second term of Eq. (5). Assuming that the angular velocit y of the precession is almost constant, the spread of the area surrounded by the separatrix results in the spread of the tim e difference between tlandtu. As a result the BSR is spread by the increase of Kas shown in Fig. 3(a) 3.2 Saturation Magnetization Dependence of the BSR Figure 4(a) shows the saturation magnetization dependence of the BSR obtained by the numerical simulation and the ana- lytical approximations. The horizontal axis represents th e sat- uration magnetization in unit of T, i.e µ0Ms. The parameters areHp=5 T,K=2.3 kJ/m3, andα=0.023. The symbols are the same as in Fig. 3(a). The lower boundary of the BSR, tl, increases as theµ0Msincreases while the upper boundary of the BSR, tu, decreases with increase of µ0Ms. Therefore, the faster switching is available for smaller Ms. Theµ0Msdepen- dence of the BSR ( tu-tl) is also shown in the inset of Fig. 4(a). The BSR decreases with increase of µ0Ms. In other words, the wider BSR is obtained for smaller Ms. In the right panel of Fig. 4(b) the separatrix and the trajec- tory withα=0 forµ0Ms=0.35 T are shown by the blue and black curves, respectively. The same plot for µ0Ms=0.92 T is shown in the left panel. As shown in these panels, the increas e ofMsdoes not change the trajectory much but decrease the separatrix significantly through the first term of Eq. (5). As - suming that the angular velocity of the precession is almost constant, the reduction of the area surrounded by the separa - trix results in the reduction of the time di fference between tl andtu. As a result the BSR decreases with increase of Msas shown in Fig. 4(a) 3J. Phys. Soc. Jpn. FULL PAPERStl, t u [ps] 0 10 20 30 40 2.03.04.05.0 K [kJ/m3]ballistic switching region (a) tltu tu - t l [ps] K [kJ/m 3]0 10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 0π π/2 0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ φ φˑˑ ˑˑ(b) K = 2.3 kJ/m3K = 9.3 kJ/m3 tltltutu Fig. 3. (Color online) (a) Anisotropy constant, K, dependence of the BSR (orange shade). Simulation results of tlandtuare plotted by the orange and blue dots, respectively. The analytical results are indica ted by the solid curves with the same color. The parameters are Hp=5 T,µ0Ms=0.92 T, andα= 0.023. In the inset the simulation and analytical results of the width of the BSR, tu-tl, are plotted by the dots and the solid curve, respectively. ( b) Typical examples of the trajectory of the magnetization (bl ack curve) and the separatrix (blue curve). The left and right panels show the r esults for K=2.3 kJ/m3andK=9.3 kJ/m3, respectively. The orange and blue stars indicate the direction at t=tlandtu, respectively. The green dots indicate the initial direction of m. 3.3 Gilbert Damping Constant Dependence of the BSR Figure 5(a) shows the simulation results of the Gilbert damping constant, α, dependence of the BSR. The width of the BSR is shown in the inset. The symbols are the same as in Fig. 3(a). The approximate values obtained by Eqs. (8) and (9) are not shown because the αis not limited toα≪1. The parameters are Hp=5 T, K=2.3 kJ/m3, andµ0Ms=0.92 T. There exists an optimal value of αthat minimizes tl. The optimum value in Fig. 5 (a) is αopt=0.35. To understand the mechanism for minimization of tlat a certain value ofαone need to consider two di fferent effects of αon the magnetization dynamics. One e ffect is the decrease of the precession angular velocity with increase of α. The pre- cession angular velocity around the e ffective field of Heffis given by (γHeff)/(1+α2), which decreases with increase of α. This effect causes the increase of tlandtu. The other effect is the increase of the energy dissipation rate with increase ofα. Let us consider the trajectory in the cases of small damping ( α=0.023) and large damping ( α=αopt). In Fig. 5 (b) the typical examples of the trajectory for the small damping are shown by the yellow and green curves and dots on the energy density contour. The pulse widths are tp=tl(=3.15 ps) and 3.14 ps. The trajectories during the pulse are represented by the solid curves and the trajectori es after the pulse are represented by the dots. The white curve shows the separatrix and the black dot indicates the initial di- tl, t u [ps] 2.03.04.05.0 0.0 0.3 0.6 0.9 1.2 μ0Ms [T]ballistic switching region tltu(a) (b) μ0Ms = 0.92 T μ0Ms = 0.35 T 0π π/2 0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ φ φˑˑ ˑˑ tl tltu tu1.5tu - t l [ps] 0123 0.0 0.3 0.6 0.9 1.2 μ0Ms [T] 1.5 Fig. 4. (Color online) (a) Saturation magnetization dependence of the BSR. The horizontal axis represents the saturation magneti zation in unit of T, i.eµ0Ms. The parameters are Hp=5 T, K=2.3 kJ/m3, andα=0.023. The symbols are the same as in Fig. 3 (a). (b) Typical examples of t he trajectory of the magnetization (black curve) and the separatrix (blue curve). The right and left panels show the results for µ0Ms=0.35 T and 0.92 T, respectively. The symbols are the same as in Fig. 3 (b). rection. The yellow and green stars indicate the points wher e the trajectories cross the separatrix surrounding the targ et and initial states, respectively. The arrows indicate the dire ction of the movement of the magnetization. For the small damp- ing, even very close to the separatrix around the target stat e at the end of the pulse, the magnetization flows to the sepatrari x around the initial state and relax to the initial state after many precessions with the slow energy dissipation. Figure 5 (c) shows the tpdependence of tSWat the large damping (α=αopt). All parameters except αare the same as in Fig. 1 (d). t′ l,tl, and tuare 0.82 ps, 1.98 ps, and 4.54 ps, respectively. t′ lis the time when for the large damping the magnetization goes across the e ffective separatrix around the initial state during the pulse duration. In Fig. 5 (d) the typ ical examples of the trajectory for the large damping are shown by the yellow ( tp=0.9 ps), green ( tp=tl=1.98 ps), and purple ( tp=4.55 ps) curves and dots on the energy density contour. The symbols are the same as in Fig. 5 (b). In the region 1 ( tp<t′ l) of Fig. 5 (c) the magnetization is located on the inside of the effective separatrix at the end of the pulse and return to the initial state. The trajectory is not shown in Fi g. 5 (d). In the region 2 ( t′ l≤tp<tl) of Fig. 5 (c) after the pulse is removed the magnetization move toward the target state unde r the effective field of Heffand goes across the separatrix. Then the magnetization finishes the switching. The typical examp le of the trajectory is shown by the yellow curve and dots in Fig. 5 (d). In the region 3 ( tl≤tp≤tu) after the pulse is removed the magnetization ballistically switches. The typical exa mple 4J. Phys. Soc. Jpn. FULL PAPERS tl, t u [ps] 2.04.06.08.0 0.0 0.2 0.4 0.6 0.8 αballistic switching region tltu(a) 0 tu - t l [ps] 0.00123 0.2 0.4 0.6 0.8 α45 αopt αopt 01 5 4 3 26E/K α = 0.023 tp = t l = 3.15 ps tp = 3.14 ps (b) 0 -π/2 π/2 π -π0π/2 π φθ < << < < ˑ ˑ (c) tp [ps] 0 1 2 3 4 5tSW [ps] 110 10 210 3 tl tu1 2 4 3 tl’ 01 5 4 3 26E/K α = 0.35 tp = 0.9 ps tp = t l = 1.98 ps tp = 4.55 ps (d) 0 -π/2 π/2 π -π0π/2 π φθ < < <ˑˑ < < < ˑ Fig. 5. (Color online) (a) Simulation results of the tlandtuas a function ofα. The parameters are Hp=5.0 T, K=2.3 kJ/m3, andµ0Ms=0.92 T. The symbols are the same as in Fig. 3 (a). (b) The trajectori es atα= 0.023 with tp=3.15 ps (yellow) and 3.14 ps (green) on the energy density contour. The trajectory during the pulse is represented by t he solid curve. The trajectory after the pulse is represented by the dots. The wh ite curve shows the separatrix.The direction of the trajectory is indicate d by the arrow. The star indicates the intersection point of the trajectory and the separatrix. The initial direction is indicated by the black dot. (c) tpdependence of tSWat α=αopt. All parameters except αare the same as in Fig. 1 (d). t′ l,tl, and tu are 0.82 ps, 1.98 ps, and 4.54 ps, respectively. tSWattl≤tp≤tuis 1.98 ps. (d) The trajectories at α=αoptwith tp=0.9 ps (yellow), 1.98 ps (green), and 4.55 ps (purple) on the energy density contour. The symbols a re the same as in Fig. 5 (b). of the trajectory is shown by the green curve and dots in Fig.5 (d). As explained in Sec. 2.1, in this study the tSWis defined as the time when mychanges the sign, in other words, the magnetization reaches the turning point ( φ=0) on the energy density contour. Therefore the tlfor the large damping is equal to the ballistic tSW, i.e. the time when under the pulse field the magnetization reaches the turning point. The ballistic tSWat the region 3 is 1.98 ps. As described above, for the large damping the magnetiza- tion can relax to the target state even at greater distances f rom the separatrix by moving under Heffwith the fast energy dis- sipation. The effect can be regarded as the e ffective spread of the separatrix and results in the decrease of tland the in- crease of tuwith increase ofα. Therefore, there exists the optimal value ofαthat minimizes tlwhile tumonotonically increases with increase of αas shown in Fig. 5(a). In the re- gion 4 ( tp>tu) after the pulse is removed the magnetization moves toward the separatrix around the initial state under Heff and relaxes to the initial state. We find that the BSR for the large damping can be explained by the anisotropic spread of the effective separatrix with increasing α, which is fundamen- tally due to the breaking of the spatial inversion symmetry o f the spin dynamics. The broken symmetry of the spatial inver- sion of the spin dynamics for the large damping can be easily confirmed by comparing Fig. 5 (c) with Fig. 1 (d). 4. Summary In summary, we study the material parameter dependence of the ballistic switching region of the in-plane magnetize d nanomagnets based on the macrospin model. The results show that the pulse width required for the ballistic switching ca n be reduced by increasing the magnetic anisotropy constant or b y decreasing the saturation magnetization. The results also re- vealed that there exists an optimal value of the Gilbert damp - ing constant that minimizes the pulse width required for the ballistic switching. The simulation results are explained by analyzing the trajectories on the energy contour. The resul ts are useful for further development of the high-speed inform a- tion processing using the ballistic switching of magnetiza tion. This work is partially supported by JSPS KAKENHI Grant Number JP20K05313. 1) L. He and W. D. Doyle: J. Appl. Phys. 79(1996) 6489. 2) Z. Z. Sun and X. R. Wang: Phys. Rev. B 71(2005) 174430. 3) D. Xiao, M. Tsoi, and Q. Niu: Journal of Applied Physics 99(2006) 013903. 4) Y . Nozaki and K. Matsuyama: Jpn. J. Appl. Phys. 45(2006) L758. 5) Y . Nozaki and K. Matsuyama: Journal of Applied Physics 100(2006) 053911. 6) Q. F. Xiao, B. C. Choi, J. Rudge, Y . K. Hong, and G. Donohoe: J ournal of Applied Physics 101(2007) 024306. 7) P. P. Horley, V . R. Vieira, P. M. Gorley, V . K. Dugaev, J. Ber akdar, and J. Barna´ s: Journal of Magnetism and Magnetic Materials 322(2010) 1373. 8) Y . B. Bazaliy: Journal of Applied Physics 110(2011) 063920. 9) T. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar, and T . Rasing: Nature 418(2002) 509. 10) I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann , J. St¨ ohr, G. Ju, B. Lu, and D. Weller: Nature 428(2004) 831. 11) H. W. Schumacher, C. Chappert, R. C. Sousa, P. P. Freitas, and J. Miltat: Phys. Rev. Lett. 90(2003) 017204. 12) W. K. Hiebert, L. Lagae, J. Das, J. Bekaert, R. Wirix-Spee tjens, and J. De Boeck: Journal of Applied Physics 93(2003) 6906. 13) W. K. Hiebert, L. Lagae, and J. De Boeck: Phys. Rev. B 68(2003) 5J. Phys. Soc. Jpn. FULL PAPERS 020402. 14) H. W. Schumacher: Appl. Phys. Lett. 87(2005) 042504. 15) N. Kikuchi, Y . Suyama, S. Okamoto, O. Kitakami, and T. Shi matsu: Appl. Phys. Lett. 104(2014) 112409. 16) K. Neeraj, M. Pancaldi, V . Scalera, S. Perna, M. d’Aquino , C. Serpico, and S. Bonetti: Phys. Rev. B 105(2022) 054415.17) K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou Hagstr ¨ om, S. S. P. K. Arekapudi, A. Semisalova, K. Lenz, B. Green, J.-C. Dein ert, I. Ilyakov, M. Chen, M. Bawatna, V . Scalera, M. d’Aquino, C. S erpico, O. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti: Nat. Ph ys.17 (2021) 245. 18) W. F. Brown: Phys. Rev. 130(1963) 1677. 6

2023-05-17

High-speed magnetization switching of a nanomagnet is necessary for faster information processing. The ballistic switching by a pulsed magnetic filed is a promising candidate for the high-speed switching. It is known that the switching speed of the ballistic switching can be increased by increasing the magnitude of the pulsed magnetic field. However it is difficult to generate a strong and short magnetic field pulse in a small device. Here we explore another direction to achieve the high-speed ballistic switching by designing material parameters such as anisotropy constant, saturation magnetization, and the Gilbert damping constant. We perform the macrospin simulations for the ballistic switching of in-plane magnetized nano magnets with varying material parameters. The results are analyzed based on the switching dynamics on the energy density contour. We show that the pulse width required for the ballistic switching can be reduced by increasing the magnetic anisotropy constant or by decreasing the saturation magnetization. We also show that there exists an optimal value of the Gilbert damping constant that minimizes the pulse width required for the ballistic switching.

Material Parameters for Faster Ballistic Switching of an In-plane Magnetized Nanomagnet

2305.10111v1

Probing spin dynamics of ultra-thin van der Waals magnets via photon-magnon coupling Christoph W. Zollitsch,1,a)Safe Khan,1Vu Thanh Trung Nam,2Ivan A. Verzhbitskiy,2Dimitrios Sagkovits,1, 3 James O’Sullivan,1Oscar W. Kennedy,1Mara Strungaru,4Elton J. G. Santos,4, 5John J. L. Morton,1, 6Goki Eda,7, 2, 8and Hidekazu Kurebayashi1, 6, 9 1)London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1 0AH, UK 2)Department of Physics, Faculty of Science, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore 3)National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK 4)Institute for Condensed Matter Physics and Complex Systems, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK 5)Higgs Centre for Theoretical Physics, The University of Edinburgh, Edinburgh EH9 3FD, UK 6)Department of Electronic & Electrical Engineering, UCL, London WC1E 7JE, United Kingdom 7)Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore 8)Department of Chemistry, Faculty of Science, National University of Singapore, 3 Science Drive 3, Singapore 117543, Singapore 9)WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1, Katahira, Sendai, 980- 8577, Japan (Dated: 1 May 2023) Layered van der Waals (vdW) magnets can maintain a magnetic order even down to the single-layer regime and hold promise for integrated spintronic devices. While the magnetic ground state of vdW magnets was extensively studied, key parameters of spin dynamics, like the Gilbert damping, crucial for designing ultra-fast spintronic de- vices, remains largely unexplored. Despite recent studies by optical excitation and detection, achieving spin wave control with microwaves is highly desirable, as modern in- tegrated information technologies predominantly are op- erated with these. The intrinsically small numbers of spins, however, poses a major challenge to this. Here, we present a hybrid approach to detect spin dynamics medi- ated by photon-magnon coupling between high-Q super- conducting resonators and ultra-thin flakes of Cr 2Ge2Te6 (CGT) as thin as 11 nm. We test and benchmark our tech- nique with 23 individual CGT flakes and extract an upper limit for the Gilbert damping parameter. These results are crucial in designing on-chip integrated circuits using vdW magnets and offer prospects for probing spin dynamics of monolayer vdW magnets. INTRODUCTION van der Waals (vdW) materials1–3consist of individual atomic layers bonded by vdW forces and can host different types of collective excitations such as plasmons, phonons and magnons. Strong coupling between these excitation modes and electromagnetic waves (i.e. photonic modes) creates con- fined light-matter hybrid modes, termed polaritons. Polaritons a)Electronic mail: c.zollitsch@ucl.ac.ukin vdW materials are an ideal model system to explore a va- riety of polaritonic states5,6, e.g. surface plasmon polaritons in graphene7,8and exciton polaritons in a monolayer MoS 2 embedded inside a dielectric microcavity9. These states can be further modified by electrostatic gating16, as well as by hetero-structuring with dissimilar vdW layers1. Numerous studies on magnon polaritons (MPs)11,12have been using macroscopic yttrium iron garnet (YIG) cou- pled to either three-dimensional cavities13or to on-chip resonators14,15, with potential applications in ultra-fast infor- mation processing, non-reciprocity or microwave to optical transduction. By reducing the number of excitations, MPs find application in the quantum regime e.g., magnon number counting via an electromagnetically coupled superconducting qubit16,17or as a building block for Bell state generation18. The rapidly developing research around polaritons and specifically MPs has so far, been little studied in magnetic vdW materials due to the relatively recent discoveries of long- range magnetic order in vdW systems at the few monolayer regime9,20,21, in addition to its technically challenging real- ization. Stable MP states are formed by strongly coupling the magnetic field oscillation of a resonant photon to the collec- tive magnetization oscillation in a magnetic material. This strong coupling is achieved when the collective coupling rate geffis larger than the average of both system loss rates. In a simplified picture, geffscales linearly with the strength of the oscillating magnetic field of a resonator and the square root number of spins14. For studies involving bulk magnetic mate- rials and low quality and large microwave resonators, strong coupling is achieved when geff=2pis in the MHz range, which is accomplished with relative ease due to the abundance of spins in bulk magnetic materials. A reduction of the bulk di- mensions down from mm to mm and nm scales, the typical lateral dimensions and thickness of vdW material monolay-arXiv:2206.02460v2 [cond-mat.mtrl-sci] 28 Apr 20232 ers, results in a decrease of the coupling strength by at least 6 orders of magnitude. Commonly used microwave resonators are not able to produce strong enough oscillating magnetic fields to compensate for such a reduction in absolute number of spins. Only by advanced resonator design and engineering the regime of strongly coupled MPs in monolayer vdW mag- netic materials can be accomplished, granting access to spin dynamic physics at a true 2d monolayer limit and research on MPs in nano-scale devices where the whole range of on-chip tuning and engineering tools, such as electric fields or device design, are available. Magnons or magnon polaritons have been observed in mag- netic vdW materials, but it had been restricted to either to the optical frequency range22,23or a large thickness limit24,25, re- spectively. Here, we present our attempt of detecting spin dynamics in ultra-thin vdW magnetic materials and the cre- ation of MPs by magnon-photon coupling in the microwave frequency range, using superconducting resonators optimized for increased magnon-photon coupling. By using microwave resonators with a small mode volume, we not only increased its oscillating magnetic field strength but also matched it more efficiently to the size of nanoscale vdW flakes. Our work presents a fundamental cornerstone for a general blueprint for designing and developing magnon-photon hybrids for any type of ultra-thin or monolayer vdW magnetic material, en- abling research on on-chip microwave applications for (quan- tum) information processing. RESULTS In this article, we report on the observation of spin dynam- ics and the creation of MPs at the onset of the high cooper- ativity regime with the vdW ferromagnet CGT of nm scale thickness, demonstrating a pathway towards stable magnon- photon polariton creation. We combine a precise transfer process of exfoliated CGT flakes and high sensitivity su- perconducting resonators, to access and study the dynami- cal response of coupled photon-magnon states in a small- volume (nm-thick and m-sized) CGT flake (illustrated in Fig. 1 (a)). High-quality-factor superconducting lumped el- ement resonators are chosen to be the counterpart due to their extremely small mode volume ( 6000m3) and con- sequently strong oscillating magnetic fields ( B125nT, see SI for resonator quality-factors and B1-field distributions), re- sulting in high spin sensitivities4,26. At cryogenic temper- atures, we perform low-power microwave spectroscopy on multiple resonator-vdW-flake hybrids, covering a frequency range from 12GHz to 18GHz for a variety of thickness. Sam- ples consist of up to 12 resonators on a single chip, all capac- itively coupled to a common microwave transmission line for read-out (see SI for details). Multiple peaks of spin-wave res- onances are observed for each CGT flake measured. The spin- wave modes are closely spaced in frequency and show a large overlap. We employ a semi-optimized fitting model to pro- duce a good estimate for the collective coupling strength and magnetic linewidth. By taking the resonance value of the most prominent peak of each spectrum, we find that all measuredpoints can be fitted very well by a single curve calculated by the Kittel formula with bulk CGT parameters. Furthermore, we extracted the linewidth for the thinnest CGT flake inves- tigated, 11nm or 15 monolayers (ML), the only device ex- hibiting well separated spin-wave modes. This allowed a fully quantitative analysis and we determined an upper limit of the Gilbert damping parameter of 0 :02. This value is comparable to the damping reported for 3d transition metal ferromagnets, suggesting that magnetic vdW flakes have the potential for the fabrication of functional spintronic devices. We investigate the dynamics of nm-thick CGT flakes, us- ing superconducting lumped element resonators made of NbN (see methods for fabrication details and SI and Ref. [28] for more performance details). The advantages of a lumped ele- ment design are the spatial separation of the oscillating mag- netic field B1and electric field E1and the concentration of B1within a narrow wire section of the resonators, as indi- cated in Fig. 1 (a). Additionally, the B1field distribution is homogeneous along the length of the narrow wire section (see finite element simulations in SI). This magnetic-field concen- tration is our primary reason to use this type of resonator in order to reduce the photon mode volume as well as achieve a considerable mode overlap between the resonator photon mode and CGT magnon mode, and consequently, a large cou- pling strength. We therefore transfer CGT flakes onto these 5 μm B0CGTB1a bcE1 0 4 8 x (μm)y (nm) 102030 1240 MCGT CrGeTe B1,extent ≈ 2 μm FIG. 1. Magnon-photon coupling between thin CGT and a super- conducting resonator. a Schematic of a resonator shows the design in detail, indicating the areas of high E1-field (yellow) and B1-field (green) intensities, as well as the orientation of the externally applied field B0. Finally, a schematic zoom in of the section loaded with a CGT flake is shown. The collective coupling between a microwave photon and the magnetization of the CGT is illustrated, as well as the approximate extent of the microwave B1-field. bMicrograph image of a CGT flake transferred onto the narrow section of a resonator. c AFM image of the CGT flake together with a height profile along the blue solid line in the AFM image. The red solid line is a fit to the flake thickness. The results of this resonator are presented in Fig. 2.3 12.8112.8212.83 ω/2π (GHz) 560 580 600 620 Magnetic Field (mT)640 66012.841.0 0.9 0.8|S21|20.7 |S21|21.0 0.9 0.8a b c 580 600 Magnetic Field (mT)620234 κeff/2π (MHz)0510 ωres/2π (MHz) d 640 + 12820 MHz 0.7550 mT 598 mT 614 mT 670 mT 12.81 12.82 12.83 ω/2π (GHz)12.84 FIG. 2. Magnon-photon coupling observed in resonator microwave transmission. a jS21j2as a function static magnetic field B0and frequency, with the microwave transmission encoded in the color. The results are obtained from the resonator shown in Fig 1 (b) and (c), featuring a loaded quality factor of QL=4600. bjS21j2as a function of frequency at fixed magnetic fields, indicated in aby dashed vertical lines. canddResonance frequency wresand effective loss rate keffas a function of magnetic field. Note the multiple resonance peaks, indicating multiple CGT FMRs. The dashed orange lines are results from the semi-optimized fit. dexemplary includes the individual peaks of which the orange dashed lines consists. The green bar in canddhighlights the main mode. narrow sections (Fig. 1 (b)). Details of CGT flake transfers are described in the methods section. Optical imaging and atomic force microscopy (AFM) measurements are used to characterise the size and thickness of the CGT flakes (see Fig. 1 (c)). Measured thicknesses range from 153 23nm down to 111:8nm (15 ML), enabling a thickness dependent study of CGT flakes and their coupling to the resonators. We measured the microwave transmission jS21j2as a func- tion of frequency and externally applied magnetic field B0for each resonator at a temperature of 1 :8K, using a microwave power of approximately 80dBm at the resonator chip. Fig- ure 2 (a) shows the resulting 2D plot of jS21j2for a resonator loaded with a 17nm 0:8nm thick CGT flake (see Fig. 1 (b) and (c) for the respective micrograph and AFM images). A resonator peak can be clearly observed for each magnetic field, with its resonance frequency wresdecreasing with in- creasing magnetic field. The reduction of the frequency is a result of a slow degradation of the superconductivity by B0, which in general exhibits a parabolic dependence29. For 580mTB0630mT the resonator prominence is reduced, highlighted byjS21j2as a function of frequency for four con- stant B0values in Fig. 2 (b). Within this field range, the mode resonance has been modified due to its hybridization with the magnetic modes of the CGT flake. To further quantify the in- teraction, we fit each jS21j2profile by a Fano resonance line- shape (solid orange lines in Fig. 2 (b)) to account for an asym- metric resonance peak due to additional microwave interfer- ence in the circuitry30,31, jS21j2=S0+A(qkeff=2+wwres)2 (keff=2)2+ (wwres)2: (1) Here, S0is the microwave transmission baseline, Athe peak amplitude, qdescribes the asymmetry of the lineshape and keffrepresents the effective loss rate of the hybrid system (seeSI for resonator parameters before and after CGT transfer for all resonators). Figure 2 (c) shows wresof the hybrid system as a function of B0.wresexperiences a dispersive shift when the photon mode and the magnon mode hybridize, indicating an onset of a strong interaction between the two individual systems14,17,32–34. We observe multiple shifts in wres, suggest- ing an interaction of several magnon modes with the resonator in our experiment. Signatures of the resonator–CGT-flake coupling are also characterised by keffof the hybrid system (Fig. 2 (d)). keffis enhanced from the value of the resonator loss rate k0due to an additional loss introduced by the magnon system charac- terized by the loss rate g14,32,35. Consistent with the B0de- pendence of wres,keffshows a rich structure, having its main peak at 598mT, together with less prominent peaks distributed around it. Based on a formalism for coupled-harmonic- oscillator systems in the high cooperativity regime32–34, we use the following to analyse our experimental results with multiple peaks: wres=wres;0+mB2 0++n å k=ng2 eff;kDk D2 k+g2; (2) keff=k0++n å k=ng2 eff;kg D2 k+g2: (3) with the detuning factor for each resonance as Dk= gCGTmB ¯h B0BFMR ;k
. Here, wres;0is the resonator resonance frequency at B0=0T and mrepresents the curvature of the resonance frequency decrease due to the applied magnetic field. BFMR ;kis the CGT FMR field, gCGT the g-factor of CGT and geff;kgives the collective coupling strength between photon and magnon mode. The summation is over all reso- nance modes kpresent on the low or high field (frequency)4 side of the main resonance mode, where ngives the number of modes on one side. For simplicity, we assume a symmet- ric distribution of modes about the main mode. The large number of multiple modes and their strong overlap prevent a reliable application of a fully optimized fit to the data, due to the large number of free parameters required. In an ef- fort to gain a good estimate of the model parameters we ap- ply the model functions Eq. (2) and (3) in a two-step semi- optimized fashion (see SI for details). With this approach, we arrive at a model in good agreement with wresandkeff(see orange dashed lines in Fig. 2 (c), (d), exemplary showing the individual peaks of the orange dashed line in Fig. 2 (d) and the SI for additional results and data). We can reproduce the data using g=2p=94:035:95MHz and a collective cou- pling strength of the main mode of 13 :251MHz. Together with k0=2p=1:40:02MHz the system resides at the onset of the high cooperativity regime, classified by the cooperativ- ityC=g2 eff=k0g=1:3>113,32. In this regime, magnon polari- tons are created and coherently exchange excitations between magnons and resonator photons on a rate given by geff. The created MPs are, however, short lived and the excitations pre- 100 200 500 700 Resonance Field BFMR (mT)300 400 600 0051015ωFMR/2π (GHz)500 Resonance Field BFMR (mT)600 7001518ωFMR/2π (GHz) 12a b 11 31 51 71 91 111 131151Flake Thickness (nm) FIG. 3. Summary of CGT-FMR conditions. a Extracted CGT res- onance fields and frequencies from the set of resonators loaded with CGT flakes of different thickness. Resonance values are taken from the most prominent peaks in keff. The solid curve is calculated us- ing the Kittel formalism presented in10, using same parameters, with gCGT =2:18,m0Ms=211:4mT and Ku=3:84104J=m3.bWider magnetic field range of awhere the CGT flake thickness for the dif- ferent symbols is indicated by the color gradient given in a.dominately dissipate in the magnonic system, as geffg. Our analysis suggests that the separation of the different FMR modes is of the same order of magnitude as the loss rate (see SI for additional data). We consider that these are from standing spin wave resonances, commonly observed for thin magnetic films12and with one reported observation in bulk of the vdW material CrI 338. In thin-film magnets under a static magnetic field applied in-plane, the magnetic-dipole interac- tion generates two prominent spin wave branches for an in- plane momentum, the backward volume spin wave (BVMSW) and magnetostatic surface spin wave (MSSW) modes39,40. These spin wave modes have different dispersion relations, having higher (MSSWs) and lower (BVMSWs) resonance frequencies with respect to that of the uniform FMR mode. We calculate the distance of these standing spin-wave modes based on magnetic parameters of bulk CGT as well as the lat- eral dimensions of the flakes (see SI for more details). We can find spin waves having a frequency separation within 100MHz and 200MHz (3 :3mT to 6 :6mT in magnetic field units), which are consistent with our experimental observa- tion in terms of its mode separation. However, the irregular shape of the CGT flakes renders exact calculations of spin wave mode frequencies very challenging. We also consid- ered a possibility that each layer of CGT might have different magnetic parameters (e.g. chemical inhomogeneity), and thus producing different individual resonance modes. Our numer- ical simulations based on atomistic spin dynamics14,15rule out this possibility, as resonance modes from individual lay- ers average to a single mode as soon as a fraction of 10% of inter-layer exchange coupling is introduced (see SI for more details). Therefore, we speculate that the multiple mode na- ture we observe in our experiments is likely originating from intrinsic properties of the CGT flakes. Figure 3 shows the extracted wFRM as a function of BFMR for each resonator–CGT-flake hybrid. The experimental val- ues are in excellent agreement with a curve calculated by the Kittel equation with magnetic parameters for bulk CGT10, from which the data exhibits a standard deviation of less than 5%. This agreement, achieved by independent characteri- zations of 23 CGT flakes measured by superconducting res- onators, is experimental evidence that the magnetic parame- ters that determine the dispersion of wFRM (BFMR), i.e. the CGT g-factor gCGT, saturation magnetization Msand uniaxial anisotropy Ku, exhibit little thickness dependence in exfoli- ated CGT flakes, and are not disturbed by the transfer onto the resonator structure. We note, that this demonstrates that vdW magnetic materials are particularly attractive for device applications, as they are less prone for contamination from exfoliation. Finally, we present our analysis of kefffor a resonator with a 111:8nm CGT flake in Fig.4. With the thickness of a single layer of CGT being 0 :7nm9, this flake consists of 15 monolayers and is the thinnest in our series. Figure 4 (a) and (b) show wresandkeffas a function of B0, respectively. While the response of the CGT flake shows a prominent signature inkeff, the CGT FMR is considerably more subtle in wres. This highlights the excellent sensitivity of the high-Q super- conducting resonators in our study. kefffeatures five well-5 separated peaks with the main peak at B0=547mT, which enables us to perform a single-peak fully optimized analy- sis for each, in contrast to our multi-step analysis for the re- mainder of the devices. We assume the additional peaks are BVMSW modes, as discussed in the previous section. How- ever, the splitting is about four times larger than compared to all other investigated devices, which would result in a signifi- cantly shorter wavelength. Thickness steps can lead to a wave- length down-conversion13, however, due to the irregular shape andB1inhomogeneities it is difficult to exactly calculate the spin wave frequencies (see SI for further details). From the main peak profile, we extract geff=2p=3:610:09MHz, g=2p=126:268:5MHz and k0=2p=0:920:05MHz. We compare the experimental value of geffwith a numerically cal- culated geff;simu, using the dimensions of the CGT flake de- termined by AFM measurements (see SI for details). The calculation yields geff;simu=2p=8:94MHz, lying within the same order of magnitude. The overestimation is likely due to in-perfect experimental conditions, like non-optimal place- ment of the flake, uncertainties in the thickness and dimen- sion determination as well as excluding the additional modes in the calculation (see SI). With ggeffandC=0:11, the hybrid system is in the weak coupling regime13, but due to the highly sensitive resonator with its small k0the response from the magnon system can still be detected. With the ex- tracted g=2pwe can give an upper limit of the Gilbert damp- ing in CGT, by calculating aupper =g=wFMR. We find aupper as 0:0210:002, which is comparable to other transition metal magnetic materials44, and is in very good agreement with a previously reported effective Gilbert damping parameter de- termined by laser induced magnetization dynamics45. Here, we emphasise that the actual Gilbert damping value is lower due to a finite, extrinsic inhomogeneous broadening contribu- tion. We further use these results to benchmark the sensitivity of our measurement techniques. The detection limit is given by comparing the main peak height characterised by g2 eff=g and the median noise amplitude which is 18kHz in Fig. 4 (b) where g2 eff=2pg= 103 kHz. By assuming the same lateral di- mensions and scale the thickness down to a single monolayer, while keeping gconstant, we calculate the expected signal re- duction numerically by geff;simu;1ML=geff;simu;15ML to 0.26. We obtain (0:26geff)2=2pg=7kHz for the monolayer limit. Al- though this suggests the noise amplitude is greater than the expected peak amplitude, we can overturn this condition by improving the coupling strength by optimising the resonator design, enhancing the exfoliation and flake transfer as well as by reducing the noise level by averaging a number of mul- tiple scans. Superconducting resonators with mode volumes of about 10 m3have been realised46, a reduction of 2 orders of magnitude compared to our current design. This would translate to an order of magnitude improvement in geff. Fur- thermore, this flake covers about 4% of the resonator. By assuming maximised coverage a 5 times enhancement of geff can be achieved. Both approaches would make the detection of monolayer flakes possible. In summary, we provide the first demonstration of photon- magnon coupling between a superconducting resonator and 520 540 600 6400.900.951.001.05 κeff/2π (MHz) Magnetic Field (mT)560 580 620 ωres/2π (MHz) 122801229012300a bFIG. 4. Magnon-photon coupling for the thinnest CGT flake. a Resonance frequency wresandbeffective loss rate keffas a func- tion of magnetic field of a resonator loaded with the thinnest CGT, consisting of 15 ML. The resonator’s loaded quality factor is 6938. The solid orange lines are results a fit to Eq.(2) and (3), respectively. The errorbars in brepresent the standard deviation from the Fano resonance lineshape fit to the resonator transmission. nm-thick vdW flakes of CGT, using a total of 23 devices with different CGT flakes of thickness from 153nm down to 11nm. By employing a coupled-harmonic-oscillator model, we extract the coupling strength, magnetic resonance field and relaxation rates for both photon and magnon modes in our devices. From our semi-broadband experiments, we find that the magnetic properties of exfoliated CGT flakes are ro- bust against the transfer process, with a standard deviation of less than 5% to expected resonance values from bulk param- eters. Notably, this suggests that vdW magnetic materials can be pre-screened at bulk to identify the most promising mate- rial for few layer device fabrication. The upper limit of the Gilbert damping in the 15 ML thick CGT flake is determined to be 0 :021, which is comparable to commonly used ferro- magnetic thin-films such as NiFe and CoFeB and thus mak- ing CGT attractive for similar device applications. We high- light that the damping parameter is key in precessional mag- netisation switching47,48, auto-oscillations by dc currents49,50, and comprehensive spin-orbit transport in vdW magnetic sys- tems51. The presented techniques are readily transferable to other vdW magnetic systems to study spin dynamics in atomically-thin crystalline materials. While creating stable magnon polaritons is still an open challenge due to the large loss rate gof the CGT magnon system, this work offers an important approach towards its achievement. There are still potential improvements to the measurement sensitivity such as resonator mode volume reduction by introducing nm scale constrictions52,53and use of exfoliation/transfer techniques to produce larger flakes to enhance the mode overlap (hence cou-6 pling strength)54,55. With concerted efforts, the formation of magnon polaritons in few layers vdW materials will become feasible. METHODS Superconducting Resonators: The resonators were fab- ricated by direct laser writing and a metal lift-off process. The individual 5mm 5mm chips are scribed from an in- trinsic, high resistivity ( r>5000Wcm) n-type silicon wafer of 250 m thickness. For a well defined lift-off, we use a double photoresist layer of LOR and SR1805. The resonator structures are transferred into the resist by a Heidelberg Di- rect Writer system. After development, 50nm NbN are de- posited by magnetron sputtering in a SVS6000 chamber, at a base pressure of 7 107mbar, using a sputter power of 200W in an 50:50 Ar/N atmosphere held at 5 103mbar, with both gas flows set to 50 SCCM28. Finally, the lift-off is done in a 1165 solvent to release the resonator structures. CGT Crystal Growth: CGT crystals used in this study were grown via chemical vapour transport. To this end, high- purity elemental precursors of Cr (chips, 99:995%), Ge (powder,99:999%), and Te (shots, 99 :999%) were mixed in the molar weight ratio Cr:Ge:Te = 10:13.5:76.5, loaded into a thick-wall quartz ampule and sealed under the vacuum of 105mbar. Then, the ampule was loaded into a two-zone furnace, heated up and kept at 950C for 1 week to homog- enize the precursors. To ensure high-quality growth, the am- pule was slowly cooled (0 :4C=h) maintaining a small tem- perature gradient between the opposite ends of the ampule. Once the ampule reached 500C, the furnace was turned off allowing the ampule to cool down to room temperature nat- urally. The large ( 1cm) single-crystalline flakes were ex- tracted from the excess tellurium and stored in the inert envi- ronment. CGT Flake Transfer: Devices for this study were made via transfer of single-crystalline thin flakes on top of the super- conducting resonators. The flakes were first exfoliated from bulk crystals on the clean surface of a home-cured PDMS (polydimethylsiloxane, Sylgard 184) substrate. The thickness of the CGT flakes on PDMS was estimated through the con- trast variation with transmission optical microscopy. Then, the selected flake was transferred to a resonator. The trans- fer was performed in air at room temperature. To minimize the air exposure, the entire process of exfoliation, inspection and transfer was reduced to 10-15 min per resonator. 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ACKNOWLEDGMENT This study is supported by EPSRC on EP/T006749/1 and EP/V035630/1. G.E. acknowledges support from the Min- istry of Education (MOE), Singapore, under AcRF Tier 3 (MOE2018-T3-1-005) and the Singapore National Research Foundation for funding the research under medium-sized cen- tre program. E.J.G.S. acknowledges computational resources through CIRRUS Tier-2 HPC Service (ec131 Cirrus Project) at EPCC (http://www.cirrus.ac.uk) funded by the University of Edinburgh and EPSRC (EP/P020267/1); ARCHER UK Na- tional Supercomputing Service (http://www.archer.ac.uk) via Project d429. E.J.G.S acknowledges the Spanish Ministry of Science’s grant program “Europa-Excelencia” under grant number EUR2020-112238, the EPSRC Early Career Fellow- ship (EP/T021578/1), and the University of Edinburgh for funding support. D.S. acknowledges EPSRC funding through the Centre for Doctoral Training in Advanced Characteri- sation of Materials (EP/L015277/1) and European Union’s Horizon 2020 Research and Innovation program under grantagreement GrapheneCore3, number 881603 and the Depart- ment for Business, Energy and Industrial Strategy through the NPL Quantum Program. AUTHOR CONTRIBUTION C.W.Z, S.K. and H.K. conceived the experimental project. Resonator design and optimization was done by J.O’S., O.W.K, C.W.Z and supervised by J.J.L.M. Resonator fabri- cation and characterization was done by C.W.Z. CGT crystals were grown by I.A.V . and exfoliated and transferred by I.A.V . and N.V .T.T. and supervised by E.G.. D.S. measured AFM on the CGT flakes on the resonators. C.W.Z. performed the ex- periments and the data analysis with input from S.K. and H.K. Atomistic spin dynamics simulations were carried out by M.S. supervised by E.J.G.S.. C.W.Z., M.S., I.A.V . and H.K. wrote the manuscript with input from all authors. COMPETING INTERESTS The Authors declare no conflict of interests. Supplemental Material - Probing spin dynamics of ultra-thin van der Waals magnets via photon-magnon coupling I. MICROWAVE SETUP AND MEASUREMENT VNA MW out MW in -20 dB+32 dB DUT B0Cryostat FIG. S1. Microwave delivery and detection setup. Schematic of the microwave delivery and detection circuit. The image shows the coplanar waveguide transmission line. A resonator chip is placed on top of the transmission line for read out. On the right, a schematic layout of the resonators on a single chip is shown.2 Figure S1 shows a schematic of the used microwave measurement setup. We are using a Keysight E5071C vector network analyzer (VNA) to deliver and detect microwaves. The VNA is connected to a low temperature probe, fitted into a closed cycle helium cryostat and cooled to a base temperature of about 1 :8K. The microwave signal is transmitted into the cryostat and is attenuated by 20dB. The attenuator is positioned just before the sample box and provides a thermal anchoring for the center conductor of the coaxial cable to minimize the thermal load onto the sample. The output line is equipped with a Low Noise Factory LNC6_20C cryogenic amplifier, operating between 6 20GHz with an average amplification of +32dB. The transmitted and amplified signal is finally detected by the VNA. Figure S1 also shows an image of the coplanar waveguide transmission line PCB, loaded with a resonator ship, of which a schematic shows the resonator layout on a single chip. The resonators on the chip are capacitively coupled to the transmission line PCB. Upon resonance the transmission through the PCB is reduced, indicating the resonator resonance. The cryostat is equipped with a mechanical rotation stage and prior to the measurements the superconducting resonators are carefully aligned to the externally applied static magnetic field B0, such that the field is in the plane of the superconductor and along the narrow section of the resonators. Figure S2 shows the raw uncalibrated microwave transmission, ranging from 10GHz to 18GHz. The transmission is domi- nated by imperfections in our microwave circuitry, masking the small signals from the superconducting resonators. Thus, we performed a simple thru calibration of the microwave transmission to remove contributions from the setup, prior each magnetic field dependent measurement. Here, we exploit the magnetic field tunability of our superconducting resonators. Before calibra- tion, we set the frequency range of the measurement. We change the applied magnetic field such that the resonator’s resonance frequency is tuned out of the set frequency range. With a frequency window just showing the transmission of the setup we perform the thru calibration. After calibration we set the magnetic field back to its starting value, resulting in a background corrected spectrum with just the resonator feature on it. 10 11 12 13 14 15 16 17 18-1010 0 Raw Transmission |S21| (dB) Frequency (GHz) FIG. S2. Raw broadband microwave transmission signal. Logarithmic microwave transmission jS21jas a function of frequency between 10GHz and 18GHz at a temperature of 1 :8K. II. RESONATOR CHARACTERIZATION In this study, we fabricate twelve superconducting lumped element resonators on each of three resonator chips were fabricated using the same design (see schematic Fig. 1 (a) in the main text). Prior to transfer of the CGT flakes, we characterized the res- onators at a temperature of 1 :8K and zero applied magnetic field, using microwave powers of about 80dBm at the resonators, which is well below the bifurcation limit starting above 60dBm. Due to finite fabrication tolerances the resonator parameters have some variation, while some didn’t work at all. However, the targeted resonance frequencies are well reproducible and very similar for the 3 different chips. We compare the resonator parameters before and after transfer of the CGT flakes and collate the parameters in Tab. I. Note, the resonator parameters with the CGT flakes on were obtained with a static magnetic field applied in the plane of the superconductor, but far detuned from the CGT FMR. In addition, we add the respective thickness of the flake on each resonator, acquired from AFM measurements. Here, we give the values of the thickest region of a given flake on a resonator, as the thickest region will dominate the FMR signal. Due to the arbitrary shape of exfoliated flakes, some exhibit regions of different thickness, as seen e.g. in Fig. S5 (h) and (i).3 TABLE I. Resonator Parameters Chip Number wres;before (MHz) QL;before wres;after(MHz) QL;after CGT Thickness (nm) 1 12165 1978 12063 5733 16.21.3 1 13303 7357 13177 4950 - 1 13968 5575 13860 4679 49.43.5 1 14184 6492 14048 5627 153.123.3 1 16648 6606 16470 5021 23.52.5 1 17431 3215 17237 6826 23.86.4 1 17959 7595 17790 3963 26.24.1 2 12285 360 12153 7135 49.19.1 2 12669 3600 12548 6693 102.85.6 2 12782 3448 12648 6557 105.93.9 2 13393 4643 13244 4501 34.44.1 2 13760 6858 13620 5488 95.95.9 2 14395 9048 14201 4139 36.74.3 2 16075 7283 - - - 2 17048 6541 16811 4241 75.55.4 3 12043 6114 11899 6044 59.732.8 3 12456 2716 12314 6938 11.41.8 3 12996 5828 12848 4600 170.8 3 13422 6517 13272 5461 89.87.5 3 13719 6800 13582 6608 - 3 14238 9184 14064 5420 73.58.4 3 15390 8680 15219 6030 30.54.2 3 15821 2386 15604 4769 33.19.9 3 16430 7518 16193 5780 30.138.1 3 17308 6521 17054 5569 137.93.4 3 18111 3542 17870 4643 50.26.9 III. RESONATOR AND COUPLING SIMULATION We use finite element and numerical simulations to optimize our resonator design. Key requirements of our resonators are a strong resilience to externally applied static magnetic fields and a small mode volume. To achieve a large field resilience we reduced the area of the resonator to minimize effects of the magnetic field on the superconducting film. Further, we designed the resonators such that they act as lumped element resonators. Here, the resonance frequency is given by the total capacitance and inductance of the structure, with wres=1=p LC, analogues to a parallel LC circuit. This allows us to locally separate oscillating electric and magnetic fields and also to concentrate the magnetic fields in more confined regions, resulting in very small mode volumes. To verify the lumped element nature of our resonators we performed finite element simulations, using CST Microwave Studio. Figure S3 shows the resulting magnitude of the E-field (left side) and H-field (right side) distribution along the resonator structure for the resonator design producing the results shown in Fig. 2 in the main text. The E-field is concentrated along the parallel running wire sections, with its strength approaching zero along the narrow wire section. The opposite is the case for the H-field, where it is zero along the parallel wire sections and strongly concentrated along the narrow wire section. Note, that the H-field magnitude is homogeneous along the whole of the narrow wire section. The CST Microwave Studio at hand allowed us a simulation with perfect electric conductors. This is sufficient to model the general electric and magnetic energy distributions and resonance frequencies, however, not to simulate the corresponding oscillating magnetic field distribution, created by a superconducting rectangular wire. To this end, we numerically solve the Biot-Savart law for a rectangular wire cross-sectionS1, assuming a superconducting current distribution Jx;zS2, B1;x;z=m0 2pZw=2 w=2Zd=2 d=2Jr (xx0)2+ (zz0)2dx0dz0; (S1) with the vectors as J= (0;J(x;z);0)Tandr= (xx0;0;zz0)Tandm0being the magnetic constant. The integration is performed over the cross-section of the wire, of width wand thickness d. We define the wire cross-section in the x-z-plane, with win x- direction and din z-direction. The length of the wire is along the y-direction. For a superconducting wire, the current is not homogeneously distributed over the cross-section of the wire. Current is only flowing on the surface and is exponentially decaying towards the center of the wire. The characteristic length scale is given by the London penetration depth lL. We use the4 FIG. S3. Finite element simulations of resonator. CST Microwave Studio simulation of the distribution of E-fields and H-fields across the resonator structure. The color encoded fields represent the magnitude values. following expression for the current distributionS2 J(x;z) =J1 coshz0=lL coshd=lL" Ccoshx0=l1 coshw=l1+1cosh x0=l2=coshw=l2p 1(x0=w)2# +J2 J1coshx0=lL coshw=lL! ; (S2) where J2 J1=1:008 coshd=lLs w=l? 4l?=lL0:08301lL=l?; C= 0:506p w=2l?0:75 ; l1=lLp 2lL=l?; l2=0:774 l2 L=l?+0:5152l?; l?=lL=2d: The prefactors J1andJ2define the amplitude of the current density and hence the absolute value of the oscillating magnetic field B1. We define J1by normalizing the vacuum B1field to the energy density stored in the resonatorS3,S4 1 2¯hwres 2=1 2m0Z B2 1dV=1 2m0B2 1Vm; (S3) with Vmrepresenting the resonator mode volume. The additional factor of1=2on the left hand side of S3 takes into account that only half of the total energy is stored in the magnetic fieldS5. As our resonator design is a quasi 1-dimensional structure we have to define boundaries for the mode volume in the x- and z-direction. A common assumption is to use the width of the conductor wire wS6. For simplicity, we approximate the x-z-area of the mode distribution with the area of an ellipse. For the last dimension we use the length of the narrow wire section, supported by the CST Microwave Studio simulations (see Fig. S3). In total we find5 the mode volume to be Vm= ((p3:0m2:025m)wd)300m=5696m3. Figure S4 shows the resulting distribution of the oscillating magnetic field for the cross-section of the rectangular wire of width w=2m and thickness d=50nm. The magnitudejB1;x;zjis encoded in the color and the arrows indicate the B1;xandB1;zcomponents of the oscillating field. -2000200z (nm)400 -400 x (μm)0 -1 1 2 3 -2 -320 15 10 5 |B| (nT) 30 25 FIG. S4. Cross-section of resonator magnetic field distribution. Calculated magnitude of the magnetic field distribution around the cross- section of a rectangular superconducting wire. The wire cross-section lies in the center, indicated by the grey rectangular. The red arrows show the direction of the magnetic field. With the simulated B1field distribution we can calculate the position dependent single photon - single spin coupling strength g0(r)S3,S4for each magnetic moment per unit cell of CGT (ab-plane 0 :68nmS7,S8, along the c-axis 0 :7nmS9). Summation over all CGT unit cells Nwithin the mode volume of the resonator results in the collective coupling strength geff=s N å i=1jg0(ri)j2=gCGTmB 2¯hs N å i=1jB1(ri)j2=gCGTmB 2¯hNys N å i=1h (B2 x;i+B2 z;i)i : (S4) Here, mBis the Bohr magneton, Nyis the number of unit cells along the y-direction and gCGTis the g-factor for CGT for which a value of 2 :18S10is used. Note, we give the collective coupling strength for spin1=2and for linear polarized microwavesS3. For the calculation of gefffor the resonator loaded with 15 monolayers of CGT we extracted its lateral dimension from the AFM measurements (see Fig. S4 (g)) to 2 m along the x-direction and 12 m along the y-direction. The flake is assumed to lie directly on top of the superconducting wire without any gap in between. For these values the simulation yields geff=2p=8:94MHz, which is about a factor 2 :5 larger than the experimentally determined value of 3 :61MHz. The overestimation of the simulation most likely results from non-ideal conditions in the experiment. The corresponding flake lies at the top end of the resonators narrow wire section (see Fig. S5 (g)), where B1is concentrated. The finite element simulations show that in this area the field strength is already declining, resulting in a reduced coupling strength. Further, AFM can overestimate the thickness of a flake slightly for when there is a gap between resonator surface and flakeS9. The calculation also not includes the multiple peaks observed in the experiment, which - depending on their real nature - can distribute the magnon density over all resonant peaks. Nevertheless, we can use the simulation to estimate the signal reduction by scaling down the thickness of the flake to a single monolayer. Reducing the simulation to a single monolayer, while keeping the lateral dimensions, results in geff=2p=2:33MHz, a reduction by a factor of 0 :26. IV. AFM MEASUREMENTS ON CGT FLAKES After the transfer of the CGT flakes onto the individual resonators and after measuring FMR, we characterized the thickness of the flakes by AFM. Figure S5 shows a selection of height profile maps from the three resonator chips, including a height profile along the inductor wire of the resonator (blue line in the AFM profile images in Fig. S5). To extract the thickness we fit the steps in the height profile (red or green lines in the height profiles in Fig. S5). Note, the height values are relative values with an arbitrary offset. Figure S5 (g) shows the thinnest flake of this study, where the processed FMR data is shown in Fig. 4 in the main text.6 10 μm 5 μm0 5 x (μm)y (nm)40506070 5 μmy (nm) 02040 0 4 x (μm)2 6 5 μmy (nm) 506070 0 4 x (μm)2 680 8 5 μm y (nm) 02040 0 10 x (μm)5 1560 2080 y (nm)050100 0 10 x (μm)5 5 μm 5 μm y (nm) 02040 0 10 x (μm)5 15 5 μm 10 μmy (nm)101520 0 6 x (μm)325 y (nm) 03060 0 20 x (μm)1090 y (nm)50100 0 6 x (μm)3150 120 30(c) (b) (a) (f) (e) (d) (i) (h) (g)9 FIG. S5. AFM measurements. AFM profile images with respective height profile (above) along the resonator inductor wire (blue and purple lines in profile images, with the arrow indicating scan direction). a-cfigures for resonator chip 1 (refer to Tab. I), having resonance frequencies with CGT of 17237MHz, 17790MHz and 16470MHz, respectively. d-ffigures for resonator chip 2 (refer to Tab. I), having resonance frequencies with CGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iimages for resonator chip 3 (refer to Tab. I), having resonance frequencies with CGT of 12314MHz, 13272MHz and 17054 ;MHz, respectively. The red and green solid lines are fits to the height profiles. V. ANALYSIS AND ADDITIONAL FMR DATA We analyze our experimental data, using the model functions (2) and (3) from the main text in a two-step semi-optimized fashion. The main intention for this approach is to minimize the number of free parameters in our model functions. In a first7 coarse step, we match the collective coupling strength geff;kto fit the experimental data, assume a constant separation between the individual magnon modes at BFMR ;kand the same magnon loss rate gfor all modes and determine the resonator loss rate k0from the resonator transmission far detuned from the FMR with the CGT flakes. This results in 3 free parameters for the first stage of our analysis, the magnon loss rate g,BFMR of the main mode and the constant separation between the BFMR ;k. After this first step we arrive at a best fit to the envelope of the experimental data, however with not matching amplitudes. In a consecutive second step, we manually optimize the geff;kto arrive at a model in good agreement with wresandkeff(see dashed lines in Fig. S6). Fig. S6 shows additional results from the corresponding FMR measurements performed on the in Fig. S5 showed resonators. As described in the main text, the measurements were performed at a temperature of 1 :8K and recording the microwave trans- missionjS21j2as a function of the static magnetic field. Analyzing the microwave transmission by fitting a Fano resonance lineshape to it we extract the effective loss rate of the resonator, interacting with the CGT keff. Figure S6 shows the resulting keffas a function of the magnetic field. In general, the response of the CGT FMR is complex and varies for the different res- onators. The resonance lineshape is not well described by just a single Lorentzian and requires multiple peaks to produce a good agreement. For some resonators, keffexhibits obvious peaks, residing on a broader spectrum (see Fig. S6 (c), (f) and (i)). Together with the observation of well and clearly separated peaks for the resonator loaded with the thinnest CGT flake of 11nm, we motivating the multiple peak analysis as presented in the main text. However, as the individual peaks are overlapping for the remainder of the resonators we only applied a qualitative analysis. κeff/2π (MHz)580 600 Magnetic Field (mT)62036912 525 550 Magnetic Field (mT)5751.61.82.0 600 625 675 700 Magnetic Field (mT)725246 7508 10 500 525 Magnetic Field (mT)5501.251.301.35 5751.40 520 560 Magnetic Field (mT)6000.900.951.00 6401.05 650 675 Magnetic Field (mT)7004812 72516 675 700 Magnetic Field (mT)7251.41.61.8 κeff/2π (MHz) 750 650 700 Magnetic Field (mT)750369 800 12 500 520 Magnetic Field (mT)540123 560 (c) (b) (a) (f) (e) (d) (i) (h) (g) κeff/2π (MHz) κeff/2π (MHz) κeff/2π (MHz) κeff/2π (MHz) κeff/2π (MHz) κeff/2π (MHz) κeff/2π (MHz) FIG. S6. Additional data on magnon-photon coupling of CGT-resonator devices. Results from FMR measurements with effective loss rate keff=2pas a function of the static magnetic field. a-cresults for resonator chip 1 (refer to Tab. I), having resonance frequencies with CGT of 17237MHz, 17790MHz and 16470MHz, respectively. d-fresults for resonator chip 2 (refer to Tab. I), having resonance frequencies with CGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iresults for resonator chip 3 (refer to Tab. I), having resonance frequencies with CGT of 12314MHz, 13272MHz and 11899MHz, respectively. The orange solid lines are semi-optimized fits, as described in the main text. The errorbars in the figures represent the standard deviation from the Fano resonance lineshape fit to the respective resonator transmission. Figure S7 shows the extracted collective coupling strength geffas a function of the square root of the FMR active volume. We define the active volume as the overlap of the oscillating magnetic field B1and the CGT flake lying on the resonator. The B1 field distribution, discussed in Sec. III, is used to estimate the extend of the B1and is taken as 2 m. From AFM measurements and microscope images we extract the thickness and lateral dimensions of the flakes to calculate the final active volume. As the collective coupling is proportional to the square root of the number of magnetic momentsS3, which are interacting with the resonator field, it follows that geffscales linearly with the square root of the active volume. This linear trend is highlighted by the orange solid line in Fig. S7. The majority of the extracted data follows this linear trend very well, corroborating our analysis. Only 3 data points deviate strongly from the rest of the data, which we attribute to significant inhomogeneities in the CGT-flakes, making the volume estimation inaccurate. These data points are highlighted in red in Fig. S7.8 0 1 2 3 4 51030 20Collective Coupling geff/2π (MHz) Square Root of Active FMR Volume ( μm3/2) FIG. S7. Scaling of the collective coupling. Collective coupling strength geffas a function of the FMR active CGT-flake volume. The orange line highlights the linear trend of geffwith increasing volume. The red symbols are regarded as outliers, as these flakes show inhomogeneities, leading to inaccurate volume estimations. The star symbol represents data from the thinnest flake (see data in Fig. 4 in the main text) and the pentagon symbol data from the 17nm flake (see data in Fig. 2 in the main text) The errorbars give confidence values for the extracted values. VI. MAGNETO-STATIC SPIN-WAVE DISPERSION IN THIN-FILM MAGNETS WITH PERPENDICULAR ANISOTROPY Here we describe the spin-wave mode frequency in a thin-film magnet with perpendicular anisotropy along the film normal. We consider this at the magnetic-dipole limit where the wavelength is relatively large and the exchange interaction contribution to the spin-wave dispersion is neglected. Furthermore, standing spin-wave modes along the thickness direction are also ruled out since these modes only appear at much higher frequencies than the main mode, where we consistently observe additional peaks at both higher and lower frequencies from the main mode. The mode (angular) frequency ( w) for wavevector k=0 when we apply a magnetic field Balong one of the film plane directions can be given by Eq. 3d in Ref.S11as: w g2 =B B+m0Ms2Ku Ms : (S5) Here, g,MsandKuare the gyromagnetic ratio, saturation magnetization and the perpendicular anisotropy energy density, respec- tively. Note, that the total field within m0Ms2Ku Msis negative for perpendicularly-magnetized films which we consider in this section. Within the magnetic-dipole limit, the demagnetization term m0Msis modified for spin-waves with finite k, depending on the relative orientation between the Msandkdirections. Here we follow the expression given in Serga et al.S12. For pure backward volume magnetostatic modes where kkMs(illustrated in Fig. S8), the mode frequency becomes: wBVMSW g2 =B B+m0Ms1ekt kt 2Ku Ms ; (S6) where tis the thickness of the magnet. Note, that this expression is only valid for the case where Msis colinear to B, meaning that jBj>jm0Ms2Ku Msj. To the limit of k!0, the term (1ekt)=ktis reduced to unity, consistent to Eq. (S5). When kis nonzero, we can observe that wBVMSW becomes smaller than that for k= 0, exhibiting a negative group velocity for this spin-wave mode. As the opposite extreme where k?Ms(illustrated in Fig. S8), the resonance frequency becomes larger than that for k= 0 and is called magneto-static surface spin-wave mode. The mode frequency expression for this mode is given by: wMSSW g2 =B B+m0Ms2Ku Ms +m2 0M2 s 1e2kt : (S7) Here, m2 0M2 s 1e2kt
is the spin-wave correction term which goes to zero for k!0 (hence consistent to Eq. (S5)) and becomes positive for k>0, meaning that wMSSW becomes larger as soon as spin-waves gain momentum along this direction. We use these two expressions in an effort to explain the origin of the multiple peaks in our experiments. Figure S8 plots the calculated wBVMSW =2pandwMSSW =2pas a function of wavevector k. The range of wavevector is chosen such that the resulting resonance frequencies are within the same order of magnitude as the observed mode splittings in the experiment ( µ100MHz).9 Wavevector (μm-1)6 4 8 10 12 2 012.712.812.9Resonance Frequency (GHz)13.0 12.6 12.5 12.4Kittel BVMSW MSSWB0B0100 MHz FIG. S8. Spin-wave dispersion. Spin-wave resonance frequency for BVMSW (green solid line) and MSSW (yellow solid line) as a function of wavevector. The dashed blue line is the resonance frequency of the k=0 main mode. The parameters used are B0=598mT, gCGT =2:18, m0Ms=194:3mT and Ku=3:84104J=m3and a thickness of 17nm. The grey area highlights a 100MHz margin relative to the main mode, indicating the order of magnitude of the mode splitting observed in the experiment. The arrows on the right hand side illustrate the relative wavevector orientations of the BVMSW and MSSW spin-wave modes with respect to the static magnetic field. The corresponding wavelength to a 100MHz resonance offset to the main mode are about 2 :2m and 620nm for wBVMSW andwMSSW , respectively. These values are within a reasonable scale for our different lateral CGT flake dimensions under investigation. This suggests that spin-wave modes are likely the origin of the multiple resonance peaks observed. The thinnest CGT flake shows, however, a deviation from this behaviour. We only observe modes at lower frequencies, which would indicate to BVMSW modes. Calculating the respective shortest wavelength results in 225nm, which is significantly shorter than for the other devices. We assume that the placement and irregular shape are likely to cause this difference. First, this flake is placed at the very edge of the inductor wire, where the B1field strength is declining (see Fig. S3), reducing the FMR active area. Thickness steps can lead to a wavelength down-conversionS13, however, with the overall irregular shape of the flake it is difficult to define a length scale for a standing spin wave mode. VII. ATOMISTIC SPIN DYNAMICS SIMULATIONS OF FMR To study the ferromagnetic resonance in CGT we perform atomistic spin dynamics simulationsS14,S15. The magnetic Hamil- tonian employed in the simulations is given by: H=1 2å i;jSiJi jSjå iDi(Si·e)2å imiSi·(B0+B1) (S8) where i,jrepresent the atoms index, Ji jrepresents the exchange interaction tensor, Dithe uniaxial anisotropy, which for CGT is orientated out of plane ( e= (0;0;1)) and B0the external static magnetic field applied in-plane during the ferromagnetic resonance simulations and B1=B1sin(2pnt)the oscillating field applied perpendicular with respect to B0. The CGT system has been parameterized from first principle methodsS9, up to the third nearest neighbor intralayer and interlayer exchange. The exchange values have also been re-scaled by Gong et al.S9with a 0.72 factor to obtain the experimental TCand multiplied by S2to match the magnetic Hamiltonian. The magnetic moment or Cr is considered 3.26 mBS16and the uniaxial anisotropy has a value of 0 :05 meV as extracted from first principle methodsS9. The parameters used in the simulations are given in Table II. FMR calculations have previously been employed for atomistic models, and can reproduced well the variation of linewidth with temperature, for example, in recording media systemsS17. Hence, in the current simulations we use the same setup of frequency swept FMRS17and we obtain the spectra by performing a Fourier transform of the magnetisation component parallel to the oscillating field. Since these calculations are done close to 0K, no averaging is require to reduce the thermal noise. To excite the FMR mode, we apply a DC field in-plane of 0.9 T on x-direction and an AC field perpendicular to the DC field, on y-direction. The Fourier transform has been performed for the y-component of magnetisation for 5ns after an initial 1ns equilibration time. A thermal bath coupling has been chosen in agreement with the upper limit of the Gilbert damping observed in experiments. The system size we performed FMR on is a 4-layer CGT system, with lateral size of 6 :91nm11:97nm, periodic boundary conditions in xy and total of 1600 atoms. The small system size has been used to reduce the computational cost associated10 Quantity Symbol quantity units Timestep ts 0.1 fs Thermal bath coupling a 0.02 Gyromagnetic ratio ge 1.7608591011rad s1T1 Magnetic moment mB 3.26S16mB Uniaxial anisotropy Di 0.05S9meV/link Simulation temperature T 0.001 K Static magnetic field B0 0.9, 0.7 T Oscillating magnetic field amplitude B0 0.001 T FMR frequency n varied GHz Intralayer exchange, NN J1 2.71S9meV/link Intralayer exchange, 2NN J2 - 0.058S9meV/link Intralayer exchange, 3NN J3 0.115S9meV/link Interlayer exchange, NN Jz 1-0.036S9meV/link Interlayer exchange, 2NN Jz 20.086S9meV/link Interlayer exchange, 3NN Jz 30.27S9meV/link TABLE II. Simulation parameters for FMR on CGT system . with FMR simulations. Experiments have showed modified g-factors due to photon-magnon coupling hence hereby we propose a simple model where the properties of the individual layers have been modified to include different gyromagnetic ratio, as illustrated in Fig. S9 a. We can define the resonance frequencies for each magnetic layer using the Kittel equation in the case of in-plane applied field with perpendicular anisotropy B?u: w=gp B0(B0B?u) (S9) We next investigate the FMR signal for a few cases assuming the CGT monolayers at low or strong interlayer exchange cou- plings J0 z=0;0:1%;10%;100% Jz, where Jzcorresponds to the pristine interlayer exchange (Fig. S9 b-c). In the low interlayer exchange regime ( J0 z=0;0:1%Jz), the CGT presents multiple peaks with each frequency corresponding to the layer dependent gyromagnetic ratio, g-n(g1) =16:81GHz, n(g2) =25:22GHz, n(g3) =33:62GHz. At J0 z=0:1%J0 z(Fig. S9 b) we can still observe resonance peaks corresponding to each individual layer. However by increasing the exchange coupling to 10% J0 zor higher (Fig. S9 c) there is a single FMR peak indicating that the system behave coherently with all layers having the same FMR frequency. The single FMR frequency corresponds to the average magnetic properties of the CGT layers. Small variations of the resonance frequency as function of the inter-layer exchange coupling can be observed which these being correlated to the transition of the system from the multi-peaks regime to a coherent excitation. By calculating the damping of the highest reso- nance peaks from a Lorenzian fit, we reobtain the damping corresponding to the input thermal bath coupling, 0 :02 with a relative tinny error5%. Overall, the interlayer exchange coupling locks the dynamics of individual layers coherently together without allowing multiple frequencies at the FMR signalS18. [S1]A. E. Primenko, M. A. Osipov, and I. A. Rudnev, Technical Physics 62, 1346 (2017). [S2]L. H. Lee, T. P. Orlando, and W. G. Lyons, IEEE Transactions on Applied Superconductivity 4, 41 (1994). [S3]C. W. Zollitsch, K. Mueller, D. P. Franke, S. T. B. Goennenwein, M. S. Brandt, R. Gross, and H. Huebl, Applied Physics Letters 107, 142105 (2015). [S4]S. Weichselbaumer, P. Natzkin, C. W. Zollitsch, M. Weiler, R. Gross, and H. Huebl, Physical Review Applied 12, 024021 (2019). [S5]R. J. Schoelkopf and S. M. Girvin, Nature 451, 664 (2008). [S6]D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Nature 445, 515 (2007). [S7]Y . F. Li, W. Wang, W. Guo, C. Y . Gu, H. Y . Sun, L. He, J. Zhou, Z. B. Gu, Y . F. Nie, and X. Q. Pan, Physical Review B 98, 125127 (2018). [S8]Y . Sun, R. C. Xiao, G. T. Lin, R. R. Zhang, L. S. Ling, Z. W. Ma, X. Luo, W. J. Lu, Y . P. Sun, and Z. G. Sheng, Applied Physics Letters 112, 072409 (2018). [S9]C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y . Xia, T. Cao, W. Bao, C. Wang, Y . Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546, 265 (2017). [S10]S. Khan, C. W. Zollitsch, D. M. Arroo, H. Cheng, I. Verzhbitskiy, A. Sud, Y . P. Feng, G. Eda, and H. Kurebayashi, Physical Review B 100, 134437 (2019). [S11]M. Farle, Reports on Progress in Physics 61, 755 (1998). [S12]A. A. Serga, A. V . Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010). [S13]J. Stigloher, T. Taniguchi, M. Madami, M. Decker, H. S. Körner, T. Moriyama, G. Gubbiotti, T. Ono, and C. H. Back, Applied Physics Express 11, 053002 (2018). [S14]D. A. Wahab, M. Augustin, S. M. Valero, W. Kuang, S. Jenkins, E. Coronado, I. V . Grigorieva, I. J. Vera-Marun, E. Navarro-Moratalla, R. F. Evans, et al. , Advanced Materials 33, 2004138 (2021). [S15]A. Kartsev, M. Augustin, R. F. Evans, K. S. Novoselov, and E. J. G. Santos, npj Computational Materials 6, 1 (2020). [S16]I. A. Verzhbitskiy, H. Kurebayashi, H. Cheng, J. Zhou, S. Khan, Y . P. Feng, and G. Eda, Nature Electronics 3, 460 (2020).11 FIG. S9. Atomistic simulations. a, Schematic of the crystal structure of CGT with atoms defined by different colours. b,FMR spectra of 4 layer CGT where the layers are low interayer exchange coupled (0 ;0:1%J0z, where J0zis the pristine CGT interlayer exchange). c,Similar as b, but with the layers at a strong exchange coupling (10% ;100% J0z). The solid lines in b-crepresent a Lorenzian fit to the numerical data. [S17]M. Strungaru, S. Ruta, R. F. Evans, and R. W. Chantrell, Physical Review Applied 14, 014077 (2020). [S18]Data inputs/plots utilised for Supplementary Figure S7 (atomistic simulations) can be found at the following GitHub repository.

2022-06-06

Layered van der Waals (vdW) magnets can maintain a magnetic order even down to the single-layer regime and hold promise for integrated spintronic devices. While the magnetic ground state of vdW magnets was extensively studied, key parameters of spin dynamics, like the Gilbert damping, crucial for designing ultra-fast spintronic devices, remains largely unexplored. Despite recent studies by optical excitation and detection, achieving spin wave control with microwaves is highly desirable, as modern integrated information technologies predominantly are operated with these. The intrinsically small numbers of spins, however, poses a major challenge to this. Here, we present a hybrid approach to detect spin dynamics mediated by photon-magnon coupling between high-Q superconducting resonators and ultra-thin flakes of Cr$_2$Ge$_2$Te$_6$ (CGT) as thin as 11\,nm. We test and benchmark our technique with 23 individual CGT flakes and extract an upper limit for the Gilbert damping parameter. These results are crucial in designing on-chip integrated circuits using vdW magnets and offer prospects for probing spin dynamics of monolayer vdW magnets.

Probing spin dynamics of ultra-thin van der Waals magnets via photon-magnon coupling

2206.02460v2

Evaluating Gilbert Damping in Magnetic Insulators from First Principles Liangliang Hong,1, 2Changsong Xu,1, 2and Hongjun Xiang1, 2,∗ 1Key Laboratory of Computational Physical Sciences (Ministry of Education), Institute of Computational Physical Sciences, State Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, China 2Shanghai Qi Zhi Institute, Shanghai 200030, China (Dated: September 21, 2023) Magnetic damping has a significant impact on the performance of various magnetic and spin- tronic devices, making it a long-standing focus of research. The strength of magnetic damping is usually quantified by the Gilbert damping constant in the Landau-Lifshitz-Gilbert equation. Here we propose a first-principles based approach to evaluate the Gilbert damping constant contributed by spin-lattice coupling in magnetic insulators. The approach involves effective Hamiltonian mod- els and spin-lattice dynamics simulations. As a case study, we applied our method to Y 3Fe5O12, MnFe 2O4and Cr 2O3. Their damping constants were calculated to be 0 .8×10−4, 0.2×10−4, 2.2×10−4, respectively at a low temperature. The results for Y 3Fe5O12and Cr 2O3are in good agreement with experimental measurements, while the discrepancy in MnFe 2O4can be attributed to the inhomogeneity and small band gap in real samples. The stronger damping observed in Cr 2O3, compared to Y 3Fe5O12, essentially results from its stronger spin-lattice coupling. In addition, we confirmed a proportional relationship between damping constants and the temperature difference of subsystems, which had been reported in previous studies. These successful applications suggest that our approach serves as a promising candidate for estimating the Gilbert damping constant in magnetic insulators. I. INTRODUCTION Recent decades have witnessed rapid developments in magnetics and spintronics [1–3]. A long-time pursuit in spintronics is to actively control and manipulate the spin degrees of freedom in solid-state systems. Related fun- damental studies involve spin transport, spin dynamics and spin relaxation [4]. Within these domains, magnetic damping often plays a crucial role. Generally, stronger damping enables a faster writing rate for magnetic mem- ories, while lower damping leads to a longer propagation distance of spin waves. Therefore, it is always essential to accurately evaluate the magnetic damping in different materials. For instance, yttrium iron garnet (YIG) is a highly promising spintronic material due to its ultra-low magnetic damping [5–7]. However, the intrinsic mecha- nism behind its unique property has yet to be fully eluci- dated, which partly motivates us to carry out this work. At present, magnetic damping is typically represented by a phenomenological term in the well-known Landau- Lifshitz-Gilbert (LLG) equation, which has been widely employed to simulate magnetization dynamics [8, 9]. A basic form of this equation can be written as, ∂ ⃗ m ∂t=−γ ⃗ m×⃗B+α m⃗ m×∂ ⃗ m ∂t(1) where ⃗Brepresents the total magnetic field acting on the local dipole ⃗ m,mdenotes the norm of ⃗ m,γis the gyro- magnetic ratio, and αis the Gilbert damping constant. The second term on the right side, as we mentioned, leads ∗hxiang@fudan.edu.cndirectly to the relaxation process, in which the rate of en- ergy dissipation is determined by the damping constant. Given the importance of αin magnetization dynamics, its origin has been extensively studied in the literature [10–13]. To the best of our knowledge, both intrinsic and extrinsic mechanisms contribute to the damping. Specif- ically, the intrinsic factors include spin-lattice and spin- electron couplings, while the extrinsic contributions pri- marily involve lattice imperfections and eddy currents [14, 15]. Two types of first-principles based methods have been developed to calculate the damping constants in the past. One approach involves the breathing Fermi surface model [16, 17] and the torque correlation model [18, 19], while the other is based on the scattering theory from linear response [20–22]. These methods have demonstrated re- markable success in studying the magnetic damping in transition metals such as Fe, Co, and Ni. Despite be- ing free from complicated experiments, which are mostly based on ferromagnetic resonance, these theoretical ap- proaches still exhibit several limitations. Firstly, when dealing with complex systems, we often have to spend a significant amount of computing resources on the first- principles calculations. In addition, these methods are more suitable for calculating the electronic contribution to Gilbert damping in metallic magnets, thus rarely tak- ing the effect of spin-lattice coupling into consideration [14, 23]. Recently, spin-lattice dynamics (SLD) simulations [24] have been adopted as an alternative method to evaluate the Gilbert damping parameters. In Ref. [23], the au- thors constructed an empirically parameterized Hamil- tonian model for a cobalt cluster. They coupled a pre- heated lattice with a fully ordered spin state, then per- formed SLD simulation. During the relaxation process,arXiv:2309.11152v1 [cond-mat.mtrl-sci] 20 Sep 20232 the energy of lattice and spin subsystems were recorded and fitted to the following logistic functions, Ulat=Ulat 0−∆U0 1 + exp[ −η∆U0t−Θ](2) Umag=Umag 0+∆U0 1 + exp[ −η∆U0t−Θ](3) from which they extracted the relaxation rate Γ = η∆U0 and calculated the damping constant α=ηµS/γ. Here, µSdenotes the magnitude of magnetic moments. In Ref. [25], the authors also built an empirical potential model for a periodic bcc Fe system. They firstly applied an ex- ternal magnetic field in the z-direction and thermalized the system to a finite temperature. Then, the magnetiza- tion orientation of each atom was rotated artificially by a same angle. Afterwards, the system would relax back to equilibrium, during which the averaged z component of atomic magnetization was recorded and fitted to the following function, mz(t) = tanhα 1 +α2γBext(t+t0) (4) where αwas exactly the Gilbert damping parameter to be estimated. Since these works selected transition met- als as the research object, their results were both orders of magnitude smaller than the experimental values. In addition, the use of empirically parameterized models re- duced the accuracy of their simulated results. In this work, we combine SLD simulations with first- principles based effective Hamiltonian models to evalu- ate the damping constants in magnetic insulators, where the dominant contribution results from spin-lattice cou- plings. Compared to the previous studies, our work has made improvements mainly in two aspects. Firstly, the utilization of first-principles based Hamiltonian models in simulations enhances the reliability of our conclusions. Besides, the better choice of research objects allows for demonstrating the superiority of SLD simulations. In particular, the microscopic origin of low damping in YIG will be investigated. The paper is organized as follows. In Sec. II, we introduce our effective Hamiltonian model, parameterization methods, and a scheme for evaluating Gilbert damping parameters. Then, both the validation and application of our method are presented in Sec. III. Finally, we summarize this work and give a brief outlook in Sec. IV. II. MODEL AND METHODS This section is split into three parts. Firstly (in Sec. II A), we introduce a generic form of our effective Hamil- tonian model. Then, methods involving the calculation of model parameters are presented in Sec. II B. At the last part (Sec. II C), we propose a novel scheme to de- termine the Gilbert damping constant through dynamics simulations.A. The Hamiltonian Model Since our purpose is to evaluate the contribution of spin-lattice coupling to magnetic damping, the effective Hamiltonian model must incorporate both spin and lat- tice degrees of freedom. A concise and generic formula that meets our basic requirements consists of the three terms as follows: H=HL({ui,α}) +HS({⃗ sj}) +HSLC({ui,α,⃗ sj}) (5) where αabbreviates three orthogonal axes, ui,αrepre- sents the displacement of atom i, and ⃗ sjis a unit vector that represents the direction of spin j. The first term HLin Hamiltonian model describes the dynamical behavior of individual phonons. Technically, we take the atomic displacements as independent vari- ables and expand the Hamiltonian to the second order with Taylor series. Then, we have the form as, HL=1 2X ijX αβKij,αβui,αuj,β+1 2X i,αMi˙ui,α˙ui,α(6) where Kij,αβ denotes the force constant tensor and Mi represents the mass of atom i. Similarly, the second term HSdescribes the dynami- cal behavior of individual magnons. For simplicity but no loss of accuracy, we only considered the Heisenberg exchange interactions between neighbor magnetic atoms in this work, though more complex interactions could be taken into account in principle. Therefore, this term can be expressed as, HS=X ⟨i,j⟩Jij⃗Si·⃗Sj (7) where Jijdenotes the isotropic magnetic interaction co- efficient. The third term HSLCrepresents the coupling of spin and lattice subsystems, and is expected to describe the scattering process between phonons and magnons. As an approximation of the lowest order, this term can be written as, HSLC=X ⟨i,j⟩X kα∂Jij ∂uk,αuk,α ⃗Si·⃗Sj (8) According to the theory of quantum mechanics, this coupling term provides a fundamental description of the single-phonon scattering process, which is believed to be dominant among all scatterings in the low-temperature region. This type of relaxation mechanism in ferromag- netic resonance was systematically studied by Kasuya and LeCraw for the first time [26]. It’s worth noting that a higher order of Taylor expansion could have been con- ducted to improve the accuracy of Hamiltonian models directly. For instance, the scattering between individual phonons can be adequately described by the anharmonic terms. However, as one always has to make a trade-off3 between the precision and complexity of models, in this work we choose to neglect the high order terms since the anharmonic effects in current investigated systems are not important. In this study, we adopted the symmetry-adapted clus- ter expansion method implemented in the Property Anal- ysis and Simulation Package for Materials (PASP) [27] to build the Hamiltonian model presented above. This package can identify the nonequivalent interactions and equivalent atom clusters in a crystal system by analyz- ing its structural properties based on the group theory. A significant benefit of working with PASP is we are en- abled to describe the target system with the least number of parameters. In the next section, we will discuss how to calculate the model parameters for different materials. B. Calculation of Model Parameters Firstly, the Heisenberg exchange coefficients Jijand spin-lattice coupling constants ∂Jij/∂uk,αcan be calcu- lated with the four-state method [28, 29]. The basic flow is to construct four artificially designated spin states of the target system, calculate the corresponding energies and forces based on the density functional theory (DFT), then determine the parameters by proper combination of those results. At the last step, the following formulas will be used, Jij=E↑↑+E↓↓−E↑↓−E↓↑ 4S2(9) ∂Jij ∂uk,α=F↑↑ k,α+F↓↓ k,α−F↑↓ k,α−F↓↑ k,α 4S2(10) where Sis the spin quantum number of magnetic atoms, Eis the total energy of system and Fk,αrefers to one component of the force on atom k. The superscripts ( ↑↑, ↓↓,↑↓,↓↑) specify the constrained spin states of system in the calculation. More technical information about the four-state method can be found in the references [28, 29]. Compared to other approaches, the four-state method of- fers an obvious advantage in that no additional DFT cal- culations are needed to determine the coupling constants ∂Jij/∂uk,αonce the exchange coefficients Jijhave been obtained. This is because the energy and forces are typ- ically provided simultaneously by one DFT calculation. Since atomic masses Mican be directly obtained from the periodic table, more efforts are needed to deal with the force constant tensor Kij,αβ. Currently, there are two commonly adopted ways to calculate the force constant tensor: density functional perturbation theory (DFPT) and finite displacement method. Both of these methods are applicable to our task. However, we cannot directly take the force constant tensor obtained from first-principles calculations as the model parameter. This is because in dynamics simula- tions we usually expand crystal cells to reduce the un- desired influence of thermal fluctuations, which leads toa conflict between the periodic boundary condition and the locality (also known as nearsightedness [30, 31]) of models. To be more specific, when calculating the con- tribution of one atom or spin to the total energy, we tend to set a well designed cutoff radius and ignore the inter- actions beyond it. This step is essential when dealing with a large-scale system, otherwise we will suffer from the model complexity and the computational cost. Nev- ertheless, if we set the elements of Kij,αβ that represent out-of-range interactions to be zero and leave the others unchanged, we may violate the so-called acoustic sum- mation rules: X iKij,αβ = 0 for all j, α, β. (11) It should be pointed out that a straightforward en- forcement of the acoustic summation rules, achieved by subtracting errors uniformly from force constants, will break the inherent crystal symmetry inevitably, which is the technique employed in phonopy [32]. To address the above issues, we adopted a more appropriate method in this work. Before a detailed introduction, it’s necessary to recall that not every element of the force constant ten- sor serves as an independent variable due to the crystal symmetries. Taking the cubic cell of Y 3Fe5O12(contain- ing 160 atoms) for example, there are 230400 elements in the tensor. After symmetry analyses, we find that only 597 independent variables {pn}are needed to adequately determine all the tensor elements {Kij,αβ({pn})}, where the effect of locality is already considered. Afterwards, our method is to set a correction factor xnfor each vari- ablepnand minimize the deviation of parameters under the constraints of Eq. (11). A mathematical reformula- tion of this method can be written as, min {xn}X n(xn−1)2,with X iKij,αβ({xnpn}) = 0 for all j, α, β.(12) In the case of Y 3Fe5O12, there are only 18 linearly inde- pendent constraints, which allow the extremum problem to be solved rigorously. The modified force constant ten- sor restores positive definiteness and translational sym- metry while maintaining the crystal symmetries. There- fore, the modified tensor meets the requirements for dy- namics simulations. In Sec. III B, the effectiveness of this approximate method will be demonstrated through a spe- cific example. All the first-principles calculations mentioned in this section are carried out using the Vienna ab initial simu- lation package (VASP) [33–35]. The force constants and phonon spectra are obtained by phonopy [32]. The opti- mizations formulated in (12) are accomplished with the function optimize.minimize implemented in SciPy [36].4 C. Evaluation of Damping Constants After the construction and parameterization of Hamil- tonian models, we are finally able to perform spin-lattice dynamics simulations. Before the evaluation of Gilbert damping constants, we briefly introduce the framework of SLD to cover some relevant concepts. In practice, the motion of magnetic moments follows the stochastic Lan- dau–Lifshitz–Gilbert (SLLG) equation [14], d⃗ mi dt=−γL⃗ mi× ⃗Bi+⃗Bfl i −γLα⃗ mi |⃗ mi|×h ⃗ mi× ⃗Bi+⃗Bfl ii (13) where γLis the renormalized gyromagnetic ratio, ⃗Bi= −∂H/∂ ⃗ m iis the effective local magnetic field and ⃗Bfl i refers to a stochastic field introduced by Langevin ther- mostat. At the same time, the motion of atoms obeys the Newton’s equation, d˙ui,α dt=1 Mi ⃗Fi,α+⃗Ffl i,α −ν˙ui,α (14) where νis the damping constant and ⃗Ffl i,αrefers to a stochastic force caused by thermal fluctuations. In this work, ⃗Bfl iand⃗Ffl i,αare modeled as normally distributed noises with temperature-dependent variances, Bfl i,β∼N 0,p 2αkBTS/γ|⃗ mi|δt (15) Ffl i,β∼N 0,p 2νMikBTL/δt (16) where TSandTLrefer to the equilibrium temperature of spin and lattice subsystems respectively. During simula- tions, we can also measure the transient temperature of each subsystem with the following formulas [37], TS=P i|⃗ mi×⃗Bi|2 2kBP i⃗ mi·⃗Bi, TL=1 2kBNX i,αMi˙u2 i,α (17) In this work, the LLG equation is numerically solved with the semi-implicit SIB method proposed by Mentink et al. [38]. The Newton’s motion equation is integrated using the Grønbech-Jensen-Farago Verlet-type method [39]. To ensure the stability of those algorithms, a step length of 0 .5 or 0 .2 fs is adopted [40], where the shorter one is used in energy-conserving simulations. Based on the combination of atomistic spin dynamics (ASD) and SLD simulations, a new scheme is proposed to evaluate the damping constant in magnetic materials. Here is the basic flow of this method and more details of a specific application are presented in Sec. III B. 1. Freeze the spin degree of freedom and thermalize the lattice from 0 to TLin the simulation. 2. Fix atomic positions and raise the temperature of spin to TS> TL. Compared to TL> TS, this type of nonequilibrium state is more common in actual scenarios.3. Perform an energy-conserving SLD simulation to relax the system. Normally, the spin temperature will decrease to the same as lattice and stay there till the end. 4. Conduct a series of ASD simulations with different Gilbert damping constants. The initial states are the same as in step 3 and the equilibrium temper- atures are set to be TL. 5. Compare the cooling rates ∂TS/∂tof spin system between SLD and ASD simulations to evaluate the equivalent Gilbert damping constant contributed by spin-lattice coupling. The key point behind step 5 is that the cooling rates observed in ASD simulations are related to the assigned damping constants, while in SLD simulation the cooling rate is determined by the strength of spin-lattice cou- pling. Note that the former relation can be viewed as a natural deduction of the LLG equation, ∂TS ∂t=1 CV∂Emag ∂t∝ −1 CV∂ ⃗ m ∂t·⃗B ∝ −1 CVα m⃗ m×∂ ⃗ m ∂t ·⃗B ∝α (18) where we have used Eq. (1) and simplified the formula of magnetic energy as Emag∝ −⃗ m·⃗B. III. RESULTS This section is divided into four parts. In Sec. III A, several test results are presented to validate the accu- racy of SLD simulations, which are implemented in the PASP package. Subsequently, detailed calculations on three magnetic materials, namely Y 3Fe5O12, MnFe 2O4 and Cr 2O3, are discussed in the rest parts. A. Validations In order to guarantee the reliability of our conclusions obtained from dynamics simulations, a series of pretests were carried out. We select some representative results and present them in Fig. 1, where Cr 2O3is taken as the object to be studied. Firstly, we set the ground state of Cr 2O3as the ini- tial state and performed a NVT simulation with Tset= 150K. As shown in Fig. 1(a), the temperature of spin and lattice subsystems increased to 150 Kin less than 5 ps and stayed there till the end. Since we can approxi- mate Ek= 0.5ELandEp= 0.5EL+ES, Fig. 1(b) also indicates that the contribution of phonons and magnons to the excited state energy is around 87.5% and 12.5% respectively. This result could be verified from another perspective. Note that there are totally 10 atoms in the5 FIG. 1. NVT and NVE relaxations of a spin-lattice coupled system (Cr 2O3) within the framework of spin-lattice dynamics. The top row plots the time evolution of temperatures and the bottom row shows the variation of potential, kinetic and total energies. (a) & (b): NVT thermalization from TL=TS= 0KtoTL=TS= 150 K. (c) & (d): NVE relaxation with TL= 30K, TS= 175 Kinitially. (e) & (f): NVE relaxation with TL= 180 K,TS= 30Kinitially. unit cell of Cr 2O3, which contribute 30 kBto the heat ca- pacity. Meanwhile, the 4 magnetic atoms will contribute another 4 kBin the low temperature region. Therefore, we can estimate that the contribution of magnons to the total heat capacity is close to 11.8%, which is consistent with the result from dynamics simulations. In Figs. 1(c) & 1(d), the initial state was set to be a nonequilibrium state with TL= 30KandTS= 175 K. As we expected, the total energy was well conserved when the system evolved to equilibrium. In addition, the final temperature fell within the range of 48 K∼55K, which agrees with our previous analysis of the heat capacities. Lastly, we simulated the relaxation process using an- other nonequilibrium excited state with TL= 180 Kand TS= 30Kas the initial state. As shown in Figs. 1(e) & 1(f), the temperature of spin system increased gradually to equilibrium with the total energy conserved through- out the simulation. Also, the final temperature is around 160K, which matches well with our analysis. It should be pointed out that there exist two notable differences be- tween this case and the previous. Firstly, the subsystems ultimately evolved to a same temperature in a finite time,which alleviated our concerns about the accuracy of SLD simulations. Besides, the relaxation time ( τ2) was much longer than that ( τ1) in Fig. 1(c). For this phenomenon, a qualitative explanation is presented below. Based on the theory of second quantization, the Hamil- tonian model presented in Sec. II A can be expressed in the following form [41, 42], HL=X qpℏωqp(b† qpbqp+ 1/2) (19) HS=X λϵλa† λaλ+Const. (20) HSLC=X λ,qpMλ,qpa† λ−qaλ b† qp−b−qp
(21) where bqpdenotes the annihilation operator of phonons with wave vector qin branch p, and aλrepresents the an- nihilation operator of magnons with wave vector λ. All the parameters, namely ωqp,ϵλandMλ,qp, can be deter- mined from the effective Hamiltonian model in principle. According to the Fermi’s golden rule, we have W{nλ−q, nλ, Nqp→nλ−q+ 1, nλ−1, Nqp+ 1}=2π ℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(Nqp+ 1)δ(ϵλ−q−ϵλ+ℏωqp) (22) W{nλ−q, nλ, N−qp→nλ−q+ 1, nλ−1, N−qp−1}=2π ℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(N−qp)δ(ϵλ−q−ϵλ−ℏω−qp) (23)6 FIG. 2. (a) The primitive cell of Y 3Fe5O12. The golden balls represent iron atoms, the cyan ball stand for yttrium atoms, and the red balls refer to oxygen atoms. (b) The magnetic ground state of YIG. The arrows of different colors represent the spin directions of Fe atoms. (c) The density of states ob- tained by DFT calculations. (d) The temperature dependence of average magnetization measured in MC and ASD simula- tions. For YIG, the phase transition point from ferrimagnetic to paramagnetic lies in 530 K approximately. where Wrepresents the probability of one-phonon emis- sion or absorption, nλdenotes the occupation number of magnons and Nqpstands for phonons. Both nλandNqp can be evaluated approximately using the Bose–Einstein distribution. According to the above formulas, the scat- tering rate Wgrows linearly with Nand quadratically with n. Compared to Fig. 1(c), there are more phonons but fewer magnons in the case of Fig. 1(e), thus leading to a lower transition probability and a longer relaxation time. More technical details about the second quantiza- tion of interactions between phonons and magnons can be found in Ref. [41, 42]. B. Damping constants in Y 3Fe5O12 In the field of spintronics, Y 3Fe5O12(yttrium iron gar- net, YIG) has gained much attention due to its ultra-low magnetic damping [5–7]. The unique property of this material motivated us to investigate the intrinsic mecha- nism behind it. The crystal structure of YIG is presented in Fig. 3(a). There are totally 80 atoms in the primitive cell, of which 12 Fe ions are located in the center of oxy- gen tetrahedrons while the other 8 Fe ions are sited in oxygen octahedrons. The magnetic ground state of YIG is illustrated in Fig. 3(b). The Fe ions situated in differ- ent chemical environments contribute spins in opposite directions, which makes YIG a typical ferrimagnetic ma- terial.TABLE I. The Heisenberg exchange coefficients J of YIG, where an effective spin S= 1 is adopted. For the FeO−FeO pairs, the Greek letters ( α&β) refer to different chemical environments. All the results are calculated with the four- state method. Spin Pair. Distance (Angst) J (meV) 1NN FeT−FeO3.445 47.414 1NN FeT−FeT3.774 2.399 1NN FeO−FeO(α) 5.337 0.538 1NN FeO−FeO(β) 5.337 5.055 2NN FeT−FeO5.555 0.285 2NN FeT−FeT5.765 3.437 In order to evaluate the Gilbert damping constants in YIG, our first step is to prepare an effective Hamilto- nian model. Considering the balance between precision and efficiency, the cutoff radius of interactions was set to be 11.0 Bohr for atomic pairs and 6.7 Bohr for 3- body clusters. After symmetry analyses, we identified 612 nonequivalent interactions in total, which included 6 Heisenberg exchange terms and 9 spin-lattice coupling terms. To determine the interaction parameters, we carried out a series of first-principles calculations, where a cu- bic cell was adopted to reduce the interference between adjacent cells caused by periodic boundary conditions. Following the settings in Ref. [43], we utilized the pro- jector augmented-wave (PAW) method [44] and revised Perdew-Burke-Ernzerhof exchange-correlation functional for solids (PBEsol) [45] in our calculations. Besides, the DFT+U method in its simplified form [46] was employed where the effective Hubbard U parameter was set to be 4 eV for the 3 delectrons of Fe ions. In addition, a cutoff energy of 520 eV for plane wave basis and a Γ-centered 2×2×2 mesh of k-points were used in the DFT calcu- lations. In Figure 2(c), we present the density of states (DOS) for YIG. With a band gap of 1.863 eV, there is hardly any electric current occurring in the low temperature re- gion. Moreover, the Heisenberg exchange coefficients of YIG is listed in Table I. To verify the accuracy of these parameters, we conducted both Monte Carlo (MC) and ASD simulations. The temperature dependence of aver- age magnetization is shown in Fig. 2(d), which reveals the critical temperature of YIG to be 530 K. This result is slightly lower than the measured Curie temperature, TC= 560 K[5], but falls within our tolerance. The cal- culated results of coupling constants are provided in the supplementary material. Next, we come to deal with the force constant tensor. In order to demonstrate the impact of locality and val- idate the effectiveness of our optimization method, we present some results pertaining to the tensor of YIG in Table II. Here we use “VASP” to tag the original tensor7 TABLE II. The force constant tensor of YIG. The columns labeled by A represent the sorted absolute values ofP iKij,αβ and the columns labeled by B list the sorted eigenvalues of Kij,αβ. For the cubic cell of YIG, we obtained the original tensor with the VASP package. Then, we eliminated the el- ements that represent interactions beyond the cutoff radius. This step was done by PASP. Finally, the tensor was modified to meet the requirement of translational symmetry through the optimization formulated in (12). VASP PASP Modified No. A B A B A B 1 0.000 0.000 1.587 -0.102 0.000 0.000 2 0.000 0.000 1.587 -0.102 0.000 0.000 3 0.000 0.000 1.587 -0.102 0.000 0.000 4 0.000 1.065 1.587 0.643 0.000 0.444 5 0.000 1.065 1.587 0.643 0.000 0.444 6 0.000 1.065 1.587 0.643 0.000 0.444 obtained from DFT calculations, “PASP” to label the modified tensor in which interactions beyond the cutoff radius are eliminated, and “Modified” to label the tensor after optimization of independent variables. As shown in Table II, the “PASP” tensor violated the acoustic sum rule and was not positive semi-definite, whereas these is- sues were resolved for the “Modified” tensor. Although an obvious difference existed between the “PASP” and “Modified” tensor in terms of their eigenvalues, we still assumed the target system could be reasonably described by the “Modified” tensor and the validity of this assump- tion would be verified by the calculated results of damp- ing constants. Additional details regarding the selection of tensor elements and the deviation of phonon spectra are provided in Fig. 3. According to figure 3(b) and 3(c), the major deviation in phonon spectra resulted from the elimination of tensor elements, rather than the subse- quent modification. Completing the preparation of Hamiltonian model, we applied the scheme proposed in Sec. II C to our first ob- ject, Y 3Fe5O12. An instance is presented in Figure 4. We setTL= 30K,TS= 180 Kfor the initial nonequilibrium state and adopted an expanded supercell which contains 12800 atoms in the simulation. Fig. 4(a) shows the time evolution of spin temperature in different types of simu- lations. By comparing the curves, we could roughly esti- mate that the equivalent damping constant in SLD simu- lation fell within the range of 10−3∼10−4. To make the estimation more precise, we calculated the initial cool- ing rates ∂TS/∂t|t=0through polynomial (or exponen- tial) fittings and plotted them in Fig. 4(b). Afterwards, a linear regression was performed to determine the quan- titative relation between lg( −∂TS/∂t|t=0) and lg( α). As we expected, the cooling rates in ASD simulations were proportional to the assigned damping constants. Then, we combined the results of SLD and ASD simulations toevaluate the equivalent damping constant. This step was accomplished by identifying the intersection of red and blue lines in Figure 4(b). Finally, the damping constant was determined to be αf= (2.87±0.13)×10−4in this case. To verify our method and result, we present a com- parison between SLD and ASD (where we set α=αf) simulations in Fig. 4(c). The curves agree well with each other in the initial stage but deviate in the second half. This phenomenon is within our expectation, because in the SLD simulation the lattice heats up as the spin cools down, thereby slowing the energy transfer between two subsystems. In addition to the above case, we have measured the equivalent damping constants under different conditions to investigate the temperature dependence of magnetic damping. The final results are summarized in Figure 5. Details about the estimation of uncertainties are given in the supplementary material. For Y 3Fe5O12, the damping constants at different temperatures stay on the order of 10−4, which is in good agreement with the experimental results (3 .2×10−4[47], 2 .2×10−4[48], 1 .2–1.7×10−4 [49]). For example, the damping constant in bulk YIG was reported as 0 .4×10−4in Ref. [50]. Meanwhile, our calculations yielded α= (2.8±0.3)×10−5at ∆T= 15 K and α= (7.0±0.7)×10−5at ∆T= 30 K, where both TL= 0 K. Therefore, the experimental value corresponds roughly to the temperature region of ∆ T= 15∼30 K in our study. We believe such extent of thermal excitation is quite common in all kinds of spintronics experiments. Moreover, Fig. 5 indicates that αis approximately pro- portional to the temperature difference between subsys- tems. This outcome is also consistent with some com- putational works in the past [23, 25]. By comparing the subfigures in Figure 5, we found that αhas little depen- dence on the lattice temperature, although here TLcould be viewed to some extent as the ambient temperature of the spin system. As a supplement to Sec. III A, we further validate our simulations by analyzing the measured cooling rates in Fig. 5(a). By subtracting Eq. (23) from Eq. (22), the transfer rate of energy between magnon and phonon sys- tems can be expressed as, ˙Q=X qpℏωqp⟨˙Nqp⟩=X λ,qpTλ,qp (24) where Tλ,qpdenotes different transfer channels, Tλ,qp∝(nλ−nλ−q)Nqp+nλ−qnλ+ 1 (25) According to the Bose–Einstein distribution, the number of magnons and phonons can be expressed as, nλ=1 eϵλ/kBTS−1, Nqp=1 eℏωqp/kBTL−1(26) When TSis high enough and TLis close to zero, we can approximate nλ=kBTS/ϵλ∝TSandNqpclose to zero. Under these conditions, we have ˙Q∝T2 S. This relation8 FIG. 3. (a) The selection of force constant tensor elements for the cubic cell of YIG. An 160 ×160 zero-one matrix is used to show the result of selection, in which ’1’ denotes the interactions within cutoff radius and ’0’ represents the elements that are artificially eliminated. (b) The phonon spectrum calculated from the force constant tensor before and after the elimination of tensor elements. (c) The phonon spectrum calculated from the force constant tensor before and after the optimization of independent variables. FIG. 4. (a) The time evolution of spin temperature in SLD and ASD simulations. The gray line represents the SLD simulation while the others refer to the ASD simulations with different damping constants. (b) The initial cooling rates ∂TS/∂t|t=0with respect to the damping constants α, where the scaling of axis is set to be logarithm. The gray squares refer to the results of ASD simulations and the blue line acts as the linear regression. The red circle is plotted by intersection of the blue line and the horizontal red dash line, which represents the initial cooling rate in the SLD simulation. Then we can obtain the equivalent damping constant from the abscissa of the red circle. (c) The comparison between ASD and SLD simulations. In the ASD simulation, the Gilbert damping constant is set to be α= 2.87×10−4, which is exactly the result of our evaluation from the SLD simulation. FIG. 5. The temperature dependence of Gilbert damping constants for Y 3Fe5O12. The label of abscissa axis ∆ Trefers to TS−TLof the initial state in dynamical simulations. Measurements on the magnetic damping are performed under different initial conditions of the lattice temperature: (a) TL= 0, (b) TL= 30K, (c)TL= 60K.9 FIG. 6. The relation between damping constants αand spin- lattice coupling constants ∂Jij/∂uk,αin YIG. Through a lin- ear fitting, the slope is determined to be 2 .01, which agrees well with our theoretical predictions. is well verified by linear regressions and the details are provided in the supplementary material. Furthermore, the accuracy of our simulations can also be proved from another perspective. According to Eqs. (22) and (23), the scattering rate Wgrows quadratically with the coupling parameters Mλ,qp. Based on the theory of second quantization, Mλ,qpshall be proportional to the coupling constants ∂Jij/∂uk,α. Therefore, under a definite condition of temperature, we have: α∝˙Q∝∆W∝M2 λ,qp∝(∂Jij/∂uk,α)2(27) In order to verify this relation, we adjusted the spin- lattice coupling constants of YIG coherently while keep- ing the other model parameters unchanged. Then, SLD simulations were carried out to evaluate the correspond- ing damping constants. The result is plotted in Fig. 6, where the x-label “slcc” stands for the spin-lattice cou- pling constants and the subscript “0” refers to the orig- inal situation. From a linear fitting, the slope is deter- mined to be 2 .01, which agrees well with our prediction. C. Damping constants in MnFe 2O4 After the calculation on YIG, we applied our method to MnFe 2O4(MFO), which was reported to possess a large Gilbert damping constant in the literature [13, 51]. As shown in Fig. 7(a), MnFe 2O4has a typical structure of spinels, where A sites are surrounded by four oxygen atoms and B sites are located in octahedrons. Generally, spinels can be classified into normal and inverse struc- tures according to the distribution of divalent and triva- lent cations between A/B sites. In experiments, MFO usually crystallizes into a mixed phase where the normal structure occupies the major part (80% in bulk MFO [52]). Here, we only considered its normal structure in this work. Also, the magnetic ground state of MFO is shown in Fig. 22(b), where the magnetic moments are antiparallel between A/B sites. FIG. 7. (a) The cubic cell of MnFe 2O4. The purple balls rep- resent manganese atoms, the golden balls refer to iron atoms, and the red balls stand for oxygen atoms. (b) The magnetic ground state of MFO. The arrows of different colors repre- sent the spin directions of Mn and Fe atoms separately. (c) The density of states obtained by DFT calculations. (d) The temperature dependence of average magnetization measured in MC and ASD simulations. For MnFe 2O4, the phase tran- sition point from ferrimagnetic to paramagnetic lies in 730K approximately. Firstly, we started to construct an effective Hamilto- nian model for MFO. With the same cutoff settings for YIG, we found 105 nonequivalent interactions, including 4 Heisenberg exchange terms and 10 spin-lattice coupling terms. Subsequently, DFT calculations were carried out to determine the interaction parameters. In these calcu- lations, we adopted a cubic cell containing 56 atoms and a Γ-centered 4 ×4×4 grid mesh in the reciprocal space. Besides, UMn= 3.3 eV and UFe= 3.6 eV were used as the effective Hubbard parameters [52]. With the exception of aforementioned settings, all the relevant first-principles calculations were performed under the same conditions as in Sec. III B. The DOS of MnFe 2O4is plotted in Fig. 7(c), yielding a calculated band gap of 0.612 eV. This value does not match with the result of transport experiments, which re- ported a much smaller band gap (0 .04–0.06 eV) [53]. In addition, MC and ASD simulations were performed using the Heisenberg exchange coefficients listed in Table III. The temperature dependence of average magnetization, shown in Fig. 7(d), suggests the critical temperature to be around 730 K. This result is significantly higher than the measured value of 573 K [54]. Both of the above dis- crepancies may be attributed to the inevitable difference between the ideal normal spinel structure in calculations and the partially disordered samples in reality. Despite this problem, we proceeded to describe the target system with our Hamiltonian model and expected to see how far the calculated results of damping constants would differ10 TABLE III. The exchange coefficients J of MnFe 2O4, where an effective spin S= 1 is adopted. Spin Pair. Distance (Angst) J (meV) 1NN Fe-Fe 3.003 6.835 1NN Mn-Fe 3.521 33.224 1NN Mn-Mn 3.667 3.956 2NN Fe-Fe 5.201 0.929 from experimental values. After the preparation of Hamiltonian model, we con- ducted dynamics simulations to evaluate the equivalent damping parameters in MFO at different temperatures. A supercell containing 13440 atoms was adopted in the simulation, and the results are summarized in Fig. 10. The average of calculated damping constants is around 8×10−5, which is much smaller than the measured value, 1.0×10−2[13, 51]. Two factors may account for this in- consistency. Firstly, the inhomogeneity in real MnFe 2O4 samples greatly enhances the scattering of magnons and phonons, thereby increasing the damping constants. Ad- ditionally, due to the narrow band gap observed in ex- periments, eddy currents can arise at finite temperatures, which leads to a rapid loss of energy in the form of joule heat. As the result of these factors, we failed to obtain a reasonable estimation of Gilbert damping constants for MnFe 2O4with our methodology. On the other side, the contribution of different relaxation mechanisms to FMR linewidth has been studied comprehensively for MnFe 2O4 in Ref. [53], which further confirms our analyses. D. Damping constants in Cr 2O3 Chromia (Cr 2O3) is a well-known collinear magneto- electric antiferromagnet, which holds great prospects in the field of spintronics [55–57]. As shown in Fig. 8(a), the primitive cell of Cr 2O3contains 10 atoms, with each chromium atom bonded to the six oxygen atoms around it. Additionally, Fig. 8(b) displays the magnetic ground state of Cr 2O3, where the spins of two nearest neighbor- ing Cr atoms are oriented in opposite directions. As a preliminary step in constructing the Hamiltonian model, we set the cutoff radius of interactions to be 11.0 Bohr for atomic pairs and 7.0 Bohr for 3-body clusters. Through symmetry analyses, we identified 319 nonequiv- alent interactions, including 5 Heisenberg exchange terms and 21 spin-lattice coupling terms. Afterwards, a series of first-principles calculations were performed to determine the model parameters. Following the settings in Ref. [58], we adopted a hexagonal cell of Cr2O3which contained a total of 90 atoms in the calcula- tions. Additionally, we used the LSDA+U method in its full spherically symmetric form [59]. As to the Hubbard parameters, Jwas fixed at its recommended value of 0.6 FIG. 8. (a) The primitive cell of Cr 2O3. The dark blue balls represent chromium atoms, and the red balls stand for oxygen atoms. (b) The magnetic ground state. The arrows of differ- ent colors represent the spin directions of Cr atoms. (c) The density of states obtained by DFT calculations. (d) The tem- perature dependence of sublattice magnetization measured in MC and ASD simulations. For Cr 2O3, the phase transition point from ferrimagnetic to paramagnetic lies in 310K approx- imately. TABLE IV. The exchange coefficients J of Cr 2O3, in which an effective spin S= 1 is adopted. Spin Pair. Distance (Angst) J (meV) 1NN Cr-Cr 2.640 44.778 2NN Cr-Cr 2.873 29.269 3NN Cr-Cr 3.411 -0.182 4NN Cr-Cr 3.635 0.007 5NN Cr-Cr 4.137 -0.500 eV, and Uwas adjusted to fit the N´ eel temperature ob- served in experiments [60]. We found U= 2.0 eV was the optimal value for 3 delectrons of Cr ions. Except for the settings specified above, all the DFT calculations were conducted under the same conditions as in Sec. III C. The DOS of Cr 2O3is plotted in Fig. 8(c), which yields a calculated band gap of 1.935 eV. This value indicates that the energy dissipation of electric currents can be ne- glected in this system. Additionally, we list the Heisen- berg exchange coefficients of chromia in Table IV. Both MC and ASD simulations were performed to investigate the temperature dependence of sublattice magnetization. According to Fig. 8(d), the critical point was determined to be 310 K approximately, which was quite consistent with experimental observations. Also, the force constants of Cr 2O3went through the modification formulated in Sec. II B, and the spin-lattice coupling parameters are provided in the supplementary material. After the construction of Hamiltonian model, we con- ducted a series of dynamics simulations to evaluate the11 FIG. 9. (a) The 1NN FeT-FeOpair in Y 3Fe5O12. (b) The 1NN Cr-Cr pair in Cr 2O3. The steel blue arrow stands for the orientation of ∂J/∂u and the red number along with it represents the magnitude in unit of meV/Angst. equivalent damping parameters in Cr 2O3. An expanded hexagonal cell containing 14400 atoms was adopted for the simulation, and the results are summarized in Fig. 11. As two specific cases, our calculation yielded α= (1.31± 0.14)×10−4at ∆T= 15 K and α= (2.7±0.3)×10−4 at ∆T= 30 K, where both TL= 0 K. Therefore, the calculated damping constants within ∆ T= 15∼30 K are quite close to 2 ×10−4, which is the estimated value reported in Ref. [61]. Furthermore, the damping constants in Cr 2O3exhibit a significant non-linear relation with the temperature dif- ference of subsystems. Through logarithmic fittings, we calculated the power exponents for Figures 11(a) to 11(c), and the results were 1.17, 1.62, 1.38. If we disregard the difference between ∆ TandTfor the moment, these val- ues are in good agreement with the theoretical prediction of Kasuya and LeCraw [26]. According to their study, the relaxation rate varies as Tnwhere n= 1∼2 while n= 2 corresponds to a larger regime of temperatures. Compared to YIG, the greater magnetic damping ob- served in chromia can be attributed to its significantly stronger spin-lattice coupling. As shown in Fig. 9, the magnitude of principal spin-lattice coupling constant in Cr2O3is two or three times larger than that in YIG. This could be explained by the fact that direct exchange in- teraction between two magnetic atoms decreases rapidlywith their distance [62]. Therefore, owing to the shorter distance of Cr-Cr pair, the direct exchange interaction between neighboring Cr atoms is believed to have a great contribution to the spin-lattice coupling in Cr 2O3. IV. CONCLUSIONS In summary, we propose a scheme to evaluate the con- tribution of spin-lattice coupling to the Gilbert damp- ing in insulating magnetic materials. Our methodology involves first-principles based Hamiltonian models and spin-lattice dynamics simulations. Following a series of validations, we applied our method to three magnetic ma- terials, namely Y 3Fe5O12, MnFe 2O4and Cr 2O3. Their damping constants were estimated separately, and the results show that, in general, αis approximately propor- tional to the temperature difference between spin and lattice subsystems. Under the condition of ∆ T= 30 K, the calculated damping constants are averaged to be 0.8×10−4for YIG, 0 .2×10−4for MFO and 2 .2×10−4 for Cr 2O3. The results for YIG and Cr 2O3are in good agreement with experimental measurements, while the discrepancy for MFO can be attributed to the inhomo- geneity and small band gap in real samples. Overall, the approach presented in this work holds great promise for accurately predicting the Gilbert damping constants for magnetic insulators. 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H. Kim, G. Jakob, M. Kl¨ aui, A. V. Chumak, B. Hillebrands, and C. A. Ross, Pulsed laser deposition of epitaxial yttrium iron garnet films with low gilbert damping and bulk-like magnetiza- tion, APL Materials 2, 106102 (2014). [49] C. Dubs, O. Surzhenko, R. Linke, A. Danilewsky, U. Br¨ uckner, and J. Dellith, Sub-micrometer yttrium iron garnet lpe films with low ferromagnetic resonance losses, Journal of Physics D: Applied Physics 50, 204005 (2017). [50] P. R¨ oschmann and W. Tolksdorf, Epitaxial growth and annealing control of fmr properties of thick homogeneous ga substituted yttrium iron garnet films, Materials Re- search Bulletin 18, 449 (1983). [51] J. Goodenough, W. Gr¨ aper, F. Holtzberg, D. Huber, R. Lefever, J. Longo, T. McGuire, and S. Methfessel, Magnetic and other properties of oxides and related com- pounds (Springer, 1970). [52] J.-R. Huang and C. Cheng, Cation and magnetic orders in mnfe2o4 from density functional calculations, Journal of Applied Physics 113, 033912 (2013). [53] A. G. Flores, V. Raposo, L. Torres, and J. I˜ niguez, Two- magnon processes and ferrimagnetic linewidth calcula- tion in manganese ferrite, Phys. Rev. B 59, 9447 (1999). [54] P. V. Reddy, K. Pratap, and T. S. Rao, Electrical con- ductivity of some mixed ferrites at curie point, Crystal Research and Technology 22, 977 (1987). [55] N. Hedrich, K. Wagner, O. V. Pylypovskyi, B. J. Shields, T. Kosub, D. D. Sheka, D. Makarov, and P. Maletinsky, Nanoscale mechanics of antiferromagnetic domain walls, Nature Physics 17, 574 (2021). [56] W.-Z. Xiao, Y.-W. Zhang, L.-L. Wang, and C.-P. Cheng, Newtype two-dimensional cr2o3 monolayer with half-metallicity, high curie temperature, and magnetic anisotropy, Journal of Magnetism and Magnetic Mate- rials543, 168657 (2022). [57] P. Makushko, T. Kosub, O. V. Pylypovskyi, N. Hedrich, J. Li, A. Pashkin, S. Avdoshenko, R. 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2023-09-20

Magnetic damping has a significant impact on the performance of various magnetic and spintronic devices, making it a long-standing focus of research. The strength of magnetic damping is usually quantified by the Gilbert damping constant in the Landau-Lifshitz-Gilbert equation. Here we propose a first-principles based approach to evaluate the Gilbert damping constant contributed by spin-lattice coupling in magnetic insulators. The approach involves effective Hamiltonian models and spin-lattice dynamics simulations. As a case study, we applied our method to Y$_3$Fe$_5$O$_{12}$, MnFe$_2$O$_4$ and Cr$_2$O$_3$. Their damping constants were calculated to be $0.8\times10^{-4}$, $0.2\times10^{-4}$, $2.2\times 10^{-4}$, respectively at a low temperature. The results for Y$_3$Fe$_5$O$_{12}$ and Cr$_2$O$_3$ are in good agreement with experimental measurements, while the discrepancy in MnFe$_2$O$_4$ can be attributed to the inhomogeneity and small band gap in real samples. The stronger damping observed in Cr$_2$O$_3$, compared to Y$_3$Fe$_5$O$_{12}$, essentially results from its stronger spin-lattice coupling. In addition, we confirmed a proportional relationship between damping constants and the temperature difference of subsystems, which had been reported in previous studies. These successful applications suggest that our approach serves as a promising candidate for estimating the Gilbert damping constant in magnetic insulators.

Evaluating Gilbert Damping in Magnetic Insulators from First Principles

2309.11152v1

Enhanc ed Gilbert Damping in Re doped FeCo Films – a combined experimental and theoretical study S. Akansel1, A. Kumar1, V.A.Venugopal2, R.Banerjee3, C. Autieri3, R.Brucas1, N. Behera1, M. A. Sortica3, D. Primetzhofer3, S. Basu2, M.A. Gubbins2, B. Sanyal3, and P. Svedlindh1 1Department of Engineering Sciences , Uppsala University, Box 534, SE -751 21 Uppsala, Sweden 2Seagate Technology, BT48 0BF, Londonderry, United Kingdom 3Department of Physics and Astronomy, Uppsala University, Box 516, SE -751 20 Uppsala, Sweden The effect s of rhenium doping in the range 0 – 10 at% on the static and dynamic magnetic properties of Fe65Co35 thin films have been studied experimentally as well as with first principles electronic structure calculations focussing on the change of the saturation magnetization (𝑀𝑠) and the Gilbert damping parameter ( 𝛼) Both experiment al and theoretical results show that 𝑀𝑠 decreases with increasing Re doping level, while at the same time 𝛼 increases. The experimental low temperature saturation magnetic induction exhibits a 2 9% decrease, from 2.3 1T to 1. 64T, in the investigated doping concentration range , which is more than predicted by the theoretical calculations. The room temperature value of the damping parameter obtained from ferromagnetic resonance measurements , correcting for extrinsic contributions to the damping, is for the undoped sample 2.7×10−3, which is close to the theoretically calculated Gilbert damping parameter . With 10 at% Re doping , the damping parameter increases to 9.0×10−3, which is in good agreement with the theoretical value of 7.3×10−3. The increase in damping parameter with Re doping is explained by the increase in density of states at Fermi level, mostly contributed by the s pin-up channel of Re. Moreover, both experimental and theoretical values for the da mping parameter are observed to be weakly decreas ing with decreasing temperature . 1. INTRODUCTION During the last decades , thin films of soft magnetic alloys such as NiFe and FeCo have been in focus due to possible use in applications such as spin valves ,1,2 magnetic tunneling junctions ,3,4,5 spin injectors ,6 magnetic storage technologies and in particular in magnetic recording write heads .7 Beside s spintronic and magnetic memory devices , such materials are useful for shielding applications that are necessary in order to reduce the effect of electromagnetic fields created by electronic devices. The magnetic damping parameter of the material play s a critical role for the performance of such spintronic and memory devices as well as for shielding applications. On the one hand, a low damping parameter is desired in order to get low critical switching current in spintronic devices .8,9,10 On the other hand , a high damping parameter is necessary in order to reduce the magetization switching time in magnetic memory devices and to be able to operate devices at high speeds .11 FeCo alloys are promising materials for high frequency spintronic applications and magnetic recording devices due to their high saturation magnetization (𝑀𝑠), high permeability, thermal stability and comparably high resistivity .12,13,14 One possible drawback is that FeCo alloy s exhibit high coercivity (𝐻𝑐), which is not favorable for such applications , however this problem can be solved by thin film growth on suitable buffer layer s.15,16,12 Except coercivity problems, the damping parameter of these materials should be increased to make them com patible for high speed devices . Dynamic properties of magnetic materials are highly dependent on the damping parameter. This parameter is composed of both intrinsic and extrinsic contributions. The intrinsic contribution is called the Gilbert damping and depends primarily on the spin-orbit coupling .17 Intrinsic damping is explained as scattering of electrons by phonons and magnons .18,19 Beside s electron scattering , due to the close relation between magnetocrystalline anisotropy and spin-orbit coupling , it can be assumed that the intrinsic damping is also related to the magnetocrystalline anisotropy constant .20 Regarding extrinsic damping , there can be a number of different contributions. The most common contribution originates from two magnon scattering (TMS) .21 However , this contribution vanishes when ferromagnetic resonance (FMR) measurements are performed by applying the static magnetic field along the film normal in inplane anisotropic thin films .22 Beside s TMS , there are some other extrinsic contributions to the damping that are not possible to get rid of by changing the measurement configuration . One of these contributions is radiative damping , which arises from inductive coupling between the precessing magnetization and the waveguide used for FMR measurem ents.23 Another contribution for metallic ferromagnetic films is the eddy current damping related to microwave magnetic field induced eddy currents in the thin film s during measurement s.23,24 In order to make a soft magnetic thin film suitible for a specific applica tion, taking into account requirements set by the device application , its damping paramete r should be tailored. As mentioned above , an increased damping parameter is necesssary for devices requiring high switching speed . Several efforts have been made for enhanching the damping parameter of soft magnetic materials. NiFe alloys constitu te one of the most studied systems in this respect . The most common way to enhance the intrinsic damping of an all oy is to dope it with differ ent elements . Rare earth elements with large spin-orbit coupling have revealed promising results as dopant s in terms of increas ed dampin g parameter .25,26,27 3d, 4d and 5d transition metals dopants have also been studied experimentally , revealing an increase of the damping parameter .28,29 Beside s experimental results , theoretical calculations support the idea that transition metals and especially 5d elements can enhance the damping parameter of NiF e alloys due to scattering in presence of chemical disorde r , as well as due to the effect of spin -orbit coupling .30 Although NiFe alloys have been the focus in several extensive studies, FeCo alloys have so far not been studied to the same extent . Attempts have been made to dope FeCo with Yb,20 Dy,31 Gd,32 and Si ,33 where in all cases an increase of the damping parameter was observed . Apart from doping of alloys , the addition of adjacent layers to NiFe and CoFe has also been studied . In particular , adding layers consisting of rare earth elem ents with large orbital moment s gave positive results in terms of increased damping parameter .34 Fe65Co35 alloy s are attractive material s because of high 𝑀𝑠 and reduced 𝐻𝑐 values. However , not much is known about the magnetic damping mechanism s for this composition . Since it is of interest for high data rate magnetic memory devices, the damping parameter should be increased in order to make the magnetic switching faster. To the best of our knowledge , systematic doping of Fe 65Co35 with 5d elements has not been studied so far experimentally . Some of us have found from ab initio calculations that 5d transition metal dopants can increase the damping parameter and Re is one of the potential candidates.35 Re is particularly interesting as it has a nice compromise of having not so much reduced saturation magnetization and a quite enh anced damping parameter. In this work, we have perfomed a systematic ab initio study of Fe65Co35 doped with increasing Re concentration to find an increasing damping parameter . The theoretical prediction s are confirmed by results obtained from temperature dependent FMR measurements performed on Re doped Fe65Co35 films. 2. EXPERIMENTAL AND THEORETIC AL METHOD S Rhenium doped Fe 65Co35 samples were prepared by varying the Re concentration from 0 to 10.23 at%. All samples were deposited using DC magnetron sputtering on Si/SiO 2 substrate s. First a 3 nm thick Ru seed layer was deposited on the Si/SiO 2 substrate followed by room temperature deposition of 20 nm and 40 nm thick Re -doped Fe65Co35 films by co -sputtering between Fe 65Co35 and Re target s. Finally, a 3 nm thick Ru layer was deposited as a capping layer over the Re -doped Fe65Co35 film. The nominal Re concentration was derived from the calibrated deposition rate used in the deposition system. The nominal Re doping concentration s of the Fe65Co35 samples are as follows ; 0, 2.62, 5.45 and 10.23 at%. The crystalline structure of the fims were investigated by utilizing grazing incident X -Ray diffraction (GIXRD). The i ncidence angle was fixed at 1o during GIXRD measurements and a CuKα source was used. Accurate values for film thickness and interface roughness were determined by X -ray reflectivity (XRR) measurements. Beside XRD , composition and areal density of the films were deduced by Rutherford backscattering spectrometry36 (RBS) with ion beams of 2 MeV 4He+ and 10 MeV 12C+. The beams were provided by a 5 MV 15SDH -2 tandem accelerator at the Tandem Laboratory at Uppsala University. The experiments were performed with the incident beam at 5° with respect to the surface normal and scattering angles of 170° and 120° . The experimental data was evaluated with the SIMNRA program .37 In-plane magnetic hysteresis measurments were performed using a Magnetic Property Measurement System (MPMS, Quantum Design) . Ferromagnetic resonance measurements were performed using two different techniques. First in- plane X -band (9.8 GHz) cavity FMR measurements were performed . The setup is equipped with a goniometer making it possible to rotate the sample with respect to the applied magnetic field; in this way the in -plane anisotropy fields of the different samples have been determine d. Beside s cavity FMR studies , a setup for broadband out-of-plane FMR measurements have been utilized . For out -of-plane measurements a vector network analyzer (VNA) was used. Two ports of the VNA were connected to a coplanar waveguide (CPW) mounted on a Ph ysical Property Measurement System (PPMS, Quantum Design) multi -function probe . The PPMS is equipped with a 9 T superconducting magnet, which is needed to saturate Fe65Co35 films out -plane and to detect the FMR signal. The broadband FMR measurements were carried out a t a fixed microwave frequency using the field -swept mode, repeating the measurement for different f requencies in the range 15 – 30GHz. The theoretical calculations are based on spin -polarized relativistic m ultiple scattering theory using the Korringa -Kohn -Rostoker (KKR) formalism implemented in the spin polarized relativistic KKR code (SPR-KKR) . The Perdew -Burke -Ernzerhof (PBE) exchange -correlation functional within generalized gradient approximation was used. The equilibrium lattice parameter s were obtained by energy minimization for each composition. Substitutional disorder was treated within the Coherent Potential Approximation (CPA). The damping parameters were calcu lated by the method proposed by Mankovsky et al.,38 based on the ab initio Green's function technique and linear res ponse formalism where one takes into consi deration scattering processes as well as spin - orbit coupling built in Dirac's relativistic formulation. The calculations of Gilbert damping parameters at finite temperatures were done using an alloy -analo gy model of atomic displacements corresponding to the thermal average of the root mean square displacement at a given temperature. 3. RESULTS AND DISCUSSION Re concentrations and layer thickness (areal densities) of the 20 nm doped films were obtained by RBS experiments. RBS employing a beam of 2 MeV He primary ions was used to deduce the areal concentration of each layer. Additional measurements with 10 MeV C probing particles permit to resolve the atomic fractions of Fe, Co and Re. The spectra for the samples with different Re concentration are shown in Fig. A1 . The measured Re concentrations are 3.0±0.1 at%, 6.6±0.3 at% and 12.6±0.5 at%. Moreover, the results for Fe and Co atomic fractions show that there is no preferential replacement by Re , implying that the two elements are replaced according to their respective concentration . Figure 1 (a) shows GIXRD spectra in the 2𝜃-range from 20o to 120o for the Fe65Co35 films with different Re concentration. Diffraction peaks corresponding to the body centered cubic Fe 65Co35 structure have been indexed in the figure; no other diffraction peaks appear in the different spectra. Depending on the Re -dopant concentration shi fts in the peak positions are observed, the diffraction peaks are suppressed to lower 2𝜃-values with increasing dopant concentration . The shift for the (110) peak for the different dopant concentrations is given as an inset in Fig. 1 (a). Similar shifts are observed for the other diffraction peaks. This trend in peak shift is an experimental evidence of an increasing amount of Re dopant within the deposited thin films. Since the peaks are shifted towards lower 2𝜃-values with increasing amount of Re dopant , the lattice parameter increases with increasing Re concentration.39 Figure 1 (b) shows the experimental as well as theoretically calculated lattice parameter versus Re concentration. The qualitative agreement between theory and experiment is obtained. However, t he rate of lattice parameter increase with increasing Re concentration is larger for the theoretically calculated lattice parameter. This is not unexpected as the generalized gradie nt approximation for the exchange -correlation potential has a tendency to overestimate the lattice parameter. Another possible explanation for the difference in lattice parameter is that the increase of the lattice parameter for the Re -doped Fe 65Co35 films is held back by the compressive strain due to lattice mismatch with Si/SiO 2/Ru. XRR measurements revealed that the surface roughness of the Fe 65Co35 films is less than 1 nm , which cannot affect static and magnetic properties drastically. Results from XRR measurements are given in table 1. Room temperature normalized magnetization curves for the Re-doped Fe 65Co35 films are shown in Fig. 2 (a) . The coercivity for all films is in the range of 2 mT and all films, except for the 1 2.6 at% Re doped film that show a slightly rounded hysteresis loop, exhibit rectangular hysteresis loops. The low value for the coercivity is expected for seed layer grown films .15 The experimentally determined low temperature saturation magnetization together with the theoretically calculated magnetization versus Re concentra tion are shown in Fig. 2 (b). As expected, both experimental and theoretical r esults show that the saturation magnetization decreases with increasing Re concentration . A linear decrease in magnetization is observed in the theoretical calculations whereas a non -linear behavior is seen in the experimental data. Angle resolved cavity FMR measurements were used to study the in -plane magnetic anisotropy . The angular -dependent resonant field ( 𝐻𝑟) data was analyzed using the following equation ,40 𝑓=µ0𝛾 2𝜋[{𝐻𝑟cos(𝜙𝐻−𝜙𝑀)+𝐻𝑐 2cos4(𝜙𝑀−𝜙𝐶)+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}{𝐻𝑟cos(𝜙𝐻− 𝜙𝑀)+𝑀𝑒𝑓𝑓+𝐻𝑐 8(3+cos4(𝜙𝑀−𝜙𝐶))+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}]12⁄ , (1) where 𝑓 is the cavity resonance frequency and 𝛾 is the gyromagnetic ratio . 𝜙𝐻, 𝜙𝑀, 𝜙𝑢 and 𝜙𝐶 are the in -plane directions for the magnetic field, magnetization, uniaxial anisotropy and cubic anisotropy, respectively, with respect to the [100 ] direction of the Si substrate. 𝐻𝑢=2𝐾𝑢 µ0𝑀𝑠 and 𝐻𝑐=4𝐾𝑐 µ0𝑀𝑠 are the uniaxial and cubic anisotropy fields, where 𝐾𝑢 and 𝐾𝑐 are the uniaxial and cubic magnetic anisotropy constants , and 𝑀𝑒𝑓𝑓 is the effective magnetization. Fitting parameters were limited to 𝑀𝑒𝑓𝑓, 𝛾 and 𝐻𝑢, since the Hr versus ϕH curves did not give any indication of a cubic anisotropy. Figure 3 shows 𝐻𝑟 versus 𝜙𝐻 extracted from the angular -dependent FMR measurements together with fits according to Eq. (1), clearly revealing dominant twofold uniaxial in -plane magnetic anisotropy. Extracted anisotropy field and effective magnetization values are given in Table 2 . The results show that 𝐻𝑢 is within the accuracy of the experiment independent of the Re concentration . Temperature dependent o ut-of-plane FMR measurements were performed in the temperature range 50 K to 300 K recording the complex transmission coefficient 𝑆21. Typical field -swept results for the r eal and imaginary components of 𝑆21 for the undoped and 1 2.6 at% Re-doped samples are shown in Fig. 4. The field -dependent 𝑆21 data was fitted to the following set of equations,41 𝑆21(𝐻,𝑡)=𝑆210+𝐷𝑡+𝜒(𝐻) 𝜒̃0 𝜒(𝐻)=𝑀𝑒𝑓𝑓(𝐻−𝑀𝑒𝑓𝑓) (𝐻−𝑀𝑒𝑓𝑓)2−𝐻𝑒𝑓𝑓2−𝑖𝛥𝐻 (𝐻−𝑀𝑒𝑓𝑓) . (2) In these equations 𝑆210 corresponds to the non-magnetic contribution to the complex transmission signal , 𝜒̃0 is an imaginary function of the microwave frequency and film thickness and 𝜒(𝐻) is the complex susceptibility of the magnetic film. The term 𝐷𝑡 accounts for a linear drift of the recorded 𝑆21 signal. 𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠, where 𝐻𝑘⫠ is the perpendicular anisotropy field and 𝐻𝑒𝑓𝑓=2𝑓 𝛾µ0. The 𝑆21 spectra were fitted to Eq. (2 ) in order to extract the linewidth 𝛥𝐻 and 𝐻𝑟 values. Fits t o Eq. (2) are shown as solid lines in Fig. 4. The experimentally measured total d amping parameter ( 𝛼𝑡𝑜𝑡𝑎𝑙 ), including both the intrinsic contribution (Gilbert damping) and extrinsic contributions , was extracted by fitting 𝛥𝐻 versus frequency to the following expression, 41 µ0𝛥𝐻=4𝛼𝑡𝑜𝑡𝑎𝑙 𝑓 𝛾+µ0𝛥𝐻0 , (3) where 𝛥𝐻0 is the frequency independent linewidth broadening due to sample inhomogeneity . Beside s 𝛼𝑡𝑜𝑡𝑎𝑙 , 𝑀𝑒𝑓𝑓 can also be extracted by fitting the 𝐻𝑟 versus frequency results to the expression µ0𝐻𝑟=2𝜋𝑓 𝛾+µ0𝑀𝑒𝑓𝑓 . (4) Typical temperature dependent results for 𝑓 versus 𝐻𝑟 and 𝛥𝐻 versus 𝑓 are shown in Fig. 5 for the 1 2.6 at% Re -doped Fe65Co35 film. Extracted values of 𝑀𝑒𝑓𝑓 at different temperatures are given in Table 3 for all samples . As expected, the results show that 𝑀𝑒𝑓𝑓 decreas es with increasing dopant concentration. Since 𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠ and the film thickness is large enough to make a possible contribution from out -of-plane anisotropy negligible one can make the justified assumption that 𝑀𝑒𝑓𝑓≈𝑀𝑠. The analysis using Eqs. (2) – (4) also give values for the Land é 𝑔- factor ( 𝛾=𝑔µ𝐵 ħ), yielding 2.064 and 2.075 for the undoped and 12.6 at% doped samples, respectively (similar values are obtained at all temperatures). As indicated above, the d amping parameters extracte d from FMR measurements ( 𝛼𝑡𝑜𝑡𝑎𝑙) include both intrinsic and extrinsic contributions. One of the most common extrinsic contribution s is TMS , which is avoided in this study by measuring FMR with the magnetic field applied out of the film plane. Except TMS , extrinsic contributions such as eddy curr ent damping and radiative damping are expected to contribute the measured damping . In a metallic ferromagnet, which is placed on top of a CPW , precession of spin waves induces AC currents in the ferromagnet ic film, thereby dissipating energy . The radiative damping has similar origin as the eddy current damping, but here the precession of the magnetization induces microwave -frequency currents in the CPW where energy is dissipated. Thus, there are two extrinsic contributions to the measured damping ; one that is caused by eddy currents in the ferromagnet ic film (𝛼𝑒𝑑𝑑𝑦) and another one caused by eddy currents in the CPW ( 𝛼𝑟𝑎𝑑).23 In order to obtain the reduced damping of the films (𝛼𝑟𝑒𝑑), which we expect to be close to the intrinsic damping of the films, the extrinsic contributions should be subtracted from 𝛼𝑡𝑜𝑡. We have neglected any contribution to the measured damping originating from spin -pumping into seed and capping layers. However, since spin -pumping in low spin -orbit coupling materials like Ru with thickness quite less than the spin -diffusion length is quit e small, the assumption of negligible contribution from spin -pumping is justified. The t otal damping can thus be given as 𝛼𝑡𝑜𝑡=𝛼𝑟𝑒𝑑+𝛼𝑟𝑎𝑑+𝛼𝑒𝑑𝑑𝑦 . When the precession of the magnetization is assumed to be uniform in the sample , the expression for radiative damping can be given as23 𝛼𝑟𝑎𝑑=𝜂𝛾µ02𝑀𝑠𝛿𝑙 2𝑍0𝑤 , (5) where 𝑍0 =50 Ω is the waveguide impedance, 𝑤=240 µm is the width of the CPW center conductor , 𝜂 is a dimensionless parameter that accounts for FMR mode profile, δ is the thickness and 𝑙 is the length of the sample. The l ength of all samples were 4mm and the thickness 20nm for the undoped and 1 2.6 at% Re-doped films and 40nm for the 3.0 at% and 6.6 at% Re-doped films. Temperature dependent radiative damping contributions for all Fe 65Co35 films are given in Table 4. Beside s 𝛼𝑟𝑎𝑑, the 𝛼𝑒𝑑𝑑𝑦 contribution should also be calculated and extracted from 𝛼𝑡𝑜𝑡𝑎𝑙 to extract the reduced damping parameter. 𝛼𝑒𝑑𝑑𝑦 can be estimated by the expression23 𝛼𝑒𝑑𝑑𝑦 =𝐶𝛾µ02𝑀𝑠𝛿2 16𝜌 , (6) where 𝐶 is a parameter describing the distribution of eddy current s within the films and its value is 0.5 in our studied samples and 𝜌 is the resistivity of the films. Resistivity is measured for all films with different dopant concentrations at different temperatures. It is in the range of 8.2×10-8 to 5.6 ×10-8 𝛺𝑚 for undoped, 5.7 ×10-7 to 5.3 ×10-7 𝛺𝑚 for 3.0 at% doped , 6.9 ×10-7 to 6.1 ×10- 7 𝛺𝑚 for 6.6 at% doped and 3.9×10-7 to 3.6 ×10-7 𝛺𝑚 for 12.6 at% doped films. Temperature dependent eddy current damping contributions , which are negligible, for all Fe 65Co35 films are given in Table 5. 𝛼𝑡𝑜𝑡 (filled symbols) and 𝛼𝑟𝑒𝑑 (open symbols) versus temperature for the differently Re -doped Fe65Co35 films are shown in Fig. 6 . Both damping parameter s slowly decrease with decreasing tempera ture. Moreover, the damping parameter increases with increasing Re concentration; the damping parameter is 4 times as large for the 12.6 at% Re -doped sample compared to the undoped sample . Since the damping parameter depends both on disorder induced scattering and spin-orbit coupling, the observed enhanc ement of the damping parameter can emerge from the electronic structure of the alloy and large spin -orbit coupling of Re. A c omparison between temperature dependent experimental 𝛼𝑡𝑜𝑡 and 𝛼𝑟𝑒𝑑 values and theoretically calculated intrinsic damping parameters is shown in Fig. 7 for the undoped and 12.6 at% Re -doped Fe 65Co35 films. In agreement with the experimental results, the theoretically calculated damping parameters decrease in magnitude with decreasing temperature . It has been argued by Schoen et al., 42 that the contribution to the intrinsic Gilbert damping parameter comes primarily from the strong electron -phonon coupling at high temperatures due to interband transition whereas at a low temperature, density of states at Fermi level (𝑛(𝐸𝐹)) and spin -orbit coupling give rise to intraband transition. In Fig. 8, we show the correspondence between the calculated damping parameter at 10 K with the density of states (spin up +spin down) at Fermi level for varying Re concentration. The increasing trend in both properties is obviously seen. The increase in DOS mainly comes from increasing DOS at Re sites in the spin -up channel. In the inset, the calculated spin -polarization as a function of Re concentration is shown. Spin polarization is defined as 𝜁=𝑛(𝐸𝐹)↑−𝑛(𝐸𝐹)↓ 𝑛(𝐸𝐹)↑+𝑛(𝐸𝐹)↓ where the contribution from both spin channels are seen. It is clearly observed that Re doping decreases the spin polarization. One should note that a quantitative comparison between theory and experiment requires more rigoro us theoretical considerations. The difference between experimental and theoretical results for the damping parameter may be explained by the incompleteness of the model used to calculate the Gilbert damping parameter by neglecting several complex scatterin g processes. Firstly, the effect of spin fluctuations was neglected, which in principle could be considered in the present methodology if the temperature dependent magnetization and hence information about the fluctuations of atomic moments were available from Monte -Carlo simulations. Other effects such as non-local damping and more sophisticated treatment of atomic displac ements in terms of phonon self -energies40 that may contribute to the relaxation of the magnetization in magnetic thin film materials have been neglected . Nevertheless, a qualitative agreement has been achieved where both experimental and theoretical results show that there is a significant increase of the damping parameter with increasing concentration of Re. 4. CONCLUSION Static and dynamic magnetic properties of rhenium doped Fe 65Co35 thin films have been investigated and clarified in a combined experimental and theoretical study. Results from first principles theoretical calculations show that the saturation magnetization gradually decreases with increasing Re concentration, from 2.3T for the undoped sample to 1.95T for the 10% Re -doped sample. The experimental results for the dependence of the saturation magnetization on the Re - doping are in agreement with the theoretical results, although indicating a more pronounced decrease of the saturation magnetization for the largest doping concentrations. The theoretical calculations show that the intrinsic Gilbert damping increases with increasing Re concentration; at room temperature the damping parameter is 2.8×10−3, which increases to 7.3×10−3 for the 10 at% Re -doped sample. Moreover, temperature dependent calculations of the Gilbert damping parameter reveal a weak decrease of the value with decreasing temperature . At a low temperature, our theoretical analysis showed the prominence of intra band contribution arising from an increase in the density of states at Fermi level. The experimental results for the damping parameter were corrected for radiative and eddy current contributions to the measured damping parameter and reveal similar trends as observed in the theoretical results; the damping parameter increases with increasing Re concentration and the damping parameter value decreases with decreasing temperature. The room temperature value for the reduced damping paramet er was 2.7×10−3 for the undoped sample, which increased to 9.0×10−3 for the 1 2.6 at% Re -doped film. The possibility to e nhanc e the damping parameter for Fe65Co35 thin films is a promising result since these materials are used in magnetic memory applications and higher data rates are achievable if the damping parameter of the material is increased. ACKNOWLEDGEMENT This work is supported by the Knut and Alice Wallenberg (KAW) Fou ndation, Grant No. KAW 2012.0031 and by the Marie Curie Action “Industry -Academia Partnership and Pathways” (ref. 612170, FP7 -PEOPLE -2013 -IAPP). The authors acknowledge financial support from Swedish Research Council (grant no. 2016 -05366) and Carl Tryggers Stiftelse (grant no. CTS 12:419 and 13:413). The simulations were performed on resources provided by the Swedish National Infrastructure (SNIC) at National Supercomputer Centre at Link öping University (NSC). M. 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Sanyal, unpublished. 36 W. K. Chu, J. W. Mayer, and M. -A. Nicolet, Backscattering Spectrometry, Academic Press , Orlando , 1978 . 37 M. Mayer, SIMNRA User's Guide , Report IPP 9/113, Max -Planck -Institut für Plasmaphysik, Garching, Germany (1997) 38 S. Mankovsky, D. Ködderitzsch, G. Woltersdorf, and H. Ebert, Phys. Rev. B 87, 014430 (2013). 39 W. Wang, Y. Chen, G.H. Yue, K. Sumiyama, T. Hihara, and D.L. Peng, J. Appl. Phys. 106, 013912 (2009). 40 A. Kumar, F. Pan, S. Husain, S. Akansel, R. Brucas, L. Bergqvist, S. Chaudhary, and P. Svedlindh, Phys. Rev. B 96, 224425 (2017). 41 H.T. Nembach, T.J. Silva, J.M. Shaw, M.L. Schneider, M.J. Carey, S. Maat, and J.R. Childress, Phys. Rev. B 84, 054424 (2011). 42 M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J. Silva, H. T. Nembach, O. Eriksson, O. Karis, and J. M. Shaw, Nature Physics, 12, 839 –842 (2016). Figure 1 (a) GIXRD plot for Fe 65Co35 films with dif ferent Re concentrations. S hift of (110) peak diffraction peak with Re concentration is given as insert . (b) Lattice parameter versus Re concentration. Circles are lattice parameters extracted from XRD measurements and squares are calculated th eoretical values. Line s are guide to the eye. Figure 2 (a) Normalized room temperature magnetization versus magnetic field for Fe 65Co35 films with different Re concentration . (b) Low temperature saturation magnetization versus Re concentration. Circles are experimental data and squares corresponding calculated results. Experimental 𝝁𝟎𝑴𝒔 values were extracted from temperature dependent FMR results. Lines are guide s to the eye . Figure 3 𝝁𝟎𝑯𝒓 versus in -plane angle of magnetic field 𝝓𝑯 for different dopant concentrations of Re. Black squares are experimental data and red line s are fits to Eq. (1). Figure 4 Room temperature real (a and c) and imaginary (b and d) 𝑺𝟐𝟏 components versus out - of-plane magnetic field for Fe65Co35 thin films with 0% and 12.6 at% Re recorded at 20GHz . Black squares are data points and red lines are fit s to Eq. (2). Figure 5 (a) Frequency versus 𝝁𝟎𝑯𝒓 values at different temperatures for the Fe65Co35 thin film with 12.6 at% Re. Coloured lines correspond to fits to Eq. ( 4). (b) Linewidth 𝝁𝟎∆𝑯 versus frequency at different temperatures for the same Re doping concentration. Coloured lines correspond to fits to Eq. ( 3). Symbols represent experimental data. Figur e 6 𝜶𝒕𝒐𝒕 versus temperature for Fe 65Co35 thin films with different concentration of Re. Besides showing 𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of radiative damping and eddy current damping contributions from 𝜶𝒕𝒐𝒕. Error bars are given for measured 𝜶𝒕𝒐𝒕 (same size as symbol size ). Figur e 7 𝜶𝒕𝒐𝒕 versus temperature for Fe 65Co35 thin films with 0 at% and 1 2.6 at% concentration of Re. Beside s showing 𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of radiative damping and eddy current damping contribution s from 𝜶𝒕𝒐𝒕. In addition to experimental results theoretically calculated intrinsic damping parameters are given for the similar concentrations of Re . Error bars are given for measured 𝜶𝒕𝒐𝒕 (same size as symbol size) . Figure 8 Calculated density of states at Fermi level (left axis) and damping parameter (right axis) are shown as a function of Re concentration. In the inset, spin -polarization is shown as a function of Re concentration. 0 0.03 0.06 0.09 0.12 Re concentration0.90.951DOS at EF (States/eV) 0 0.03 0.06 0.09 0.12 Re concentration0.350.40.450.50.55Spin polarization 0123456 Damping parameter (x 10-3)Re (at%) 𝑡𝑅𝑢,𝑐𝑎𝑝 (nm) 𝜎 (nm) 𝑡𝐹𝑒𝐶𝑜 (nm) 𝜎 (nm) 𝑡𝑅𝑢,𝑠𝑒𝑒𝑑 (nm) (nm) 0 2.46 1.89 39.71 0.67 2.74 0.66 3.0 2.47 1.80 37.47 0.59 2.45 1.03 6.6 1.85 0.50 37.47 0.51 2.13 0.90 12.6 2.15 1.49 37.38 0.64 1.89 1.03 Table 1 Thickness and roughness (𝝈) values for different layers in films extracted from XRR data. Error margin is 0.02nm for all thickness and roughness values. Re (at%) 𝜇0𝐻𝑢 (mT) 𝜇0𝑀𝑒𝑓𝑓 (T) 0 2.20 2.31 3.0 2.10 2.12 6.6 2.30 1.95 12.6 2.20 1.64 Table 2 Room temperature 𝝁𝟎𝑴𝒆𝒇𝒇 and 𝝁𝟎𝑯𝒖 values for Fe 65Co35 films with different concentration of Re extracted by fitting the angle dependent cavity FMR data to Eq. (1). Temperature (K) 0% Re 3.0 at% Re 6.6 at% Re 12.6 at% Re 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 300 2.29 2.16 1.99 1.61 200 2.31 2.16 2.04 1.67 150 2.33 2.24 2.06 1.70 100 2.36 2.25 2.07 1.72 50 2.36 2.27 2.08 1.74 Table 3 Temperature dependent 𝝁𝟎𝑴𝒆𝒇𝒇 values for Fe65Co35 films with different concentrati on of Re extracted by fitting broadband out -of-plane FMR data to Eq. (4). Error margin is about 10 mT. Temperature(K) 𝛼𝑟𝑎𝑑 (×10-3) 0% Re 3.0 at% Re 6.6 at% Re 12.6 at% Re 300 0.218 0.482 0.438 0.154 200 0.222 0.494 0.450 0.160 150 0.216 0.499 0.454 0.162 100 0.225 0.502 0.456 0.219 50 0.221 0.505 0.458 0.166 Table 4 Temperature dependent r adiative damping contribution to total damping parameter for Fe65Co35 films with different concentration of Re calculated using Eq. (5). Temperature(K) 𝛼𝑒𝑑𝑑𝑦 (×10-3) 0% Re 3.3 at% Re 6.6 at% Re 12.6 at% Re 300 0.038 0.077 0.064 0.006 200 0.047 0.081 0.067 0.006 150 0.050 0.084 0.070 0.006 100 0.055 0.084 0.073 0.007 50 0.058 0.086 0.075 0.007 Table 5 Temperature dependent eddy current damping contribution to total damping parameter for Fe 65Co35 films with different concentration of Re calculated using Eq. ( 6). Figure A1 RBS spectra for the Re -doped Fe 65Co35 films.

2019-02-26

The effects of rhenium doping in the range 0 to 10 atomic percent on the static and dynamic magnetic properties of Fe65Co35 thin films have been studied experimentally as well as with first principles electronic structure calculations focusing on the change of the saturation magnetization and the Gilbert damping parameter. Both experimental and theoretical results show that the saturation magnetization decreases with increasing Re doping level, while at the same time Gilbert damping parameter increases. The experimental low temperature saturation magnetic induction exhibits a 29 percent decrease, from 2.31 T to 1.64 T, in the investigated doping concentration range, which is more than predicted by the theoretical calculations. The room temperature value of the damping parameter obtained from ferromagnetic resonance measurements, correcting for extrinsic contributions to the damping, is for the undoped sample 0.0027, which is close to the theoretically calculated Gilbert damping parameter. With 10 atomic percent Re doping, the damping parameter increases to 0.0090, which is in good agreement with the theoretical value of 0.0073. The increase in damping parameter with Re doping is explained by the increase in density of states at Fermi level, mostly contributed by the spin-up channel of Re. Moreover, both experimental and theoretical values for the damping parameter are observed to be weakly decreasing with decreasing temperature.

Enhanced Gilbert Damping in Re doped FeCo Films: A Combined Experimental and Theoretical Study

1902.09896v1

arXiv:1408.3499v1 [math.AP] 15 Aug 2014Linear hyperbolic equations with time-dependent propagation speed and strong damping Marina Ghisi Universit` a degli Studi di Pisa Dipartimento di Matematica PISA (Italy) e-mail:ghisi@dm.unipi.itMassimo Gobbino Universit` a degli Studi di Pisa Dipartimento di Matematica PISA (Italy) e-mail:m.gobbino@dma.unipi.itAbstract We consider a second order linear equation with a time-dependent co efficientc(t) in front of the “elastic” operator. For these equations it is well-know n that a higher space- regularity of initial data compensates a lower time-regularity of c(t). In this paper we investigate the influence of a strong dissipation, na mely a friction term which depends on a power of the elastic operator. What we discover is a threshold effect. When the exponent of the ela stic operator in the friction term is greater than 1/2, the damping prevails and the equation behaves as if the coefficient c(t) were constant. When the exponent is less than 1/2, the time- regularity of c(t) comes into play. If c(t) is regular enough, once again the damping prevails. On the contrary, when c(t) is not regular enough the damping might be ineffective, and there are examples in which the dissipative equation b ehaves as the non-dissipative one. As expected, the stronger is the damping, th e lower is the time- regularity threshold. We also provide counterexamples showing the optimality of our result s. Mathematics Subject Classification 2010 (MSC2010): 35L20, 35L80, 35L90. Key words: linear hyperbolic equation, dissipative hyperbolic equation, strong d amp- ing, fractional damping, time-dependent coefficients, well-posedn ess, Gevrey spaces.1 Introduction LetHbe a separable real Hilbert space. For every xandyinH,|x|denotes the norm ofx, and/a\}⌊ra⌋ketle{tx,y/a\}⌊ra⌋ketri}htdenotes the scalar product of xandy. LetAbe a self-adjoint linear operator on Hwith dense domain D(A). We assume that Ais nonnegative, namely /a\}⌊ra⌋ketle{tAx,x/a\}⌊ra⌋ketri}ht ≥0 for every x∈D(A), so that for every α≥0 the power Aαxis defined provided that xlies in a suitable domain D(Aα). We consider the second order linear evolution equation u′′(t)+2δAσu′(t)+c(t)Au(t) = 0, (1.1) with initial data u(0) =u0, u′(0) =u1. (1.2) As far as we know, this equation has been considered in the literatur e either in the case where δ= 0, or in the case where δ >0 but the coefficient c(t) is constant. Let us give a brief outline of the previous literature which is closely related to our results. The non-dissipative case Whenδ= 0, equation (1.1) reduces to u′′(t)+c(t)Au(t) = 0. (1.3) This is the abstract setting of a wave equation in which c(t) represents the square of the propagation speed. If the coefficient c(t) is Lipschitz continuous and satisfies the strict hyperbolicity condition 0<µ1≤c(t)≤µ2, (1.4) then it is well-know that problem (1.3)–(1.2) is well-posed in the classic e nergy space D(A1/2)×H(see for example the classic reference [14]). If the coefficient is not Lipschitz continuous, things are more comple x, even if (1.4) still holds true. This problem was addressed by F. Colombini, E. De Gior gi and S. Spag- nolo in the seminal paper [6]. Their results can be summed up as follows ( we refer to section 2 below for the precise functional setting and rigorous sta tements). (1) Problem (1.3)–(1.2) has always a unique solution, up to admitting t hat this solu- tion takes its values in a very large Hilbert space (ultradistributions) . This is true for initial data in the energy space D(A1/2)×H, but also for less regular data, such as distributions or ultradistributions. (2) If initial data are regular enough, then the solution is regular as well. How much regularity is required depends on the time-regularity of c(t). Classic examples are the following. If c(t) is just measurable, problem (1.3)–(1.2) is well-posed in the class of analytic functions. If c(t) isα-H¨ older continuous for some α∈(0,1), problem (1.3)–(1.2) is well-posed in the Gevrey space of order (1 −α)−1. 1(3) If initial data are not regular enough, then the solution may exh ibit a severe derivative loss for all positive times. For example, for every α∈(0,1) there exist a coefficientc(t) which isα-H¨ older continuous, and initial data ( u0,u1) which are in the Gevrey class of order βfor everyβ >(1−α)−1, such that the corresponding solution to (1.3)–(1.2) (which exists in the weak sense of point (1)) is not even a distribution for every t>0. In the sequel we call (DGCS)-phenomenon the instantaneous loss of regularity de- scribed in point (3) above. The dissipative case with constant coefficients Ifδ >0 andc(t) is a constant function (equal to 1 without loss of generality), equation (1.1) reduces to u′′(t)+2δAσu′(t)+Au(t) = 0. (1.5) Mathematical models with damping terms of this form were proposed in [1], and then rigorously analyzed by many authors from different points of v iew. The first papers [2, 3, 4], and the more recent [10], are devoted to analyticity properties of the semigroup associated to (1.5). The classic assumptions in these pap ers are that the operatorAis strictly positive, σ∈[0,1], and the phase space is D(A1/2)×H. On a different side, the community working on dispersive equations consid ered equation (1.5) intheconcretecasewhere σ∈[0,1]andAu=−∆uinRnorspecialclassesofunbounded domains. They proved energy decay and dispersive estimates, but exploiting in an essential way the spectral properties of the Laplacian in those do mains. The interested reader is referred to [11, 12, 13, 19] and to the references quot ed therein. Finally, equation (1.5) was considered in [9] in full generality, namely fo r every σ≥0 and every nonnegative self-adjoint operator A. Two different regimes appeared. In the subcritical regime σ∈[0,1/2], problem (1.5)–(1.2) is well-posed in the classic energy space D(A1/2)×Hor more generally in D(Aα+1/2)×D(Aα) withα≥0. In the supercritical regime σ≥1/2, problem (1.5)–(1.2) is well-posed in D(Aα)×D(Aβ) if and only if 1−σ≤α−β≤σ. (1.6) This means that in the supercritical regime different choices of the p hase space are possible, even with α−β/\e}atio\slash= 1/2. The dissipative case with time-dependent coefficients As far as we know, the case of a dissipative equation with a time-dependent propagation speed had n ot been considered yet. The main question we address in this paper is the extent to which the dissipative term added in (1.1) prevents the (DGCS)-phenomenon of (1.3) fro m happening. We discover a composite picture, depending on σ. •In the subcritical regime σ∈[0,1/2], if the strict hyperbolicity assumption (1.4) is satisfied, well-posedness results do depend on the time-regularit y ofc(t) (see Theorem 3.2). Classic examples are the following. 2–Ifc(t) isα-H¨ older continuous for some exponent α >1−2σ, then the dis- sipation prevails, and problem (1.1)–(1.2) is well-posed in the classic en ergy spaceD(A1/2)×Hor more generally in D(Aβ+1/2)×D(Aβ) withβ≥0. –Ifc(t) is no more than α-H¨ older continuous for some exponent α <1−2σ, thenthedissipationcanbeneglected, sothat(1.1)behavesexact lyasthenon- dissipative equation (1.3). This means well-posedness in the Gevrey s pace of order (1−α)−1and the possibility to produce the (DGCS)-phenomenon for less regular data (see Theorem 3.10). –The case with α= 1−2σis critical and also the size of the H¨ older constant ofc(t) compared with δcomes into play. •In the supercritical regime σ >1/2 the dissipation prevails in an overwhelming way. In Theorem 3.1 we prove that, if c(t) is just measurable and satisfies just the degenerate hyperbolicity condition 0≤c(t)≤µ2, (1.7) then (1.1) behaves as (1.5). This means that problem (1.1)–(1.2) is w ell-posed in D(Aα)×D(Aβ) if and only if (1.6) is satisfied, the same result obtained in [9] in the case of a constant coefficient. The second issue we address in this paper is the further space-reg ularity of solutions for positive times, since a strong dissipation is expected to have a re gularizing effect similar to parabolic equations. This turns out to be true provided tha t the assumptions of our well-posedness results are satisfied, and in addition σ∈(0,1). Indeed, we prove that in this regime u(t) lies in the Gevrey space of order (2min {σ,1−σ})−1for every t>0. We refer to Theorem 3.8 and Theorem 3.9 for the details. This effec t had already been observed in [15] in the dispersive case. We point out that the regularizing effect is maximum when σ= 1/2 (the only case in which solutions become analytic with respect to space variables) and disappears when σ≥1, meaning that a stronger overdamping prevents smoothing. Overview of the technique The spectral theory reduces the problem to an analysis of the family of ordinary differential equations u′′ λ(t)+2δλ2σu′ λ(t)+λ2c(t)uλ(t) = 0. (1.8) Whenδ= 0, a coefficient c(t) which oscillates with a suitable period can produce a resonance effect so that (1.8) admits a solution whose oscillations h ave an amplitude which grows exponentially with time. This is the primordial origin of the ( DGCS)- phenomenon for non-dissipative equations. When δ >0, the damping term causes an exponential decay of the amplitude of oscillations. The competition between the 3exponential energy growth due to resonance and the exponent ial energy decay due to dissipation originates the threshold effect we observed. Whenc(t) is constant, equation (1.8) can be explicitly integrated, and the ex plicit formulae for solutions led to the sharp results of [9]. Here we need th e same sharp estimates, but without relying on explicit solutions. To this end, we int roduce suitable energy estimates. In the supercritical regime σ≥1/2 we exploit the following σ-adapted “Kovaleskyan energy” E(t) :=|u′ λ(t)+δλ2σuλ(t)|2+δ2λ4σ|uλ(t)|2. (1.9) In the subcritical regime σ≤1/2 we exploit the so-called “approximated hyperbolic energies” Eε(t) :=|u′ λ(t)+δλ2σuλ(t)|2+δ2λ4σ|uλ(t)|2+λ2cε(t)|uλ(t)|2,(1.10) obtained by adding to (1.9) an “hyperbolic term” depending on a suita ble smooth ap- proximation cε(t) ofc(t), which in turn is chosen in a λ-dependent way. Terms of this type are the key tool introduced in [6] for the non-dissipative equa tion. Future extensions We hope that this paper could represent a first step in the theory of dissipative hyperbolic equations with variable coefficients, both line ar and nonlinear. Next steps could be considering a coefficient c(x,t) depending both on time and space variables, and finally quasilinear equations. This could lead to improve t he classic results by K. Nishihara [16, 17] for Kirchhoff equations, whose linear ization has a time- dependent coefficient, and finally to consider more general local no nlinearities, in which case the linearization involves a coefficient c(x,t) depending on both variables. Inadifferent direction, thesubcritical case σ∈[0,1/2]withdegeneratehyperbolicity assumptions remains open and could be the subject of further res earch, in the same way as [7] was the follow-up of [6]. On the other side, we hope that our counterexamples could finally dis pel the dif- fuse misconception according to which dissipative hyperbolic equatio ns are more stable, and hence definitely easier to handle. Now we know that a friction ter m below a suit- able threshold is substantially ineffective, opening the door to patho logies such as the (DGCS)-phenomenon, exactly as in the non-dissipative case. Structure of the paper This paper is organized as follows. In section 2 we introduce the functional setting and we recall the classic existence results f rom [6]. In section 3 we state our main results. In section 4 we provide a heuristic descriptio n of the competition between resonance and decay. In section 5 we prove our existenc e and regularity results. In section 6 we present our examples of (DGCS)-phenomenon. 42 Notation and previous results Functional spaces LetHbe a separable Hilbert space. Let us assume that Hadmits a countable complete orthonormal system {ek}k∈Nmade by eigenvectors of A. We denote the corresponding eigenvalues by λ2 k(with the agreement that λk≥0), so that Aek=λ2 kekfor everyk∈N. In this case every u∈Hcan be written in a unique way in the form u=/summationtext∞ k=0ukek, whereuk=/a\}⌊ra⌋ketle{tu,ek/a\}⌊ra⌋ketri}htare the Fourier components of u. In other words, the Hilbert space Hcan be identified with the set of sequences {uk}of real numbers such that/summationtext∞ k=0u2 k<+∞. We stress that this is just a simplifying assumption, with substantially no loss of generality. Indeed, according to the spectral theorem in its gene ral form (see for ex- ample Theorem VIII.4 in [18]), one can always identify HwithL2(M,µ) for a suitable measure space ( M,µ), in such a way that under this identification the operator Aacts as a multiplication operator by some measurable function λ2(ξ). All definitions and statements in the sequel, with the exception of the counterexamp les of Theorem 3.10, can be easily extended to the general setting just by replacing the sequence {λ2 k}with the function λ2(ξ), and the sequence {uk}of Fourier components of uwith the element /hatwideu(ξ) ofL2(M,µ) corresponding to uunder the identification of HwithL2(M,µ). The usual functional spaces can be characterized in terms of Fou rier components as follows. Definition 2.1. Letube a sequence {uk}of real numbers. •Sobolev spaces . For every α≥0 it turns out that u∈D(Aα) if /⌊ard⌊lu/⌊ard⌊l2 D(Aα):=∞/summationdisplay k=0(1+λk)4αu2 k<+∞. (2.1) •Distributions . We say that u∈D(A−α) for someα≥0 if /⌊ard⌊lu/⌊ard⌊l2 D(A−α):=∞/summationdisplay k=0(1+λk)−4αu2 k<+∞. (2.2) •Generalized Gevrey spaces . Letϕ: [0,+∞)→[0,+∞) be any function, let r≥0, and letα∈R. We say that u∈ Gϕ,r,α(A) if /⌊ard⌊lu/⌊ard⌊l2 ϕ,r,α:=∞/summationdisplay k=0(1+λk)4αu2 kexp/parenleftbig 2rϕ(λk)/parenrightbig <+∞. (2.3) •Generalized Gevrey ultradistributions . Letψ: [0,+∞)→[0,+∞)beanyfunction, letR≥0, and letα∈R. We say that u∈ G−ψ,R,α(A) if /⌊ard⌊lu/⌊ard⌊l2 −ψ,R,α:=∞/summationdisplay k=0(1+λk)4αu2 kexp/parenleftbig −2Rψ(λk)/parenrightbig <+∞. (2.4) 5Remark 2.2. Ifϕ1(x) =ϕ2(x) for every x >0, thenGϕ1,r,α(A) =Gϕ2,r,α(A) for every admissible value of randα. For this reason, with a little abuse of notation, we consider the spaces Gϕ,r,α(A) even when ϕ(x) is defined only for x >0. The same comment applies also to the spaces G−ψ,R,α(A). The quantities defined in (2.1) through (2.4) are actually norms which induce a Hilbert space structure on D(Aα),Gϕ,r,α(A),G−ψ,R,α(A), respectively. The standard inclusions Gϕ,r,α(A)⊆D(Aα)⊆H⊆D(A−α)⊆ G−ψ,R,−α(A) hold true for every α≥0 and every admissible choice of ϕ,ψ,r,R. All inclusions are strict if α,randRare positive, and the sequences {λk},{ϕ(λk)}, and{ψ(λk)}are unbounded. We observe that Gϕ,r,α(A) is actually a so-called scale of Hilbert spaces with respect to theparameter r, withlarger values of rcorresponding to smaller spaces. Analogously, G−ψ,R,α(A) is a scale of Hilbert spaces with respect to the parameter R, but with larger values ofRcorresponding to larger spaces. Remark 2.3. Let us consider the concrete case where I⊆Ris an open interval, H=L2(I), andAu=−uxx, with periodic boundary conditions. For every α≥0, the spaceD(Aα) is actually the usual Sobolev space H2α(I), andD(A−α) is the usual space of distributions of order 2 α. Whenϕ(x) :=x1/sfor somes>0, elements of Gϕ,r,0(A) withr>0 are usually called Gevrey functions of order s, the cases= 1 corresponding to analytic functions. When ψ(x) :=x1/sforsomes>0, elements of G−ψ,R,0(A)withR>0areusually called Gevrey ultradistributions of order s, the cases= 1 corresponding to analytic functionals. In this case the parameter αis substantially irrelevant because the exponential term is dominant both in (2.3) and in (2.4). For the sake of consistency, with a little abuse of notation we use th e same terms (Gevrey functions, Gevrey ultradistributions, analytic functions and analytic function- als) in order to denote the same spaces also in the general abstrac t framework. To be more precise, we should always add “with respect to the operator A”, or even better “with respect to the operator A1/2”. Continuity moduli Throughout this paper we call continuity modulus any continuous functionω: [0,+∞)→[0,+∞) such that ω(0) = 0,ω(x)>0 for every x >0, and moreover x→ω(x) is a nondecreasing function , (2.5) x→x ω(x)is a nondecreasing function. (2.6) A function c: [0,+∞)→Ris said to be ω-continuous if |c(a)−c(b)| ≤ω(|a−b|)∀a≥0,∀b≥0. (2.7) 6More generally, a function c:X→R(withX⊆R) is said to be ω-continuous if it satisfies the same inequality for every aandbinX. Previous results We are now ready to recall the classic results concerning existence , uniqueness, and regularity for solutions to problem (1.1)–(1.2). We state them using our notations which allow general continuity moduli and general spaces of Gevrey functions or ultradistributions. Proofs are a straightforward application of the approximated ene rgy estimates in- troduced in [6]. In that paper only the case δ= 0 is considered, but when δ≥0 all new terms have the “right sign” in those estimates. The first result concerns existence and uniqueness in huge spaces such as analytic functionals, with minimal assumptions on c(t). Theorem A (see [6, Theorem 1]) .Let us consider problem (1.1)–(1.2) under the fol- lowing assumptions: •Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH, •c∈L1((0,T))for everyT >0(without sign conditions), •σ≥0andδ≥0are two real numbers, •initial conditions satisfy (u0,u1)∈ G−ψ,R0,1/2(A)×G−ψ,R0,0(A) for someR0>0and someψ: (0,+∞)→(0,+∞)such that limsup x→+∞x ψ(x)<+∞. Then there exists a nondecreasing function R: [0,+∞)→[0,+∞), withR(0) =R0, such that problem (1.1)–(1.2) admits a unique solution u∈C0/parenleftbig [0,+∞);G−ψ,R(t),1/2(A)/parenrightbig ∩C1/parenleftbig [0,+∞);G−ψ,R(t),0(A)/parenrightbig .(2.8) Condition (2.8), with the range space increasing with time, simply mean s that u∈C0/parenleftbig [0,τ];G−ψ,R(τ),1/2(A)/parenrightbig ∩C1/parenleftbig [0,τ];G−ψ,R(τ),0(A)/parenrightbig ∀τ≥0. This amounts to say that scales of Hilbert spaces, rather than fixe d Hilbert spaces, are the natural setting for this problem. Inthesecondresultweassumestricthyperbolicityand ω-continuityofthecoefficient, and we obtain well-posedness in a suitable class of Gevrey ultradistrib utions. 7Theorem B (see [6, Theorem 3]) .Let us consider problem (1.1)–(1.2) under the fol- lowing assumptions: •Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH, •the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4) and theω-continuity assumption (2.7) for some continuity modulus ω(x), •σ≥0andδ≥0are two real numbers, •initial conditions satisfy (u0,u1)∈ G−ψ,R0,1/2(A)×G−ψ,R0,0(A) for someR0>0and some function ψ: (0,+∞)→(0,+∞)such that limsup x→+∞x ψ(x)ω/parenleftbigg1 x/parenrightbigg <+∞. (2.9) Letube the unique solution to the problem provided by Theorem A. Then there exists R>0such that u∈C0/parenleftbig [0,+∞),G−ψ,R0+Rt,1/2(A)/parenrightbig ∩C1([0,+∞),G−ψ,R0+Rt,0(A)). The third result we recall concerns existence of regular solutions. The assumptions onc(t) are the same as in Theorem B, but initial data are significantly more r egular (Gevrey spaces instead of Gevrey ultradistributions). Theorem C (see [6, Theorem 2]) .Let us consider problem (1.1)–(1.2) under the fol- lowing assumptions: •Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH, •the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4) and theω-continuity assumption (2.7) for some continuity modulus ω(x), •σ≥0andδ≥0are two real numbers, •initial conditions satisfy (u0,u1)∈ Gϕ,r0,1/2(A)×Gϕ,r0,0(A) for somer0>0and some function ϕ: (0,+∞)→(0,+∞)such that limsup x→+∞x ϕ(x)ω/parenleftbigg1 x/parenrightbigg <+∞. (2.10) 8Letube the unique solution to the problem provided by Theorem A. Then there exist T >0andr>0such thatrT <r 0and u∈C0/parenleftbig [0,T],Gϕ,r0−rt,1/2(A)/parenrightbig ∩C1([0,T],Gϕ,r0−rt,0(A)). (2.11) Remark 2.4. The key assumptions of Theorem B and Theorem C are (2.9) and (2.10 ), respectively, representing the exact compensation between spa ce-regularity of initial dataandtime-regularityofthecoefficient c(t)requiredinordertoobtainwell-posedness. These conditions do not appear explicitly in [6], where they are replace d by suitable specific choices of ω,ϕ,ψ, which of course satisfy the same relations. To our knowledge, those conditions were stated for the first time in [8], thus unifying se veral papers that in thelast 30 years hadbeendevoted tospecial cases (see forexam ple [5] andthe references quoted therein). Remark 2.5. The standard example of application of Theorem B and Theorem C is the following. Let us assume that c(t) isα-H¨ older continuous for some α∈(0,1), namelyω(x) =Mxαfor a suitable constant M. Then (2.9) and (2.10) hold true with ψ(x) =ϕ(x) :=x1−α. This leads to well-posedness both in the large space of Gevrey ultradistributions of order (1 −α)−1, and in the small space of Gevrey functions of the same order. Remark 2.6. The choice of ultradistributions in Theorem B is not motivated by the searchforgenerality, butitisinsomesense theonlypossibleonebec auseofthe(DGCS)- phenomenon exhibited in [6], at least in the non-dissipative case. When δ= 0, if initial data are taken in Sobolev spaces or in any space larger than the Gev rey spaces of Theorem C, then it may happen that for all positive times the solution lies in the space of ultradistributions specified in Theorem B, and nothing more. In ot her words, for δ= 0 there is no well-posedness result in between the Gevrey spaces o f Theorem C and the Gevrey ultradistributions of Theorem B, and conditions (2.9 ) and (2.10) are optimal. The aim of this paper is to provide an optimal picture for the case δ >0. 3 Main results In this section we state our main regularity results for solutions to ( 1.1)–(1.2). To this end, we need some further notation. Given any ν≥0, we write Has an orthogonal direct sum H:=Hν,−⊕Hν,+, (3.1) whereHν,−is the closure of the subspace generated by all eigenvectors of Arelative to eigenvalues λk<ν, andHν,+is the closure of the subspace generated by all eigenvectors ofArelative to eigenvalues λk≥ν. For every vector u∈H, we writeuν,−anduν,+ to denote its components with respect to the decomposition (3.1). We point out that 9Hν,−andHν,+areA-invariant subspaces of H, and thatAis a bounded operator when restricted to Hν,−, and a coercive operator when restricted to Hν,+ifν >0. In the following statements we provide separate estimates for low- frequency compo- nentsuν,−(t) and high-frequency components uν,+(t) of solutions to (1.1). This is due to the fact that the energy of uν,−(t) can be unbounded as t→+∞, while in many cases we are able to prove that the energy of uν,+(t) is bounded in time. 3.1 Existence results in Sobolev spaces The first result concerns the supercritical regime σ≥1/2, in which case the dissipation always dominates the time-dependent coefficient. Theorem 3.1 (Supercritical dissipation) .Let us consider problem (1.1)–(1.2) under the following assumptions: •Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH, •the coefficient c: [0,+∞)→Ris measurable and satisfies the degenerate hyper- bolicity assumption (1.7), •σandδare two positive real numbers such that either σ >1/2, orσ= 1/2and 4δ2≥µ2, •(u0,u1)∈D(Aα)×D(Aβ)for some real numbers αandβsatisfying (1.6). Letube the unique solution to the problem provided by Theorem A. Thenuactually satisfies (u,u′)∈C0/parenleftbig [0,+∞),D(Aα)×D(Aβ)/parenrightbig . (3.2) Moreover, for every ν≥1such that 4δ2ν4σ−2≥µ2, it turns out that |Aβu′ ν,+(t)|2+|Aαuν,+(t)|2≤/parenleftbigg 2+2 δ2+µ2 2 δ4/parenrightbigg |Aβu1,ν,+|2+3/parenleftbigg 1+µ2 2 2δ2/parenrightbigg |Aαu0,ν,+|2(3.3) for everyt≥0. Our second result concerns the subcritical regime σ∈[0,1/2], in which case the time-regularity of c(t) competes with the exponent σ. Theorem 3.2 (Subcritical dissipation) .Let us consider problem (1.1)–(1.2) under the following assumptions: •Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH, •the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4) and theω-continuity assumption (2.7) for some continuity modulus ω(x), 10•σ∈[0,1/2]andδ>0are two real numbers such that 4δ2µ1>Λ2 ∞+2δΛ∞, (3.4) where we set Λ∞:= limsup ε→0+ω(ε) ε1−2σ, (3.5) •(u0,u1)∈D(A1/2)×H. Letube the unique solution to the problem provided by Theorem A. Thenuactually satisfies u∈C0/parenleftbig [0,+∞),D(A1/2)/parenrightbig ∩C1([0,+∞),H). Moreover, for every ν≥1such that 4δ2µ1≥/bracketleftbigg λ1−2σω/parenleftbigg1 λ/parenrightbigg/bracketrightbigg2 +2δ/bracketleftbigg λ1−2σω/parenleftbigg1 λ/parenrightbigg/bracketrightbigg (3.6) for everyλ≥ν, it turns out that |u′ ν,+(t)|2+2µ1|A1/2uν,+(t)|2≤4|u1,ν,+|2+2(3δ2+µ2)|A1/2u0,ν,+|2(3.7) for everyt≥0. Let us make a few comments on the first two statements. Remark 3.3. Inbothresultsweprovedthatasuitablehigh-frequencycompone nt ofthe solution can be uniformly bounded in terms of initial data. Low-frequ ency components might in general diverge as t→+∞. Nevertheless, they can always be estimated as follows. Let us just assume that c∈L1((0,T)) for every T >0. Then for every ν≥0 the component uν,−(t) satisfies |u′ ν,−(t)|2+|A1/2uν,−(t)|2≤/parenleftbig |u1,ν,−|2+|A1/2u0,ν,−|2/parenrightbig exp/parenleftbigg νt+ν/integraldisplayt 0|c(s)|ds/parenrightbigg (3.8) for everyt≥0. Indeed, let F(t) denote the left-hand side of (3.8). Then F′(t) =−4δ|Aσ/2u′ ν,−(t)|2+2(1−c(t))/a\}⌊ra⌋ketle{tu′ ν,−(t),Auν,−(t)/a\}⌊ra⌋ketri}ht ≤2(1+|c(t)|)·|u′ ν,−(t)|·ν|A1/2uν,−(t)| ≤ν(1+|c(t)|)F(t) for almost every t≥0, so that (3.8) follows by integrating this differential inequality. 11Remark 3.4. The phase spaces involved in Theorem 3.1 and Theorem 3.2 are exactly the same which are known to be optimal when c(t) is constant (see [9]). In particular, the only possible choice in the subcritical regime is the classic energy s paceD(A1/2)×H, or more generally D(Aα+1/2)×D(Aα). This “gap1/2” between the powers of Ainvolved in the phase space is typical of hyperbolic problems, and it is the same which appears in the classic results of section 2. On the contrary, in the supercritical regime there is an interval of possible gaps, described by (1.6). This interval is always centered in 1/2, but also d ifferent values are allowed, including negative ones when σ>1. Remark 3.5. The classic example of application of Theorem 3.2 is the following. Let us assume that c(t) isα-H¨ older continuous for some α∈(0,1), namely ω(x) =Mxαfor some constant M. Then problem (1.1)–(1.2) is well-posed in the energy space provided that either α>1−2σ, orα= 1−2σandMis small enough. Indeed, for α>1−2σwe get Λ∞= 0, and hence (3.4) is automatically satisfied. For α= 1−2σwe get Λ ∞=M, so that (3.4) is satisfied provided that Mis small enough. In all other cases, namely when either α <1−2σ, orα= 1−2σandMis large enough, only Theorem B applies to initial data in Sobolev spaces, prov iding global existence just in the sense of Gevrey ultradistributions of order ( 1−α)−1. Remark 3.6. Let us examine the limit case σ= 0, which falls in the subcritical regime. Whenσ= 0, assumption (3.4) is satisfied if and only if c(t) is Lipschitz continuous and its Lipschitz constant is small enough. On the other hand, in the Lipschitz case it is a classic result that problem (1.1)–(1.2) is well-posed in the energy s pace, regardless of the Lipschitz constant. Therefore, the result stated in Theor em 3.2 is non-optimal whenσ= 0 andc(t) is Lipschitz continuous. A simple refinement of our argument would lead to the full result also in this case, but unfortunately it would be useless in all other limit cases in which c(t) isα-H¨ older continuous with α= 1−2σandσ∈(0,1/2]. We refer to section 4 for further details. Remark 3.7. Let us examine the limit case σ= 1/2, which falls both in the subcritical and in the supercritical regime, so that the conclusions of Theorem 3.1 and Theorem 3.2 coexist. Both of them provide well-posedness in the energy space, but with different assumptions. Theorem 3.1 needs less assumptions on c(t), which is only required to be measurable and to satisfy the degenerate hyperbolicity assumption (1.7), but it requires δto be large enough so that 4 δ2≥µ2. On the contrary, Theorem 3.2 needs less assumptions on δ, which is only required to be positive, but it requires c(t) to be continuous and to satisfy the strict hyperbolicity assumption (1.4). Indeed, inequality (3.4) is automatically satisfied in the caseσ= 1/2 because Λ ∞= 0. The existence of two different sets of assumptions leading to the sa me conclusion suggests the existence of a unifying statement, which could proba bly deserve further investigation. 123.2 Gevrey regularity for positive times A strong dissipation in the range σ∈(0,1) has a regularizing effect on initial data, provided that the solution exists in Sobolev spaces. In the following t wo statements we quantify this effect in terms of scales of Gevrey spaces. Both results can be summed up by saying that the solution lies, for po sitive times, in Gevrey spaces of order (2min {σ,1−σ})−1. It is not difficult to show that this order is optimal, even in the case where c(t) is constant. Theorem 3.8 (Supercritical dissipation) .Let us consider problem (1.1)–(1.2) under the same assumptions of Theorem 3.1, and let ube the unique solution to the problem provided by Theorem A. Let us assume in addition that either σ∈(1/2,1), orσ= 1/2and4δ2>µ2. Let us setϕ(x) :=x2(1−σ), and C(t) :=/integraldisplayt 0c(s)ds. (3.9) Then there exists r>0such that (u,u′)∈C0/parenleftbig (0,+∞),Gϕ,α,rC(t)(A)×Gϕ,β,rC(t)(A)/parenrightbig , (3.10) and there exist ν≥1andK >0such that /⌊ard⌊lu′ ν,+(t)/⌊ard⌊l2 ϕ,β,rC(t)+/⌊ard⌊luν,+(t)/⌊ard⌊l2 ϕ,α,rC(t)≤K/parenleftbig |Aβu1,ν,+|2+|Aαu0,ν,+|2/parenrightbig (3.11) for everyt>0. The constants r,ν, andKdepend only on δ,µ2, andσ. Of course, (3.10) and (3.11) are nontrivial only if C(t)>0, which is equivalent to saying that the coefficient c(t) is not identically 0 in [0 ,t]. On the other hand, this weak form of hyperbolicity is necessary, since no regularizing effect on u(t) can be expected as long asc(t) vanishes. Theorem 3.9 (Subcritical dissipation) .Let us consider problem (1.1)–(1.2) under the same assumptions of Theorem 3.2, and let ube the unique solution to the problem provided by Theorem A. Let us assume in addition that σ∈(0,1/2](instead of σ∈[0,1/2]), and let us set ϕ(x) :=x2σ. Then there exists r>0such that u∈C0/parenleftbig (0,+∞),Gϕ,1/2,rt(A)/parenrightbig ∩C1((0,+∞),Gϕ,0,rt(A)), and there exist ν≥1andK >0such that /⌊ard⌊lu′ ν,+(t)/⌊ard⌊l2 ϕ,0,rt+/⌊ard⌊luν,+(t)/⌊ard⌊l2 ϕ,1/2,rt≤K/parenleftbig |u1,ν,+|2+|A1/2u0,ν,+|2/parenrightbig (3.12) for everyt>0. The constants r,ν, andKdepend only on δ,µ1,µ2,σandω. TheestimateswhichprovideGevreyregularityofhigh-frequencyc omponentsprovide also the decay of the same components as t→+∞. We refer to Lemma 5.1 and Lemma 5.2 for further details. 133.3 Counterexamples The following result shows that even strongly dissipative hyperbolic e quations can ex- hibit the (DGCS)-phenomenon, provided that we are in the subcritic al regime. Theorem 3.10 ((DGCS)-phenomenon) .LetAbe a linear operator on a Hilbert space H. Let us assume that there exists a countable (not necessaril y complete) orthonormal system{ek}inH, and an unbounded sequence {λk}of positive real numbers such that Aek=λ2 kekfor everyk∈N. Letσ∈[0,1/2)andδ>0be real numbers. Letω: [0,+∞)→[0,+∞)be a continuity modulus such that lim ε→0+ω(ε) ε1−2σ= +∞. (3.13) Letϕ: (0,+∞)→(0,+∞)andψ: (0,+∞)→(0,+∞)be two functions such that lim x→+∞x ϕ(x)ω/parenleftbigg1 x/parenrightbigg = lim x→+∞x ψ(x)ω/parenleftbigg1 x/parenrightbigg = +∞. (3.14) Then there exist a function c:R→Rsuch that 1 2≤c(t)≤3 2∀t∈R, (3.15) |c(t)−c(s)| ≤ω(|t−s|)∀(t,s)∈R2, (3.16) and a solution u(t)to equation (1.1) such that (u(0),u′(0))∈ Gϕ,r,1/2(A)×Gϕ,r,0(A)∀r>0, (3.17) (u(t),u′(t))/\e}atio\slash∈ G−ψ,R,1/2(A)×G−ψ,R,0(A)∀R>0,∀t>0.(3.18) Remark 3.11. Due to (3.15), (3.16), and (3.17), the function u(t) provided by Theo- rem 3.10 is a solution to (1.1) in the sense of Theorem A with ψ(x) :=x, or even better in the sense of Theorem B with ψ(x) :=xω(1/x). Remark 3.12. Assumption (3.13) is equivalent to saying that Λ ∞defined by (3.5) is equal to + ∞, so that (3.4) can not be satisfied. In other words, Theorem 3.2 giv es well-posedness in the energy space if Λ ∞is 0 or small, while Theorem 3.10 provides the (DGCS)-phenomenon if Λ ∞= +∞. The case where Λ ∞is finite but large remains open. We suspect that the (DGCS)-phenomenon is still possible, bu t our construction does not work. We comment on this issue in the first part of section 6 . Finally, Theorem 3.10 shows that assumptions (2.9) and (2.10) of The orems B and C are optimal also in the subcritical dissipative case with Λ ∞= +∞. If initial data are in the Gevrey space with ϕ(x) =xω(1/x), solutions remain in the same space. If initial are in a Gevrey space corresponding to some ϕ(x)≪xω(1/x), then it may happen that for positive times the solution lies in the space of ultradistributions with ψ(x) :=xω(1/x), but not in the space of ultradistributions corresponding to any give nψ(x)≪xω(1/x). 144 Heuristics Thefollowingpicturessummarizeroughlytheresultsofthispaper. I nthehorizontalaxis we represent the time-regularity of c(t). With some abuse of notation, values α∈(0,1) mean that c(t) isα-H¨ older continuous, α= 1 means that it is Lipschitz continuous, α >1 means even more regular. In the vertical axis we represent the s pace-regularity of initial data, where the value sstands for the Gevrey space of order s(so that higher values ofsmean lower regularity). The curve is s= (1−α)−1. α 1s 1 δ= 0Potential (DGCS)-phenomenon Well-posedness α 1−2σ/Bullets 1 δ >0,0<σ<1/2α 1s 1 δ >0, σ>1/2 Forδ= 0 we have the situation described in Remark 2.5 and Remark 2.6, name ly well-posedness provided that either c(t) is Lipschitz continuous or c(t) isα-H¨ older con- tinuous and initial data are in Gevrey spaces of order less than or eq ual to (1 −α)−1, and (DGCS)-phenomenon otherwise. The same picture applies if δ >0 andσ= 0. Whenδ >0 and 0< σ <1/2, the full strip with α >1−2σfalls in the well- posedness region, as stated in Theorem 3.2. The region with α <1−2σis divided as in the non-dissipative case. Indeed, Theorem C still provides well-po sedness below the curve and on the curve, while Theorem 3.10 provides the (DGCS)-ph enomenon above the curve. What happens on the vertical half-line which separates the two regions is less clear (it is the region where Λ ∞is positive and finite, see Remark 3.12). Finally, when δ >0 andσ>1/2 well-posedness dominates because of Theorem 3.1, even in the degenerate hyperbolic case. Now we present a rough justification of this threshold effect. As alr eady observed, existence results for problem (1.1)–(1.2) are related to estimates for solutions to the family of ordinary differential equations (1.8). Let us consider the simplest energy function E(t) :=|u′ λ(t)|2+λ2|uλ(t)|2, whose time-derivative is E′(t) =−4δλ2σ|u′ λ(t)|2+2λ2(1−c(t))uλ(t)u′ λ(t) ≤ −4δλ2σ|u′ λ(t)|2+λ(1+|c(t)|)E(t). (4.1) 15Sinceδ≥0, a simple integration gives that E(t)≤ E(0)exp/parenleftbigg λt+λ/integraldisplayt 0|c(s)|ds/parenrightbigg , (4.2) which is almost enough to establish Theorem A. Ifinaddition c(t)isω-continuousandsatisfiesthestricthyperbolicitycondition(1.4), then (4.2) can be improved to E(t)≤M1E(0)exp(M2λω(1/λ)t) (4.3) for suitable constants M1andM2. Estimates of this kind are the key point in the proof of both Theorem B and Theorem C. Moreover, the (DGCS)-phenom enon is equivalent to saying that the term λω(1/λ) in (4.3) is optimal. Let us assume now that δ >0. Ifσ >1/2, orσ= 1/2 andδis large enough, then it is reasonable to expect that the first (negative) term in the right-hand side of (4.1) dominates the second one, and hence E(t)≤ E(0), which is enough to establish well-posedness in Sobolev spaces. Theorem 3.1 confirms this intuition . Ifσ≤1/2 andc(t) is constant, then (1.8) can be explicitly integrated, obtaining that E(t)≤ E(0)exp/parenleftbig −2δλ2σt/parenrightbig . (4.4) Ifc(t) isω-continuous and satisfies the strict hyperbolicity assumption (1.4) , then we expect a superposition of the effects of the coefficient, repres ented by (4.3), and the effects of the damping, represented by (4.4). We end up with E(t)≤M1E(0)exp/parenleftbig [M2λω(1/λ)−2δλ2σ]t/parenrightbig . (4.5) Therefore, it is reasonable to expect that E(t) satisfies an estimate independent of λ, which guarantees well-posedness in Sobolev spaces, provided tha tλω(1/λ)≪λ2σ, or λω(1/λ)∼λ2σandδis large enough. Theorem 3.2 confirms this intuition. The same argument applies if σ= 0 andω(x) =Lx, independently of L(see Remark 3.6). On the contrary, in all other cases the right-hand side of (4.5) dive rges asλ→ +∞, opening the door to the (DGCS)-phenomenon. We are able to show that it does happen provided that λω(1/λ)≫λ2σ. We refer to the first part of section 6 for further comments. 5 Proofs of well-posedness and regularity results All proofs of our main results concerning well-posedness and regula rity rely on suitable estimates for solutions to the ordinary differential equation (1.8) w ith initial data uλ(0) =u0, u′ λ(0) =u1. (5.1) For the sake of simplicity in the sequel we write u(t) instead of uλ(t). 165.1 Supercritical dissipation Let us consider the case σ≥1/2. The key tool is the following. Lemma 5.1. Let us consider problem (1.8)–(5.1) under the following ass umptions: •the coefficient c: [0,+∞)→Ris measurable and satisfies the degenerate hyper- bolicity assumption (1.7), •δ,λ,σare positive real numbers such that 4δ2λ4σ−2≥µ2. (5.2) Then the solution u(t)satisfies the following estimates. (1) For every t≥0it turns out that |u(t)|2≤2 δ2λ4σu2 1+3u2 0, (5.3) |u′(t)|2≤/parenleftbigg 2+µ2 2 δ4λ8σ−4/parenrightbigg u2 1+3µ2 2 2δ2λ4σ−4u2 0. (5.4) (2) Let us assume in addition that λ≥1andσ≥1/2, and letαandβbe two real numbers satisfying (1.6). Then for every t≥0it turns out that λ4β|u′(t)|2+λ4α|u(t)|2≤/parenleftbigg 2+2 δ2+µ2 2 δ4/parenrightbigg λ4βu2 1+3/parenleftbigg 1+µ2 2 2δ2/parenrightbigg λ4αu2 0.(5.5) (3) In addition to the assumptions of the statement (2), let u s assume also that there existsr>0satisfying the following three inequalities: δλ4σ−2>rµ2,2δr≤1,4δ2λ4σ−2≥(1+2rδ)µ2.(5.6) Then for every t≥0it turns out that λ4β|u′(t)|2+λ4α|u(t)|2≤/bracketleftbigg 2/parenleftbigg 1+2µ2 2 δ4+1 δ2/parenrightbigg λ4βu2 1+3/parenleftbigg 1+2µ2 2 δ2/parenrightbigg λ4αu2 0/bracketrightbigg × ×exp/parenleftbigg −2rλ2(1−σ)/integraldisplayt 0c(s)ds/parenrightbigg . (5.7) 17ProofLet us consider the energy E(t) defined in (1.9). Since −3 4|u′(t)|2−4 3δ2λ4σ|u(t)|2≤2δλ2σu(t)u′(t)≤ |u′(t)|2+δ2λ4σ|u(t)|2, we easily deduce that 1 4|u′(t)|2+2 3δ2λ4σ|u(t)|2≤E(t)≤2|u′(t)|2+3δ2λ4σ|u(t)|2∀t≥0.(5.8) Statement (1) The time-derivative of E(t) is E′(t) =−2/parenleftbig δλ2σ|u′(t)|2+δλ2σ+2c(t)|u(t)|2+λ2c(t)u(t)u′(t)/parenrightbig .(5.9) The right-hand side is a quadratic form in u(t) andu′(t). The coefficient of |u′(t)|2 is negative. Therefore, this quadratic form is less than or equal to 0 for all values of u(t) andu′(t) if and only if 4δ2λ4σ−2c(t)≥c2(t), and this is always true because of (1.7) and (5.2). It follows that E′(t)≤0 for (almost) everyt≥0, and hence δ2λ4σ|u(t)|2≤E(t)≤E(0)≤2u2 1+3δ2λ4σu2 0, (5.10) which is equivalent to (5.3). In order to estimate u′(t), we rewrite (1.8) in the form u′′(t)+2δλ2σu′(t) =−λ2c(t)u(t), which we interpret as a first order linear equation with constant coe fficients in the unknownu′(t), with the right-hand side as a forcing term. Integrating this differ ential equation in u′(t), we obtain that u′(t) =u1exp/parenleftbig −2δλ2σt/parenrightbig −/integraldisplayt 0λ2c(s)u(s)exp/parenleftbig −2δλ2σ(t−s)/parenrightbig ds. (5.11) From (1.7) and (5.3) it follows that |u′(t)| ≤ |u1|+µ2λ2·max t∈[0,T]|u(t)|·/integraldisplayt 0e−2δλ2σ(t−s)ds ≤ |u1|+µ2λ2 2δλ2σ/parenleftbigg2 δ2λ4σu2 1+3u2 0/parenrightbigg1/2 , and therefore |u′(t)|2≤2|u1|2+µ2 2λ4 2δ2λ4σ/parenleftbigg2 δ2λ4σu2 1+3u2 0/parenrightbigg , which is equivalent to (5.4). 18Statement (2) Exploiting (5.3) and (5.4), with some simple algebra we obtain that λ4β|u′(t)|2+λ4α|u(t)|2≤/parenleftbigg 2+µ2 2 δ4·1 λ4(2σ−1)+2 δ2·1 λ4(β+σ−α)/parenrightbigg λ4βu2 1 +3/parenleftbigg 1+µ2 2 2δ2·1 λ4(α−β+σ−1)/parenrightbigg λ4αu2 0. All exponents of λ’s in denominators are nonnegative owing to (1.6). Therefore, sinceλ≥1, all those fractions can be estimated with 1. This leads to (5.5). Statement (3) Let us define C(t) as in (3.9). To begin with, we prove that in this case the function E(t) satisfies the stronger differential inequality E′(t)≤ −2rλ2(1−σ)c(t)E(t), (5.12) and hence E(t)≤E(0)exp/parenleftbig −2rλ2(1−σ)C(t)/parenrightbig ∀t≥0. (5.13) Coming back to (5.9), inequality (5.12) is equivalent to λ2σ/parenleftbig δ−rλ2−4σc(t)/parenrightbig |u′(t)|2+δλ2σ+2(1−2rδ)c(t)|u(t)|2+λ2(1−2rδ)c(t)u(t)u′(t)≥0. As in the proof of statement (1), we consider the whole left-hand s ide as a quadratic form inu(t) andu′(t). Sincec(t)≤µ2, from the first inequality in (5.6) it follows that δλ4σ−2>rµ2≥rc(t), which is equivalent to saying that the coefficient of |u′(t)|2is positive. Therefore, the quadratic form is nonnegative for all values of u(t) andu′(t) if and only if 4λ2σ/parenleftbig δ−rλ2−4σc(t)/parenrightbig ·δλ2σ+2c(t)(1−2rδ)≥λ4c2(t)(1−2rδ)2, hence if and only if (1−2rδ)c(t)/bracketleftbig 4δ2λ4σ−2−(1+2rδ)c(t)/bracketrightbig ≥0, and this follows from (1.7) and from the last two inequalities in (5.6). Now from (5.13) it follows that δ2λ4σ|u(t)|2≤E(t)≤E(0)exp/parenleftbig −2rλ2(1−σ)C(t)/parenrightbig , (5.14) which provides an estimate for |u(t)|. In order to estimate u′(t), we write it in the form (5.11), and we estimate the two terms separately. The third inequa lity in (5.6) implies that 2δλ4σ−2≥rµ2. SinceC(t)≤µ2t, it follows that 2δλ2σt≥rλ2−2σµ2t≥rλ2−2σC(t), 19and hence /vextendsingle/vextendsingleu1exp/parenleftbig −2δλ2σt/parenrightbig/vextendsingle/vextendsingle≤ |u1|exp/parenleftbig −2δλ2σt/parenrightbig ≤ |u1|exp/parenleftbig −rλ2(1−σ)C(t)/parenrightbig .(5.15) As for the second terms in (5.11), we exploit (5.14) and we obtain tha t /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0λ2c(s)u(s)exp/parenleftbig −2δλ2σ(t−s)/parenrightbig ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ2µ2/integraldisplayt 0|u(s)|exp/parenleftbig −2δλ2σ(t−s)/parenrightbig ds ≤µ2[E(0)]1/2 δλ2σ−2exp/parenleftbig −2δλ2σt/parenrightbig/integraldisplayt 0exp/parenleftbig −rλ2(1−σ)C(s)+2δλ2σs/parenrightbig ds. From the first inequality in (5.6) it follows that 2δλ2σ−rλ2(1−σ)c(s)≥2δλ2σ−rλ2(1−σ)µ2≥δλ2σ, hence /integraldisplayt 0exp/parenleftbig −rλ2(1−σ)C(s)+2δλ2σs/parenrightbig ds ≤1 δλ2σ/integraldisplayt 0/parenleftbig 2δλ2σ−rλ2(1−σ)c(s)/parenrightbig exp/parenleftbig 2δλ2σs−rλ2(1−σ)C(s)/parenrightbig ds ≤1 δλ2σexp/parenleftbig 2δλ2σt−rλ2(1−σ)C(t)/parenrightbig , and therefore /vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt 0λ2c(s)u(s)exp/parenleftbig −2δλ2σ(t−s)/parenrightbig ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤µ2[E(0)]1/2 δ2λ4σ−2exp/parenleftbig −rλ2(1−σ)C(t)/parenrightbig .(5.16) From (5.11), (5.15) and (5.16) we deduce that |u′(t)| ≤/parenleftbigg |u1|+µ2[E(0)]1/2 δ2λ4σ−2/parenrightbigg exp/parenleftbig −rλ2(1−σ)C(t)/parenrightbig , and hence |u′(t)|2≤/parenleftbigg 2|u1|2+2µ2 2E(0) δ4λ8σ−4/parenrightbigg exp/parenleftbig −2rλ2(1−σ)C(t)/parenrightbig . (5.17) Finally, we estimate E(0) as in (5.10). At this point, estimate (5.7) follows from (5.17) and (5.14) with some simple algebra (we need to exploit that λ≥1 and assump- tion (1.6) exactly as in the proof of statement (2)). /square 205.1.1 Proof of Theorem 3.1 Let us fix a real number ν≥1 such that 4 δ2ν4σ−2≥µ2(such a number exists because of ourassumptions on δandσ). Letusconsiderthecomponents uk(t)ofu(t)corresponding to eigenvalues λk≥ν. Sinceλk≥1 and 4δ2λ4σ−2 k≥µ2, we can apply statement (2) of Lemma 5.1 to these components. If u0kandu1kdenote the corresponding components of initial data, estimate (5.5) read as λ4β k|u′ k(t)|2+λ4α k|uk(t)|2≤/parenleftbigg 2+2 δ2+µ2 2 δ4/parenrightbigg λ4β k|u1,k|2+3/parenleftbigg 1+µ2 2 2δ2/parenrightbigg λ4α k|u0,k|2. Summing over all λk≥νwe obtain exactly (3.3). This proves that uν,+(t) is bounded with values in D(Aα) andu′ ν,+(t) is bounded with values in D(Aβ). The same estimate guarantees the uniform convergence in the whole half-line t≥0 of the series defining Aαuν,+(t) andAβu′ ν,+(t). Since all summands are continuous, and the convergence is uniform, the sum is continu ous as well. Since low-frequency components uν,−(t) andu′ ν,−(t) are continuous (see Remark 3.3), (3.2) is proved. /square 5.1.2 Proof of Theorem 3.8 Let us fix a real number ν≥1 such that 4 δ2ν4σ−2>µ2(such a number exists because of our assumptions on δandσ). Then there exists r>0 such that the three inequalities in (5.6) hold true for every λ≥ν. Therefore, we can apply statement (3) of Lemma 5.1 to all components uk(t) ofu(t) corresponding to eigenvalues λk≥ν. Ifu0kandu1kdenote the corresponding components of initial data, estimate (5.7) read as /parenleftBig λ4β k|u′ k(t)|2+λ4α k|uk(t)|2/parenrightBig exp/parenleftbigg 2rλ2(1−σ) k/integraldisplayt 0c(s)ds/parenrightbigg ≤K/parenleftBig λ4β k|u1k|2+λ4α k|u0k|2/parenrightBig for everyt≥0, whereKis a suitable constant depending only on µ2andδ. Summing over allλk≥νwe obtain exactly (3.11). The continuity of u(t) andu′(t) with values in the suitable spaces follows from the uniform convergence of serie s as in the proof of Theorem 3.1. /square 5.2 Subcritical dissipation Let us consider the case 0 ≤σ≤1/2. The key tool is the following. Lemma 5.2. Let us consider problem (1.8)–(5.1) under the following ass umptions: •the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4) and theω-continuity assumption (2.7) for some continuity modulus ω(x), •δ>0,λ>0, andσ≥0are real numbers satisfying (3.6). 21Then the solution u(t)satisfies the following estimates. (1) It turns out that |u′(t)|2+2λ2µ1|u(t)|2≤4u2 1+2/parenleftbig 3δ2λ4σ+λ2µ2/parenrightbig u2 0∀t≥0.(5.18) (2) Let us assume in addition that λ≥1,σ∈[0,1/2], and there exists a constant r∈(0,δ)such that 4(δ−r)(δµ1−rµ2)≥/bracketleftbigg λ1−2σω/parenleftbigg1 λ/parenrightbigg/bracketrightbigg2 +2δ(1+2r)/bracketleftbigg λ1−2σω/parenleftbigg1 λ/parenrightbigg/bracketrightbigg +8rδ3.(5.19) Then for every t≥0it turns out that |u′(t)|2+2λ2µ1|u(t)|2≤/bracketleftbig 4u2 1+2/parenleftbig 3δ2λ4σ+λ2µ2/parenrightbig u2 0/bracketrightbig exp/parenleftbig −2rλ2σt/parenrightbig .(5.20) ProofFor everyε>0 we introduce the regularized coefficient cε(t) :=1 ε/integraldisplayt+ε tc(s)ds∀t≥0. It is easy to see that cε∈C1([0,+∞),R) and satisfies the following estimates: µ1≤cε(t)≤µ2∀t≥0, (5.21) |c(t)−cε(t)| ≤ω(ε)∀t≥0, (5.22) |c′ ε(t)| ≤ω(ε) ε∀t≥0. (5.23) Approximated energy For everyε >0 we consider the approximated hyperbolic energyEε(t) defined in (1.10). Since −1 2|u′(t)|2−2δ2λ4σ|u(t)|2≤2δλ2σu(t)u′(t)≤ |u′(t)|2+δ2λ4σ|u(t)|2, we deduce that 1 2|u′(t)|2+µ1λ2|u(t)|2≤Eε(t)≤2|u′(t)|2+(3δ2λ4σ+λ2µ2)|u(t)|2(5.24) for everyε>0 andt≥0. The time-derivative of Eε(t) is E′ ε(t) =−2δλ2σ|u′(t)|2−2δλ2σ+2c(t)|u(t)|2 −2λ2(c(t)−cε(t))u(t)u′(t)+λ2c′ ε(t)|u(t)|2, (5.25) hence E′ ε(t)≤ −2δλ2σ|u′(t)|2−/parenleftbig 2δλ2σ+2c(t)−λ2|c′ ε(t)|/parenrightbig |u(t)|2 +2λ2|c(t)−cε(t)|·|u(t)|·|u′(t)|. (5.26) 22Statement (1) We claim that, for a suitable choice of ε, it turns out that E′ ε(t)≤0∀t≥0. (5.27) If we prove this claim, then we apply (5.24) with that particular value o fεand we obtain that 1 2|u′(t)|2+µ1λ2|u(t)|2≤Eε(t)≤Eε(0)≤2u2 1+(3δ2λ4σ+λ2µ2)u2 0, which is equivalent to (5.18). Inordertoprove(5.27),weconsiderthewholeright-handsideof( 5.26)asaquadratic form in|u(t)|and|u′(t)|. Since the coefficient of |u′(t)|2is negative, this quadratic form is less than or equal to 0 for all values of |u(t)|and|u′(t)|if and only if 2δλ2σ·/parenleftbig 2δλ2σ+2c(t)−λ2|c′ ε(t)|/parenrightbig −λ4|c(t)−cε(t)|2≥0, hence if and only if 4δ2λ4σ−2c(t)≥ |c(t)−cε(t)|2+2δλ2σ−2|c′ ε(t)|. (5.28) Now in the left-hand side we estimate c(t) from below with µ1, and we estimate from above the terms in the right-hand side as in (5.22) and (5.23). We obt ain that (5.28) holds true whenever 4δ2µ1≥ω2(ε) λ4σ−2+2δω(ε) λ2σε. This condition is true when ε:= 1/λthanks to assumption (3.6). This completes the proof of (5.18). Statement (2) Let us assume now that λ≥1 and that (5.19) holds true for some r∈(0,δ). In this case we claim that, for a suitable choice of ε>0, the stronger estimate E′ ε(t)≤ −2rλ2σEε(t)∀t≥0 (5.29) holds true, hence Eε(t)≤Eε(0)exp/parenleftbig −2rλ2σt/parenrightbig ∀t≥0. Due to (5.24), this is enough to deduce (5.20). So it remains to prove (5.29). Owing to (5.25), inequality (5.29) is equivalent to 2λ2σ(δ−r)|u′(t)|2+/bracketleftbig 2λ2σ+2(δc(t)−rcε(t))−λ2c′ ε(t)−4rδ2λ6σ/bracketrightbig |u(t)|2 +2/bracketleftbig λ2(c(t)−cε(t))−2rδλ4σ/bracketrightbig u(t)u′(t)≥0. Keeping (1.4) and (5.21) into account, the last inequality is proved if w e show that 2λ2σ(δ−r)|u′(t)|2+/bracketleftbig 2λ2σ+2(δµ1−rµ2)−λ2|c′ ε(t)|−4rδ2λ6σ/bracketrightbig |u(t)|2 23−2/bracketleftbig λ2|c(t)−cε(t)|+2rδλ4σ/bracketrightbig |u(t)|·|u′(t)| ≥0. As in the proof of the first statement, we consider the whole left-h and side as a quadratic form in |u(t)|and|u′(t)|. The coefficient of |u′(t)|is positive because r < δ. Therefore, this quadratic form is nonnegative for all values of |u(t)|and|u′(t)|if and only if 2λ2σ(δ−r)·/bracketleftbig 2λ2σ+2(δµ1−rµ2)−λ2|c′ ε(t)|−4rδ2λ6σ/bracketrightbig ≥/bracketleftbig λ2|c(t)−cε(t)|+2rδλ4σ/bracketrightbig2. Now we rearrange the terms, and we exploit (5.22) and (5.23). We ob tain that the last inequality is proved if we show that 4(δ−r)(δµ1−rµ2)≥λ2−4σω2(ε)+2δ/parenleftbig 1+2rελ2σ/parenrightbigω(ε) ελ2σ+8rδ3 λ2−4σ.(5.30) Finally, we choose ε:= 1/λ, so that (5.30) becomes 4(δ−r)(δµ1−rµ2)≥/bracketleftbigg λ1−2σω/parenleftbigg1 λ/parenrightbigg/bracketrightbigg2 +2δ/parenleftbigg 1+2r λ1−2σ/parenrightbigg/bracketleftbigg λ1−2σω/parenleftbigg1 λ/parenrightbigg/bracketrightbigg +8rδ3 λ2−4σ. Sinceλ≥1 andσ≤1/2, this inequality follows from assumption (5.19). /square 5.2.1 Proof of Theorem 3.2 Let us rewrite (3.5) in the form Λ∞= limsup λ→+∞λ1−2σω/parenleftbigg1 λ/parenrightbigg . (5.31) Due to (3.4), there exists ν≥1 such that (3.6) holds true for every λ≥ν. Therefore, we can apply statement (1) of Lemma 5.2 to the components uk(t) ofu(t) corresponding to eigenvalues λk≥ν. Ifu0kandu1kdenote the corresponding components of initial data, estimate (5.18) read as |u′ k(t)|2+2λ2 kµ1|uk(t)|2≤4|u1k|2+2/parenleftbig 3δ2λ4σ k+λ2 kµ2/parenrightbig |u0k|2. Sinceσ≤1/2 and we chose ν≥1, this implies that |u′ k(t)|2+2λ2 kµ1|uk(t)|2≤4|u1k|2+2/parenleftbig 3δ2+µ2/parenrightbig λ2 k|u0k|2. Summing over all λk≥νwe obtain exactly (3.7). This proves that uν,+(t) is bounded with values in D(A1/2) andu′ ν,+(t) is bounded with values in H. The continuity of u(t) andu′(t) with values in the same spaces follows from the uniform convergence of series as in the proof of Theorem 3.1./square 245.2.2 Proof of Theorem 3.9 Let us rewrite (3.5) in the form (5.31). Due to (3.4), there exists r >0 andν≥1 such that (5.19) holds true for every λ≥ν. Therefore, we can apply statement (2) of Lemma 5.2 to the components uk(t) ofu(t) corresponding to eigenvalues λk≥ν. Ifu0k andu1kdenote the corresponding components of initial data, estimate (5 .20) reads as /parenleftbig |u′ k(t)|2+2λ2 kµ1|uk(t)|2/parenrightbig exp/parenleftbig 2rλ2σ kt/parenrightbig ≤4|u1k|2+2/parenleftbig 3δ2λ4σ k+λ2 kµ2/parenrightbig |u0k|2. Sinceσ≤1/2 and we chose ν≥1, this implies that /parenleftbig |u′ k(t)|2+2λ2 kµ1|uk(t)|2/parenrightbig exp/parenleftbig 2rλ2σ kt/parenrightbig ≤4|u1k|2+2/parenleftbig 3δ2+µ2/parenrightbig λ2 k|u0k|2 for everyt≥0. Summing over all λk≥νwe obtain (3.12) with a constant Kdepending only onµ1,µ2, andδ. The continuity of u(t) andu′(t) with values in the suitable spaces follows from the uniform convergence of series as in the proof of Th eorem 3.1. /square 6 The (DGCS)-phenomenon In this section we prove Theorem 3.10. Let us describe the strateg y before entering into details. Roughly speaking, what we need is a solution u(t) whose components uk(t) are small at time t= 0 and huge at time t>0. The starting point is given by the following functions b(ε,λ,t) := (2ελ−δλ2σ)t−εsin(2λt), w(ε,λ,t) := sin(λt)exp(b(ε,λ,t)), (6.1) γ(ε,λ,t) := 1+δ2 λ2−4σ−16ε2sin4(λt)−8εsin(2λt). (6.2) With some computations it turns out that w′′(ε,λ,t)+2δλ2σw′(ε,λ,t)+λ2γ(ε,λ,t)w(ε,λ,t) = 0 ∀t∈R, where “primes” denote differentiation with respect to t. As a consequence, if we set c(t) :=γ(ε,λ,t) andε:=ω(1/λ), the function u(t) :=w(ε,λ,t) turns out to be a solution to (1.8) which grows as the right-hand side of (4.5). Unfort unately this is not enough, because we need to realize a similar growth for countably ma ny components with the same coefficient c(t). To this end, we argue as in [6]. We introduce a suitable decreasing sequ encetk→0+, and in the interval [ tk,tk−1] we design the coefficient c(t) so thatuk(tk) is small and uk(tk−1) is huge. Then we check that the piecewise defined coefficient c(t) has the required time-regularity, and that uk(t) remains small for t∈[0,tk] and remains huge fort≥tk−1. This completes the proof. Roughly speaking, the coefficient c(t) plays on infinitely many time-scales in order to “activate” countably many components, but these countably m any actions take place 25onebyoneindisjointtimeintervals. Ofcoursethismeansthatthelen gthstk−1−tkofthe “activationintervals”tendto0as k→+∞. Inordertoobtainenoughgrowth, despiteof the vanishing length of activationintervals, we areforced to assum e thatλω(1/λ)≫λ2σ asλ→+∞. In addition, components do not grow exactly as exp( λω(1/λ)t), but just more than exp( ϕ(λ)t) and exp(ψ(λ)t). This is the reason why this strategy does not work when λω(1/λ)∼λ2σandδ is small. In this case one would need components growing exactly as ex p(λω(1/λ)t), but this requires activation intervals of non-vanishing length, which are thus forced to overlap. In a certain sense, the coefficient c(t) should work once againoninfinitely many time-scales, but now the countably many actions should take place in the same time. Definition of sequences From (3.13) and (3.14) it follows that lim x→+∞x1−2σω/parenleftbigg1 x/parenrightbigg = +∞, (6.3) lim x→+∞1 x1−2σω(1/x)+ϕ(x) xω(1/x)+ψ(x) xω(1/x)= 0, (6.4) and a fortiori lim x→+∞x1+2σω/parenleftbigg1 x/parenrightbigg = +∞, (6.5) lim x→+∞x2σ+ϕ(x)+ψ(x) xω/parenleftbigg1 x/parenrightbigg = 0. (6.6) Let us consider the sequence {λk}, which we assumed to be unbounded. Due to (6.5) and (6.4) we can assume, up to passing to a subsequence (not relabeled), that the following inequalities hold true for every k≥1: λk>4λk−1, (6.7) λ1+2σ kω/parenleftbigg1 λk/parenrightbigg ≥δ4 210π21 λ2−8σ k−1+4k2 π2λ2 k−1, (6.8) λ1+2σ kω/parenleftbigg1 λk/parenrightbigg ≥4k2 π2λ3 k−1/parenleftbig λ2σ k−1+ϕ(λk−1)+ψ(λk−1)/parenrightbig ω/parenleftbigg1 λk−1/parenrightbigg ,(6.9) λ1+2σ kω/parenleftbigg1 λk/parenrightbigg ≥λk−1/parenleftbig λ2σ k−1+ϕ(λk−1)+ψ(λk−1)/parenrightbig ω/parenleftbigg1 λk−1/parenrightbigg ,(6.10) 1 λ1−2σ kω(1/λk)+ϕ(λk) λkω(1/λk)+ψ(λk) λkω(1/λk)≤π2 4k21 λ2 k−1. (6.11) Now let us set tk:=4π λk, s k:=π λk/floorleftbigg 2λk λk−1/floorrightbigg , (6.12) 26where⌊α⌋denotes the largest integer less than or equal to α, and εk:=/braceleftbiggλ2σ k+ϕ(λk)+ψ(λk) λkω/parenleftbigg1 λk/parenrightbigg/bracerightbigg1/2 . Properties of the sequences We collect in this section of the proof all the properties of the sequences which are needed in the sequel. First of all, it is clear thatλk→+∞, hencetk→0 andεk→0 (because of (6.6)). Moreover it turns out that tk−1 4=π λk−1≤sk≤2π λk−1=tk−1 2. (6.13) Keeping (6.7) into account, it follows that tk<sk<tk−1∀k≥1, and in particular also sk→0. In addition, it turns out that sin(λktk) = sin(λksk) = 0 (6.14) and |cos(λktk)|=|cos(λksk)|= 1. (6.15) Sinceσ <1/2,λk→+∞,εk→0,tk→0, keeping (6.3) and (6.4) into account, we deduce that the following seven inequalities are satisfied provided th atkis large enough: δ2 λ2−4σ k+16ε2 k+8εk≤1 2, (6.16) εk≤1 4, (6.17) 16πεk+16πδ λ1−2σ k≤2π, (6.18) 1 λ1−2σ kω(1/λk)+ϕ(λk) λkω(1/λk)+ψ(λk) λkω(1/λk)≤1 52·210·π2, (6.19) δ2 (4π)2−4σ(2tk)1−2σsup/braceleftbiggx1−2σ ω(x):x∈(0,tk)/bracerightbigg ≤1 5, (6.20) λ1−2σ kω/parenleftbigg1 λk/parenrightbigg ≥δ2, (6.21) 2δ2 λ2−4σ k−1ω(1/λk−1)≤1 5. (6.22) 27Letk0∈Nbe a positive integer such that (6.16) through (6.22) hold true for e very k≥k0. From (6.21) it follows that εkλk≥δλ2σ k∀k≥k0. (6.23) From (6.19) it follows that 32πεk ω(1/λk)≤1 5∀k≥k0. (6.24) Sincesk≥π/λk−1(see the estimate from below in (6.13)), from (6.8) it follows that εkλksk≥δ2 321 λ2−4σ k−1∀k≥k0, (6.25) εkλksk≥2k∀k≥k0, (6.26) from (6.9) it follows that εkλksk≥2kεk−1λk−1∀k≥k0, (6.27) and from (6.11) it follows that εkλksk≥2k/parenleftbig λ2σ k+ϕ(λk)+ψ(λk)/parenrightbig ∀k≥k0. (6.28) As a consequence of (6.26) through (6.28) it turns out that 2εkλksk≥kεk−1λk−1+2k/parenleftbig λ2σ k+ϕ(λk)+ψ(λk)/parenrightbig +k∀k≥k0.(6.29) Finally, from (6.10) it follows that εkλk≥εk−1λk−1∀k≥k0. (6.30) Definition of c(t)andu(t) For every k≥1, letℓk:R→Rbe defined by ℓk(t) :=δ2 tk−1−sk/parenleftbigg1 λ2−4σ k−1−1 λ2−4σ k/parenrightbigg (t−sk)+1+δ2 λ2−4σ k∀t∈R. Thanks to (6.14), ℓk(t) is the affine function such that ℓk(sk) =γ(εk,λk,sk) and ℓk(tk−1) =γ(εk−1,λk−1,tk−1). Letk0∈Nbe such that (6.16) through (6.22) hold true for every k≥k0. Let us set c(t) := 1 if t≤0, γ(εk,λk,t) ift∈[tk,sk] for somek≥k0, ℓk(t) if t∈[sk,tk−1] for somek≥k0+1, γ(εk0,λk0,sk0) ift≥sk0. 28The following picture describes this definition. The coefficient c(t) is constant for t≤0 and fort≥sk0. In the intervals [ tk,sk] it coincides with γ(εk,λk,t), hence it oscillates, with period of order λ−1 kand amplitude of order εk, around a value which tends to 1. In the intervals [ sk,tk−1] it is just the affine interpolation of the values at the endpoints. /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet sk+2tk+1sk+1tksk tk−1sk−1period∼λ−1 kperiod∼λ−1 k−1 1+δ2 λ2−4σ k∼εk∼εk−1 For everyk≥k0, let us consider the solution uk(t) to the Cauchy problem u′′ k(t)+2δλ2σ ku′ k(t)+λ2 kc(t)uk(t) = 0, with “initial” data uk(tk) = 0, u′ k(tk) =λkexp/parenleftbig (2εkλk−δλ2σ k)tk/parenrightbig . (6.31) Then we set ak:=1 kλkexp(−kϕ(λk)), (6.32) and finally u(t) :=∞/summationdisplay k=k0akuk(t)ek. Weclaim that c(t) satisfies (3.15)and(3.16), andthat u(t)satisfies (3.17)and(3.18). The rest of the proof is a verification of these claims. Estimate and continuity of c(t) We prove that |c(t)−1| ≤1 2∀t≥0, (6.33) 29which is equivalent to (3.15), and that c(t) is continuous on the whole real line. To this end, it is enough to check (6.33) in the intervals [ tk,sk], because in the intervals [sk,tk−1]the function c(t) isjust aninterpolationofthevalues at theendpoints, and it is constant for t≤0 and fort≥sk0. In the intervals [ tk,sk] the function c(t) coincides with γ(εk,λk,t), hence from (6.2) it turns out that |c(t)−1|=|γ(εk,λk,t)−1| ≤δ2 λ2−4σ k+16ε2 k+8εk, (6.34) so that (6.33) follows immediately from (6.16). Since the right-hand side of (6.34) tends to 0 as k→+∞, the same estimate shows also thatc(t)→1 ast→0+, which proves the continuity of c(t) int= 0, the only point in which continuity was nontrivial. Estimate on c′(t) We prove that |c′(t)| ≤32εkλk∀t∈(tk,sk),∀k≥k0, (6.35) |c′(t)| ≤32εkλk∀t∈(sk,tk−1),∀k≥k0+1. (6.36) Indeed in the interval ( tk,sk) it turns out that |c′(t)|=|γ′(εk,λk,t)|=/vextendsingle/vextendsingle−64ε2 kλksin3(λkt)cos(λkt)−16εkλkcos(2λkt)/vextendsingle/vextendsingle ≤64ε2 kλk+16εkλk= 16εkλk(4εk+1), so that (6.35) follows from (6.17). In the interval ( sk,tk−1) it turns out that |c′(t)|=δ2 tk−1−sk/parenleftbigg1 λ2−4σ k−1−1 λ2−4σ k/parenrightbigg ≤δ2 tk−1−sk·1 λ2−4σ k−1≤δ2 sk·1 λ2−4σ k−1, where the last inequality follows from the estimate from above in (6.13 ). At this point (6.36) is equivalent to (6.25). Modulus of continuity of c(t) Let us prove that c(t) satisfies (3.16). Since c(t) is continuous, and constant for t≤0 andt≥sk0, it is enough to verify its ω-continuity in (0,sk0]. In turn, the ω-continuity in (0 ,sk0] is proved if we show that |c(ti)−c(tj)| ≤1 5ω(|ti−tj|)∀i≥k0,∀j≥k0, (6.37) |c(a)−c(b)| ≤1 5ω(|a−b|)∀(a,b)∈[tk,sk]2,∀k≥k0, (6.38) 30|c(a)−c(b)| ≤1 5ω(|a−b|)∀(a,b)∈[sk,tk−1]2,∀k≥k0+1. (6.39) Indeed, any interval [ s,t]⊆(0,sk0] can be decomposed as the union of at most 5 intervals whose endpoints fit in one of the 3 possibilities above. Let us prove (6.37). From (6.14) it turns out that |c(ti)−c(tj)|=δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 λ2−4σ i−1 λ2−4σ j/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 λ2 i−1 λ2 j/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−2σ , where the inequality follows from the fact that the function x→x1−2σis (1−2σ)-H¨ older continuous with constant equal to 1. Now from (6.12) it follows that δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 λ2 i−1 λ2 j/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−2σ =δ2 (4π)2−4σ|t2 i−t2 j|1−2σ=δ2 (4π)2−4σ|ti+tj|1−2σ|ti−tj|1−2σ ω(|ti−tj|)ω(|ti−tj|). Since|ti+tj| ≤2tk0and|ti−tj| ≤tk0, we conclude that |c(ti)−c(tj)| ≤δ2 (4π)2−4σ(2tk0)1−2σsup/braceleftbiggx1−2σ ω(x):x∈(0,tk0)/bracerightbigg ω(|ti−tj|), so that (6.37) follows from (6.20). Let us prove (6.38). Since c(t) isπ/λkperiodic in [ tk,sk], for every ( a,b)∈[tk,sk]2 there exists ( a1,b1)∈[tk,sk]2such thatc(a) =c(a1),c(b) =c(b1), and|a1−b1| ≤π/λk. Thus from (6.35) it follows that |c(a)−c(b)|=|c(a1)−c(b1)| ≤32εkλk|a1−b1|= 32εkλk|a1−b1| ω(|a1−b1|)ω(|a1−b1|), so that we are left to prove that 32εkλk|a1−b1| ω(|a1−b1|)≤1 5. (6.40) Due to (2.6), (2.5), and the fact that |a1−b1| ≤π/λk, it turns out that |a1−b1| ω(|a1−b1|)≤π/λk ω(π/λk)≤π λkω(1/λk), so that now (6.40) follows from (6.24). Let us prove (6.39). Since c(t) is affine in [ sk,tk−1], for every aandbin this interval it turns out that |c(a)−c(b)|=δ2 tk−1−sk/parenleftbigg1 λ2−4σ k−1−1 λ2−4σ k/parenrightbigg |a−b|. 31Sincesk≤tk−1/2, it follows that |c(a)−c(b)| ≤2δ2 tk−11 λ2−4σ k−1·|a−b|=2δ2 tk−11 λ2−4σ k−1·|a−b| ω(|a−b|)·ω(|a−b|). Due to (2.6), (2.5), and the fact that |a−b| ≤tk−1, it turns out that |a−b| ω(|a−b|)≤tk−1 ω(tk−1)≤tk−1 ω(1/λk−1), so that now (6.39) is a simple consequence of (6.22). Energy functions Let us introduce the classic energy functions Ek(t) :=|u′ k(t)|2+λ2 k|uk(t)|2, Fk(t) :=|u′ k(t)|2+λ2 kc(t)|uk(t)|2. Due to (3.15), they are equivalent in the sense that 1 2Ek(t)≤Fk(t)≤3 2Ek(t)∀t∈R. Therefore, proving (3.17) is equivalent to showing that ∞/summationdisplay k=k0a2 kEk(0)exp(2rϕ(λk))<+∞ ∀r>0, (6.41) while proving (3.18) is equivalent to showing that ∞/summationdisplay k=k0a2 kFk(t)exp(−2Rψ(λk)) = +∞ ∀R>0,∀t>0. (6.42) We are thus left to estimating Ek(0) andFk(t). Estimates in [0,tk] We prove that Ek(0)≤λ2 kexp(4π)∀k≥k0. (6.43) To this end, we begin by estimating Ek(tk). From (6.31) we obtain that uk(tk) = 0 and |u′ k(tk)| ≤λkexp(2εkλktk) =λkexp(8πεk), so that Ek(tk)≤λ2 kexp(16πεk). (6.44) 32Now the time-derivative of Ek(t) is E′ k(t) =−4δλ2σ k|u′ k(t)|2−2λ2 k(c(t)−1)u′ k(t)uk(t)∀t∈R. Therefore, from (3.15) it follows that E′ k(t)≥ −4δλ2σ kEk(t)−λk|c(t)−1|·2|u′ k(t)|·λk|uk(t)| ≥ −/parenleftbigg 4δλ2σ k+λk 2/parenrightbigg Ek(t) for everyt∈R. Integrating this differential inequality in [0 ,tk] we deduce that Ek(0)≤Ek(tk)exp/bracketleftbigg/parenleftbigg 4δλ2σ k+λk 2/parenrightbigg tk/bracketrightbigg . Keeping (6.44) and (6.12) into account, we conclude that Ek(0)≤λ2 kexp/parenleftbigg 16πεk+16πδ λ1−2σ k+2π/parenrightbigg , so that (6.43) follows immediately from (6.18). Estimates in [tk,sk] In this interval it turns out that uk(t) :=w(εk,λk,t), where w(ε,λ,t) is the function defined in (6.1). Keeping (6.14) and (6.15) into accou nt, we obtain that uk(sk) = 0 and |u′ k(sk)|=λkexp(b(εk,λk,sk)) =λkexp/parenleftbig (2εkλk−δλ2σ k)sk/parenrightbig . Therefore, from (6.23) it follows that |u′ k(sk)| ≥λkexp(εkλksk), and hence Fk(sk) =Ek(sk)≥λ2 kexp(2εkλksk). (6.45) Estimates in [sk,tk−1] We prove that Fk(tk−1)≥λ2 kexp(2εkλksk−4δλ2σ ktk−1). (6.46) Indeed the time-derivative of Fk(t) is F′ k(t) =−4δλ2σ k|u′ k(t)|2+λ2 kc′(t)|uk(t)|2∀t∈(sk,tk−1). Sincec′(t)>0 in (sk,tk−1), it follows that F′ k(t)≥ −4δλ2σ k|u′ k(t)|2≥ −4δλ2σ kFk(t)∀t∈(sk,tk−1), and hence Fk(tk−1)≥Fk(sk)exp/parenleftbig −4δλ2σ k(tk−1−sk)/parenrightbig ≥Fk(sk)exp/parenleftbig −4δλ2σ ktk−1/parenrightbig . Keeping (6.45) into account, we have proved (6.46). 33Estimates in [tk−1,+∞) We prove that Fk(t)≥λ2 kexp/parenleftbig 2εkλksk−8δλ2σ kt−64εk−1λk−1t/parenrightbig ∀t≥tk−1.(6.47) To this end, let us set Ik:= [tk−1,+∞)\k−1/uniondisplay i=k0{ti,si}. First of all, we observe that |c′(t)| ≤32εk−1λk−1∀t∈Ik (6.48) Indeed we know from (6.35) and (6.36) that |c′(t)| ≤32εiλi∀t∈(ti,si)∪(si,ti−1), and of course c′(t) = 0 for every t>sk0. Now it is enough to observe that Ik= (tk0,sk0)∪(sk0,+∞)∪k−1/uniondisplay i=k0+1[(ti,si)∪(si,ti−1)], and thatεiλiis a nondecreasing sequence because of (6.30). Now we observe that the function t→Fk(t) is continuous in [ tk−1,+∞) and differ- entiable in Ik, with F′ k(t) =−4δλ2σ k|u′ k(t)|2+λ2 kc′(t)|uk(t)|2 ≥ −4δλ2σ k|u′ k(t)|2−|c′(t)| c(t)·λ2 kc(t)|uk(t)|2 ≥ −/parenleftbigg 4δλ2σ k+|c′(t)| c(t)/parenrightbigg Fk(t). Therefore, from (6.48) and (3.15) it follows that F′ k(t)≥ −/parenleftbig 4δλ2σ k+64εk−1λk−1/parenrightbig Fk(t)∀t∈Ik, and hence Fk(t)≥Fk(tk−1)exp/bracketleftbig −/parenleftbig 4δλ2σ k+64εk−1λk−1/parenrightbig (t−tk−1)/bracketrightbig ≥Fk(tk−1)exp/bracketleftbig −/parenleftbig 4δλ2σ k+64εk−1λk−1/parenrightbig t/bracketrightbig for everyt≥tk−1. Keeping (6.46) into account, we finally obtain that Fk(t)≥λ2 kexp/parenleftbig 2εkλksk−4δλ2σ ktk−1−4δλ2σ kt−64εk−1λk−1t/parenrightbig , from which (6.47) follows by simply remarking that t≥tk−1. 34Conclusion We are now ready to verify (6.41) and (6.42). Indeed from (6.32) an d (6.43) it turns out that a2 kEk(0)exp(2rϕ(λk))≤1 k2λ2 kexp(−2kϕ(λk))·λ2 kexp(4π)·exp(2rϕ(λk)) =1 k2exp(4π+2(r−k)ϕ(λk)). The argument of the exponential is less than 4 πwhenkis large enough, and hence the series in (6.41) converges. Let us consider now (6.42). For every t>0 it turns out that t≥tk−1whenkis large enough. For every such kwe can apply (6.47) and obtain that a2 kFk(t)exp(−2Rψ(λk)) ≥1 k2exp/parenleftbig −2kϕ(λk)−2Rψ(λk)+2εkλksk−8δλ2σ kt−64εk−1λk−1t/parenrightbig . Keeping (6.29) into account, it follows that a2 kFk(t)exp(−2Rψ(λk)) ≥1 k2exp/parenleftbig (k−64t)εk−1λk−1+2(k−R)ψ(λk)+(2k−8δt)λ2σ k+k/parenrightbig ≥1 k2exp(k) whenkis large enough. This proves that the series in (6.42) diverges. /square References [1]G. Chen, D. L. Russell ; A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39(1981/82), no. 4, 433–454. [2]S. P. Chen, R. Triggiani ; Proof of extensions of two conjectures on structural damping for elastic systems. Pacific J. Math. 136(1989), no. 1, 15–55. [3]S. P. Chen, R. Triggiani ; Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications. J. Differential Equa- tions88(1990), no. 2, 279–293. [4]S. P. Chen, R. Triggiani ; Gevrey class semigroups arising from elastic systems with gentle dissipation: the case 0 <α<1/2.Proc. Amer. Math. Soc. 110(1990), no. 2, 401–415. [5]F. Colombini ; Quasianalytic and nonquasianalytic solutions for a class of weakly hyperbolic Cauchy problems. J. Differential Equations 241(2007), no. 2, 293–304. 35[6]F. Colombini, E. De Giorgi, S. Spagnolo ; Sur le ´ equations hyperboliques avec des coefficients qui ne d´ ependent que du temp. (French) Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6(1979), no. 3, 511–559. [7]F. Colombini, E. Jannelli, S. Spagnolo ; Well-posedness intheGevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coe fficients de- pending on time. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)10(1983), no. 2, 291– 312. [8]M. Ghisi, M. Gobbino ; DerivativelossforKirchhoffequationswithnon-Lipschitz nonlinear term. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 8(2009), no. 4, 613–646. [9]M. Ghisi, M. Gobbino, H. Haraux ; Local and global smoothing effects for some linear hyperbolic equations with a strong dissipation. Trans. Amer. Math. Soc.To appear. Preprint arXiv:1402.6595 [math.AP] . [10]A. Haraux, M. ˆOtani; Analyticity and regularity for a class of second order evolution equation. Evol. Equat. Contr. Theor. 2(2013), no. 1, 101–117. [11]R. Ikehata ; Decayestimatesofsolutionsforthewaveequationswithstrongd amp- ing terms in unbounded domains. Math. Methods Appl. Sci. 24(2001), no. 9, 659– 670. [12]R. Ikehata, M. Natsume ; Energy decay estimates for wave equations with a fractional damping. Differential Integral Equations 25(2012), no. 9-10, 939–956. [13]R. Ikehata, G. Todorova, B. Yordanov ; Wave equations with strong damp- ing in Hilbert spaces. J. Differential Equations 254(2013), no. 8, 3352–3368. [14]J.-L. Lions, E. Magenes , Probl` emes aux limites non homog` enes et applications. Vol. 3. (French) Travaux et Recherches Mathmatiques, No. 20. D unod, Paris, 1970. [15]S. Matthes, M. Reissig ; Qualitative properties of structural damped wave mod- els.Eurasian Math. J. 4(2013), no. 3, 84–106. [16]K. Nishihara ; Degenerate quasilinear hyperbolic equation with strong damping. Funkcial. Ekvac. 27(1984), no. 1, 125–145. [17]K. Nishihara ; Decay properties of solutions of some quasilinear hyperbolic equa- tions with strong damping. Nonlinear Anal. 21(1993), no. 1, 17–21. [18]M. Reed, B. Simon ;Methods of Modern Mathematical Physics, I: Functional Analysis. Second edition . Academic Press, New York, 1980. [19]Y. Shibata ; Onthe rate of decay of solutions to linear viscoelastic equation. Math. Methods Appl. Sci. 23(2000), no. 3, 203–226. 36

2014-08-15

We consider a second order linear equation with a time-dependent coefficient c(t) in front of the "elastic" operator. For these equations it is well-known that a higher space-regularity of initial data compensates a lower time-regularity of c(t). In this paper we investigate the influence of a strong dissipation, namely a friction term which depends on a power of the elastic operator. What we discover is a threshold effect. When the exponent of the elastic operator in the friction term is greater than 1/2, the damping prevails and the equation behaves as if the coefficient c(t) were constant. When the exponent is less than 1/2, the time-regularity of c(t) comes into play. If c(t) is regular enough, once again the damping prevails. On the contrary, when c(t) is not regular enough the damping might be ineffective, and there are examples in which the dissipative equation behaves as the non-dissipative one. As expected, the stronger is the damping, the lower is the time-regularity threshold. We also provide counterexamples showing the optimality of our results.

Linear hyperbolic equations with time-dependent propagation speed and strong damping

1408.3499v1

arXiv:1311.3518v1 [math.GT] 14 Nov 2013THE DIMENSION OF THE LEAFWISE REDUCED COHOMOLOGY JES´US A.´ALVAREZ L ´OPEZ AND GILBERT HECTOR Abstract. Geometric conditions are given so that the leafwise reduced co- homology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the int ersection number of subfoliations with “appropriate coefficients”. The leafw ise reduced cohomol- ogy is also described for homogeneous foliations with dense leaves on closed nilmanifolds. 1.Introduction LetFbe aC∞foliation on a manifold M. Theleafwise de Rham complex (Ω·(F),dF) is the restriction to the leaves of the de Rham complex of M; i.e., Ω(F) is the space of differential forms on the leaves that are C∞on the ambient manifoldM, anddFis the de Rham derivative on the leaves. We use the notation Ω(F) =C∞(/logicalandtextT∗F) meaning C∞sections on M. The cohomology H·(F) = H·(Ω(F),dF) is called the leafwise cohomology ofF. It is well known that H·(F) canalso be defined as the cohomologyof Mwith coefficients in the sheafofgerms of C∞functions which are locally constant on the leaves, but we do not use this. The (weak)C∞topology on Ω( F) induces a topology on H·(F), which is non-Hausdorff in general [15]. The quotient space of H·(F) over the closure of its trivial subspace is called the leafwise reduced cohomology ofF, and denoted by H·(F). Similarly, we can also define Ω· c(F),H· c(F) andH· c(F) by considering compactly supported C∞sections of/logicalandtextTF∗. For degree zero we have that H0(F) =H0(F) is the space of C∞functions on Mthat are constant on each leaf—the so called (smooth) basic functions ; thus H0(F)∼=Rif the leaves are dense. Though density of the leaves seems to yield strong restrictions on the leafwise cohomology also for higher degr ee, this cohomol- ogy may be of infinite dimension when leaves are dense and Mis closed. In fact, for dense linear flows on the two-dimensional torus, we have dim H1(F) = 1 when the slope of the leaves is a diophantine irrational number [18], but dim H1(F) =∞ if the slope is a Liouville’s irrational number [30]. Nevertheless H1(F)∼=Rin both cases. This computation was later generalized to the case of linear f oliations on tori of arbitrary dimension [20, 8]. Other known properties of the leafwise cohomology are the following ones. The leafwisecohomologyofdegreeonewithcoefficientsinthenormalbun dleisrelatedto the infinitesimal deformations of the foliation [18]. For p= dimF, the dual space Hp c(F)′is canonically isomorphic to the space of holonomy invariant transver se distributions [15]—recall that for a topological vector space V, the dual space V′is 1991Mathematics Subject Classification. 57R30. Partially supported by Xunta de Galicia (Spain). 12 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR the space of continuous linear maps V→R.H·(F) is invariant by leaf preserving homotopies, and Mayer-Vietoris arguments can be applied [10], whic h was used to computeH·(F) for someexamples. For an arbitraryflow Fon the two-torus, it was proved that dim H1(F) =∞ifFis not minimal, and dim H1(F) = 1 if and only if FisC∞conjugate to a Diophantine linear flow [7]. The triviality of H1(F) implies the triviality of the linear holonomy [10], and is equivalent to Thurston’s stability if codim F= 1 andMis closed [6]. However, more general relations between the leafwise cohomology and the geometry of the foliation remain rather unknown. The above examples of linear foliations on tori could wrongly suggest thatH·(F) maybe offinite dimension if Mis closedand the leavesaredense. In fact, S. Hurder and the first author gave examples of foliations with dense leaves on closed Rie- mannian manifolds with an infinite dimensional space of leafwise harmon ic forms that areC∞on the ambient manifold [4], and this space is canonically injected in the leafwise reduced cohomology; indeed this injection is an isomorph ism at least for the so called Riemannian foliations [5]. So a natural problem is the fo llowing: Give geometric properties characterizing C∞foliations whose leafwise reduced co- homology is of finite dimension; specially for foliations wi th dense leaves on closed manifolds . The aim of this paper is to give an approach to this problem. The first a nd main geometric idea we use is the intersection number of subfoliations with “appropriate coefficients”. To explain it, consider the simplest example where M=T×Lwith thefoliation Fwhoseleavesarethe slices {∗}×L,whereT,Lareclosedmanifolds of dimensions q,p. Let (Ω·(L),dL) be the de Rhamcomplex of L, and letH·(L),H·(L) denote the homology and cohomology of Lwith real coefficients. Then Ω·(F) is theC∞closure ofC∞(T)⊗Ω·(L), wheredF= 1⊗dL. So Hk(F) =Hk(F)≡C∞(T)⊗Hk(L) becauseH·(L) is of finite dimension. Assume Lis oriented for simplicity. Then recall that Poincar´ e duality and integration of differential forms e stablish canonical isomorphisms Hk(L)∼=Hq−k(L), Hk(L)′∼=Hk(L), such that the canonical pairing between Hk(L)⊗Hk(L)′→Rcorresponds to the intersection pairing Hp−k(L)⊗Hk(L)→R[32]. Hence Hk(F)∼=C∞(T)⊗Hp−k(L), (1) Hk(F)′∼=C∞(T)′⊗Hk(L), (2) such that the canonical pairing Hk(F)⊗Hk(F)′→Rcorresponds to the product of the evaluation of distributions on C∞functions and the intersection pairing. Now observe that, according to [32], the right hand side spaces in (1 ) and (2) are respectively generated by elements of the form f⊗[K1] andD⊗[K2], wherefis aC∞function on T,Dis a distribution on T, andK1,K2⊂Lare closed oriented submanifolds of dimensions p−k,k. Hence dim Hk(F) =∞is equivalent to the existence of sequences of elements fm⊗[K1] andDn⊗[K2] as above so that K1,K2 havenon-trivial intersection number and Dn(fn)/\e}atio\slash= 0 if and only if m=n; of course this holds just when Hk(L)/\e}atio\slash= 0 andq>0. Now consider each element fm⊗[K1] as the family of homology classes fm(t)[{t}×K1]∈Hp−k({t}×L), t∈T ,LEAFWISE REDUCED COHOMOLOGY 3 determined by the family of closed oriented submanifolds {t} ×K1of the leaves ofFand the family of coefficients fm(t). The elements Dn⊗[K2] have a similar interpretationby considering distributions as generalized function s. A key property here is that the families {t}×K1and{t}×K2depend smoothly on t, determining C∞subfoliations F1,F2ofF. Other key properties are the C∞dependence of the coefficients fm(t) ont, and the distributional dependence of the generalized coefficients Dn(t) ont. This means that the fmareC∞basic functions of F1 and theDnare “distributional basic functions” of F2; i.e., theDnare holonomy invariant transverse distributions of F2. It turns out that these key properties are enough to generalize the above ideas in a way that can be applied even when the leavesaredense, obtainingourfirstmaintheoremthatroughlyass ertsthefollowing: For aC∞oriented foliation Fof dimension p, we have dimHk c(F) =∞whenF has oriented subfoliations F1,F2of dimensions k−p,p, and there is a sequence of basic functions fmofF1and a sequence of transverse invariant distributions Dnof F2, such that the corresponding “intersection numbers” are no n-trivial if and only if m=n—certain simple conditions are also required for the “inter section numbers” to be defined . We do not know whether such conditions form a characterization of the cases where dim Hk c(F) =∞; this depends on whether it is possible to “smoothen” the representatives of classes in certain leafwise hom ologies introduced in [3]. Indeed the above fmandDnplay the rˆ ole of coefficients in homology, assigning a number to each leaf of the subfoliations; the way these n umbers vary from leaf to leaf is what makes these coefficients appropriate. Though these conditions are difficult to check in general, this result h as many easy to apply corollaries. For instance, suppose an oriented foliatio nFisRiemann- ian—in the sense that all of its holonomy transformations are local isom etries for some Riemannian metric on local transversals [29, 25]. Then dim H· c(F) =∞if Fis of positive codimension and some leaf of Fcontains homology classes with non-trivial intersection. These conditions are quite simple to verify . In this case, the infinitely many linearly independent classes obtained in H· c(F) can be consid- ered as “transverse diffusions” of the homology classes in the leaf. This diffusion idea is inspired by the unpublished preprints [19, 4]. Indeed [19] is the g erminal work about the relation of the analysis on the leaves and on the ambie nt manifold obtained by transverse diffusion. Other consequences of the above general theorem hold when Fis asuspension foliation. That is, the ambient manifold of Fis the total space of a fiber bundle M→Bwith the leaves transverse to the fibers, and such that the restr iction of the bundle projection to each leaf is a regular covering of the bas eB. Now dimH· c(F) =∞whenBis oriented and has homology classes with non-trivial intersection satisfying additional properties with respect to the h olonomy of F. In this case the leaves may not contain homology classes with non-trivia l intersection, and thus the idea of “transverse diffusion” of homology classes in th e leaves may fail. In fact we shall see that the infinite dimension of H1 c(F) may be more related to the number of ends of the leaves. To explain another theorem of this paper, recall that a foliation Fon a manifold Mis aLie foliation when it has a complete transversal diffeomorphic to an open subset of a Lie group Gso that holonomy transformationson this transversalcorre- spond to restrictions of left translations on G—this type of foliations play a central rˆ ole in the study of Riemannian foliations [25]. The Lie algebra gofGis called the4 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR structural Lie algebra ofF; we may also simply say that Fis a Lieg-foliation. In this case, if Mis closed and oriented, and gis compact semisimple, then we obtain that dim H·(F) =∞when some additional hypotheses are satisfied. Again we use homology classes with non-trivial intersection in the hypotheses, b ut now they live in the homology of M. The proof of this result is reduced to the case of suspension foliations to apply what we already know. This reduction process con tains rather delicate arguments based on the work [2] of the first author. The above results are negative in the sense that all of them give con ditions for the nonexistence of finite Betti numbers for the reduced leafwise cohomology. In contrast, our final theorem shows that the reduced leafwise coh omology of the so calledhomogeneous foliations with dense leaves in closed nilmanifolds is isomorphic to the cohomology of the Lie algebra defining the foliation. This has be en also proved by X. Masa with different techniques. Acknowledgment . We wish to thank F. Alcalde for many helpful conversations. The first author would like to thank the hospitality of the Institut de Math´ ema- tiques et d’Informatique of the University Claude Bernard of Lyon s everal times during the preparation of this work. We would like also to thank the re feree for important corrections. 2.Main results For the sake of simplicity, all manifolds, foliations, maps, functions, differential forms and actions will be assumed to be C∞from now on, unless the contrary is explicitly stated. LetFbe a foliation on a manifold M. For any closed saturated subset S⊂M, let Ω· S(F)⊂Ω·(F) be the subcomplex of leafwise differential forms whose support has compact intersection with S. Consider the topology on Ω· S(F) determined as follows: A sequence αn∈Ω· S(F) converges to zero if it converges to zero in Ω·(F) and there is a compact subset K⊂Ssuch thatS∩suppαn⊂Kfor alln. We have the corresponding cohomology H· S(F), and reduced cohomology H· S(F). With this notations, observe that Ω·(F) = Ω· ∅(F) and Ω· c(F) = Ω· M(F) as topological vector spaces. Letf: (M1,F1)→(M2,F2) be a map of foliated manifolds, and let Si⊂Mi, i= 1,2, beclosedsaturatedsubsetssuchthat the restriction f:S1→S2is aproper map. Then f∗(Ω· S2(F2))⊂Ω· S1(F1), yielding a homomorphism f∗:H· S2(F2)→ H· S1(F1). In particular we get f∗:H· c(F2)→ H· S1(F1) iff:S1→M2is proper. The following is what we need to define the intersection number of sub foliations with “appropriate coefficients”: •An oriented foliation Fon a manifold M, and two immersed oriented sub- foliationsιi: (Mi,Fi)→(M,F),i= 1,2. •dimF= dimF1+dimF2, and codim F= codim F1. •Eachιiistransversely regular in the sense that it defines embeddings of small enough local transversals of Fiinto local transversals of F; i.e. the homomorphism defined by the differential of ιibetween the normal bundles ofFiandFis injective on the fibers. •A compactly supported basic function fofF1. •A holonomy invariant transverse distribution DofF2such that the map ι2: suppD→Mis proper.LEAFWISE REDUCED COHOMOLOGY 5 •The restrictions ι1|suppfandι2|suppDintersect transversely inFin the sense that, for all leaves LiofFiandLofFsuch thatL1⊂suppf,L2⊂ suppDandι1(L1)∪ι2(L2)⊂L, the immersed submanifolds ιi:Li→L intersect transversely in L. Observe that there are open neighborhoods, N1of suppfandN2of suppD, such that theιi|Niintersect transversely in F. Consider the pull-back diagram Tσ1− −−− →N1 σ2/arrowbt/arrowbtι1 N2ι2− −−− →M . Here T={(x1,x2)∈N1×N2|ι1(x1) =ι2(x2)}, and theσiare restrictions of the factor projections. It is easy to check th atι1×ι2: N1×N2→M×Mis transverse to the diagonal ∆, and thus Tis a manifold with dimT= codim F2. Moreover the σiare immersions, and σ2is transverse toF2. SoDdefines a distribution on T, which will be denoted by DT. We also have the locally constant intersection function ε:T→ {±1}, whereε(x1,x2) =±1 depending on whether the identity Tιi(xi)F ≡ι1∗Tx1F1⊕ι2∗Tx2F2 is orientation preserving or orientation reversing. On the other ha nd (ι1(suppf)×ι2(suppD))∩∆ iscompactbecauseit isaclosedsubsetofthe compactspace ι1(suppf)×ι1(suppf). So suppσ∗ 1f∩suppDT= (ι1×ι2)−1((ι1(suppf)×ι2(suppD))∩∆) is a compact subspace of Tsinceι1×ι2: suppf×suppD→M×Mis a proper map. Thus the following definition makes sense. Definition 2.1. With the above notations, the intersection number of (ι1,f) and (ι2,D), denoted by /a\}b∇acketle{t(ι1,f),(ι2,D)/a\}b∇acket∇i}ht, is defined as DT(g) for any compactly sup- ported function gonTwhich is equal to the product εσ∗ 1fon some neighborhood of suppσ∗ 1f∩suppDT. Now our first main theorem is the following. Theorem 2.2. LetFbe an oriented foliation on a manifold M, andιi: (Mi,Fi)→ (M,F),i= 1,2, transversely regular immersed oriented subfoliations. S uppose dimF= dimF1+ dimF2, andcodimF= codim F1. Letfmbe a sequence of compactly supported basic functions of F1, andDna sequence of holonomy invari- ant transverse distributions of F2such that each restriction ι2: suppDn→Mis a proper map. Suppose each pair ι1|suppfmandι2|suppDnintersect transversely in F, and/a\}b∇acketle{t(ι1,fm),(ι2,Dn)/a\}b∇acket∇i}ht /\e}atio\slash= 0if and only if m=n. ThendimHk c(F) =∞for k= dimF2. The following two corollaries are the first type of consequences of T heorem 2.2; the second corollary follows directly from the first one.6 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR Corollary 2.3. LetFbe an oriented foliation of codimension q >0,La leaf of F, andh:π1(L)→G∞ qits holonomy representation, where G∞ qis the group of germs at the origin of local diffeomorphisms of Rqwith the origin as fixed point. Let ιi:Ki→L,i= 1,2, be smooth immersions of closed oriented manifolds of com- plementary dimension and nontrivial intersection. Suppos e there is a Riemannian metric on Rqso that the elements in the image of the composites (3) π1(Ki)π1(ιi)− −−− →π1(L)h− −−− →G∞ q are germs of local isometries. Then dimHki c(F) =∞forki= dimKi,i= 1,2. Corollary 2.4. LetFbe an oriented Riemannian foliation of positive codimensio n. Suppose some leaf of Fhas homology classes of complementary degrees, k1andk2, with non-trivial intersection. Then dimHki c(F) =∞,i= 1,2. BeforestatingthenexttypeofcorollariesofTheorem2.2, recallt hatasuspension foliation Fis given as follows. Let π:L→Bbe a regular covering map of an oriented manifold, and let Γ be its group of deck transformations. F or any effective action of Γ on some manifold T, consider the right diagonal action of Γ on L×T: (z,t)γ= (zγ,γ−1t) forγ∈Γ and (z,t)∈L×T. Then Fis the foliation on M= (L×T)/Γwhoseleavesaretheprojectionsofthesubmanifolds L×{∗} ⊂L×T. The element in Mdefined by each ( z,t)∈L×Twill be denoted by [ z,t]. The mapρ:M→Bgiven byρ([z,t]) =π(z) is a fiber bundle projection with typical fiberT. The leaves of Fare transverse to the fibers of ρ, and define coverings of B. The leaf that contains [ z,t] can be canonically identified to L/Γt, where Γ tis the isotropy subgroup of Γ at t. This leaf is dense if and only if the Γ-orbit of tis dense inT. Corollary 2.5. With the above notation, let h:π1(B)→Γbe the surjective homomorphism defined by the regular covering LofB, and letιi:Ki→B,i= 1,2, be immersions of connected oriented manifolds of complemen tary dimension in B. SupposeK1is a closed manifold, ι2a proper map, and the homology class defined byι1has non-trivial intersection with the locally finite homolo gy class defined by ι2. For each i, letΓi⊂Γbe the image of the composite π1(Ki)π1(ιi)− −−− →π1(B)h− −−− →Γ. Letfmbe a sequence of compactly supported Γ1-invariant functions on T, andDn a sequence of Γ2-invariant distributions on Tsuch thatDn(fm)/\e}atio\slash= 0if and only if m=n. ThendimHk c(F) =∞fork= dimK2. Corollary 2.6. LetB,L,h,Γ,T,F,Ki,ιiandΓibe as in Corollary 2.5. Let µbe aΓ2-invariant measure on T. Suppose the closure of the image of Γ1in the topological group of diffeomorphisms of T(with the weak C∞topology)is a compact Lie group, and there is an infinite sequence of Γ1-invariant open subsets of Twith non-trivial µ-measure and pairwise disjoint Γ2-saturations. Then dimHk c(F) =∞ fork= dimK2. Observe that, in Corollary 2.6, the infinite sequence of Γ 1-invariant open sets may not be Γ 2-invariant, and their Γ 2-saturations may not be Γ 1-invariant. Corollary 2.7. LetB,L,h,Γ,TandFbe as in Corollary 2.5. Suppose that there is a loopc:S1→Bwith a lift to Lthat joins two distinct points of the end set of L. Leta=h([c])∈Γ, where[c]is the element of π1(B)represented by c, and assumeLEAFWISE REDUCED COHOMOLOGY 7 that the closure Hof the image of /a\}b∇acketle{ta/a\}b∇acket∇i}htin the topological group of diffeomorphisms ofT(with the weak C∞topology)is a compact Lie group. Suppose also that there is an infinite sequence of disjoint non-trivial H-invariant open subsets of T. Then dimH1 c(F) =∞. In Corollaries 2.5, 2.6 and 2.7, if Bis compact, then the leaves of Fcan only be dense when Lhas either one end or a Cantor space of ends, as follows from the following. Proposition 2.8. LetΓbe a finitely generated group with two ends, and C⊂Γ an infinite subgroup. Suppose Γacts continuously on some connected T1topological spaceX. Then the Γ-orbits are dense in Xif and only if so are the C-orbits. Now letFbe a Lie g-foliation on a closed manifold M. The following property characterizes such a type of foliations [11, 24, 25]. Let /tildewiderMbe the universal covering ofM,/tildewideFthe lift of Fto/tildewiderM, andGthe simply connected Lie group with Lie algebrag. Then the leaves of /tildewideFare the fibers of a fiber bundle /tildewiderM→G, which is equivariant with respect to some homomorphism h:π1(M)→G, where we consider the right action of π1(M) on/tildewiderMby deck transformations and the right action ofGon itself by right translations. This hand its image are respectively called the holonomy homomorphism andholonomy group ofF. Observe that the fibers ofDare connected because Gis simply connected (a connected coveringof G is given by the quotient of /tildewiderMwhose points are the connected components of these fibers). Theorem 2.9. With the above notation, suppose that Mis oriented and the struc- tural Lie algebra gofFis compact semisimple. Let ιi:Ki→M,i= 1,2, be immersions of closed oriented manifolds of complementar y dimension defining homology classes of Mwith non-trivial intersection. Let Γibe the image of the composite π1(Ki)π1(ιi)− −−− →π1(M)h− −−− →G. Suppose the group generated by Γ1∪Γ2is not dense in G. Letk= dimK2, and suppose either 1≤k≤2orι1is transverse to F. ThendimHk(F) =∞. The following is our final theorem. Theorem 2.10. LetHbe a simply connected nilpotent Lie group, K⊂Ha normal connected subgroup, and Γ⊂Ha discrete uniform subgroup whose projection to H/Kis dense. Let Fbe the foliation of the closed nilmanifold Γ\Hdefined as the quotient of the foliation on Hwhose leaves are the translates of K. Then there is a canonical isomorphism H·(F)∼=H·(k), wherekis the Lie algebra of K. The following two examples are of different nature. In both of them t here are infinitely many linearly independent leafwise reduced cohomology class es of degree one. But these classes are induced by the handles in the leaves in Exa mple 2.11, whereasthey areinduced by the “branches”of the leavesthat de fine a Cantorspace of ends in Example 2.12. Example 2.11 ([4]).LetLbe aZ-covering of the compact oriented surface of genus two; i.e., Lis a cylinder with infinitely many handles attached to it. Each handle contains two circles defining homology classes with non-trivial intersection. Hence for any injection of Zinto then-torusRn/Zn, the corresponding suspension8 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR foliation fulfills the hypotheses of Corollary 2.6, and thus has infinite d imensional reduced leafwise cohomologyof degree one. We could also use Corolla ry 2.4 instead of Corollary 2.6. Example 2.12. Let Γ be the free group with two generators, and La Γ-covering of the compact orientable surface of genus two. This Lhas a Cantor space of ends. Hence, for any injective homomorphism of Γ in a compact Lie group, t he reduced leafwise cohomology of degree one of the corresponding suspensio n foliation is of infinite dimension by Corollary 2.7. 3.Leafwise reduced cohomology and subfoliations This section is devoted to the proof of Theorem 2.2. With the notatio ns in- troduced in Section 2, the idea of the proof is the following. The ( ι1,fm) yield elementsζm∈ Hr c(F) by “leafwise Poincar´ eduality”. On the other hand, the argu- mentsin[15]showthateach Dncanbeconsideredasanelementin Hr Sn(F2)′, where Sn= suppDn⊂M. Moreover there are homomorphisms ι∗ 2:H· c(F)→ H· Sn(F2) since theι2:Sn→Mare proper maps. Then the result follows by verifying /a\}b∇acketle{t(ι1,fm),(ι2,Dn)/a\}b∇acket∇i}ht=Dn(ι∗ 2ζm). We first explain the way “leafwise Poincar´ e duality” works. Consider thetrans- verse complex Ω· c(TrF) introduced in [15], which will be only used for degree zero. For any representative Hof the holonomy pseudogroup of F, defined on some manifoldT, Ω0 c(TrF) is defined as the quotient of C∞c(T) over the subspace gen- erated by the functions of the type φ−h∗φ, whereh∈ Handφ∈C∞c(T) with suppφ⊂domh. As a topological vector space, Ω0 c(TrF) is independent of chosen representative of the holonomy pseudogroup. From the definition it easily follows that the dualspaceΩ0 c(TrF)′canbe canonicallyidentified tothe spaceofholonomy invariant transverse distributions of F. Now consider the representative Hof the holonomy pseudogroup induced by an appropriate locally finite covering of Mby foliation patches Ui; that is, iffi:Ui→ Tiis the localquotient map whosefibers arethe plaques in Ui, then appropriateness of this covering means that each equality fj=hi,jfionUi∩Ujdetermines a diffeomorphisms hi,j:fi(Ui∩Uj)→fj(Ui∩Uj), and the collection of all of these diffeomorphisms generate the pseudogroup HonT=/unionsqtext iTi. Fix also a partition of unityφisubordinated to the covering Ui. With these data we have a map Ωp c(F)→Ω· c(T) given by α/mapsto→/summationtext i/integraltext fiφiα, wherep= dimFand/integraltext fidenotes integration along the fibers of fi. This “integration along the leaves” induces an isomorphism Hp(F)∼=Ω0 c(TrF) of topological vector spaces, which is independent of the choice of the Uiandφi[15,§3.3]. So Hp c(F)′≡Hp c(F)′∼=Ω0 c(TrF)′; i.e., any holonomy invariant distribution Dcan be canonically considered as an element in Hp c(F)′. Moreover Dcan be also considered as an element in Hp S(F)′≡ Hp S(F)′forS= suppDasfollowsfromthe followingargument. Forany α∈Ωp c(F), it is easily verified that D/parenleftBig/summationtext i/integraltext fiφiα/parenrightBig depends only on the restriction of αto any neighborhood of the support of DinM. Therefore, if ζ∈Hp S(F),α∈Ωp S(F) is any representative of ζ, andβ∈Ωp c(F) has the same restriction as αto some neighborhood of S, thenD/parenleftBig/summationtext i/integraltext fiφiβ/parenrightBig does not depend on the choices of αand β, and thus this is a good definition of D(ζ).LEAFWISE REDUCED COHOMOLOGY 9 Theorem 2.2 will follow easily from the following result, which will be prove d in Section 4. Proposition 3.1. LetFbe an oriented foliation on a manifold M. Letι1: (M1,F1)→(M,F)be a transversely regular immersed oriented subfoliation w ith codimF= codim F1, andfa compactly supported basic function of F1. Then there is a class ζ∈ Hk c(F),k= dimF −dimF1, such that (4) /a\}b∇acketle{t(ι1,f),(ι2,D)/a\}b∇acket∇i}ht=D(ι∗ 2ζ) for any subfoliation ι2: (M2,F2)→(M,F)and any holonomy invariant transverse distribution DofF2so that the left hand side of (4)is defined. In the right hand side of(4),Dis considered as an element of Hk S(F2)′forS= suppD, andι∗ 2 denotes the homomorphism Hk c(F)→ Hk S(F2)induced byι2, which is defined since ι2:S→Mis a proper map. We do not know whether (4) completely determines ζ. If so,ζcould be called theleafwise Poincar´ e dual class of (ι1,f). Proof of Theorem 2.2. Letζm∈ Hk c(F) be the classes defined by the ( ι1,fm) ac- cording to Proposition 3.1. If Pn∈ Hk c(F)′is given by the composite Hk c(F)ι∗ 2− −−− → Hk Sn(F2)Dn− −−− →R, we havePn(ζm)/\e}atio\slash= 0 if and only if m=nby Proposition 3.1, yielding the linear independence of the ζm. /square 4.Leafwise Poincar ´e duality This section will be devoted to the proof of Proposition 3.1. 4.1.On the Thom class of a vector bundle. Thefollowinglemma isatechnical step in the proof of Proposition 3.1, which will be proved in Section 4.2. Lemma 4.1. LetMbe a manifold and π:E→Man oriented vector bundle. IdentifyMto the image of the zero section, whose normal bundle is canon ically oriented. There is a sequence Φnof representatives of the Thom class of Esuch that, iffis any function on M, V is any neighborhood of MinE,K⊂Mis any compact subset, and φ:V→Eis any map which restricts to the identity on M and its differential induces an orientation preserving auto morphism of the normal bundle ofM, thenπ−1(K)∩φ−1(suppΦ n)is compact for large enough n, and the sequence of functions/integraltext πφ∗(π∗fΦn)converges to foverKwith respect to the C∞ topology. Corollary 4.2. Letπ:E→Mbe an oriented vector bundle, and ι:N→Man immersion. Let πN:ι∗E→Nbe the pull-back vector bundle, and ˜ι:ι∗E→E the canonical homomorphism. Identify MandNto the image of the zero sections ofEandι∗E, respectively, and consider the induced orientations on th eir normal bundles. Let Vbe an open neighborhood of Ninι∗E, andh:V→Ean extension ofιsuch that the homomorphism between the normal bundles of NandM, defined by the differential of h, restricts to orientation preserving isomorphisms betwee n the fibers. Let Φnbe the forms on Egiven by Lemma 4.1, K⊂Na compact subset, and fa function on M. Thenπ−1 N(K)∩h−1(suppΦ n)is compact for large enoughn, and the sequence of functions/integraltext πNh∗(π∗fΦn)converge to ι∗foverK with respect to the C∞topology.10 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR Proof.LetU1,...,U mbe a finite open cover of Ksuch that each ι:Ui→Mis an embedding. For each i, there is a compactly supported function fionMwhich is supportedinsometubularneighborhood Wiofι(Ui), andsuchthat f=f1+···+fm on some neighborhood of ι(K). Then, taking a neighborhood Viof eachUiinVso thath:Vi→Eis an embedding, we get (5)/integraldisplay πNh∗(π∗fΦn) =/summationdisplay i/integraldisplay πN|π−1 N(Ui)(h|Vi)∗(π∗fiΦn) aroundK, yielding the result if each term in the right hand side of (5) converge s toι∗fi. Therefore we can assume ι, ˜ιandhare embeddings. With this assumption, there is an open disk bundle DoverV, and extensions ˜ι′,h′:D→π−1(V) of ˜ιandh, respectively, which are diffeomorphisms onto open subsets ofE. Letφdenote the composite ˜ι′(D)(˜ι′)−1 − −−− →Dh′ − −−− →π−1(V). Clearly,φsatisfies the conditions of Lemma 4.1, and we can suppose fis supported in ˜ι′(D). So/integraltext πφ∗(π∗fΦn) converges to fover any compact subset of Vwith respect to the C∞topology. But ι∗/integraldisplay πφ∗(π∗fΦn) =/integraldisplay πN((˜ι′)∗φ∗(π∗fΦn)|V) =/integraldisplay πNh∗(π∗fΦn), and the result follows. /square Observe that Lemma 4.1 is a particular case of Corollary 4.2. The coro llary could be proved directly with the arguments of the lemma, but the no tation would become more complicated. 4.2.Proof of Lemma 4.1. The following easy observations will be used to prove Lemma 4.1. Remark 1.LetEandFbe vector bundles over the manifolds MandN, respec- tively. Suppose f:E→Fis a homomorphism which restricts to isomorphisms on the fibers, and let g:M→Nbe the map induced by f. Thus the homomor- phismE→g∗F, canonically defined by f, is an isomorphism. Therefore there is a composite of homeomorphisms C∞(F)→C∞(g∗F)→C∞(E). Here, the first homomorphism is canonically defined by the pull-back d iagram of g∗F, and the second one is induced by the inverse of E→g∗F. Ifs/mapsto→s′by the above composite, then s′is determined by f(s′(x)) =s(g(x)) forx∈M. Remark 2.SetE=Rn×Rk, and letπi,i= 1,2, denote the factor projections of EontoRnandRk, respectively. Let Kbe a compact subset of Rn, andφ:V→W a diffeomorphism between open neighborhoods of Rn×{0}. Supposeφrestricts to the identity on Rn× {0}. For anyr >0, letBr,Sr⊂Rkrespectively denote the Euclidean ball and the Euclidean sphere of radius r centered at the o rigin. Then there is anR>0 and an open neighborhood UofKsuch that, for every x∈UandLEAFWISE REDUCED COHOMOLOGY 11 everyy∈BR,{x} ×Rkintersects transversely φ−1(Rn× {y}) at just one point. Moreover, the map σ:U×BR→(U×Rk)∩φ−1(Rn×BR), determined by {σ(x,y)}= ({x}×Rk)∩φ−1(Rn×{y}), is a diffeomorphism. Indeed σis smooth because each ( U×Rk)∩φ−1(Rn×{y}) can be given as the graph of a map ψy:U→Rkdepending smoothly on y∈BR, and σ(x,y) = (x,ψy(x)). It also has a smooth inverse since ( x,y) = (x,π2φσ(x,y)). Therefore, for r≤R,π1: (U×Rk)∩φ−1(Rn×Sr)→Uis a sphere bundle, whose fibers are of volume uniformly bounded by Crk−1for someC >0 ifUandRare small enough. To begin with the proof of Lemma 4.1, fix a Riemannian structure on E, and letBr,Sr⊂Erespectively denote the corresponding open disk bundle and spher e bundle of radius r. SetS=S1. Letψbe a global angular form of S[9,§11]. (IfE is of rankk,ψis a differential form of degree k−1 restricting to unitary volume forms on the fibers and so that dψ=−π∗e, whereerepresents the Euler class of S.) Letr:E→Rdenote the radius function, and h:E\M→Sthe deformation retraction given by h(v) =v/r(v). For each n, let alsoρnbe a function on [0 ,∞) such that −1≤ρn≤0,ρ′ n≥0,ρn≡ −1 on a neighborhood of 0, and ρn≡0 on [1/n,∞). Then each Φn=d(ρn(r)h∗ψ) =ρ′ n(r)dr∧h∗ψ−ρn(r)π∗e represents the Thom class of E[9,§12]. Local orthonormal frames canonically define isomorphisms of trivia lity ofE which restrict to local isomorphisms between restrictions of each Srand trivial sphere bundles with typical fiber the Euclidean sphere of radius r. So Remark 2 and the conditions satisfied by φyield the existence of some R,C >0 and some relatively compact open neighborhood UofKinMso that •π−1(U)∩φ−1(BR)⊂V, •the map φ:π−1(U)∩φ−1(BR)→φπ−1(U)∩BR is a diffeomorphism whose differential is of fiberwise uniformly bounded norm, and •for 0< r≤R,φ−1(Sr) is transverse to the fibers of πoverUandπ: π−1(U)∩φ−1(Sr)→Uis a sphere bundle whose fibers are of volume uniformly bounded by Crk−1. Theφ∗Φnalso represent the Thom class of EoverUforn>1/R. Hence f−/integraldisplay πφ∗(π∗fΦn) =/integraldisplay π(π∗f−φ∗π∗f)φ∗Φn =/integraldisplay1/n 0ρ′ n(r)dr/integraldisplay π|π−1(U)∩φ−1(Sr)(π∗f−φ∗π∗f)φ∗h∗ψ (6) −/integraldisplay π(π∗f−φ∗π∗f)ρn(φ∗r)φ∗π∗e. (7)12 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR We have to prove that (6) and (7) converge uniformly to zero on Kasn→ ∞, as well as all of its derivatives of any order. Take a Riemannian metric on M, and a splitting TE=V ⊕H, whereVis the vertical bundle of πandHthe horizontal bundle of any Riemannian connection. This yields a Riemannian structure on TEdefined in the standard way by using the canonical isomorphisms V∼=π∗EandH∼=π∗TM. We also have TM=H|M, H|Sr⊂TSr, and (8) TSr= (V ∩TSr)⊕(H|Sr). Finally, we can assume (9) V ∩φ−1 ∗(H) = 0 over π−1(U)∩φ−1(BR) by the properties of φ. By the conditions on φ, the supremum of |π∗f−φ∗π∗f|overπ−1(U)∩φ−1(B1/n) convergestozeroas n→ ∞. Alsothepointwisenormof φ∗π∗eisuniformlybounded onπ−1(U)∩φ−1(B1/n), thus (7) converges uniformly to zero as n→ ∞. On the otherhand, because the fiberwise normof each h∗:TSr→TSisr−1, the pointwise norm ofφ∗h∗ψis uniformly bounded on π−1(U)∩φ−1(Sr) byC1r−k+1withC1>0 independent of r≤R. So (6) also converges uniformly to zero on Uasn→ ∞by the estimate on the volume of the fibers of πonπ−1(U)∩φ−1(Sr). Now fix vector fields X1,...,X monU. By (8) and (9) the Xihave liftings Yi which are sections of φ−1 ∗Hoverπ−1(U)∩φ−1(BR). For any subset I⊂ {1,...,m}, letθIdenote the composite of Lie derivatives θY1···θYlifI={i1,...il}with i1< i2<···< il, and letθ∅be the identity homomorphism. Then the order m derivativeX1···Xmover (6) and (7) is respectively given by (10)/summationdisplay I,J/integraldisplay1/n 0ρ′ n(r)dr/integraldisplay π|π−1(U)∩φ−1(Sr)θI(π∗f−φ∗π∗f)θJφ∗h∗ψ, and (11) −/summationdisplay I,J/integraldisplay πθI(π∗f−φ∗π∗f)ρn(φ∗r)θJφ∗π∗e, whereI,Jruns over the partitions of {1,...,m}. By the properties of Handφ, the supremum of the |θI(π∗f−φ∗π∗f)|onπ−1(U)∩φ−1(B1/n) converges to zero asn→ ∞. Hence (11) converges uniformly to zero on Kbecause the pointwise normofthe θJφ∗π∗ecanbe uniformlybounded on π−1(K)∩φ−1(BR). The uniform convergenceof (10)tozerofollowsbyestimatingthepointwisenor moftheθJφ∗h∗ψ onπ−1(K)∩φ−1(Sr) byC2r−k+1for someC2>0 independent of r. This in turn follows by proving a similar estimate for the pointwise norm of θ′ Jh∗ψon φπ−1(K)∩Sr, where the θ′ Jare defined in the same way as the θJby using the Y′ i=φ∗Yiinstead of the Yi. To do this, consider the multiplication map µ: [0,R]×S→BR. Since µ∗: [0,1]×(H|S)⊂T([0,1]×S)→ H restricts to isomorphisms on the fibers, by Remark 1 there are smo oth sections Y′′ i of [0,1]×(H|S) so thatµ∗(Y′′ i(r,v)) =Y′ i(rv). Also because the composite (0,R]×Sµ− −−− →BR\Mh− −−− →SLEAFWISE REDUCED COHOMOLOGY 13 is the second factor projection, µ∗h∗ψis the form canonically defined by ψon (0,R]×S, which extends smoothly to [0 ,R]×S. So, ifθ′′ Jis defined in the same way as theθJby using the Y′′ iinstead of the Yi, the pointwise norm of the θ′′ Jµ∗h∗ψ is uniformly bounded. Then the desired estimation of the pointwise no rm of the θ′ Jh∗ψfollows by observing that the fiberwise norm of µ∗:{r}×TS→TSrisr. 4.3.Proof of Proposition 3.1. Recall that any local diffeomorphism φ:M→N induces a homomorphism of complexes, φ∗: Ωc(M)→Ωc(N), defined as follows. For anyα∈Ωc(M), choose a finite open cover U1,...,U nof suppαsuch that each restriction φ:Ui→φ(Ui) is a diffeomorphism. There is a decomposition α=α1+···+αnso that supp αi⊂Ui. For eachi, there is a unique βi∈Ωc(N) supported in φ(Ui) such that βi|φ(Ui)corresponds to αi|Uibyφ. Defineφ∗α= β1+···+βn. This definition is easily checked to be independent of the choices involved and compatible with the differential maps. If φ: (M,F)→(N,G) is a local diffeomorphism of foliated manifolds, we similarly have a homomor phism φ∗: Ωc(F)→Ωc(G) which is compatible with the leafwise de Rham derivative. Moreoverφ∗is surjective if so is φ. Now Proposition 3.1 can be proved as follows. There is a canonical injection of TF1as vector subbundle of ι∗ 1TF. LetE= ι∗ 1TF/TF1, andπ:E→M1the bundle projection. Identify M1with the image of the zero section of E. Fixing any Riemannian metric on M, there are induced Riemannian metrics on the Mi, and an induced Riemannian structure on E. For eachr >0, letBr⊂Edenote the open disk bundle of radius roverM1. Then there is an R >0 and an open neighborhood Uof the support of finM1such that, ifV=π−1(U)∩BR, the restriction of ι1toUcan be extended to a map of foliated manifolds, ˜ ι1: (V,π∗F1|V)→(M,F), which is defined over each x∈M1 as a composite of the restriction of the canonical homomorphism (ι∗ 1TF/TF1)x→Tι1(x)F/ι1∗TxF1≡(ι1∗TxF1)⊥∩TxF, and the exponential map of the leaves of Fdefined on the ball of radius Rcentered at zero in (ι1∗TxF1)⊥∩TxF. By elementary properties of the exponential map and sinceι1is transversely regular with codim F1= codim F,Rcan be chosen so that ˜ι1is a local diffeomorphism and ˜ ι∗ 1F=π∗F1|V. Eis of rankk, and with an induced orientation. The representatives Φ nof its Thom class, given by Lemma 4.1, can be assumed to be supported in BR. The Φ n are of degree k, closed and compactly supported in the vertical direction, i.e. with compactly supported restrictions to the fibers. Moreover all the Φn|BRare pairwise cohomologousin the complex of forms in Ω·(BR) which are compactly supported in the vertical direction. On the other hand, fis basic and compactly supported. So theπ∗fΦnrestrict to leafwise closed forms αn∈Ωk c(π∗F1|V) which are pairwise cohomologous. Thus the ˜ ι1∗αn∈Ωk c(F) are leafwise closed and define the same classζ∈ Hk c(F). LetU1,...,U m, be an open cover of the support of finUsuch that each ι1:Uj→Mis an embedding, j= 1,...,m. The above Rcan be chosen small enough so that the ˜ ι1:Vj=π−1(Uj)∩BR→˜ι1(Vj) are diffeomorphisms. Take a decomposition f=f1+···+fmwith eachfjcompactly supported in Uj, and let αn,j∈Ωk c(π∗F1|Vj) be the restriction to the leaves of π∗fjΦn. Then, by definition, ˜ι1∗αn=βn,1+···+βn,m,14 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR where each βn,j∈Ωk c(F) is the extension by zero of the forms in Ωk c(F|˜ι1(Vj)) which correspond to the αn,j|Vjby ˜ι1. Givenι2: (M2,F2)→(M,F), we use the same notation as in the preamble of Definition 2.1. We can clearly assume the Ujare contained in N1. Letι=ι1σ1= ι2σ2:T→M. There is a canonical isomorphism ι∗TF∼=σ∗ 1TF1⊕σ∗ 2TF2 because the ιi|Niintersect transversely in F. Soσ∗ 1E∼=σ∗ 2TF2canonically. This isomorphism will be considered as an identity. LetπT:σ∗ 1E→Tbe the pull-back vector bundle projection, and ˜ σ1:σ∗ 1E→E the canonical homomorphism. Identify Tto the image of the zero section of σ∗ 1E. For eachj, take a relatively compact open subset Oj⊂σ−1 1(Uj) containing the compact set supp σ∗ 1fj∩suppDT. The above Rcan be chosen small enough so that σ2:Oj→N2hasan extensionto alocaldiffeomorphism ˜ σ2:π−1 T(Oj)∩˜σ−1 1(BR)→ N2defined as the composite of the restriction of the canonical homom orphism σ∗ 1E≡σ∗ 2TF2→TF2, and the exponential map of the leaves of F2defined on the tubular neighborhood of radius Rof the zero section in TF2. In this way, ˜ σ2maps each fiber of πTinto a leaf of F2. Observe that the diagram ˜σ−1 1(BR)∩π−1 T(Oj)˜σ1− −−− →Vj ˜σ2/arrowbt/arrowbt˜ι1 N2ι2− −−− →M is obviously non-commutative in general. This is the main technical diffic ulty. To solve it, we have chosen the Φ nso that their supports concentrate around M1and satisfy the needed properties at the limit (Lemma 4.1 and Corollary 4.2 ). We need the observation that (12) σ2σ−1 1(A) = (ι2|N2)−1ι1(A) for any subset A⊂N1, as can be easily checked. Using the compactness of BR∩π−1(suppfj) and since suppfj=/intersectiondisplay 0<r<RBr∩π−1(suppfj), we easily get ι1(suppfj) =/intersectiondisplay 0<r<R˜ι1/parenleftbig Br∩π−1(suppfj)/parenrightbig . Therefore/intersectiondisplay 0<r<RsuppD∩ι−1 2˜ι1/parenleftbig Br∩π−1(suppfj)/parenrightbig = suppD∩ι−1 2ι1(suppfj) = suppD∩σ2σ−1 1(suppfj) =σ2(σ−1 2(suppD)∩σ−1 1(suppfj)) =σ2(suppDT∩suppσ∗ 1fj), where the second equality follows by (12). Then, since the suppD∩ι−1 2˜ι1/parenleftbig Br∩π−1(suppfj)/parenrightbig are compact, and since ˜σ2/parenleftbig ˜σ−1 1(BR)∩π−1 T(Oj)/parenrightbigLEAFWISE REDUCED COHOMOLOGY 15 is an open neighborhood of σ2(suppDT∩suppσ∗ 1fj), there is an r<Rsuch that suppD∩ι−1 2˜ι1/parenleftbig Br∩π−1(suppfj)/parenrightbig ⊂˜σ2/parenleftbig ˜σ−1 1(BR)∩π−1 T(Oj)/parenrightbig . So suppD∩suppι∗ 2βn,j⊂˜σ2(Wj) for large enough n, where Wj= ˜σ−1 2ι−1 2˜ι1(Vj)∩˜σ−1 1(BR)∩π−1 T(Oj). We can assume this holds for every n. Hence there is some ωn,j∈Ωk c(F2) which is supported in ˜ σ(Wj) and has the same restriction to some neighborhood of supp D asι∗ 2βn,j. IfFπTis the foliation on σ∗ 1Edefined by the fibers of πT, there is some γn,j∈Ωk c(FπT|Wj) such that (˜ σ2|Wj)∗ωn,j=ι∗ 2βn,j. We get (13) D(ι∗ 2ζ) =/summationdisplay jDT/parenleftBigg/integraldisplay πT|Wjγn,j/parenrightBigg by definition. Let hj:Wj→Ebe the immersion given by the composite Wj˜σ2− −−− →ι−1 2˜ι1(Vj)ι2− −−− →˜ι1(Vj)˜ι−1 1− −−− →Vj⊂E . Clearlyhjis an extension of σ1:Oj→N1⊂M1⊂E, andγn,j=h∗ j(π∗fjΦn) aroundWj∩π−1 T(σ−1 1(suppfj)∩suppDT). Moreover the homomorphism between the normal bundles of OjandM1, defined by the differential of hj, restricts to isomorphismsonthefibers. Theseisomorphismsareorientationpre servingonfibers over points with ε= 1, and orientation reversing on fibers over points with ε=−1. Therefore/integraltext πT|Wjγn,jconverges to εσ∗ 1fjonσ−1 1(suppfj)∩suppDTwith respect to theC∞topology by Corollary 4.2. Hence (13) is equal to /a\}b∇acketle{t(ι1,f),(ι2,D)/a\}b∇acket∇i}ht, and the proof is complete. Remark3.Observe that, in Proposition 3.1, ζhas representativessupported in any neighborhood of ι1(M1). Thus, in Theorem 2.2, the linearly independent classes ζm∈ H· c(F) also have representatives supported in any neighborhood of ι1(M1). 5.Case where the leaves have homology classes with non-trivia l intersection This section will be devoted to the proof of Corollary 2.3. LetMbetheambientmanifoldof F. Letπi:ι∗ i(TM/TF)→Kidenotethepull- backvectorbundle projection, andidentify Kitothe imageofitszerosection. Fix a Riemannian metric on Mand, for some R>0, letMibe the tubular neighborhood of radiusRaroundKiinι∗ i(TM/TF). SuchRcan be chosen so that the maps ˜ιi:Mi→Mare well defined as composites of the restrictions of the canonical homomorphisms ι∗ i(TM/TF)→TM/TF ≡(TF)⊥, and the restriction of the exponential map of Mto the tubular neighborhood of radius Rof the zero section ofTF⊥. ChooseRsmall enough so that ˜ ιi:π−1 i(xi)∩Mi→˜ιi(π−1 i(xi)∩Mi) is an embedded transversal of Ffor eachiand eachxi∈Mi. Observe that ˜ι1(π−1 1(x1)∩M1) = ˜ι2(π−1 2(x2)∩M2) ifι1(x1) =ι2(x2). The ˜ιiare thus transverse toF, and the Fi= ˜ι∗ iFhave the same codimension as F. Then ˜ιiare transversely16 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR regular immersions of foliated manifolds. By deforming the ιiif needed, we can supposethe ιiintersecteachothertransversely,andthusthe˜ ιiintersecttransversely inF. Moreover the orientations of the Kiinduce orientations of the Fi. EachKiis a closed leaf of Fiwhose holonomy representation is given by the composite (3). So the holonomy group of Kiis given by germs of local isometries. HenceFiisaRiemannianfoliationaround Ki, asfollowseasilyfrom[14, Theorem2 in Chapter IV]. (See also [12, Theorem 2.29 in Chapter II] or [17].) We ca n assume the whole Fiis Riemannian, which can thus be described as follows [16, 26]. Fix a transverse Riemannian structure of Fi. LetQibe theO(q)-principal bundle overMiof transverse orthonormal frames of Fiwith the transverse Levi-Civita connection, and ˆFithe horizontal lifting of FitoQi[24, 25]. Let Pibe a leaf closure of ˆFioverKi. ThenPiis anHi-principal bundle over Kifor some closed subgroupHi⊂O(q). For the open disk B⊂Rqof radiusRcentered at the origin, we can assume Mi≡(Pi×B)/Hias fiber bundles over Ki, where the Hi-action on P2×Bis the diagonal one; i.e. ( z,v)h= (zh,h−1v) for (z,h)∈Pi×Bandh∈Hi. Moreover the above identity can be chosen so that Fiis identified to the foliation whose leaves are the projections of products of leaves of ˆFiinPiand points in B. (This description is simpler than the one in [16] and [26] because the le af closure Kiis just a compact leaf.) Consider the transverse Riemannian structure of each Fidefined by the Eu- clidean metric on Busing the above description. Since the elements in the image of the composites (3) are germs of local isometries for the same me tric onRq, the composite B≡π−1 2(x2)∩M2˜ι2−→˜ι2(π−1 2(x2)∩M2) = ˜ι1(π−1 1(x1)∩M1)˜ι−1 1−→π−1 1(x1)∩M1≡B is an isometry around the origin for all ( x1,x2)∈K1×K2withι1(x1) =ι2(x2). We can assume such composite is an isometry on the whole B, which will be denoted byφx2,x1. With the above description, any compactly supported basic functio nfofF1can be canonically considered as an H1-invariant compactly supported function on B, and any compactly supported holonomy invariant transverse distr ibutionDofF2 can be canonically considered as a compactly supported H2-invariant distribution onB. For suchfandD, we clearly have (14) /a\}b∇acketle{t(˜ι1,f),(˜ι2,D)/a\}b∇acket∇i}ht=/summationdisplay ε(x1,x2)D(φ∗ x2,x1f), where the sum runs over the pairs ( x1,x2)∈K1×K2withι1(x1) =ι2(x2). Here ε(x1,x2) =±1 depending on whether the identity Tιi(xi)B≡ι1∗Tx1K1⊕ι2∗Tx2K2 isorientationpreservingororientationreversing. Let fmbeasequenceofcompactly supportedO(q)-invariant functions in Bwith integral equal to one and pairwise disjoint supports, and let µmbe the restriction of the Euclidean measure to the support offm. Then /a\}b∇acketle{t(˜ι1,fm),(˜ι2,µn)/a\}b∇acket∇i}ht=/a\}b∇acketle{tι1,ι2/a\}b∇acket∇i}ht/integraldisplay Bfmdµn by (14), where /a\}b∇acketle{tι1,ι2/a\}b∇acket∇i}htis theintersectionnumberof ι1andι2inB. Sodim Hk2 c(F) = ∞by Theorem 2.2. Similarly, dim Hk1 c(F) =∞, which completes the proof.LEAFWISE REDUCED COHOMOLOGY 17 6.Case of suspension foliations Proof of Corollary 2.5. Recall the notation used for suspension foliations in the statement of Corollary 2.5, and consider the fiber bundles Mi=ι∗ iMoverKi. Each canonical map ˜ ιi:Mi→Mis transverse to F, and let Fi= ˜ι∗ iF. Then ˜ιiare transverselyregularimmersionsoffoliatedmanifolds. Bydeforming theιiifneeded, we can suppose the ιiintersect each other transversely, thus the ˜ ιiintersect each other transversely in F. Moreover the orientations of the Kiinduce orientations of theFi. The group of deck transformations of each pull-back covering map ι∗ iL→Kiis isomorphic to Γ i, andFiis canonically isomorphic to the corresponding suspension foliation given by the restriction to Γ iof the Γ-action on T. Hence the fmcan be canonically considered as compactly supported basic functions of F1, and theDn can be canonically considered as holonomy invariant transverse dist ributions of F2. The ˜ι2: suppDn→Mare clearly proper, and we easily get /a\}b∇acketle{t(˜ι1,fm),(˜ι2,Dn)/a\}b∇acket∇i}ht=/a\}b∇acketle{tι1,ι2/a\}b∇acket∇i}htDn(fm). Therefore the result follows from Theorem 2.2. /square Proof of Corollary 2.6. LetAnbe a sequence of Γ 1-saturated open subsets of T with non-trivial µ-measure and pairwise disjoint Γ 2-saturations. Clearly, there are open sets BnofTwith positive µ-measure and such that Bn⊂An. Since the closure of Γ 1in the group of diffeomorphisms of Tis a compact Lie group, there exists a sequence of non-negative Γ 1-invariant functions fnonTsuch that Bn⊂suppfn⊂An. Letµnbe the Γ 2-invariant measure on Tdefined as the product of µand the characteristic function of the closure of the Γ 2-saturation of suppfn. Then/integraltext Tfmdµn/\e}atio\slash= 0 if and only if m=n, and the result follows by Corollary 2.5. /square Proof of Corollary 2.7. Since some lift of ctoLjoins two distinct points of its end set,Lis disconnected by some codimension one immersed closed submanifold , ι:K→L, such that candπιdefine homology classes of Bwith non-trivial intersection. Clearly, the composite π1(K)π1(πι)− −−− →π1(B)h− −−− →Γ is trivial, and the image of the composite π1(S1)π1(c)− −−− →π1(B)h− −−− →Γ is/a\}b∇acketle{ta/a\}b∇acket∇i}ht. Take a sequence Anof disjoint non-trivial H-invariant open subsets of T. SinceHis an abelian compact Lie group (a torus), there is an H-invariant prob- abilistic measure supported in any H-orbit inT. Take thus one of such measures µnsupported in each An. Then the result follows from Corollary 2.6 by taking as µthe sum of the µn. /square To prove Proposition 2.8, we use the following. Lemma 6.1. LetΓbe a finitely generated group, and Xa connected T1topological space. For any continuous action of ΓonX, a finite union of orbits is dense if and only if so is each orbit in the union.18 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR Proof.Takex1,...,x n∈Xsuch that X=Γx1∪...∪Γxn=Γx1∪...∪Γxn. Each orbit closure Γxican be decomposed as a disjoint union of sets Li=/intersectiondisplay F(Γ\F)xi, Ii=Γxi\Li, whereFruns over the finite subsets of Γ. We have X=L∪I, whereL=/uniontextn i=1Li andI=/uniontextn i=1Ii. Moreover, since Lis saturated we have L∩I=∅. SoI=∅ becauseXisT1and connected. (If we had I/\e}atio\slash=∅, for anyy∈I,{y}would be closed inXbecauseXisT1. But since Lis closed and I=X\Lis discrete, {y} would be also open in X. ThusXwould not be connected.) Therefore X=Land Li=Γxifor eachi. But each Liis closed in X, andLi∩Lj/\e}atio\slash=∅impliesLi=Lj, obtainingX=Lifor everyiby the connectedness of X. /square Proof of Proposition 2.8. Clearly,ifthe C-orbitsaredensein X,soaretheΓ-orbits. Reciprocally, suppose the Γ-orbits are dense. By a theorem of Sta llings [31], there is a finite normal subgroup F⊂Γ such that Γ 1= Γ/Fis isomorphic either toZor to the diedric group Z2∗Z2. The action of Γ on Xdefines an action of Γ1on the connected T1spaceX1=X/Fwith dense orbits. Since Cis infinite, so is its projection C1to Γ1, and any infinite subgroup of such Γ 1is of finite index. ThereforeanyΓ 1-orbitinX1isafiniteunionof C1-orbits,andthusthe C1-orbitsare dense inX1by Lemma 6.1. This implies the density of the CF-orbits inXbecause the canonical projection of XontoX1is open and continuous. But any CF-orbit is a finite union of C-orbits. Hence the C-orbits are dense by Lemma 6.1. /square 7.Case of Lie foliations with compact semisimple structural L ie algebra Theorem 2.9 will be proved in this section (Corollaries 7.16 and 7.17). 7.1.Construction of a spectral sequence for an arbitrary Lie fol iation on a closed manifold. LetFbe a Lie foliationwith dense leaveson a closedmanifold M. Letgbe the structural Lie algebra of F, andGthe simply connected Lie group with Lie algebra g. Letπ:/tildewiderM→Mbe the universal covering map. Then the leaves of/tildewideF=π∗Fare the fibers of a fiber bundle D:/tildewiderM→G. It will be convenient to consider the right action of π1(M) on/tildewiderMby deck transformations and the left action ofGon itself by left translations. Thus Dis anti-equivariant with respect to the holonomy homomorphism h:π1(M)→G; i.e.,D(˜xγ) =h(γ)−1D(˜x) [11]. The density ofthe leavesimplies the density of Γ = h(π1(M)) inG. The homomorphism hdefines an action of π1(M) onGby left translations, yielding the corresponding suspension foliation GonN=/parenleftBig /tildewiderM×G/parenrightBig /π1(M) (defined as in Section 6). Gis a Lie foliation with the same transverse structure as F, given by ( G,Γ). The section (id ,D) :/tildewiderM→/tildewiderM×Gisπ1(M)-equivariant: (id,D)(˜xγ) = (˜xγ,D(˜xγ)) = (˜xγ,γ−1D(˜x)) = (˜x,D(˜x))γ . Thus (id,D) defines a section s:M→N, andNis trivial as principal G-bundle overM. Clearlysis transverse to G, ands∗G=F.LEAFWISE REDUCED COHOMOLOGY 19 LetD:/tildewiderM×G→Gbe defined by D(˜x,g) =g−1D(˜x). SuchDisπ1(M)- invariant: D((˜x,g)a) =D(˜xa,h(a)−1g) =g−1h(a)D(˜xa) =g−1D(˜x). SoDdefines a map DN:N→G. ClearlyDNs= const e, whereeis the identity element inG. Moreover DNisG-anti-equivariant: DN([˜x,g]g′) =DN([˜x,gg′]) = (gg′)−1D(˜x) = (g′)−1DN([˜x,g]). ThereforeDNis the composite of the second factor projection of the trivializatio n ofN→Mdefined bysand the inversion map on G. Let/tildewideFalso denote the foliation on Ndefined by the lifting of Fto all the leaves ofG./tildewideFis a subfoliation of Gwhose leaves are the intersections of the leaves of G with all the translations of s(M). Letν⊂TGbe aG-invariant subbundle so that TG=ν⊕T/tildewideF. We get /logicalanddisplay TG∗=/logicalanddisplay ν∗⊗/logicalanddisplay T/tildewideF∗, and thus there is a bigrading of Ω = Ω( G) defined by Ωu,v=C∞/parenleftBiggu/logicalanddisplay ν∗⊗v/logicalanddisplay T/tildewideF∗/parenrightBigg , u,v∈Z. For simplicity, dGwill be denoted by d. There is a decomposition of das sum of bihomogeneous components, d=d0,1+d1,0+d2,−1, where each double subindex denotes the corresponding bidegree. From d2= 0 we get (15) d2 0,1=d2 2,−1=d0,1d1,0+d1,0d0,1= 0, (16) d1,0d2,−1+d2,−1d1,0=d2 1,0+d0,1d2,−1+d2,−1d0,1= 0. The decreasing filtration of (Ω ,d) by the differential ideals (17) Fl= Ωl,·∧Ω, depends only on/parenleftBig G,/tildewideF/parenrightBig ; it could be defined without using ν. So we get a spectral sequence (Ei,di) which converges to H·(G). As for the spectral sequence of a foliation (see e.g. [1]), in this case there are canonical identities (18) ( E0,d0)≡(Ω,d0,1),(E1,d1)≡(H(Ω,d0,1),d1,0∗). TheC∞topology on the space of differential forms induces a topology on ea chEi which is not Hausdorff in general. At eachz∈Nwe have DN∗:νz∼=− −−− →TDN(z)G. So for each X∈gthere is a well defined vector field Xν∈C∞(ν) which isDN- projectable and such that DN∗Xν=X. SuchXνisG-invariant since Xν zg∈νzg and DN∗(Xν zg) =g−1DN∗Xν z=g−1XDN(z)=Xg−1DN(z)=XDN(zg). LetθXandiXrespectively denote the Lie derivative and interior product on Ω with respect to Xν. (We are considering θXandiXas operators on the leaves of G, but preserving smoothness on N.) By comparing bidegrees in the usual formulas20 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR that relate Lie derivatives, interior products and the de Rham deriv ative, we easily get d0,1iX+iXd0,1= 0, (θX)0,0d0,1=d0,1(θX)0,0, i[X,Y]= (θX)0,0iY−iY(θX)0,0, (θX)0,0=d1,0iX+iXd1,0, (θ[X,Y])0,0= (θXθY−θYθX)0,0−d0,1iΞ(X∧Y)−iΞ(X∧Y)d0,1, where Ξ :/logicalandtext2g→C∞/parenleftBig T/tildewideF/parenrightBig is given by Ξ(X∧Y) = [Xν,Yν]−[X,Y]ν. Therefore we get the operation ( g,i1,θ1,E1,d1), wherei1X≡iX∗andθ1X≡ (θX)0,0∗according to (18), and the algebraic connection D∗ N:g∗→E1,0 1⊂Ω1,0 [13]. Then Eu,v 2∼=Hu(g;θ1:g→End(E0,v 1)). Letφ:N×g→Nbe defined by φ(z,X) =Xν 1(z), whereXν tdenotes the uni- parametric group of transformations defined by Xν, considered as group of trans- formations of the leaves of Gpreserving smoothness on N. Then the following diagram is commutative N×gφ− −−− →N DN×exp/arrowbt/arrowbtDN G×G− −−− →G, where the lowest map denotes the operation on G. (This follows because Xt= Rexp(tX)for allX∈g.) 7.2.Tensor product decomposition of E2whengis compact semisimple. From now on suppose gis compact semisimple, and thus Gis compact [28]. Theorem 7.1. With the above notations, Eu,v 2∼=Hu(g)⊗E0,v 2=Hu(g)⊗(E0,v 1)θ1=0. The resultfollowswith the sametypeofargumentsasin thosegivenin Sections2 and 3 of [2] to prove Theorem 3.5 in [2]. We will indicate the main steps in th e proof because some of them will be needed later. Consider the canonical biinvariant metric on G[28, Chapter 6], and let C⊂G andC∗⊂gbe the cut locus and tangential cut locus corresponding to the iden tity elemente∈G. LetB∗be the radial domain in gbounded by C∗, and letB= exp(B∗). From the general properties of the cut locus we have C=∂B=G\B, exp :B∗→Bis a diffeomorphism, CandC∗have Lebesgue measure zero, and B∗ is compact (since so is G) [22, 21]. Consider the compact space F={(X,Y,Z)∈B∗3: exp(X) exp(Y) = exp(Z)} ⊂g3, and for each X∈B∗the compact slice FX={(Y,Z)∈g2: (X,Y,Z)∈F} ⊂g2. Smoothness on FandFXwill refer to the smoothness obtained by considering these spaces as subspaces of g3andg2, respectively.LEAFWISE REDUCED COHOMOLOGY 21 Letι:g2→g2be the involution ( Y,Z)/mapsto→(Z,Y). Fora= exp(X) we also have the smooth map JX:B∩L−1 aB→FXgiven by JX(g) = (log(g),log(ag)), where log = exp−1:B→B∗. LetWX=JX(B∩L−1 aB)⊂FX. Lemma 7.2 ([2, Proposition 2.2]) .We have: (i)WXis open inFXandJX:B∩L−1 aB→WXis a diffeomorphism. (ii)ι(FX) =F−X, and the diagram B∩L−1 aBJX− −−− →FX La/arrowbt/arrowbtι B∩LaBJ−X− −−− →F−X is commutative. ForX,Y∈B∗letWX,Y=JX(B∩L−1 aB∩L−1 bB)⊂FX, wherea= exp(X) andb= exp(Y). We have the diffeomorphism JX,Y=JYJ−1 X:WX,Y→WY,X. Let ∆ be the unique biinvariant volume form on Gsuch that/integraltext G∆ = 1, which defines a Haar measure µonG. Then for each X∈B∗letµXbe the Borel measure onFX, concentrated on WX, where it corresponds to µbyJX. Corollary 7.3 ([2, Proposition 2.3]) .We have: (i)µX(FX) =µX(WX) =µX(WX,Y) =µ(B∩L−1 aB∩L−1 bB) =µ(B∩L−1 aB) =µ(G) = 1 (ii)µXcorresponds to µ−Xbyι:FX→F−X. (iii)µXcorresponds to µYbyJX,Y:WX,Y→WY,X. LetI= [0,1], and define continuous maps σ,η:F×I→Gby setting σ(ξ,t) = exp(tZ), η(ξ,t) =/braceleftBigg exp(2tX) if t∈I1= [0,1/2] exp(X) exp((2t−1)Y) ift∈I2= [1/2,1], whereξ= (X,Y,Z)∈F. The map σis smooth, and so are the restrictions of ηto eachF×Ii(i= 1,2). Lemma 7.4 ([2, page 178]) .There is a finite open cover Q1,...,Q kofF, and continuous maps Hj:Qj×I×I→Gwith smooth restrictions to each Qj×Ii×I, i= 1,2,j= 1,...,k, so that Hj(·,·,0) =σ|Qj×I,Hj(·,·,1) =η|Qj×I, Hj(ξ,0,s) =efor alls∈Iandξ∈Qj, Hj(ξ,1,s) = exp(Z)for alls∈Iandξ= (X,Y,Z)∈Qj. Lemma 7.5. For eachj= 1,...,kthere exists a unique continuous map Hj:N×Qj×I×I→N with smooth restrictions to each N×Qj×Ii×I,i= 1,2, such that (i)DNHj(z,ξ,t,s) =DN(z)Hj(ξ,t,s), (ii)Hj(z,ξ,0,s) =z, (iii) (d/dt)Hj(z,ξ,t,s)∈νfort/\e}atio\slash= 1/2. Moreover for ξ= (X,Y,Z)∈Qjwe have22 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR (iv)Hj(·,ξ,1,0) =φZ, (v)Hj(·,ξ,1,1) =φYφX (vi)Hj(z,ξ,1,s)∈D−1 N(D(z) exp(Z))for allz∈Nand alls∈I. Proof.It is completely similar to the proofs of Lemmas 3.1 and 3.2 in [2]. /square Therefore, for all ξ= (X,Y,Z)∈Qj,Hj(·,ξ,1,·) :N×I→Nis an/tildewideF-integrable homotopy of φZtoφYφX[10]. Hence the corresponding homotopy operator in Ω preservesthefiltration, andthusits(0 ,−1)-bihomogeneouscomponent kj,ξ: Ω→Ω satisfies (φ∗ Xφ∗ Y−φ∗ Z)0,0=d0,1kj,ξ+kj,ξd0,1. Define the operators ρ,λ: Ω→Ω by setting ρ(α) =/integraldisplay B∗φ∗ Xα∆∗(X), λ(α) =/integraldisplay B∗ΦXα∆∗(X), where ∆∗= exp∗∆ and Φ Xis the homogeneous operator of degree −1 on Ω associ- atedtothehomotopy φtX(t∈I) [9]. Theoperators ρandλarelinearhomogeneous of degrees 0 and −1, respectively, satisfying ρ−id =dλ+λd. Moreover, since φtX preserves the pair of foliations/parenleftBig G,/tildewideF/parenrightBig (becauseXνis an infinitesimal transforma- tion of/parenleftBig G,/tildewideF/parenrightBig ), ΦXreduces the filtration at most by a unit. Therefore the bihomo- geneous operators ρ1≡ρ0,0∗andλ1≡λ−1,0∗onE1satisfyρ1−id =d1λ1+λ1d1. Forα∈Ω andX∈B∗, by Lemma 7.2 and Corollary 7.3 we have φ∗ Xρ(α) =/integraldisplay FXφ∗ Xφ∗ YαdµX(Y,Z), ρ(α) =/integraldisplay WX,−Xφ∗ YαdµX(Y,Z) =/integraldisplay W−X,Xφ∗ Yαdµ−X(Y,Z) =/integraldisplay F−Xφ∗ Yαdµ−X(Y,Z) =/integraldisplay FXφ∗ ZαdµX(Y,Z). So (19) ( φ∗ Xρ−ρ)α=/integraldisplay FX(φ∗ Xφ∗ Y−φ∗ Z)αdµX(Y,Z). Take a smooth partition of unity f1,...,f kofFsubordinated to the open cover Q1,...,Q k. Then the fj(X,·,·) form a partition of unity of FXsubordinated to the open cover given by the slices Qj,X={(Y,Z)∈g2: (X,Y,Z)∈Qj}. Let ΨX: Ω→Ω be the (0 ,−1)-bihomogeneous linear operator given by ΨXα=k/summationdisplay j=1/integraldisplay Qj,Xkj,ξαfj(ξ)dµX(Y,Z), whereξ= (X,Y,Z) for each (Y,Z)∈Qj,X. From (19) we get (20) ( φ∗ Xρ−ρ)0,0=d0,1ΨX+ΨXd0,1.LEAFWISE REDUCED COHOMOLOGY 23 Lemma 7.6. ΨXαdepends continuously on X∈B∗for eachα∈Ωfixed. Proof.It is completely analogous to the proof of Lemma 3.3 in [2]. /square Lemma 7.7. Forα∈Ω,X∈gandt∈Rwe have φ∗ tXα=α+/integraldisplayt 0φ∗ sXθXαds=α+θX/integraldisplayt 0φ∗ sXαds. Proof.It is completely analogous to the proof of Lemma 3.4 in [2]. /square Lemma 7.8. ρ1(E1) = (E1)θ1=0, and ρ1∗:E2∼=− −−− →H((E1)θ1=0). Proof.First, we shall prove that ρ1(E1)⊂(E1)θ1=0. Take any α∈ker(d0,1) defining [α]∈E1. If [α]∈ρ1(E1), we can suppose α=ρ0,0βfor someβ∈ker(d0,1). Then (21) ( φ∗ X)0,0α−α=d0,1ΨXβfor allX∈B∗ by (20). Thus Lemmas 7.6 and 7.7 yield (θX)0,0α=d0,1/parenleftbigg ΨXβ−(θX)0,0/integraldisplay1 0ΨsXβds/parenrightbigg as in [2, page 181]. Therefore ρ1([α])∈(E1)θ1=0. Letι: (E1)θ1=0→E1be the inclusion map. If [ α]∈(E1)θ1=0, since (θX)0,0 depends linearly on X∈g, there is a linear map X/mapsto→βXofgto Ω so that (θX)0,0α=d0,1βXfor allX∈g. Thus by Lemma 7.7 we get ρ0,0α=α+d0,1/integraldisplay B∗/integraldisplay1 0(φ∗ sX)0,0βXds∆∗(X), yieldingρ1ι= id. In particular ρ1(E1) = (E1)θ1=0. We also have ιρ1−id = d1λ1+λ1d1, and the result follows. /square End of the proof of Theorem 7.1. SinceGiscompact,therepresentation θgissemisim- ple [13, Sections 4.4 and 5.12]. So H((E1)θ1=0)∼=H(g)⊗(E0,· 1)θ1=0 by [13, Theorem V in Section 4.11, and Section 5.26]. The result now follo ws from Lemma 7.8. /square 7.3.Relation between H·(F)andE2. Theorem 7.9. With the above notations, H·(F)∼=E0,· 2. To begin with the proof of Theorem 7.9, the section s:M→Ndefines a homomorphism ( s∗)1:E0,· 1→H·(F) sinces∗d0,1=dFs∗. By restricting ( s∗)1, we get (s∗)2:E0,· 2= (E0,· 1)θ1=0→H·(F). We will prove that ( s∗)2is an isomorphism. For anyX∈gsetsX=φXs:M→N, which is an embedding, but not a section ofπNin general. Nevertheless sX(M) =s(M) exp(X). Analogously to s, the mapsXalso defines ( s∗ X)1:E0,· 1→H·(F). LetUXbe the neighborhood of sX(M) given by UX=/uniondisplay Y∈B∗φYsX(M) =sX(M)B=D−1 N(exp(X)B).24 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR For eachX∈gand eachx∈M,sXdefines an isomorphism sX∗:TxF∼=− −−− →TsX(x)/tildewideF. So s∗ X:/logicalandtextTsX(x)/tildewideF∗ ∼=− −−− →/logicalandtextTxF∗. For eachω∈Ω(F), letωXbe the unique smooth section of/logicalandtextT/tildewideF∗oversX(M) such thats∗ XωX=ω. DefineTXω∈Ω0,·(G|UX) by setting (TXω)(φYsX(x)) = (φ∗ −Y)0,0ωX(sX(x)) forY∈B∗andx∈M. This is well defined since ( x,Y)/mapsto→φYsX(x) is a dif- feomorphism of M×B∗ontoUX. Moreover d0,1TX=TXdFsinced0,1≡d/tildewideFon Ω0,·≡Ω·/parenleftBig /tildewideF/parenrightBig , and (φYsX)∗/tildewideF=Ffor allX,Y∈g. Therefore TXdefines a map TX∗:H·(F)→E0,· 1(G|UX). The inclusion map ιX:UX→Ninduces (ι∗ X)1:E0,· 1→E0,· 1(G|UX). Lemma 7.10. Ifζ∈(E0,· 1)θ1=0, thenTX∗(s∗ X)1ζ= (ι∗ X)1ζ. Proof.By Lemma 7.8 we have ρ1ζ=ζ. We thus can choose forms α,γ∈Ω0,·such thatd0,1α= 0,ζ= [α], andα=ρ0,0α+d0,1γ. Then (21) yields (φ∗ Y)0,0(α−d0,1γ)−(α−d0,1γ) =d0,1ΨYα for anyY∈B∗. So (22) ( φ∗ Y)0,0α−α=d0,1(ΨYα+(φ∗ Y)0,0γ−γ). Clearly (s∗ Xα)X=α|sX(M). Hence (TXs∗ Xα)(φYsX(x)) = (φ∗ −Y)0,0(α(sX(x))) = (α+d0,1(Ψ−Yα+(φ∗ −Y)0,0γ−γ))(φYsX(x)) by (22). But since each φYsX(M) is/tildewideF-saturated, d0,1≡d/tildewideFcommutes with the restriction to each φYsX(M). Therefore we get TXs∗ Xα=α+d0,1ηX onUX, whereηXis the (0,·)-form onUXdefined by ηX(φYsX(x)) = (Ψ −Yα+(φ∗ −Y)0,0γ−γ)(φYsX(x)), which finishes the proof. /square SinceGis compact, there is a finite sequence 0 = X1,X2,...,X lof elements of gsuch that G=B∪exp(X2)B∪···∪exp(Xl)B . LetUj=UXjTj=TXj,sj=sXjandιj=ιXjforj= 1,...,l. ThenN= U1∪···∪Ul. Leth1,...,h lbe a smooth partition of unity of Gsubordinated to the open cover exp( X1)B,...,exp(Xl)Bso thath1(e) = 1. Then D∗ Nh1,...,D∗ Nhlis a partition of unity of N subordinated to U1,...,U l. Forω∈Ω(F), defineTω∈Ω0,·by setting Tω=l/summationdisplay j=1D∗ NhjTjω.LEAFWISE REDUCED COHOMOLOGY 25 Since each D∗ Nhjis constant along the leaves of /tildewideF, we getd0,1T=TdF.SoT defines a map T∗:H·(F)→E0,· 1. Lemma 7.11. Ifζ∈(E0,· 1)θ1=0, thenT∗(s∗)1ζ=ζ. Proof.For eachX∈g, let (φ∗ X)1:E1→E1be the homomorphism defined by φ∗ X ((φ∗ X)1≡(φ∗ X)0,0∗). SincesX=φXs, by (21) we have (s∗ X)1ζ=s∗ 1(φ∗ X)1ζ=s∗ 1ζ . Therefore, by Lemma 7.10, (ι∗ j)1ζ=Tj∗(s∗ j)1ζ=Tj∗(s∗)1ζ forj= 1,...,l. So, ifζ= [α] forα∈Ω0,·withd0,1α= 0, there is some βj∈Ω0,· for eachjsuch thatα−Tjs∗α=d0,1βjoverUj. Let β=l/summationdisplay j=1D∗ Nhjβj∈Ω0,·. Since eachD∗ Nhjis constant on the leaves of /tildewideFandd0,1≡d/tildewideF, we get d0,1β=l/summationdisplay j=1D∗ Nhjd0,1βj =l/summationdisplay j=1D∗ Nhj(α−Tjs∗α) =α−Ts∗α, and the proof is complete. /square Lemma 7.12. (s∗)2:E0,· 2→H·(F)is surjective. Proof.Take anyω∈Ω(F) withdFω= 0, and take any function f≥0 compactly supported in Bsuch that/integraltext Bf(g)∆(g) = 1. Then α=D∗ NfT1ωis a (0,·)-form compactly supported in U1and satisfying d0,1α= 0. Soαdefines a class ζ∈E0,· 1. We shall prove that ( s∗)1ρ1ζ= [ω]. Forx∈MandY∈B∗we have α(φYs(x)) =f(exp(Y))(φ∗ −Y)0,0(ωX1(s(x))). So ((φ∗ Y)0,0α)(s(x)) =f(exp(Y))ωX1(s(x)), yielding (ρ0,0α)(s(x)) =/integraldisplay B∗((φ∗ Y)0,0α)(s(x))∆∗(Y) =ωX1(s(x))/integraldisplay B∗f(exp(Y))∆∗(Y) =ωX1(s(x))/integraldisplay Gf(g)∆(g) =ωX1(s(x)). Therefores∗ρ0,0α=s∗ωX1=ω, and the proof follows. /square Corollary 7.13. T∗(H·(F))⊂E0,· 2.26 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR Proof.It follows directly from Lemmas 7.11 and 7.12. /square End of the proof of Theorem 7.9. By Corollary 7.13 we can consider T∗:H·(F)→ E0,· 2. By Lemma 7.11 we have T∗(s∗)2= id. On the other hand, ( s∗)2T∗= id because (D∗ Nh1)(s(x)) = 1 for all x∈Msinceh1(e) = 1. So ( s∗)2is an isomor- phism. /square Corollary 7.14. H1(F)∼=H1(G)andH1(F)∼=H1(G). Proof.Theorem 7.1 yields E2,0 2∼=H2(g)⊗E0,0 2= 0 since gis compact semisimple. SoE0,1 2=E0,1 ∞∼=H1(G) canonically. Then H1(F)∼=H1(G) as topological vector spaces by Theorem 7.9, obtaining also H1(F)∼=H1(G). /square Corollary 7.15. H2(F)andH2(F)are of finite dimension if and only if so are H2(G)andH2(G), respectively. Proof.The leaves of Gare dense since so are the leaves of F. ThusH0(G)∼=R, yieldingE·,0 2∼=H·(g)byTheorem7.1. Ontheotherhand, H1(g) =H2(g) = 0since gis compact semisimple [28]. So E1,· i=E2,· i= 0 for 2 ≤i≤ ∞by Theorem 7.1. HenceE0,2 3=E0,2 2∼=H2(F) (using Theorem 7.9), and E3,0 3=E3,0 2∼=H3(g). Therefore, since E0,2 ∞=E0,2 4= ker(d3:E0,2 3→E3,0 3), H2(G)∼=E0,2 ∞can be identified to the kernel of some continuous homomorphism of H2(F) toH3(g), and the result follows. /square Corollary 7.16. SupposeMis oriented. Let ιi:Ki→M,i= 1,2, be smooth immersions of closed oriented manifolds of complementary d imension which define homology classes of Mwith non-trivial intersection. Let Γibe the image of the composite π1(Ki)π1(ιi)− −−− →π1(M)h− −−− →G. Suppose the group generated by Γ1∪Γ2is not dense in G. If1≤k= dimK2≤2, thendimHk(F) =∞. Proof.The result follows directly applying Corollaries 7.14, 7.15, and 2.6 to G./square Corollary 7.17. SupposeMis oriented. Let ιi:Ki→M,i= 1,2, be smooth immersions of closed oriented manifolds of complementary d imension which define homology classes of Mwith non-trivial intersection. Let Γibe the image of the composite π1(Ki)π1(ιi)− −−− →π1(M)h− −−− →G. Suppose the group generated by Γ1∪Γ2is not dense in G. Ifι1is transverse to F, thendimHk(F) =∞fork= dimK2. Proof.By Corollary 7.16, we can assume k >2. LetFlH·(G) andFlH·(G),l= 0,1,2,..., be the filtrations of H·(G) andH·(G) induced by (17). We have H·(G)/F1H·(G)∼=E0,· ∞⊂E0,· 2∼=H·(F), where both isomorphisms preserve the topologies, and E0,· ∞is a closed subspace of E0,· 2. (The last isomorphism follows from Theorem 7.9.) So H·(G)/F1H·(G) can be injected into H·(F), and it is enough to prove that Hk(G)/F1Hk(G) is of infinite dimension.LEAFWISE REDUCED COHOMOLOGY 27 This is a special case of the setting of Theorem 2.2 and Corollaries 2.5 a nd 2.6. The proofs of those results yield linearly independent classes ζm∈ Hk(G). In this case, we shall prove that the ζmare also linearly independent modulo F1Hk(G). Consider the pull-back bundles ι∗ iNoverKi. The canonical maps ˜ ιi:ι∗ iN→N areimmersionstransverseto G, whichthuscanbeconsideredastransverselyregular immersionsof ( ι∗ iN,Gi) into (N,G), where Gi= ˜ι∗ iG. We can assume the ιiintersect each other transversely, and thus the ˜ ιiintersect transversely in G. Let/tildewideH·⊂H·(G) and/tildewideH·⊂ H·(G) be the subspaces given by the classes that have representatives supported in π−1 N(U) for any open subset U⊂Mcontaining ι1(K1). SetF1/tildewideH·=/tildewideH·∩F1H·(G) andF1/tildewideH·=/tildewideH·∩F1H·(G). Sinceζm∈/tildewideHkby Remark3, it is enoughto provethat the ζmarelinearly independent modulo F1/tildewideHk. Hence, according to the proof of Theorem 2.2, it is enough to prove thatι2can be chosen so that ˜ ι∗ 2/parenleftBig F1/tildewideHk/parenrightBig = 0 where ˜ ι∗ 2:H·(G)→ H·(G2). In fact we shall prove the stronger property that the choice of ι2can be made so that ˜ ι∗ 2/parenleftBig F1/tildewideHk/parenrightBig = 0 for ˜ι∗ 2:H·(G)→H·(G2). Sinceι1is transverseto F, we can choose ι2such that, for some open subset U⊂ Mcontainingι1(K1), each connected component of ι2(K2)∩Uis contained in some leaf ofF. So, for every leaf L2ofG2, the connected components of ˜ ι2(L2)∩π−1 N(U) are contained in leaves of /tildewideF, yielding ˜ι∗ 2α= 0 over˜ι−1 2π−1 N(U) for anyα∈F1Ω·(G). MoreoverUandι2can be chosen so that the connected components of ι−1 2(U) are contractible;thus˜ ι−1 2π−1 N(U)≡ι−1 2(U)×Gcanonically,wheretheslices ι−1 2(U)×{∗} are the leaves of the restriction G2,UofG2to ˜ι−1 2π−1 N(U). HenceHl(G2,U) = 0 for l>0. Finally, the abovechoices can be made so that, for some open sub setV⊂M, we haveι1(K1)∩V=∅,U∪V=M, and each connected component of ι−1 2(U∩V) is contractible. Thus, as above, Hl(G2,U∩V) = 0 forl >0, where G2,U∩Vis the restriction of G2to ˜ι−1 2π−1 N(U∩V). Therefore, by using the Mayer-Vietoris type spectral sequence (cf. [10]) ··· →Hl−1(G2,U∩V)→Hl(G2)→Hl(G2,U)⊕Hl(G2,V)→Hl(G2,U∩V)→ ··· and sincek>2, we get (23) Hk(G2)∼=Hk(G2,V) by the restriction homomorphism. Nowanyξ∈F1/tildewideHkcanbedefinedbyaleafwiseclosedform α∈Ωk(G) supported inM\Vwithα+dGβ∈F1Ωk(G) for some β∈Ωk−1(G). Then ˜ι∗ 2(α+dGβ) is supported in ˜ ι−1 2π−1 N(V), where it is the G2-leafwise derivative of ˜ ι∗ 2β. So ˜ι∗ 2ξis mapped to zeroin Hk(G2,V), and thus ˜ ι∗ 2ξ= 0 by (23), which finishes the proof. /square 8.Case of foliations on nilmanifolds Γ\Hdefined by normal subgroups of H The goal of this section is to prove Theorem 2.10. It will be done by ind uction, which needs leafwise reduced cohomology with coefficients in a vector bundle with a flatF-partial connection. Thus we shall prove a more general theorem by taking arbitrary coefficients. Forafoliation Fonamanifold Mandavectorbundle VoverM, aflatF-partial connection on Vcan be defined as a flat connection on the restriction of Vto the28 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR leaves whose local coefficients are smooth on each foliation chart of FonM. So the corresponding de Rham derivative dFwith coefficients in Vpreserves smoothness onM; i.e.dFpreservesΩ( F,V) =C∞(/logicalandtextTF∗⊗V). ThenH·(F,V) canbe defined in the same way as H·(F) by using (Ω( F,V),dF) instead of (Ω( F),dF). Consider the following particular case. Let Hbe a simply connected nilpotent Lie group, K⊂Ha normal connected subgroup, and Γ ⊂Ha discrete uniform subgroup whose projection to H/Kis dense. Then let Fbe the foliation on the nilmanifold M= Γ\Hdefined as the quotient of the foliation /tildewideFonHwhose leaves are the translates of K. In this case, Mis closed and the leaves of Fare dense. Let/tildewideVbe anH×K-vector bundle over Hfor the left action of H×KonHgiven by (h,k)h′=hh′k−1, (h,k)∈H×Kandh′∈H. We also consider the induced left actions of HandKonH. The space of H-invariant sections of /tildewideVwill be denoted by C∞/parenleftBig /tildewideV/parenrightBig H, and the subspaces of invariant sections will be denoted in a similar way for other actions. Suppose /tildewideVis endowed with an H×K-invariant flat/tildewideF-partial connection, and let Vbe the induced vector bundle on Mwith the induced flat F-partial connection. The structure of H×K-vector bundle on V canonically defines an action of konC∞/parenleftBig /tildewideV/parenrightBig H, wherekis the Lie algebra of K. Moreover the induced differential map on/logicalandtextk∗⊗C∞/parenleftBig /tildewideV/parenrightBig Hcorresponds to d/tildewideFby the canonical injection of this space in Ω( F,V). Theorem 8.1. With the above notations, H·(F,V)∼=H·/parenleftBig k,C∞/parenleftBig /tildewideV/parenrightBig H/parenrightBig . The result will follow by induction on the codimension qofF. Forq= 0 andVthe trivial line bundle, this is just a well known theorem of K. Nomizu [27]. If q= 0 andVis arbitrary, the result still follows with the obvious adaptation of the arguments in [27]. Supposeq >0 and the result is true for foliations of codimension less than q. The proof has two cases. Case1.AssumeK∩Γ = 1. ThegroupΓisnilpotentsincesois H, thusthecenterof Γ is non-trivial. Let abe a non-trivial element in the center of Γ. By the universal property of Mal’cev’s completion [23], there exists a one dimensional c onnected subgroupLof the center of Hcontaining /a\}b∇acketle{ta/a\}b∇acket∇i}htas a discrete uniform subgroup. L is isomorphic to RsinceHis simply connected. Let H1=H/L, and Γ 1= Γ//a\}b∇acketle{ta/a\}b∇acket∇i}ht. Clearly Γ 1is canonically injected in H1as a discrete uniform subgroup. We get L∩K= 1 because /a\}b∇acketle{ta/a\}b∇acket∇i}ht∩K= 1, and thus there is a canonical injection of KintoH1 as a normal subgroup, defining a foliation F1on the nilmanifold M1= Γ1\H1.F1 is a foliationofthe type considered in the statement ofthis theorem , ofcodimension q−1, but observe that the canonical injection of KintoH1may not have trivial intersection with Γ 1. The projection H//a\}b∇acketle{ta/a\}b∇acket∇i}ht →H1is canonically an S1-principal bundle (considering S1≡L//a\}b∇acketle{ta/a\}b∇acket∇i}ht), so the induced map π:M→M1is also an S1-principal bundle in a canonical way. Then Vcanonically is an S1-vector bundle so that the partial connection is invariant, and thus induces the ve ctor bundle V1=V/S1overM1with the corresponding flat F1-partial connection. The lifting ofV1toH1is/tildewideV1=/tildewideV/L, which satisfies the same properties as /tildewideVwith respect to K1instead ofK.LEAFWISE REDUCED COHOMOLOGY 29 For eachx∈M1and eachm∈Z, define Cm,x={f∈C∞(π−1(x),C) :f(yθ) =f(y)e2πmθi for ally∈π−1(x) and allθ∈S1≡R/Z}. It is easy to see that Cm=/unionsqdisplay x∈M1Cm,x is a one-dimensional C-vectorbundle over M1in a canonical way. For m∈Z, define also Ω(F,V⊗C)m={α∈Ω(F,V⊗C) :α(yθ) =α(y)e2πmθi for ally∈π−1(x) and allθ∈S1}, andsimilarlydefine C∞/parenleftBig/parenleftBig /tildewideV//a\}b∇acketle{ta/a\}b∇acket∇i}ht/parenrightBig ⊗C/parenrightBigm consideringthe S1-principalbundle H//a\}b∇acketle{ta/a\}b∇acket∇i}ht → H1. By the Fourier series expression for functions on S1, we get that Ω( F,V⊗C) is theC∞closure of /circleplusdisplay m∈ZΩ(F,V⊗C)m. It can be easily seen that there is a canonical isomorphism (24) Ω( F1,V1⊗Cm)∼=Ω(F,V⊗C)m defined byπ∗and the canonical identity C∞(Cm)≡C∞(M,C)m. SinceFis preserved by the S1-action onM,dFpreserves each Ω( F,V⊗C)mand corresponds to dF1by (24). By induction H·(F1,V1⊗Cm)∼=H·/parenleftbigg k,C∞/parenleftBig /tildewideV1⊗˜Cm/parenrightBig H1/parenrightbigg . But C∞/parenleftBig /tildewideV1⊗/tildewideCm/parenrightBig H1∼=C∞/parenleftBig/parenleftBig /tildewideV//a\}b∇acketle{ta/a\}b∇acket∇i}ht/parenrightBig ⊗C/parenrightBigm H//angbracketlefta/angbracketright canonically, which is obviously trivial if m/\e}atio\slash= 0. ButC0is the trivial complex line bundle, so H·(F,V⊗C)∼=H·(F1,V1⊗C0) ∼=H·/parenleftbigg k,C∞/parenleftBig /tildewideV1⊗C/parenrightBig H1/parenrightbigg ∼=H·/parenleftBig k,C∞/parenleftBig /tildewideV⊗C/parenrightBig H/parenrightBig . Case2.In the general case, let G=H/Kand Γ 1the projection of Γ to G. We use Mal’cev’s construction for the pair ( G,Γ1). It yields a simply connected nilpo- tent Lie group H1containing Γ 1as a discrete uniform subgroup, and a surjective homomorphism D1:H1→Gwhich is the identity on Γ 1. The kernel K1ofD1 defines a foliation Gof codimension qon the nilmanifold M1= Γ1\H1, and we have K1∩Γ1= 1. SoGis the type of foliation we have considered in Case 1. Gis the classifying foliation for foliations with transverse structure g iven by (G,Γ1). So there is a smooth map f:M→M1which is transverse to Gand so thatF=f∗G. In this particular case, fcan be constructed in the following way. By the universal property of Mal’cev’s construction, the surject ive homomorphism30 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR of Γ to Γ 1can be uniquely extended to a surjective homomorphism ˜f:H→H1, which defines a map f:M→M1. We have D1˜f=D. SoKis projected onto K1, and thus F=f∗F1. Moreover fis a locally trivial bundle with fiber the nilmanifold P/(P∩Γ), wherePis the kernel of ˜f. Fix a vector subbundle ν⊂TFwhich is complementary to the subbundle τ⊂TFof vectors that are tangent to the fibers of f. Then we get a canoni- cal isomorphism/logicalanddisplay TF∗⊗V∼=/logicalanddisplay ν∗⊗/logicalanddisplay τ∗⊗V , yielding a bigrading of Ω( F,V) given by Ωu,v(F,V) =C∞/parenleftBiggu/logicalanddisplay ν∗⊗v/logicalanddisplay τ∗⊗V/parenrightBigg . Consider the filtration of Ω( F,V) given by the differential subspaces FkΩ(F,V) =/circleplusdisplay u≥kΩu,·(F,V), which depend only on FandV; in fact they could be defined without using ν. This filtration induces a spectral sequence ( Ei,di) converging to H·(F,V), whose terms (E0,d0) and (E1,d1) can be described as follows. The derivative dFdecomposes as sum of bihomogeneous operators dF,0,1,dF,1,0anddF,2,−1, where each double subindex indicates the corresponding bidegree. These operators satisfy identities which are similar to those in (15) and (16), yielding (E0,d0)≡(Ω(F,V),dF,0,1), (E1,d1)≡(H(Ω(F,V),dF,0,1),dF,1,0∗). Letk1be the Lie algebra of K1. EachX∈k1canonically defines a vector field X1onM1which is tangent to the leaves of F1. LetXνbe the unique vector field onMwhich is a section of νand projects to X1. Forα∈Ω0,v(F) ands∈C∞(V), defineθX(α⊗s) to be the (0 ,·)-component of θXνα⊗s+α⊗∇Xνs, where∇denotes the flat F-partial connection of V. It can be easily checked that θXdF,0,1=dF,0,1θX. SoθXdefines an operator, also denoted by θX, onE0,· 1. In this way, we get a representation θofk1onE0,· 1, and a canonical isomorphism Eu,v 2∼=Hu(k1,θ). Define V1,y=H·/parenleftbig f−1(y),V|f−1(y)/parenrightbig , y∈M1, V1=/unionsqdisplay y∈M1V1,y, and let/tildewideV1be the lifting of V1toH1. It is easy to see that /tildewideV1canonically is a H1×K1-vector bundle over the H1×K1-manifoldH1with anH1×K1-invariant flat/tildewideF1-partial connection. (The fibers of /tildewideV1are of finite dimension since the fibers offare compact.) It is also easily seen that there is a canonical isomorph ism C∞(V1)∼=E0,· 1. Moreover the representation of k1onE0,· 1corresponds to the representation of k1onC∞/parenleftBig /tildewideV1/parenrightBig defined by the flat partial connection of /tildewideV1. So Eu,· 2∼=Hu/parenleftBig k1,C∞/parenleftBig /tildewideV1/parenrightBig/parenrightBig ∼=Hu(F1,V1).LEAFWISE REDUCED COHOMOLOGY 31 LetEibe the quotient of Eiover the closure 0iof its trivial subspace. Then Eu,· 2∼=Hu(F1,V1)∼=Hu/parenleftbigg k1,C∞/parenleftBig /tildewideV1/parenrightBig H1/parenrightbigg by Case 1. If the above filtration is restricted to the space of differential for ms in Ω( F,V) whose lifting to HisH-left invariant, we get a spectral sequence ( Ei,di) converging toH·/parenleftBig k1,C∞/parenleftBig /tildewideV/parenrightBig H/parenrightBig , and there is a canonical homomorphism ( Ei,di)→(Ei,di) of spectral sequences. Analogously, we have a canonical isomorp hism Eu,· 2∼=Hu(k1,C∞(V1)H1). So the composite E2→E2→ E2is an isomorphism, and thus E2∼=E2⊕02as differential complexes. Then E3∼=E3⊕H(02,d2), yielding H(02,d2)∼=03, and the above decomposition is of differential complexes. We get E4∼=E4⊕H(03,d3). Continuing with these arguments, we finally obtain Ei∼=Ei⊕0ias topological differential complexes for i≥2, and thus H·(F,V)∼=E∞∼=E∞⊕0∞. Hence H·(F,V)∼=E∞∼=H·(k,C∞(V)H) as desired. Remark 4.For general Lie foliations with dense leaves and nilpotent structura l Lie algebra,theclassifyingfoliationsareofthetypeconsideredinTheo rem2.10. Onthe one hand, if the ambient manifold is closed and the classifying map can b e chosen to be a fiber bundle, then a spectral sequence argument shows th at the leafwise reduced cohomology is of finite dimension. On the other hand, if the c lassifying map has unavoidable singularities, then they should correspond to h andles on the leavesandthe leafwisereducedcohomologyisofinfinite dimensionbyC orollary2.4. References [1] J. A. ´Alvarez L´ opez. A finiteness theorem for the spectral sequen ce of a Riemannian foliation. Illinois J. of Math. , 33:79–92, 1989. [2] J. A. ´Alvarez L´ opez. A decomposition theorem for the spectral se quence of Lie foliations. Trans. Amer. Math. Soc. , 329:173–184, 1992. [3] J. A. ´Alvarez L´ opez and G. Hector. Leafwise homologies, leafwis e cohomology, and subfo- liations. In E. Mac´ ıas-Virg´ os X. M. Masa and J. A. ´Alvarez L´ opez, editors, Analysis and Geometry in Foliated Manifolds , pages 1–12, Singapore, 1995. Proceedings of the VII Inter- national Colloquium on Differential Geometry, Santiago de C ompostela, 26–30 July, 1994, World Scientific. [4] J. A. ´Alvarez L´ opez and S. Hurder. Pure-point spectrum for folia tion geometric operators. Preprint, 1994. [5] J. A. ´Alvarez L´ opez and Y. A. Kordyukov. Long time behavior of lea fwise heat flow for Riemannian foliations. Compositio Math. , 125: 129–153, 2001. [6] P. Andrade and G. Hector. Foliated cohomology and Thurst on’s stability. Preprint, 1993. [7] P. Andrade and M. do S. Pereira. On the cohomology of one-d imensional foliated manifolds. Bol. Soc. Brasil. Mat. , 21:79–89, 1990. [8] J.L. Arraut and N. M. dos Santos. Linear foliations of Tn.Bol. Soc. Brasil. Mat. , 21:189–204, 1991. [9] R. Bott and L. W. Tu. Differential Forms in Algebraic Topology , volume 82 of Graduate Texts in Math. Springer-Verlag, New York, 1982. [10] A. El Kacimi-Alaoui. Sur la cohomologie feuillet´ ee. Compositio Math. , 49:195–215, 1983.32 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR [11] E. Fedida. Feuilletages de Lie, feuilletages du plan. I nLect. Notes Math. , volume 352, pages 183–195. Springer-Verlag, 1973. [12] C.Godbillon. Feuilletages: ´Etudes G´ eom´ etriques , volume 98 of Progress in Math. Birkh¨ auser, Boston, Basel and Stuttgart, 1991. [13] W. Greub, S. Halperin, and R. Vanstone. Connections, Curvature, and Cohomology. Vol. III. Academic Press, New York, San Francisco and London, 1975. [14] A. Haefliger. Structures feuillet´ ees et cohomologie ` a valeurs dans un faisceau de groupo¨ ıdes. Comment. Math. Helv. , 32:248–329, 1958. [15] A. Haefliger. Some remarks on foliations with minimal le aves.J. Differential Geom. , 15:269– 384, 1980. [16] A. Haefliger. Leaf closures in Riemannian foliations. I nA Fˆ ete on Topology , pages 3–32, New York, 1988. Academic Press. [17] G. Hector and U. Hirsch. Introduction to the Geometry of Foliations, Part B , volume E3 of Aspects of Mathematics . Friedr. Vieweg and Sohn, Braunschweig, 1983. [18] J. L. Heitsch. A cohomology for foliated manifolds. Comment. Math. Helv. , 50:197–218, 1975. [19] S. Hurder. Spectral theory of foliation geometric oper ators. Preprint, 1992. [20] A. El Kacimi-Alaoui and A. Tihami. Cohomologie bigradu ´ ee de certains feuilletages. Bull. Soc. Math. Belge , s´ erie B-38, 1986. [21] W. Klingenberg. Riemannian Geometry , volume 1 of De Gruyter Studies in Mathematics . De Gruyter, Berlin, 1982. [22] S. Kobayashi. On conjugate and cut loci. In Studies in Global Geometry and Analysis , pages 96–122. Math. Assoc. of Amer., 1967. [23] A.I. Mal’cev. On a class of homogeneous spaces. Transl. Amer. Math. Soc. , 39:276–307, 1951. [24] P. Molino. G´ eom´ etrie globale des feuilletages riema nniens.Nederl. Akad. Wetensch. Indag. Math., 44:45–76, 1982. [25] P. Molino. Riemannian Foliations , volume 73 of Progress in Math. Birkh¨ auser Boston Inc., Boston, MA, 1988. [26] P. Molino and M. Pierrot. Th´ eor´ emes des slice et holon omie des feuilletages Riemanniens. Annales Inst. Fourier, Grenoble , 37:207–223, 1987. [27] K. Nomizu. On the cohomology of compact homogeneous spa ces of nilpotent lie groups. Ann. of Math. , 59:531–538, 1954. [28] W. A. Poor. Differential Geometric Structures . McGraw-Hill, New York, 1982. [29] B. L. Reinhart. Foliated manifolds with bundle-like me trics.Ann. of Math. , 69:119–132, 1959. [30] C. Roger. M´ ethodes Homotopiques et Cohomologiques en Th´ eorie de Fe uilletages . Universit´ e de Paris XI, Paris, 1976. [31] J. Stallings. Group Theory and Three Dimensional Manifolds . Yale University Press, Yale, 1971. [32] R. Thom. Quelques propri´ et´ es globales des vari´ et´ e s diff´ erentiables. Comment. Math. Helv. , 28:17–86, 1954. Departamento de Xeometr ´ıa e Topolox ´ıa, Facultade de Matem ´aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Sp ain E-mail address :jesus.alvarez@usc.es Institut Girard Desargues, UPRESA 5028, 43, boulevard du 11 No vembre 1918, Uni- versit´e Claude Bernard-Lyon I, 69622 Villeurbanne Cedex, France E-mail address :hector@geometrie.univ-lyon1.fr

2013-11-14

Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described for homogeneous foliations with dense leaves on closed nilmanifolds.

The dimension of the leafwise reduced cohomology

1311.3518v1

arXiv:2210.16931v1 [math.AP] 30 Oct 2022Intrinsic polynomial squeezing for Balakrishnan-Taylor beam models E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vicen te Abstract We explore the energy decay properties related to a model in e xtensible beams with the so-called energy damping . We investigate the influence of the non- loncal damping coefficient in the stability of the model. We pr ove, for the first time, that the corresponding energy functional is squeezed by pol ynomial-like functions involving the power of the damping coefficient, which arises i ntrinsically from the Balakrishnan-Taylor beam models. As a consequence, it is sh own that such models with nonlocal energy damping are never exponentially stabl e in its essence. 1 Introduction In 1989 Balakrishnan and Taylor [3] derived some prototypes of vibrating exten- sible beams with the so-called energy damping . Accordingly, the following one dimensional beam equation is proposed /u1D715/u1D461/u1D461/u1D462−2/u1D701√ /u1D706/u1D715/u1D465/u1D465/u1D462+/u1D706/u1D715/u1D465/u1D465/u1D465/u1D465/u1D462−/u1D6FC/bracketleftbigg/uni222B.dsp/u1D43F −/u1D43F/parenleftbig/u1D706|/u1D715/u1D465/u1D465/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightbig/u1D451/u1D465/bracketrightbigg/u1D45E /u1D715/u1D465/u1D465/u1D461/u1D462=0,(1) where/u1D462=/u1D462(/u1D465,/u1D461)represents the transversal deflection of a beam with length 2 /u1D43F >0 in the rest position, /u1D6FC > 0 is a damping coefficient, /u1D701is a constant appearing in Krylov-Bogoliubov’s approximation, /u1D706 > 0 is related to mode frequency and spectral density of external forces, and /u1D45E=2(/u1D45B+/u1D6FD) +1 with/u1D45B∈Nand 0≤/u1D6FD<1 2. E. H. Gomes Tavares State University of Londrina, 86057-970, Londrina, PR, Bra zil, e-mail:eduardogomes7107@gmail.com M. A. Jorge Silva State University of Londrina, 86057-970, Londrina, PR, Bra zil. e-mail:marcioajs@uel.br V. Narciso State University of Mato Grosso do Sul, 79804-970, Dourados , MS, Brazil. e-mail:vnarciso@uems.br A. Vicente Western Paraná State University, 85819-110, Cascavel, PR, Brazil. e-mail:andre.vicente@unioeste.br 12 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte We still refer to [3, Sect. 4] for several other beam equation s taking into account nonlocal energy damping coefficients, as well as [2, 4, 6, 7, 12 , 17, 18] for associated models. A normalized /u1D45B-dimensional equation corresponding to (1) can be seen as follows /u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462−/u1D6FC/bracketleftbigg/uni222B.dsp Ω/parenleftBig |Δ/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightBig /u1D451/u1D465/bracketrightbigg/u1D45E Δ/u1D715/u1D461/u1D462=0, (2) where we denote /u1D706=1 and/u1D705=2/u1D701;Ωmay represent an open bounded of R/u1D45B; and the symbols ΔandΔ2stand for the usual Laplacian and Bi-harmonic operators, respectively. Additionally, in order to see the problem wit hin the frictional context of dampers, we rely on materials whose viscosity can be essen tially seen as friction between moving solids. In this way, besides reflecting on a mo re challenging model (at least) from the stability point of view, one may metaphys ically supersede the viscous damping in (2) by a nonlocal frictional one so that we cast the model /u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp Ω/parenleftBig |Δ/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightBig /u1D451/u1D465/bracketrightbigg/u1D45E /u1D715/u1D461/u1D462=0. (3) The main goal of this paper is to explore the influence of the no nloncal damping coefficient in the stability of problem (3). Unlike the existi ng literature on extensible beams with full viscous or frictional damping, we are going t o see for the first time that the feature of the energy damping coefficient E/u1D45E(/u1D461):=E/u1D45E(/u1D462,/u1D462/u1D461)(/u1D461)=/bracketleftbigg/uni222B.dsp Ω/parenleftBig |Δ/u1D462(/u1D461)|2+ |/u1D715/u1D461/u1D462(/u1D461)|2/parenrightBig /u1D451/u1D465/bracketrightbigg/u1D45E , /u1D45E > 0, (4) not only prevents exponential decay, but also gives us a poly nomial range in terms of/u1D45Ewhose energy is squeezed and goes to zero polynomially when t ime goes to infinity. More precisely, by noting that the corresponding e nergy functional is given by /u1D438/u1D705(/u1D461):=/u1D438/u1D705(/u1D462,/u1D462/u1D461)(/u1D461)=/uni222B.dsp Ω/parenleftBig |Δ/u1D462(/u1D461)|2+ |/u1D715/u1D461/u1D462(/u1D461)|2+/u1D705|∇/u1D462(/u1D461)|2/parenrightBig /u1D451/u1D465, /u1D705≥0,(5) then it belongs to an area of variation between upper and lowe r polynomial limits as follows /u1D4500/u1D461−1 /u1D45E/lessorsimilar/u1D438/u1D705(/u1D461)/lessorsimilar/u1D4360/u1D461−1 /u1D45E, /u1D461→ +∞, (6) for some constants 0 < /u1D450 0≤/u1D4360depending on the initial energy /u1D438/u1D705(0), /u1D705≥0. Indeed, such a claim corresponds to an intrinsic polynomial range of (uniform) stability and will follow as a consequence of a more general r esult that is rigorous stated in Theorem 2. See also Corollary 1. In particular, we c an conclude that (3) is not exponentially stable when dealing with weak initial dat a, that is, with solution in the standard energy space. See Corollary 2. In conclusion, Theorem 2 truly reveals the stability of the a ssociated energy /u1D438/u1D705(/u1D461), which leads us to the concrete conclusions provided by Coro llaries 1-2, being pioneering results on the subject. Due to technicalities in the well-posedness process,Intrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 3 we shall work with /u1D45E≥1/2. In Section 2 we prepare all notations and initial results. Then, all precise details on the stability results shall be g iven in Section 3. 1.1 Previous literature, comparisons and highlights In what follows, we are going to highlight that our approach a nd results are different or else provide generalized results, besides keeping more p hysical consistency in working exactly with (4) instead of modified versions of it. I ndeed, there are at least three mathematical ways of attacking the energy damping coe fficient (4) along the equation (3) (or (2)), namely: 1. Keeping the potential energy in (4), but neglecting the ki netic one; 2. Keeping the kinetic energy in (4), but neglecting the pote ntial one; 3. Keeping both potential and kinetic energies, but conside ring them under the action of a strictly (or not) positive function /u1D440(·)as a non-degenerate (or possibility degenerate) damping coefficient. In the first case, equation (3) becomes to /u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp Ω|Δ/u1D462|2/u1D451/u1D465/bracketrightbigg/u1D45E /u1D715/u1D461/u1D462=0 inΩ× (0,∞). (7) This is, for sure, the most challenging case once the damping coefficient becomes now to a real degenerate coefficient. In [5, Theorem 3.1], work ing on a bounded domainΩwith clamped boundary condition, it is proved the following with/u1D45E=1 in (7): for every/u1D445 >0, there exist constants /u1D436/u1D445=/u1D436(/u1D445)>0and/u1D6FE/u1D445=/u1D6FE(/u1D445)>0 depending on /u1D445such that /u1D438/u1D705(/u1D461) ≤/u1D436/u1D445/u1D438/u1D705(0)/u1D452−/u1D6FE/u1D445/u1D461, /u1D461 > 0, (8) only holds for every regular solution /u1D462of(3)with initial data (/u1D4620,/u1D4621)satisfying /ba∇dbl(/u1D4620,/u1D4621)/ba∇dbl(/u1D43B4(Ω)∩/u1D43B2 0(Ω))×/u1D43B2 0(Ω)≤/u1D445. (9) We stress that (8) only represents a local stability result since it holds on every ball with radius /u1D445 > 0 in the strong topology (/u1D43B4(Ω) ∩/u1D43B2 0(Ω)) ×/u1D43B2 0(Ω),but they are not independent of the initial data. Moreover, as ob served by the authors in [5], the drawback of (8)-(9) is that it could not be proved i n the weak topology /u1D43B2 0(Ω) ×/u1D43F2(Ω), even taking initial data uniformly bounded in /u1D43B2 0(Ω) ×/u1D43F2(Ω). Although we recognized that our results for (3) can not be fai rly compared to such a result, we do can conclude by means of the upper and lowe r polynomial bounds (6) that the estimate (8) will never be reached for wea k initial data given in /u1D43B2 0(Ω) ×/u1D43F2(Ω). Therefore, our results act as complementary conclusions t o [5] by clarifying such drawback raised therein, and yet giving a di fferent point of view of stability by means of (6) and its consequences concerning pr oblem (3).4 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte In the second case, equation (3) falls into /u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp Ω|/u1D715/u1D461/u1D462|2/u1D451/u1D465/bracketrightbigg/u1D45E /u1D715/u1D461/u1D462=0 inΩ× (0,∞). (10) Unlike the first case, here we have an easier setting because t he kinetic damping coefficient provides a kind of monotonous (polynomial) dampi ng whose computa- tions to achieve (6) remain unchanged (and with less calcula tions). This means that all results highlighted previously still hold for this part icular case. In addition, they clarify what is precisely the stability result related to pr oblems addressed in [19, 20], which in turn represent particular models of abstract dampi ng given by [1, Section 8]. In other words, in terms of stability, our methodology pr ovides a way to show the existence of absorbing sets with polynomial rate (and no t faster than polynomial rate depending on /u1D45E) when dealing with generalized problems relate to (10), sub ject that is not addressed in [19, 20]. Finally, in the third case let us see equations (2)-(3) as fol lows /u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D440/parenleftbigg/uni222B.dsp Ω/parenleftBig |Δ/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightBig /u1D451/u1D465/parenrightbigg /u1D434/u1D715/u1D461/u1D462=0 inΩ× (0,∞),(11) where operator /u1D434represents the Laplacian operator /u1D434=−Δor else the identity one /u1D434=/u1D43C. Thus, here we clearly have two subcases, namely, when /u1D440(·) ≥ 0 is a non- degenerate or possibly degenerate function. For instance, when/u1D440(/u1D460)=/u1D6FC/u1D460/u1D45E, /u1D460≥0, and/u1D434=−Δ, then we go back to problem (2). For this (degenerate) nonloc al strong damping situation with /u1D45E≥1, it is considered in [11, Theorem 3.1] an upper polynomial stability for the corresponding energy, which a lso involves a standard nonlinear source term. Nonetheless, we call the attention t o the following prediction result provided in [11, Theorem 4.1] for (2) addressed on a bo unded domain Ωwith clamped boundary condition and /u1D45E≥1:By taking finite initial energy 0</u1D438/u1D705(0)< ∞, then/u1D438/u1D705(/u1D461)given in (5)satisfies /u1D438/u1D705(/u1D461) ≤3/u1D438/u1D705(0)/u1D452−/u1D6FF∫/u1D461 0/ba∇dbl/u1D462(/u1D460)/ba∇dbl2/u1D45E/u1D451/u1D460, /u1D461 > 0, (12) where/u1D6FF=/u1D6FF(1 /u1D438/u1D705(0))>0is a constant proportional to 1//u1D438/u1D705(0). Although the estimate (12) provides a new result with an exponential face , it does not mean any kind of stability result. Indeed, it is only a peculiar estimate indicating that prevents exponential decay patterns as rem arked in [11, Section 4]. In addition, it is worth pointing out that our computations t o reach the stability result for problem (3) can be easily adjusted to (2), even for /u1D45E≥1/2 thanks to a inequality provided in [1, Lemma 2.2]. Therefore, through the polynomi al range (6) we provide here a much more accurate stability result than the estimate expressed by (12), by concluding indeed that both problems (2) and (3) are never ex ponentially stable in the topology of the energy space. On the other hand, in the non-degenerate case /u1D440(/u1D460)>0, /u1D460≥0, but still taking /u1D434=−Δ, a generalized version of (11) has been recently approached by [16] in a context of strong attractors , that is, the existence of attractors in the topology ofIntrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 5 more regular space than the weak phase space. In this occasio n, the/u1D4361-regularity for/u1D440 > 0 brings out the non-degeneracy of the damping coefficient, wh ich in turn allowed them to reach interesting results on well-posednes s, regularity and long-time behavior of solutions over more regular spaces. Such assump tion of positiveness for the damping coefficient has been also addressed by other au thors for related problems, see e.g. [8, 9, 10]. From our point of view, in spite of representing a nice case, the latter does not portray the current situation of th is paper so that we do not provide more detailed comparisons with such a non-degenera te problems, but we refer to [5, 8, 9, 10, 11, 16] for a nice survey on this kind of no n-degenerate damping coefficients. Additionally, we note that the suitable case of non-degenerate damping coefficient/u1D440(/u1D460)>0, /u1D460≥0, and/u1D434=/u1D43Cin (11) has not been considered in the literature so far and shall be concerned in another work by th e authors in the future. At light of the above statements, one sees e.g. when /u1D440(/u1D460)=/u1D6FC/u1D460/u1D45E, /u1D460≥0,and /u1D434=/u1D43C, then problem (11) falls into (3), being a problem not yet add ressed in the literature that brings out a new branch of studies for such a n onlocal (possibly degenerate) damped problems, and also justifies all new stab ility results previously specified. 2 The problem and well-posedness Let us consider again the beam model with energy damping /u1D715/u1D461/u1D461/u1D462+Δ2/u1D462−/u1D705Δ/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp Ω/parenleftBig |/u1D715/u1D461/u1D462|2+ |Δ/u1D462|2/parenrightBig /u1D451/u1D466/bracketrightbigg/u1D45E /u1D715/u1D461/u1D462=0 inΩ× (0,∞),(13) with clamped boundary condition /u1D462=/u1D715/u1D462 /u1D715/u1D708=0 on/u1D715Ω× [0,∞), (14) and initial data /u1D462(/u1D465,0)=/u1D4620(/u1D465), /u1D715/u1D461/u1D462(/u1D465,0)=/u1D4621(/u1D465), /u1D465∈Ω. (15) To address problem (13)-(15), we introduce the Hilbert phas e space (still called energy space ) H:=/u1D43B2 0(Ω) ×/u1D43F2(Ω), equipped with the inner product/angbracketleftbig /u1D4671,/u1D4672/angbracketrightbig H:=/angbracketleftbig Δ/u1D4621,Δ/u1D4622/angbracketrightbig +/angbracketleftbig /u1D463.alt1,/u1D463.alt2/angbracketrightbig for/u1D467/u1D456=(/u1D462/u1D456,/u1D463.alt/u1D456) ∈ H, /u1D456=1,2,and norm /ba∇dbl/u1D467/ba∇dblH=/parenleftbig/ba∇dblΔ/u1D462/ba∇dbl2+ /ba∇dbl/u1D463.alt/ba∇dbl2/parenrightbig1/2,for/u1D467=(/u1D462,/u1D463.alt) ∈ H,where /an}b∇acke⊔le{⊔/u1D462,/u1D463.alt/an}b∇acke⊔∇i}h⊔:=/uni222B.dsp Ω/u1D462/u1D463.alt/u1D451/u1D465 ,/ba∇dbl/u1D462/ba∇dbl2:=/an}b∇acke⊔le{⊔/u1D462,/u1D462/an}b∇acke⊔∇i}h⊔and/ba∇dbl/u1D467/ba∇dbl2 H:=/an}b∇acke⊔le{⊔/u1D467,/u1D467/an}b∇acke⊔∇i}h⊔H. In order to stablish the well-posedness of (13)-(15), we defi ne the vector-valued function/u1D467(/u1D461):=(/u1D462(/u1D461),/u1D463.alt(/u1D461)),/u1D461≥0,with/u1D463.alt=/u1D715/u1D461/u1D462. Then we can rewrite system (13)-(15) as the following first order abstract problem6 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte /braceleftBigg/u1D715/u1D461/u1D467=A/u1D467+M(/u1D467), /u1D461 > 0, /u1D467(0)=(/u1D4620,/u1D4621):=/u1D4670,(16) whereA:D(A) ⊂ H → H is the linear operator given by A/u1D467=(/u1D463.alt,−Δ2/u1D462),D(A) :=/u1D43B4(Ω) ∩/u1D43B2 0(Ω), (17) andM:H → H is the nonlinear operator M(/u1D467)=(0,/u1D705Δ/u1D462−/u1D6FC/ba∇dbl/u1D467/ba∇dbl2/u1D45E H/u1D463.alt), /u1D467=(/u1D462,/u1D463.alt) ∈ H. (18) Therefore, the existence and uniqueness of solution to the s ystem (13)-(15) relies on the study of problem (16). Accordingly, we have the follow ing well-posedness result. Theorem 1. Let/u1D705,/u1D6FC≥0and/u1D45E≥1 2be given constants. If /u1D4670∈ H, then (16)has a unique mild solution /u1D467in the class/u1D467∈/u1D436([0,∞),H). In addition, if /u1D4670∈ D(A) , then/u1D467is a regular solution lying in the class /u1D467∈/u1D436([0,∞),D(A)) ∩/u1D4361([0,∞),H). Proof. To show the local version of the first statement, it is enough t o prove that A given in (17) is the infinitesimal generator of a /u1D4360-semigroup of contractions /u1D452A/u1D461 (which is very standard) and Mset in (18) is locally Lipschitz on Hwhich will be done next. Indeed, let /u1D45F >0 and/u1D4671,/u1D4672∈ Hsuch that max {/ba∇dbl/u1D4671/ba∇dblH,/ba∇dbl/u1D4672/ba∇dblH} ≤/u1D45F. We note that /bardblex/bardblex/bardblex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E H/u1D463.alt1− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E H/u1D463.alt2/bardblex/bardblex/bardblex≤/bracketleftBig /ba∇dbl/u1D4671/ba∇dbl2/u1D45E H+ /ba∇dbl/u1D4672/ba∇dbl2/u1D45E H/bracketrightBig /ba∇dbl/u1D463.alt1−/u1D463.alt2/ba∇dbl+/barex/barex/barex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E H− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E H/barex/barex/barex/ba∇dbl/u1D463.alt1+/u1D463.alt2/ba∇dbl. (19) The first term on the right side of (19) can be estimated by /bracketleftBig /ba∇dbl/u1D4671/ba∇dbl2/u1D45E H+ /ba∇dbl/u1D4672/ba∇dbl2/u1D45E H/bracketrightBig /ba∇dbl/u1D463.alt1−/u1D463.alt2/ba∇dbl ≤2/u1D45F2/u1D45E/ba∇dbl/u1D4671−/u1D4672/ba∇dblH. Now, from a suitable inequality provided in [1] /one.supwe estimate the second term as follows /barex/barex/barex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E H− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E H/barex/barex/barex/ba∇dbl/u1D463.alt1+/u1D463.alt2/ba∇dbl ≤4/u1D45E/u1D45F2/u1D45E/ba∇dbl/u1D4671−/u1D4672/ba∇dblH. Plugging the two last estimates in (19), we obtain /bardblex/bardblex/bardblex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E H/u1D463.alt1− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E H/u1D463.alt2/bardblex/bardblex/bardblex H≤2(2/u1D45E+1)/u1D45F2/u1D45E/ba∇dbl/u1D4671−/u1D4672/ba∇dblH. Thus, /one.supSee [1, Lemma 2.2]: Let/u1D44Bbe a normed space with norm /ba∇dbl · /ba∇dbl/u1D44B. Then, for any /u1D460≥1we have /barex/barex/ba∇dbl/u1D462/ba∇dbl/u1D460 /u1D44B− /ba∇dbl/u1D463.alt/ba∇dbl/u1D460 /u1D44B/barex/barex≤/u1D460max{/ba∇dbl/u1D462/ba∇dbl/u1D44B,/ba∇dbl/u1D463.alt/ba∇dbl/u1D44B}/u1D460−1/ba∇dbl/u1D462−/u1D463.alt/ba∇dbl/u1D44B,∀/u1D462,/u1D463.alt∈/u1D44B. (20)Intrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 7 /ba∇dblM(/u1D4671) −M(/u1D4672)/ba∇dblH≤/parenleftBig /u1D705+2(2/u1D45E+1)/u1D6FC/u1D45F2/u1D45E/parenrightBig /ba∇dbl/u1D4671−/u1D4672/ba∇dblH, andMis locally Lipschitz in H. Hence, according to Pazy [15, Chapter 6], if /u1D4670∈ H(/u1D4670∈/u1D437(A)), there exists a time/u1D461max∈ (0,+∞]such that (16) has a unique mild (regular) solution /u1D467∈/u1D436([0,/u1D461max),H) (/u1D467∈/u1D436([0,/u1D461max),/u1D437(A)) ∩/u1D4361([0,/u1D461max),H)). Moreover, such time /u1D461maxsatisfies either the conditions /u1D461max=+∞or else/u1D461max<+∞ with lim /u1D461→/u1D461−max/ba∇dbl/u1D467(/u1D461)/ba∇dblH=+∞. (21) In order to show that /u1D461max=+∞, we consider /u1D4670∈/u1D437(A)and the corresponding regular solution /u1D467of (16). Taking the inner product in Hof (16) with /u1D467, we obtain 1 2/u1D451 /u1D451/u1D461/bracketleftbig /ba∇dbl/u1D467(/u1D461)/ba∇dbl2 H+/u1D705/ba∇dbl∇/u1D462(/u1D461)/ba∇dbl2/bracketrightbig +/u1D6FC/ba∇dbl/u1D467(/u1D461)/ba∇dbl2/u1D45E H/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2=0/u1D461∈ [0,/u1D461max).(22) Integrating (22) over (0,/u1D461), /u1D461∈ [0,/u1D461max), we get /ba∇dbl/u1D467(/u1D461)/ba∇dblH≤ (1+/u1D450′/u1D705)1/2/ba∇dbl/u1D4670/ba∇dblH, /u1D461∈ [0,/u1D461max). Here, the constant /u1D450′>0 comes from the embedding /u1D43B2 0(Ω)↩→/u1D43B1 0(Ω). The last estimate contradicts (21). Hence, /u1D461/u1D45A/u1D44E/u1D465=+∞. Using a limit process, one can conclude the same result for mild solutions. The proof of Theorem 1 is then complete. 3 Lower-upper polynomial energy’s bounds By means of the notations introduced in Section 2, we recall t hat the energy functional corresponding to problem (13)-(15) can be expressed by /u1D438/u1D705(/u1D461)=1 2/bracketleftbig /ba∇dbl(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))/ba∇dbl2 H+/u1D705/ba∇dbl∇/u1D462(/u1D461)/ba∇dbl2/bracketrightbig , /u1D461≥0. (23) Our main stability result reveals that /u1D438/u1D705(/u1D461)is squeezed by decreasing polynomial functions as follows. Theorem 2. Under the assumptions of Theorem 1, there exists an increasing function J:R+→R+such that the energy /u1D438/u1D705(/u1D461)satisfies /bracketleftbig 2/u1D45E+1/u1D6FC/u1D45E/u1D461+/bracketleftbig /u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbig−1//u1D45E≤/u1D438/u1D705(/u1D461) ≤/bracketleftbigg/u1D45E J (/u1D438/u1D705(0))(/u1D461−1)++/bracketleftbig /u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbigg−1//u1D45E ,(24) for all/u1D461 >0, where we use the standard notation /u1D460+:=(/u1D460+ |/u1D460|)/2. Proof. Taking the scalar product in /u1D43F2(Ω)of (13) with /u1D715/u1D461/u1D462, we obtain8 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte /u1D451 /u1D451/u1D461/u1D438/u1D705(/u1D461)=−/u1D6FC||(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))||2/u1D45E H/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2, /u1D461 > 0. (25) Let us prove the lower and upper estimates in (24) in the seque l. Lower bound. We first note that ||(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))||2/u1D45E H/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2≤2/u1D45E+1[/u1D438/u1D705(/u1D461)]/u1D45E+1, and replacing it in (25), we get /u1D451 /u1D451/u1D461/u1D438/u1D705(/u1D461) ≥ − 2/u1D45E+1/u1D6FC[/u1D438/u1D705(/u1D461)]/u1D45E+1, /u1D461 > 0. (26) Thus, integrating (26) and proceeding a straightforward co mputation, we reach the first inequality in (24). Upper bound. Now, we are going to prove the second inequality of (24). To do so, we provide some proper estimates and then apply a Nakao’s res ult (cf. [13, 14]). We start by noting that ||(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461)||2/u1D45E H/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2≥ /ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2(/u1D45E+1), (27) and replacing (27) in (25), we get /u1D451 /u1D451/u1D461/u1D438/u1D705(/u1D461) +/u1D6FC/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2(/u1D45E+1)≤0, /u1D461 > 0, (28) which implies that /u1D438/u1D705(/u1D461)is non-increasing with /u1D438/u1D705(/u1D461) ≤/u1D438/u1D705(0)for every/u1D461 >0. Also, integrating (28) from /u1D461to/u1D461+1, we obtain /u1D6FC/uni222B.dsp/u1D461+1 /u1D461/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2(/u1D45E+1)/u1D451/u1D460≤/u1D438/u1D705(/u1D461) −/u1D438/u1D705(/u1D461+1):=[/u1D437(/u1D461)]2. (29) Using Hölder’s inequality with/u1D45E /u1D45E+1+1 /u1D45E+1=1 and (29), we infer /uni222B.dsp/u1D461+1 /u1D461/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2/u1D451/u1D460≤1 /u1D6FC1 /u1D45E+1[/u1D437(/u1D461)]2 /u1D45E+1. (30) From the Mean Value Theorem for integrals, there exist /u1D4611∈ [/u1D461,/u1D461+1 4]and/u1D4612∈ [/u1D461+3 4,/u1D461+1]such that /ba∇dbl/u1D715/u1D461/u1D462(/u1D461/u1D456)/ba∇dbl2≤4/uni222B.dsp/u1D461+1 /u1D461/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2/u1D451/u1D460≤4 /u1D6FC1 /u1D45E+1[/u1D437(/u1D461)]2 /u1D45E+1, /u1D456=1,2. (31) On the other hand, taking the scalar product in /u1D43F2(Ω)of (13) with /u1D462and inte- grating the result over [/u1D4611,/u1D4612], we haveIntrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 9 /uni222B.dsp/u1D4612 /u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460=/uni222B.dsp/u1D4612 /u1D4611/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2/u1D451/u1D460+1 2[(/u1D715/u1D461/u1D462(/u1D4611),/u1D462(/u1D4611)) − (/u1D715/u1D461/u1D462(/u1D4612),/u1D462(/u1D4612))] −/u1D6FC 2/uni222B.dsp/u1D4612 /u1D4611||(/u1D462(/u1D460),/u1D715/u1D461/u1D462(/u1D460))||2/u1D45E H(/u1D715/u1D461/u1D462(/u1D460),/u1D462(/u1D460))/u1D451/u1D460. (32) Let us estimate the terms in the right side of (32). Firstly, w e note that through Hölder’s inequality, (31) and Young’s inequality, we obtai n |(/u1D715/u1D461/u1D462(/u1D4611),/u1D462(/u1D4611)) − (/u1D715/u1D461/u1D462(/u1D4612),/u1D462(/u1D4612))| ≤/u1D4512/summationdisplay.1 /u1D456=1/ba∇dbl/u1D715/u1D461/u1D462(/u1D461/u1D456)/ba∇dbl/ba∇dblΔ/u1D462(/u1D461/u1D456)/ba∇dbl ≤8/u1D451 /u1D6FC1 2(/u1D45E+1)[/u1D437(/u1D461)]1 /u1D45E+1sup /u1D4611≤/u1D460≤/u1D4612[/u1D438/u1D705(/u1D460)]1/2 ≤128/u1D4512 /u1D6FC1 /u1D45E+1[/u1D437(/u1D461)]2 /u1D45E+1+1 8sup /u1D4611≤/u1D460≤/u1D4612/u1D438/u1D705(/u1D460), where the constant /u1D451 >0 comes from the embedding /u1D43B2 0(Ω)↩→/u1D43F2(Ω). Addition- ally, using that /u1D438/u1D705(/u1D461) ≤/u1D438/u1D705(0), we have /ba∇dbl(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))/ba∇dbl2/u1D45E H≤2/u1D45E[/u1D438/u1D705(/u1D461)]/u1D45E≤2/u1D45E[/u1D438/u1D705(0)]/u1D45E. From this and (30) we also get /barex/barex/barex/barex/uni222B.dsp/u1D4612 /u1D4611||(/u1D462(/u1D460),/u1D715/u1D461/u1D462(/u1D460))||2/u1D45E H(/u1D715/u1D461/u1D462(/u1D460),/u1D462(/u1D460))/u1D451/u1D460/barex/barex/barex/barex≤22/u1D45E+3/u1D4512[/u1D438/u1D705(0)]2/u1D45E /u1D6FC−/u1D45E /u1D45E+1[/u1D437(/u1D461)]2 /u1D45E+1 +1 8/u1D6FCsup /u1D4611≤/u1D460≤/u1D4612/u1D438/u1D705(/u1D460). Regarding again (30) and replacing the above estimates in (3 2), we obtain /uni222B.dsp/u1D4612 /u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460≤ K (/u1D438/u1D705(0)) [/u1D437(/u1D461)]2 /u1D45E+1+1 8sup /u1D4611≤/u1D460≤/u1D4612/u1D438/u1D705(/u1D460), (33) where we set the function Kas K(/u1D460):=/bracketleftbigg64/u1D4512+1 /u1D6FC1 /u1D45E+1+2(/u1D45E+1)/u1D4512/u1D6FC2/u1D45E+1 /u1D45E+1/u1D4602/u1D45E/bracketrightbigg >0. Using once more the Mean Value Theorem for integrals and the f act that/u1D438/u1D705(/u1D461)is non-increasing, there exists /u1D701∈ [/u1D4611,/u1D4612]such that /uni222B.dsp/u1D4612 /u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460=/u1D438/u1D705(/u1D701)(/u1D4612−/u1D4611) ≥1 2/u1D438/u1D705(/u1D461+1), and then10 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vic ente sup /u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460)=/u1D438/u1D705(/u1D461)=/u1D438/u1D705(/u1D461+1) + [/u1D437(/u1D461)]2≤2/uni222B.dsp/u1D4612 /u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460+ [/u1D437(/u1D461)]2. Thus, from this and (33), we arrive at sup /u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460) ≤ [/u1D437(/u1D461)]2+2/uni222B.dsp/u1D4612 /u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460 ≤ [/u1D437(/u1D461)]2+2K (/u1D438/u1D705(0)) [/u1D437(/u1D461)]2 /u1D45E+1+1 4sup /u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460), and since 0<2 /u1D45E+1≤2, we obtain sup /u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460) ≤4 3[/u1D437(/u1D461)]2 /u1D45E+1/bracketleftBig [/u1D437(/u1D461)]2/u1D45E /u1D45E+1+2K (/u1D438/u1D705(0))/bracketrightBig . (34) Observing that [/u1D437(/u1D461)]2/u1D45E /u1D45E+1≤ [/u1D438/u1D705(/u1D461) +/u1D438/u1D705(/u1D461+1)]/u1D45E /u1D45E+1≤2/u1D45E /u1D45E+1[/u1D438/u1D705(0)]/u1D45E /u1D45E+1,and de- noting by J(/u1D460):=/parenleftbigg4 3/parenrightbigg/u1D45E+1/bracketleftBig (2/u1D460)/u1D45E /u1D45E+1+2K(/u1D460)/bracketrightBig/u1D45E+1 >0, (35) and also recalling the definition of [/u1D437(/u1D461)]2in (29), we obtain from (34) that sup /u1D461≤/u1D460≤/u1D461+1[/u1D438/u1D705(/u1D460)]/u1D45E+1≤ J (/u1D438/u1D705(0)) [/u1D438/u1D705(/u1D461) −/u1D438/u1D705(/u1D461+1)]. Hence, applying e.g. Lemma 2.1 of [14] with /u1D438/u1D705=/u1D719,J (/u1D438/u1D705(0))=/u1D4360,and/u1D43E=0, we conclude /u1D438/u1D705(/u1D461) ≤/bracketleftbigg /u1D45E J(/u1D438/u1D705(0))(/u1D461−1)++1/bracketleftbig /u1D438/u1D705(0)/bracketrightbig/u1D45E/bracketrightbigg−1//u1D45E ,which ends the proof of the second inequality in (24). The proof of Theorem 2 is therefore complete. Remark 1. It is worth point out that we always have /bracketleftbig 22/u1D45E+1/u1D6FC/u1D45E/u1D461+/bracketleftbig /u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbig−1//u1D45E≤/bracketleftbigg/u1D45E J (/u1D438/u1D705(0))(/u1D461−1)++/bracketleftbig /u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbigg−1//u1D45E ,(36) so that it makes sense to express /u1D438/u1D705(/u1D461)between the inequalities in (24). Indeed, from the definition Jin (35) one easily sees that J (/u1D438/u1D705(0)) ≥1 22/u1D45E+1/u1D6FC,from where one concludes (36) promptly. Corollary 1. (Polynomial Range of Decay). Under the assumptions of Theorem 2, the energy functional /u1D438/u1D705(/u1D461)defined in (23)decays squeezed as follows /u1D4500/u1D461−1 /u1D45E/lessorsimilar/u1D438/u1D705(/u1D461)/lessorsimilar/u1D4360/u1D461−1 /u1D45Eas/u1D461→ +∞, (37) for some constants 0</u1D4500≤/u1D4360depending on the initial energy /u1D438/u1D705(0).Intrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 11 In other words, /u1D438/u1D705(/u1D461)decays polynomially at rate /u1D461−1//u1D45E(/u1D45E≥1/2) as/u1D461→ +∞ . ⊓ ⊔ Corollary 2. (Non-Exponential Stability). Under the assumptions of Theorem 2, the energy/u1D438/u1D705(/u1D461)set in (23)never decays exponentially as /u1D452−/u1D44E/u1D461(/u1D44E >0) as/u1D461→ +∞ . ⊓ ⊔ References 1. F. Aloui, I. Ben Hassen, A. Haraux, Compactness of traject ories to some nonlinear second order evolution equations and applications, J. Math. Pures Appl. 100 (2013), no. 3, 295-326. 2. A. V. Balakrishnan, A theory of nonlinear damping in flexib le structures. Stabilization of flexible structures, p. 1-12, 1988. 3. A. V. Balakrishnan and L. W. Taylor, Distributed paramete r nonlinear damping models for flight structures, in: Proceedings Daming 89, Flight Dynami cs Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989. 4. R. W. Bass and D. Zes, Spillover, Nonlinearity, and flexibl e structures, in The Fourth NASA Workshop on Computational Control of Flexible Aerospace Sy stems, NASA Conference Publication 10065 ed. L.W.Taylor (1991) 1-14. 5. M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Si lva, V. Narciso, Stability for extensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type, J. of Differential Equations, 290 (2021), 197-222. 6. E. H. Dowell, Aeroelasticity of plates and shells, Gronin ger, NL, Noordhoff Int. Publishing Co. (1975). 7. T. J. Hughes and J. E. Marsden, Mathematical foundation of elasticity, Englewood C. Prentice- Hall, 1983. 8. M. A. Jorge Silva and V. Narciso, Long-time behavior for a p late equation with nonlocal weak damping, Differential Integral Equations 27 (2014), no. 9-1 0, 931-948. 9. M. A. Jorge Silva and V. Narciso, Attractors and their prop erties for a class of nonlocal extensible beams, Discrete Contin. Dyn. Syst. 35 (2015) no. 3, 985-1008. 10. M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping, Evol. Equ. Control Theory 6 (20 17), no. 3, 437-470. 11. M. A. Jorge Silva, V. Narciso and A. Vicente, On a beam mode l related to flight structures with nonlocal energy damping, Discrete and Continuous Dyna mical Systems Series B. 24 (2019), 3281-3298. 12. C. Mu, J. Ma, On a system of nonlinear wave equations with B alakrishnan-Taylor damping, Z. Angew. Math. Phys. 65 (2014) 91-113. 13. M. Nakao, Convergence of solutions of the wave equation w ith a nonlinear dissipative term to the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (19 76), 257-265. 14. M. Nakao, A difference inequality and its application to n onlinear evolution equations, J. Math. Soc. Japan 30 4 (1978) 747-762. 15. A. Pazy, Semigroups of linear operators and application s to partial differential equations, vol. 44, Springer-Verland,1983. 16. Y. Sun, Z. Yang, Strong attractors and their robustness f or an extensible beam model with energy damping, Discrete and Continuous Dynamic al Systems, 2021. (doi: 10.3934/dcdsb.2021175) 17. Y. You, Inertial manifolds and stabilization of nonline ar beam equations with Balakrishnan- Taylor damping, Abstr. Appl. Anal. 1(1) (1996) 83-102. 18. W. Zhang, Nonlinear Damping Model: Response to Random Ex citation in: 5th Annual NASA Spacecraft Control Laboratory Experiment (SCOLE) Worksho p (1988) 27-38.12 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vic ente 19. C. Zhao, C. Zhao, C. Zhong, The global attractor for a clas s of extensible beams with nonlocal weak damping, Discrete Contin. Dyn. Syst. - B, 25 (2020), 935 -955. 20. C. Zhao, S. Ma, C. Zhong, Long-time behavior for a class of extensible beams with nonlocal weak damping and critical nonlinearity Journal of Mathemat ical Physics 61 (2020), p. 032701.

2022-10-30

We explore the energy decay properties related to a model in extensible beams with the so-called energy damping. We investigate the influence of the nonloncal damping coefficient in the stability of the model. We prove, for the first time, that the corresponding energy functional is squeezed by polynomial-like functions involving the power of the damping coefficient, which arises intrinsically from the Balakrishnan-Taylor beam models. As a consequence, it is shown that such models with nonlocal energy damping are never exponentially stable in its essence.

Intrinsic polynomial squeezing for Balakrishnan-Taylor beam models

2210.16931v1

Transport properties of spin superfluids—comparing easy-plane ferro- and antiferromagnets Martin Evers and Ulrich Nowak Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany (Dated: December 2, 2019) We present a study on spin-superfluid transport based on an atomistic, classical spin model. Easy- plane ferro- as well as antiferromagnets are considered, which allows for a direct comparison of these two material classes based on the same model assumptions. We find a spin-superfluid transport which is robust against variations of the boundary conditions, thermal fluctuations, and dissipation modeled via Gilbert damping. Though the spin accumulations is smaller for antiferromagnets the range of the spin-superfluid transport turns out to be identical for ferro- and antiferromagnets. Fi- nally, we calculate and explore the role of the driving frequency and especially the critical frequency, where phase slips occur and the spin accumulation breaks down. I. INTRODUCTION Spin transport in magnetic insulators [1, 2] has been intensively studied beacause of the fundamental interest in the various physical phenomena that occur in these materials and because of their potential for future appli- cations. Magnetic insulators do not exhibit Joule heat- ing [3] as no electron transport is involved and many of these are oxides with exceptionally low magnetic damp- ing [4], which hopefully allows for energy efficient trans- port properties. It has even been shown that the realiza- tion of logic elements is possible [5], such that devices are compatible and integratable with CMOS technology [6]. Studies on transport in this material class focuses mostly on transport of magnons [7], i.e. quanta of spin waves— the elementary excitations of the magnetic ground state. As magnons are quasi particles, their number is not con- served and each magnon mode shows an exponential de- cay upon transport through the system on a length scale ξcalled magnon propagation length [8–13]. This is even true at zero temperature and in a clean system without any disorder due to the coupling of the magnons to elec- tronic and phononic degrees of freedom, a fact which is described phenomenologically via Gilbert damping in the equation of motion as will be explained below. In contrast to this damped magnonic transport, a pro- posal for spin transport was made that carries the name spin superfluidity. The original idea is in fact quite old [14, 15] and rests on a similarity of the magnetic or- der parameter—either the magnetization of a ferromag- net or the Néel vector of an antiferromagnet—compared to the order parameter of superfluidity—the macroscopic wave function—as it occurs for He-4 below the lambda transition. For instance, in easy-plane ferromagnets the magnetizationfeaturesaspontaneouslybrokenrotational symmetry in the easy plane ( SO(2)symmetry) that is equivalent to the spontaneously broken gauge invariance of the macroscopic wave function ( U(1)symmetry). This symmetry leads in both cases to currents that are sta- ble against small deviations—the supercurrents. [16] One striking difference of spin-superfluid transport to spin- wave transport is its distance dependence: for spin su-perfluidity it is expected to be non-exponential, pushing the limit of the range of magnonic transport. The first experimental realizations of a spin superfluid was achieved in a system of nuclear spins of He-3 atoms [17]—a model system which is not in a solid state. Only recently the physics of spin superfluidity has drawn again attention for the case of solid magnets [18–23], including a proposed dissipationless transport in metallic magnets [18]. However, König et al. neglected spin-orbit inter- action in their model for the electrons, which is one of the reasons for Gilbert damping in magnets [24]. But ev- ery known material exhibits spin-orbit interaction—since spinandangularmomentumofanatomareneverexactly zero—and therefore also magnetic damping, even if it is small. Consequently, spinsuperfluidsdoalwaysshowdis- sipation in contrast to their conventional counterparts. Recent theoretical work has focused on insulators rather than metals, usually based of phenomenological modelsincluding theLandau-Lifshitz-Gilbertequationof motion for both ferro- and antiferromagnets. [16, 19, 20] The experimental detection of spin superfluidity in solid- state magnets has been reported for magnon condensates [25], where the origin of the spin-superfluid order param- eter is different to the cases described above, and also in antiferromagnetic solids [23]. However, the interpre- tation of the experimental findings is still controversially discussed [16, 26–28]. In the following, we will investigate and compare spin superfluidityinferro-andantiferromagneticmodels. The geometry of our model resembles that of an experimen- tal non-local spin-transport investigation as sketched in fig. 1. In the corresponding experiments [29] at one side (here on the left) a spin current is injected into the mag- net viathe spin-Halleffect causedby an electricalcurrent through an attached heavy-metal stripe. The resulting spin current is detected using the inverse spin-Hall ef- fect at another position (here the right-hand side). In our model we avoid the details of the excitation mech- anism and model the effect of the injected spin current by an appropriate boundary condition that triggers the dynamics of the spin systems that we investigate. This is done from the perspective of an atomistic, classicalarXiv:1911.12786v1 [cond-mat.mes-hall] 28 Nov 20192 Figure 1. Basic concept of non-local spin transport as in an experimental setup: heavy metal stripes are attached to the magnet to inject a spin current via the spin-Hall effect (here on the left hand side). The spin current in a certain distance (here at the right end) is detected via inverse spin-Hall effect. spin model, which has some advantages: the approach is not restricted to small deviations from the ground state, finite temperatures can be investigated and our calcu- lations are not limited to the steady state only. Fur- thermore, we are able to compare ferro- and antiferro- magnetic systems. Their behavior turns out to be very similar, except for the resulting spin accumulation that is muchlowerforthelatter. However,fromanexperimental point of view antiferromagnets are much more promising, since these are not prone to a breakdown of spin super- fluidity as a consequence of dipolar interactions, which is hard to avoid in ferromagnets. [22] II. ATOMISTIC SPIN MODEL We consider the following classical, atomistic spin model of Heisenberg type [30], comprising Nnormal- ized magnetic moments Sl=µl/µSon regular lattice sitesrl. We assume a simple cubic lattice with lattice constanta. The Hamiltonian for these moments, in the following called “spins”, is given by H=−J 2/summationdisplay /angbracketleftn,m/angbracketrightSn·Sm−dz/summationdisplay n(Sn z)2,(1) taking into account Heisenberg exchange interaction of nearest neighbors quantified by the exchange constant J, where each spin has Nnbnearest neighbors. Further- more, a uniaxial anisotropy with respect to the zdirec- tion with anisotropy constant dzis included. In this work we consider the easy-plane case dz<0, where the mag- nets ground state readsgSl=±(cos(gϕ),sin(gϕ),0)with some arbitrary, but uniform anglegϕ∈[0,2π](SO(2) symmetry) and an alternating sign ±in case of antifer- romagnetic order ( J <0). The time evolution of the spins Slis governed by the stochastic Landau-Lifshitz-Gilbert (LLG) equationof motion [31–33] dSl dt=−γ µS(1 +α2)/bracketleftbig Sl×/parenleftbig Hl+αSl×Hl/parenrightbig/bracketrightbig (2) Hl=−∂H ∂Sl+ξl /angbracketleftbig ξl β(t)/angbracketrightbig = 0,/angbracketleftBig ξl β(t)ξl/prime η(t/prime)/angbracketrightBig =δll/primeδβηδ(t−t/prime)2µSαkBT γ describing the motion of a spin in its effective field Hl, whereγisthegyromagneticratio, αtheGilbertdamping constant,kBthe Boltzmann constant and Tthe absolute temperature. The properties of the thermal noise ξlare chosen such that the dissipation-fluctuation theorem is satisfied [34]. The material parameters define our sys- tem of units,|J|for the energy, tJ:=µS/γ|J|for the time,afor the distance. Numerically the LLG equation is solved either by the classical Runge-Kutta method in case of zero temperature, or at finite temperature using stochastic Heun’s method. At zero temperature the dis- sipated power per spin due to Gilbert damping follows directly from the time evolution of the spins Sl(t)[35]: Pdiss=1 NdH dt=1 N/summationdisplay n∂H ∂Sn/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright eff.field·∂Sn ∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright LLG.(3) We study a magnetic wire extended along xdirection. The system size for our numerical simulations is given byN=Nx×Ny×Nzspins along x-,y- andzdirec- tion, where Nx/greatermuchNy,Nz. For transverse directions we use periodic boundary conditions if not noted otherwise. Boundary spins at x=Nxa(the right-hand side) are denotedSl/vextendsingle/vextendsingle rightand at this side an open boundary con- dition is applied, Sl/vextendsingle/vextendsingle right= 0. At the opposite side, at x= 0, we use a time-dependent boundary condition, Sl/vextendsingle/vextendsingle left=±(cos(ω0t),sin(ω0t),0), (4) in form of an in-plane precession with frequency ω0that injects a spin current from this side. The alternating sign (±) is used only for antiferromagnetic systems, according to the sublattices with antiparallel spin orientation. The use of this boundary condition creates an ex- citation with well-defined frequency ω0. Alternatively, we also assumed an externally given spin accumulation µ=µezat the left-hand side that causes additional torques on the spins and drives them out of equilibrium, which directly maps an experimental implementation us- ing a spin-Hall-generated spin accumulation to the model utilized here. This method has been used for instance in [22]. In appendix B we calculate how this spin accumula- tion maps to the excitation frequency ω0and we further- more confirmed numerically that both mechanisms lead to the same response for ferro- and antiferromagnets. Although an atomistic picture—comprising discrete degrees of freedom—is studied numerically, the micro- magnetic approximation is of particular value for analyt- ical considerations of ferromagnets. This approximation3 assumes that spatial variations of magnetic structures are small compared to the atomic distance a. In this case differences can be approximated as derivatives and the spins form a continuous field S(r,t). It is handy to use cylindrical coordinates S=/parenleftBig/radicalbig 1−S2zcosϕ,/radicalbig 1−S2zsinϕ, Sz/parenrightBig ,where definitions Sz(rl) :=Sl zandϕ(rl) :=ϕllink the atomistic picture to the micromagnetics. Note that for a spin superfluid Szis considered as the spin-superfluid density and ϕits phase. The use of the micromagnetic approximationforferromagnetsallowstoreformulatethe LLGequationintermsofdifferentialequationsfor Szand ϕthat read µS γ˙ϕ=Ja2/bracketleftBigg 1 1−S2z∆Sz+Sz|∇Sz|2 (1−S2z)2+Sz|∇ϕ|2/bracketrightBigg + 2dzSz−αµS γ˙Sz 1−S2z(5) µS γ˙Sz=−Ja2/bracketleftbig/parenleftbig 1−S2 z/parenrightbig ∆ϕ−2Sz∇Sz·∇ϕ/bracketrightbig +α/parenleftbig 1−S2 z/parenrightbigµS γ˙ϕ. (6) These two equations are strictly equivalent to the LLG equation eq. (2) for zero temperature with the only as- sumption of the micromagnetic approximation. If one expands these equations in lowest order in ∇ϕ,∆ϕ,∇Sz, and∆Szfor an easy-plane magnet, which implies espe- cially assuming|Sz|/lessmuch1, but keeping|∇ϕ|2, one ends up with µS γ˙ϕ=Ja2∆Sz+Ja2Sz|∇ϕ|2+ 2dzSz−αµS γ˙Sz(7) µS γ˙Sz=−Ja2∆ϕ+αµS γ˙ϕ. (8) Importantly, keeping the |∇ϕ|2term is actually required if the damping takes relatively high values, a fact which we checked numerically. Furthermore, these equations are very similar to others already reported in [19, 21], but not exactly equivalent. Ref. [19] uses more approxima- tions, especially neglecting the |∇ϕ|2-term, and ref. [21] considers a different starting point, namely a quantum theory at low temperatures, where this term has a dif- ferentSz-dependence. Because of this difference, the re- sult from [21] does not exactly match our numerical re- sults of the atomistic spin model, nor does it match the classical micromagnetic theory. Hence, we use eqs. (7) and (8) that do describe the atomistic spin simulations well. However, eqs. (7) and (8) can be solved in steady state for a special case: a ferromagnet that is of length L alongxdirection and exhibits translational invariance in y- andzdirection as carried out in appendix A. Steady state means a coherent precession of all spins with a fre- quency ˙ϕ=ω0and a stationary profile Sz(x). This so- lution of eqs. (7) and (8) reads: sϕ(x,t) =α 2µSω0 γJ(x−L)2 a2+ω0t+ϕ0(9) sSz(x) =sSz(L) 1 +µ2 Sω2 0 2γ2Jdzα2/parenleftbigx−L a/parenrightbig2, (10) with a spin accumulation at the right end of the sys- tem (atx=Nxa=:L) ofsSz(L) =µSω0/2γdz, avalue which is independent of L—one of the striking fea- tures of spin superfluidity. Another feature is the mono- tone increase of ϕwhich implies the formation of an in- plane spin spiral with winding number Nw, which reads 2πNw=/integraltext dϕ=ϕ(L)−ϕ(0). Note furthermore, that an open boundary condition at the right end is an assump- tion that leads to solutions eqs. (9) and (10), correspond- ing to a Neumann condition ∇ϕ/vextendsingle/vextendsingle right= 0, which must be justified as a realistic choice. For the numerical study of eq. (2) we assume an open boundary at the right end. Equation (10) assumes the same and results in a finite Szatx= 0, which contra- dicts the numerical driving boundary at this side, eq. (4), that forces Sz(x=0) = 0. Furthermore, in an experiment an open boundary at the right end might not be feasi- ble because of outflowing spin currents, for example into an attached heavy metal. Thus, the real behavior at the boundaries for sure deviates from the ideal solution eq. (10) and raises the question how strong that devia- tion is and in how far the boundary conditions influence the overall bulk behavior of the spin transport. This is examined numerically from the full model eq. (2) by varying the boundary conditions on the left and right. One example of the variations we tested is an absorbing boundary condition on the right, modeling an outflow- ing spin current by an enhanced damping. As result we observe the profile Sl zto show only little change in that case compared to an open boundary and also that in all cases the numerical profiles well follow eq. (10) (see in the following fig. 2 a) as example). Other variations of the boundary condition which we tested have also hardly any impact on the magnets overall response. III. EASY-PLANE FERROMAGNET In a first step of the numerical investigation, we con- sider a collinear ferromagnet as most simple case, with parameters J > 0for the ferromagnetic state and dz= −0.01Jas in-plane anisotropy. Let us describe the phe-4 0 1000 2000 3000 4000 5000012345610-3 010203040506070 0 0.5 1 1.5 2 2.5 10-300.010.020.03 5 6 10-40.0240.026 Figure 2. Spin superfluidity in a 1D ferromagnet at T= 0in the steady state: a)depicts the spin accumulation Szand the in-plane angle ϕforω0tJ=−2×10−4; numerical data (blue and red symbols) follow perfectly the theoretical curve eqs. (9) and (10) (black, dashed lines), except for the vicinity of the left boundary. This is an artifact of the boundary condition, eq. (4), used for the numerics. b)shows the spin accumulation at the right end of the system SN zversus driving frequency ω0; for small driving frequencies up to a critical value ωcritthe numerical data follow the analytical curvesSz(L); for larger frequencies the spin accumulation breaks down, deviating form the theoretical curve, due to phase slips and spin wave excitations. nomenology of the spin superfluid in a 1D system of size Nx×Ny×Nz= 5000×1×1at temperature T= 0. This model is equivalent to a 3D system with transla- tional invariance in y- andzdirection. Furthermore, we setα= 0.05andω0tJ=−2×10−4. Starting from a uniform ferromagnet as initial condi- tion, the boundary spin starts to rotate and due to ex- change the next spin will follow this rotation and ac- cordingly drive its neighbor and so on. But since a spin cannot immediately follow the dynamics of its neighbor, there is a certain phase difference Dϕbetween the spins, i.e., the neighbor to the right is lagging behind. In the micromagnetic approximation this effect is described by a phase gradient ∇ϕ≈Dϕ/a. The rotation of the spins speeds up, until it reaches the final precession frequency, given by the driving frequency ω0. At the same time theout-of-planecomponent Sz—thespinaccumulation— increases until it reaches a steady state profile. The time scale of this transient phase for reaching a steady state can be quantified: ˙ϕ(t)andSz(t)follow a limited expo- nential growth on a characteristic time τt≈5×105tJ for the parameters used here. τtscales positively with system size Nxand damping α. Eventually, the numerical time evolution reaches a steady state as shown in fig. 2 a). This steady state verifies the analytical solution eqs. (9) and (10) in bulk with a deviation only at the left boundary as anticipated and described above. Note that the finite spin accumu- lationSzas a consequence of this type of dynamics has importantfeatures: itisalong-rangespintransportsince it decays non-exponential and it would allow to measure spin transport by means of the inverse spin-Hall effect. Furthermore, it could also be addressed, for instance, by magneto-optical measurements—if sensitive to the out- of-plane magnetization for a geometry as studied here. For a further investigation, we vary the frequency ω0 and find two different regimes, one for sufficiently smallω0, where the system is able to follow the excitation without disturbance, and one for large ω0where the sys- tems response deviates from the theoretical expectation. Thesetworegimes, whichwewillcalllinearandnonlinear regime in the following, are sharply separated by a crit- ical frequency ωcrit. The existence of these two regimes can be seen from the data depicted in fig. 2 b). Here, as a measure, we consider the spin accumulation of the last spinSN zat the right end of the system. Below ωcritwe find just the analytical valuesSz(L), see eq. (10), which scales linearly with ω0. Atωcritthis behavior breaks down and the spin accumulation SN zdecreases with in- creasing pumping frequency. This breakdown can be un- derstood in terms of the phase gradient ∇ϕwhich scales linearly with the driving frequency ω0, see eq. (9). How- ever, one can expect a maximum phase gradient ∇ϕfor a spin-superfluid state given by the Landau criterion [36]: if the phase gradient exceeds locally a critical value, it is energetically favorable for the spins at this position to rotate out of the x-yplane and return to the plane by unwinding the spiral. Hence, the winding number Nw decreases by one—an effect which is called a phase slip. The Landau criterion for the stability of a spin superfluid with respect to phase slips reads [36] |∇ϕ|</radicalbigg −2dz Ja2. (11) Note that this relation is not exact as a uniform Szis assumed for its derivation. Nevertheless do we observe these phase slips numerically. In the linear regime the winding number is constant in the steady state, whereas in the nonlinear regime it relaxes by one at a regular rate Γpsas shown in fig. 3 at the example of ω0tJ= −6.5×10−4. Theω0dependence of the phase-slip rate Γpsis de- picted in fig. 4. Each phase slip is accompanied by the excitation of a broad spin-wave spectrum on top of the5 4 4.005 4.01 4.015 4.02 10748.54949.550 00.511.510-7 Figure 3. Winding number Nwand dissipated power Pdiss in the steady state of a driven ferromagnet for ω0tJ= −6.5×10−4, well in the nonlinear regime. At a rate of Γps phase slips relax the winding number of the in-plane spiral. For each such event the dissipated power spikes. spin superfluid. These spin waves are visible as oscil- lations ofSzaround the spin-superfluid magnitude and, hence, there is strictly speaking no steady state any more as the phase slips and the spin-wave excitation are def- initely time dependent. In particular, for systems with low Gilbert damping this dynamics leads to determinis- ticchaos, thoughthe spin-superfluidbackground remains visible. These findings have some severe implications as there is a maximum spin accumulation, which is achieved right at the edge between the linear and the nonlinear regime. Furthermore, driving the system in the nonlin- ear regime means also to waste energy to the phase slips and the excitation of incoherent spin waves. For the parameters here the critical frequency takes the valueωcrittJ≈−5.15×10−4, which is determined from fig. 2 b). We also tested different parameters, vary- ingαandL(data not shown), and find that ωcritscales negatively with αandL. Our numerical result can be compared to the analytical prediction above, eq. (11). From eq. (9) follows that the maximum phase gradient is given by∇sϕ(0) =αµSω0L/γJa2, which, inserted into eq. (11), implies |ωcrit|=γJa αµSL/radicalbigg −2dz J. (12) For our parameter set this takes value 4×10−41/tJ, which is slightly lower compared to the numerical value above. This discrepancy is probably due to the fact that eq. (11) ignores the spatial dependence of Sz. Further- more, a test for very low damping α= 10−4showed that eq. (11) is even more inaccurate in that case. Another important quantity is the dissipated power given by eq. (3), which takes negative sign as it lowers the total energy. Figure 4 depicts its dependence on ω0. Belowthecriticalfrequency, inthesteadystate, itistime independent as the dynamics is completely stationary. In this regime it scales quadratically with the excitation frequency,Pdiss∝ω2 0, a result which has already been 10-410-310-210-1010-910-810-7 10-610-410-2Figure 4. Time-averaged dissipated power /angbracketleftPdiss/angbracketrightand phase- slip rate Γpsin the steady state versus driving frequency ω0comparing ferro- (FM) and antiferromagnets (AFM). The perpendicular dash-dotted lines mark ωcritfor the FM and the AFM, where the latter takes on the higher value. For ω0< ω critthe dissipated power scales as /angbracketleftPdiss/angbracketright∝ω2 0and is identical for FMs and AFMs. Above ωcritthe increase slows down and the curve flattens for very high ω0. In this regime, /angbracketleftPdiss/angbracketrightis higher for AFMs as compared to FMs. For the phase-slip rate we find Γps= 0for|ω0|<|ωcrit|and similar, increasing values for the FM and the AFM above ωcrit. The deviation between FM and AFM near ωcritis due to the dif- ferent critical frequency, i.e. the data almost coincide when plotted versus ω0−ωcrit. reported before [37]. This behavior changes above ωcrit. Thetimeevolutionofthedissipationinthisregimeshows that the phase slips notably contribute to the dissipated power, i.e. for each phase slip Pdissspikes as shown in fig. 3. Because of this time-dependence of Pdiss, we have to consider an average over time for the evaluation of the dissipated power. Still, the dissipated power increases furtherwith ω0butlessthanlinearandthecurvenotably flattens. So far, our results were obtained from zero- temperature simulations. In the following we address the robustness of spin-superfluid transport at finite tem- perature. For this, we consider a finite cross section Ny×Nz>1andNx= 2000 and vary the tempera- ture. An average over Navrealizations of thermal noise is carried out and, furthermore, data are averaged over thecrosssectioninordertoreducethenoise. Thespecific choice of parameters in provided is table I. Figure 5 presents the numerical results for the exam- ple ofkBT/J = 10−2forSzandϕ. The spin-superfluid transport remains in tact but, in particular, the spin ac- cumulation Szshows strong thermal fluctuations despite the averages taken over the cross section and the Navre- alizations. However, on average the spin accumulation clearly deviates from its equilibrium value, which is zero. To quantify the influence of the temperature we calculate the spatial average over the xdirection/angbracketleftSz/angbracketrightxand com- parethistothezero-temperaturevalue, givenbyeq.(10). The results are included in Table I. Furthermore, the in- plane angle/angbracketleftϕ/angbracketrightNavshows only little fluctuations and its6 00.020.04 0 500 1000 1500 200001020 Figure 5. Spin superfluidity in a ferromagnet at finite tem- peraturekBT/J = 10−2and forω0tJ=−2×10−4: shown is the spin accumulation Szand the in-plane angle ϕ. Blue lines represent the numerical data, black dash-dotted lines the an- alytical solution at zero-temperature. The spin accumulation is subjected to strong thermal fluctuations but still has a fi- niteaveragevalue /angbracketleftSz/angbracketrightx=/summationtext nSn z/Nx, depictedasreddashed line. Its value is only slightly lower than the zero-temperature value. Thermal fluctuations are much less pronounced for the in-plane angle. Table I. Averaged spin accumulation of a ferromagnet driven withω0tJ=−2×10−4for different temperatures. The cor- responding zero-temperature value is /angbracketleftSz/angbracketright= 0.01, from which no significant deviation is observed. kBT/JNx×NyNav/angbracketleftSz/angbracketrightx 10−44×4 38 0.010 10−24×4 15 0.009 0.05 8×8 5 0.010 0.10 8×8 4 0.011 0.20 14×145 0.012 spatial profile shows hardly any deviation from the zero- temperature behavior, given by eq. (9). Overall, we find no significant difference to the zero temperature case. We also checked whether phase slips due to thermal ac- tivation can be observed, but from the available data we could not observe a single one with the conclusion that ΓpstJ<4×10−5. Hence, spin superfluidity is very robust against thermal fluctuations, even though these fluctuations are a problem in our simulations in terms of the signal-to-noise ratio. IV. EASY-PLANE ANTIFERROMAGNETS For antiferromagnets, the magnetic unit cells comprise twoatoms—denotedAandBinthefollowing—thatform two sublattices. We write all properties using this label- ing so thatASlandBSlare spins of the corresponding sublattices. In the ground state both sublattices have opposite orientation,ASl=−BSl. The field equations, eqs. (5) and (6), do not hold as these require a small in-plane angle difference between two neighboring spins Dϕ, which is obviously not true in this case. However,it is reasonable to define phase differences and gradients within each sublattice, i.e.ADϕas phase difference be- tween a spin of sublattice A and its next-nearest neigh- bor, which is the nearest neighbor within sublattice A. Accordingly,BDϕdefines the phase difference of sublat- ticeB. Assuming sufficiently weak excitation, spatial variationswithineachsublatticearesmallsuchthatami- cromagnetic approximation inside the sublattices reads ∇A,Bϕ≈A,BDϕ/2a. Interestingly, numerical results re- veal that the antiferromagnetic system in bulk fulfills field equation (8), applied separately to each sublattice. The other eq. (7) is not valid, as has been reported before [20] for a phenomenological model for antiferromagnets. Consequently, the antiferromagnet is expected to exhibit the same in-plane angleA,Bϕ(up to phase difference of π betweensublattices)asaferromagnetwithcorresponding parameters, but not the same spin accumulationA,BSz. Before we discuss the numerical results in detail, let us first introduce two differences to the ferromagnet that are essential for understanding the following results: the roleofexchangeand(interlinkedwiththis)thetransverse geometry. Just as in a ferromagnet, a spin-superfluid dy- namics imposes a finite spin accumulationA,BSzwhich, remarkably, carries the same sign for both sublattices leading to a small out-of-plane magnetization. But this is of course antagonized by the antiferromagnetic exchange that favors antiparallel orientation of all components be- tween sublattices. Consequently, the exchange interac- tions must lower the spin accumulation Sztremendously as compared to the ferromagnet (compare fig. 6 a) and fig. 2 a)). This also implies that the behavior of Szis determined by the number of nearest neighbors Nnbof a spin as more neighbors imply stronger exchange cou- pling. Consequently, a 1D spin chain is less prone to this exchange reduction than a 3D system. We checked this numerically by comparing 1D, 2D and 3D models and, indeed, the spin accumulation of the spin superfluid Sz scales linearly with Nnb. There is another implication: at a boundary the num- berofneighborsislocallyreduced—andthereforetheim- portance of the exchange—, resulting in deviations of the sublattice componentsA,BSz, see fig. 6 a) for a 1D setup (the effect is less pronounced in 3D). This 1D setup owns only boundaries along the xdirection and the question whether for finite cross section Ny×Nz>1these devi- ations aty- andzboundaries significantly influence the bulk behavior has also been tested numerically. Fortu- nately, deviations at transversal boundaries quickly fall off with distance to the boundary over a few lattice con- stants. The bulk then behaves qualitatively and quan- titatively just as a 1D system, except for the reduced spin accumulation due to the number of neighbors as discussed above. The study of 1D systems is preferable to keep computational costs feasible. We turn now to the presentation of the numerical data for a 1D system. The model parameters are the same as given above for the ferromagnet, except for the exchange constant which is now negative. Similarly to the ferro-7 050100024610-5 10002000300040004900 5000 0 0.5 1 1.5 2 2.5 10-300.511.510-4 5.566.5 10-41.31.41.510-4 Figure 6. Spin superfluidity in antiferromagnetic spin chains: a)the spin accumulation in the stead state resolved for the two sublattices A and B. In the bulk both take the same value, leading to a finite total spin accumulation, which is two orders of magnitude lower as compared to the ferromagnet. At the boundaries the profiles show deviations from bulk behavior because of the broken exchange right at the boundary. b) the spin accumulation at the right end of the system as function of driving frequencyω0; as for the ferromagnet there are two regimes separated by a critical frequency ωcrit. magnet, the system reaches a steady state after a tran- sientphasecharacterizedbyalimitedexponentialgrowth on a time scale τt, which is roughly the same as for the ferromagnet. In the steady state the sublattice-resolved in-plane anglesA,Bϕboth follow exactly the same profile as the ferromagnet, i.e. eq. (9), but with a phase differ- ence ofπbetween the two sublattices because of the an- tiferromagnetic order (data for the antiferromagnet not shown). The spin accumulation deviates from the behavior of a ferromagnet as depicted in fig. 6 a). The bulk profiles (away from boundaries at x= 0andx=Nxa) are iden- tical in the two sublattices,ASz=BSz. Hence, a measur- able spin accumulation is present, but it is two orders of magnitude lower than in comparable ferromagnetic cases. This is the aforementioned exchange reduction. If we consider the spin accumulation Szin bulk, in the data in fig. 6 a) hardly a space dependence is observed in contrast to the ferromagnet, where Sl zhas a finite slope. The antiferromagnet exhibits this in the same way, but it is also much smaller and the profile becomes roughly constant. Contrary to the ferromagnet, there are distur- bances at the boundaries in the profile of Szwhich we already discussed before. Driving the antiferromagnet with the time-dependent boundary condition eq. (4) at frequency ω0leads to the very same two different regimes as for ferromagnets, a linear regime up to a critical frequency ωcritand above— in the nonlinear regime—phase slips occur. These phase slips reduce the winding number, lead to the excitation of spin waves, and a further increase of the spin accu- mulation is not possible. We quantify this behavior in a similar way as for the ferromagnet. It is, however, not possible to use the spin accumulation of the last spin SN zas a measure because of the deviating profile at the boundary. Instead, we take the spin accumulation at the end of the bulk in form of a spatial average over the spins in the range xl/a∈[4900,4920],Send z:=/angbracketleftbig Sl z/angbracketrightbig [4900,4920].This range is chosen such that it is sufficiently separated from the boundary. The data for the ω0dependence of thespinaccumulationareshowninfig.6, panelb): These show that critical frequencies takes roughly same values for ferro- and antiferromagnets, a result which has been tested and confirmed for another parameter set with dif- ferentNx,α, anddz. For the data set shown here the value isωcrittJ≈−5.75×10−4. However, the decrease of the spin accumulation Send zwith increasing driving fre- quencyω0in the nonlinear regime is less pronounced for antiferromagnets. We also calculated the ω0dependence of the time-averaged dissipated power /angbracketleftPdiss/angbracketrightand of the phase-slip rate Γps, both shown in fig. 4. Similar to other features these properties behave for the antiferromagnets very much as for ferromagnets: below ωcritthe dissipated power shows exactly the same dependence and above it is dominated by phase slips. However, a difference is that aboveωcritthe dissipated power increases faster with ω0. One reason for this might be the dynamics of spin waves that very much differ between ferro- and antiferromag- nets. The phase-slip rate differs slightly, however, this seems to be solely due to the fact that ωcritdiffers for ferro- and antiferromagnets. When Γpsis plotted versus ω0−ωcrit, both curves match almost. The next step is to consider finite temperature. Again this requires a finite cross section for which we use Nx×Ny×Nz= 2000×4×4and we test two temper- atures,kBT/J = 10−2andkBT/J = 10−4. As before, the magnetic response is very similar to that of a ferro- magnet: the in-plane angles follow the zero-temperature profiles, as well as does the average spin accumulation for the lower of the two temperatures. The only major difference is the ratio of the spin-superfluid spin accumu- lation to the thermal fluctuations, which is two orders of magnitude smaller as a result of the lower spin-superfluid signal and an equal strength of the fluctuations. For the higher temperature, this even leads to an average Szthat is essentially zero. This does not mean that there is no8 spin-superfluid spin accumulation, but rather that the available numerical data are not sufficient to resolve it and more averaging is needed. Note that the in-plane angle is not affected by this—it is as robust against the fluctuations just as for the ferromagnet. V. DISCUSSION AND CONCLUSION Our comparative study addresses spin superfluidity in ferro- and antiferromagnets, where one should bear in mind that the former are less promising for spin super- fluidity as the latter because of the negative influence of the stray field [22]. Nevertheless, the former can help to understand the behavior of the latter, which we utilize in this work. One of the striking features of spin super- fluidity is the transport range that leads to a spin ac- cumulation at the end of the system Sz(L)(see eqs. (9) and (10)) that does depend on the system length L— a completely different situation compared to spin-wave transport where the intensity decays exponentially with the distance. However, this non-exponential decay does not imply the possibility of an infinite transport range since with increasing system size the critical frequency lowers until no undisturbed spin superfluid is possible anymore. We present a full analytical solution for the steady state of the ferromagnet, which slightly deviates from the analytical theory reported before [19, 21]. This theory is tested numerically by the full atomistic model, which allows to test the robustness of the spin-superfluid trans- port against varying boundary conditions, against high excitation frequencies and finite temperature. We show that this kind of transport is remarkably robust: bound- ary conditions and also elevated temperature hardly hamper the magnets spin-superfluid response. Furthermore, we identify the critical frequency ωcrit— a manifestation of the Landau criterion—as the limiting factor for the range of this transport. Above this critical frequency phase slips occur, which also sets a limit to the spin accumulation that can be achieved. In ref. [38] another limitation on the spin current of such a spin su- perfluid is discussed, which rests on the fact that |Sz| is bounded above. But the estimated values would re- quire an out-of-plane component that takes quite large values|Sz|>0.1, which our simulations reveal to be hardly possible even for low damping. This is in particu-lar true for the case for antiferromagnets and, therefore, we conclude that the critical frequency—and therefore the phase slips—is a more relevant limitation on spin su- perfluid transport. The direct comparison of antiferromagnets to ferro- magnets shows that both exhibit the very same behavior: Driven by an in-plane rotation, both form an in-plane spin spiral that exhibits exactly the same behavior, in- cluding a spin accumulation in form of an out-of-plane magnetization. Antiferromagnets show in principle the same transport range as ferromagnets with a spin accu- mulation at the end of the system independent of the system length, provided the excitation frequency ω0is kept constant ( ω0itself depends on the magnets geome- try in experimental setups, see eq. (B12)). Furthermore, the critical frequency takes very similar value for the two types of magnets. This general accordance of spin super- fluidity for both types of magnets is in contrast to spin- wavetransportthatisknowntobedifferentforferro-and antiferromagnets[39]. Yetthereisamajordeviation: the antiferromagnetic exchange lowers tremendously the spin accumulation. Ourstudyalsocoversanexaminationofthedissipation ofaspinsuperfluidandoftheeffectoffinitetemperature. We proof the principle robustness of spin superfluidity against thermal fluctuations, i.e. that quite high temper- atures are required before thermal phase slips start to hamper the transport. But the fluctuations are a prob- lem from the numerical side as these require integration overalargeamountofdatainordertoidentifyanon-zero mean spin accumulation. The signal-to-noise ratio might be a problem in experimental setups as well and it could be more promising to measure rather the in-plane an- gleϕ, which is more robust against thermal fluctuations and which always delivers a clear signal in the cases we investigated here. A measurement of ϕcan be done in two ways: either by its time evolution, i.e. the preces- sion frequency ω0, or spatially resolved by measuring the formation of the in-plane spin spiral. 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Melkov, A. Pomyalov, V. S. L’vov, and B. Hillebrands, Nature Physics 12, 1057 (2016). [26] E. B. Sonin, “Comment on "supercurrent in a room tem- perature Bose-Einstein magnon condensate",” (2016), arXiv:1607.04720. [27] D. A. Bozhko, A. A. Serga, P. Clausen, V. I. Vasyuchka, G. A. Melkov, V. S. L’vov, and B. Hillebrands, “On supercurrents in Bose-Einstein magnon condensates in YIG ferrimagnet,” (2016), arXiv:1608.01813. [28] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and M. Kläui, Nature 561, 222 (2018). [29] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and B. J. van Wees, Nature Physics 11, 1022 (2015). [30] U. Nowak, “Classical spin models,” in Handbook of Mag- netism and Advanced Magnetic Materials , Vol. 2, edited by H. Kronmüller and S. Parkin (John Wiley & Sons, Ltd, 2007) available under http://nbn-resolving.de/ urn:nbn:de:bsz:352-opus-92390 . [31] L. Landau and E. Lifshitz, Phys. Z. 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Appendix A: Analytical theory for a 1D ferromagnet The ferromagnet in the micromagnetic approximation under the assumption of small out-of-plane component, |Sz|/lessmuch1, is described by the LLG equation in cylindrical coordinates, eqs. (7) and (8). Assuming translational invariancealong y-andzdirectionleadstoa1Dproblem: µS γ˙ϕ=Ja2∂2 xSz+Ja2Sz(∂xϕ)2+ 2dzSz−αµS γ˙Sz (A1) µS γ˙Sz=−Ja2∂2 xϕ+αµS γ˙ϕ. (A2) Steady state means ˙ϕ=ω0and ˙Sz= 0. This allows to integrate the latter equation, sϕ(x,t) =α 2µSω0 γJ/parenleftbiggx−L a/parenrightbigg2 +ω0t+ϕ0,(A3) where the first integration constant follows from the Neu- mann boundary condition at the right end, ∂xϕ(L) = 0 (no outflow of spin current), and the second one satisfies the condition ˙ϕ=ω0and allows for an arbitrary phase ϕ0. This is inserted in the first equation, which then reads −Ja2∂2 xSz=−µSω0 γ+µ2 Sω2 0 γ2J/parenleftbigg αx−L a/parenrightbigg2 Sz+ 2dzSz. (A4) We argue that the second-derivative term can be ne- glected−Ja2∂2 xSz≈0. This is justified in a twofold manner: first we compared the relevance of all terms in that equation numerically by calculating those three terms from simulations of the full atomistic spin model, eq. (2). Indeed the result is that in steady state the second-derivative term is several orders of magnitude smaller compared to the other two. The second reason follows a-posteriori from the calculated solution and is10 spin injector (using SHE) spins not subjected to SHEspins driven by SHEspins not subjected to SHE Figure 7. 1D setup for calculation of the excitation frequency ω0of a magnet driven by a spin injector utilizing the spin-Hall effecttoexertexternaltorquesonthespins. Thesetorquesare applied in the region [l1,l2]and vanish outside. Furthermore, the Gilbert damping in [l1,l2]is enhanced by αd. The ground stateSis in-plane, the spin accumulation µperpendicular. explainedbelow. From −Ja2∂2 xSz≈0followsthesteady- state solution for Sz: sSz=µSω0 2γdz 1 +µ2 Sω2 0 2γ2Jdz/parenleftbig αx−L a/parenrightbig2. (A5) This solution does not fulfill eq. (A4), however, we can insert it and calculate the deviation by calculating ∂2 xsSz=−2µSω0 γJα2 a2sS2 z+ 4/parenleftbiggµSω0 γJ/parenrightbigg2α4(x−L)2 a4sS3 z =O/parenleftbig S2 z/parenrightbig . This allows the conclusion that the correction by taking the second derivative into account is of higher order in Szand neglecting this is consistent with the original as- sumption|Sz|/lessmuch1. Hence, eqs. (A3) and (A5) form the analytical solution for a 1D setup. Appendix B: Frequency of a spin superfluid The usual excitation of a spin current in a magnet rests on a spin accumulation µat an interface between the magnet and a heavy metal, which is created by an electrical current. Normally for that the spin-Hall effect is utilized. The aim of this appendix is to calculate the resulting excitation frequency ω0of a spin superfluid. We assume here that the spin accumulation is per- pendicular to the magnets ground state, i.e. µ∝ez. Consequently, there is an additional damping-like torque [22, 40] in the LLG equation (here written as viscousdamping): ˙Sl=−γ µSSl×Hl+αlSl×˙Sl+α/prime lSl×/parenleftbigg Sl×µl ~/parenrightbigg . (B1) A subsetVdof the total volume of the magnet is driven, i.e. subjected to the additional torques and the driving also creates an enhanced damping α/prime lwithinVd: µl=/braceleftbigg µdezforrl∈Vd 0else(B2) αl=α0+α/prime lwithα/prime l=/braceleftbigg αdforrl∈Vd 0else.(B3) α0is the intrinsic Gilbert damping of the magnet. To proceed we consider the LLG equation in the fol- lowing form, resolved for the time derivative: ˙Sl=−γ µS(1 +α2 l)Sl×/parenleftbig Hl+αlSl×Hl/parenrightbig +Tl 1Sl×Al+Tl 2Sl×/parenleftbig Sl×Al/parenrightbig .(B4) Tl 1andTl 2parameterize arbitrary additional torques with respecttoanaxis Alandforthespecificchoice Al=µl/~, Tl 1=αlα/prime l/(1+α2 l)andTl 2=−α/prime l/(1+α2 l)eq. (B4) is equiva- lent to eq. (B1). However, for the sake of generality we consider for the calculation eq. (B4). Assuming Al∝ez and using cylindrical coordinates and again the micro- magnetic approximation, this form of the LLG reads µS γ˙ϕ=Ja2Sz|∇ϕ|2+ 2dzSz−αµS γ˙Sz −µS γAz(T1+αT2) (B5) µS γ˙Sz=−Ja2∆ϕ+αµS γ˙ϕ+µS γAz(αT1−T2),(B6) an extension of eqs. (7) and (8). In the same spirit as in appendix A we can solve these equations in one di- mension in steady-state (assuming ˙Sz= 0and ˙ϕ=ω0), where the geometry depicted in fig. 7 is assumed. We ap- ply the external spin accumulation in the interval [l1,l2], whereas the total magnet expands over [0,L]. Therefore, T1,2(x) =/braceleftbigg Td 1,2forx∈[l1,l2] 0else A(x) =/braceleftbigg Ad zezforx∈[l1,l2] 0else.11 In the 1D setup eq. (B6) reads ∂2 xϕ=α(x)µS γJa2ω0+µS γJa2Az(x) [α(x)T1(x)−T2(x)] = =:¯ω0/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright α0µS γJa2ω0 forx∈[0,l1] (α0+αd)µS γJa2ω0 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =:¯ω/prime 0+µS γJa2Ad z/bracketleftbig (α0+αd)Td 1−Td 2/bracketrightbig /bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright =:tforx∈[l1,l2] α0µS γJa2ω0 forx∈[l2,L], (B7) which can be integrated. There are six boundary conditions to consider, each one at the left and right end of the magnet, where we assume a Neumann condition ∂xϕ(0) =∂xϕ(L) = 0, i.e. no outflow of spin currents. Furthermore, ϕ and∂xϕmust be continuous at l1andl2, delivering four internal boundary conditions. But there is another condition, a gauge condition for ϕ, which allows to add an arbitrary constant phase to ϕ(x)without altering the physics. (In practice this gauge phase depends on the prehistory of the magnet, i.e. on how it had reached its steady state, and also which exact instant in time is considered.) As gauge we use ϕ(0) = 0. Altogether there are 6 integration constants and the unknown frequency ω0in combination with 6 boundary conditions and a gauge, such that the problem has a unique solution. As result we obtain ϕ= 1 2¯ω0x2forx∈[0,l1] 1 2(¯ω/prime 0+t)x2+ (¯ω0−¯ω/prime 0−t)l1x+1 2(¯ω/prime 0−¯ω0+t)l2 1forx∈[l1,l2] 1 2¯ω0x2+ (¯ω/prime 0−¯ω0+t)/bracketleftbig (l2−l1)x+1 2(l2 1−l2 2)/bracketrightbig forx∈[l2,L](B8) Sz=µS γω0+Az(x) [T1(x) +α(x)T2(x)] Ja2(∂xϕ)2+ 2dz(B9) and, importantly, we also gain ω0=−Ad z/bracketleftbig (α0+αd)Td 1−Td 2/bracketrightbig (l2−l1) α0L+αd(l2−l1). (B10) This holds true for arbitrary torques taking form eq. (B4). If the specific case of the spin injector utiliz- ing the spin-Hall effect is considered, then inserting the parameters T1,T2andAreads ω0=−µd ~αd /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =:τ·l2−l1 α0L+αd(l2−l1).(B11) The former factor τis the strength of the spin-Hall effect on the magnet [40]: τ=γ Ms~ 2eηϑSHjel1 d, with spin transparency of the interface η, spin-Hall an- gleϑSH, saturation magnetization Msand thickness d of the magnet. jelis the electric current density. The latter factor in eq. (B11) is a geometric factor that is ba- sically the ratio between the driven volume l2−l1and the total volume L, weighted with the total damping of the magnet, where the Gilbert-damping enhancement can beexpressed as [22] αd=g⊥~2 2e2γ Msd, with transverse spin mixing conductance g⊥of the in- terface. This rigorous derivation holds only true for 1D ferromagnets, however, the natural extension to 2D and 3D is given by ω0=−τ·Vd α0V+αdVd, (B12) whereVis the magnets total and Vdthe driven volume. The validity of this expression has been checked numeri- cally for 1D and 2D systems using various geometries by investigating the full atomistic LLG eq. (B4). As a result we obtain very good agreement with the analytical calcu- lation except for two cases. First, when the assumption |Sz|/lessmuch 1is violated and second if the setup is not ef- fectively one dimensional, i.e. if the system is not driven over the entire transverse width. However, such a mis- match in usually small for realistic experimental setups. We furthermore did not only simulate ferromagnets, but12 also antiferromagnets with same parameters except for the sign of J. These simulations result in exactly the same frequencies ω0as the corresponding ferromagnetsand thus eqs. (B10) to (B12) are also valid for antiferro- magnets, even though note that the resulting spin accu- mulation deviates.

2019-11-28

We present a study on spin-superfluid transport based on an atomistic, classical spin model. Easy-plane ferro- as well as antiferromagnets are considered, which allows for a direct comparison of these two material classes based on the same model assumptions. We find a spin-superfluid transport which is robust against variations of the boundary conditions, thermal fluctuations, and dissipation modeled via Gilbert damping. Though the spin accumulations is smaller for antiferromagnets the range of the spin-superfluid transport turns out to be identical for ferro- and antiferromagnets. Finally, we calculate and explore the role of the driving frequency and especially the critical frequency, where phase slips occur and the spin accumulation breaks down.

Transport properties of spin superfluids: comparing easy-plane ferro- and antiferromagnets

1911.12786v1

Determination of the damping coecient of electrons in optically transparent glasses at the true resonance frequency in the ultraviolet from an analysis of the Lorentz-Maxwell model of dispersion Surajit Chakrabarti (Ramakrishna Mission Vidyamandira) Howrah, India The Lorentz-Maxwell model of dispersion of light has been analyzed in this paper to determine the true resonance frequency in the ultraviolet for the electrons in optically transparent glasses and the damping coecient at this frequency. For this we needed the refractive indices of glass in the optical frequency range. We argue that the true resonance condition in the absorption region prevails when the frequency at which the absorption coecient is maximum is the same as the frequency at which the average energy per cycle of the electrons is also a max- imum. We have simultaneously solved the two equations obtained from the two maxima conditions numerically to arrive at a unique solution for the true resonance frequency and the damping coecient at this frequency. Assuming the damping coecient to be constant over a small frequency range in the absorption region, we have determined the frequencies at which the extinction coecient and the re- ectance are maxima. These frequencies match very well with the published data for silica glasses available from the literature. 1arXiv:1907.04499v1 [physics.optics] 10 Jul 20191 Introduction The Lorentz-Maxwell model of dispersion of electromagnetic waves in matter is very successful in describing the properties of matter under the action of electro- magnetic waves over its whole spectrum where the wavelength is large compared to the interatomic distances. The model is generally studied in the optical frequency range where only the oscillation of electrons bound to atoms and molecules is rel- evant for the study of dispersion. Two important parameters of the model namely the natural oscillation frequency and the plasma frequency of the electrons in a dielectric medium like glass can be easily determined from the refractive indices of a glass prism measured in the optical band [1] where glass is transparent. In a condensed system like glass one has to include the eect of the local eld on the electrons apart from the eld of the incident wave. This leads to another fre- quency which is conventionally known as the resonance frequency and is related to the plasma and the natural oscillation frequencies of the electron [2]. Though it is called the resonance frequency, there is no proof that the absorption coecient is maximum at this frequency. In order to study the absorption of EM waves in matter, a phenomenologi- cal variable called the damping coecient is introduced in the Lorentz-Maxwell model. Glass is opaque in the ultraviolet indicating that it has a strong absorption there. In scientic literature, there are innumerable experimental works which have studied the interaction of silica glasses with electromagnetic waves over its whole spectrum. A summary of these works can be found in Kitamura et al. [3]. From the experimental data on the extinction coecient for silica glass in the ultraviolet, we can nd the frequency at which this coecient is maximum. However, as far as we are aware, there has been no theoretical study so far which has determined this frequency by an analysis of the Lorentz-Maxwell model of dispersion. The main problem with the theoretical analysis is the fact that it has not been possible so far to determine the value of the damping coecient theoretically. 2In this work we have determined the damping coecient at the true resonance frequency which we dene to be the frequency at which the absorption coecient for the energy of the electromagnetic eld in the medium is maximum. We have done this theoretically by taking the natural oscillation frequency and the plasma frequency determined from the refractive indices of glass in the optical region as two known parameters of the Lorentz-Maxwell model. We have formed two algebraic equations containing the true resonance frequency and the damping coecient as two unknown variables. We have solved these two equations simultaneously by numerical method to nd a unique solution for the two variables. With the value of the damping coecient known, we have explored the anomalous dispersion region in the ultraviolet for glass. It is well known that the Kramers-Kronig relations [3] allow us to determine the imaginary part of the dielectric constant from an integral of the real part over the whole range of frequencies and vice versa. The theory is based on a very general causality argument and a linear response of the medium to an external perturbation. We have, on the other hand, determined the damping coecient of the Lorentz-Maxwell model of dispersion starting from the refractive indices in the optical region corresponding to the real value of the dielectric constant. From this we have extracted the information about the absorptive region in the ultraviolet corresponding to the imaginary part of the dielectric constant. In section 2 we give the outline of the Lorentz-Maxwell model. In section 3 we oer our physical argument for the method adopted to determine the damping coecient and the true resonance frequency. The next four sections are just an execution of these ideas. We conclude with a summary of the work. 32 Lorentz-Maxwell model of dispersion In the Lorentz model [4] of dispersion of light in a dense medium like a solid or liquid, electrons execute forced simple harmonic oscillations with damping in the combined eld of the incident electromagnetic wave of frequency !and the local eld. The local eld arises as a result of the interaction of the electron with the elds of other atoms close by. Without any loss of generality we can assume that the direction in which the electron is oscillating is the ydirection. We can write the equation of motion as y+ _y+!2 0y=qE0 0 mei!t: (1) whereE0 0is the amplitude of the eective eld acting on the electrons. Here !0is the natural oscillation frequency and is the damping coecient of the electron. In the steady state the electron will oscillate at a frequency !of the incident wave though shifted in phase. E0 0is related to the amplitude of the eld ( Ei0) outside from where it is incident on the medium as E0 0=1 + 3 1 +DEi0: (2) The 3term in equation (2) arises as a result of the eect of local eld in the Lorentz-Lorenz theory of dielectric polarizability valid for an isotropic medium [5] whereis the electric susceptibility. Dis the depolarisation factor, a dimensionless number of the order of unity [6]. The dielectric function of the medium is given by = 1 +: (3) Using Maxwell's phenomenological relation =n2 cwherencis the complex refrac- tive index and the Lorentz-Lorenz equation [5], we arrive at the following equation for a number of resonance regions [7,8]. n2 c1 n2 c+ 2=Nq2 30mX jfj !2 0j!2i j!: (4) 4Herefjis the fraction of electrons that have a natural oscillation frequency !0j and damping constant jwith fj= 1.Nis the density of electrons taking part in dispersion. It is a common practice to assume a single dominant absorption frequency which is true in many practical cases and which makes the analysis simpler [9]. With this assumption fj= 1 and equation (4) can be written as n2 c= 1 +!2 p !2 n!2i !(5) where the plasma frequency !pis given by !2 p=Nq2 0m(6) and we dene !2 n=!2 0!2 p 3: (7) In scientic literature [2], !0is known as the natural oscillation frequency of the electrons and !nis known conventionally as the resonance frequency. So far, au- thors have used some chosen values of the damping coecient and the plasma frequency which mimic the absorptive properties of dielectric materials, in order to carry out model analysis [9]. We have actually determined the damping coe- cient from a prior knowledge of the natural oscillation frequency and the plasma frequency of a glass medium. In the optical limit where the absorption in glass is negligible we take = 0. In this limit the refractive index is real and equation (5) reduces to n2= 1 +!2 p !2 n!2: (8) which is essentially the Sellmeier's formula [7] for dispersion in the frequency do- main with one absorption band. If we have a set of measurements of refractive indices of a glass prism for several optical wavelengths, we can determine !nand !pusing equation (8) [1].The resonance wavelength which falls in the ultraviolet region, has been determined in a similar work [7]. Once !nand!pare known, !0 can be determined using equation (7). 5In the absorptive region the dielectric function picks up an imaginary part given by n2 c==1+i2: (9) The refractive index ( n) and the extinction coecient ( ) known as optical con- stants are written as nc=n+i (10) whererepresents the attenuation factor of the amplitude of the electromagnetic wave in an absorptive medium. Using the last two equations and equation (5) we obtain for the real and imaginary parts of the complex dielectric function [10], 1=n22= 1 +!2 p(!2 n!2) (!2 n!2)2+ 2!2(11) and 2= 2n=!2 p ! (!2 n!2)2+ 2!2: (12) We can express n2and2as functions of frequency !using equations (11) and (12). The details and the nal expressions have been shown in Appendix A. The absorption coecient of the incident EM wave is given by [7,10], =2! c (13) wherecis the speed of light in vacuum. gives the attenuation coecient of the intensity of the incident wave. Intensity is the rate of ow of energy per unit area normal to a surface. will be a maximum at the frequency at which the absorption of energy by the electrons from the EM eld is maximum. This gives the condition of resonance. It is a general practice to consider !ndened in equation (7) as the resonance frequency though there is no proof that the energy absorption is maximum at this frequency. So we do not assume a priori !nto be the resonance frequency. In the next section we will describe our strategy to nd the true resonance frequency and in the results section we will see that the true resonance frequency is dierent from both !0and!nand lies between them. There 6is no real reason to call !nthe resonance frequency. We treat the true resonance frequency as an unknown variable to be found from our analysis. The damping coecient is introduced in the Lorentz model to explain absorp- tion. We model such that it is zero in the optical band and upto the frequency !n. In the absorption band we assume that is constant from frequency !nto !0. Above!0, falls down and rises again to another constant value of in the next resonance region if the material under study has one. With this model for in mind we can extrapolate equation (8) to nd !nand!p. In the next section we will explain how to get the constant value of in the absorption region and the true resonance frequency. Even if the system under study may have several absorption bands, we can study it with the assumption of a single resonance region. The optical waves os- cillate the outermost electrons of atoms and molecules having the lowest natural frequencies and as a result we get the phenomenon of refraction. With an anal- ysis of the refractive indices in the optical region under this assumption of single resonance, we are most likely to nd information about the absorption band with the lowest natural oscillation frequency in the ultraviolet closest to the optical band. This will of course depend on the strength of the resonance. The justica- tion of the single resonance calculations with the chosen model for can be found from the results of our theoretical calculations which will be found to match the experimental results very well. 73 Physical argument for the method adopted to determine the damping coecient at the true resonance frequency From various experiments on the absorption of EM waves in matter, we know that the absorption coecient ( ) attains a maximum value at a characteristic frequency. We try to nd this frequency where is maximum. We dierentiate with respect to frequency and equating the derivative to zero get one equation. However, we have two unknown variables in the theory - the damping coecient and the true resonance frequency. We look for a second equation. The incident EM wave interacts with the electrons bound to the atoms and molecules.The electrons execute a forced simple harmonic oscillation with damp- ing. The total energy of the electron is time dependent, as the electron is being perturbed by a time dependent harmonic force.The average energy of the electron per cycle can be worked out easily [11]. We nd the frequency at which this av- erage energy per cycle is maximum. This leads to another equation involving the two unknown variables. When the frequency at which is maximum is the same as the frequency at which the average energy per cycle of the electron is also a maximum, the electromagnetic wave will share its energy most with the electrons and will be attenuated most. This will constitute the true condition of resonance. By solving the two equations simultaneously using numerical method, we nd both the variables. We call the characteristic frequency, the true resonance frequency !tand the damping coecient at the true resonance frequency t. Heitler [12] has proposed a quantum theory of the phenomenon of damping. According to this theory the damping coecient is dependent on frequency though of a very slowly varying nature near resonances. This gives support to our earlier assumption that the damping coecient is a constant within a small frequency range about the resonance frequency. However, it can be taken as zero in the 8optical band where glass is transparent and absorption is negligible. 4 Condition for the maximum of the absorption coecient as a function of frequency Our aim in this section is to nd the frequency at which is maximum. We rst dierentiate with respect to !assuming constant. In order to nd the derivative of we rst dierentiate equations (11) and (12) with respect to !. We nd two algebraic equations involvingdn d!andd d!. By eliminatingdn d!from the two equations, we get the expression ford d!and henced d!using equation (13). We have shown the dierentiations in Appendix B. Eliminatingdn d!between equations (B.2) and (B.3) we get 2d d!(n+2 n) =AB C(14) where A=!2 p(!2 n!2)2[ 2 n!] (15) and B=!2 p! [! 24(!2 n!2)!2 n!2 n ] (16) and C= [(!2 n!2)2+ 2!2]2: (17) From this we get d d!= c[2 +n ! n2+2AB C]: (18) Ifis maximum thend d!should be zero. So we write at the maximum !(AB) C=2 n(n2+2): (19) It is to be noted that two sides of equation (19) are dimensionless and they will be compared later numerically to nd the solution for the true resonance frequency and the damping coecient. 95 Condition for the maximum of the average en- ergy per cycle of the electron as a function of frequency In the steady state the electron will oscillate at a frequency !as given by the steady state solution of equation (1) and the total energy of the system averaged over a period is given by [11], E(!) =1 4(qE0 0)2 m(!2+!2 0) [(!2 0!2)2+ (! )2]=1 4(qE0 0)2 mg(!) (20) where g(!) =(!2+!2 0) [(!2 0!2)2+ (! )2]: (21) Equation (2) shows the relationship between the incident electric eld and the eld acting on an electron. With the variation of frequency in the ultraviolet we can imagine that the amplitude of the incident eld is kept constant. However, the amplitudeE0 0is dependent on which is frequency dependent. Lorentz theory is based on the assumption that the response of the medium to the external eld is small [13]. In equation (2), appears both in the numerator as well as in the denominator. With the depolarization factor Dpositive, any variation of in the numerator will be oset to some extent by the variation in in the denominator. So we neglect the variation of the term E0 0with frequency and assume it to be constant. To nd the derivative of the average energy per cycle E(!), it is sucient to nd the derivative of the function g(!) given by equation (21) with respect to !. Equating the derivative to zero, we nd the condition at which the average energy per cycle is maximum. It turns out that the frequency is given by !=!0[r 4( !0)21]1 2: (22) If the incident electromagnetic wave can oscillate the bound electrons steadily at frequency!given by the last equation, then the wave has to deliver maximum energy per cycle and its absorption will be maximum. 10It is clear from equation (22) that for real values of !we should have the ratio f= !0<p 3. By trial we take several positive values of fupto its maximum of p 3. For each value of fwe nd the values of and!using equation (22) with the known value of !0. With these values we determine the refractive index nand the extinction coecient in the resonance region using equations (A.3) and (A.4) respectively. We put these values in equation (19) and try to see for which value offthis equation is satised. From fwhich satises equation (19) we calculate t and!tusing equation (22). 6 Determination of the damping coecient and the true resonance frequency The results of an experiment performed with a prism made of int glass have been reported by Chakrabarti [1]. In this experiment1 n21has been plotted against the inverse wavelength squared at optical range following equation (8). From this plot we have determined the values of !n,!p:The value of !0has been estimated using equation (7). The errors in these frequencies are less than 1% :Refractive indices as a function of wavelengths have been shown in table 1 and the parameters needed for this work have been shown in table 2. It has been shown in the discussion following equation (22) that the maximum value off= !0isp 3. So we take trial values of flike 0.1, 0.2 upto 1.7. For all these values offwe determine and then!using equation (22). We then determine n andusing equations (A.3) and (A.4) respectively and calculate A,B,Caccording to equations (15),(16),(17) respectively. With these values we try to see whether they satisfy equation (19) or not. The algorithm for this is as follows: suppose we call the left side of equation (19), Y1 and the right side, Y2. Now we form a parameter, Y=Y1Y2. For all values of ffrom 0.1 to 1.7 we calculate Y and nd between which two values of f, the sign of Ychanges. In our case there 11was only one sign change in Ybetweenfvalues 0.6 and 0.7. Now we check this region more closely, that is between 0.60 to 0.70 at an interval of 0.01. We nd that forf= 0:65,Y=0:001 and for f= 0:66,Y= 0:053. So there is a zero crossing between fvalues 0.65 and 0.66. Proceeding similarly we nally nd that forf= 0:6501,Y=0:0005 and for f= 0:6502,Y= 0:000055. So the solution lies between 0.6501 and 0.6502. We take the solution as f= 0:65015 with an error of 0:00005 which is just 0:008%. From this value of fwe get tsince we know the value of !0. We get the value of !tby putting the value of fin equation (22). These values are shown in table 3. The values of the damping coecient and the true resonance frequency turn out to be t= 11:61015rad/s with an error of only 0 :76% and !t= 16:81015rad/s with an error similar to the error in !0.!tlies between !nand!0. This solution is unique. We nd that !tdiers signicantly from !n. We have presented the parameters at true resonance frequency in table 4. The maximum value of the absorption coecientat the true resonance frequency has come out to be 8 :18107m1 corresponding to an attenuation length 12 :2 nm. Once we determine the damping coecient, we can determine the absorption coecient in a small frequency range about !twhere we assume is constant and equal to t. In gure 1 we plot as a function of x=! !0betweenx= !n !0= 0:81 to 1. We assume that within the frequency range !nto!0, remains constant. We clearly nd that attains a maximum value at x=!t !0= 0:94: Jackson [14] has given the frequency range for absorption coecient in the ultraviolet as well as the plasma frequency of water. In the ultraviolet, we are concerned with electronic oscillations as the most important component responsible 12for dispersion. So the properties of glass will not be too dierent from water at these frequencies. This work [14] shows that the maximum value of is around 108m1which is approximately the same as the maximum value that we have obtained. The frequency at which the maximum occurs is also the same in order of magnitude as !t:The full width at half maximum of the absorption curve can be read o approximately. It is determined to be approximately 15 1015rad/s which is of the same order of magnitude that we have obtained for the value of t. The damping coecient determined in this work is rather large.This broad ab- sorption in the ultraviolet is due to the outer electrons in the atoms and molecules of the solid which take part in the process of dispersion. The outer electrons are aected by the collisions and the electric elds of the neighbouring atoms. Con- sequently an extensive region of continuous absorption is obtained in solids and liquids [8,15]. So this large value of tis expected. This value of tis valid only in the resonance region. 7 Some other results We assume the damping coecient tis constant within the small frequency range from!nto!0in which!tlies. We now nd the frequency !at which the extinction coecientis maximum, assuming !is close to!t. Forto be maximum,d d! should be zero and from equation (14) this should occur when A=B:Here we scan a parameter! !0from 0.1 to 1.0. For each value of !and known twe calculate nandusing equations (A.3) and (A.4) respectively. We then check whether the condition A=Bis satised. Once again we nd a unique solution with ! !0= 0:84843:This corresponds to an angular frequency != 15:11015rad/s. From this we get = 2:401015Hz 13corresponding to a wavelength = 0:125m: In gure 2 we have plotted determined by our theoretical analysis, as a function of ! !0:This gure clearly shows that has attained a maximum value. The maximum value ofthat we have obtained at !=!is 0.769. This is of the same order of magnitude as the maximum value shown in gure 2 of Kitamura et al.[3] which gives the maximum of in the ultraviolet at a wavelength very near to 0 :12m. This is very close to the wavelength that we have obtained. In the last column of table 3 we show the value of !.that we have determined may be seen from gure 1 of [3] to be the nearest to the optical wavelengths on long wavelength side as we claimed in last paragraph of section 2. The data in gure 1 of [3] may be showing other resonance/s at shorter wavelengths or larger frequencies. In gure 3 we show a plot of re ectance Ras a function of frequency in the absorption region where Ris the normal re ectivity dened in [3] as R=(n1)2+2 (n+ 1)2+2: (23) We nd that this function has a peak at! !0= 0:908 with the maximum value of R= 0:118. The peak position corresponds to != 16:21015rad/s:The re ectance peak position corresponds to an energy 10.6 eV. Experimentally, silica glasses show the lowest frequency re ectance peak in the ultraviolet region at 10.2 eV [16] which is very close to what we have found theoretically. This is once again as we claimed in the last paragraph of section 2. Sigel has shown in gure 3 [16] that the re ectance peak position is the same for two glassy materials though their re ectance values at this frequency are dierent. Similarly the actual values of refractive indices and extinction coecients at the same frequency can vary signicantly due to glass manufacturing process [3]. Data for these coecients in the absorption region for int glasses were not available in the literature. However, the order of magnitude of these coecients is similar to that of other silica glasses. Since we are getting 14the peak positions of the extinction coecient and the re ectance data very close to the published data for silica glasses, we conclude that the value of the damping coecient obtained by us is correct. We have shown in table 5 the values of n,,, the absorption length1 and g(!) at frequencies close to !t. They have been calculated at three frequencies !0, !and!nfor the same t. Comparing tables 4 and 5 we nd that andg(!) are indeed maxima at !t. The anomalous nature of variation of the refractive index is evident from the values of nat frequencies around !t. In section 4 we determined the condition for to be maximum. In a similar way we can nd the condition for the refractive index nto be an extremum by equating its derivative to zero. This derivative can be obtained by eliminatingd d! from equations (B.2) and (B.3). Interestingly, this condition gives two solutions for the frequency. The refractive index is maximum at one and minimum at the other frequency. We nd that the Lorentz-Maxwell model of dispersion reproduces all the features of anomalous dispersion in the absorption region as observed in actual experiments [3]. 8 Conclusions The Lorentz-Maxwell model of dispersion of electromagnetic waves in matter has been studied in this paper with an analysis of the phenomenon of absorption in the ultraviolet in dielectrics like int glass. We have shown that if we know the refractive indices of glass fairly accurately in the optical frequencies, we can explore the anomalous dispersion region in the ultraviolet quantitatively. The key nding of this work is that the damping coecient of the model can be determined by a simple argument. We also determine the frequency at which the absorption coecient is maximum. We call this the true resonance frequency. In the optical region where glass is transparent, the damping coecient can be assumed to be 15zero. In the absorptive part the damping coecient has been taken to be a constant within a short range of frequencies. The value of the damping coecient matches in order of magnitude with the experimental width of the absorption coecient data for water available from literature. Once the damping coecient is determined, we can nd the frequencies at which the extinction coecient and the re ectance are maxima. These frequencies match very well with the experimental data available in the literature for silica glasses. This indirectly shows that the value of determined by us is correct. Our assumption of a single resonance should give us the information of the absorption band closest to the optical frequencies. We actually observe this by comparing the peak positions of the extinction coecient and re ectance data obtained by us with that found from literature. Refractive indices estimated at dierent frequencies close to the true resonance frequency in the absorption region reveal the anomalous nature of dispersion. All the features of dispersion by a dielectric like glass in the ultraviolet absorption region have been reproduced from our theoretical analysis of the Lorentz-Maxwell model. 9 Acknowledgment The author would like to thank Prof. Jayanta Kumar Bhattacharjee for some helpful discussions. Thanks are due to Prof. Debashis Mukherjee for making some helpful comments on the paper. 16Appendix A Equations (11) and (12) can be easily inverted and we get [2], n2=1 2[(2 1+2 2)1 2+1] (A.1) and 2=1 2[(2 1+2 2)1 21]: (A.2) We express 1and2as functions of frequency using equations (11) and (12) respectively. After a fairly straightforward algebra we arrive at the nal expressions forn2and2as functions of frequency !. n2=[!4 p 2!2+ (!04+ 2!2+!2 p!02)2]1 2+ (!04+ 2!2+!2 p!02) 2(!04+ 2!2)(A.3) 2=[!4 p 2!2+ (!04+ 2!2+!2 p!02)2]1 2(!04+ 2!2+!2 p!02) 2(!04+ 2!2)(A.4) where !02=!2 n!2: (A.5) These equations are exact and will be used for determining nandin the resonance region in the ultraviolet. It can be easily checked that in the limit tending to zero, becomes zero at all frequencies and vice versa. Thus the medium is transparent in the optical frequencies as it should. In this limit the refractive index nsatises a relation which has been used in the rst place to get the parameters !nand!p[1]. Appendix B We rst nd the derivative of1 (!2n!2)2+ 2!2with respect to !and get d d![1 (!2 n!2)2+ 2!2] =2![2(!2 n!2) 2] [(!2 n!2)2+ 2!2]2(B.1) 17Dierentiating equation (11) with respect to !we get ndn d!d d!=!!2 p[(!2 n!2)2+ 2!2] +!!2 p(!2 n!2)[2(!2 n!2) 2] [(!2 n!2)2+ 2!2]2 =!!2 p[(!2 n!2)2!2 n 2] [(!2 n!2)2+ 2!2]2(B.2) Similarly dierentiating equation (12) with respect to !we get 2dn d!+ 2nd d!=!2 p [(!2 n!2)2+ 2!2] + 2!2!2 p [2(!2 n!2) 2] [(!2 n!2)2+ 2!2]2 =!2 p (!2 n!2)[(!2 n!2) + 4!2]!2!2 p 3 [(!2 n!2)2+ 2!2]2:(B.3) 18REFERENCES 1. Chakrabarti S 2006 Phys. Educ. 23167- 175 (New Delhi, India: South Asian Publishers PVT LTD) 2. Christy R W 1972 Am.J.Phys. 401403-1419 3. Kitamura R, Pilon L and Jonasz M 2007 Applied Optics 46(33) 8118-8133 4. Almog I F, Bradley M S and Bulovic V 2011 The Lorentz Oscillator and its Applications ( MIT OpenCourseWare, MIT6-007S11/lorentz) 5. Born M, Wolf E 1980 Principles of Optics ( New York:Pergamon Press, Sixth.Edn.) p 85 6. Kittel C 1976 Introduction To Solid State Physics (New Delhi:WileyEastern Limited, 5th Edn.) p 405 7. Hecht E 2002 Optics ( Delhi,India:Pearson Education, 4th Edn.) p 71, 85,70,128 8. Feynman R P, Leighton R B, Sands M 2003 The Feynman Lectures on Physics, 2nd Volume (New Delhi,India:Narosa Publishing House) p 1211,1210 9. Oughstun K E, Cartwright N A 2003 Optics Express 11(13) 1541-1546 10. Tanner D B 2013 Optical eects in solids (www.phys.u .edu/tanner/notes.pdf ) 11. Kleppner D, Kolenkow R J 1973 An Introduction To Mechanics (New Delhi :Tata Mcgraw Hill ) p 426 12. Heitler W 1954 The quantum theory of radiation (Oxford University Press, 3rd. Edn.) p 163 1913. Seitz F 1940 The Modern Theory of Solids (McGraw-Hill ,International Series In Pure And Applied Physics) p 629 14. Jackson J D 1999 Classical Electrodynamics 3rd edn. (John Wiley & Sons, Inc) p 314 15. Jenkins F A , White H E 1957 Fundamentals of Optics (Mcgraw-Hill book company,Inc, 3rd Edn.) p 486 16. Sigel G H Jr 1973/74 Journal of Non-Crystalline Solids 13372-398 20Table 1: Refractive indices as a function of wavelengths for the int glass prism [1] wavelength refractive index (nm) n 706.544 1.6087 667.815 1.6108 587.574 1.6167 504.774 1.6259 501.567 1.6264 492.193 1.6277 471.314 1.6311 447.148 1.6358 438.793 1.6377 21Table 2: Parameters obtained from tting of data of refractive indices to Lorentz model[1] !n N ! p !0 rad/s m3rad/s rad/s 14:510151:02102918:0101517:81015 Table 3: Table for t,!tand! f= t !0 t !t ! rad/s rad/s rad/s 0.65015 11 :6101516:8101515:11015 Table 4: Parameters at the true resonance frequency frequency n  1 g(!) =4mE(!) (qE0 0)2 ! m1m s2=rad2 !t 0.995 0.729 8 :181070.0122 0:1531031 Table 5: Values of some parameters in the ultraviolet region frequency n  1 g(!) =4mE(!) (qE0 0)2 ! m1m s2=rad2 !0 0.906 0.678 8 :061070.0124 0:1491031 ! 1.18 0.769 7 :761070.0129 0:1411031 !n 1.26 0.763 7 :391070.0135 0:1331031 22 7.2 7.4 7.6 7.8 8 8.2 8.4 0.8 0.85 0.9 0.95 1 1.05α(107 m-1) ω/ω0Figure 1: Distribution of the absorption coecient in the absorption region 23 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.75 0.8 0.85 0.9 0.95 1κ ω/ω0Figure 2: Distribution of the extinction coecient in the absorption region 24 0.115 0.1155 0.116 0.1165 0.117 0.1175 0.118 0.1185 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02R ω/ω0Figure 3: Distribution of Re ectance Rin the absorption region 25

2019-07-10

The Lorentz-Maxwell model of dispersion of light has been analyzed in this paper to determine the true resonance frequency in the ultraviolet for the electrons in optically transparent glasses and the damping coefficient at this frequency. For this we needed the refractive indices of glass in the optical frequency range. We argue that the true resonance condition in the absorption region prevails when the frequency at which the absorption coefficient is maximum is the same as the frequency at which the average energy per cycle of the electrons is also a maximum. We have simultaneously solved the two equations obtained from the two maxima conditions numerically to arrive at a unique solution for the true resonance frequency and the damping coefficient at this frequency. Assuming the damping coefficient to be constant over a small frequency range in the absorption region, we have determined the frequencies at which the extinction coefficient and the reflectance are maxima. These frequencies match very well with the published data for silica glasses available from the literature.

Determination of the damping co-efficient of electrons in optically transparent glasses at the true resonance frequency in the ultraviolet from an analysis of the Lorentz-Maxwell model of dispersion

1907.04499v1

DRAFT VERSION JANUARY 24, 2022 Typeset using L ATEXpreprint2 style in AASTeX63 Cosmic ray streaming in the turbulent interstellar medium SIYAO XU1AND ALEX LAZARIAN2, 3 1Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA; sxu@ias.edua 2Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA; lazarian@astro.wisc.edu 3Centro de Investigaci ´on en Astronom ´ıa, Universidad Bernardo O’Higgins, Santiago, General Gana 1760, 8370993,Chile ABSTRACT We study the streaming instability of GeV 100GeV cosmic rays (CRs) and its damping in the turbulent interstellar medium (ISM). We find that the damping of streaming instability is dominated by ion-neutral col- lisional damping in weakly ionized molecular clouds, turbulent damping in the highly ionized warm medium, and nonlinear Landau damping in the Galactic halo. Only in the Galactic halo, is the streaming speed of CRs close to the Alfv ´en speed. Alfv ´enic turbulence plays an important role in both suppressing the streaming in- stability and regulating the diffusion of streaming CRs via magnetic field line tangling, with the effective mean free path of streaming CRs in the observer frame determined by the Alfv ´enic scale in super-Alfv ´enic turbu- lence. The resulting diffusion coefficient is sensitive to Alfv ´en Mach number, which has a large range of values in the multi-phase ISM. Super-Alfv ´enic turbulence contributes to additional confinement of streaming CRs, irrespective of the dominant damping mechanism. 1.INTRODUCTION The resonant streaming instability (Wentzel 1974; Kul- srud & Pearce 1969; Wentzel 1969; Skilling 1971) is im- portant for confining cosmic rays (CRs) with energies up to 100GeV in the Galaxy (Farmer & Goldreich 2004). It has many astrophysical implications on, e.g., shock accel- eration (Bell 1978), heating of intracluster media (Guo & Oh 2008; Brunetti & Jones 2014), launching galactic winds (Ipavich 1975; Wiener et al. 2017; Mao & Ostriker 2018; Holguin et al. 2019; Quataert et al. 2021), transport of CRs in starburst galaxies (Krumholz et al. 2020) and around CR sources (Marcowith et al. 2021), and explaining PAMELA and AMS-02 observations at Earth (Blasi et al. 2012; Amato & Casanova 2021). The self-generated Alfv ´en waves by CRs via the streaming instability are subject to various damping effects, including ion-neutral collisional damping in a partially ionized medium (Kulsrud & Pearce 1969; Plotnikov et al. 2021; Armillotta et al. 2021), nonlinear Landau damping in a collisionless medium (Kulsrud 2005), as well as turbulent damping by background Alfv ´enic turbulence (Lazarian 2016). Unlike other damping mechanisms depending on plasma conditions, turbulent damping depends on properties of magnetohydro- dynamic (MHD) turbulence. Measurements in different in- terstellar phases reveal a large range of turbulence parame- ters, e.g., Alfv ´en Mach number MAthat characterizes the magnetization level of turbulence (Lazarian et al. 2018; Hu et al. 2019). Based on the theoretical understanding of MHD turbulence developed since Goldreich & Sridhar (1995) and Lazarian & Vishniac (1999), Farmer & Goldreich (2004) first formulated aHubble Fellowthe turbulent damping rate for trans-Alfv ´enic (MA= 1) tur- bulence. Lazarian (2016) further provided a detailed analysis on turbulent damping in both super-Alfv ´enic (MA>1) and sub-Alfv ´enic (MA<1) turbulence. When the growth of streaming instability is limited by turbulent damping, the re- sulting streaming speed of CRs can deviate from the Alfv ´en speed and is sensitive to turbulence parameters. In addition, due to the magnetic field line tangling in super-Alfv ´enic tur- bulence, CRs streaming along turbulent magnetic fields have an effective mean free path determined by the Alfv ´enic scale lA=LM3 A(Lazarian 2006; Brunetti & Lazarian 2007), whereLis the injection scale of turbulence, and an isotropic distribution on scales larger than lA. The above effect on the spatial diffusion of streaming CRs has not been addressed in previous studies. In this work, we focus on the effect of Alfv ´enic turbulence on the streaming speed and diffusive propagation of stream- ing CRs in the energy range GeV 100GeV in different tur- bulence regimes. We also examine the relative importance between turbulent damping and other damping mechanisms of streaming instability in various interstellar phases. In par- ticular, in a partially ionized medium, as MHD turbulence is also subject to ion-neutral collisional damping (Xu et al. 2015, 2016; Xu & Lazarian 2017a), the relative importance between turbulent damping and ion-neutral collisional damp- ing of CR-driven Alfv ´en waves depends on the ionization fraction and the coupling state between ions and neutrals in different ranges of length scales. The paper is organized as follows. The description on streaming instability and different damping effects is pre- sented in Section 2. In Section 3, we compare turbulent damping and ion-neutral collisional damping in both weakly and highly ionized media, and we derive the correspond- ing streaming speed and diffusion coefficient in different regimes. The comparison between turbulent damping andarXiv:2112.06941v2 [astro-ph.HE] 20 Jan 20222 nonlinear Landau damping in the Galactic halo is carried out in Section 4. Discussion and our summary are in Section 5 and Section 6, respectivley. 2.GROWTH AND DAMPING OF CR-DRIVEN ALFV ´EN WA VES 2.1. Growth of Alfv ´en waves The same resonance condition, rL, applies to both gyroresonant scattering of CRs by Alfv ´en waves and genera- tion of Alfv ´en waves via the CR resonant streaming instabil- ity, whereis the wavelength of Alfv ´en waves, and rLis the Larmor radius of CRs. For CRs streaming from a source to a sink, when their bulk drift velocity, i.e., streaming velocity vD, is larger than the Alfv ´en speedVA, the Alfv ´en waves ex- cited by streaming CRs become unstable. The wave growth rate is (Kulsrud & Pearce 1969) CR= 0nCR(>rL) nvD VA1 ; (1) when neutrals and ions are strongly coupled together with the Alfv ´en wave frequency r1 LVAmuch smaller than the neutral-ion collisional frequency ni= diin a weakly ion- ized medium or the ion-neutral collisional frequency in= dnin a highly ionized medium. Here dis the drag coeffi- cient (Shu 1992), iandnare the ion and neutral mass den- sities, 0=eB0=(mc)is the nonrelativistic gyrofrequency, eandmare the proton electric charge and mass, cis the light speed,nCR(> rL)is the number density of CRs with the Larmor radius larger than rL,nis the total number den- sity of gas,vDVAis the drift velocity in the wave frame, VA=B0=p4,B0is the mean magnetic field strength, and=i+nis the total mass density. When neutrals and ions are weakly coupled with r1 LVAi> inin a partially ionized medium, where VAi= B0=p4iis the Alfv ´en speed in ions, or in a fully ionized medium, the growth rate is CR= 0nCR(>rL) nivD VAi1 : (2) Hereniis the ion number density. The CR-generated Alfv ´en waves in turn scatter the CRs. The quasilinear gyroresonant scattering of CRs in the wave frame regulates vDVA(i). In a steady state, the ampli- tude of CR-driven Alfv ´en waves is stabilized by the balance between CRand the damping rate of Alfv ´en waves. The pitch-angle scattering corresponding to this wave amplitude is also in balance with the net streaming (Kulsrud 2005). The net drift velocity in the wave frame in a steady state is (Kul- srud 2005; Wiener et al. 2013) vDVA(i)=1 3vrL HB2 0 B(rL)2; (3) wherevcfor relativistic CRs, His the distance from the source to the sink, and B(rL)2=B2 0is the relative magnetic fluctuation energy of the resonant Alfv ´en waves.The damping of streaming instability depends on both properties of the background MHD turbulence and plasma conditions of the surrounding medium. Next we will discuss different damping mechanisms. 2.2. Turbulent damping Turbulent damping was first mentioned in Yan & Lazar- ian (2002) and later studied in detail by Farmer & Goldre- ich (2004) for trans-Alfv ´enic turbulence and Lazarian (2016) in various turbulence regimes for a more general astrophysi- cal application. For strong MHD turbulence with the critical balance (Goldreich & Sridhar 1995) between the turbulent motion in the direction perpendicular to the local magnetic field and the wave-like motion along the local magnetic field (Lazarian & Vishniac 1999), i.e., x? ux=xk VA; (4) wherex?andxkare the length scales of a turbulent eddy perpendicular and parallel to the local magnetic field, and ux=Vst(x?=Lst)1 3 (5) is the turbulent velocity at x?. The corresponding turbulent cascading rate, i.e., eddy turnover rate, is uxx1 ?=VstL1 3 stx2 3 ?: (6) Here Vst=VA; Lst=lA=LM3 A; (7) for super-Alfv ´enic turbulence with the Alfv ´en Mach number MA=VL=VA>1,lAis the Alfv ´enic scale, and Vst=VLMA; Lst=ltran=LM2 A; (8) for sub-Alfv ´enic turbulence with MA<1, whereVLis the turbulent velocity at the injection scale Lof turbulence. We follow the analysis in Lazarian (2016) to derive the tur- bulent damping rate. The CR-driven Alfv ´en waves propagate along the local magnetic field. For the Alfv ´en waves with the wavelength , the distortion by the turbulent motion at the resonant perpendicular scale x?is most efficient. andx? are related by x? VA= ux: (9) The scaling relations in Eqs. (4) and (9) are illustrated in Fig. 1, and they give =ux VAx?=u2 x V2 Axk: (10) By inserting Eq. (5) into Eq. (9), one finds x?=3 4VA Vst3 4L1 4 st: (11)3 Figure 1. Sketch of the relation between xkandx?for strong anisotropic MHD turbulence and the relation between x?andfor turbulent damping of CR-driven Alfv ´en waves. The turbulent damping rate is determined by the eddy turnover rate at x?(Eqs. (6) and (11)), st=ux x?=V1 2 AV3 2 stL1 2 st1 2: (12) Note thatx?should lie within the range of strong MHD tur- bulence, i.e., [xmin;?;Lst], wherexmin;?is the perpendicular damping scale of MHD turbulence and determined by micro- scopic plasma effects. The corresponding range of rLis (Eq. (11)), Vst VAL1 3 stx4 3 min;?<rL<Vst VALst: (13) Eqs. (12) and (13) become (Eq. (7)) st=VAL1 2M3 2 A1 2=VLL1 2M1 2 A1 2; (14) and l1 3 Ax4 3 min;?<rL<lA; (15) for super-Alfv ´enic turbulence, and (Eq. (8)) st=VAL1 2M2 A1 2=VLL1 2MA1 2; (16) and L1 3M4 3 Ax4 3 min;?<rL<LM4 A; (17) for sub-Alfv ´enic turbulence. We see that stincreases with MA. Naturally, a larger amplitude of turbulence can result in a more efficient turbulent damping. For the same reason, stof sub-Alfv ´enic turbulence is smaller than that of super- Alfv ´enic turbulence under the same physical condition. 2.3. Ion-neutral collisional damping in a partially ionized medium Alfv ´en waves propagating in the partially ionized interstel- lar medium (ISM) with a wide range of ionization fractions, e.g., from weakly ionized molecular clouds (MCs) to highlyionized warm phases, are subject to the damping effect due to the collisional friction between ions and neutrals. In a weakly ionized medium with ni< in, when ions and neutrals are strongly coupled together with the wave fre- quency!=VAkk< ni, the ion-neutral collisional (IN) damping rate is (Piddington 1956; Kulsrud & Pearce 1969) IN=nV2 Ak2 k 2ni; (18) wherekkis the wavevector component parallel to the mag- netic field, and n=n=. When neutrals and ions are de- coupled from each other, i.e., in the weak coupling regime with!=VAikk>in, there is IN=in 2: (19) MHD turbulent cascade in a weakly ionized medium is also subject to IN damping (Xu et al. 2015, 2016; Xu & Lazarian 2017a). We consider that the driving of turbulence occurs in the strong coupling regime. MHD turbulence is damped when INin Eq. (18) equalizes with the turbulent cascading rate ukk?, whereukis the turbulent velocity at wavenumber k, andk?is the wavevector component perpen- dicular to the magnetic field. For strong MHD turbulence, k? andkkare related by the critical balance relation (see Section 2.2) k?uk=kkVA: (20) The corresponding IN damping scale of MHD turbulence is (Xu et al. 2015, 2016) xmin;?=2ni n3 2L1 2 stV3 2 st; (21) which gives the smallest perpendicular scale of MHD turbu- lent cascade. It becomes xmin;?=2ni n3 2L1 2V3 2 L (22) for super-Alfv ´enic turbulence, and xmin;?=2ni n3 2L1 2V3 2 LM1 2 A (23) for sub-Alfv ´enic turbulence. With ukk?=VAkk<ni<in; (24) and nV2 Ak2 k 2ni<nni 2<ni 2<in 2; (25) strong MHD turbulence injected in the strong coupling regime cannot cascade into the weak coupling regime, and INof Alfv ´en waves in the weak coupling regime is larger thanINand the eddy turnover rate of MHD turbulence in the strong coupling regime (Xu et al. 2016).4 In a highly ionized medium with in< ni, in the strong coupling regime with VAkk<in,INis given by Eq. (18). When ions are decoupled from neutrals with VAikk> in, there is (Xu et al. 2016) IN=niV2 Aik2 k 2 (1 +)22 ni+V2 Aik2 k; (26) where=n=i. When neutrals and ions are decoupled from each other with VAikk>ni, the above expression can be reduced to Eq. (19). As ukk?=VAkk(orVAikk)>IN in both strong and weak coupling regimes, MHD turbulence in a highly ionized medium is not damped by IN damping. Briefly, IN damping is sensitive to the ionization fraction, and the damping effect in a weakly ionized medium is much stronger than that in a highly ionized medium. 2.4. Nonlinear Landau damping In the fully ionized gaseous Galactic halo or corona (Spitzer 1990; McKee 1993), Alfv ´en waves are subject to nonlinear Landau (NL) damping due to the resonant interac- tions of thermal ions with the beat waves produced by cou- ples of Alfven waves (Lee & V ¨olk 1973; Kulsrud 1978). The damping rate is (Kulsrud 1978) NL=1 2 21 2vth cB(rL)2 B2 0 ; (27) where =eB0=( mc)c=rLis the gyrofrequency of rel- ativistic CRs with the Lorentz factor ,vth=p kBTi=mi is the average thermal ion speed, kBis the Boltzmann con- stant,Tiis ion temperature, and miis ion mass. Unlike st andIN,NLdepends on the amplitude of CR-generated Alfv ´en waves. 3.TURBULENT DAMPING VS. IN DAMPING Depending on the driving condition of MHD turbulence and the plasma condition in different interstellar phases, the dominant damping mechanism of streaming instability varies. We first compare turbulent damping with IN damp- ing in weakly and highly ionized media, and then compare turbulent damping with NL damping in a fully ionized hot medium (see Section 4). As the streaming instability and wave damping together determine vD, a proper description of the damping effect in different regimes is important for determining the diffusion coefficient of CRs and understand- ing their confinement in the Galaxy. 3.1. Dominant damping mechanism in different regimes (1) Weakly ionized medium. We first consider the case when both MHD turbulence and CR-driven Alfv ´en waves are in the strong coupling regime, i.e., r1 LVA< ni. If the turbulent damping is the dominant damping mechanism, we should have (i):st(x?)>IN(xk); (28) so that MHD turbulence is not damped at x?, and (ii):st(x?)>IN(rL): (29)We easily see rL<x?<xk (30) based on the relation in Eq. (10), meaning IN(rL)>IN(xk): (31) Therefore, if condition (ii) is satisfied, then condition (i) is naturally satisfied. As an example, using the following parameters, we have VA rLni =0:07B0 1G2nH 100cm33 2ne=nH 0:11ECR 10GeV1 <1; (32) wherene=nHis the ionization fraction, neandnHare num- ber densities of electrons and atomic hydrogen, mi=mn= mH,mnis neutral mass, mHis hydrogen atomic mass, d= 5:51014cm3g1s1(Shu 1992), and ECRis the energy of CR protons. The values used here do not represent the typical conditions of MCs, but are still considered as a possibility given the large variety of interstellar conditions. Condition (ii) in Eq. (29) can be rewritten as (Eqs. (14) and (18)) MA>n 2niVAL1 2r3 2 L2 3 = 2B0 1G5 3nH 100cm31ne=nH 0:12 3 L 0:1pc1 3ECR 10GeV1(33) for super-Alfv ´enic turbulence driven on small length scales, e.g., near supernova shocks when the shock and shock pre- cursor interact with interstellar or circumstellar density inho- mogeneities (e.g., Xu & Lazarian 2017b, 2021). We note that the outer scale of this turbulence is determined by the size of density clumps. For instance, the typical size of ubiquitous HI clouds in the ISM is 0:1pc (Inoue et al. 2009). As this scale is much larger than rLof low-energy CRs considered here, the CR-induced Alfv ´en waves are subject to turbulent damping in this scenario. With the above parameters used, in Fig. 2(a), the shaded area shows the ranges of MAandne=nHfor turbulent damp- ing to dominate over IN damping. The solid line represents MA=n 2niVAL1 2r3 2 L2 3; (34) below which, IN damping dominates over turbulent damping. In the area above the solid line, as MHD turbulence is also subject to IN damping, to ensure that the condition in Eq. (15) is also satisfied, other constraints on MAindicated in Fig. 2(a) are MA<2ni nL VA1 3; (35)5 corresponding to (Eq. (22)) xmin;?<lA; (36) MA<h2ni n2 V2 ALrLi1 3; (37) corresponding to (Eqs. (11) and (22)) st(x?)>IN(xk); (38) and MA<L rL1 3; (39) corresponding to rL<lA: (40) In addition, the vertical dashed line indicates the ne=nH value corresponding to r1 LVA=ni. Toward a larger ne=nH, the Alfv ´en waves are in the strong coupling regime. In Fig. 2(b), using the MAvalue given by Eq. (34), we illustrate the relation between different length scales. For the regime of interest, we have rL;min<VA ni<rL<x?<xk<lA; (41) whererL;min=l1 3 Ax4 3 min;?is given in Eq. (15). In typical MC conditions, we find that CR-driven Alfv ´en waves are in the weak coupling regime with VAi rLin2103B0 10G2nH 100cm33 2ne=nH 1041 2 ECR 1GeV1 1: (42) For MHD turbulence injected at a large scale in the strong coupling regime, there is always (see Section 2.3) st(x?)<IN(rL) =in 2: (43) Therefore, the damping of CR-driven Alfv ´en waves in MCs predominantly comes from ion-neutral collisions. (2) Highly ionized medium. At a high ionization fraction, when both MHD turbulence and CR-generated Alfv ´en waves are in the strong coupling regime, i.e., r1 LVA<in, similar to the analysis for the strong coupling regime in a weakly ion- ized medium, st(x?)should be compared with IN(rL)to determine the relative importance between the two damping effects. When MHD turbulence at x?is in the strong cou- pling regime, but CR-generated Alfv ´en waves are in the weak coupling regime and also have r1 LVAi> ni, IN damping is more important than turbulent damping. When MHD tur- bulence atx?is also in the weak coupling regime, there is always st(x?)>IN(rL) =in 2; (44) and MHD turbulence dominates the wave damping.By using the typical parameters of the warm ionized medium (WIM) (Reynolds 1992), we find that CR-generated Alfv ´en waves are in the weak coupling regime and further have VAi rLni =7:6103B0 1G2ni 0:1cm33 2ECR 1GeV1 1: (45) As discussed above, under the condition st(x?) IN(rL)=st(x?) in 2>1; (46) turbulent damping dominates over IN damping. The above condition can be rewritten as (Eq. (14)) MA>in 2V1 AiL1 2r1 2 L2 3 = 0:2B0 1G1ni 0:1cm31 3nn 0:01cm32 3 L 100pc1 3ECR 1GeV1 3(47) for super-Alfv ´enic turbulence, which is naturally satisfied, and (Eq. (16)) MA>in 2V1 AiL1 2r1 2 L1 2 = 0:3B0 1G3 4ni 0:1cm31 4nn 0:01cm31 2 L 100pc1 4ECR 1GeV1 4(48) for sub-Alfv ´enic turbulence, where nnis the neutral num- ber density, and L100pc is the typical injection scale of interstellar turbulence driven by supernova explosions. We note thatVAVAican be used for estimating MAof MHD turbulence injected in the strong coupling regime in a highly ionized medium. As the above constraints on MAcan be eas- ily satisfied in the WIM, turbulent damping is likely to be the dominant damping effect for CR-generated Alfv ´en waves in the WIM. 3.2.vDin different regimes Knowing the dominant damping mechanism in different coupling regimes and at different ionization fractions, we can further determine vDat the balance between wave growth and damping. (1) Weakly ionized medium. In the strong coupling regime, when MHD turbulence dominates the wave damping, at the balance between growth and damping rates of Alfv ´en waves (Eqs. (1) and (12)), we find vD VA= 1 + 1 0nCR(>rL) n1 V1 2 AV3 2 stL1 2 str1 2 L;(49)6 10-210-1ne/nH100101102MAst (x) = IN (rL)st(x) = IN(x||)xmin, = lArL = lAst(x) > IN(rL) (a) 10-210-1ne/nH1081010101210141016length scales [cm]lAVA/nirL,minrLxx|| (b) Figure 2. (a) Ranges of MAandne=nHfor turbulent damping to dominate over IN damping (shaded area above the solid line) and for IN damping to dominate over turbulent damping (below the solid line) in a weakly ionized medium, where super-Alfv ´enic turbulence driven on a small length scale ( 0:1pc) is considered. Other limits on MAare indicated by other lines as explained in the text. (b) Relation between different length scales, where the MAvalue corresponding to the solid line in (a) is used. The vertical dashed lines in (a) and (b) correspond to r1 LVA=niwithne=nH= 0:007. which is (Eq. (7)) vD VA1 + 1:7105B0 1G1nH 100cm35 4 L 0:1pc1 2VL 1km s13 2ECR 10GeV1:1(50) for super-Alfv ´enic turbulence, where we adopt the integral number density of CRs near the Sun (Wentzel 1974) nCR(>rL) = 21010 1:6cm3: (51) When ion-neutral collisions dominate the wave damping, there is (Eqs. (1) and (18)) vD VA= 1 + 1 0nCR(>rL) n1nV2 Ar2 L 2ni 1 + 4:9104B0 1G3nH 100cm31 ne=nH 0:11ECR 10GeV0:4 :(52) We see that with the parameters adopted here, in the strong coupling regime there is vDVAdue to the strong damping of CR-generated Alfv ´en waves irrespective of the dominant damping mechanism. In a typical MC environment, when CR-driven Alfv ´en waves are in the weak coupling regime and mainly subjectto IN damping, we have (Eqs. (2) and (19)) vD VAi= 1 +in 2 1 0nCR(>rL) ni1 1 + 26:5B0 10G1nH 100cm32 ne=nH 104ECR 1GeV1:6 :(53) We see that vDis significantly larger than VAidue to the damping effect. (2) Highly ionized medium. In the WIM, CR-driven Alfv ´en waves are in the weak coupling regime and mainly subject to turbulent damping. CR= stgives (Eqs. (2), (7), (8), (12)) vD VAi=1 + 1 0nCR(>rL) ni1 V1 2 AiV3 2 stL1 2 str1 2 L 1 + 3:2B0 1G1ni 0:1cm35 4ECR 1GeV1:1 VL 10km s13 2L 100pc1 2(54) for super-Alfv ´enic turbulence, and vD VAi1 + 0:9B0 1G3 2ni 0:1cm33 2ECR 1GeV1:1 VL 5km s12L 100pc1 2 (55) for sub-Alfv ´enic turbulence, where we consider VAVAi andninH, andVL10km s1is the typical turbu- lent velocity for supernova-driven turbulence (Chamandy &7 Shukurov 2020). The second term in Eqs. (54) and (55) be- comes the dominant term at higher CR energies, and vDis energy dependent. The larger vDin Eq. (54) is caused by the stronger turbulent damping in super-Alfv ´enic turbulence than in sub-Alfv ´enic turbulence (see Section 2.2). 3.3. Diffusion coefficient in different regimes The diffusion coefficient Dof streaming CRs depends on bothvDand the characteristic scale of turbulent magnetic fields. In super-Alfv ´enic turbulence, lAis the characteris- tic tangling scale of turbulent magnetic fields, at which the turbulent and magnetic energies are in equipartition. Over lAthe field line changes its orientation in a random walk manner. Therefore, lAis the effective mean free path of CRs streaming along turbulent magnetic field lines (Brunetti & Lazarian 2007). In sub-Alfv ´enic turbulence, magnetic fields are weakly perturbed with an insignificant change of magnetic field orientation on all length scales. So the mag- netic field structure cannot provide additional confinement for streaming CRs. In this case, streaming CRs do not have a diffusive propagation in the observer frame, but we still intro- duce a diffusion coefficient to quantify the CR confinement and adopt the CR gradient scale length Hfor calculating D. In a weakly ionized medium, e.g., MCs, by using Eq. (53) and considering super-Alfv ´enic turbulence, we have D=vDlA=VAivD VAiLM3 A 1:81028cm2s1nH 100cm33 2ne=nH 1041 2 ECR 1GeV1:6L 10pc M3 A;(56) where the factor M3 Acan be much smaller than unity. Here we consider L10pc for turbulence in MCs, and we note that as MHD turbulence is in the strong coupling regime, VA should be used when calculating MA. AsD/M3 A, a slow diffusion with a small Dis expected at a large MA. In a highly ionized medium, e.g., the WIM, we have D=vDlA (57) for super-Alfv ´enic turbulence, and D=vDH (58) for sub-Alfv ´enic turbulence, where vDis given in Eq. (54) and Eq. (55), respectively, and H1kpc as the scale height of the WIM. In Fig. 3, we present Das a function of ECR for both super- and sub-Alfv ´enic turbulence with MA= 1:4 and0:7in the WIM. The smaller Din super-Alfv ´enic turbu- lence is caused by the tangling of turbulent magnetic fields. We seeD/E1:1 CRin both turbulence regimes. This steep en- ergy scaling can be important for explaining the CR spectrum observed at Earth below 100 GeV (Blasi et al. 2012). 4.TURBULENT DAMPING VS. NL DAMPING 100101102ECR [GeV]1026102710281029D [cm2 s-1]MA = 0.7MA = 1.4 ECR1.1Figure 3. Diffusion coefficient vs. ECRin super- and sub-Alfv ´enic turbulence in the WIM. In the Galactic halo, if NL damping is the dominant damp- ing mechanism of CR-generated Alfv ´en waves, at the bal- ance CR= NL; (59) by combining Eqs. (2), (3), and (27), one can obtain vD=VAi+pc 3H1 2" 2 3r 2  0 vthnCR(>rL) ni1 VAi#1 2 ; (60) and B(rL)2 B2 0=" 2 3r 2  0 vthnCR(>rL) nic VAir2 L H#1 2 :(61) Inserting Eq. (61) into Eq. (27) yields NL= 21 41 61 2v1 2 th" 0nCR(>rL) nic VAi1 H#1 2 ; (62) which becomes smaller at a larger Hwith a smaller CR gra- dient. To be consistent with our assumption that NL damping dominates over turbulent damping, the condition (Eqs. (12) and (62)) NL st=" 21 21 6Lst HnCR(>rL) nivth Vstc Vst 0rL Vst#1 2 >1 (63) should be satisfied. If turbulent damping dominates over NL damping, the bal- ance (Eqs. (2) and (12)) CR= st (64) gives (see also Eq. (54)) vD=VAi+ 1 0nCR(>rL) ni1 V1 2 AiV3 2 stL1 2 str1 2 L:(65)8 Then inserting the above expression into Eq. (3) gives B(rL)2 B2 0=c 3H 0nCR(>rL) niV1 2 AiV3 2 stL1 2 str3 2 L:(66) Moreover, NLcorresponding to the above relative magnetic fluctuation energy is (Eqs. (27) and (66)) NL=1 6 21 2c H 0vthnCR(>rL) niV1 2 AiV3 2 str1 2 LL1 2 st: (67) Under the assumption of dominant turbulent damping, there should be NL st= 21 21 6Lst HnCR(>rL) nivth Vstc Vst 0rL Vst<1: (68) Comparing the conditions in Eqs. (63) and (68), we see that at NL= st, there is  21 21 6Lst HnCR(>rL) nivth Vstc Vst 0rL Vst= 1: (69) Using the typical parameters in the Galactic halo (Farmer & Goldreich 2004), we find that the turbulence in this low- density environment has MA0:1VL 10km s1B0 1G1ni 103cm31 2:(70) For sub-Alfv ´enic turbulence, Eq. (69) can be rewritten as MA=h 21 21 6L HnCR(>rL) nivth VAic VAi 0rL VAii1 4;(71) which is shown as the solid line in Fig. 4. Other parameters areTi= 106K,L= 100 pc, andECR= 1 GeV . The area above and below the solid line corresponds to the parame- ter space for turbulent damping and NL damping to be the dominant damping mechanism, respectively. When turbulent damping is dominant, another constraint on MAis (Eq. (17)) MA>rL L1 40:008; (72) which is naturally satisfied in this situation. Given the small MAof MHD turbulence in the Galactic halo, the wave damping is more likely to be dominated by NL damping. Using Eq. (60), we find vD VAi1+0:02H 5kpc1 2B0 1G1ni 103cm33 4 Ti 106K1 4ECR 1GeV0:8 : (73) vDis very close to VAi, indicative of the insignificant wave damping in the Galactic halo. Therefore, GeV CRs can be confined due to streaming instability. For CRs with ECR< 12345H [kpc]00.20.40.60.81MAFigure 4. Comparison between turbulent damping and NL damping for Alfv ´en waves generated by GeV CRs in the Galactic halo. The shaded area shows the ranges of MAandHfor turbulent damping to dominate over NL damping. The solid line represents the relation in Eq. (71). 100GeV , there is approximately DVAiH = 1:11029cm2s1B0 1Gni 103cm31 2H 5kpc ; (74) which is energy independent. 5.DISCUSSION Effect of MHD turbulence on diffusion of streaming CRs. In cases when MHD turbulence dominates the damping of streaming instability, e.g., in the WIM, MHD turbulence is important for setting vD. Super-Alfv ´enic turbulence also provides additional confinement to streaming CRs due to the field line tangling at lA, irrespective of the dominant damping mechanism. In addition, the non-resonant mirroring interac- tion of CRs with slow and fast modes in MHD turbulence can also suppress the diffusion of CRs in the vicinity of CR sources (Lazarian & Xu 2021; Xu 2021). The relative im- portance between mirroring and streaming instability in af- fecting CR diffusion near CR sources will be investigated in our future work. In this work we do not consider gyroreso-9 nant scattering and resonance-broadened transit time damp- ing (TTD) by fast modes of MHD turbulence (Xu & Lazarian 2018, 2020), as fast modes are damped at a large scale due to IN damping in a weakly ionized medium (Xu et al. 2016) and their energy fraction is small at a small MA(Hu et al. 2021). Moreover, the energy scaling of diffusion coefficient corre- sponding to scattering by fast modes is incompatible with AMS-02 observations at CR energies .103GeV (Kempski & Quataert 2021). Cutoff range of Alfv ´en waves in a weakly ionized medium. In a weakly ionized medium, within the cutoff range of kk there is no propagation of Alfv ´en waves due to the severe IN damping (Kulsrud & Pearce 1969). The boundary [k+ c;k;k c;k] of the cutoff range is set by != INin both strong and weak coupling regimes (Xu et al. 2015), k+ c;k=2ni VAn; k c;k=in 2VAi: (75) If CR-driven Alfv ´en waves fall in the cutoff range with k+ c;k<r1 L<k c;k; (76) the streaming instability cannot occur. We note that for GeV CRs in a typical MC environment, the CR-driven Alfv ´en waves are in the weak coupling regime with r1 Lk c;k(see Eq. (42)). Microscopic vs. macroscopic diffusion. By “microscopic diffusion”, we refer to the diffusion in the wave frame caused by the gyroresonant scattering of CRs by CR-amplified Alfv ´en waves, while the “macroscopic diffusion” in the ob- server frame is a result of both streaming of CRs and tangling of turbulent magnetic fields on scales much larger than rL. The former was included in calculating the total diffusion co- efficient in earlier studies, e.g., Hopkins et al. (2021). In this work we only consider the latter as it can be directly com- pared with observations in the observer frame, which was also adopted in Krumholz et al. (2020) for studying CR trans- port in starburst galaxies. Coupling between CRs and gas. In the Galactic disk with super-Alfv ´enic turbulence (Hu et al. 2019), due to the strong IN damping and turbulent damping, CRs have fast streaming and do not suffer significant energy loss via wave genera- tion. The coupling between CRs and gas is caused by field line tangling. This coupling can result in additional pres- sure support and suppression of star formation. By contrast, in the Galactic halo with sub-Alfv ´enic turbulence, due to the weak NL damping, CRs are well self-confined and coupled to the gas via streaming instability, and thus effectively transfer momentum to the gas. Both wave damping and turbulent tan- gling can significantly affect the transport of streaming CRs and their coupling with the gas, and thus are important for studying CR-driven galactic winds. Turbulence and CRs at phase transition. In the multi-phase ISM, the transition from hot/warm to cold gas can be driven by, e.g., passage of shock waves (Inutsuka et al. 2015). The phase transition induces various instabilities and turbulence.The turbulent mixing layers at the interfaces between dif- ferent gas phases have been recently studied in detail by Ji et al. (2019). As the transport of CRs is sensitive to the turbulent magnetic field structure, when CRs interact with the shock-compressed magnetic field, they can be reflected off the shock surface (Xu & Lazarian 2021) and undergo an abrupt change of trajectory. 6.SUMMARY We study the damping of streaming instability of GeV-100 GeV CRs and the resulting diffusion coefficients in different MHD turbulence regimes and interstellar phases. In a partially ionized medium, both CR-generated Alfv ´en waves and MHD turbulence are subject to IN damping. The damping rate depends on the ionization fraction and the cou- pling state between ions and neutrals. In both weakly ion- ized MCs and highly ionized WIM, CR-generated Alfv ´en waves are in the weak coupling regime. In a weakly ionized medium, IN damping is strong and dominates the damping of both MHD turbulence and CR-amplified Alfv ´en waves. In a highly ionized medium, IN damping is so weak that MHD turbulence injected in the strong coupling regime can cas- cade into the weak coupling regime and dominates the wave damping. Both IN damping in MCs and turbulent damping in the WIM act to suppress the streaming instability, leading to a streaming speed of CRs larger than the Alfv ´en speed. The resulting diffusion coefficient is thus dependent on CR en- ergies. The steep energy scaling of diffusion coefficient in the WIM (see Fig. 3) is important for explaining the CR spectrum observed at Earth, as the turbulence properties mea- sured in the nearby ( .1kpc) ISM (Armstrong et al. 1995) are similar to that in the WIM (Chepurnov & Lazarian 2010). We find that MHD turbulence not only can affect the CR streaming speed by turbulent damping but also causes the diffusive propagation of streaming CRs by the field line tan- gling. The latter effect was not considered in most earlier studies on CR streaming. Because of the field line tangling in super-Alfv ´enic turbulence at lA=LM3 A, CRs streaming along turbulent field lines have an effective mean free path given bylA. At a large MAin, e.g., cold interstellar phases, a significant reduction of the diffusion coefficient by M3 A is expected. The slow diffusion of streaming CRs in star- forming MCs can have an important influence on the Galac- tic disk structure and star formation (Semenov et al. 2021). In the multi-phase ISM with a large variety of MAof inter- stellar turbulence, measuring MAwith new techniques (e.g., Lazarian et al. 2018; Xu & Hu 2021) is necessary for realistic modeling of diffusion coefficients of streaming CRs. In the diffuse Galactic halo, MHD turbulence is sub- Alfv ´enic with a small MA, and NL damping is a more impor- tant mechanism for damping CR-generated Alfv ´en waves. This finding is different from that in Lazarian (2016). The resulting streaming speed is basically given by Alfv ´en speed, and CRs are confined mainly due to streaming instability. In the WIM and Galactic halo, the global pressure gradient formed by streaming CRs plays an important dynamical role10 in driving galactic outflows and affecting galaxy evolution (Padovani et al. 2020). In addition to the interstellar turbulence injected on  10100pc, we also considered a special case with small- scale (.0:1pc) preshock turbulence in supernova rem- nants, which is driven by the interaction between the CR precursor and upstream density inhomogeneities. When CR- amplified Alfv ´en waves are in the strong coupling regime at a low ionization fraction, we find the condition and parameter space for turbulent damping to dominate over IN damping of streaming instability. ACKNOWLEDGMENTS S.X. acknowledges the support for this work provided by NASA through the NASA Hubble Fellowship grant # HST- HF2-51473.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA con- tract NAS5-26555. A.L. acknowledges the support of NASA ATP AAH7546. 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2021-12-13

We study the streaming instability of GeV$-100~$GeV cosmic rays (CRs) and its damping in the turbulent interstellar medium (ISM). We find that the damping of streaming instability is dominated by ion-neutral collisional damping in weakly ionized molecular clouds, turbulent damping in the highly ionized warm medium, and nonlinear Landau damping in the Galactic halo. Only in the Galactic halo, is the streaming speed of CRs close to the Alfv\'{e}n speed. Alfv\'{e}nic turbulence plays an important role in both suppressing the streaming instability and regulating the diffusion of streaming CRs via magnetic field line tangling, with the effective mean free path of streaming CRs in the observer frame determined by the Alfv\'{e}nic scale in super-Alfv\'{e}nic turbulence. The resulting diffusion coefficient is sensitive to Alfv\'{e}n Mach number, which has a large range of values in the multi-phase ISM. Super-Alfv\'{e}nic turbulence contributes to additional confinement of streaming CRs, irrespective of the dominant damping mechanism.

Cosmic ray streaming in the turbulent interstellar medium

2112.06941v2

arXiv:1109.4964v1 [cond-mat.mtrl-sci] 22 Sep 2011Hole spin relaxation and coefficients in Landau-Lifshitz-Gi lbert equation in ferromagnetic GaMnAs K. Shen and M. W. Wu∗ Hefei National Laboratory for Physical Sciences at Microsc ale and Department of Physics, University of Science and Technology of China, Hefei, Anhui , 230026, China (Dated: April 19, 2022) We investigate the temperature dependence of the coefficient s in the Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs by employing the Zener mod el. We first calculate the hole spin relaxation time based on the microscopic kinetic equation. We find that the hole spin relaxation time is typically several tens femtoseconds and can present a nonmonotonic temperature dependence due to the variation of the interband spin mixing, influenced by the temperature related Zeeman splitting. With the hole spin relaxation time, we are able to calculate the coefficients in the Landau- Lifshitz-Gilbert equation, such as the Gilbert damping, no nadiabatic spin torque, spin stiffness and vertical spin stiffness coefficients. We find that the nonadiab atic spin torque coefficient βis around 0.1∼0.3 at low temperature, which is consistent with the experimen t [Adam et al., Phys. Rev. B 80, 193204 (2009)]. As the temperature increases, βmonotonically increases and can exceed one in the vicinity of the Curie temperature. In the low temperat ure regime with β <1, the Gilbert damping coefficient αincreases with temperature, showing good agreement with th e experiments [Sinovaet al., Phys. Rev. B 69, 085209 (2004); Khazen et al.,ibid.78, 195210 (2008)]. Furthermore, we predict that αdecreases with increasing temperature once β >1 near the Curie temperature. We also find that the spin stiffness decreases with increasing temperature, especially near the Curie temperature due to the modification of the finite β. Similar to the Gilbert damping, the vertical spin stiffness coefficient is also found to be nonmonotonicall y dependent on the temperature. PACS numbers: 72.25.Rb, 75.50.Pp, 72.25.Dc, 75.30.Gw I. INTRODUCTION The ferromagnetic semiconductor, GaMnAs, has been proposed to be a promising candidate to realize all- semiconductor spintronic devices,1,2where the existence of the ferromagnetic phase in the heavily doped sample sustains seamless spin injection and detection in normal non-magneticsemiconductors.3,4Oneimportant issuefor such applications lies in the efficiency of the manipula- tion of the macroscopic magnetization, which relies on properties of the magnetization dynamics. Theoretically, the magnetization dynamics can be described by the ex- tended Landau-Lifshitz-Gilbert (LLG) equation,5–10 ˙n=−γn×Heff+αn×˙n−(1−βn×)(vs·∇)n −γ Mdn×(Ass−Av ssn×)∇2n, (1) withnandMdstanding for the direction and magni- tude of the magnetization, respectively. Heffis the ef- fective magnetic field and/or the external field. The sec- ond term on the right hand side of the equation is the Gilbert damping torque with αdenoting the damping coefficient.5,6The third one describes the spin-transfer torque induced by the spin current vs.7,8As reported, the out-of-plane contribution of the spin-transfer torque, measured by the nonadiabatic torque coefficient β, can significantly ease the domain wall motion.7,8In Eq.(1), thespinstiffnessandverticalspinstiffnesscoefficientsare evaluated by AssandAv ssrespectively, which are essen- tially important for the static structure of the magnetic domain wall.10Therefore, for a thorough understandingof properties of the magnetization dynamics, the exact values of the above coefficients are required. In the past decade, the Gilbert damping and nonadia- batic torque coefficients have been derived via many mi- croscopic approaches, such as the Blotzmann equation,11 diagrammatic calculation,12,13Fermi-surface breathing model14–16andkineticspinBlochequations.10,17Accord- ing to these works, the spin lifetime of the carriers was found to be critical to both αandβ. However, to the best of our knowledge, the microscopic calculation of the hole spin lifetime in ferromagnetic GaMnAs is still ab- sent in the literature, which prevents the determination ofthe values of αandβfrom the analyticalformulas. Al- ternatively, Sinova et al.18identified the Gilbert damp- ing from the susceptibility diagram of the linear-response theory and calculated αas function of the quasiparticle lifetime and the hole density. Similar microscopic calcu- lation on βwas later given by Garate et al..19In those works, the quasiparticle lifetime was also treated as a parameter instead of explicit calculation. Actually, the accurate calculation of the hole spin and/or quasiparti- cle lifetime in ferromagnetic GaMnAs is difficult due to the complex band structure of the valence bands. In the presentwork,weemploythe microscopickineticequation tocalculatethespinlifetimeoftheholegasandtheneval- uateαandβin ferromagnetic GaMnAs. For the velocity of the domain-wall motion due to the spin current, the ratioβ/αisanimportantparameter,whichhasattracted much attention.12,19,20Recently, a huge ratio ( ∼100) in nanowire was predicted from the calculation of the scat- tering matrix by Hals et al..20By calculating αandβ, we2 are able to supply detailed information of this interesting ratio in bulk material. Moreover, the peak-to-peak fer- romagnetic resonance measurement revealed pronounced temperature and sample preparation dependences of the Gilbert damping coefficient.18,21,22For example, in an- nealed samples, αcan present an increase in the vicinity of the Curie temperature,18,21which has not been stud- ied theoretically in the literature. Here, we expect to uncover the underlying physics of these features. In ad- dition, the nonadiabatic torque coefficient βin GaMnAs has been experimentally determined from the domain- wall motion and quite different values were reported by different groups, from 0.01 to 0.36,23,24which need to be verified by the microscopic calculation also. Moreover, to the best of our knowledge, the temperature depen- dence of βhas not been studied theoretically. We will also address this issue in the present work. In the literature, the spin stiffness in GaMnAs was studied by K¨ onig et al.,25,26who found that Assincreases with hole density due to the stronger carrier-mediated interaction between magnetic ions, i.e., Ass=Nh/(4m∗) withNhandm∗being the density and effective mass of hole gas, separately. However, as shown in our pre- vious work, the stiffness should be modified as Ass∼ Nh/[4m∗(1+β2)] in ferromagnetic GaMnAs with a finite β.10As a result, Assas well as the vertical spin stiffness Av ss=βAssmay show a temperature dependence intro- duced by β. This is also a goal of the present work. For a microscopic investigation of the hole dynamics, the valence band structure is required for the descrip- tion of the occupied carrier states. In the literature, the Zener model27based on the mean-field theory has been widely used for itinerant holes in GaMnAs,28–31 where the valence bands split due to the mean-field p- dexchange interaction. In the present work, we utilize this model to calculate the band structure with the ef- fective Mn concentration from the experimental value of the low-temperaturesaturatemagnetization in GaMnAs. The thermal effect on the band structure is introduced viathe temperaturedependence ofthe magnetizationfol- lowing the Brillouin function. Then we obtain the hole spin relaxation time by numerically solving the micro- scopic kinetic equations with the relevant hole-impurity and hole-phonon scatterings. The carrier-carrier scatter- ing is neglected here by considering the strongly degen- erate distribution of the hole gas below the Curie tem- perature. We find that the hole spin relaxation time decreases/increases with increasing temperature in the small/large Zeeman splitting regime, which mainly re- sults from the variation of the interband spin mixing. Then we study the temperature dependence of the co- efficients in the LLG equation, i.e., α,β,AssandAv ss, by using the analytical formulas derived in our previous works.10,17Specifically, we find that βincreases with in- creasing temperature and can exceed one in the vicinity of the critical point, resulting in very interesting behav- iors of other coefficients. For example, αcan present an interesting nonmonotonic temperature dependence withthe crossoveroccurringat β∼1. Specifically, αincreases withtemperatureinthelowtemperatureregime,whichis consistent with the experiments. Near the Curie temper- ature, an opposite temperature dependence of αis pre- dicted. Similar nonmonotonic behavior is also predicted in the temperature dependence of Av ss. Our results of β andAssalso show good agreement with the experiments. This work is organized as follows. In Sec.II, we setup our model and lay out the formulism. Then we show the band structure from the Zener model and the hole spin relaxation time from microscopic kinetic equations in Sec.III. The temperature dependence of the Gilbert damping, nonadiabatic spin torque, spin stiffness and vertical spin stiffness coefficients are also shown in this section. Finally, we summarize in Sec.IV. II. MODEL AND FORMULISM Inthesp-dmodel, theHamiltonianofholegasinGaM- nAs is given by31 H=Hp+Hpd, (2) withHpdescribing the itinerant holes. Hpdis thesp-d exchange coupling. By assuming that the momentum k is still a good quantum number for itinerant hole states, one employsthe Zenermodel and utilizes the k·ppertur- bation Hamiltonian to describe the valence band states. Specifically, we take the eight-band Kane Hamiltonian HK(k) (Ref.32) in the present work. The sp-dexchange interaction reads Hpd=−1 N0V/summationdisplay l/summationdisplay mm′kJmm′ exSl·∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htc† mkcm′k, (3) withN0andVstanding for the density of cation sites and the volume, respectively. The cation density N0= 2.22×1022cm−3. The eight-band spin operator can be written as ˆJ= (1 2σ)⊕J3/2⊕J1/2, where1 2σ,J3/2and J1/2represent the total angular momentum operators of the conduction band, Γ 8valence band and Γ 7valence band, respectively. Jmm′ exstands for the matrix element of the exchange coupling, with {m}and{m′}being the basis defined as the eigenstates of the angular momen- tum operators ˆJ. The summation of “ l” is through all localized Mn spins Sl(atrl). Then we treat the localized Mn spin in a mean-field approximation and obtain ¯Hpd=−xeff∝an}b∇acketle{tS∝an}b∇acket∇i}ht·/parenleftBigg/summationdisplay mm′kJmm′ ex∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htc† mkcm′k/parenrightBigg , (4) where∝an}b∇acketle{tS∝an}b∇acket∇i}htrepresents the average spin polarization of Mn atoms with uncompensated doping density NMn= xeffN0. Obviously, ¯Hpdcan be reduced into three blocks asˆJ, i.e.,¯Hmm′ pd(k) = ∆mmn·∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htwith the Zee- man splitting of the m-band ∆mm=−xeffSdJmm exM(T) M(0).3 Here,nis the direction of ∝an}b∇acketle{tS∝an}b∇acket∇i}ht. For a manganese ion, the total spin Sd= 5/2. The temperature-dependent spon- taneous magnetization M(T) can be obtained from the following equation of the Brillouin function33 BSd(y) =Sd+1 3SdT Tcy, (5) wherey=3Sd Sd+1M(T) M(0)Tc TwithTcbeing the Curie temperature. Here, BSd(y) =2Sd+1 2Sdcoth(2Sd+1 2Sdy)− 1 2Sdcoth(1 2Sdy). The Schr¨ odinger equation of the single particle Hamil- tonian is then written as /bracketleftbig HK(k)+¯Hpd(k)/bracketrightbig |µ,k∝an}b∇acket∇i}ht=Eµk|µ,k∝an}b∇acket∇i}ht.(6) One obtains the band structure and wave functions from the diagonalization scheme. In the presence of a finite Zeemansplitting, thestructureofthe valencebandsdevi- ates from the parabolic dispersion and becomes strongly anisotropicaswewillshowinthenextsection. Moreover, the valence bands at Fermi surface are well separated in ferromagnetic GaMnAs because of the high hole density (>1020cm−3) and Zeeman splitting, suggesting that the Fermi golden rule can be used to calculate the lifetime of the quasiparticlestates. For example, the contribution of the hole-impurity scattering on the µth-band state with energyǫcan be expressed by [τhi µ,p(ǫ)]−1= 2π/summationdisplay νni Dµ(ǫ)/integraldisplayd3k (2π)3/integraldisplayd3q (2π)2δ(ǫ−ǫµk) ×δ(ǫµk−ǫνq)U2 k−q|∝an}b∇acketle{tµk|νq∝an}b∇acket∇i}ht|2f(ǫµk)[1−f(ǫνq)],(7) whereDµ(ǫ) stands for the density of states of the µth band.f(ǫµk) satisfies the Fermi distribution in the equi- librium state. The hole-impurity scattering matrix ele- mentU2 q=Z2e4/[κ0(q2+κ2)]2withZ= 1.κ0and κdenote the static dielectric constant and the screening constant under the random-phase approximation,34re- spectively. Similar expression can also be obtained for the hole-phonon scattering. However, it is very complicated to carry out the multi- fold integrals in Eq.(7) numerically for an anisotropic dispersion. Also the lifetime of the quasiparticle is not equivalent to the spin lifetime of the whole system, which is required to calculate the LLG coefficients according to our previous work.10,17Therefore, we extend our ki- netic spin Bloch equation approach35to the current sys- tem to study the relaxation of the total spin polarization as follows. By taking into account the finite separation between different bands, one neglects the interband co- herence and focuses on the carrier dynamics of the non- equilibrium population. The microscopic kinetic equa- tion is then given by ∂tnµ,k=∂tnµ,k/vextendsingle/vextendsinglehi+∂tnµ,k/vextendsingle/vextendsinglehp, (8) withnµ,kbeing the carrier occupation factor at the µth band with momentum k. The first and second terms onthe right hand side stand for the hole-impurity and hole- phonon scatterings, respectively. Their expressions can be written as ∂tnµ,k/vextendsingle/vextendsinglehi=−2πni/summationdisplay ν,k′U2 k−k′(nµk−nνk′)|∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2 ×δ(Eµk−Eνk′), (9) and ∂tnµ,k/vextendsingle/vextendsinglehp=−2π/summationdisplay λ,±,ν,k′|Mλ k−k′|2δ(Eνk′−Eµk±ωλ,q) ×[N± λ,q(1−nνk′)nµk−N∓ λ,qnνk′(1−nµk)]|∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2,(10) withN± λ,q= [exp(ωλ,q/kBT)−1]−1+1 2±1 2. Thedetailsof the hole-phonon scattering elements |Mλ q|2can be found in Refs.36–38. From an initial condition with a small non-equilibrium distribution, the temporal evolution of the hole spin polarization is carried out by J(t) =1 Nh/summationdisplay µ,k∝an}b∇acketle{tµk|ˆJ|µk∝an}b∇acket∇i}htnµ,k(t),(11) from the numerical solution of Eq.(8). The hole spin relaxation time can be extracted from the exponential fitting of Jwith respect to time. One further calculates the concerned coefficients such as α,β,AssandAv ss. III. NUMERICAL RESULTS In the Zener model, the sp-dexchange interaction con- stantsJmm exareimportant parametersfor the band struc- ture. In the experimental works, the p-dexchange cou- pling constant Jpp exwas reported to vary from −1 eV to 2.5 eV, depending on the doping density.39–41In ferromagnetic samples, Jpp exis believed to be negative, which was demonstrated by theoretical estimation Jpp ex≈ −0.3 eV (Ref.42). In our calculation, the antiferromag- neticp-dinteraction Jpp exis chosen to be −0.5 eV or −1.0 eV. The ferromagnetic s-dexchange coupling con- stant is taken to be Jss ex= 0.2 eV (Ref.31). Another important quantity for determining the Zee- man splitting is the macroscopic magnetization or the effective concentration of the Mn atoms. As deduced from the low-temperature saturate magnetization, only around 50% Mn atoms can contribute to the ferromag- netic magnetization, which hasbeen recognizedasthe in- fluence of the compensation effect due to the deep donors (e.g., Asantisites)ortheformationofsixfold-coordinated centers defect Mn6As(Ref.43). As only the uncompen- sated Mn atoms can supply holes and contribute to the ferromagneticmagnetization,44one can also estimate the total hole density from the saturate magnetization.45 However, the density of the itinerant hole can be smaller than the effective Mn concentration because of the local- ized effect in such disordered material. It was reported4 Tc Ms NMn (K) (emu ·cm−3) (1020cm−3) Aa130 38 8 Ba157 47 10 Cb114 33 6.9 Dc110 – – Ed139 53.5 11.5 aRef. 21,bRef. 23,cRef. 18,dRef. 45 TABLE I: The parameters obtained from the experiments fordifferentsamples: A:Ga 0.93Mn0.07As/Ga 0.902In0.098As; B: Ga0.93Mn0.07As/GaAs; C: Ga 0.93Mn0.07As/Ga 1−yInyAs; D: Ga0.92Mn0.08As; E: Ga 0.896Mn0.104As0.93P0.07.Msstands for the saturate magnetization at zero temperature M(0). that the hole density is only 15-30% of the total concen- tration of the Mn atoms.43 In our calculation, the magnetization lies along the principle axis chosen as [001]-direction.31The conven- tional parameters are mainly taken from those of GaAs in Refs.46 and 47. Other sample-dependent parame- ters such as the Curie temperature and effective Mn concentration are picked up from the experimental works.18,21,23,45For sample A, B and E (C), only the saturate magnetization at 4 (104) K was given in the references. Nevertheless, one can extrapolate the zero temperature magnetization Msfrom Eq.(5). The effec- tive Mn concentrations listed in TableI are derived from NMn=Ms/(gµBSd). It is clearto see that all of these ef- fectiveMnconcentrationsaremuchsmallerthanthedop- ing density ( ≥1.5×1021cm−3) due to the compensation effect as discussed above. Since the saturate magneti- zation of sample D is unavailable, we treat the effective Mn concentration as a parameter in this case. More- over, since the exact values of the itinerant hole densities are unclear in such strongly disordered samples, we treat them as parameters. Two typical values are chosen in our numerical calculation, i.e., Nh= 3×1020cm−3and 5×1020cm−3. The effective impurity density is taken to be equal to the itinerant hole density. For numerical calculation of the hole spin dynamics, the momentum space is partitioned into blocks. Com- pared to the isotropic parabolic dispersion, the band structure in ferromagnetic GaMnAs is much more com- plex as we mentioned above [referred to Figs.1(b) and 4]. Therefore, we need to extend the partition scheme used in isotropic parabolic dispersion48into anisotropic case. In our scheme, the radial partition is still carried out with respect to the equal-energy shells, while the an- gular partition is done by following Ref.48. In contrast to the isotropic case, the number of states in one block is generally different from that in another block even both of them are on the same equal-energy shell. We calculate the number of states of each block from its volume inmomentum space. 0 10 20 30 40 50 60 0.2 0.4 0.6 0.8 1∆pp (meV)T/Tc (a) 6.9×1020 cm-3 8×1020 cm-3 1×1021 cm-3 -0.4-0.3-0.2-0.1 0 0.1-0.1 0 0.1 0.2 E (eV)k (2π/a ) [111] [001] (b) 0 51015 -0.1 -0.05 0 0.05 0.1 0.15 0.2DOS (1020/eVcm3) E (eV)T = 0.1 TcNMn = 8×1020 cm-3 (c) -0.05 0 0.05 0.1 0.15 0.2 0 5 10 15 DOS (1020/eVcm3) E (eV)T = 0.99 Tc (d) FIG. 1: (Color online) (a) Zeeman energy as function of tem- perature. (b)Thevalencebandstructurewith∆pp= 45meV. The blue dashed curve illustrates the Fermi level for the hol e densityNh= 3×1020cm−3, while the green dotted one gives Nh= 5×1020cm−3. The density of states as function of energy at (c) T/Tc= 0.1 and (d) T/Tc= 0.99 for the uncom- pensated Mn density NMn= 8×1021cm−3. In (d), the blue dashed curve stands for the upper heavy hole band from the spherical approximation and the corresponding DOS from the analytical formula (√ 2E[√ m∗/(2π/planckover2pi1)]3) is given as the green dotted curve. Here, Jpp ex=−0.5 eV. A. Density of states By solving Eq.(5), one obtains the magnetization at finite temperature M(T) and the corresponding Zeeman energy ∆pp. In Fig.1(a), the Zeeman splitting from Jpp ex=−0.5 eV is plotted as function of the temperature. It is seen that the Zeeman energy is tens of milli-electron volts at low temperature and decreases sharply near the Curie temperature due to the decrease of the magnetiza- tion. To show the anisotropicnonparabolicfeature of the band structure in the presence of the Zeeman splitting, we illustrate the valence bands along [001]- and [111]- directions in Fig.1(b), which are obtained from Eq.(6) atT/Tc= 0.1 forNMn= 8×1020cm−3. In this case, the Zeeman splitting ∆pp= 45 meV. The Fermi levels for the hole densities Nh= 3×1020cm−3and 5×1020cm−3are shown as blue dashed and green dotted curves, respec- tively. As one can see that all of the four upper bands can be occupied and the effective mass approximation obviously breaks down. By integrating over the volume of each equal-energy shell, one obtains the density of states (DOS) of each band as function of energy in Fig.1(c) and (d). Here the energy is defined in the hole picture so that the sign of5 the energy is opposite to that in Fig.1(b). It is seen that the DOS of the upper two bands are much larger than those of the other bands, regardless of the magnitude of the Zeeman splitting. For T/Tc= 0.99, the systems ap- proaches the paramagnetic phase and the nonparabolic effect is still clearly seen from the DOS in Fig.1(d), es- pecially in the high energy regime. Moreover, the pro- nounced discrepancy of the DOS for the two heavy hole bands suggests the finite splitting between them. We find that these features are closely connected with the anisotropy of the valence bands, corresponding to the Luttinger parameters γ2∝ne}ationslash=γ3in GaAs.49In our calcu- lation, we take γ1= 6.85,γ2= 2.1 andγ3= 2.9 from Ref.47. As a comparison, we apply a spherical approx- imation ( γ1= 6.85 andγ2=γ3= ¯γ= 2.5) and find that the two heavy hole bands become approximately degenerate.38The DOS of the upper heavy hole band is shown as the blue dashed curve in Fig.1(d), where we also plot the corresponding DOS from the analyti- cal expression, i.e.,√ 2E[√ m∗/(2π/planckover2pi1)]3, as the green dot- ted curve. Here, we use the heavy-hole effective mass m∗=m0/(γ1−2¯γ) withm0denoting the free electron mass. The perfect agreement between the analytical and our numerical results under the spherical approximation suggests the good precision of our numerical scheme. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 10 20 30 40 50 60Equilibrium Hole Polarization ∆pp (meV)A: Nh=3×1020 cm-3 5×1020 cm-3 B: Nh=3×1020 cm-3 5×1020 cm-3 FIG. 2: (Color online) The equilibrium hole spin polarizati on as function of Zeeman splitting for sample A and B. Here, Jpp ex=−0.5 eV. B. Hole spin relaxation In this part, we investigate the hole spin dynamics by numericallysolvingthe microscopickinetic equation, i.e., Eq.(8). By taking into account the equilibrium hole spin polarization, we fit the temporal evolution of the total spin polarization along [001]-direction by Jz(t) =J0 z+J′ ze−t/τs, (12) whereJ0 zandJ′ zcorrespondto the equilibrium and non- equilibrium spin polarizations, respectively. τsisthe hole spin relaxation time.In all the cases of the present work, the equilibrium hole spin polarization for a fixed hole density is found to be approximately linearly dependent on the Zeeman splitting. In Fig.2, J0 zin samples A and B (similar be- haviorforothers) areplotted asfunction ofZeemansplit- ting, where the exchange coupling constant Jpp exis taken to be−0.5 eV. One notices that the average spin polar- izationbecomessmallerwith the increaseofthe holeden- sity, reflecting the large interband mixing for the states in the high energy regime. 30 40 50 60 70 80 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1τs (fs) T/Tc(a)Jexpp = -0.5 eV A: Nh=3×1020 cm-3 5×1020 cm-3 B: Nh=3×1020 cm-3 5×1020 cm-3 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 0 20 40 60 80 100 120τs (fs) T/Tc∆pp (meV) (b) Jexpp = -1 eV0.4 Tc0.99 Tc B: Nh=3×1020 cm-3 5×1020 cm-3 FIG. 3: (Color online) (a) Spin relaxation time as function o f temperaturewith Jpp ex=−0.5eVfor sampleAandB.(b)Spin relaxation time as function of temperature and Zeeman split - ting obtained from the calculation with Jpp ex=−1 eV for sam- ple B. The inset at the left (right) upper corner illustrates the band structure from [001]-direction to [111]-direction [r efer to Fig.1(b)] for T/Tc= 0.4 (0.99) and ∆pp= 105 (16.7) meV. The Fermi levels of Nh= 3×1020cm−3and 5×1020cm−3 are shown as the blue dashed and green dotted curves in the insets, separately. The temperature dependence of the hole spin relax- ation time in samples A and B with Jpp ex=−0.5 eV is shown in Fig.3(a), where the spin relaxation time mono- tonically decreases with increasing temperature. This feature can be understood from the enhancement of the interband mixing as the Zeeman splitting decreases (shownbelow).50Togainacompletepicture ofthe roleof the Zeeman splitting on the hole spin relaxation in fer-6 romagnetic GaMnAs, we also carry out the calculation with the exchange constant Jpp ex=−1 eV.31,39Very in- terestingly, onefinds that the holespin relaxationtime at low temperature increases with increasing temperature, resulting in a nonmonotonic temperature dependence of the hole spin relaxation time in sample B. The results in this case are shown as solid curves in Fig.3(b), where we also plot the Zeeman splitting dependence of the hole spin relaxation time as dashed curves. It is seen that the hole spin relaxation time for the hole density Nh= 3×1020cm−3first increases with increasing temperature (alternativelyspeaking,decreasingZeemansplitting)and starts to decrease at around 0 .8Tcwhere the Zeeman splitting ∆pp= 70 meV. To understand this feature, we showthetypicalbandstructureinthe increase(decrease) regime of the hole relaxation time at T/Tc= 0.4 (0.99), corresponding to ∆pp= 105 (16.7) meV, in the inset at the left (right) upper corner. The Fermi levels of the hole density 3 ×1020cm−3are labeled by blue dashed curves. One finds that the carrier occupations in the increase and decrease regimes are quite different. Specif- ically, all of the four upper bands are occupied in the decrease regime while only three valence bands are rele- vant in the increase regime. One may naturally expect that the increase regime originates from the contribution of the fourth band via the inclusion of the additional scattering channels or the modification of the screening. However, we rule out this possibilitythroughthecomputationwiththefourthband artificially excluded, where the results are qualitatively the same as those in Fig.3(b). Moreover, the variations of the screening and the equilibrium distribution at fi- nite temperature are also demonstrated to be irrelevant to the present nonmonotonic dependence by our calcula- tion (not shown here). Therefore, the interesting feature has to be attributed to the variations of the band dis- tortion and spin mixing due to the exchange interaction. This is supported by our numerical calculation, where the nonmonotonic behavior disappears once the effect of the interband mixing is excluded by removing the wave- function overlaps |∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2in Eqs.(9) and (10) (not shown here). For a qualitative understanding of the nonmonotonic temperature dependence of the hole spin relaxation time, we plot the Fermi surface in the kx-kz(ky= 0) and kx- ky(kz= 0) planes at Nh= 3×1020cm−3in Fig.4. We choose typical Zeeman splittings in the increase regime (∆pp= 105meV), the decreaseregime(∆pp= 16.7meV) and also the crossover regime (∆pp= 70 meV). One no- tices that the Fermi surfacesin Fig.4(a) and (d) arecom- posed of three closed curves, meaning that only three bands are occupied for ∆pp= 105 meV [also see the in- set of Fig.3(b)]. For the others with smaller Zeeman splittings, all of the four upper bands are occupied. The spin expectation of each state at Fermi surface is repre- sented by the color coding. Note that the spin expecta- tion of the innermost band for ∆pp= 70 meV is close to −1.5 [see Fig.4(b) and (e)], suggesting that this band is-1.5-1-0.5 0 0.5 1 1.5ξ∆pp = 105 meV 70 meV 16.7 meV (a)-0.2-0.10.00.10.2kz (2π/a) -1.5-1-0.5 0 0.5 1 1.5 (b) -1.5-1-0.5 0 0.5 1 1.5 (c) -1.5-1-0.5 0 0.5 1 1.5 (d) -0.2 -0.1 0.0 0.1 0.2-0.2-0.10.00.10.2ky (2π/a)-1.5-1-0.5 0 0.5 1 1.5 (e) -0.2 -0.1 0.0 0.1 0.2 kx (2π/a)-1.5-1-0.5 0 0.5 1 1.5 (f) -0.2 -0.1 0.0 0.1 0.2 FIG.4: (Color online)TheFermisurface inthe kx-kz(ky= 0) andkx-ky(kz= 0) planes with ∆pp=105 meV (a,d), 70 meV (b,e) and 16.7 meV (c,f). The color coding represents the spin expectation of each state, ξ=/angbracketleftµ|Jz|µ/angbracketright. Here, Nh= 3×1020cm−3. the spin-down heavy hole band and the mixing of other spin components in this band is marginal. Therefore, the spin-flip scattering related to this band is weak and can not result in the increase of the hole spin relaxation time mentioned above. By comparing the results with ∆pp= 105 meV and 70 meV, one notices that the spin expectation of the Fermi surface of the outermost band is insensitive to the Zeeman splitting. Therefore, this band can not be the reason of the increase regime also. Moreover, for the second and third bands in Fig.4(a) and (b), from the comparable color coding between the two figures in this regime [also see Fig.4(d) and (e) with kz= 0], one finds that the spin expectation for the states with small kzis also insensitive to the Zeeman splitting. However, for the states with large kz, the spin expectation of the spin-down states ( ξ <0) approaches a large magnitude ( −1.5) with decreasing Zeeman split- ting, suggestingthe decrease ofthe mixing from the spin- up states. As a result, the interband spin-flip scattering from/to these states becomes weak and the hole spin re- laxation time increases. In the decrease regime of the hole spin relaxation time, Fig.4(c) and (f) show that the two outer/inner bands approach each other, leading to a strong and anisotropic spin mixing. Therefore, the spin- flip scattering becomes more efficient in this regime and the spin relaxation time decreases. One may suppose that the nonmonotonic temperature dependence of the hole spin relaxation time can also arise from the varia- tionofthe shapeofthe Fermisurface, accordingtoFig.4. However, this variation itself is not the key of the non- monotonic behavior, because the calculation with this effect but without band mixing can not recover the non- monotonic feature as mentioned in the previous para- graph. For the hole density Nh= 5×1020cm−3, the structures of the Fermi surface at ∆pp= 105 meV are similartothoseinFig.4(b)and(e). Thisexplainstheab- sence of the increase regime for this density in Fig.3(b). Moreover,weshouldpoint outthat the increaseregime7 of the hole spin relaxation time in sample A for Jpp ex= −1 eV is much narrower than that in sample B. The reason lies in the fact of lower effective Mn density in sample A, leading to the smaller maximal Zeeman split- ting∼90 meV, only slightly larger than the crossover value 70 meV at Nh= 3×1020cm−3. As a summary of this part, we find different temper- ature dependences of the hole spin relaxation time due to the different values of effective Mn concentration, hole density and exchange coupling constant Jpp ex. In the case with large coupling constant and high effective Mn con- centration, the interband spin mixing can resultin a non- monotonic temperature dependence of the hole spin re- laxation time. Our results suggest a possible way to esti- mate the exchange coupling constant with the knowledge of itinerant hole density, i.e., by measuring the temper- ature dependence of the hole spin relaxation time. Al- ternatively, the discrepancy between the hole relaxation time from different hole densities in Fig.3(b) suggests that one can also estimate the itinerant hole density if the exchange coupling constant has been measured from other methods. C. Gilbert damping and non-adiabatic torque coefficients Facilitated with the knowledge of the hole spin re- laxation time, we can calculate the coefficients in the LLG equation. According to our previous works,10,17 the Gilbert damping and nonadiabatic spin torque co- efficients can be expressed as α=Jh/[NMn|∝an}b∇acketle{tS∝an}b∇acket∇i}ht|(β+1/β)], (13) and β= 1/(2τs∆pp), (14) respectively. In Eq.(13), Jhrepresents the total equi- librium spin polarization of the itinerant hole gas, i.e., Jh=NhJ0 zwithJ0 zbeing the one defined in Eq.(12) in our study. The average spin polarization of a single Mn ion is given by |∝an}b∇acketle{tS∝an}b∇acket∇i}ht|=SdM(T)/M(0). In Fig.5(a), (c) and (e), the nonadiabatic spin torque coefficients βin sample A-C are plotted as function of temperature with Jpp ex=−0.5 eV and −1.0 eV. Our re- sults in sample C show good agreement with the experi- mental data (plotted as the brown square) in Fig.5(e).23 At low temperature, the value of βis around 0.1 ∼0.3, which is also comparable with the previous theoretical calculation.19Very interestingly, one finds that βsharply increases when the temperature approaches the Curie temperature. This can be easily understood from the pronounced decreases of the spin relaxation time and the Zeeman splitting in this regime [see Figs.1(a) and 3]. By comparing the results with different values of the exchange coupling constant, one finds that βfrom Jpp ex=−1 eV is generally about one half of that ob- tained from Jpp ex=−0.5 eV because of the larger Zeeman 0 0.5 1 1.5 2 2.5 3 20 40 60 80 100 120β T (K)Sample A(a)-0.5 eV, Nh=3×1020 cm-3 5×1020 cm-3 -1.0 eV, Nh=3×1020 cm-3 5×1020 cm-3 0 0.01 0.02 0.03 0.04 20 40 60 80 100 120α T (K)(b) 0 0.5 1 1.5 2 2.5 3 20 40 60 80 100 120 140 160β T (K)Sample B(c) 0 0.01 0.02 0.03 0.04 20 40 60 80 100 120 140 160α T (K)(d) 0 0.5 1 1.5 2 2.5 3 20 40 60 80 100 120β T (K)Sample C(e) 0 0.01 0.02 0.03 0.04 0.05 20 40 60 80 100 120α T (K)(f) FIG. 5: (Color online) βandαas function of temperature withJpp ex=−0.5 eV and −1.0 eV in sample A-C. In (b) and (d), the dots represent the experimental data from ferromag - netic resonance measurement for [001] (brown solid upper tr i- angles), [110] (orange solid circles), [100] (green open sq uares) and [1-10] (black open lower triangles) dc magnetic-field or i- entations (Ref.21). The brown solid square in (e) stands for the experimental result from domain-wall motion measure- ment (Ref.23). splitting. Moreover, one notices that the nonmonotonic temperature dependence of the hole spin relaxation time in Fig.3(b) is not reflected in βdue to the influence of the Zeeman splitting. In all cases, the values of βcan exceed one very near the Curie temperature. The results of the Gilbert damping coefficient from Eq.(13) are shown as curves in Fig.5(b), (d) and (f). The dots in these figures are the reported experimental data from the ferromagnetic resonance along different magnetic-field orientations.21Both the magnitude and thetemperaturedependenceofourresultsagreewellwith the experimental data. From Fig.2, one can conclude that the prefactor in Eq.(13), Jh/(NMn|∝an}b∇acketle{tS∝an}b∇acket∇i}ht|), is almost independent of temperature. Therefore, the temperature dependence of αmainly results from the nonadiabatic spin torque coefficient β. Specifically, αis insensitive to the temperature in the low temperature regime and it gradually increases with increasing temperature due8 to the increase of β. Moreover, we predict that αbe- gins to decrease with increasing temperature once βex- ceeds one. This crossover lying at β≈1 can be expected from Eq.(13). By comparing the results with different values of Jpp ex, one finds that the value of αis robust against the exchange coupling constant in the low tem- perature regime. In this regime, β≪1 and one can sim- plify the expression of the Gilbert damping coefficient as α≈Nh NMnSdJ0 z (τs∆pp). Since the total hole spin polariza- tion is proportional to the Zeeman splitting (see Fig.2) andτsis only weakly dependent on the Zeeman split- ting (see Fig.3) in this regime, the increase of Jpp exdoes not show significant effect on α. However, at high tem- perature, the scenario is quite different. For example, one has the maximum of the Gilbert damping coefficient αm≈Nh 2NMn|/angbracketleftS/angbracketright|J0 z∝Jpp exatβ= 1. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 20 40 60 80 100β T (K)Sample D(a)-0.5 eV, Nh=3×1020 cm-3 5×1020 cm-3 -1.0 eV, Nh=3×1020 cm-3 5×1020 cm-3 0 0.02 0.04 0.06 0.08 0.1 20 40 60 80 100α T (K)(b) FIG. 6: (Color online) βandαas function of temperature by takingNMn= 5×1020cm−3withJpp ex=−0.5 eV and −1.0 eV in sample D. The dots are from ferromagnetic resonance mea- surement (Ref.18) for [001] (brown solid upper triangles) a nd [110] (orange solid circles) dc magnetic-field orientation s. Since the effective Mn concentration of sample D is unavailable as mentioned above, we here take NMn= 5×1020cm−3. The results are plotted in Fig.6. It is seen that the Gilbert damping coefficients from our cal- culation with Jpp ex=−1 eV agree with the experiment very well. As reported, the damping coefficient in this sample is much larger ( ∼0.1) before annealing.18The large Gilbert damping coefficient in the as-grown sample may result from the direct spin-flip scattering between the holes and the random Mn spins, existing in low qual- ity samples. In the presence of this additional spin-flip channel, the hole spin relaxation time becomes shorter and results in an enhancement of αandβ(forβ <1). Moreover, in the low temperature regime, a decrease of the Gilbert damping coefficient was observed by increas- ing temperature,18which is absent in our results. This may originate from the complicated localization or cor- relation effects in such a disordered situation. The quan- titatively microscopic study in this case is beyond the scope of the present work. In addition, one notices that βin Ref.24 was deter- mined to be around 0.01, which is one order of magni- tude smaller than our result. The reason is because of the incorrectparameterused in that work, aspointed outby Adam et al..23 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 140Ass(v) (pJ/m) T (K)Nh=3×1020 cm-3, 1.0m0 1×1020 cm-3, 1.0m0 1×1020 cm-3, 0.5m0 FIG. 7: (Color online) Spin stiffness (vertical spin stiffnes s) coefficient as function of temperature is plotted as curves with (without) symbols. The calculation is carried out with Jpp ex=−0.5 eV in sample E. The effective mass is taken to be 1.0 (0.5) m0as labeled in the figure. The brown solid (from the period of the domains) and open (from the hysteresis cy- cle) squares are the experimental data of spin stiffness from Ref.45. D. Spin stiffness and vertical spin stiffness In this subsection, we calculate the spin stiffness and vertical spin stiffness coefficients according to our previ- ous derivation10 Ass=Nh/[4m∗(1+β2)] (15) and Av ss=Nhβ/[4m∗(1+β2)]. (16) Since the effective mass m∗is a rough description for the anisotropic valence bands in the presence of a large Zee- man splitting, it is difficult to obtain the accurate value of the stiffness coefficients from these formulas. Nev- ertheless, one can still estimate these coefficients with the effective mass taken as a parameter. The results are plotted in Fig.7. By fitting the DOS of the occupied hole states,wefind m∗≈m0, whichisconsistentwiththepre- vious work.31The spin stiffness and vertical spin stiffness coefficients with Nh= 3×1020cm−3(1×1020cm−3) are plotted as the red solid (blue dashed) curves with and without symbols, respectively. The sudden decrease of Assoriginates from the increase of βin the vicinity of the Curie temperature (see Fig.5). Our results are com- parable with the previous theoretical work from 6-band model.26As a comparison, we take m∗= 0.5m0, which is widely used to describe the heavy hole in the low en- ergy regime in the absence of the Zeeman splitting.51 The spin stiffness becomes two times larger. Moreover,9 Av ssis found to present a nonmonotonic behavior in the temperature dependence as predicted by Eq.(16). In Fig.7, we also plot the experimental data of the spin stiffness coefficient from Ref.45. It is seen that these values of Assare comparable with our results and show a decrease as the temperature increases. However, one notices that the experimental data is more sensitive to the temperature especially for those determined from the domain period in the low temperature regime. This may originate from the strong anisotropic interband mixing and inhomogeneity in the real material. In Ref.10, we have shown that the vertical spin stiff- ness can lead to the magnetization rotated around the easy axis within the domain wall structure by ∆ ϕ= (/radicalbig 1+β2−1)/βin the absence of the demagnetization field. For β= 1, ∆ϕ≈0.13π, while ∆ ϕ=β/2→0 for β≪1. As illustrated above, βis always larger than 0.1. Therefore, the vertical spin stiffness can present observ- able modification of the domain wall structure in GaM- nAs system.10 IV. SUMMARY In summary, we theoretically investigate the tempera- ture dependence of the LLG coefficients in ferromagnetic GaMnAs, based on the microscopic calculation of the hole spin relaxation time. In our calculation, we employ the Zener model with the band structure carried out by diagonalizing the 8 ×8 Kane Hamiltonian together with the Zeeman energy due to the sp-dexchange interaction. We find that the hole spin relaxation time can present different temperature dependences, depending on the ef-fective Mn concentration, hole density and exchangecou- pling constant. In the case with high Mn concentra- tion and large exchange coupling constant, the hole spin relaxation time can be nonmonotonically dependent on temperature, resulting from the different interband spin mixings in the large and small Zeeman splitting regimes. These features are proposed to be for the estimation of the exchange coupling constant or itinerant hole density. Bysubstituting the hole relaxationtime, we calculatethe temperature dependence of the Gilbert damping, nona- diabatic spin torque, spin stiffness, and vertical spin stiff- ness coefficients. We obtain the nonadiabatic spin torque coefficient around 0 .1∼0.3 at low temperature, which is consistent with the experiment. As the temperature increases, this coefficient shows a monotonic increase. In the low temperature regime, the Gilbert damping co- efficient increases with temperature, which shows good agreement with the experiments. We predict that the Gilbert damping coefficient can decrease with increasing temperatureoncethenonadiabaticspintorquecoefficient exceed one in the vicinity of the Curie temperature. We also find that the spin stiffness decreases with increasing temperature and the vertical spin stiffness can present a nonmonotonic temperature dependence, similar to the Gilbert damping. 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J. von Bardeleben, L. Theve- nard, A. Lemaˆ ıtre, and C. Gourdon, Phys. Rev. B 82, 041301 (2010). 46Semiconductors , edited by O. Madelung (Springer-Verlag, Berlin, 1987), Vol. 17a. 47R. Winkler, Spin-Orbit Coupling Effects in Two- Dimensional Electron and Hole Systems (Springer, Berlin, 2003). 48J. H. Jiang and M. W. Wu, Phys. Rev. B 79, 125206 (2009). 49J. M. Luttinger, Phys. Rev. 102, 1030 (1956). 50The hole-phonon scattering is rather weak compared with the hole-impurity scattering in this low tempera- tureregime. Therefore thepronouncedtemperaturedepen- dence of the hole spin relaxation time can not be explained by this scattering mechanism. 51K. Leo, W. W. R¨ uhle, and K. Ploog, Phys. Rev. B 38, 1947 (1988).

2011-09-22

We investigate the temperature dependence of the coefficients in the Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs by employing the Zener model. We first calculate the hole spin relaxation time based on the microscopic kinetic equation. We find that the hole spin relaxation time is typically several tens femtoseconds and can present a nonmonotonic temperature dependence due to the variation of the interband spin mixing, influenced by the temperature related Zeeman splitting. With the hole spin relaxation time, we are able to calculate the coefficients in the Landau-Lifshitz-Gilbert equation, such as the Gilbert damping, nonadiabatic spin torque, spin stiffness and vertical spin stiffness coefficients. We find that the nonadiabatic spin torque coefficient $\beta$ is around $0.1\sim 0.3$ at low temperature, which is consistent with the experiment [Adam {\em et al.}, Phys. Rev. B {\bf 80}, 193204 (2009)]. As the temperature increases, $\beta$ monotonically increases and can exceed one in the vicinity of the Curie temperature. In the low temperature regime with $\beta<1$, the Gilbert damping coefficient $\alpha$ increases with temperature, showing good agreement with the experiments [Sinova {\em et al.}, Phys. Rev. B {\bf 69}, 085209 (2004); Khazen {\em et al.}, {\em ibid.} {\bf 78}, 195210 (2008)]. Furthermore, we predict that $\alpha$ decreases with increasing temperature once $\beta>1$ near the Curie temperature. We also find that the spin stiffness decreases with increasing temperature, especially near the Curie temperature due to the modification of the finite $\beta$. Similar to the Gilbert damping, the vertical spin stiffness coefficient is also found to be nonmonotonically dependent on the temperature.

Hole spin relaxation and coefficients in Landau-Lifshitz-Gilbert equation in ferromagnetic GaMnAs

1109.4964v1

arXiv:1605.05063v1 [math.AP] 17 May 2016Simultaneous Identification of Damping Coefficient and Initial Value for PDEs from Boundary Measurement Zhi-Xue Zhaoa,bM.K. Bandab, and Bao-Zhu Guoc,d∗ aSchool of Mathematical Sciences, Tianjin Normal University, Tianjin 300387, China bDepartment of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa cAcademy of Mathematics and Systems Science, Academia Sinica, Beijing 100190, China, dSchool of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Afri ca Abstract In this paper, the simultaneous identification of damping or anti-dam ping coefficient and initial value for some PDEs is considered. An identification algorithm is p roposed based on the fact that the output of system happens to be decomposed into a p roduct of an exponential func- tion and a periodic function. The former contains information of the damping coefficient, while the latter does not. The convergence and error analysis are also d eveloped. Three examples, namely an anti-stable wave equation with boundary anti-damping, th e Schr¨ odinger equation with internal anti-damping, and two connected strings with middle jo int anti-damping, are in- vestigated and demonstrated by numerical simulations to show the effectiveness of the proposed algorithm. Keywords: Identification; damping coefficient; anti-stable PDEs; anti-damping coefficient. AMS subject classifications: 35K05, 35R30, 65M32, 65N21, 15A22. 1 Introduction LetHbe a Hilbert space with the inner product /an}bracketle{t·,·/an}bracketri}htand inner product induced norm /bardbl·/bardbl, and letY=R(orC). Consider the dynamic system in H: /braceleftBigg ˙x(t) =A(q)x(t), x(0) =x0, y(t) =Cx(t)+d(t),(1.1) ∗Corresponding author. Email: bzguo@iss.ac.cn 1whereA(q) :D(A(q))⊂H→His the system operator depending on the coefficient q, which is assumed to be a generator of C0-semigroup Tq= (Tq(t))t∈R+onH,C:H→Yis the admissible observation operator for Tq([20]),x0∈His the initial value, and d(t) is the external disturbance. Various PDE control systems with damping mechanism can be fo rmulated into system (1.1), whereqis the damping coefficient. For a physical system, if the dampi ng is produced by material itself that dissipates the energy stored in system, then the system keeps stable. The identification of damping coefficient has been well considered for distribut ed parameter systems like Kelvin-Voigt viscoelastic damping coefficient in Euler-Bernoulli beam in vestigated in [4], and a more general theoretical framework for various classes of parameter est imation problems presented in [5]. In these works, the inverse problems are formulated as least sq uare problems and are solved by finite dimensionalization. For more revelent works, we can refer t o the monograph [6]. Sometimes, however, the source of instability may arise from the negati ve damping. One example is the thermoacoustic instability in duct combustion dynamics an d the other is the stick-slip instability phenomenon in deep oil drilling, see, for instance, [7] and t he references therein. In such cases, the negative damping will result in all the eigenvalues loca ted in the right-half complex plane, and the open-loop plant is hence “anti-stable” (exponentially stable in negative time) and the qin such kind of system is said to be the anti-damping coefficient. Awidely investigated probleminrecent years isstabilizat ion foranti-stable systemsbyimposing feedback controls. A breakthrough on stabilization for an a nti-stable wave equation was first reached in [19] where a backstepping transformation is prop osed to design the boundary state feedback control. By the backstepping method, [11] general izes [19] to two connected anti-stable strings with joint anti-damping. Very recently, [12, 13] in vestigate stabilization for anti-stable wave equation subject to external disturbance coming throu gh the boundary input, where the sliding mode control and active disturbance rejection cont rol technology are employed. It is worth pointing out that in all aforementioned works, the anti-dam ping coefficients are always supposed to be known. On the other hand, a few stabilization results for anti-stab le systems with unknown anti- damping coefficients are also available. In [16], a full state feedback adaptive control is designed for an anti-stable wave equation. By converting thewave equati on into acascade of two delay elements, an adaptive output feedback control and parameter estimato r are designed in [7]. Unfortunately, no convergence of the parameter update law is provided in the se works. It can be seen in [7, 16] that it is the uncertainty of the anti- damping coefficient that leads to complicated design for adaptive control and parameter upda te law. This comes naturally with the identification of unknown anti-damping coefficient. To the be st of our knowledge, there are few studies on this regard. Our focus in the present paper is on si multaneous identification for both anti-damping (or damping) coefficient and initial value for s ystem (1.1), where the coefficient qis assumed to be in a prior parameter set Q= [q,q] (qorqmay be infinity) and the initial value is supposed to be nonzero. We proceed as follows. In Section 2, we propose an algorithm t o identify simultaneously the 2coefficient and initial value through the measured observati on. The system may not suffer from disturbance or it may suffer from general bounded disturbance . In Section 3, a wave equation with anti-damping term in the boundary is discussed. A Schr¨ odinger equation with internal anti- damping term is investigated in Section 4. Section 5 is devot ed to coupled strings with middle joint anti-damping. In all these sections, numerical simulation s are presented to verify the performance of the proposed algorithms. Some concluding remarks are pre sented in Section 6. 2 Identification algorithm Before giving the main results, we introduce the following w ell known Ingham’s theorem [14, 15, 23] as Lemma 2.1. Lemma 2.1. Assume that the strictly increasing sequence {ωk}k∈Zof real numbers satisfies the gap condtion ωk+1−ωk≥γfor allk∈Z, (2.1) for someγ >0. Then, for all T >2π/γ, there exist two positive constants C1andC2, depending only onγandT, such that C1/summationdisplay k∈Z|ak|2≤/integraldisplayT 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay k∈Zakeiωkt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dt≤C2/summationdisplay k∈Z|ak|2, (2.2) for every complex sequence (ak)k∈Z∈ℓ2, where C1=2T π/parenleftbigg 1−4π2 T2γ2/parenrightbigg , C2=8T π/parenleftbigg 1+4π2 T2γ2/parenrightbigg . (2.3) To begin with, we suppose that there is no external disturban ce in system (1.1), that is, /braceleftBigg ˙x(t) =A(q)x(t), x(0) =x0, y(t) =Cx(t).(2.4) The succeeding Theorem 2.1 indicates that identification of the coefficient qand initial value x0 can be achieved exactly simultaneously without error for A(q) with some structure. Theorem 2.1. LetA(q)in system (2.4) generate a C0-semigroup Tq= (Tq(t))t∈R+and suppose thatA(q)andCsatisfy the following conditions: (i).A(q)has a compact resolvent and all its eigenvalues {λn}n∈N(or{λn}n∈Z) admit the following expansion: λn=f(q)+iµn,···<µn<µn+1<···, (2.5) wheref:Q→Ris invertible, µnis independent of q, and there exists an L>0such that µnL 2π∈Zfor alln∈N. (2.6) (ii). The corresponding eigenvectors {φn}n∈Nform a Riesz basis for H. 3(iii). There exist two positive numbers κandKsuch thatκ≤ |κn| ≤Kfor alln∈N, where κn:=Cφn, n∈N. (2.7) Then both coefficient qand initial value x0can be uniquely determined by the output y(t),t∈[0,T], whereT >2L. Precisely, q=f−1/parenleftBigg 1 Lln/bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−L,T2−L)/parenrightBigg , (2.8) for anyL<T1<T2−L, and x0=1 L/summationdisplay n∈N1 κn/parenleftbigg/integraldisplayL 0y(t)e−λntdt/parenrightbigg φn. (2.9) Proof.Since{φn}n∈Nforms a Riesz basis for H, there exists a sequence {ψn}n∈Nof eigenvectors ofA(q)∗, which is biorthogonal to {φn}n∈N, that is, /an}bracketle{tφn,ψm/an}bracketri}ht=δnm. In this way, we can express the initial value x0∈Hasx0=/summationtext n∈N/an}bracketle{tx0,ψn/an}bracketri}htφn,and the solution of system (2.4) as x(t) =Tq(t)x0=/summationdisplay n∈Neλnt/an}bracketle{tx0,ψn/an}bracketri}htφn. (2.10) By (2.6), there exists an increasing sequence {Kn} ⊂Zsuch that µn=2πKn L, n∈N, (2.11) which implies that {µn}satisfies the following gap condition µn+1−µn=2π(Kn+1−Kn) L≥2π L/definesγ. (2.12) In addition, by the assumption that {φn}forms a Riesz basis for Hand|Cφn|is uniformly bounded with respect to n, it follows from Proposition 2 of [8] or Theorem 2 of [9] that Cis admissible for Tq. So generally, we have y(t) =Cx(t) =/summationdisplay n∈Neλnt/an}bracketle{tx0,ψn/an}bracketri}htCφn=ef(q)t/summationdisplay n∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/definesef(q)tPL(t),(2.13) wherePL(t) =/summationtext n∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφnis well-defined and it follows from (2.11) that PL(t) is a Y-valued function of period L. For anyT2−L>T1>L, /integraldisplayT2 T1|y(t)|2dt=e2f(q)L/integraldisplayT2−L T1−L|y(t)|2dt, (2.14) that is, /bardbly/bardblL2(T1,T2)=ef(q)L/bardbly/bardblL2(T1−L,T2−L). (2.15) To obtain (2.8), we need to show that /bardbly/bardblL2(T1,T2)/ne}ationslash= 0 forT2−T1> L. Actually, it follows from (2.13) that /bardbly/bardbl2 L2(T1,T2)=/integraldisplayT2 T1/vextendsingle/vextendsingle/vextendsingleef(q)tPL(t)/vextendsingle/vextendsingle/vextendsingle2 dt≥C3/integraldisplayT2 T1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay n∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dt, (2.16) 4whereC3= min/braceleftbig e2T1f(q),e2T2f(q)/bracerightbig >0. By Lemma 2.1 and the gap condition (2.12), it follows that forT2−T1>2π γ=L, /integraldisplayT2 T1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay n∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dt≥C1κ2/summationdisplay n∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2, (2.17) where C1=2(T2−T1) π/parenleftbigg 1−L2 (T2−T1)2/parenrightbigg >0 forT2−T1>L. The inequality (2.16) together with (2.17) gives /bardbly/bardbl2 L2(T1,T2)≥C1C3κ2/summationdisplay n∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2. (2.18) Notice that {φn}n∈Nforms a Riesz basis for Hand so does {ψn}n∈NforH, there are two positive numbersM1andM2such that M1/summationdisplay n∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2≤ /bardblx0/bardbl2≤M2/summationdisplay n∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2. (2.19) Combining (2.18) with (2.19) yields /bardbly/bardblL2(T1,T2)≥C/bardblx0/bardbl>0, (2.20) whereC=κ/radicalBig C1C3 M2>0. The identity (2.8) then follows from (2.15). The inequality (2.20) means that system (2.4) is exactly obs ervable for T2−T1> L. So the initial value x0can be uniquely determined by the output y(t),t∈[T1,T2]. We show next how to reconstruct the initial value from the output. Actually, it follows from (2.11) that 1 L/integraldisplayL 0ei(µm−µn)tdt=δnm, (2.21) Hence,/integraldisplayL 0y(t)e−λntdt=/integraldisplayL 0/parenleftBigg/summationdisplay m∈Nei(µm−µn)t/an}bracketle{tx0,ψm/an}bracketri}htCφm/parenrightBigg dt=κnL·/an}bracketle{tx0,ψn/an}bracketri}ht,(2.22) Therefore the initial value x0can be reconstructed by x0=/summationdisplay n∈N/an}bracketle{tx0,ψn/an}bracketri}htφn=1 L/summationdisplay n∈N1 κn/parenleftbigg/integraldisplayL 0y(t)e−λntdt/parenrightbigg φn. (2.23) This completes the proof of the theorem. Remark 2.1. Clearly, (2.8) and (2.9) provide an algorithm to reconstruc tqandx0from the output. It seems that the condition (2.6) is restrictive but it is satisfied by some physical systems discussed in Sections 3-5. Condition (2.6) is only for ident ification of q. For identification of initial value only, this condition can be removed. From numerical st andpoint, the function PL(t) in (2.13) 5can be approximated by the finite series in (2.13) with the firs tNterms for sufficiently large N. Hence condition (2.6) can be relaxed in numerical algorithm to be C1.There exists an Lsuch that: everyµnL 2πis equal to (or close to) some integer for n∈ {1,2,···,N},for some sufficiently large N. Obviously, the relaxed condition C1 can still ensure that PL(t) is close to a function of period L. In this case, some points µnmay be very close to each other and the corresponding Riesz ba sis property of the family of divided differences of exponentials eiµntdeveloped in [1, Section II.4] and [2, 3] can be used. For the third condition, |Cφn| ≤Kimplies that Cis admissible for Tqwhich ensures that the output belongs to L2 loc(0,∞;Y), and|Cφn| ≥κimplies that system (2.4) is exactly observable which ensures the unique determination of the in itial value. It is easily seen from (2.15) that the coefficient qcan always be identified as long as /bardbly/bardblL2(T1,T2)/ne}ationslash= 0 for some time interval [T1,T2], which shows that the identifiability of coefficient qdoes not rely on the exact observability yet approximate observability. Remark 2.2. The condition T2−T1>Lin Theorem 2.1 is only used in application of Ingham’s inequalityin(2.17)toensurethat /bardbly/bardblL2(T1,T2)/ne}ationslash= 0. Inpracticalapplications, however, thiscondition is not always necessary. Actually, any L<T1<T2is applicable in (2.8) as long as /bardbly/bardblL2(T1,T2)/ne}ationslash= 0. Similar remark also applies for Theorem 2.2 below. Remark 2.3. It should be noted that for identification of damping coefficie nt in [4, 5, 6], the distributed observations are always required. In Theorem 2 .1, however, we use only boundary measurement and our identification algorithm utilizes phys ics of the system that the anti-damping coefficient can make the measurement have an exponential term in (2.13). We should also point out that identification of the damping or anti-damping coeffic ientqin Theorem 2.1 does not rely on the knowledge of the initial value. Actually, after qbeing estimated, there are various methods for initial value reconstruction, see, e.g. [18, 21] and the ref erences therein. The idea of the algorithm for reconstruction of the initial value here is borrowed fro m the Riesz basis approach proposed in [21]. Now we come to the system with external disturbance which is i nevitable in many situations. Suppose that system (1.1) is corrupted by an unknown general bounded disturbance d(t) in obser- vation. It should be noted that system (1.1) is supposed to be anti-stable in Theorem 2.2 below whereas in Theorem 2.1, there is no constraint on the stabili ty of system. Theorem 2.2. Suppose that system (1.1) is anti-stable and all the conditi ons in Theorem 2.1 are satisfied. If the inverse of f(q)is continuous and the disturbance d(t)is bounded, i.e. |d(t)| ≤M for someM >0and allt≥0, then for any T2−L>T1>L, lim T1→+∞qT1=q,lim T1→+∞/bardblˆx0T1−x0/bardbl= 0, (2.24) where qT1=f−1/parenleftBigg 1 Lln/bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−L,T2−L)/parenrightBigg , (2.25) 6and ˆx0T1=1 L/summationdisplay n∈N1 κn/parenleftbigg/integraldisplayT1+L T1y(t)e−λntdt/parenrightbigg φn, T1≥0. (2.26) Moreover, for sufficiently large T1, the errors |f(qT1)−f(q)|and/bardblˆx0T1−x0/bardblsatisfy |f(qT1)−f(q)|<4 LM√T2−T1 /bardbly/bardblL2(T1−L,T2−L)−M√T2−T1, (2.27) and /bardblˆx0T1−x0/bardbl ≤CM κ√ Le−f(q)T1for someC >0. (2.28) Proof.Introduce ye(t) =CTq(t)x0=y(t)−d(t) =ef(q)tPL(t), (2.29) wherePL(t) is defined in (2.13). We first show that lim T1→+∞/bardblye/bardblL2(T1,T2)= +∞. (2.30) Since system (1.1) is anti-stable, the real part of the eigen valuesf(q)>0. It then follows from (2.29) that /bardblye/bardbl2 L2(T1,T2)=/integraldisplayT2 T1/vextendsingle/vextendsingle/vextendsingleef(q)tPL(t)/vextendsingle/vextendsingle/vextendsingle2 dt≥e2f(q)T1/integraldisplayT2 T1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay n∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 dt. (2.31) Using the same arguments as (2.16)-(2.20) in the proof of The orem 2.1, we have /bardblye/bardblL2(T1,T2)≥Cef(q)T1/bardblx0/bardbl, (2.32) whereC=κ/radicalBig C1 M2>0. Sincef(q)>0, x0/ne}ationslash= 0,(2.30) holds. Therefore for sufficiently large T1, /bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−L,T2−L)=/bardblye+d/bardblL2(T1,T2) /bardblye+d/bardblL2(T1−L,T2−L)≤/bardblye/bardblL2(T1,T2)+/bardbld/bardblL2(T1,T2) /bardblye/bardblL2(T1−L,T2−L)−/bardbld/bardblL2(T1−L,T2−L).(2.33) Since|d(t)| ≤M, for any finite time interval I, /bardbld/bardblL2(I)=/parenleftbigg/integraldisplay I|d(t)|2dt/parenrightbigg1 2 ≤M/radicalbig |I|, (2.34) where|I|represents the length of the time interval I. Hence /bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−L,T2−L)≤eLf(q)+ε(T1,T2) 1−ε(T1,T2), (2.35) where ε(T1,T2) =M√T2−T1 /bardblye/bardblL2(T1−L,T2−L). (2.36) Similarly, /bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−L,T2−L)≥eLf(q)−ε(T1,T2) 1+ε(T1,T2). (2.37) 7It is clear from (2.30) and (2.36) that lim T1→+∞ε(T1,T2) = 0. This together with (2.35) and (2.37) gives lim T1→+∞/bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−L,T2−L)=eLf(q). (2.38) Sincef−1(q) is continuous, lim T1→+∞qT1=f−1/parenleftBigg 1 Lln lim T1→+∞/bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−L,T2−L)/parenrightBigg =q. We next show convergence of the initial value. Similarly wit h the arguments (2.21)-(2.23) in the proof of Theorem 2.1, we have x0=1 L/summationdisplay n∈N1 κn/parenleftbigg/integraldisplayT1+L T1ye(t)e−λntdt/parenrightbigg φn,∀T1≥0. It then follows from (2.26) that for arbitrary T1≥0, ˆx0T1−x0=1 L/summationdisplay n∈N1 κn/parenleftbigg/integraldisplayT1+L T1d(t)e−λntdt/parenrightbigg φn. (2.39) In view of the Riesz basis property of {φn}, it follows that /bardblˆx0T1−x0/bardbl2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 L/summationdisplay n∈N1 κn/parenleftbigg/integraldisplayT1+L T1d(t)e−λntdt/parenrightbigg φn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 ≤M2 L2κ2e−2f(q)T1/summationdisplay n∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayL 0/parenleftBig d(t+T1)e−f(q)t/parenrightBig e−iµntdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ,(2.40) whereM2>0 is introduced in (2.19). To estimate the last series in (2.4 0), we need the Riesz basis (sequence) property of the exponential system Λ :=/braceleftbig fn=eiµnt/bracerightbig n∈N. There are two cases according to the relation between the sets {Kn}n∈Nintroduced in (2.11) and integers Z: Case 1: {Kn}n∈N=Z, that is, Λ =/braceleftBig ei2nπ Lt/bracerightBig n∈Z. In this case, since/braceleftbig eint/bracerightbig n∈Zforms a Riesz basis forL2[−π,π], Λ forms a Riesz basis for L2[−L 2,L 2]. Case 2: {Kn}n∈N/subsetnoteql Z. In this case, it is noted that the exponential system/braceleftbig eiµnt/bracerightbig n∈Nforms a Riesz sequence in L2[−L 2,L 2]. In each case above, by properties of Riesz basis and Riesz seq uence (see, e.g., [23, p. 32-35, p.154]), there exists a positive constant C4>0 such that /summationdisplay n∈N|(g,fn)|2≤C4/bardblg/bardbl2 L2[−L 2,L 2], (2.41) for allg∈L2[−L 2,L 2], where ( ·,·) denotes the inner product in L2[−L 2,L 2]. We return to the estimation of /bardblˆx0T1−x0/bardbl. By variable substitution of t=L 2−sin (2.40), together with (2.41), we have /bardblˆx0T1−x0/bardbl2≤M2 L2κ2e−2f(q)T1/summationdisplay n∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayL 2 −L 2/bracketleftbigg d/parenleftbigg T1+L 2−s/parenrightbigg ef(q)(s−L 2)e−iµnL 2/bracketrightbigg eiµnsds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ≤M2C4 L2κ2e−2f(q)T1/integraldisplayL 2 −L 2/vextendsingle/vextendsingle/vextendsingle/vextendsingled/parenleftbigg T1+L 2−s/parenrightbigg ef(q)(s−L 2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 ds ≤M2M2C4 Lκ2e−2f(q)T1. 8Therefore, /bardblˆx0T1−x0/bardbl ≤/radicalbigg M2C4 LM κe−f(q)T1, (2.42) which implies that /bardblˆx0T1−x0/bardblwill tend to zero as T1→+∞forf(q)>0. The inequality (2.28) with the positive number C=√M2C4is also concluded. Finally, we estimate |f(qT1)−f(q)|. SettingT1large enough so that ε(T1,T2)<1, it follows from (2.25) and (2.35) that Lf(qT1)≤lneLf(q)+ε(T1,T2) 1−ε(T1,T2)<Lf(q)+2ε(T1,T2) 1−ε(T1,T2). (2.43) Similarly, for T1large enough so that ε(T1,T2)≤1 4, it follows from (2.25) and (2.37) that Lf(qT1)>Lf(q)−4ε(T1,T2) 1+ε(T1,T2). (2.44) Combining (2.43) and (2.44), and setting T1large enough so that ε(T1,T2)≤1 4, we have |f(qT1)−f(q)|<4ε(T1,T2) L. The error estimation (2.27) comes from the fact ε(T1,T2)≤M√T2−T1 /bardbly/bardblL2(T1−L,T2−L)−M√T2−T1. (2.45) We thus complete the proof of the theorem. Remark 2.4. Theorem 2.2 shows that when system (1.1) is anti-stable, the nqT1defined in (2.25) can be regarded as an approximation of the coefficient qwhenT1is sufficiently large. Roughly speaking, the ε(T1,T2) defined in (2.36) reflects the ratio of the energy, in L2norm, of the distur- banced(t) which is an unwanted signal, with the energy of the real outp ut signalye(t). We may regard 1/ε(T1,T2) as signal-to-noise ratio (SNR) which is well known in signa l analysis. Theorem 2.2 indicates that qT1defined in (2.25) is an approximation of the coefficient qwhen SNR is large enough. However, if system (1.1) is stable, i.e.f(q)<0, similar analysis shows that the output will be exponentially decaying oscillation, which implies that the unknown disturbance will account for a large proportion in observation and the SNR can not be to o large. In this case, it is difficult to extract enough useful information from the corrupted obs ervation as that with large SNR. Remark 2.5. Theanti-stability assumptionin Theorem 2.2 is almost nece ssary since otherwise, we may have the case of y(t) =Cx(t)+d(t)≡0 for which we cannot obtain anything for identification. Remark 2.6. It is well known that the inverse problems are usually ill-po sed in the sense of Hadamard, that is, arbitrarily small error in the measureme nt data may lead to large error in solution. Theorem 2.2 shows that if system (1.1) is anti-sta ble, our algorithm is robust against bounded unknown disturbance in measurement data. Actually , similar to the analysis in Theorem 2.2, it can be shown that when system (1.1) is not anti-stable , the algorithm in Theorem 2.1 is also numerically stable in the presence of small perturbations i n the measurement data, as long as the perturbation is relatively small in comparison to the outpu t. Some numerical simulations validate this also in Example 3.1 in Section 3. 93 Application to wave equation In this section, we apply the algorithm proposed in previous section to identification of the anti- damping coefficient and initial values for a one-dimensional vibrating string equation described by ([7, 16]) utt=uxx, 0<x<1, t>0, u(0,t) = 0, ux(1,t) =qut(1,t), t≥0, y(t) =ux(0,t)+d(t), t ≥0, u(x,0) =u0(x), ut(x,0) =u1(x),0≤x≤1,(3.1) wherexdenotes theposition, tthetime, 0 <q/ne}ationslash= 1theunknownanti-dampingcoefficient, u0(x) and u1(x) the unknown initial displacement and initial velocity, re spectively, and y(t) is the boundary measured output corrupted by the disturbance d(t). LetH=H1 E(0,1)×L2(0,1), whereH1 E(0,1) ={f∈H1(0,1)|f(0) = 0}, equipped with the inner product /an}bracketle{t·,·/an}bracketri}htand the inner product induced norm /bardbl(f,g)/bardbl2=/integraldisplay1 0/bracketleftbig |f′(x)|2+|g(x)|2/bracketrightbig dx. Define the system operator A:D(A)(⊂ H)→ Has /braceleftBigg A(f,g) = (g,f′′), D(A) =/braceleftbig (f,g)∈H2(0,1)×H1 E(0,1)|f(0) = 0,f′(1) =qg(1)/bracerightbig ,(3.2) and the observation operator CfromHtoCas C(f,g) =f′(0),(f,g)∈D(A). (3.3) It is indicated in [21] that the operator Agenerates a C0-group on H. Lemma 3.1. [21] LetAbe defined by (3.2) and let q/ne}ationslash= 1. Then the spectrum of Aconsists of all isolated eigenvalues given by λn=1 2ln1+q q−1+inπ, n∈Z, if q> 1, (3.4) or λn=1 2ln1+q 1−q+i2n+1 2π, n∈Z, if0<q<1, (3.5) and the corresponding eigenfunctions Φn(x)are given by Φn(x) =/parenleftbiggsinhλnx λn,sinhλnx/parenrightbigg ,∀n∈Z. (3.6) Moreover, {Φn(x)}n∈Zforms a Riesz basis for H. Lemma 3.2. [21] Let Abe defined by (3.2) and let q/ne}ationslash= 1. Then the adjoint operator A∗ofAis given by /braceleftBigg A∗(v,h) =−(h,v′′), D(A∗) =/braceleftbig (v,h)∈H2(0,1)×H1(0,1)|v(0) = 0,v′(1)+qh(1) = 0/bracerightbig ,(3.7) 10andσ(A∗) =σ(A). The eigenvector Ψn(x)ofA∗corresponding to λnis given by Ψn(x) =/parenleftbiggsinhλnx λn,−sinhλnx/parenrightbigg ,∀n∈Z. (3.8) It is easy to verify that for any n,m∈Z,/an}bracketle{tΦn,Ψm/an}bracketri}ht=δnm.System (3.1) can be written as the following evolutionary equation in H: dX(t) dt=AX(t), t>0, X(0) = (u0,u1), (3.9) whereX(t) = (u(·,t),ut(·,t)), and the solution of (3.9) is given by X(t) =/summationdisplay n∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htΦn. (3.10) Thus y(t) =/summationdisplay n∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}ht+d(t). (3.11) It can be seen from Lemma 3.1 that when q= 1, the real part of the eigenvalues is + ∞, while for 0<q/ne}ationslash= 1, the real part is finite positive. Hence, we suppose 1 /∈Qas usual (see, e.g., [7, 16]), whereQ= [q,q] is the prior parameter set. We takeq∈Q= (1,+∞) as an example to illustrate how to apply the algorithms prop osed in previous section to simultaneous identification for the ant i-damping coefficient qand initial values. The following Corollaries 3.1-3.2 are the direct consequen ces of Theorem 2.1 and Theorem 2.2, respectively, by noticing that for system (3.1), the releva nt function and parameters now are f(q) =1 2lnq+1 q−1, µn=nπ, L= 2, κn= 1. (3.12) Corollary 3.1. Suppose that d(t) = 0in system (3.1). Then both the coefficient qand initial values u0(x)andu1(x)can be uniquely determined by the output y(t),t∈[0,T], whereT >4. Specifically, qcan be recovered exactly from q=/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2) /bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2),2≤T1<T2−2, (3.13) and the initial values u0(x)andu1(x)can be reconstructed from u0(x) =1 2/summationdisplay n∈Z/parenleftbigg/integraldisplay2 0y(t)e−λntdt/parenrightbiggsinhλnx λn, u1(x) =1 2/summationdisplay n∈Z/parenleftbigg/integraldisplay2 0y(t)e−λntdt/parenrightbigg sinhλnx.(3.14) Notice that in (3.14), the observation interval [0 ,2] is the minimal time interval for observation to identify the initial values for any identification algori thm. Corollary 3.2. Suppose that q∈Q= (1,+∞)in system (3.1) and the disturbance is bounded, i.e. |d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−2>T1≥2, lim T1→+∞qT1=q,lim T1→+∞/bardbl(ˆu0T1,ˆu1T1)−(u0,u1)/bardbl= 0, (3.15) 11where qT1=/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2) /bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2), (3.16) and ˆu0T1(x) =1 2/summationdisplay n∈Z/parenleftbigg/integraldisplayT1+2 T1y(t)e−λntdt/parenrightbiggsinhλnx λn, ˆu1T1(x) =1 2/summationdisplay n∈Z/parenleftbigg/integraldisplayT1+2 T1y(t)e−λntdt/parenrightbigg sinhλnx.(3.17) To end this section, we present some numerical simulations f or system (3.1) to illustrate the performance of the algorithm. Example 3.1. The observation with random noises when system (1.1) is stabl e. A simple spectral analysis together with Theorem 2.1 shows t hat Corollary 3.1 is also valid for q∈Q= (−∞,−1). In this example, the damping coefficient qand initial values u0(x),u1(x) are chosen as q=−3, u0(x) =−3sinπx, u 1(x) =πcosπx. (3.18) In this case, the output can be obtained from (3.11) (with d(t) = 0), where the infinite series is approximated by a finite one, that is, {n∈Z}is replaced by {n∈Z|−5000≤n≤5000}. Some random noises are added to the measurement data and we use the se data to test the algorithm proposed in Corollary 3.1. LetT1= 2,T2= 2.5. Then the damping coefficient qcan be recovered from (3.13), and the initial values u0(x) andu1(x) can be reconstructed from (3.14). Table 1 lists the numeric al results for the damping coefficients (the second column in Table 1) and Figure 1(a)-1(c) for the initial values in various cases of noise levels. In Table 1, the absol ute errors of the real damping coefficient and the recovered ones, and the L2-norm of the differences between the exact initial values and t he reconstructed ones are also shown. It is worth pointing out that in reconstruction of the initia l values from (3.14), the infinite series is approximated by a finite one once again, that is, {n∈Z}is replaced by {n∈Z| |n| ≤1000}, which accounts for the zero value of the reconstructed initi al velocity at the left end. This is also the reason that the errors of the initial velocity (the last c olumn in Table 1) are relatively large even if there is no random noise in the measured data. Table 1: Absolute errors with different noise levels Noise Level Recovered qErrors forqErrors foru0(x) Errors for u1(x) 0 -3.0000 9.3259E-15 1.1744E-08 2.2215E-01 1% -2.9994 6.2498E-04 1.1618E-03 2.2748E-01 3% -2.9979 2.0904E-03 3.5604E-03 2.6662E-01 120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3í2.5í2í1.5í1í0.50 xinitial displacementreal u0(x) recovered 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234 xinitial velocityreal u1(x) recovered (a) without random noise0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3.5í3í2.5í2í1.5í1í0.50 xinitial displacementreal u0(x) recovered 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234 xinitial velocityreal u1(x) recovered (b) with 1% random error 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3.5í3í2.5í2í1.5í1í0.50 xinitial displacementreal u0(x) recovered 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234 xinitial velocityreal u1(x) recovered (c) with 3% random error Figure 1: The initial values: initial displacement (upper) and initial velocity (lower) Example 3.2. The observation with general bounded disturbance when syste m (1.1) is anti-stable. The anti-damping coefficient and initial values are chosen as q= 3, u0(x) = 3sinπx, u 1(x) =πcosπx. (3.19) and the observation is corrupted by the bounded disturbance : d(t) = 2sin1 1+t+3cos10t. (3.20) The relevant parameters in Corollary 3.2 are chosen to be T2=T1+3, and let T1be different values increasing from 2 to 10. The corresponding anti-damping coe fficientsqT1recovered from (3.16) are depicted in Figure 2. It is seen that qT1converges to the real value q= 3 asT1increases. Setting T1= 0,3,7 in (3.17) and reconstructing the initial values produce re sults in Figure 2 from which we can see that the reconstructed initial values become clos er to the real ones as T1increases. 132 3 4 5 6 7 8 9 102.9533.053.13.153.2 T1Real and recovered antidamping coefficient 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 101234 xInitial dispacement u0(x) real u0(x) T1=0 T1=3 T1=7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í505 xInitial velocity u1(x) real u1(x) T1=0 T1=3 T1=7 Figure 2: anti-damping coefficient qand initial values u0, u1 4 Application to Schr¨ odinger equation In this section, we consider a quantum system described by th e following Schr¨ odinger equation: ut=−iuxx+qu, 0<x<1, t>0, ux(0,t) = 0, u(1,t) = 0, t≥0, y(t) =u(0,t)+d(t), t ≥0, u(x,0) =u0(x), 0≤x≤1.(4.1) whereu(x,t) is the complex-valued state, iis the imaginary unit, and the potential q>0 andu0(x) are the unknown anti-damping coefficient and initial value, r espectively. LetH=L2(0,1) be equipped with the usual inner product /an}bracketle{t·,·/an}bracketri}htand the inner product induced norm/bardbl·/bardbl.Introduce the operator Adefined by /braceleftBigg Aφ=−iφ′′+qφ, D(A) =/braceleftbig φ∈H2(0,1)|φ′(0) =φ(1) = 0/bracerightbig .(4.2) A straightforward verification shows that such defined Agenerates a C0-semigroup on H. Lemma 4.1. [17] Let Abe defined by (4.2). Then the spectrum of Aconsists of all isolated eigenvalues given by λn=q+i/parenleftbigg n−1 2/parenrightbigg2 π2, n∈N, (4.3) 14and the corresponding eigenfunctions φn(x)are given by φn(x) =√ 2cos/parenleftbigg n−1 2/parenrightbigg πx, n∈N. (4.4) In addition, {φn(x)}n∈Nforms an orthonormal basis for H. System (4.1) can be rewritten as the following evolutionary equation in H: dX(t) dt=AX(t), t>0, X(0) =u0, (4.5) and the solution of (4.5) is given by X(t) =/summationdisplay n∈Neλnt/an}bracketle{tX(0),φn/an}bracketri}htφn. (4.6) Thus y(t) =√ 2/summationdisplay n∈Neλnt/an}bracketle{tu0,φn/an}bracketri}ht+d(t). (4.7) The relevant function and parameters in Theorems 2.1-2.2 fo r system (4.1) are f(q) =q, µn=/parenleftbigg n−1 2/parenrightbigg2 π2, L=8 π, κn=√ 2. Parallel toSection 3, wehavetwocorollaries correspondin gtotheexact observation andobservation with general bounded disturbance, respectively, for syste m (4.1). Here we only list the latter one and the former is omitted. Corollary 4.1. Suppose that q∈Q= (0,+∞)in system (4.1) and the disturbance is bounded, i.e. |d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−8 π>T1>8 π, lim T1→+∞qT1=q,lim T1→+∞/bardblˆu0T1−u0/bardbl= 0, (4.8) where qT1=π 8ln/bardbly/bardblL2(T1,T2) /bardbly/bardblL2(T1−8 π,T2−8 π),8 π<T1<T2−8 π, (4.9) and ˆu0T1(x) =π 8/summationdisplay n∈Z/parenleftBigg/integraldisplayT1+8 π T1y(t)e−λntdt/parenrightBigg cos/parenleftbigg n−1 2/parenrightbigg πx. (4.10) We also give a numerical simulation to test the algorithm pro posed in Corollary 4.1 for system (4.1), where the anti-damping coefficient qand initial value u0(x) are chosen as q= 0.7, u0(x) = sinπx+icosπx, (4.11) and the observation is corrupted by the disturbance d(t) = 2sint 10+t+3icos20t. (4.12) 15The observation can be obtained from (4.7) by a finite series a pproximation, that is, {n∈N} is replaced by {n∈N|n≤5000}. The relevant parameters in Corollary 4.1 are chosen to be T2=T1+1, andT1increasing from 2 .55 to 10. The corresponding anti-damping coefficients qT1 recovered from (4.9) are shown in Figure 3. It is obvious that qT1is convergent to the real value q= 0.7 asT1increases. Setting T1= 0,3,7, respectively, in (4.10), the reconstructed initial valu es are shown in Figure 3 from which it is seen that the errors betw een the reconstructed initial values and the real ones become smaller as T1increases. 3 4 5 6 7 8 9 100.60.650.7 T1Real and recovered antidamping coefficient 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í0.500.511.5 xReal part of initial value Re(u0) T1=0 T1=3 T1=7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í1.5í1í0.500.511.5 xImaginary part of initial value Im(u0) T1=0 T1=3 T1=7 Figure 3: coefficient (upper), real part (middle) and imagina ry part (lower) of initial value u0(x) 5 Application to coupled strings equation In this section, we consider the following two connected ant i-stable strings with joint anti-damping described by utt(x,t) =uxx(x,t),x∈(0,1 2)∪(1 2,1), t>0, u/parenleftBig 1 2−,t/parenrightBig =u/parenleftBig 1 2+,t/parenrightBig , t≥0, ux/parenleftBig 1 2−,t/parenrightBig −ux/parenleftBig 1 2+,t/parenrightBig =qut(1,t), t≥0, u(0,t) =ux(1,t) = 0, t≥0, u(x,0) =u0(x),ut(x,0) =u1(x),0≤x≤1, y(t) =ux(0,t)+d(t), t≥0,(5.1) whereq >0,q/ne}ationslash= 2 is the unknown anti-damping constant. System (5.1) model s two connected strings with joint vertical force anti-damping, see [10, 11 , 22] for more details. 16LetH=H1 E(0,1)×L2(0,1) be equipped with the inner product /an}bracketle{t·,·/an}bracketri}htand its induced norm /bardbl(u,v)/bardbl2=/integraldisplay1 0/bracketleftbig |u′(x)|2+|v(x)|2/bracketrightbig dx, whereH1 E(0,1) =/braceleftbig u|u∈H1(0,1),u(0) = 0/bracerightbig . Then system (5.1) can be rewritten as an evolution- ary equation in Has follows: d dtX(t) =AX(t), (5.2) whereX(t) = (u(·,t),ut(·,t))∈ HandAis defined by A(u,v) = (v(x),u′′(x)), (5.3) with the domain D(A) = (u,v)∈H1(0,1)×H1 E(0,1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(0) =u′(1) = 0, u|[0,1 2]∈H2(0,1 2), u|[1 2,1]∈H2(1 2,1), u′(1 2−)−u′(1 2+) =qv(1 2), ,(5.4) whereu|[a,b]denotes the function u(x) confined to [ a,b]. We assume without loss of generality that the prior paramete r set forqisQ= (2,+∞) since the case for Q= (0,2) is very similar. Lemma 5.1. [22] LetAbe defined by (5.3)-(5.4) and q∈Q= (2,+∞). ThenA−1is compact on Hand the eigenvalues of Aare algebraically simple and separated, given by λn=1 2lnq+2 q−2+inπ, n∈Z. (5.5) The corresponding eigenfunctions Φn(x)are given by Φn(x) = (φn(x),λnφn(x)),∀n∈Z, (5.6) where φn(x) = √ 2 λncoshλn 2sinhλnx, 0<x<1 2, √ 2 λnsinhλn 2coshλn(1−x),1 2<x<1. and{Φn(x)}n∈Zforms a Riesz basis for H. In addition, Agenerates a C0-semigroup on H. Lemma 5.2. [22] LetAbe defined by (5.3)-(5.4) and q∈Q= (2,+∞). Then the adjoint operator A∗ofAis given by A∗(u,v) =−(v,u′′), (5.7) with the domain D(A∗) = (u,v)∈H1(0,1)×H1 E(0,1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(0) =u′(1) = 0, u|[0,1 2]∈H2(0,1 2), u|[1 2,1]∈H2(1 2,1), u′(1 2−)−u′(1 2+) =−qv(1 2) , andσ(A∗) =σ(A). The eigenfunctions Ψn(x)ofA∗corresponding to λnare given by Ψn(x) =/parenleftbig φn(x),−λnφn(x)/parenrightbig ,∀n∈Z. (5.8) 17A direct calculation shows that {Ψn(x)}is biorthogonal to {Φn(x)}. Hence, the solution of (5.2) can be expressed as X(t) =/summationdisplay n∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htΦn. (5.9) Define the observation operator CfromHtoCto be C(f,g) =f′(0),(f,g)∈D(A). (5.10) Then y(t) =/summationdisplay n∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htCΦn+d(t). (5.11) The succeeding Corollary 5.1 is a direct consequence of Theo rem 2.2 by noticing that for system (5.1), the relevant function and parameters now are f(q) =1 2lnq+2 q−2, µn=nπ, L= 2, κn=√ 2coshλn 2, and a simple calculation shows that √ 2 2/bracketleftBigg/parenleftbiggq+2 q−2/parenrightbigg1 4 −/parenleftbiggq−2 q+2/parenrightbigg1 4/bracketrightBigg ≤ |κn| ≤√ 2 2/bracketleftBigg/parenleftbiggq+2 q−2/parenrightbigg1 4 +/parenleftbiggq−2 q+2/parenrightbigg1 4/bracketrightBigg ,∀n∈Z. The corollaries corresponding to Theorem 2.1 are omitted he re. Corollary 5.1. Suppose that q∈Q= (2,+∞)in system (5.1) and the disturbance is bounded, i.e. |d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−2>T1≥2, lim T1→+∞qT1=q,lim T1→+∞/bardbl(ˆu0T1,ˆu1T1)−(u0,u1)/bardbl= 0, where qT1=2/parenleftbig /bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2)/parenrightbig /bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2), (5.12) and ˆu0T1(x) = 1 2/summationdisplay n∈Z/parenleftbigg/integraldisplayT1+2 T1y(t)e−λntdt/parenrightbiggsinhλnx λn, 0<x<1 2, 1 2/summationdisplay n∈Z/parenleftbigg/integraldisplayT1+2 T1y(t)e−λntdt/parenrightbigg tanhλn 2coshλn(1−x) λn,1 2<x<1. ˆu1T1(x) = 1 2/summationdisplay n∈Z/parenleftbigg/integraldisplayT1+2 T1y(t)e−λntdt/parenrightbigg sinhλnx, 0<x<1 2, 1 2/summationdisplay n∈Z/parenleftbigg/integraldisplayT1+2 T1y(t)e−λntdt/parenrightbigg tanhλn 2coshλn(1−x),1 2<x<1.(5.13) As before, we present some numerical simulations for system (5.1) to showcase the effectiveness of the algorithm proposed in Corollary 5.1. The anti-dampin g coefficient qand initial values u0(x) andu1(x) are chosen to be q= 3, u0(x) = sinx, u1(x) = cosx, (5.14) 18and the observation is corrupted by the disturbance d(t) = sint2 10+t+cos10t. (5.15) Theobservation is obtained from(5.11) by afiniteseries app roximation, that is, {n∈N}is replaced by{n∈N| |n| ≤5000}. The relevant parameters in Corollary 5.1 are chosen to be T2=T1+1, and T1increases from 2 to 8. The corresponding anti-damping coeffic ientsqT1recovered from (5.12) are plotted in Figure 4. It can be seen that qT1converges to the real value q= 3 asT1increases. LetT1= 0,3,7 in (5.13), and the reconstructed initial values are shown i n Figure 4, from which it is seen that the reconstructed initial values become clos er to the real ones as T1increases. 2 3 4 5 6 7 82.9533.053.13.15 T1Real and recovered antidamping coefficient 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.81 xInitial dispacement u0(x) real u0(x) T1=0 T1=3 T1=7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í10123 xInitial velocity u1(x) real u1(x) T1=0 T1=3 T1=7 Figure 4: anti-damping coefficient q(upper), initial values u0(middle) and u1(lower) 6 Concluding remarks In this paper, we propose an algorithm to reconstruct simult aneously an anti-damping coefficient and an initial value for some anti-stable PDEs. When the meas ured output is exact, the recovered values are exact whereas if the measured output suffers from bo unded unknown disturbance, the approximated values of the anti-damping coefficient and init ial value can also be obtained. Some numericalexamplesarecarriedouttovalidatetheeffectiven ess ofthealgorithm. Itisverypromising to apply the algorithm presented here to stabilization of an ti-stable systems with an unknown anti- damping coefficient. 19Acknowledgements This work was supported by the National Natural Science Foun dation of China and the National Research Foundation of South Africa. References [1] S.A. Avdonin and S.A. Ivanov, Families of Exponentials: The Method of Moments in Control- lability Problems for Distributed Parameter Systems , Cambridge University Press, 1995. [2] S.A. Avdonin and W. Moran, Ingham-type inequalities and Riesz bases of divided differences, Int. J. Appl. Math. Comput. Sci. , 11(2001), 803–820. [3] S.A. Avdonin and S.A. Ivanov, Riesz bases of exponential s and divided differences, St. Peters- burg Math. J. , 13(2002), 339–351. [4] H.T. Banks and I.G. Rosen, Computational methods for the identification of spatially varying stiffness and damping in beams, Control Theory Adv. Tech. , 3(1987), 1–32. [5] H.T. Banks and K. Ito, A unified framework for approximati on in inverse problems for dis- tributed parameter systems, Control Theory Adv. Tech. , 4(1988), 73–90. [6] H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter System , Birkh¨ auser, Boston, 1989. [7] D. Bresch-Pietri and M. Krstic, Output-feedback adapti ve control of a wave PDE with bound- ary anti-damping, Automatica , 50(2014), 1407–1415. [8] B.Z. Guo and Y.H. Luo, Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator, Systems Control Lett. , 46(2002), 45–65. [9] B.Z. Guo and Y.H. Luo, Riesz basis property of a second-or der hyperbolic system with collo- cated scalar input-output, IEEE Trans. Autom. Control , 47(2002), 693–698. [10] B.Z. Guo and W.D. Zhu, On the energy decay of two coupled s trings through a joint damper, J. Sound Vib. , 203(1997), 447–455. [11] B.Z. Guo and F.F. Jin, Arbitrary decay rate for two conne cted strings with joint anti-damping by boundary output feedback, Automatica , 46(2010), 1203–1209. [12] B.Z. Guo and F.F. Jin, Sliding mode and active disturban ce rejection control to stabilization of one-dimensional anti-stable wave equations subject to d isturbance in boundary input, IEEE Trans. Autom. Control , 58(2013), 1269–1274. [13] B.Z. Guo and F.F. Jin, Output feedback stabilization fo r one-dimensional wave equation sub- ject to boundary disturbance, IEEE Trans. Autom. Control , 60(2015), 824–830. 20[14] A.E. Ingham, Sometrigonometrical inequalities with a pplications in thetheory of series, Math. Z., 41(1936), 367–379. [15] V. Komornik and P. Loreti, Fourier Series in Control Theory , Springer-Verlag, New York, 2005. [16] M. Krstic, Adaptive control of an anti-stable wave PDE, Dyn. Contin. Discrete Impuls. Syst. Ser. A. Math. Anal. , 17(2010), 853–882. [17] M. Krstic, B.Z. Guo, and A. Smyshlyaev, Boundary contro llers and observers for the linearized Schr¨ odinger equation, SIAM J. Control Optim. , 49(2011), 1479–1497. [18] K. Ramdani, M. Tucsnak, and G. Weiss, Recovering the ini tial state of an infinite-dimensional system using observers, Automatica, 46(2010), 1616–1625. [19] A. Smyshlyaev and M. Krstic, Boundary control of an anti -stable wave equation with anti- damping on the uncontrolled boundary, Systems Control Lett. , 58(2009), 617–623. [20] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math. , 65(1989), 17-43. [21] G.Q. Xu, State reconstruction of a distributed paramet er system with exact observability, J. Math. Anal. Appl. , 409(2014), 168–179. [22] G.Q. Xu and B.Z. Guo, Riesz basis property of evolution e quations in Hilbert spaces and application to a coupled string equation, SIAM J. Control Optim. , 42(2003), 966–984. [23] R. M. Young, An Introduction to Nonharmonic Fourier Series , Academic Press, New York, 1980. 21

2016-05-17

In this paper, the simultaneous identification of damping or anti-damping coefficient and initial value for some PDEs is considered. An identification algorithm is proposed based on the fact that the output of system happens to be decomposed into a product of an exponential function and a periodic function. The former contains information of the damping coefficient, while the latter does not. The convergence and error analysis are also developed. Three examples, namely an anti-stable wave equation with boundary anti-damping, the Schr\"odinger equation with internal anti-damping, and two connected strings with middle joint anti-damping, are investigated and demonstrated by numerical simulations to show the effectiveness of the proposed algorithm.

Simultaneous Identification of Damping Coefficient and Initial Value in PDEs from boundary measurement

1605.05063v1

arXiv:1606.09326v2 [cond-mat.mes-hall] 6 Dec 2016Skyrmion dynamics in a chiral magnet driven by periodically varying spin currents Rui Zhu*and Yin-Yan Zhang Department of Physics, South China University of Technolog y, Guangzhou 510641, People’s Republic of China Abstract In this work, we investigated the spin dynamics in a slab of ch iral magnets induced by an alternating (ac) spin current. Periodic trajectories of th e skyrmion in real space are discovered under the ac current as a result of the Magnus and viscous forc es, which originate from the Gilbert damping, the spin transfer torque, and the β-nonadiabatic torque effects. The results are obtained by numerically solving the Landau-Lifshitz-Gilbert equat ion and can be explained by the Thiele equation characterizing the skyrmion core motion. PACS numbers: 75.78.-n, 72.25.-b, 71.70.-d *Corresponding author. Electronic address: rzhu@scut.edu.cn 1I. INTRODUCTION The skyrmion spin texture is a kind of topologically-nontrivial magnet ic vortex formed most typically in the bulk chiral magnets (CMs) and magnetic thin films1–3. In CMs it is believed that the spin-orbit coupling induced Dzyaloshinskii-Moriya int eraction (DMI) gov- erns the spin twisting1. Recently the magnetic skyrmion structure attracts intensive fo cus, both in the fundamental theoretic aspect and in its potential applic ation in the information technology4–7. In the magnetic skyrmion state the emergent electrodynamic effe ct originates from its nontrivial spin topology and gives rise to the topological Hall effect and a remark- able current-driven spin transfer torque effect1,8–14. The so-called skyrmionics makes use of the skyrmion as a memory unit favored by its topologically protect ed long lifetime and ultralow driving current, which is five or six orders smaller than that f or driving a magnetic domain wall1,15. Although the current-driven spin dynamics in the CMs with DMI has be en intensively studied recently, less work of an alternating current (ac) driven s kyrmion dynamics was reported. The skyrmion-motion-induced ac current generation h as been predicted, which shares the reversed effect of our consideration16. In this work, we investigated the ac- spin-current driven skyrmion dynamics with the DMI, Gilbert damping17, adiabatic and nonadiabatic spin torques, and different current profiles taken int o account. Our proposition is inspired by the following several aspects. Firstly, it is both theore tically and technically interesting to know the behavior of a skyrmion when an external ac current is applied. Secondly, ahigh-speedlow-powermodulationofaskyrmionisfavora bleforpotentialmemory processing. Lastly but not least, we noticed the mathematical sign ificance of the solution of the Landau-Lifshitz-Gilbert (LLG) equation of a collinear magnet wit h periodically varying spin-currents applied, in which chaos is observed18,19. The topological property of a spin texture can be described by the surface integral of the solid angle of the unitary spin-field vector n(r). The skyrmion number is so defined as S= 1 4π/integraltext n·/parenleftBig ∂n ∂x×∂n ∂y/parenrightBig d2rcounting how many times the spin field wraps the unit sphere. More specific topological properties of a skyrmion can be considered by a nalyzing its radial and whirling symmetric pattern1,20–22. In the continuum field theory, as a result of topological protection, the skyrmion cannot be generated from a topologically trivial magnetic state such as a ferromagnet or a helimagnet by variation without a topolog ically nontrivial force 2such as a spatially nontrivial spin current11,12, geometrical constriction13, domain wall pair source5,6, the edge spin configuration22, etc., and vise versa. It is predicted by simulation that the skyrmion can be generated from a quasi-ferromagnetic a nd helimagnetic state by external Lorentzian and radial spin current12and that transformation is possible between different topologically-nontrivial states such as that between the domain-wall pair and the skyrmion5,6. The local current flowing from the scanning tunneling microscope t o generate the skyrmion in experiment can be approximated by a radial spin curr ent, which imbues nontrivial topology into the helimagnet11. Also an artificial magnetic skyrmion can be tailored by an external magnetic field with nontrivial geometric distr ibution23. When the boundary geometry of the material is tailored such as by a notch in a long plate, a skyrmion can be generated by a collinear spin current13. In this case, the nontrivial constriction topology contributes to the formation of the skyrmion. The unifor m current can move and rotate a skyrmion without changing its topology15,24. In this work we will show that these topological behaviors of the skyrmion are retained in the spin dynam ics driving by an ac spin current. Almost all kinds of ferromagnetic and vortex spin dynamics can be de scribed by the LLG equation. The behavior of the LLG equation is of importance in both t he physical and mathematical sciences18. It has been shown by previous works that the spin torque effect driven by a periodic varying spin current can be described as well by t he LLG equation with the original time-independent current replaced by the time-de pendent current in the spin torque term18,19. Although chaotic behaviors are predicted in the spatially-uniform a c spin-current driven collinear ferromagnetic spin structure18,19, which is well described by the single-spin LLG equation, no similar phenomenon is reported in a spatia lly-nonuniform spin lattice, the latter of which can be attributed to the relaxation proc esses of the inter-site scattering. Even if some sort of chaotic behavior occurs after a lo ng time of evolution, it is workable to restore the original state by applying a magnetic field a fter some time. The influence of it on the skyrmionics exploitation is not large. In this work , we use a matrix- based fourth-order Runge-Kutta method to solve the LLG equat ion with both the adiabatic and nonadiabatic spin torques taken into account. Analytical solut ion of the generalized Thiele equation1,13,24reproduces our numerical results. 3II. THEORETIC FORMALISM We consider a thin slab of CM modulated by a constant magnetic field an d an ac spin current. The strong DMI makes the material a skyrmion-host. In the continuum approxi- mation, the Hamiltonian of the localized magnetic spin in a CM can be desc ribed as1,12,13 H=−J/summationtext rMr·/parenleftbig Mr+ex+Mr+ey/parenrightbig −D/summationtext r/parenleftbig Mr×Mr+ex·ex+Mr×Mr+ey·ey/parenrightbig −B·/summationtext rMr,(1) withJandDthe ferromagnetic and Dzyaloshinskii-Moriya (DM) exchange energ ies, re- spectively. The dimensionless local magnetic moments Mrare defined as Mr≡ −Sr//planckover2pi1, whereSris the local spin at rand/planckover2pi1is the plank constant divided by 2 π. We assume that the length of the vector |Mr|=Mis fixed, therefore Mr=Mn(r) withn(r) the unitary spin field vector. The unit-cell dimension is taken to be unity. An exte rnal magnetic field Bis applied perpendicular to the slab plane to stabilize the skyrmion confi guration. The Bohr magneton µBis absorbed into Bto have it in the unit of energy. The typical DMI D= 0.18Jis used throughout this work13. This DM exchange strength corresponds to the critical magnetic fields Bc1= 0.0075Jbetween the helical and skyrmion-crystal phases and Bc2= 0.0252Jbetween the skyrmion-crystal and ferromagnetic phases, resp ectively. We adoptB= (0,0,0.01J) in our numerical considerations with J= 1 meV. TheextendedformoftheLLGequationthattakesintoaccountth eDMIandtheadiabatic and nonadiabatic spin torque effects can be expressed in the followin g formula1,12,13,25 dMr dt=−γMr×Beff r+α MMr×dMr dt+pa3 2eM[j(r,t)·∇]Mr −pa3β 2eM2{Mr×[j(r,t)·∇]Mr}.(2) By assuming that the energy of a magnet with the local magnetizatio nMrin a spatially varying magnetic field Beff rhas the form of H=−γ/planckover2pi1/summationtext rMr·Beff r, we have Beff r=−1 /planckover2pi1γ∂H ∂Mr, (3) and therefore the first term in the right hand side of Eq. (2) is −γMr×Beff r=−J /planckover2pi1Mr×/parenleftbig Mr+ex+Mr+ey+Mr−ex+Mr−ey/parenrightbig −1 /planckover2pi1(Mr×B) −D /planckover2pi1Mr× /parenleftbig Mr−ey,z−Mr+ey,z/parenrightbig ex+(Mr+ex,z−Mr−ex,z)ey +/parenleftbig Mr+ey,x−Mr+ex,y−Mr−ey,x+Mr−ex,y/parenrightbig ez .(4) 4The second to the last terms of Eq. (2) sequentially correspond to the effect of the Gilbert damping, the time-dependent spin current j(t) =jesin(ωt)-induced adiabatic and nonadia- batic spin torques, respectively. pmeasures the polarization of the conduction electrons, eis the positive electron charge, and ais the average in-plane lattice constant of the CM. In our considerations the frequency of the ac spin current ωis small enough in comparison of the magnetization evolution rate. Therefore, the spin torques can be satisfactorily described by using the time-dependent current in thestandard torque expres sion, which has been justified by previous studies18,19. Here, the unit of time is set to be t0=/planckover2pi1/J≈6.6×10−13s. A phenomenologically expectedvalueof α= 0.1isusedinafterwardsnumerical considerations. By looking deep into Eqs. (2) and (4), we can make some predictions o f the behavior of the local magnetization. We know that the effect of the magnetic fie ld together with the Gilbert term is to precess the magnetic spin into the direction of the e xternal field. The first term in the right hand side of Eq. (4) is that the effective magnetic fie ld is in the direction of neighboring spins. Therefore the evolution tends to form a ferr omagnet. This contributes to the centripetal force of the magnetization in the direction of Mr×Mr′, which results in the precession of one around the other. The effective field in the DM term in Eq. (4) is −∇×Mrwith unitary lattice constants. The integral counterpart of the c url is/contintegraltext Mr′·dl. Whentheneighboringspinsformaring, theenergyisthelowest, hen cegeneratingaspiraling force to the CM. It helps our understanding if we analogize all the ot her terms in the right hand side of Eq. (2) to the effect of a magnetic field. The local “magn etic field” of the phenomenological Gilbert damping force is proportional to −dMr/dtin the standard linear- response damping form, proportional to the velocity and pointing o ppositely to it. While the local spin is precessing, the direction of Mr×dMr/dtpoints to the precession axis of Mradding a force swaying to that axis. The last two terms are the effec t of the current- induced spin torques. For convenience of interpretation, we discu ss the case that j(r,t) is spatially-uniform and along the x-direction. Then [ j(r,t)·∇]Mr=jx(t)∂Mr/∂x. In the case of the adiabatic torque, this term adds a velocity to Mrmaking it sway to the direction ofMr+ex, andMr+extoMr+2ex, and etc. if jx(t) is positive. Therefore, the complete spin texture moves along the direction of the external spin current like a relay race no matter it is a skyrmion or a domain wall. In a periodic magnetic structure such a s a ferromagnet and a helimagnet the “relay race” goes back to itself and hence no sp in structure movement occurs. Following this physical picture, the local “magnetic field” of the nonadiabatic spin 5torque is along the direction of jx(t)∂Mr/∂x. It exerts a velocity perpendicular to that originates from the adiabatic spin torque. Its result is the motion of the spin texture in the direction perpendicular to the spin current. We have already analyz ed the mechanisms of theLLGequation termby term. However, they affectsthe system collaboratively. While the adiabatic spin torque moves the skyrmion along the spin current, th e Gilbert damping force contributes a velocity in the direction of Mr×dMr/dtandtherefore the effect is a transverse motion of the skyrmion, which is the so-called Hall-like motion12. Also it is noticeable that the transverse velocity resulting from the Gilbert damping and the n onadiabatic spin torque is opposite to each other. In real situations, both αandβare much less than 1. The adiabatic spin torque makes the main contribution to the motion of th e skyrmion. And when the two transverse force is equal, the motion of the skyrmion is straightly along the direction of the spin current. Therefore, periodic trajectories o f the skyrmion in real space can be predicted under the influence of a spatially uniform ac spin cur rent. The previous discussions are well expressed in the Thiele equation de scribing the motion of the center of mass of a skyrmion as1,13,24,26. G×[−j(t)−vd]+κ[−βj(t)−αvd]−∇U(r) =0, (5) wherevd=dR/dt=/parenleftBig ˙X,˙Y/parenrightBig withR= (X,Y) the center of mass coordinates, κis a dimensionless constant of the order of unity, and G= 2πSezis the gyrovector with ezin the direction perpendicular to the CM plane. The minus sign before j(t) is because of that the direction of the motion of conduction electrons is opposite to that o f the current. The Thiele equation (5) describes5coaction of the Magnus force Fg=G×[−j(t)−vd], the viscous forceFv=κ[−βj(t)−αvd], and the confining force Fp=−∇U(r). In our considerations, periodic boundary conditions are used to justify an infinite two-dime nsional model. The applied magnetic field is spatially uniform and the impurity effect is neglec ted. Therefore ∇U≈0. The analytical result of Eq. (5) assuming S=−1 andj(t) =jesin(ωt)excan be obtained as X=αβκ2+4π2 (α2κ2+4π2)ωjecos(ωt), Y=2πκ(β−α) (α2κ2+4π2)ωjecos(ωt).(6) Since the spin current is time dependent, FgandFvinstantaneously change their direc- tion with the motion of the skyrmion core and simultaneously react on the motion of the skyrmion, giving rise to the trigonometric trajectory of the skyrm ion shown in Eq. (6), 6which agrees with the simulation results. Because the skyrmion vort ex moves in the relay fashion under the effect of the spin torque shown by the LLG equat ion, there is a π/2 phase lag between its core motion and the sinusoidally varying spin current. III. NUMERICAL RESULTS AND INTERPRETATIONS By multiplying ˜ α−1with ˜α= 1−α 0−(Mr)z(Mr)y (Mr)z0−(Mr)x −(Mr)y(Mr)x0 , (7) from the left to Eq. (2), the matrix-based Runge-Kutta method is developed. In Figs. 1 to 3, numerical results of our simulations are given. We set M= 1,p= 0.5, anda= 4˚A. The integral step h= 0.1t0is used and its convergence is justified by comparison with the result s ofh= 0.01t0. WithD= 0.18J, the natural helimagnet wavevector Q= 2π/λ=D/J with the diameter of the skyrmion λ=D/J≈35 in the unit of a. A 30×30 square lattice is considered which approximately sustains a single skyrmion. P eriodic boundary condition is used to allow the considered patch to fit into an infinite plan e. While part of the skyrmion moves out of the slab, complementary part enters from t he outside as the natural ground state of a CM is the skyrmion crystal. We use the theoretica lly perfect skyrmion profilen(r) = [cosΦ( ϕ)sinΘ(r),sinΦ(ϕ)sinΘ(r),cosΘ(r)] with Θ( r) =π(1−r/λ) and Φ(ϕ) =ϕin the polar coordinates as the initial state and it would change into a n atural skyrmion in less than one current period. The skyrmion number for t his state S=−1. The spatially-uniform ac spin current is applied in the x-direction as j(t) =jesin(ωt)exwithin the CM plane. Variationofthe skyrmion number intimedriven bythe acspincurrent is shown inFig. 1. Itcanbeseen thatcosinusoidal variationof Soriginatesfromthesinusoidal j(t)withexactly the same period. Fig. 2 shows the snapshots of the spin profile at th e bottoms and peeks of thecosinusoidal variationof SandFig. 3shows thetrajectoriesof thecenter oftheskyrmion (see Ref. 27 for Supplementary Movie). The skyrmion number is a de monstration of the motion pattern of the skyrmion. While the skyrmion moves to one side of the CM slab, only part of a skyrmion is within the view and hence the skyrmion number is r educed. Previous authors have found that the velocity of the skyrmion increases line arly with the increase of 7the current amplitude and that thedynamical threshold current t o move a skyrmion isin the sameorderofthatneededforadomainwall1. Herewehavereobtainedthetwo points. Itcan beseeninFig. 1(a)thatthepeakhight oftheskyrmionnumber incre ases withtheamplitude of the current density and it becomes almost invisible when jeis as small as 1010Am−2. In Fig. 1(b), the evolutions of Sfor different ac periods are shown. The frequency of the ac current is in the order of GHz, which is sufficiently adiabatic as the rat e of the spin dynamics is in the order of 10−12s. We can see that the periodic pattern of Sis better kept with larger amplitudes for smaller ac frequencies. It shows that the phenomen on is a good adiabatic one. Within our numerical capacity, it can be predicted that strong cosinusoidal variation can occur at MHz or smaller ac frequencies, which promises experime ntal realization. The variation of Sis the result of the motion of the skyrmion. The periodic translation o f skyrmion is the result of the coaction of the instantaneous Magnus and viscous forces. The spin current gives rise to the drift velocity of the spin texture. As a combined result of the Gilbert damping, the DMI, the adiabatic and nonadiabatic spin torque s, the skyrmion Hall effect, namely, the transverse motion of the skyrmion perpendicu lar to the spin current, is observed in topologically-nontrivial spin textures. As shown in Fig. 2 , in spite of its motion, the topological properties of the skyrmion are conserved becaus e the initial skyrmion state and the natural skyrmion ground state share similar topology and n o topology-breaking source such as an in-plane magnetic field is present. When part of th e skyrmion moves out of the CM slab, only the remaining part contributes to the skyrmion n umber and hence Sis decreased. The cosinusoidal variation of Sdirectly reflects the oscillating trajectory of the skyrmion Shown in Figs. 2 and 3. We can see that the skyrmion change s from the initial artificial skyrmion state into the natural skyrmion state with S=−1 conserved, as shown in Fig. 2(a) and (b). At the times of integer periods the skyrmion is at the center of the CM slab and at the times of half-integer periods the skyrmion moves to t he left side as shown in Fig. 2 (c) to (f). As predicted by the Thiele equation, the trajectory of the skyrmio n follows a cosinusoidal pattern expressed in Eq. (6). It is interesting that the trajecto ry of the skyrmion results from the competition between the drift motion of the skyrmion and t he skyrmion Hall effect under the influence of the adiabatic and nonadiabatic spin torque eff ects. The adiabatic spin torque effect exerts a force to align the spin at each site to its + x-direction neighbor while j(t) is in the exdirection, which results in the motion of the spin pattern to the + xdirection 8in a relay fashion. The Gilbert damping effect and the nonadiabatic spin torque add a transverse velocity to the moving skyrmion perpendicular to its orig inal velocity. These two forces are in opposite directions when αandβare both positive. Therefore the transverse motion is determined by the sign and relative strength of these two e ffects. From Eq. (6) we can see that when β−α >0 the skyrmion’s y-direction motion is in a cosinusoidal form and when β−α <0 it is in a negative cosinusoidal form. For the x-direction motion of the skyrmion, the direction is the same in the two cases and the magnitud e is slightly smaller for the latter because |4π2| ≫ |αβκ2|holds for all physical parameter settings. And physically it is because the x-direction motion of the skyrmion is mainly determined by the adiabatic spin torque, which is the prerequisite for any motion of the skyrmion . Our simulation results of the skyrmion trajectories for β= 0.5α,α, and 2αwith fixed α= 0.1 are shown in Fig. 3. Good agreement with the prediction by the Thiele equation is obtained. In the three cases, Xevolves cosinusoidally with the initial position ( X,Y) = (15,15) at the center of the CM slab. For β= 0.5α,Yevolves minus-cosinusoidally; for β=α,Yis constant at 15; for β= 2α,Yevolves cosinusoidally. As the difference between βandαis small in Fig. 3 (a) and (c), the cosinusoidal pattern shrinks into a s tep jump. Besides the oscillation, a tiny linear velocity of the skyrmion can be see n in Fig. 3 (a) and (c). And the directions of this velocity are different in the two cases . We attribute this linear velocity to the whirling of the skyrmion from the artificial initial p rofile to the natural profile sustained by the real CM. Because at this whirling step, the G ilbert damping and the adiabatic and nonadiabatic torques are already in effect, the init ial linear velocities are different in the two cases. IV. CONCLUSIONS In this work, we have investigated the dynamics of the skyrmion in a C M driven by periodically varying spin currents by replacing the static current in t he LLG equation by an adiabatic time-dependent current. Oscillating trajectories of t he skyrmion are found by numerical simulations, which are in good agreement with the analyt ical solution of the Thiele equation. In the paper, physical behaviors of the general L LG equation with the Gilbert damping and the adiabatic and nonadiabatic spin torques coex istent are elucidated. Especially, the effect of the nonadiabatic spin torque is interpreted both physically and 9numerically. V. AUTHOR CONTRIBUTION STATEMENT R.Z. wrote the program and the paper. Y.Y.Z. made the simulation. VI. ACKNOWLEDGEMENTS R.Z. would like to thank Pak Ming Hui for stimulation and encouragemen t of the work. This project was supported by the National Natural Science Foun dation of China (No. 11004063) and the Fundamental Research Funds for the Centra l Universities, SCUT (No. 2014ZG0044). 101N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013). 2S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A . Neubauer, R. Georgii, and P. B¨ oni, Science 323, 915 (2009). 3S.Heinze, K. V. Bergmann, M. Menzel, J. Brede, A. Kubetzka, R . Wiesendanger, G. Bihlmayer, and S. Bl¨ ugel, Nat. Phys. 7, 713 (2011). 4X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 9400 (2015). 5X. Xing, P. W. T. Pong, and Y. Zhou, Phys. Rev. B 94, 054408 (2016). 6Y. Zhou and M. Ezawa, Nat. Commun. 5, 4652 (2014). 7J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat . Nanotechnol. 8, 839 (2013). 8K. Hamamoto, M. Ezawa, and N. Nagaosa, Phys. Rev. B 92, 115417 (2015). 9A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Ni klowitz, and P. B¨ oni, Phys. Rev. Lett. 102, 186602 (2009). 10T. Schulz, R. Ritz, A. Bauer, M. Halder, M. Wagner, C. Franz, C . Pfleiderer, K. Everschor, M. Garst, and A. Rosch, Nat. Phys. 8, 301 (2012). 11N. Romming, C. Hanneken, M. Menzel, J. E. Bickel, B. Wolter, K . V. Bergmann, A. Kubetzka, and R. Wiesendanger, Science 341, 636 (2013). 12Y. Tchoe and J. H. Han, Phys. Rev. B 85, 174416 (2012). 13J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotechnol. 8, 742 (2013). 14D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008). 15F. Jonietz, S. M¨ uhlbauer, C. Pfleiderer, A. Neubauer, W. M¨ u nzer, A. Bauer, T. Adams, R. Georgii, P. B¨ oni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Science 330, 1648 (2010). 16S.-Z. Lin, C. D. Batista, C. Reichhardt, and A. Saxena, Phys. Rev. Lett. 112, 187203 (2014). 17K. M. D. Hals and A. Brataas, Phys. Rev. B 89, 064426 (2014). 18M. Lakshmanan, Phil. Trans. R. Soc. A 369, 1280 (2011). 19Z. Yang, S. Zhang, and Y. C. Li, Phys. Rev. Lett. 99, 134101 (2007). 20X. Zhang, Y. Zhou, and M. Ezawa, Phys. Rev. B 93, 024415 (2016). 21A. Bogdanov and A. Hubert, J. Magn. Magn. Mater. 195, 182 (1999). 22Y.-Y. Zhang and R. Zhu, Phys. Lett. A 380, 3756 (2016). 23J. Li, A. Tan, K.W. Moon, A. Doran, M.A. Marcus, A.T. Young, E. Arenholz, S. Ma, R.F. 11Yang, C. Hwang, and Z.Q. Qiu, Nat. Commun. 5, 4704 (2014). 24K. Everschor, M. Garst, B. Binz, F. Jonietz, S. M¨ uhlbauer, C . Pfleiderer, and A. Rosch, Phys. Rev. B86, 054432 (2012). 25I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Piz zini, J. Vogel, and P. Gambardella, Nat. Mater. 9, 230 (2010). 26A. A. Thiele, Phys. Rev. Lett. 30, 230 (1972). 27Supplementary Movie. 12/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s48/s46/s56/s48/s45/s48/s46/s55/s53 /s116/s83/s32/s106 /s101/s61/s48 /s32/s84/s61/s52/s48/s48 /s32/s84/s61/s53/s48/s48 /s32/s84/s61/s54/s48/s48/s40/s98/s41/s83/s32/s106 /s101/s61/s49/s48/s49/s48 /s65/s109/s45/s50 /s32/s106 /s101/s61/s49/s48/s49/s49 /s65/s109/s45/s50 /s32/s106 /s101/s61/s50 /s49/s48/s49/s49 /s65/s109/s45/s50/s40/s97/s41/s32 FIG. 1: Variation of the skyrmion number S in time (a) for differ ent current amplitudes and (b) for different ac current frequencies. The time tand ac spin current period Tare in the unit of t0≈6.6×10−13s.β= 0.05. In panel (a), T= 500. In panel (b), je= 1011Am−2. 13(0,−0.91763) (a) −1−0.500.51(750,−0.77054) (b)(1000,−0.97276) (c) (1250,−0.76819) (d)(1500,−0.97019) (e)(1750,−0.76716) (f) FIG. 2: Snapshots of the dynamical spin configurations at the bottoms and peaks of the skyrmion number shown in Fig. 1. The in-plane components of the magnet ic moments are represented by arrows and their z-components are represented by the color plot. The paramete rs areje= 2×1011 Am−2,T= 500, and β= 0.05. On the top of each panel are the ( t,S) values. 14/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53/s49/s54 /s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53 /s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53 /s32/s88 /s32/s89/s40/s97/s41/s32 /s61/s48/s46/s48/s53 /s40/s98/s41/s32 /s61/s48/s46/s49 /s116/s40/s99/s41/s32 /s61/s48/s46/s50 FIG. 3: Variation of the skyrmion center coordinates ( X,Y) in time (a) for β= 0.05, (b) for β= 0.1, and (c) for β= 0.2. Other parameters are the same as Fig. 2. 15

2016-06-30

In this work, we investigated the spin dynamics in a slab of chiral magnets induced by an alternating (ac) spin current. Periodic trajectories of the skyrmion in real space are discovered under the ac current as a result of the Magnus and viscous forces, which originate from the Gilbert damping, the spin transfer torque, and the $ \beta $-nonadiabatic torque effects. The results are obtained by numerically solving the Landau-Lifshitz-Gilbert equation and can be explained by the Thiele equation characterizing the skyrmion core motion.

Skyrmion dynamics in a chiral magnet driven by periodically varying spin currents

1606.09326v2

arXiv:1303.1192v1 [cond-mat.mtrl-sci] 5 Mar 2013Angle-Dependent Spin-Wave Resonance Spectroscopy of (Ga, Mn)As Films L. Dreher,1,∗C. Bihler,1E. Peiner,2A. Waag,2W. Schoch,3W. Limmer,3S.T.B. Goennenwein,4and M.S. Brandt1 1Walter Schottky Institut, Technische Universit¨ at M¨ unch en, Am Coulombwall 4, 85748 Garching, Germany 2Institut f¨ ur Halbleitertechnik, Technische Universit¨ a t Braunschweig, Hans-Sommer-Straße 66, 38023 Braunschweig, Germany 3Institut f¨ ur Quantenmaterie, Universit¨ at Ulm, 89069 Ulm , Germany 4Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften, Walther-Meißner-Straße 8, 85748 Garching, Germany (Dated: October 31, 2018) A modeling approach for standing spin-wave resonances base d on a finite-difference formulation of the Landau-Lifshitz-Gilbert equation is presented. In c ontrast to a previous study [Bihler et al., Phys. Rev. B 79, 045205 (2009)], this formalism accounts for elliptical ma gnetization precession and magnetic properties arbitrarily varying across the layer t hickness, including the magnetic anisotropy parameters, the exchange stiffness, the Gilbert damping, an d the saturation magnetization. To demonstrate the usefulness of our modeling approach, we exp erimentally study a set of (Ga,Mn)As samples grown by low-temperature molecular-beam epitaxy b y means of angle-dependent stand- ing spin-wave resonance spectroscopy and electrochemical capacitance-voltage measurements. By applying our modeling approach, the angle dependence of the spin-wave resonance data can be re- produced in a simulation with one set of simulation paramete rs for all external field orientations. We find that the approximately linear gradient in the out-of- plane magnetic anisotropy is related to a linear gradient in the hole concentrations of the sample s. PACS numbers: 75.50.Pp, 76.50.+g, 75.70.-i, 75.30.Ds Keywords: (Ga,Mn)As; spin wave resonance; magnetic anisot ropy I. INTRODUCTION Due to their particular magnetic properties, in- cluding magnetic anisotropy,1–3anisotropic magneto- resistance4,5and magneto-thermopower,6in the past years ferromagnetic semiconductors have continued to be of great scientific interest in exploring new physics and conceptual spintronic devices.7–11The most promi- nentferromagneticsemiconductoris(Ga,Mn)As, wherea small percentage of Mn atoms on Ga sites introduces lo- calizedmagneticmomentsaswellasitinerantholeswhich mediate the ferromagneticinteractionofthe Mn spins ( p- dexchange interaction).12Both theoretical and experi- mental studies have shown that the magnetic anisotropy, i.e., the dependence of the free energy of the ferromagnet on the magnetization orientation, depends on the elas- tic strain and the hole concentration in the (Ga,Mn)As layer,12,13openingupseveralpathwaystomanipulatethe magnetic anisotropy of (Ga,Mn)As.14–16 A common spectroscopic method to probe the mag- netic anisotropy of ferromagnets and in particular (Ga,Mn)As, is angle-dependent ferromagnetic resonance (FMR),17–23where FMR spectra are taken as a func- tion of the orientation of the external magnetic field. If the magnetic properties of the ferromagnet are homo- geneous, a zero wave vector ( k= 0) mode of collec- tively, uniformally precessing magnetic moments couples to the microwavemagnetic field, e.g., in a microwavecav- ity, allowing for a detection of the magnetization preces- sion. The resonance field of this mode, referred to as uniform resonance magnetic field, depends on the em- ployedmicrowavefrequency and the magnetic anisotropyparameters. Thus, by recording FMR spectra at differ- ent orientations of the external field with respect to the crystal axes, the anisotropy parameters can be deduced from the experiment. However, if the magnetic prop- erties of a ferromagnetic layer are non-homogeneous or the spins at the surface and interface of the layer are pinned, non-propagating modes with k/negationslash= 0, referred to as standing spin-wave resonances (SWR), can be excited by the cavity field and thus be detected in an FMR ex- periment. On one hand this can hamper the derivation of anisotropy parameters, on the other hand a detailed analysis of these modes can elucidate the anisotropy pro- file of the layer and the nature of spin pinning condi- tions. Furthermore, the excitation of spin waves is of topical interest in combination with spin-pumping,24–27 i.e., the generation of pure spin currents by a precessing magnetization.28–30In this context, the exact knowledge of the magnetization precession amplitude as a function of the position coordinate within the ferromagnet is of particular importance.24 Several publications report on SWR modes in (Ga,Mn)As with a mode spacing deviating from what is expected according to the Kittel model for magnetically homogeneous films with pinned spins at the surface.31–36 These results have been attributed to an out-of-plane anisotropyfieldlinearly31,36orquadraticallyvarying33–35 as a function of the depth into the layer, as well as to specific spin pinning conditions at the surface and at the interface to the substrate.35While most of these studies have focused on the spacings of the resonancefields when modeling SWR measurements, in Ref. 36 a more sophis- ticated approach, based on a normal mode analysis,37,38 was employed to model resonance fields as well as rela-2 tive mode intensities for the external field oriented along high-symmetry directions, assuming a circularly precess- ing magnetization. In this work, we present a more general modeling ap- proach for SWR, based on a finite-difference formulation of the Landau-Lifshitz-Gilbert (LLG) equation. This ap- proach holds for any orientation of the external mag- netic field and accounts for elliptical magnetization pre- cession [Sec. II]. It allows for a simulation of arbitrar- ily varying profiles of the magnetic properties across the thickness of the film, including vatiations of the mag- netic anisotropy parameters, the exchange stiffness, and the Gilbert damping parameter. As the result ofthe sim- ulation, we obtain the Polder susceptibility tensor as a function of the depth within the ferromagnet. Based on this result, the absorbedpowerupon spin waveresonance andthe magnetizationprecessionamplitude asafunction of the depth can be calculated for any orientation of the external magnetic field. We apply our modeling approach to a set of four (Ga,Mn)As samples epitaxially grown with different V/III flux ratios [Sec. III], motivated by the obser- vation that V/III flux ratios of /lessorsimilar3 lead to a gra- dient in the hole concentration p[Ref. 39], which in turn is expected to cause non-homogeneous magnetic anisotropyparameters.31,36Electrochemicalcapacitance- voltage (ECV) measurements revealed a nearly linear gradient in pacross the thickness of the layers investi- gated. To show that our modeling approach is capa- ble of simulating SWR spectra for arbitrary magnetic field orientations, angle-dependent SWR data were taken and compared with the model using one set of magnetic parameters for each sample, revealing gradients in the uniform resonance magnetic fields. We discuss the in- fluence of the gradient in pon the observed uniform resonance field gradients as well as possible influences of strain and saturation magnetization gradients on the observed out-of-plane anisotropy profile. It should be emphasized, however, that the objective of this work is to show the usefulness of our modeling approach, while a detailed investigation of the origin of the gradient in the out-of-plane magnetic anisotropy profile and there- fore a detailed understanding of the particular materials physics of (Ga,Mn)As is beyond the scope of this study. Finally, we summarize our results and discuss further po- tential applications of this work [Sec. IV]. II. THEORETICAL CONSIDERATIONS In this section, we provide the theoretical framework necessary to describe the full angle dependence of the spin-wave resonance spectra presented in Sec. III. Refer- ring to the coordinate system depicted in Fig. 1, we start from the canonical expression for the free enthalpy den- sity (normalized to the saturation magnetization M) forφ0Θ0 123m2 m1m3≈1 m x||[100]y||[010]z||[001] SubstrateFerromag net FIG. 1: (color online) Relation between the two coordinate systems employed. The ( x,y,z) frame of reference is spanned by the cubic crystal axes, while the (1 ,2,3) coordinate sys- tem is determined by the equilibrium orientation of the mag- netization (3-direction) and two transverse directions, t he 2- direction being parallel to the film plane; the latter system is zandµ0Hdependent, as described in the text. a tetragonally distorted (Ga,Mn)As film13,20,40,41 G= const −µ0H·m+B001m2 z+B4⊥m4 z +B4/bardbl(m4 x+m4 y)+1 2B1¯10(mx−my)2.(1) Here,µ0His a static external magnetic field, B001 is a uniaxial out-of-plane anisotropy parameter, re- flecting shape and second-order crystalline anisotropy,13 B4⊥,B4/bardbl, andB1¯10are fourth-order crystalline and second-order uniaxial in-plane anisotropy parameters, respectively;1mx,my,mzdenote the components of the normalized magnetization vector m(z) =M(z)/M(z) along the cubic axes [100], [010], and [001], respectively. We assume the magnetic properties of the layer to be ho- mogeneouslaterally(within the xyplane) and inhomoge- neous vertically (along z); the anisotropy parameters in Eq. (1) and the magnetization are consequently a func- tion of the spatial variable z. To obtain the anisotropy parameters from Eq. (1) in units of energy density, it would therefore be required to know the zdependence and the absolute value of M. The minimum of Eq. (1) determines the equilibrium orientation of the magnetization, given by the angles θ0=θ0(z) andφ0=φ0(z), cf. Fig. 1. To describe the magnetization dynamics, we introduce a new frame of reference (1 ,2,3) shown in Fig. 1, in which the equilib- rium orientation of the magnetization m0coincides with the axis 3. For small perturbations, the magnetization precesses around its equilibrium with finite transverse components of the magnetization mi(i= 1,2) as illus- trated in the inset in Fig. 1. The transformation between the two coordinate systems is given in the Appendix A by Eqs. (A1) and (A2). We write for the (normalized) magnetization3 m= 0 0 1 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright m0+ m1 m2 0 +O(m2 1,m2 2).(2) The evolution ofthe magnetizationunder the influence of an effective magnetic field µ0Heffis described by the LLG equation42,43 ∂tm=−γm×µ0Heff+αm×∂tm,(3) whereγis the gyromagnetic ratio and αa phenomeno- logical damping parameter. The effective magnetic field is given by36 µ0Heff=−∇mG+Ds M∇2M+µ0h(t),(4) where∇m= (∂m1,∂m2,∂m3) isthe vectordifferentialop- eratorwith respect to the componentsof m,Ds= 2A/M is the exchange stiffness with the exchange constant A, ∇2is the spatial differential operator ∇2=∂2 x+∂2 y+∂2 z, andh(t) =h0e−iωtis the externally applied microwave magnetic field with the angular frequency ω;h(t) is ori- ented perpendicularly to µ0H. Since the magnetic prop- erties are independent of xandy, Eq. (3) simplifies to ∂tm=−γm×[−∇mG+Dsm′′+µ0h(t)]+αm×∂tm, (5) withm′′=∂2 zm, neglecting terms of the order of m2 i(for i= 1,2). By definition of the (1 ,2,3) coordinate system, the only non-vanishing component of ∇mGin the equi- librium is along the 3-direction. For small deviations of mfrom the equilibrium we find44 ∇mG= G11m1+G21m2 G12m1+G22m2 G3 , (6) where we have introduced the abbreviations Gi= ∂miG|m=m0andGij=∂mi∂mjG|m=m0; the explicit ex- pressions for these derivatives are given in the Appendix A. Inthefollowing,wecalculatethetransversemagnetiza- tion components assuming a harmonic time dependence mi=mi,0e−iωt. The linearized LLG equation, consider- ing only the transverse components, reads as /parenleftbigg H11H12 H21H22/parenrightbigg/parenleftbigg m1 m2/parenrightbigg −Ds/parenleftbigg m′′ 1 m′′ 2/parenrightbigg =µ0/parenleftbigg h1 h2/parenrightbigg ,(7) wherewe haveintroducedthe abbreviations H11=G11− G3−iαω/γ,H12=H∗ 21=G12+iω/γ, andH22=G22− G3−iαω/γ. We have dropped all terms which are non- linear inmiand products of miwith the driving field. Resonant uniform precession of the magnetization (m′′ i= 0) occurs at the so called uniform resonance fieldµ0Huni(z), which is found by solving the homogeneous (h= 0) equation H11(z)H22(z)−H12(z)H21(z) = 0 ⇔(G11−G3)(G22−G3)−G2 12=/parenleftbiggω γ/parenrightbigg2 (8) forµ0H, neglecting the Gilbert damping ( α= 0). Equa- tion(8)canbeusedtoderiveanisotropyparametersfrom angle-dependent FMR spectra. As extensively discussed by Baselgia et al.44, using Eq. (8) is equivalent to using the method of Smit and Beljers, which employs second derivatives of the free enthalpy with respect to the spher- ical coordinates.41,45,46 To illustrate the role of the uniform resonance field in the context of spin-wave resonances, we consider the spe- cial case where magnetization is aligned along the [001] crystal axis ( θ0= 0), before we deal with the general case of arbitrary field orientations. Neglecting the uniax- ial in-plane anisotropy( B1¯10= 0) since this anisotropyis typically weaker than all other anisotropies,13,41we find G3=−µ0H+2B001+4B4⊥andG11=G22=G12= 0, resulting in the uniform resonance field µ0H001 uni(z) =ω/γ+2B001(z)+4B4⊥(z).(9) To find the eigenmodes of the system, we consider the unperturbed and undamped case, i.e., α= 0 and h= 0 in Eq. (7). With m2=im1= ˜mwe find the spin-wave equation Ds˜m′′+µ0H001 uni(z)˜m=µ0H˜m (10) in agreementwith Ref. 36. The relationofthe anisotropy parameters defined in Ref. 36 to the ones used here is given by B001=K100 eff/M+B1¯10,B1¯10=−K011 u/M, 2B4⊥=−K⊥ c1/M, and 2B4/bardbl=−K/bardbl c1/M. Equation (10) is mathematically equivalent to the one-dimensional time-independent Schr¨ odinger equation, where the uniform resonance field corresponds to the potential, ˜mto the wave function, µ0Hto the energy, and Dsis proportional to the inverse mass. To calculate the actual precession amplitude of the magnetization, the coupling of the eigenmodes of Eq. (10) to the driving field is relevant, which is proportional to the net magnetic moment of the mode.36,38In analogy to a particle in a box, the geometry of the uniform resonance field as well as the boundary conditions determine the resonance fields and the spatial form of the precession amplitude. For the remainder of this work, we assume the spins to exhibit natural freedom at the boundaries of the film, i.e.,∂z˜m= ˜m′= 0 at the interfaces,36,47since these boundary conditions have been shown to describe the out-of-plane SWR data of similar samples well.36 To graphically illustrate the influence of the uniform resonance field on the SWR modes, we consider in Fig. 2 a ferromagnetic layer with a thickness of 50 nm with constant magnetic properties across the layer (a) and with a linearly varying uniform resonance field (c); in4 (a) (b) (c) (d) m (arb. u.)~ m (arb. u.)~ FIG. 2: Simulation to demonstrate the influence of the uni- form resonance field µ0H001 union theSWRmodes for m0||[001], assuming circular precession. In (a), µ0H001 uniis set to be con- stant across the layer, while in (c) it varies linearly (blue , dashed lines), in analogy to a square potential and a trian- gular potential, respectively. The dotted black lines are t he resonance fields, calculated assuming boundary conditions of natural freedom, see text. The solid red lines show the eigen - modes of the system, i.e., the precession amplitude ˜ mof the magnetization; for each mode the dotted line corresponds to ˜m= 0. As can be seen in (a), for a constant uniform reso- nancefieldthefirstmodeoccursattheuniformresonance field and exhibits a constant precession amplitude across the lay er, i.e., an FMR mode. The second and third mode (higher-order modes are not shown) exhibit a non-uniform magnetization profile. In order to couple to the driving field the modes need to have a finite net magnetic moment. As can be seen in (a), the positive and negative areas of the second and third mode are equal, thus these modes are not visible in the SWR spectrum (b). This is in contrast to the case of the linearly varying uniform resonance field (c) where the mode profile is given by Airy functions, which have a nonzero net magnetic moment also for the second and third mode, resulting in a finite SWR intensity of these modes (d). The spectra in (b) and (d) were calculated by integrating over the eigenmodes ˜mand convoluting the square of the result with Lorentzians. both cases we assume Ds= 13 Tnm−2, a similar value as obtained in previous studies.36For these conditions, we numerically solve Eq. (10) by the finite difference method described in the Appendix B1, in orderto obtain the resonance fields (eigenvalues) and the zdependence of the transverse magnetic moments (eigenfunctions). To which amount a mode couples to the driving field is determined by the net magnetic moment of the mode, which is found by integrating ˜ m(z) over the thickness of the film. For the magnetically homogeneous layer, the only mode that couples to the driving field is the uniformprecession mode at µ0H001 uni, since modes of higher order have a zero net magnetic moment [Fig. 2 (a)], resulting in one resonance at the uniform resonance field, cf. Fig. 2 (b). For the non-uniform layer, with µ0H001 uni(z) linearly varying across the film, the mode profile is given by Airy functions31,36,38and various non-uniform modes couple to the driving field, resulting in several spin-wave reso- nances with their amplitude proportional to the square of the net magnetic moment36,38of the corresponding mode, cf. Fig. 2 (c) and (d). We now turn to the general case of arbitrary field ori- entations. Due to the magnetic anisotropy profile, the magnetizationorientationisaprioriunknownandafunc- tion ofzandµ0H. Furthermore, the assumption of a circularly precessing magnetization is not generally jus- tified. To solve Eq. (7) for arbitrary field orientations, we employ a finite difference method as outlined in the Appendix B2. BysolvingEq.(7), weobtainthe zdepen- dent generalized Polder susceptibility tensor ¯ χ(µ0H,z), which relates the transverse magnetization components Mi(z) =M(z)mi(z) with the components of the driving field by /parenleftbigg M1 M2/parenrightbigg = ¯χ(µ0H,z)/parenleftbigg h1 h2/parenrightbigg . (11) Inamicrowaveabsorptionmeasurement, thecomponents Miwhich are out-of-phase with the driving field are de- tected. The absorbed power density is related to the imaginary part of ¯ χ(µ0H,z) and can be calculated by48 P=ωµ0 2z0Im/braceleftbigg/integraldisplay0 −z0/bracketleftbigg/parenleftbigh∗ 1,h∗ 2/parenrightbig ¯χ(µ0H,z)/parenleftbigg h1 h2/parenrightbigg/bracketrightbigg dz/bracerightbigg , (12) wherez0is the thickness ofthe ferromagneticlayer. Note that the position coordinate zis negative in the film, cf. Fig. 1. To obtain an impression of how gradients in differ- ent anisotropy parameters influence the SWR spectra, we plot in Fig. 3 simulated SWR spectra together with the magnetization precession cone as a function of depth in the ferromagnetic layer. We assume a constant sat- uration magnetization (its value is not relevant for the outcome of the simulation), a constant exchange stiff- nessDs= 35 Tnm2unless otherwise specified, α= 0.09, andB001= 90 mT, B4||=−50 mT,B4⊥= 15 mT. In Fig. 3 (a), we assume B001to vary across the layer thickness according to B001(z) =B001−b001×zwith b001=−0.8 mT/nm. Figure 3 (a i) shows the simu- lated SWR spectra calculated by taking the first deriva- tive of Eq. (12) with respect to µ0Hfor different angles ψdefined in the inset in Fig. 3 (c iv). We observe sev- eral SWR modes for µ0H||[001] which become less as µ0His tilted away from [001]. At ψ= 40◦only one mode is visible while for ψ= 0◦we again observe mul- tiple SWR modes. This observation can be understood by considering the uniform resonance fields as a func- tion of the depth for these orientations. In Fig. 3 (a5 90 03060 90 03060 90 03060 SWR Intensi ty (arb . units)ψ (deg.) ψ (deg.) ψ (deg.) 200 600 400 µ0H (mT)ψµ0H [110][001](a i) (a ii) (a iii) (a iv) (b i) (b ii) (b iii) (b iv) (c i) (cii) (ciii) (c iv)Im(m1m2-m1m2) (arb. u.) * * FIG. 3: Atlas illustrating the influence of gradients in the a nisotropy parameters on SWR spectra. In (a) all anisotropy parameters are kept constant with the values given in the tex t, exceptB001which is varied linearly. Correspondingly, in (b) and (c)B4⊥andB4||were varied linearly, respectively. Panels (i) show the firs t derivative of simulations using Eq. (12) with respect toµ0Hand panels (ii)-(iv) show the precession cone Im( m∗ 1m2−m1m∗ 2) in a color plot together with the uniform resonance field µ0Huni(z) (dashed blue lines) at three different external field orient ations; the black dotted lines indicate the resonance field positions of the modes. Panel (a i) additiona lly shows the influence of a linear gradient in the exchange st iffness parameter on the spin-wave spectra, see text for further det ails and discussion. ii)-(a iv), we show the uniform resonance field (dashed blue line) for ψ= 0◦,ψ= 30◦, andψ= 90◦, respec- tively, together with the magnetization precession cone Im(m∗ 1m2−m1m∗ 2) in a contour plot as a function of depth andµ0H. Atψ= 90◦, the uniform resonance field varies strongly across the film, which can be understood by considering Eq. (9). This results in several spin wave modes with their resonance fields indicated by dotted lines. For other field orientations, the formula for the uni- form resonance field can also be derived but results in a longer, more complex equation than Eq. (9). Important in this context is that positive values of B001lead to an increase(decrease) ofthe resonancefield for the magneti- zation oriented perpendicular (parallel) to the film plane, accountingfor the reversedsign ofthe slopesof µ0Huniin Fig. 3 (a ii) and (a iv). Consequently, in between those two extreme cases µ0Hunimust be constant across the layer for some field orientation, in our case for ψ= 30◦, resulting in a single SWR mode, cf. Fig. 3 (a i) and (a iii). In addition to the SWR simulations with constantDs, weplotinFig.3(ai)simulatedSWRspectrawith Ds varying linearly across the film with Ds= 35−65 Tnm2 (blue, dotted lines) and Ds= 35−5 Tnm2(green, dotted lines). A decreasing Dsleads to a decreasing spacing in the modes and vice versa for an increasing Dsas can be seen, e.g., for µ0H||[001]. In Fig. 3 (b), we consider the case where all magnetic parameters are constant with the values given above, ex- ceptB4⊥(z) =B4⊥−b4⊥×zwithb4⊥=−0.4 mT/nm. As evident from Eq. (9), this results in the same slope of µ0Huniforψ= 90◦as in the case above where we varied B001only, cf. Fig. 3 (a iv) and (b iv). In contrast to the casedepictedin(a), however,herefor ψ= 0◦theuniform resonancefield is constant. This can be understood when evaluating the parametersthat enter in the calculation of the uniform resonance field [Eq. (8)]. If mis in the film plane, none of the parameters in Eqs. (A4)-(A6) depends onB4⊥, resulting in a constant uniform resonance field forψ= 0◦. Asmis tilted away from the film plane, B4⊥enters in some of the terms Eqs. (A4)-(A6). As a consequence, µ0Hunivaries, first such that it increases6 [cf. Fig. 3 (b iii)] and finally, such that it decreases as a function of depth [cf. Fig. (b iv)]. Finally, we discuss the case where all parameters are constant except B4||(z) =B4||−b4||×zwithb4||= −0.4 mT/nm [Fig. 3 (c)]. Here, µ0Huniis constant for ψ= 90◦aspredictedbyEq.(9). As mistiltedawayfrom [001] a varying B4||leads to a varying uniform resonance field as shown in Fig. 3 (c ii) and (c iii). Here, a sign reversal of the slope as it was the case in Fig. 3 (a) and (b) does not take place and multiple resonances occur, starting from ψ= 60◦[Fig. 3 (c i)]. III. EXPERIMENTAL RESULTS AND DISCUSSION (Ga,Mn)As samples with a nominal Mn concentration of≈4% were grown on (001)-oriented GaAs substrates by low-temperature molecular-beam epitaxy at a sub- strate temperature of 220◦C using V/III flux ratios of 1.1, 1.3, 1.5, and 3.5, referred to as sample A, B, C, and D, respectively. The layer thickness was 210-280 nm as determined from the ECV measurements, cf. Fig. 4. For samples with V/III flux ratios /lessorsimilar3 a gradient in the hole concentration has been reported,39hence this set of sam- ples was chosen to study the influence of a gradient in p on the out-of-plane magnetic anisotropy. Further details on the sample growth can be found in Refs. 39 and 41. The hole concentration profile of the as-grown (Ga,Mn)As layers were determined by ECV profiling us- ing a BioRad PN4400 profiler with a 250 ml aqueous solution of 2.0 g NaOH+9.3 g EDTA as the electrolyte. For further details on the ECV analysis see Ref. 39. The results of the ECV measurements for the layers investi- gated areshown in Fig. 4(a). Except for the sample with V/III=3.5, they reveala nearlylinearly varying hole con- centration across the layer thickness with different slopes and with the absolute value of the hole concentration at the surface of the layer varying by about 20%. The profiles are reproducible within an uncertainty of about 15%. To investigate the magnetic anisotropy profiles of the samples, weperformedcavity-basedFMRmeasurements, using a Bruker ESP300 spectrometer operating at a mi- crowave frequency of 9.265 GHz ( X-band) with a mi- crowave power of 2 mW at T= 5 K; we used magnetic field modulation at a frequency of 100 kHz and an ampli- tude of3.2mT. Since wearemainlyinterested in the out- of-plane magnetic anisotropy, we recorded spectra for ex- ternalmagneticfieldorientationswithinthe crystalplane spanned by the [110] and [001] crystal axes in 5◦steps, cf. the inset in Fig. 5. For each orientation, the field was ramped to 1 T in order to saturate the magnetization and then swept from 650 mT to 250 mT; the spectra for the samples investigated are shown in Fig. 5. We start by discussing qualitative differences in the spectra. The samples A and B exhibit several pro- nounced resonances for the external field oriented along(a) (b) FIG.4: (a)Theholeconcentration inthedifferent(Ga,Mn)As samples is shown as a function of the depth within the layers as determined by ECV profiling. (b) The uniform resonance fieldsµ0H001 uni(z) for the four samples obtained from the sim- ulations for the out-of-plane orientation of the external fi eld (ψ= 90◦) as a function of the depth. [001], which we attribute to standing spin-wave reso- nances [Fig. 5 (a) and (b)]. For these samples, the [001] direction is the magnetically hardest axis since at this orientation the resonance field of the fundamental spin- wavemode is largerthan at all other orientations. As the external field is rotated into the film plane, the resonance position of this mode gradually shifts to lower field val- ues as expected for a pronounced out-of-plane hard axis. In contrast, the samples C and D exhibit the largest res- onance fields for a field orientation of 50-60◦[Fig. 5 (c) and (d)] pointing to an interplay of second- and fourth- order out-of-plane anisotropy with different signs of the corresponding anisotropy parameters. These samples ex- hibit spin-wave resonances as well, however they are less pronounced than for samples A and B. To quantitatively model the spin-wave spectra we nu- merically solve for each magnetic field orientation the spin-wave equation (7) by the finite difference method as outlined in the Appendix B2. Although this method al- lows for the modeling of the SWR for arbitrary profiles of the anisotropy parameters, the exchange stiffness, the Gilbert damping parameter, and the saturation magne- tization, we assume the parameters to vary linearly as a function of z. This approach is motivated by the lin- ear gradient in the hole concentration, which in first ap- proximation is assumed to cause a linear gradient in the7 ψ[001] [110]µ0Hext SimulationExperiment(a) (b) (d) (c) V/III=1.5 V/III=3.5V/III=1.3 V/III=1.1 FIG. 5: The spin-wave resonance data (dotted, blue lines) ar e shown together with simulations (red, solid lines) using t he numerical procedure described in the text and in the Appendi x B2. The data were obtained as a function of the external magnetic field orientation and magnitude for samples with a V /III flux ratio of (a) 1.1, (b) 1.3, (c) 1.5, and (d) 3.5. The rotation angle ψis defined in the inset and the parameters used for the simulat ions are summarized in Tab. I.8 anisotropy parameters, resulting in the spin-wave reso- nances observed in the samples.31,36In Tab. I, we have summarized the parameters used in the simulation for the different samples. The parameters in capital let- ters denote the value at the surface of the sample while the ones in lower-case letters denote the slope of this parameter; e.g., the zdependence of the second-order, uniaxial out-of-plane anisotropy parameter is given by B001(z) =B001−b001×z. The layer thickness used for the simulationcanbe inferredfromFig. 4(a) andwasde- termined from the ECV data under the assumption that at the position where the hole concentration rapidly de- creases the magnetic properties of the layer abruptly un- dergo a transition from ferromagnetic to paramagnetic. For the simulations, we divided each film into n= 100 layers with constant magnetic properties within each layer. For the gyromagnetic ratio we used γ=gµB//planckover2pi1 withg= 2.21 As result of the simulation we obtain the Polder sus- ceptibility tensor ¯ χ(µ0H,z) and the transverse magneti- zationcomponentsasafunctionof zandµ0H. Addition- ally, weobtainthe zdependenceoftheuniformresonance field by solving Eq. (8) for each field orientation. In an SWR absorption experiment with magnetic field mod- ulation, the obtained signal is proportional to the first derivative of the absorbed microwave power with respect to the magnetic field. Thus, we calculate the absorbed powerusingthesimulatedsusceptibilityandEq.(12) and numerically differentiate the result in order to compare the simulated SWR spectra with the experiment. Addi- tionally, we use a global scaling factor, accounting, e.g., for the modulation amplitude, which is the same for all field orientations, and we multiply all the simulated data with this factor. In Fig. 5, we plot the experimental data together with the simulations using the parameters given in Tab. I, demonstrating that a reasonableagreementbe- tweentheoryandexperimentcanbefoundwithonesetof simulation parameters for all magnetic field orientations for each sample. We will now exemplarily discuss the angle dependence of the SWR spectrum of sample A shown in Fig. 5(a) based on the uniform resonance field and the resulting magnetizationmodeprofileobtainedfromthesimulation. To this end, we plot in Fig. 6 (a)-(c) the magnetization precession amplitude Im( m∗ 1m2−m1m∗ 2) for selected ex- ternal field orientations as a function of depth and ex- ternal magnetic field in a contour plot, together with the corresponding uniform resonance field. In Fig. 6 (d)-(f), we show for each external field orientation a magnifica- tion of the corresponding SWR spectrum together with thesimulation. Notethatincontrasttothenormal-mode approach (Appendix B1) used to calculate the modes in Fig. 2, where the coupling of each mode to the cavity field has to be found by integration, the approach elabo- rated in the Appendix B2 directly yields the transverse magnetization components, already accounting for the coupling efficiency and the linewidth. Further, the ap- proach presented in the Appendix B2 is also valid whenthe differencein the resonancefields oftwomodesis com- parable with or smaller than their linewidth, in contrast to the normal-mode approach38. If the external field is parallel to the surface normal (ψ= 90◦) the uniform resonance field varies by about 350 mT across the film thickness [cf. the dashed line in Fig. 6 (a)], resulting in several well-resolved stand- ing spin-wave modes. The spin-wave resonance fields are plotted as dotted lines in Fig. 6 (a); since the spacing of the resonance fields is larger than the SWR linewidth, the modes are clearly resolved, cf. Fig. 6 (a) and (d). In the simulation two regions with different b001values wereused in orderto reproducethe spacingofthe higher- order spin-wave modes found in the experiment. Using the same slope as in the first 100 nm for the entire layer would lead to a smaller spacing between the third and higher order modes. Instead of defining two regions with different slopes b001, a gradient in the exchange stiffness with positiveslopecouldalsobe usedto modelthe exper- imentally found mode spacing as discussed in the context of Fig. 3. Since the exchange interaction in (Ga,Mn)As is mediated by holes12andpdecreases across the layer, we refrainfrom modeling ourresultswith a positivegradient inDs. Further, the results in Ref. 36 rather point to a negative gradient in Dsin a similar sample. However, a decreasing Mn concentration as a function of the depth could lead to an increase of Ds.34 Finally, we note, since B1¯10= 0 in the simulations, the magnetization precesses circularly for ψ= 90◦and thus Im(m∗ 1m2−m1m∗ 2) = 2sin2τ,49with the precession cone angle τ. For all other orientations, mprecesses elliptically which is accounted for in our simulations. In the simulations of the precession amplitudes, we have assumed an externally applied microwave magnetic field withµ0h= 0.1 mT. At an external field orientation of ψ= 50◦the uni- form resonance field is nearly constant across the layer, and consequently only one SWR mode is observed with an almost uniform magnetization precession across the layer, cf. Fig. 6 (b). The precession amplitude is a mea- sureforthe SWR intensity. While the fundamental mode atψ= 90◦exhibits a larger precession cone at the in- terface, it rapidly decays as a function of the depth, in contrast to the nearly uniform precession amplitude for ψ= 50◦. Since the entire layer contributes to the power absorption, consequently, the SWR mode at ψ= 50◦is more intense than the fundamental mode for ψ= 90◦, which is indeed observed in the experiment [cf. Fig. 6 (d) and (e)]. For the magnetic field within the film plane [ ψ= 0◦, cf.Fig.6(c)], the uniformresonancefieldagainvarieslin- early across the film, however in a less pronounced way than for the out-of-plane field orientation and with an opposite sign of the slope. The sign reversal of the slope can be understood in terms of the uniaxial out-of-plane anisotropy parameter B001: positive values of these pa- rameters lead to an increase (decrease) of the resonance field for the magnetization oriented perpendicular (par-9 TABLE I: Simulation parameters and their zdependence of the samples under study as obtained by fitting t he simulations to the SWR measurements. For the anisotropy parameters the cap ital letters denote the value at the surface of the film and the lower case letters the slope as described in the text. For sam ple A, the first value of b001was used for the first 100 nm and the second one for the remaining layer. In addition to the anisot ropy parameters, the saturation magnetization is also assu med to vary linearly across the layer, while its absolute value is u nknown and not important for the SWR simulations. Sample V/III B001 b001 B4/bardblb4/bardblB4⊥b4⊥Dsα∂M(z) ∂zM(0) (mT) (mT nm) (mT) (mT nm) (mT) (mT nm) (Tnm2) (1 µm) A 1.1 90 -0.1, -0.3 -50 0.05 25 -0.3 35 0.09 -3 B 1.3 130 -0.5 -50 0 0 0 20 0.06 -4 C 1.5 75 -0.4 -55 -0.04 -15 0 40 0.11 -4 D 3.5 91 -0.3 -55 -0.04 -15 0 20 0.09 -3 allel) to the film plane, accounting for the slopes of the uniform resonance fields in Fig. 6. Since the gradient in the uniform resonance field is less pronounced for ψ= 0◦ than forψ= 90◦, the spin-wave modes are not resolved forψ= 0◦, since their spacing is smaller than the SWR linewidth, leading to one rather broad line [cf. Fig. 6 (c) and (f)]. A steeper gradient in B4||in combination with a different Gilbert damping (or with an additional inho- mogeneous damping parameter) and amplitude scaling factor, could improve the agreement of simulation and experiment in the in-plane configuration, as discussed later. A detailed study of the in-plane anisotropy pro- file is however beyond the scope of this work. Given that the presented simulations were obtained with one set of parameters, the agreement of theory and experiment is reasonably good also for the in-plane configuration, since salient features of the SWR lineshape are reproduced in the simulation. Having discussed the angle-dependence of the SWR spectra, we turn to the zdependence of the out-of-plane anisotropy of sample A. Our simulations reveal that it is governed by the zdependence of both B001(z) and B4⊥(z). Assuming only a gradient in B001results in a reasonable agreement of theory and experiment for the external field oriented along [001] and [110], but fails to reproduce the spectra observed for the intermediate field orientations, e.g., ψ= 50◦. This is illustrated by the dashed black line in Fig. 6 (e), which represent simu- lations with a constant B4⊥(z) forψ= 50◦. As can be seen, this simulation produces several spin-wave res- onances, whereas in the experiment only one resonance is present, which is better reproduced by the simulation with bothB001(z) andB4⊥(z) varying across the layer. We will now discuss the anisotropy parameters of all samples. In contrast to sample A, the out-of-plane anisotropy profile of all other samples appears to be gov- erned by a gradient in B001(z). As already discussed qualitatively, the hard axis of the samples is determined by an interplay of B001andB4⊥. For sample A and B B4⊥is positive and zero, leading to an out-of-plane hard axis. Incontrast,sampleCandDexhibitanegative B4⊥, leading to a hard axis between out-of-plane and in-plane. TheB4||parameter is negative and of similar magnitude for all samples.Since the out-of-plane anisotropy profile of sample A is governed by B001(z) andB4⊥(z), a comparison of the out-of-plane anisotropy profile between all samples based on anisotropy parameters is difficult. We therefore com- pare the uniform resonance fields, where both anisotropy parameters enter. As evident from Fig. 6, the strongest influence of the magnetic inhomogeneity of the layers on the uniform resonance fields is observed for the exter- nal field along [001]. To compare the hole concentration profile in Fig. 4 (a) with the anisotropy profile, we there- fore plot in Fig. 4 (b) the zdependence of the uniform resonance field µ0H001 unifor this field orientation. The figure demonstrates that the gradient in µ0H001 uniis cor- related with the gradient in p. For the sample with the strongest gradient in pthe gradient in µ0H001 uniis also most distinct while the samples with a weaker gradient inpexhibit a less pronounced gradient in µ0H001 uni. How- ever, for sample D, exhibiting a nearly constant p, we stillobservestandingspinwaveresonancesfor µ0H||[001] [Fig. 5 (d)], reflected in a slight gradient of µ0H001 uni. This observation suggests that aditionally other mechanisms lead to a variation of the anisotropy profile. One possi- bilitywouldbeagradientintheelasticstrainofthelayer, due to a non-homogeneous incorporation of Mn atoms in the lattice. However, x-ray diffraction measurements of this sample, in combination with a numerical simulation based on dynamic scattering theory, reveal a variation of the vertical strain ∆ εzzas small as 3 ×10−5across the layer. According to the measurements in Ref. 13, such a variation in strain would lead to a variation of the B001 parameter by a few mT only, insufficient to account for the variation of µ0Huniby almost 100 mT across the layer. A more likely explanation seems to be a varia- tion of the saturation magnetization, which should also influence the anisotropy parameters. In the simulation, a non-homogeneous saturation was assumed, potentially explaining also the observed gradient in the anisotropy parameters and therefore in the uniform resonance field. In contrast to the out-of-plane anisotropy parameters, B4||was found to depend only weakly on z, for all sam- ples except sample B where it was constant. Addition- ally,B1¯10, typically of the order of a few mT,13might have an influence and interplay with B4||in determining the in-plane anisotropy. We here however focus on the10 0° (b) (c) (a)Im(m1m2-m1m2) (10-5) * * 0 1.2Im(m1m2-m1m2) (10-5) * * 0 0.3Im(m1m2-m1m2) (10-5) * * 0 0.53 (d) (e) (f)001arb.(a) FIG. 6: Simulated magnetization mode profile and uniform res onance field of sample A. The contour plots show the magneti- zation precession amplitude Im( m∗ 1m2−m1m∗ 2) as a function of the position within the film and the external magnetic field for the external field aligned (a) along [001], (b) at an angle of 50◦with respect to [110] (cf. the inset in Fig. 5) and (c) along [110]. The blue, dashed lines in (a)-(c) show the uniform res onance field, obtained by numerically solving Eq. (8) for eac h given field orientation. The dotted black lines in (a) indicate the resonance magnetic fields. In (d)-(f), a magnification of the data (blue dotted lines) and simulation (red solid lines) from Fi g. 5 (a) is shown using the same scale for all orientations. In (e), a simulation with a different set of parameters is shown for com parison (black, dashed line), see text.11 out-of-plane anisotropy and therefore neglect B1¯10in our simulations. An in-plane rotation of the external field would be required for a more accurate measurement of B4||andB1¯10, but is outside the scope of this work. According to the valence-band model in Ref. 12, an oscillatory behavior of the magnetic anisotropy parame- ters is expected as a function of p. Therefore, depend- ing on the absolute value of p, different values for, e.g., ∂B001/∂pare expected. In particular, there are regions where a anisotropy parameter might be nearly indepen- dent ofpand other regions with a very steep pdepen- dence. Since the absolute value of pis unknown, a quan- titative discussion of the pdependence of the obtained anisotropy parameters based on the model in Ref. 12 is not possible. In addition to p, thep-dexchange integral,12which mayalsovary asa function ofthe depth in a non-homogneous film, also influences the anisotropy parameters,12further complicating a quantitative analy- sis. For all samples, we used a constant exchange stiffness Dsin our modeling. As alluded to above, there is some ambiguity in this assumption, since the exchange stiff- ness and the gradient in the anisotropy both influence the mode spacing. For simplicity, however, we intended to keep as many simulation parameters as possible con- stant. The absolute values obtained for the exchange stiffness agree within a factor of 2 with the ones obtained in previous experiments36,50but are a factor of 2-4 larger than theoretically predicted.51For the reasons discussed above,thereisalargeuncertaintyalsointhederivationof the absolute value of Dsfrom standing spin-wave modes in layers with a gradient in the magnetic anisotropy con- stants. In order to use one parameter set for all field- orientations, the Gilbert damping parameter was as- sumed to be isotropic in the simulations. The modeling of the SWR data could be further improved by assum- ing a non-isotropic damping, its value being larger for µ0H||[110] than for µ0H||[001] [cf. Fig. 5]. This how- ever, only improves the result when assuming a field orientation-dependent scaling factor for the amplitude, which could be motivated, e.g., by the assumption that the microwave magnetic field present at the sample po- sition depends on the sample orientation within the cavity. The absolute values of αdetermined here are comparable with the ones obtained by ultra-fast opti- cal experiments,52but are larger than the typical α= 0.01...0.03 values found by frequency-dependent FMR studies.53,54As already alluded to, inhomogeneous line- broadening mechanisms may play a dominant role,54in particular for as-grown samples.55We therefore assume that the values for αobtained in this study overesti- mate the actual intrinsic Gilbert damping. A frequency- dependent SWR study would be required to determine the intrinsic α. Such a study could possibly also reveal a p-dependent αas theoretically predicted.55In our study, assuming a zdependent αdid not improve the agree- ment between simulation and experiment, corroboratingthe conjecture that inhomogeneous broadening mecha- nisms dominate the linewidth and therefore obscure a possiblezdependence of α. IV. SUMMARY Wehavepresentedafinitedifference-typemodelingap- proach for standing spin-wave resonances based on a nu- merical solution of the LLG equation. With this generic formalism, SWR spectra can be simulated accounting for elliptical magnetization precession, for arbitrary orienta- tionsofthe externalmagneticfield, andforarbitrarypro- files of all magnetic properties, including anisotropy pa- rameters, exchange stiffness, Gilbert damping, and sat- uration magnetization. The approach is applicable not only to (Ga,Mn)As but to all ferromagnets. Four(Ga,Mn)Assamples, epitaxiallygrownwithV/III flux ratios of 1.1, 1.3, 1.5, and 3.5 were investigated by ECV and spin-wave resonance spectroscopy, revealing a correlation of a linear gradient in the hole concentration with the occurrence of standing spin wave resonances, in particularfortheexternalfieldorientedout-of-plane. Us- ing the presented modeling approach, the SWR spectra could be reproduced in a simulation with one parameter set for all external field orientations. The simulation re- sults demonstrate that the profileof the out-of-planeuni- formresonancefieldiscorrelatedwiththeholeconcentra- tion profile. However, our measurements and simulations show,that anon-uniformholeconcentrationprofileisnot the only cause that leads to the observed non-uniform magnetic anisotropy; possibly, a variation in the satura- tion magnetization also influences the anisotropy param- eters. To gain a quantitative understanding of this issue, more samples with known hole concentrations would be required, where both the absolute values and the profiles ofpare varied. Such a study was, however, outside the scope of this work. Besides the modeling of SWR intensities and linewidths, the presented formalism yields the magne- tization precession amplitude as a function of the po- sition within the ferromagnet. It can therefore be used to investigate spin-pumping intensities in (Ga,Mn)As/Pt bilayers.27The spin-pumping signal, detected as a volt- age across the Pt layer, should be proportional to the magnetization precession cone in the vicinity of the (Ga,Mn)As/Pt interface. By measuring the spin- pumping signal as well as the SWR intensities of (Ga,Mn)As/Pt and by using our modeling approach, it should be possible to investigate to which extent a mag- netization mode which is localized at a certain posi- tion within the (Ga,Mn)As layer contributes to the spin- pumping signal.12 Acknowledgments This work was supported by the Deutsche Forschungs- Gemeinschaft via Grant No. SFB 631 C3 (Walter Schot- tky Institut) and Grant No. Li 988/4 (Universit¨ at Ulm). Appendix A: Coordinate Transformation and Free Enthalpy derivatives The transformation between the crystallographiccoor- dinate system ( x,y,z) and the equilibrium system (1,2,3) is given by mx my mz =T m1 m2 m3 , (A1) with T= cosθ0cosφ0−sinφ0sinθ0cosφ0 cosθ0sinφ0cosφ0sinθ0sinφ0 −sinθ00 cos θ0 .(A2) The derivatives of the free enthalpy density Eq. (6) with respect to the magnetization components are G3=∂m3G|m=m0=−µ0H3+2B001cos2θ0 +B1¯10(sinθ0cosφ0−sinθ0sinφ0)2 + 4B4⊥cos4θ0 + 4B4/bardblsin4θ0(cos4φ0+sin4φ0) (A3) G21=G12=∂m1∂m2G|m=m0 = cosθ0(1−2cos2φ0)[B1¯10 + 12B4/bardblsin2θ0cosφ0sinφ0] (A4) G11=∂m1∂m1G|m=m0= 2B001sin2θ0 + 12cos2θ0sin2θ0[B4⊥ +B4/bardbl(cos4φ0+sin4φ0)] +B1¯10cos2θ0(cosφ0−sinφ0)2(A5) G22=∂m2∂m2G|m=m0= 2B1¯10(sinφ0+cosφ0)2 + 24B4/bardblsin2θ0cos2φ0sin2φ0. (A6)Appendix B: Finite Difference Method In this Appendix, we describe how the spin-waveequa- tion can be numerically solved by the finite difference method. We start with the simple case of a circulary pre- cessing magnetization, neglecting Gilbert damping and the driving field (Sec. B1). Then we turn to the gen- eral case, where the magnetization precesses elliptically and the Gilbert damping as well as the driving field are included (Sec. B2). 1. The One-Dimensional, Homogeneous, Undamped Case Here, we describe how the resonance fields and the spin-wave modes can be found, assuming a circularly precessing magnetization m2=im1= ˜m, a constant ex- changestiffness, and a zindependent equilibrium magne- tization. This case has been considered in Ref. 36 using a semi-analytical approach to solve the spin-wave equation Eq. (10). The approach considered here, is slightly more general, as it is straight forward to determine resonance fields and eigenmodes of the system for an arbitrary z dependence of the uniform resonance field. To solve Eq. (10), we divide the ferromagnetic film into a finite numbernof layers with equal thickness land constant magnetic properties within each of these layers. The z dependence of ˜ mandµ0H001 uniis thus given by an index j= 1...n. Within each of these layers the uniform reso- nance field and ˜ m(z) are thus constant and given by the valuesµ0H001,j uni=:Kjand ˜mj, respectively. The second derivative of ˜ mis approximated by ˜m′′(z=j·l)≈˜mj−1−2˜mj+ ˜mj+1 l2.(B1) Consequently, Eq. (10) is converted to the homogeneous equation system ......... ... Kj−1+2d−d 0... ...−d Kj+2d−d ... ...0 −d Kj+1+2d ... ......... ... ˜mj−1 ˜mj ˜mj+1 ... =µ0H ... ˜mj−1 ˜mj ˜mj+1 ... , (B2) with the abbreviation d=−Ds/l2. The boundary condi- tion of natural freedom36(von Neumann boundary con-dition) reads as ˜ m0= ˜m1and ˜mn−1= ˜mnand can be incorporated in Eq. (B2). Since the matrix on the left13 hand side of Eq. (B2) is sparse, it can be efficiently diag- onalized numerically, yielding the resonancefields (eigen- values) and the corresponding modes (eigenvectors). Af- ter diagonalizingthe matrix, the relevantresonancefields are found by sorting the eigenvalues and considering only the modes with positive resonance fields, corresponding to the bound states in the particle-in-a-box analogon. The SWR amplitude of each mode is proportional to its net magnetic moment; thus, the amplitudes can be found by integrating the (normalized) eigenmodes. The mode profile, the resonance fields, and the SWR intensities are illustrated in Fig. 2 for a constant and a linearly varying uniform resonance field. The finite linewidth of the SWR modes can be accounted for by assuming a Lorentzian lineshape for each mode with a certain linewidth and with the resonance fields and intensities calculated as described above36. Note that this approach to derive resonance fields and intensities is only valid if the mode separation is large compared with the linewidth of the modes; this restriction does not apply to the model pre- sented in the Appendix B2. 2. The General Case TosolveEq.(7)forarbitrary µ0Handarbitrarilyvary- ing magnetic properties, we again divide the ferromag-netic film into a finite number nof layers with equal thicknessland constant magnetic properties within each of these layers. In contrast to the case in the Appendix B1, where only the uniform resonance field was varied across the layer, here potentially all magnetic proper- ties entering Eq. (7) can be assumed to be zdependent. Additionally, the components of the driving field µ0hi (i= 1,2), can also vary as a function of z, since the (1,2,3) frame of reference is zdependent and thus the projections of the driving field have to be calculated for each layer. The zdependence of the components mi (i= 1,2), of the parameters H11,H12,H21,H22(de- fined in Sec. II) and the exchange stiffness is thus given by the index j= 0...n; the second derivative of each of the components miis approximated as in Eq. (B1). The linearized LLG equation Eq. (7), is thus converted into the inhomogeneous equation system .................. ...Hj−1 11−2dj−1Hj−1 12dj−10 0 0 ... ... Hj−1 21Hj−1 22−2dj−10dj−10 0 ... ... dj0Hj 11−2djHj 12dj0... ... 0 djHj 21Hj 22−2dj0 dj... ... 0 0 dj+10Hj+1 11−2dj+1Hj+1 12... ... 0 0 0 dj+1Hj+1 21Hj+1 22−2dj+1... .................. ... mj−1 1 mj−1 2 mj 1 mj 2 mj+1 1 mj+1 2... =µ0 ... hj−1 1 hj−1 2 hj 1 hj 2 hj+1 1 hj+1 2... ,(B3) with the abbreviation dj=−Dj s/l2. At the boundaries of the magnetic film we again assume the spins to exhibit natural freedom m0 i=m1 iandmn i=mn+1 i. To simulate a spin-wave spectrum for a given orienta- tion of the external field and a given profile of the mag- netic properties, we numerically sweep the magnetic fieldand calculate the equilibrium magnetization orientation for all indices j= 0...nat a given external field. The inverse of the matrix in Eq. (B3), multiplied by µ0M(z), is the generalized Polder susceptibility tensor ¯ χ(µ0H,z), which relatesthe transversemagnetization with the driv- ing field, cf. Eq. (11). ∗Electronic address: dreher@wsi.tum.de 1J. Zemen, J. Kucera, K. Olejnik, and T. 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2013-03-05

A modeling approach for standing spin-wave resonances based on a finite-difference formulation of the Landau-Lifshitz-Gilbert equation is presented. In contrast to a previous study [Bihler et al., Phys. Rev. B 79, 045205 (2009)], this formalism accounts for elliptical magnetization precession and magnetic properties arbitrarily varying across the layer thickness, including the magnetic anisotropy parameters, the exchange stiffness, the Gilbert damping, and the saturation magnetization. To demonstrate the usefulness of our modeling approach, we experimentally study a set of (Ga,Mn)As samples grown by low-temperature molecular-beam epitaxy by means of electrochemical capacitance-voltage measurements and angle-dependent standing spin-wave resonance spectroscopy. By applying our modeling approach, the angle dependence of the spin-wave resonance data can be reproduced in a simulation with one set of simulation parameters for all external field orientations. We find that the approximately linear gradient in the out-of-plane magnetic anisotropy is related to a linear gradient in the hole concentrations of the samples.

Angle-Dependent Spin-Wave Resonance Spectroscopy of (Ga,Mn)As Films

1303.1192v1

Gilbert damping of magnetostatic modes in a yttrium iron garnet sphere S. Klingler,1, 2,a)H. Maier-Flaig,1, 2C. Dubs,3O. Surzhenko,3R. Gross,1, 2, 4H. Huebl,1, 2, 4 S.T.B. Goennenwein,1, 5, 6and M. Weiler1, 2 1)Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2)Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany 3)INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany 4)Nanosystems Initiative Munich, 80799 Munich, Germany 5)Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden, Germany 6)Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden, Germany (Dated: 8 December 2016) The magnetostatic mode (MSM) spectrum of a 300 m diameter single crystalline sphere of yttrium iron garnet is investigated using broadband ferromagnetic resonance (FMR). The individual MSMs are identied via their characteristic dispersion relations and the corresponding mode number tuples ( nmr) are assigned. Taking FMR data over a broad frequency and magnetic eld range allows to analyze both the Gilbert damping parameter and the inhomogeneous line broadening contribution to the total linewidth of the MSMs separately. The linewidth analysis shows that all MSMs share the same Gilbert damping parameter = 2:7(5)105irrespective of their mode index. In contrast, the inhomogeneous line broadening shows a pronounced mode dependence. This observation is modeled in terms of two-magnon scattering processes of the MSMs into the spin-wave manifold, mediated by surface and volume defects. The ferrimagnetic insulator yttrium iron garnet (YIG) has numerous applications in technology and funda- mental research due to its low intrinsic Gilbert damp- ing and large spin-wave propagation length.1It is used as prototypical material in various experiments in spin electronics2{4and spin caloritronics5,6and is indispens- able for microwave technology. Recently, YIG spheres attracted attention in the eld of quantum information technology.7{15For exam- ple, strong coupling between magnons and photons in YIG/cavity hybrid systems can be employed for the up- and down-conversion of quantum signals between mi- crowave and optical frequencies, enabling a long-range transmission of quantum information between microwave quantum circuits.14{16Here, the damping of the mag- netic excitation plays a crucial role, since it limits the time-scale in which energy and information is exchanged and stored in the magnon-photon hybrid system. One type of magnetic excitations in YIG spheres17{19 are magnetostatic modes (MSMs) which resemble stand- ing spin-wave patterns within the sphere. Although the linewidth of MSMs in YIG spheres has been studied at xed frequencies in the past,20{22the respective contri- butions of intrinsic Gilbert damping and inhomogeneous line broadening23to the total linewidth have not yet been investigated. In particular, it is not evident from the literature, whether dierent MSMs feature the same or dierent Gilbert damping.24,25 Here, we report on the study of dynamic properties of multiple MSMs for a 300 m diameter YIG sphere us- ing broadband ferromagnetic resonance. The frequency a)Electronic mail: stefan.klingler@wmi.badw.deand magnetic eld resolved FMR data allows to separate Gilbert damping and inhomogeneous line broadening of the MSMs. One and the same Gilbert damping parame- ter= 2:7(5)105is found for all MSMs, independent of their particular mode index. However, the inhomoge- neous line broadening markedly diers between the ob- served MSMs. This nding is attributed to two-magnon scattering processes of the MSMs into the spin-wave man- ifold, mediated by surface and volume defects. The MSM proles and eigenfrequencies of a magnetic sphere can be calculated in the magnetostatic approx- imationrH= 0,17{19using the Landau-Lifshitz- Gilbert equation (LLG).26,27The resonance frequencies of the MSMs are obtained by solving the characteristic equation:17{19 n+ 1 +0dPm n(0)=d0 Pmn(0)m= 0; (1) where2 0= 1 + 1=,= H= 2 H 2
,= = 2 H 2
, H=0Hi=0Msand =!=  0Ms. Here, =gJB=~is the gyromagnetic ratio, gJis the Land eg-factor,Bis the Bohr magneton, ~is the reduced Planck constant, 0is the vacuum permeability and Ms is the saturation magnetization. The angular frequency of the applied microwave eld is denoted as != 2f. The internal eld is given by Hi=H0+Hani+Hdemag , whereH0is the applied static magnet eld, Haniis the anisotropy eld, and Hdemag =Ms=3 is the demagneti- zation eld of a sphere. The mode proles of the MSMs have the form of asso- ciated Legendre polynomials Pm n, where the localization of the MSMs at the surface is related to the mode index n2N.21The indexjmjncorresponds to an angular- momentum quantum number of the MSM,28where thearXiv:1612.02360v1 [cond-mat.mtrl-sci] 7 Dec 20162 bar above the mode index mis used for indices m < 0. The index r0 enumerates the solutions of the char- acteristic equation (1) for given nandmfor increasing frequencies.18,29In total, each MSM is uniquely identi- ed by the index tuple ( nmr). For more information and plots of the MSM mode patterns, the review of Ref. 19 is recommended. The Gilbert damping parameter phenomenologically accounts for the viscous (linearly frequency-dependent) relaxation of magnetic excitations. Assuming a domi- nant Gilbert-type damping for all MSM modes, the full linewidth at half maximum (FWHM)
f(nmr)of a MSM resonance line at frequency f(nmr) res is given by:30
f(nmr)= 2f(nmr) res +
f(nmr) 0: (2) Here,
f0denotes the inhomogeneous line broadening contributions to the total linewidth. For a two-magnon scattering process mediated by volume and surface de- fects the latter can be written as:21
f(nmr) 0 =
fm-mF(nmr)+
f0 0: (3) Here,
fm-maccounts for the two-magnon scattering pro- cess of the MSMs into the spin-wave manifold.21,22,31The factorF(nmr)represents the ratio of the linewidth of a particular MSM with respect to the uniform precessing (110)-mode.21,22,32,33It therefore accounts for the surface sensitivity of the specic mode compared to the (110)- mode. The two-magnon scattering processes can be sup- pressed if a perfectly polished YIG sphere is used, due to the vanishing ability of the system to transfer linear and angular momentum from and to the lattice.21The term
f0 0represents a constant contribution to the linewidth in which all other frequency-independent broadening ef- fects are absorbed. The complete scattering theory used in this letter is presented in Ref. 21. Fig. 1 (a) shows a sketch of the measurement setup. The YIG sphere with a diameter of d= 300m is placed in a disk shaped Vespel sample holder (diameter 6 mm, not shown), which has a centered hole with a diameter of 350m. The sphere in the sample holder is exposed to a static magnetic eld in order to align the easy [111]- direction of the YIG crystal parallel to the eld direc- tion. The orientation of the sphere is subsequently xed using photoresist and the alignment is conrmed by Laue diraction. The oriented YIG sphere is placed on a 50 impedance matched coplanar waveguide (CPW) structure. The sphere is placed in the middle of the w= 300m wide center conductor, with the YIG [110]-axis aligned par- allel to the long axis of the center conductor of the CPW. Additionally, a pressed crumb of Diphenylpicryl- hydrazyl (DPPH) is glued on the center conductor, where the distance between the YIG sphere and the DPPH is l1 cm. DPPH is a spin marker with a g-factor34of gDPPH = 2:0036(3). The measurement of its resonance frequency fDPPH =gDPPHB 2~0HDPPH 0 (4) P1 P2 z, [111] y, [110] x, h xVNAelectro magnet top view (a) side view YIG DPPH CPW H0 Im ∆S 21 ,Re ∆S 21 (a.u.) -10 0 10 f-f res (MHz)(b) (530)-mode H0w laa/2 P1 P2 hrf FIG. 1. (a) The CPW with the YIG sphere and the DPPH is positioned in the homogeneous eld of an electromagnet. The CPW is connected to port 1 (P1) and port 2 (P2) of a vector network analyzer (VNA). The YIG sphere is placed on top of the center conductor of the CPW with its [111]- axis parallel to the applied magnetic eld H0inz-direction. (b) Typical normalized transmission spectrum of the (530)- mode at0H0= 0:8 T (symbols) including a t to Eq. (5) (lines). provides an independent magnetic eld reference at the sample position, in addition to Hall probe measurements. The static magnetic eld calculated from the DPPH resonance frequency is denoted as HDPPH 0 . The stray eld originating from the YIG sphere at the location of the DPPH creates a systematic measurement error of 0Hstray40T, as estimated using a dipole approxi- mation. For the broadband FMR experiments, the CPW is po- sitioned between the pole shoes of an electromagnet with a maximum eld strength of j0H0j2:25 T. The pole shoe diameter is a= 6 cm, while the pole shoe sepa- ration isa=2, to ensure a sucient homogeneity of the applied magnetic elds. The measured radial eld gra- dient creates a systematic eld measurement error of 0Hdisp= 0:3 mT forl= 1 cm displacement from the center axis. The CPW is connected to port 1 (P1) and port 2 (P2) of a vector network analyzer (VNA) and the complex scattering parameter S21is recorded as a function of H0 andf26:5 GHz. The applied microwave power is - 20 dBm to avoid non-linear eects causing additional line broadening. The microwave current owing along the center conductor generates a microwave magnetic eld predominately in the x-direction at the location of the YIG sphere. This results in an oscillating torque on the magnetization, which is aligned in parallel to the z- direction by the external static eld H0. Forf=f(nmr) res , the excited resonant precession of the magnetization re- sults in an absorption of microwave power. In order to eliminate the eect of the frequency depen- dent background transmission of the CPW, the following measurement protocol is applied: First, S21is measured for xedH0in a frequency range fDPPH1 GHz. Second, S21is measured for the same frequency range at a slightly3 FIG. 2. (a) Normalized transmission magnitude j
S21j plotted versus applied magnetic eld 0H0and microwave frequencyfrelative to the DPPH resonance fDPPH . The contrast between the dashed lines is stretched for better vis- ibility. (b) Calculated and measured dispersions of various MSMs (lines and open circles, respectively). larger magnetic eld H0+
H0, with0
H0= 100 mT. Since for this eld no YIG and DPPH resonances are present in the observed frequency range, the latter mea- surement contains the pure background transmission. Third, the normalized transmission spectra is obtained as
S21=S21(H0)=S21(H0+
H0), which corrects the magnitude and the phase of the signal. This procedure is repeated for all applied magnetic elds. The transmitted magnitude around the resonance can be expressed as:30
S21(f) =A+Bf+Z  f(nmr) res2 if2if
f(nmr):(5) Here,Ais a complex oset parameter, Bis a complex lin- ear background and Zis a complex scaling parameter.35 Fig. 1 (b) exemplary shows the real and imaginary part of
S21for the (530)-mode at 0H0= 0:8 T. In addition, a t of Eq. 5 to the data is shown, which adequately models the shape of the resonances. Fig. 2 (a) shows the normalized transmitted magnitude j
S21jas a function of H0andffDPPH on a linear color-coded scale. The frequency axis is chosen relative to the DPPH resonance frequency, so that all modes with a linear dispersion f(nmr) res/H0appear as straight lines,whereas modes with a non-linear dispersion are curved. Note, that the eld values displayed on the y-axis repre- sent the magnetic eld strength measured with the Hall probe. The dierent modes appearing in the color plot in Fig. 2 (a) can be identied in a straightforward manner. At rst, all visible resonances are tted using Eq. (5) in order to extract f(nmr) res and
f(nmr). Furthermore, the DPPH resonance line is identied as straight line at ffDPPH = 0 MHz and the resonance elds HDPPH 0 are calculated using Eq. (4). Second, the straight lines at about ffDPPH 60 MHz and ffDPPH740 MHz are identied as the (110)- and (210)-mode, respectively. A simultaneous t of the dispersion relations18 f(110) res =gYIGB 2~0(H0+Hani) (6) and f(210) res =gYIGB 2~0 H0+Hani2 15Ms (7) to the measured values of f(110) res,f(210) res and0HDPPH 0 yieldsgYIG = 2:0054(3),0Ms= 176:0(4) mT and 0Hani=2:5(4) mT. The error of gYIGis given by the systematic error introduced by the eld normalization usinggDPPH . The errors in 0Haniand0Msare given by0Hdisp+0Hstray. All values are in good agree- ment with previously reported material parameters36{40 for YIG (gYIG = 2:005(2),0Hani=5:7 mT and 0Ms= 180 mT) and, hence, justify the (110)- and (210)- mode assignments. Third, the complete MSM manifold is computed using the extracted material parameters. The mode numbers of the remaining modes are determined from the charac- teristic dispersions. Fig. 2 (b) shows the dispersions of the identied modes as function of f(nmr) resfDPPH and HDPPH 0 , with very good agreement of theory (lines) and experiment (circles). Slight deviations between model predictions and data might be attributed to a non-perfect spherical shape of the sample, which would change the boundary conditions for the magnetization dynamic in the YIG spheroid, and thus the dispersion relations. In Fig. 3 (a) the linewidth
f(nmr)of each MSM is plotted versus its resonance frequency f(nmr) res . The oset
f(nmr) 0 is magnied by a factor of 5 to emphasize the dierences in the inhomogeneous line broadening. Indi- vidual ts of all
f(nmr)to Eq. (2) yield identical slopes for all modes within a small scatter, which is also evident from the linewidth data in Fig. 3 (a). Hence, the Gilbert damping parameter and inhomogeneous line broadening are obtained from a simultaneous t of Eq. (2) to the extracted data points. Here, is a shared t parameter for all MSMs, but the inhomogeneous line broadening
f(nmr) 0 is tted separately for each mode. To avoid tting errors, the linewidths data are disregarded when a mode anti-crossing is observed, since this results in a4 5 10 15 20 25 ∆f (nmr) (MHz) fres (GHz) (a) 0(110) (440) (531) (530) (511) (631) (502) (nmr) 246810 Offset x5 (b) 0.00.51.0 1.5 2.0 ∆f0(nmr) (MHz) -500 0 500 -250 250 ∆f00=0.3 MHz (110) (440) (531) (530) (511) (631) (502) Measurement Theory fres - fDPPH (MHz) (nmr) FIG. 3. (a) Linewidth vs. resonance frequency of the measured MSMs. The Gilbert damping of all MSMs is = 2:7(5)105as evident from the same slope of all curves. The inhomogeneous line broadening is dierent for each MSM. Note that the data points are plotted with an o- set proportional to the inhomogeneous line broadening. (b) Inhomogeneous line broadening as a function of ffDPPH . pronounced increase in linewidth.41As evident from the solid t curves in Fig. 3 (a) the evolution of the linewidth with resonance frequency of all measured MSMs can be well described with a shared Gilbert damping parameter of= 2:7(5)105, independent of the mode num- ber and the mode intensity. The latter strongly sug- gests a negligible eect of radiative damping on the mea- sured linewidths.42The error in is given by the scat- ter offrom the independent ts. Other groups report Gilbert damping parameters for YIG lms43{49larger than= 6:15105, whereas for bulk YIG37,49,50values of= 4105are found. Hence, the Gilbert damp- ing parameter obtained here is the smallest experimen- tal value reported so far. The results are in agreement with the notion, that the Gilbert damping parameter is a bulk property which only depends on intrinsic damping eects. However, the inhomogeneous line broadening is indeed dierent for the various MSMs. Fig. 3 (b) shows the extracted values for the inhomo- geneous line broadening (lled dots) as a function of f(nmr) resfDPPH . The error bars indicate the variation of the inhomogeneous line broadening between global andindividual ts. In order to show the approximate posi- tion of the modes in comparison to Fig. 2, the x-scale is calculated for a magnetic eld strength of 0H= 0:5 T. Additionally, the linewidths
f(nmr) 0 for all modes are calculated using the two-magnon scattering theory, given in Eq. (4) of Ref. 21 (open circles). For the calculations of the linewidths, a pit radius R= 350 nm and a constant linewidth contribution of
f0 0= 30 kHz was assumed. Since the calculated
fm-mare slightly frequency depen- dent, the average linewidth values for the measured eld and frequency range are used and the standard deviation is indicated by the error bars of the open symbols. For most MSMs the variation is smaller than 10 kHz. Never- theless, the (440)-mode should show a prominent peak in the linewidth measurement at about f(440) res = 10 GHz in Fig. 3 (a),21which is however not observed in the experi- mental data. Additionally, the (110)-MSM shows a much larger linewidth than expected from the calculations. In a perfect sphere the (110)-mode is degenerate with the (430)-mode,18but in a real sphere this degeneracy might be lifted. If the dierence of the (110)- and (430)-mode frequencies is smaller than the linewidth of the measured resonance, an additional inhomogeneous line broadening is expected. Indeed, a careful analysis of the (110)-MSM line shape reveals a second resonance line in very close vicinity to the (110)-mode, yielding an articial inhomo- geneous line broadening of this mode. Besides these two MSMs, an excellent quantitative agreement between the two-magnon scattering model and experiment is found. In conclusion, broadband ferromagnetic resonance ex- periments on magnetostatic modes in a YIG sphere are presented and various magnetostatic modes are identi- ed. The linewidth analysis of the data allows to distin- guish between the Gilbert damping and inhomogeneous line broadening. A very small Gilbert damping parame- ter of= 2:7(5)105is found for all MSMs, indepen- dent of their mode indices. Furthermore, the inhomoge- neous line broadening diers between the various magne- tostatic modes, in agreement with the expectations due to two-magnon scattering processes of the magnetostatic modes into the spin-wave manifold. 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2016-12-07

The magnetostatic mode (MSM) spectrum of a 300$\mu$m diameter single crystalline sphere of yttrium iron garnet is investigated using broadband ferromagnetic resonance (FMR). The individual MSMs are identified via their characteristic dispersion relations and the corresponding mode number tuples $(nmr)$ are assigned. Taking FMR data over a broad frequency and magnetic field range allows to analyze both the Gilbert damping parameter~$\alpha$ and the inhomogeneous line broadening contribution to the total linewidth of the MSMs separately. The linewidth analysis shows that all MSMs share the same Gilbert damping parameter $\alpha=2.7(5) \times 10^{-5}$ irrespective of their mode index. In contrast, the inhomogeneous line broadening shows a pronounced mode dependence. This observation is modeled in terms of two-magnon scattering processes of the MSMs into the spin-wave manifold, mediated by surface and volume defects.

Gilbert damping of magnetostatic modes in a yttrium iron garnet sphere

1612.02360v1

1 Unifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni bilayers with electronic relaxation near the Fermi surface Wei Zhang, Wei He*, Xiang -Qun Zhang, and Zhao -Hua Cheng* State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China Jiao Teng Department of Materials Physics and Chemistry, University of Sci ence and Technology Beijing, Beijing 100083, P. R. China Manfred Fä hnle Max Planck Institute for Intelligent Systems, Heisenbergstra e 3, 70569 Stuttgart, Germany Abstract The ability to controllably manipulate the laser -induced ultrafast magnetic dynamics is a prerequisite for future high speed spintronic devices. The optimization of devices requires the controllability of the ultrafast demagnetization time, , and intrinsic Gilbert damping, . In previous attempts to establish the relationship between M and rint , the rare -earth doping of a permalloy film with two different demagnetization mechanism is not a suitable candidate. Here, we choose Co/Ni bilayers to investigate the relations between and by means of ti me-resolved magneto -optical Kerr effect (TRMOKE) via adjusting the thickness of the Ni layers, and obtain an approximately proportional relation between these two parameters. M intr M intr 2 The remarkable agreement between TRMOKE experiment and the prediction of breathi ng Fermi -surface model confirms that a large Elliott -Yafet spin -mixing parameter 2b is relevant to the strong spin -orbital coupling at the Co/Ni interface. More importantly, a proportional relation between M and intr in such metallic films or heterostructures with electronic relaxation near Fermi surface suggests the local spin -flip scattering domains the mechanism of ultrafast demagnetization, otherwise the spin -current mechanism domains. It is a n effective method to distinguish the dominant contributions to ultrafast magnetic quenching in metallic heterostructures by investigating both the ultrafast demagnetization time and Gilbert damping simultaneously. Our work can open a novel avenue to manip ulate the magnitude and efficiency of Terahertz emission in metallic heterostructures such as the perpendicular magnetic anisotropic Ta/Pt/ Co/Ni /Pt/Ta multi layers, and then it has an immediate implication of the design of high frequency spintronic devices . PACS numbers: 75.78.Jp, 75.40.Gb, 76.50.+g, 78.47.+p *Correspondence and requests for materials should be addressed to Z.H.C (zhcheng@iphy.ac.cn ) or W.H. ( hewei@iphy.ac.cn ) 3 Since the pioneering work on ultrafast demagnetization of Ni thin film after femtosecond laser irradiation was demonstrated in 1996 by Beaurepaire et al1, the quest for ultrafast modification of the magnetic moments has triggered a new field of research : Femtomagnetism . It leads to the dawn of a new ear for breaking the ultimate physical limit for the speed of magnetic switching and manipulation, which are relevant to current and future information storage. In the past two decades, the ultrafast dynamics in hundreds of femtoseconds have been probed with the femtosecond laser pulse using magneto -optical Kerr1 or Faraday effect2, or other time-resolved techniques such as the high -harmonic generation (HHG) of extreme ultraviolet(XUV) radiation3, magnetic cir cular dichroism4, or spin resolved two -photo photoemission5. Nevertheless, the microscopic mechanism underlying ultrafast quenching of magnetization remains elusive. Various mechanisms including electron -phonon mediated spin -flip scattering6-9, electron -electron scattering10,11, electron -magnon scattering12,13, direct angular momentum transfer from photon to electron mediated by spin-orbit coupling14,15, coherent interaction among spins electrons and photons16, were proposed to explain the ultraf ast spin dynamics. In addition, since Malinowski17 et al first proposed that the laser excited spin current transport could increase and speed up the magnetic quenching in metallic heterostructures, the laser -induced super -diffusive spin current was raise d to play an important role in determining the ultrafast demagnetization in metallic films or heterostructures18-22. However, the recent demonstration23 shows that the unpolarized hot electrons transport can 4 demagnetize a ferromagnet, indicating the local spin angular momentum dissipation is unavoidable even when super -diffusive spin transport domains in the metallic heterostructures. Moreover, even in th e similar samples, the local spin -flip scattering and nonlocal spin transport mechanism were proposed respectively by different experimental tools19, 24 to explain the ultrafast demagnetization . It is harmful for clarifying the underlying ultrafast demagne tization mechanism in such metallic heterostructures. Therefore, an effective method to distinguish the two dominant contribution s to ultrafast demagnetization in metallic heterostructures is highly desirable19,23,24. Here, we propose that investigating bo th the ultrafast demagnetization time and Gilbert damping25 simultaneously is a candidate method, although the relationship between the two parameters has never been unified successfully so far between the experiments and theoretical predictions. An inv erse relation between and was first derived by Koopmans et al. from a quantum -mechanical calculation on the basis of the Elliot t-Yalfet (EY) spin-flip scattering model6. Later, the attempted experiments have ever been carried out to demonstrate the predict ion in rare -earth -doped permalloys26,27 and amorphous TbFeCo films28. In this case, t he localized 4f electrons rather than itinerant 5 d6s electrons domain most of the large magnetic moment in rare -earth elements. Because the 4 f electrons are far from the Fermi level, their ultrafast demagnetization processes are medicated by 5 d6s electrons after laser pulse excitation7. The indirect excitation leads to the so called type_II ultrafast demagnetization behavior in rare -earth elemen ts, which is much slower than that of itinerant electrons. Therefore, it is not unexpected M intr 5 that the ultrafast demagnetization time M of permalloys increases with the doping contents of rare -earth elements increasing. Meanwhile , it happ ened that the Gilbert damping constant of permalloys is also increased by doping 4 f elements, which mainly comes from the so called “slow relaxing impurities mechanism”29. Therefore, by introducing the extra mechanism unavoidablely ，a trivial consequence wa s obtained that the ultrafast demagnetization time increases as the Gilbert damping increases in rare -earth -doped permalloys26. In hindsight, from this experiment, one can not confirm the relation between ultrafast demagnetization time and Gilbert damping due to the defects of the experimental design. A genuine relation between ultrafast demagnetization time and Gilbert damping should be explored in a clean system without extra demagnetization mechanism. So far, the explicit relationship between the two parameters has never been unified successfully between the experiments and theoretical predictions. Our work in Co/Ni bilayers with the electrons relaxing at the Fermi surface can fill in the blank. In the cas e of pure 3 d itinerant electrons relaxing near the Fermi surface after the laser excitation , both ultrafast demagnetization and Gilbert damping are determined by spin -flip scattering of itinerant electrons at quasi -particles or impurities . Based on the breathing Fermi -surface model of Gilbert damping and on the EY relation for the spin-relaxation time, a proportional relation between and was derived by Fä hnle et al30,31 for the materials with conductivity -like damping. And an inverse relation was also d erived which is similar with that proposed by B. Koopmans et al when the resistivity -type damping domains in the materials. Although the predicted M M M intr 6 single numerical values of intr/M are in good agreement with the experimental ones for Fe, Ni, or Co, for a confirmation of the explicit relation between and one has to vary the values on the two parameters systematically for one system, as we do it in our paper by changing the thickness of the films. Co/Ni bilayers with a stack of Ta (3 nm)/Pt (2 nm)/Co ( 0.8 nm)/Ni ( dNi nm)/Pt (1 nm)/Ta (3 nm) were grown on glass substrates by DC magnetron sputtering32, 33. The thickness of Ni layer changes from dNi = 0.4 nm to dNi = 2.0 nm. T heir static properties have been shown in the Part Ⅰof the Supplementary Materials34. Both and for Co/Ni bilayer systems have been achieved by using time -resolved magneto -optical Kerr effect (TRMOKE) technique21, 35. The reasons for selecting the Co/Ni bilayers are three -fold. First, Co/Ni bilayers with perpendicular magnetic anisotropy (PMA) are one of candidates for perpendicular magnetic recording (PMR) media and spintronic devices36-39. Second, the electrons in both Co and Ni are itinerant near the Fermi surface and they have the same order of magnitude of demagnetization time7,10. Without rare earth element doping in 3 d metals, one can exclude the possibility of an extra slow demagnetization accompanied by doping with 4f rare-earth metals. Third, both and in Co/Ni bilayers can be tuned by changing the Ni thickness. Therefore, Co/Ni bilayers provide an ideal system to investigate the relation between and . A nearly p roportional relationship between and was evident in Co/Ni bilayers, suggesting that the conductivity -like damping30, 31 plays a dominant role. It is distinct i n physics with previous experiments26 where the seemingly similar results have been obtained via M intr M intr M intr M intr M intr 7 introducing extra slow demagnetization mechanism. Moreover, we discussed the origin of Gilbert damping, analyzed its influence on the relation between M and intr and proposed a new approach to distinguish the intrinsic spin -flip and extrinsic spin current mechanism for ultrafast demagnetization in metallic heterostructures. The finding for this unification can provid e the possibility for manipulating the laser -induced ultrafast demagnetization via Gilbert damping in high frequency or ultrafast spintronic devices such as the Terahertz emitters . Fig. 1(a) shows time -resolved MOKE signals40 for films with various Ni lay er thickness measured with an external field Oe. The quantitative values of intrinsic Gilbert damping constant41-44 in Fig. 1(b) can be obtained by eliminating the extrinsic contributions (See the Supplementary Materials [34], PartⅡ for details). It was observed that intr decreases with increasing Ni layer thickness. On the one hand, previous investigations39, 45 have been reported that the large PMA origins from the strong spin -orbit coupling effect at Co/Ni interface. A thickness modification in Co/Ni bilayer can change the competition between interface and volume effect, and consequently the PMA. When we plot the intrinsic Gilbert damping constant as a function of effective anisotropy field in Fig. 4 in the Part Ⅱ in Supplementary Material (See the Supplementary Materials [34], PartⅡ for details), a proportional relation was confirmed in our Co/Ni bilayer system, which demonstrates that spin-orbit coupling contributes to both Gilbert damping and PMA (Also, for the achievement of effec tive anisotropy field, please see the Supplementary Materials [34] PartⅡfor details ). On the other hand, the interface between Ni and Pt maybe also 4000H 8 modified via changing Ni layer thickness. Because the Gilbert damping increases linearly when the Ni layer b ecomes thinner, it seems that the spin current dissipation is involved partly. A similar trend was observed in a Pt/CoFeB/Pt system46, in which a pure non -local spin pumping effect domains the Gilbert damping. Therefore, the total Gilbert damping equals to α=𝛼𝑖𝑛𝑡𝑟 +𝛼𝑠𝑝 , in which 𝛼𝑠𝑝 represents the contributions from spin current. Due to the low spin diffusion length of Pt, the magnetization precession in Ni layer entering the Pt layer would be absorbed completely like in the system of Py/Pt and Py/Pd47 and so on. H owever，we have to address that, i n the case of the variation of ferromagnetic layer thickness, the amount of spin current pumped out of ferromagnet is determined entirely by the parameter of interfacial mixing conductance 𝐺𝑒𝑓𝑓𝑚𝑖𝑥 48,49. It is a constant value once the normal metal thickness is fixed , although the Gilbert damping in thinner magnetic layer is enhanced. Therefore, given the spin current contributes partly to the Gilbert damping at present, the spin angular momentum transferring from Ni layer to Pt layer would be the same for various Ni lay er thickness. The central strategy of our study is to establish a direct correlation between ultrafast demagnetization time and the intrinsic Gilbert damping constant. The intrinsic Gilbert damping constant was extracted from magnetization precessi on in hundreds of ps timescale. The laser -induced ultrafast demagnetization dynamics has been measured carefully within time delay of 2.5 ps at a step of 15 fs and low laser fluence of 1 was used. Fig. 2 (a) shows the TRMOKE signals of the ultrafast demagn etization evolution after optical excitation. A rapid decrease of magnetization 2/cmmJ 9 takes place on the sub -picosecond timescale followed by a pronounced recovery. As can be seen in this figure, the ultrafast demagnetization rate is different by changing the Ni thickness. To identify the effect of the heat transport across the film thickness on demagnetization time, a numerical simulation50 was carried out to demonstrate that the demagnetization time variation induced with the thicknesses ranged from 1.2 nm to 2.8 nm is so small that can be ignored (See the Supplementary Materials [34], Part Ⅲ for details), although a relatively large error of could be resulted in when the sample thickness spans very large. According to the simulation results, the heat transport not only affects the rate of ultrafast magnetization loss but also the maximum magnetic quenching. So, in experiment we obtain the ultrafast demagnetization time for various samples with almost the sa me maximum quenching of 9% to suppress the influence of heat transport7, 21, 51 -54 as well as the non local spin current effect17. The temporal evolution of magnetization in sub -picosecond time scale was fitted by the analytic solution based on the phenome nological three temperature model (3TM)1, 17: (1) where presents the convolution product with the Gaussian laser pulse profile, whose full width at half maximum (FWHM) is . A temporal stretching of the laser pulse was introduced by the excited hot ele ctrons55, which is the trigger for the observed ultrafast demagnetization. In the fitting procedure, the demagnetization M ),()()()( 1)( 321 1 2 5.0 01 Gt M EEt M EM EtGtAteAAeAA tA MtMM M ),(GtG G 10 time we cared can be influenced by the value of , which is inter -dependence with within the three temperature model. As is shown in Table 1 in the Supplementary Material34 Part Ⅳ, was fixed at 330 fs for various samples to eliminate its relevance with . The time variable in eq. (5) corresponds to , with the free fit parameter characterizing the onset of the demagnetization dynamic s of the actual data trace, which is fixed as 100 fs for various samples. is a step function, is the Dirac delta function and are the fitting constants. The two critical time parameters are the ultrafast demagnetization time and magnetization recov ery time, respectively. The well fitted curves by 3TM are also shown as the solid lines in Fig. 3(a) from which the ultrafast demagnetization time and the magnetization recovery time were evaluated. Within 3TM model, the magnetization recovery process is affected by , charactering the electron -phonon relaxation, and , representing heat transport timescale through the substrates as well as demagnetization time . In the fitting procedure by 3TM model, we assigned a fixed value to and varies slightly to exclude the heat transport effect through thickness. Via changing the single parameter ， , we can accurately reproduce the experimental results for various samples. And the heat transport across the thickness domains within 3TM model characterized by the parameter of , which is shown in Table. 1 in Part Ⅳ of Supplementary Material34 as around 2 ps. It is about three times bigger than indicating that we are not mixing the heat transport and the electron -phonon relaxation56. Only in this case, are both th e values of and genuine. The value M G M G M 0 expt tt 0t )(t )(t 3 2 1,,AAA E M, M E E 0 M E 0 M 0 E E M 11 of indicates that the heat was transferred through the substrate in less than 3 ps in this paper, rather than what was observed by F. Busse et al57 where the heat was trapped laterally in the Gaussian profile up to 1 ns. Therefore, the lateral heat transport effect can be ignored, and hencely the modification of precessional dynamics here. As illustrated in Fig. 2(b), it can be clearly seen that decreases with increasing dNi. By replotting Fig. 1(b) and Fig. 2(b), an approximately proportional relationship between and intr was confirmed by our experimental results (Fig. 2(c)). This relationship between intr and is consistent well with the theoretical prediction based on the breathing Fermi -surface model30,31,58 for materials with conductivity -like damping contributions. On the basis of the breathing Fermi -surface model, the Elliott -Yafet spin -mixing parameter 𝑏2 in Co/Ni bilayers can be estimated from the theoretical equation30, 31 shown as the red solid l ine in Fig. 2(c): (2) where the quantity contains the derivatives of the single -electron energies with respect to the orientation e of the magnetization M=Me. p is a material -specific parameter which should be close to 4. If we use = from ab initio density functional electron theory calculation for fcc bulk Ni31, the experimental value of Elliott -Yafet spin -mixing parameter 𝑏2 = 0.28 can be estimated in Co/Ni bilayers, which is far larger than that of Co or Ni. The significant enhancement of spin -mixing 0 M M M int Mr 2pbFM elM elF J231087.1 12 parameters is related to the strong spin -orbital coupling at the Co/Ni interface since b2 is proportional to 2 in first -order perturbation theory, where is the coefficient of the spin-orbit coupling. A detailed ab initio calculation for Elliott -Yafet spin -mixing parameter in Co/Ni bilayers is highly desirable. For a derivation of eq. (2) it must be assumed that the same types of spin-flip scattering processes are relevant for the ultrafast demagnetization and for the damping. The assumption does not say anything about these detailed types. It has been shown in Ref. 9 that mere electron -phonon scatterings cannot explain the expe rimentally observed demagnetization quantitatively. In reality there are also contributions from electron -electron scatterings11, electron -magnon scatterings12 and from a combination of electron -phonon and electron -magnon scatterings13. Because both for de magnetization and for damping ， the spin angular momentum has to be transferred from the electronic spin system to the lattice, there is no reason why different types of theses spin -flip scatterings should be relevant for the two situations. Therefore , the Elliott -Yafet relation, eq. (2) should be applicable for our system. It would not be valid if non -local spin -diffusion processes would contribute a lot to demagnetization. Examples are a superdiffusive spin current in the direction perpendicular to th e film plane, or a lateral diffusion out of the spot irradiated by the laser pulse and investigated by the TRMOKE. However, we definitely found the validity of the Elliott -Yafet relation, and this shows that nonlocal spin -diffusion processes are so small t hat can be neglected in our experiment. Despite this , previous demonstrations17,19-21 show that the ultrafast spin current 13 caused by the transport of spin -majority and spin -minority electrons in the antiparallel (AP) state of magnetic multilayers after the laser pulse accelerates the ultrafast demagnetization. Similarly, as is indicated in Fig. 1(b), with the assistance of interface between FM (Ni) and NM (Pt), the spin current induced by the flow of spin -up and spin-down electrons in opposite directions59 may contribute partly to the Gilbert damping in Pt/Co/Ni/Pt mulitilayers. The femtosecond laser induced spin current lives very shortly which is in sub -picosecond timescale, while the duration of spin current triggered by spin precession is in the timescal e of nanosecond. The difference of the duration of the spin current is just related to the timescale of the perturbation of the system. One has to note that spin currents at the femtosecond time scale gives rise to a lowering of the demanetization time17, while spin pumping induced spin current gives rise to the enhancement of Gilbert damping and thus a lowering of the relaxation time. Therefore, when spin current contributes largely to both ultrafast demagnetization and spin precession dynamics, an inverse relationship between ultrafast demagnetization time and Gilbert damping could be expected. That is, t he more spin current transferred from ferromagnetic layer to normal metal, the faster ultrafast demagnetization should be. Therefore, a t present paper, to explain the experimental results the local Ellio tt-Yafet scattering theory suffices. And , the non -local spin current effect can be ignored, although it contributes partly to the fitted value of spin-mixing parameter 𝑏2 . The discussions here inspire us t o continuously clarify the various relationships between ultrafast demagnetization time and Gilbert dam ping coming from different microscopic mechanisms, which is helpful for understanding 14 the underlying physics of ultrafast spin dynamics as well as the ap plication of ultrafast spin current triggered by ultrashort laser60, 61. For instance, recently, the researchers are seeking for the potential candidates as the Terahertz waves emitters including the metallic heterostructures. Previous demonstrations show that the magnitude and efficiency of Terahertz signals in these multilayers are determined by Gilbert damping60. The investigations of the relationship between Gilbert damping and ultrafast demagnetization time will open up a new avenue to tailor the Terah ertz emission. Meanwhile , the dominant contribution to ultrafast demagnetization in metallic heterostructures, either from the localized spin -flip scattering or non -local spin transport, has been a controversial issue for a long time23. Here, a new approa ch, by establishing the relation between the demagnetization time and Gilbert damping, is proposed to distinguish the two mechanisms . The proportional relationship indicates the localized spin -flip scattering mechanism domains, otherwise the nonlocal spin current domains. In conclusion, the fast and ultrafast dynamic properties of Ta(3 nm)/Pt(2 nm)/Co(0.8 nm)/Ni( dNi nm)/Pt(1 nm)/Ta(3 nm) bilayers with the electrons relaxing near the Fermi surface have been investigated by using TRMOKE pump -probe technique. An genuine proportional relationship , contrast to previous trivial consequence induced by impurities mechanism, between ultrafast demagnetization time and Gilbert damping constant is confirmed fr om experimental results. The estimated value of spin -mixing parameter on the basis of breathing Fermi -surface 15 model is far larger than that of Co or Ni , which is originated from the strong spin-orbital coupling at the interface. More importantly, distingui shing the dominant mechanism underlying ultrafast demagnetization in metallic heterstructures has been a tough task for a long time. Here, an effective method by unification of the ultrafast demagnetization time and Gilbert damping is proposed to solve thi s task, namely that, a proportional relation between the two parameters indicates the local spin flip scattering mechanism domains, otherwise the non local spin current effect domains. 16 Acknowledgments This work was supported by the National Basic Research Program of China (973 program, Grant Nos. 2015CB921403 and 2016YFA0300701), the National Natural Sciences Foundation of China (51427801, 11374350, and 11274361). 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Macizo, Modeling of ultrafast laser -induced magnetization dynamics within the Landau -Lifshitz -Bloch approach , PhD Thesis 2012. P.78 55. B. Vodungb o, B. Tudu, J. Perron, R. Delaunay, L. Mü ller, M. H. Berntsen, G. Grü bel, G. Malinowski, C. Weier, J. Gautier, G. Lambert, P. Zeitoun, C. Gutt, E. Jal, A. H. Reid, P. W. Granitzka, N. Jaouen, G. L. Dakovski, S. Moeller, M. P. Minitti, A. Mitra, S. Carron, B. Pfau, C. von Korff Schmising, M. Schneider, S. Eisebitt, and J. Lü ning, Sci. Rep, 6, 18970 (2016). 56. F. Dalla Longa, J. T. Kohlhepp, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. B, 75, 224431 (2007). 57. F. Busse, M. Mansurova, B. Lenk, M. von der Ehe and M. Mü nzenberg, Sci. Rep. 5, 12824 (2015). 58. K. Gilmore, Y. -U. Idzerda, and M. -D. Stiles, Phys. Rev. Lett. 99, 027204 (2007). 59. I. Zutic and H. Dery, Nature Mater. 10, 647 (2011). 60. J. Shen, X. Fan, Z. Y. Chen, M. F. DeCamp, H. W. Zhang, and J. Q. Xiao, Appl. Phys. Lett. 101, 072401 (2012). 61. T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. Freimuth, A. Kronenberg, J. Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov , P. M. Oppeneer , M. Jourdan , G. Jakob , D. Turchinovich , L. M. M. Hayde n , M. Wolf , M. Mü nzenberg , M. Klä ui , and T. Kampfrath , Nat. Photonics. 10, 483 (2016). 23 Figure caption: FIG. 1 Spin precession. (a)TRMOKE signals of Co/Ni bilayers with dNi=0.4-2.0 nm in applied field H = 4000 Oe. (b) Intrinsic Gilbert damping constant as a function of dNi. FIG. 2 Ultrafast demagnetization. (a) Ultrafast demagnetization curves with various Ni layer thickness. (b) Ultrafast demagnetization time as a function of Ni layer thickness. (c) Ult rafast demagnetization time as a function of Gilbert damping constant. The red full line indicates theoretical fitting . 24 Fig. 1 (Color Online) Spin precession. (a)TRMOKE signals of Co/Ni bilayers with dNi=0.4-2.0 nm in applied field H = 4000 Oe. (b) Intrinsic Gilbert damping constant as a function of dNi. 25 Fig.2 (Color Online) Ultrafast demagnetization. (a) Ultrafast demagnetization curves with various Ni layer thickness. (b) Ultrafast demagnetization time as a function of Ni layer thickness. (c) Ultrafast demagnetization time as a function of Gilbert damping constant. The red full line indicates theoret ical fitting. 26 Supplementary Information Unifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni bilayers with electronic relaxation near the Fermi surface PartⅠ The measurements of static properties for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (dNi nm)/Pt (1 nm)/Ta (3 nm) . Fig. 1(a) shows the polar magneto -optical Kerr signal measured at room temperature with maximum applied field of 300 Oe. The static polar Kerr loops of Co/Ni bilayers were acquired using a laser diode with a wavel ength of 650 nm. All samples show very square loops with a remanence ratio of about 100% , indicating t he well-established perpendicular magnetization anisotropy ( PMA) of the samples. The measured coercivity Hc decreases with dNi from 103Oe for dNi = 0.4 nm to 37Oe for dNi =2.0 nm (Fig. 1(b)). The decrease of coercivity implies that the PMA decreases with the thickness of Ni. 27 Fig.1 Static magnetic properties of of Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (d Ni nm)/Pt (1 nm)/Ta (3 nm) bilayers. (a) Polar -MOKE loops with various thickness of Ni layer d Ni. (b) Coercivity Hc and effective anisotropy field as a function of Ni layer thickness d Ni. eff KH 28 PartⅡ The measurements of spin dynamics for Co/Ni bilayers in ns timescales and the analysis of extrinsic contributions to spin precession In this part, we show the details of spin precession experiment. For example, Fig. 2(a) illustrates the scheme for laser -induced magnetization precession. The direction of applied field is fixed at . The typical time -resolved magnetization dynamics with various applied fields for Ta(3 nm)/Pt(2 nm)/Co(0.8 nm)/Ni(0.8 nm)/Pt(1 nm)/Ta(3 nm) shown in Fig. 2(b) can be best fitted by using the damped harmonic function added to an exponential -decaying background1: (1) where A and B are the background magnitudes, and is the background recovery rate. C, , f and are magnetization precession amplitude, relaxation time, frequency and phase, respectively. From the f itting curves shown in Fig. 2(b) as the solid lines, the values of precession frequency f and relaxation time are extracted. Since the applied fields are large enough, we can obtain the Gilbert damping constant using the following relationship2 (2). 80H ( ) exp( ) exp( )sin(2 )tM t A B t C ft 1)2( f 29 In the case of films with a relatively low Gilbert damping3-7 as well as thickness larger than the optical penetration depth8, ultrafast laser may generate non -uniform spin waves and affect the relation ship between demagnetization and Gilbert damping as extrinsic contributions . In order to check the contribution of non -uniform modes, we performed a fast Fourier transform shown in Fig. 2(c). Only the uniform precession mode was excited at present Co/Ni bi layers with perpendicular magnetic anisotropy. Both and f as a function of H are plotted in Fig.3. Since the overall damping constant consists of intrinsic damping and extrinsic damping whereby the second one arises from inhomogeneities in the sample , the Gilbert damping constant decreases monotonously to a constant value as the applied field increases (Fig. 3(a)). In the low external fields range, the inhomogeneously distributed anisotropy may lead to higher values. Fortunately, the sufficient high field we used can suppress the extrinsic contributions to the magnetization precession, because for high fields the magnetization dynamics is mainly determined by the external field9. In addition, because of the interaction between femtosecond laser sourc e and the thin films, the lateral heat distribution across the film plane has to be considered as another candidate contributions to affect the processional dynamics. As is shown by F. Busse et al6, the heat was trapped as the Gaussian distribution across the film plane of CoFeB up to 1 ns due to the use of regenerative amplifier. It can enhance the laser power largely while the pump laser spot kept as large as around 90 μm. This facilitates the occurrence of the temperature profile, and consequently the sp in-waves 30 in the range of laser spot size. However, in the absent of regenerative amplifier at present, the laser spot is so small as less than 10 1,10 that one can excite the nonequilibrium state of the samples. And the laser fluence used here is around 1 , which is far weaker than that used in previous report6. Although smaller laser spot seems easier to trigger the nonuniform spin waves, the very low laser power we used here can suppress the influence of lateral heat distribution on the relaxation time o f spin dynamics at present. M oreover, the absence of non -uniform spin wave demonstrated in Fig. 2(c) in the pump laser spot confirms that the lateral heat transport can be neglected here. In fact, it is found in the main text, within the three temperature model (3TM model) describing the ultrafast demagnetization dynamics, that the heat induced by laser pulse mainly transports along the thickness direction to substrate in less than a few picoseconds. The observation of pronounced magnetization recovery aft er ultrafast demagnetization can exclude the possibility of lateral heat trap. In order to avoid the effect of extrinsic damping constant, the intrinsic damping constants were obtained by fitting the overall damping factor as the function of applied fields with the expression shown as the red line in Fig. 3(a) : (3) where and are the intrinsic and extrinsic parts of the damping factor, respectively. The intrinsic part is independent of the external field or precession frequency, while the extrinsic part is field -dependent. m 2/cmmJ 0/ int 1HH rae intr 0/ 1HHe 31 The experimental f-H relation in Fig. 3(b) can be fitted by analytic Kittel formula derived from LLG equation2: (4) where , . The equilibrium angle of magnetization was calculated from the relationship . The direction of applied field is fixed at . In the above equations, and are the effective perpendicular magnetization anisotropy and gyromagnetic ratio, respectively, wher e , . In our calculation, the Lande factor was set to 2.2 as the bulk Co value2. is the only adjustable parameter. The variation of effective field with the thickness of Ni layer was also plotted in Fig. 1(b). When we plot the intrinsic Gilbert damping constant as a function of effective anisotropy field in Fig. 4, a proportional relation was confirmed in our Co/Ni bilayer system, which demonstrates that spin -orbit coupling contributes to both Gilbert damping and PMA . 2 12HH f 2 1 cos ) cos(eff K H H H H 2cos ) cos(2eff K H H H H ) sin(22sin H eff KHH 80H eff KH seff eff KMKH2 2Bg h g eff KH 32 The numerical simulation for ultrafast demagnetization Fig. 2 (a) Scheme of TRMOKE. (b): TRMOKE signals with various applied field for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (0.8 nm)/Pt (1 nm)/Ta (3 nm) bilayers. (c): Fast Fourier transform ation s ignals. 33 Fig. 3 Gilbert damping and precession frequency. Field dependence of overall damping constant (a) and precession frequency (b) of Co/Ni bilayers with nm dnm dNi Co 8.0 ,8.0 34 Fig. 4 Dependence of intrinsic Gilbert damping constant on the effective anisotropy field. 35 PartⅢ Numerical simulation for the effect of heat transport across the film thickness on the ultrafast demagnetization time To estimate the evolution of heat transport profile in time, w e carried out a numerical simulation based on M3TM11 model, in which the heat transport12 was dominated by electrons and a temperature gradient across the film thickness was introduced. It is divided in thin slabs in the direct ion normal to the film plane, and the slabs is 0.1 nm thick. For each slab, the evolution of the electron and phonon temperatures eT and pT are determined by a set of coupled differential equations :13 )),)(coth()( 1()()()()),( )(()()),( )(( ))( ()()( zTmTzmTzTzRmdtzdmzTzTgdtz dTCzTzTg zTdtzdTzT ec cpp e epp pe p ep ez ze e (5) Where sMMm , )()( 0zTzT pe 4, 228 D atB atcBep sf EVTkgaR ,with at the atomic magnetic moment in units of Bohr magneton B , atV the atomic volume, and DE is the Debye energy. eC and pC are the heat capacities of the e and p systems respectively. )(zTez is the electron temperature gradient normal to the film . Bk is 36 the Boltzmann constant. 0k is the material dependent electronic thermal conductivity. epg is the e -p coupling constant and determines the decay of the electronic temperature until equilibrium is reached14. sfa represents the spin -flip probability11. The equations of motion for each slab thus describe heating of the electron system by a Gaussian laser pulse, heat diffusi on by electrons to neighboring slabs, e -p equilibration, and finally the evolution of the magnetization due to e -p spin -flip scattering. In the simulation, the total magneto -optical signal was obtained by the calculation of dzztzm t ) exp(),( )( . The electronic system after the action of the laser pulse is in a strongly non-equilibrium situation. Nevertheless, one can describe the electron system by use of an electron temperature. The reason is that the laser photons excite electrons, but these excite d electrons thermalize more or less instantly due to very rapid and frequent electron -electron scatterings via their Coulomb interactions. This is the assumption of the accepted Elliott -Yafet scenario which describes the effect of the laser pulse directly after the action of the laser pulse. Fig.4(a) shows the simulated ultrafast demagnetization curves for various film thicknesses. We can clearly observe that the evolution of magnetization curves looks almost identical for various film thicknesses , indicating that the effect of heat transport on the demagnetization time can be neglected. Despite this, for the remagnetization part, a deviation from the experimental curves occurs. This is mainly because that the heat diffusion can almost be neglected d uring the ultrafast demagnetization timescale, but starts playing an increasing role from ps timescale 37 onwards. The similar phenomenon was reported previously by B. Koopmans et al. Fortunately, what we should be focused on here is in the ultrafast demagnet ization timescale, in which the effect of heat transport can be neglected. In fact, as is shown in Fig. 4(b), less than 10 fs variation was induced with the thicknesses ranged from 1.2 nm to 2.8 nm . The parameters used in the simulation is given in Table.1 . 38 Fig. 4(a) Dependence of demagnetization as a function of delay time after pulsed laser heating at 0t (b) Maximum demagnetization and demagnetization time versus the sample thickness. 39 Table 1: Parameters used in the M3TM12,13,15. Parameters Value Units 5400 ) /(23KmJ pC 61033.2 ) /(3KmJ epg 181005.4 ) /(3sKmJ DE 0.036 eV at 0.62 cT 630 K 0 90.7 ) /(smKJ sfa 0.185 40 Part Ⅳ Table . 1 Values of the main fit parameters of ultrafast demagnetizations curves for various thicknesses of the samples. References : dNi (nm) 0.4 200 860 2.3 330 100 0.8 170 860 2.1 330 100 1.0 150 860 2.0 330 100 1.5 120 860 2.3 330 100 2.0 90 860 2.0 330 100 )(fsM )fsE（ )(0ps )(fsG )(0fst 41 1、W. He, B. Hu, Q. F. Zhan, X. Q. Zhang, and Z. H. Cheng, Appl. Phys. Lett. 104, 142405 (2014). 2、H. S. Song, K. D. Lee, J. W. Sohn, S. H. Yang, Stuart S. P. Parkin, C. Y. You, and S. C. Shin, Appl. Phys. Lett. 103, 022406 (2013). 3、Y. Au, M. Dvornik, T. Davison, E. Ahmad, P. S. Keatley, A. Vansteenkiste, B. Van Waeyenberge, and V.V. Kruglyak, Phys. Rev. Lett. 110, 097201 (2013). 4、C.Y. Cheng, K. K. Meng, S. F. Li, J. H. Zhao, and T. S. Lai, Appl. Phys. Lett. 103, 232406 (2013). 5、Y. Au, T. Davison, E. Ahmad, P. S. Keatley, R. J. Hicken, and V. V. Kruglyak, Appl. Phys. Lett. 98, 122506 (2011). 6、F. Busse, M. Mansurova, B. Lenk, M. von der Ehe and M. Mü nzenberg, Sci. Rep. 5, 12824 (2015). 7、B. Lenk, G. Eilers, J. Hamrle, and M. Mü nzenberg, Phys. Rev. B 82,134443 (2010). 8、M. van Kampen, C. Jozsa, J.T. Kohlhepp, P. LeClair, L. Lagae, W. J.M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 (2002). 9、S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106, 117201 (2011) 10、W. He, T. Zhu, X. -Q. Zhang, H. -T. Yang, and Z. -H. Cheng, Sci. Rep. 3, 2883 (2013). 42 11、B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fä hnle, T. Roth, M. Cinchetti and M. Aeschlimann, Nat. Mater . 9, 259265 (2010). 12、K. C. Kuiper, G. Malinowski, F. Dalla Longa, and B. Koopmans, J. Appl. Phys 109, 07D316 (2011). 13、K. C. Kuiper, T. Roth, A. J. Schellekens, O. Schmitt, B. Koopmans, M. Cinchetti, and M. Aeschlimann, Appl. Phys. Lett. 105, 202402 (2014). 14、L.I. Berger, Optical properties of selected inorganica and organic solids, in Handbook of Chemistry and Physic s, 88th Edition, p. 12 -144 - 12-159. 3.1.5, 3.4 15、U. A. Macizo, Modeling of ultrafast laser -induced magnetization dynamics within the Landau -Lifshitz -Bloch approach , PhD Thesis 2012. P.78

2017-12-20

The ability to controllably manipulate the laser-induced ultrafast magnetic dynamics is a prerequisite for future high speed spintronic devices. The optimization of devices requires the controllability of the ultrafast demagnetization time, , and intrinsic Gilbert damping, . In previous attempts to establish the relationship between and , the rare-earth doping of a permalloy film with two different demagnetization mechanism is not a suitable candidate. Here, we choose Co/Ni bilayers to investigate the relations between and by means of time-resolved magneto-optical Kerr effect (TRMOKE) via adjusting the thickness of the Ni layers, and obtain an approximately proportional relation between these two parameters. The remarkable agreement between TRMOKE experiment and the prediction of breathing Fermi-surface model confirms that a large Elliott-Yafet spin-mixing parameter is relevant to the strong spin-orbital coupling at the Co/Ni interface. More importantly, a proportional relation between and in such metallic films or heterostructures with electronic relaxation near Fermi surface suggests the local spin-flip scattering domains the mechanism of ultrafast demagnetization, otherwise the spin-current mechanism domains. It is an effective method to distinguish the dominant contributions to ultrafast magnetic quenching in metallic heterostructures by investigating both the ultrafast demagnetization time and Gilbert damping simultaneously. Our work can open a novel avenue to manipulate the magnitude and efficiency of Terahertz emission in metallic heterostructures such as the perpendicular magnetic anisotropic Ta/Pt/Co/Ni/Pt/Ta multilayers, and then it has an immediate implication of the design of high frequency spintronic devices.

Unifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni bilayers with electronic relaxation near the Fermi surface

1712.07323v1

Role of nonlinear anisotropic damping in the magnetization dynamics of topological solitons Joo-V on Kim Institut d’Electronique Fondamentale, Univ. Paris-Sud, 91405 Orsay, France and CNRS, UMR 8622, 91405 Orsay, France (Dated: May 31, 2021) The consequences of nonlinear anisotropic damping, driven by the presence of Rashba spin-orbit coupling in thin ferromagnetic metals, are examined for the dynamics of topological magnetic solitons such as domain walls, vortices, and skyrmions. The damping is found to a ect Bloch and N ´eel walls di erently in the steady state regime below Walker breakdown and leads to a monotonic increase in the wall velocity above this transition for large values of the Rashba coe cient. For vortices and skyrmions, a generalization of the damping tensor within the Thiele formalism is presented. It is found that chiral components of the damping a ect vortex- and hedgehog-like skyrmions in di erent ways, but the dominant e ect is an overall increase in the viscous-like damping. PACS numbers: 75.60.Ch, 75.70.Kw, 75.75.-c, 75.78.Fg I. INTRODUCTION Dissipation in magnetization dynamics is a longstanding problem in magnetism [1–3]. For strong ferromagnets such as cobalt, iron, nickel, and their alloys, a widely used theoretical approach to describe damping involves a local viscous form due to Gilbert for the Landau-Lifshitz equation of motion, @m @t= 0mHe+0m@m @t; (1) which appears as the second term on the right hand side, pro- portional to the damping constant 0. This equation describes the damped magnetization precession about a local e ective fieldHe=(1=0Ms)U=m, which is given by a variational derivative of the magnetic energy Uwith respect to the mag- netization field described by the unit vector m, with 0=0 being the gyromagnetic constant and Msis the saturation mag- netization. Despite the multitude of physical processes that underlie dissipation in such materials, such as the scattering of magnons with electrons, phonons, and other magnons, the form in Eq. (1) has proven to be remarkably useful for describ- ing a wide range of dynamical phenomena from ferromagnetic resonance to domain wall motion. One feature of the dissipative dynamics described in Eq. (1) is that it is local, i.e., the damping torque only depends on the local magnetization and its time dependence. With the ad- vent of magnetic heterostructures, however, this restriction of locality has been shown to be inadequate for systems such as metallic multilayers in which nonlocal processes can be important [4]. A striking example involves spin pumping, which describes how spin angular momentum can be dissi- pated in adjacent magnetic or normal metal layers through the absorption of spin currents generated by a precessing magne- tization [5, 6]. Early experimental observations of this phe- nomena involved iron films sandwiched by silver layers [7] and permalloy films in close proximity with strong spin-orbit normal metals such as palladium and platinum [8, 9], where joo-von.kim@u-psud.frferromagnetic resonance line widths were shown to depend strong on the composition and thickness of the adjacent lay- ers. Such observations also spurred other studies involving ferromagnetic multilayers separated by normal metal spacers, where spin pumping e ects can lead to a dynamic coupling between the magnetization in di erent layers [10, 11]. In the context of damping, such dynamic coupling was shown to give rise to a configuration dependent damping in spin-valve structures [12, 13]. A generalization of the spin-pumping picture in the context of dissipation was given by Zhang and Zhang, who proposed that spin currents generated within the ferromagnetic material itself can lead to an additional contribution to the damping, provided that large magnetization gradients are present [14]. This theory is based on an sdmodel in which the local mo- ments (4 d) are exchange coupled to the delocalized conduc- tion electrons (3 s), which are treated as a free electron gas. The spin current “pumped” at one point in the material by the precessing local moments are dissipated at another if the current encounters strong spatial variations in the magneti- zation such as domain walls or vortices – a mechanism that can be thought of as the reciprocal process of current-induced spin torques in magnetic textures [15–18]. For this reason, the mechanism is referred to as “feedback” damping since the pumped spin currents generated feed back into the magnetiza- tion dynamics in the form of a dissipative torque. This addi- tional contribution is predicted to be both nonlinear and non- local, and can have profound consequences for the dynamics of topological solitons such as domain walls and vortices as a result of the spatial gradients involved. Indeed, recent experi- ments on vortex wall motion in permalloy stripes indicate that such nonlinear contributions can be significant and be of the same order of magnitude as the usual Gilbert damping char- acterized by 0in Eq. (1) [19]. An extension to this feedback damping idea was proposed recently by Kim and coworkers, who considered spin pump- ing involving a conduction electron system with a Rashba spin-orbit coupling (RSOC) [20]. By building upon the Zhang-Zhang formalism, it was shown that the feedback damping can be expressed as a generalization of the Landau-arXiv:1502.04695v2 [cond-mat.mtrl-sci] 4 Jun 20152 Lifshitz equation [14, 20], @m @t= 0mHe+mD LL(m)
@m @t; (2) where the 33 matrixDLLrepresents the generalized damping tensor, which can be expressed as [20] Di j LL=0i j+X k(Fki+˜R3ki) Fk j+˜R3k j :(3) Here,0is the usual Gilbert damping constant, = gB~G0=(4e2Ms) is a constant related to the conductivity G0 of the spin bands [14], Fki=(@m=@xk)iare components of the spatial magnetization gradient, ˜ R=2Rme=~2is the scaled Rashba coe cient,i jkis the Levi-Civita symbol, and the indices ( i jk) represent the components ( xyz) in Cartesian coordinates. In addition to the nonlinearity present in the Zhang-Zhang picture, the inclusion of the Rterm results in an anisotropic contribution that is related to the underly- ing symmetry of the Rashba interaction. Numerical estimates based on realistic parameters suggest that the Rashba con- tribution can be much larger than the nonlinear contribution alone [20], which may have wide implications for soliton dynamics in ultrathin ferromagnetic films with perpendicular magnetic anisotropy, such as Pt /Co material systems, in which large spin-orbit e ects are known to be present. In this article, we explore theoretically the consequences of the nonlinear anisotropic damping given in Eq. (3) on the dynamics of topological magnetic solitons, namely domain walls, vortices, and skyrmions, in which spatial gradients can involve 180rotation of the magnetization vector over length scales of 10 nm. In particular, we examine the role of chiral- ity in the Rashba-induced contributions to the damping, which are found to a ect chiral solitons in di erent ways. This ar- ticle is organized as follows. In Section II, we discuss the eects of nonlinear anisotropic damping on the dynamics of Bloch and N ´eel domain walls, where the latter is stabilized by the Dzyaloshinskii-Moriya interaction. In Section III, we examine the consequences of this damping for vortices and skyrmions, and we derive a generalization to the damping dyadic appearing in the Thiele equation of motion. Finally, we present some discussion and concluding remarks in Sec- tion IV. II. BLOCH AND N ´EEL DOMAIN WALLS The focus of this section are domain walls in ultrathin films with perpendicular magnetic anisotropy. Consider a 180domain wall representing a boundary separating two oppositely magnetized domains along the xaxis, with zbe- ing the uniaxial anisotropy axis that is perpendicular to the film plane. We assume that the magnetization remains uni- form along the yaxis. The unit magnetization vector m(x;t) can be parametrized in spherical coordinates ( ;), such that m=(sincos;sinsin;cos). With this definition, thespherical angles for the domain wall profile can be written as (x;t)=2 tan1exp xX0(t)
! ; (x;t)=0(t); (4) where X0(t) denotes the position of the domain wall,
=pA=K0represents the wall width parameter that depends on the exchange constant Aand the e ective uniaxial anisotropy K0, and the azimuthal angle 0(t) is a dynamic variable but spatially uniform. The anisotropy constant, K0=Ku 0M2 s=2, involves the di erence between the magnetocrys- talline ( Ku) and shape anisotropies relevant for an ultrathin film. In this coordinate system, a static Bloch wall is given by 0==2, while a static N ´eel wall is given by 0=0;. A positive sign in the argument of the exponential function for in Eq. (4) describes an up-to-down domain wall profile go- ing along the +xdirection, while a negative sign represents a down-to-up wall. To determine the role of the nonlinear anisotropic damping term in Eq. (3) on the wall dynamics, it is convenient to com- pute the dissipation function W(˙X0;˙0) for the wall variables, where the notation ˙X0@tX0, etc., denotes a time derivative. The dissipation function per unit surface area is given by W˙X0;˙0 =Ms 2 Z1 1dx˙miDi j LL(m) ˙mj; (5) where mi=mixX0(t);0(t)and the Einstein summation convention is assumed. By using the domain wall ansatz (4), the integral in Eq. (5) can be evaluated exactly to give W=W0+WNL, where W0represents the usual (linear) Gilbert damping, W0=0Ms
0BBBB@˙X2 0
2+˙2 01CCCCA; (6) while WNLis the additional contribution from the nonlinear anisotropic damping, WNL=Ms
2666641 33sin20(t)˙X2 0
2 + 2 31 22cos0(t)+3cos20(t)! ˙2 0# ;(7) where1=
2,2˜R=
, and3˜2 Rare dimen- sionless nonlinear damping constants. In contrast to the linear case, the nonlinear anisotropic dissipation function exhibits a configuration-dependent dissipation rate where the prefactors of the ˙X2 0and˙2 0terms depend explicitly on 0(t). In addition to the nonlinearity a chiral damping term, pro- portional to 2, appears as a result of the Rashba interaction and is linear in the Rashba coe cientR. The sign of this term depends on the sign chosen for the polar angle in the wall profile (4). To illustrate the chiral nature of this term, we consider small fluctuations about the static configuration by writing0(t)=0+(t), where(t)is a small angle. This approximation is useful for the steady state regime below3 Walker breakdown. For up-to-down Bloch walls ( 0==2), the nonlinear part of the dissipation function to first order in (t) becomes WNL;BlochMs
2666643 3˙X2 0
2+ 21 3+Cx2 2(t)! ˙2 0377775:(8) The quantity Ci=1 is a component of the chirality vec- tor [21], C=1 Z1 1dxm@xm; (9) which characterizes the handedness of the domain wall. For a right-handed Bloch wall, 0==2 and the only nonva- nishing component is Cx=1, while for a left-handed wall (0==2) the corresponding value is Cx=1. Thus, the term proportional to 2depends explicitly on the wall chiral- ity. Similarly for up-to-down N ´eel walls, the same lineariza- tion about the static wall profile leads to WNL;NeelMs
21 3+Cy2 2+3! ˙2 0; (10) where Cy=1 for a right-handed N ´eel wall (0=0) and Cy=1 for its left-handed counterpart ( 0=). Since the fluctuation(t) is taken to be small, the chiral damping term is more pronounced for N ´eel walls in the steady-state velocity regime since it does not depend on the fluctuation amplitude (t) as in the case of Bloch walls. To better appreciate the magnitude of the chirality- dependent damping term, it is instructive to estimate numer- ically the relative magnitudes of the nonlinear damping con- stants1;2;3. Following [Ref. 20], we assume =0:2 nm2 andR=1010eV m. If we suppose
= 10 nm, which is consistent with anisotropy values measured in ultrathin films with perpendicular anisotropy [22], the damping constants can be evaluated to be 1=0:002,2=0:052, and3=1:37. Since0varies between 0.01–0.02 [23] and 0.1–0.3 [24] de- pending on the material system, the chiral term 2is compa- rable to Gilbert damping in magnitude, but remains almost an order of magnitude smaller than the nonlinear component 3 that provides the dominant contribution to the overall damp- ing. The full equations of motion for the domain wall dynam- ics can be obtained using a Lagrangian formalism that ac- counts for the dissipation given by W[25, 26]. For the sake of simplicity, we will focus on wall motion driven by mag- netic fields alone, where a spatially-uniform magnetic field Hzis applied along the +zdirection. In addition, we include the Dzyaloshinskii-Moriya interaction appropriate for the ge- ometry considered [27, 28] when considering the dynamics of N ´eel walls. From the Euler-Lagrange equations with the Rayleigh dissipation function, d dt@L @˙X0@L @X0+@W @˙X0=0; (11) with an analogous expression for 0, the equations of motion for the wall coordinates are found to be ˙0+ 0+3 3sin20˙X0
= 0Hz; (12) 0 5 10 15 20 m0Hz (mT)010 20 30 Wall Velocity (m/s) 0 0.1 0.2 0.3 0.4 aR (eV nm)0.9 0.95 1vw / v w,0 aR (eV nm)aR = 0 0.05 eV nm 0.1 eV nm 0.15 eV nm (a) (b) (c) 0 0.1 0.2 0.3 0.4 0.15 0.2 0.25 df w / pFIG. 1. (Color online) Bloch wall dynamics. (a) Steady-state domain wall velocity,h˙X0i, as a function of perpendicular applied magnetic field,0Hz, for several values of the Rashba coe cient,R. The horizontal dashed line indicates the Walker velocity and the arrows indicate the Walker transition. (b) The ratio between the Walker ve- locity, vW, to its linear damping value, vW;0, as a function of R. (c) Deviation in the wall angle from rest at the Walker velocity, W, as a function of R ˙X0
0+21 3+2 2cos0+3cos20! ˙0 = 0  2Dex 0Ms
+2K? 0Mscos0! sin0;(13) where Dexis the Dzyaloshinskii-Moriya constant [28] and K?represents a hard-axis anisotropy that results from vol- ume dipolar charges. The Dzyaloshinskii-Moriya interaction (DMI) is present in ultrathin films in contact with a strong spin-orbit coupling material [29, 30] and favors a N ´eel-type wall profile [31, 32]. The DMI itself can appear as a con- sequence of the Rashba interaction and therefore its inclu- sion here is consistent with the nonlinear anisotropic damping terms used [20, 33, 34]. Results from numerical integration of these equations of motion for Bloch and N ´eel walls are presented in Figs. 1 and 2. We used parameters consistent with ultrathin films with perpendicular anisotropy, namely 0=0:1,Ms=1 MA /m,
= 10 nm, and K?=0NxM2 s=2 with the demagnetiza- tion factor Nx=0:02 [28]. To study the dynamics of the Dzyaloshinskii (N ´eel) wall we assumed a value of Dex=1 mJ/m2, which is much stronger than the volume dipolar in- teraction represented by K?and is of the same order of mag- nitude as values determined by Brillouin light spectroscopy in Pt/Co/Al2O3films [35]. As in the discussion on numeri- cal estimates above, we assumed =0:2 nm2but considered several di erent values for the Rashba coe cientR. The steady-state domain wall velocity, h˙X0i, was computed as a function of the perpendicular applied magnetic field, Hz. In4 the precessional regime above Walker breakdown in which 0(t) becomes a periodic function in time, h˙X0iis computed by averaging the wall displacement over few hundred periods of precession. For the Bloch case [Fig. 1(a)], the Walker field is observed to increase with the Rashba coe cient, which is consistent with the overall increase in damping experienced by the do- main wall. However, there are two features that di er qual- itatively from the behavior with linear damping. First, the Walker velocity is not attained for finite R, where the peak velocity at the Walker transition is below the value reached forR=0. This is shown in more detail in Fig. 1(b), where the ratio between the Walker velocity, vW, and its linear damp- ing value, vW;0, is shown as a function of R. The Walker limit is set by the largest extent to which the wall angle 0can de- viate from its equilibrium value, 0;eq. By assuming ˙=0 in the linear regime, we can determine this limit by rearrang- ing Eqs. 12 and 13 to obtain the following relationship for the Bloch wall, 2Hz NxMs= 0+3 3sin20 sin 20: (14) The angle0=Wfor which the right hand side of this equation is an extremum determines the Walker limit. In Fig. 1(c), we present this limit in terms of the deviation an- gle,WjW0;eqj, which is shown as a function of R. As the Rashba parameter is increased, the maximum wall tilt possible in the linear regime decreases from the linear damp- ing value of =4, which results in an overall reduction in the Walker velocity. Second, the field dependence of the wall ve- locity below Walker breakdown is nonlinear and exhibits a slight convex curvature, which becomes more pronounced as Rincreases. This curvature can be understood by examining the wall mobility under fields, which can be deduced from Eq. (12) by setting ˙=0, ˙X0= 0
0+(3=3)sin20Hz: (15) Since the angle 0for Bloch walls varies from its rest value of 0;eq==2 at zero field to Wat the Walker field, the sin20 term in the denominator decreases from its maximum value of sin20;eq=1 at rest with increasing applied field and therefore an increase in the mobility is seen as Hzincreases, resulting in the convex shape of the velocity versus field relation below Walker breakdown. It is interesting to note that the nonlinear damping terms aect the Dzyaloshinskii (N ´eel) wall motion di erently. In contrast to the Bloch case, the Walker velocity for increasing Rslightly exceeds the linear damping value, which can be seen by the arrows marking the Walker transition in Fig. 2(a) and in detail in Fig. 2(b). In addition, the field dependence of the velocity exhibits a concave curvature below breakdown, which can also be understood from Eq. (15) by considering that0instead deviates from the rest value of 0;eq=0 or at zero field. As for the Bloch wall case, the deviation angle at breakdown is determined by the value of 0that gives an 0 50 100 150 200 0100 200 300 Wall Velocity (m/s) 0 0.1 0.2 0.3 0.4 11.0008 aR = 0 0.05 eV nm 0.1 eV nm 0.15 eV nm vw / v w,0 aR (eV nm) aR (eV nm)m0Hz (mT)(a) (b)(c) 0 0.1 0.2 0.3 0.4 0.5 0.51 0.52 0.53 df w / pFIG. 2. (Color online) Dzyaloshinskii (N ´eel) wall dynamics. (a) Steady-state domain wall velocity, h˙X0i, as a function of perpendic- ular applied magnetic field, 0Hz, for several values of the Rashba coecient,R. The horizontal dashed line indicates the Walker ve- locity and the arrows indicate the Walker transition. (b) The ratio between the Walker velocity, vW, to its linear damping value, vW;0, as a function of R. (c) The wall angle at the Walker velocity, W, as a function of R extremum for the right hand side of 2Hz NxMs= 0+3 3sin20 Dex 2K?
cos0+sin 20! ;(16) and is also seen to decrease with increasing Rashba coe - cient [Fig. 2(c)]. In contrast to the Bloch wall case, how- ever, changes in Whave a comparatively modest e ect on the Walker velocity. The shape of the velocity versus field curve is consistent with experimental reports of field-driven domain wall motion in the Pt /Co (0.6 nm) /Al2O3system [36], which possess a large DMI value [35] and harbors N ´eel-type domain wall profiles at equilibrium [37]. As the preceding discussion shows, the di erences in the field dependence of the wall velocity for the two profiles are a result of the DMI, rather than the chiral damping term that is proportional to 2. This was verified by setting 2=0 for the N ´eel wall case with D,0, which did not modify the overall behavior of the field dependence of the velocity. In the one-dimensional approximation for the wall dynamics, the DMI enters the equations of motion like an e ective magnetic field along the xaxis, which stabilizes the wall structure by minimizing deviations in the wall angle 0(t). III. VORTICES AND SKYRMIONS The focus of this section is on the dissipative dynamics of two-dimensional topological solitons such as vortices and skyrmions. The equilibrium magnetization profile for these5 micromagnetic objects are described by a nonlinear di er- ential equation similar to the sine-Gordon equation, where the dispersive exchange interaction is compensated by dipo- lar interactions for vortices [38, 39] and an additional uniax- ial anisotropy for skyrmions [40]. The topology of vortices and skyrmions can be characterized by the skyrmion winding number Q, Q=1 4" dxdy m
 @xm@ym : (17) While the skyrmion number for vortices ( Q=1=2) and skyrmions ( Q=1) are di erent, their dynamics are quali- tatively similar and can be described using the same formal- ism. For this reason, vortices and skyrmions will be treated on equal footing in what follows and distinctions between the two will only be drawn on the numerical values of the damp- ing parameters. A key approximation used for describing vortex or skyrmion dynamics is the rigid core assumption, where it is assumed that the spin structure of the soliton remains unper- turbed from its equilibrium state during motion. Within this approximation, the dynamics is given entirely by the position of the core in the film plane, X0(t)=[X0(t);Y0(t)], which al- lows the unit magnetization vector to be parametrized as (x;y;t)=0[kxX0(t)k]; (x;y;t)=qtan1"yY0(t) xX0(t)# +c 2; (18) where qis a topological charge and cis the chirality. An il- lustration of the magnetization field given by the azimuthal angle(x;y) is presented in Fig. 3. q=1 corresponds to a vortex or skyrmion, while q=1 represents the antivortex or antiskyrmion. The dynamics of a vortex or skyrmion in the rigid core ap- proximation is given by the Thiele equation, G˙X0+DT
˙X0=@U @X0; (19) where G=Msd " dxdy sin()(rr) (20) is the gyrovector and U(X0) is the e ective potential that is ob- tained from the magnetic Hamiltonian by integrating out the spatial dependence of the magnetization. The damping dyadic in the Thiele equation, DT, can be obtained from the dissipa- tion function in the rigid core approximation, W(˙X0), which is defined in the same way as in Eq. (5) but with the ansatz given in Eq. (18). For this system, it is more convenient to eval- uate the dyadic by performing the integration over all space after taking derivatives with respect to the core velocity. In other words, the dyadic can be obtained using the Lagrangian formulation by recognizing that DT
˙X0=Msd 2 " dxdy@ @˙X0 ˙miDi j LL(m) ˙mj : (21) c = 0 q = +1 q = –1c = 1 c = 2 c = 3 (a) (b) (c) 1 – 1 0FIG. 3. (Color online) In-plane magnetization fields for vortices and skyrmions. (a) Vector fields given by (x;y) in (18) for di erent values of qandc. (b) V ortex and (c) skyrmion for spin structure with c=1;q=1, where the arrows indicate the in-plane components (mx;y) and the color code gives the perpendicular component of the magnetization ( mz). By using polar coordinates for the spatial coordinates, ( x;y)= (rcos';rsin'), assuming translational invariance in the film plane, and integrating over ', the damping dyadic is found to be DT=Msd (0D0+1D1+3D3)I+2D2" a110 0a22#! ; (22) whereIis the 22 identity matrix and the dimensionless damping constants are defined as 1=r2 c,2˜R=rc, and 3˜2 R, in analogy with the domain wall case where the core radius rcplays the role here as the characteristic length scale. The coe cients Didepend on the core profile and are given by D0=Z1 0dr r(@r0)2+sin20 r! ; (23) D1=2r2 cZ1 0dr1 r(@r0)2sin20; (24) D2=2rcZ1 0dr1 r(@r0)sin0(r(@r0)cos0+sin0); (25) D3=Z1 0dr r(@r0)2cos20+sin20 r! ; (26) where the expression for D0is a known result but D1;D2and D3are new terms that arise from the nonlinear anisotropic damping due to RSOC. The coe cients a11anda22are configuration-dependent and represent the chiral component of the Rashba-induced damping. For vortex-type spin textures ( c=1;3 and q=1),6 TABLE I. Coe cients a11anda22of the chiral damping term in Eq. (22) for di erent vortex /skyrmion charges qand chirality c. q=1 q=1 c 0 1 2 3 0 1 2 3 a11 1 01 011 1 1 a22 1 01 0 111 1 a11=a22=0, which indicates that the 2term plays no role for such configurations. This is consistent with the result for Bloch domain walls discussed previously, since the vortex- type texture [Fig. 3(b)], particularly the vortex-type skyrmion [Fig. 3(c)], can be thought of as being analogous to a spin structure generated by a 2 revolution of a Bloch wall about an axis perpendicular to the film plane. The rigid core approx- imation implies that fluctuations about the ground state are ne- glected, which is akin to setting (t)=0 in Eq. (8). As such, no contribution from 2is expected for vortex-type textures. On the other hand, a finite contribution appears for hedgehog- type vortices and skyrmions ( q=1), where a11=a22=1 forc=0 and a11=a22=1 for c=2. This can be un- derstood with the same argument by noting that hedgehog- type textures can be generated by revolving N ´eel-type domain walls. A summary of these coe cients is given in Table I. For antivortices ( q=1), it is found that the coe cients aiiare nonzero for all winding numbers considered. We can understand this qualitatively by examining how the magneti- zation varies across the core along two orthogonal directions. For example, for c=0, the variation along the xandyaxes across the core are akin to two N ´eel-type walls of di erent chiralities, which results in nonvanishing contributions to a11 anda22but with opposite sign. The sign of these coe cients depends on how these axes are oriented in the film plane, as witnessed by the di erent chiralities cin Fig. 3. Such damping dynamics is therefore strongly anisotropic, which may have interesting consequences on the rotational motion of vortex- antivortex dipoles, for example, where the antivortex configu- ration oscillates between the di erent cvalues in time [41]. For vortex structures, we can provide numerical estimates of the di erent damping contributions iDiby using the Usov ansatz for the vortex core magnetization, cos0=8>>>><>>>>:r2 cr2 r2c+r2rrc 0 r>rc: (27) LetLrepresent the lateral system size. The coe cients Diare then found to be D0=[2+ln(L=rc)],D1=D2=14=3, andD3=[4=3+ln(L=rc)]. We note that for D0andD3, the system size Land core radius rcappear as cuto s for the divergent 1=rterm in the integral. By assuming parameters of 0=0:1,=0:05 nm2, andR=0:1 eV nm, along with typical scales of rc=10 nm and L=1m, the damping terms can be evaluated numerically to be 0D02:1,1D1 0:0073,2D20:19, and3D36:4. As for the domain walls, the Rashba term 3D3is the dominant contribution and is of the same order of magnitude as the linear damping term,while the chiral term 2D2is an order of magnitude smaller and the nonlinear term 1D1is negligible in comparison. For skyrmion configurations, a similar ansatz can be used for the core magnetization, cos0 2 =8>>>><>>>>:r2 cr2 r2c+r2rrc 0 r>rc: (28) We note that this di ers from the “linear” profiles discussed elsewhere [40], but the numerical di erences are small and do not influence the qualitative features of the dynamics. The ad- vantage of the ansatz in Eq. (28) is that the integrals for Di have simple analytical expressions. Because spatial variations in the magnetization for a skyrmion are localized only to the core, in contrast to the circulating in-plane moments of vor- tices that extend across the entire system, the damping con- stants Dihave no explicit dependence on the system size. By using Eq. (28), we find D0=D3=16=3,D1=496=15, and D2=52=5. By using the same values of 0,, andRas for the vortices in the preceding paragraph, we find 0D01:7, 1D10:052,2D20:43, and3D33:3. IV . DISCUSSION AND CONCLUDING REMARKS A clear consequence of the nonlinear anisotropic damp- ing introduced in Eq. (3) is that it provides a mechanism by which the overall damping constant, as extracted from domain wall experiments, for example, can di er from the value ob- tained using linear response methods such as ferromagnetic resonance [19]. However, the Rashba term can also a ect the ferromagnetic linewidth in a nontrivial way. To see this, we consider the e ect of the damping by evaluating the dissipa- tion function associated with a spin wave propagating in the plane of a perpendicularly magnetized system with an ampli- tude b(t) and wave vector kjj. The spin wave can be expressed asm=b(t) cos( kjj
rjj);b(t) sin(kjj
rjj);1, which results in a dissipation function per unit volume of Wsw=Ms 2 ˙b(t)2 0+3+b(t)2kkjjk2 ; (29) where a term proportional to the chiral part ˜Rspatially aver- ages out to zero. The Rashba contribution 3˜2 Rleads to an overall increase in the damping for linear excitations and plays the same role as the usual Gilbert term 0in this ap- proximation, which allows us to assimilate the two terms as an eective FMR damping constant, FMR0+3. On the other hand, the nonlinear feedback term proportional to is only important for large spin wave amplitudes and de- pends quadratically on the wave vector. This is consistent with recent experiments on permalloy films (in the absence of RSOC) in which the linear Gilbert damping was recovered in ferromagnetic resonance while nonlinear contributions were only seen for domain wall motion [19]. This result also sug- gests that the large damping constant in ultrathin Pt /Co/Al2O3 films as determined by similar time-resolved magneto-optical7 microscopy experiments, where it is found that FMR=0:1– 0:3 [24], may partly be due to the RSOC mechanism described here (although dissipation resulting from spin pumping into the platinum underlayer is also likely to be important [42]). Incidentally, the nonlinear term b(t)2may provide a physi- cal basis for the phenomenological nonlinear damping model proposed in the context of spin-torque nano-oscillators [43]. For vortices and skyrmions, the increase in the overall damping due to the Rashba term 3can have important con- sequences for their dynamics. The gyrotropic response to any force, as described by the Thiele equation in Eq. (19), depends on the overall strength of the damping term. This response can be characterized by a deflection angle, H, that describes the degree to which the resulting displacement is noncollinear with an applied force. This is analogous to a Hall e ect. By neglecting the chiral term 2D2, the deflection or Hall angle can be deduced from Eq. (19) to be tanH=G0 0D0+1D1+3D3; (30) where G0=2for vortices and G0=4for skyrmions. Con- sider the skyrmion profile and the magnetic parameters dis- cussed in Section III. With only the linear Gilbert damping term (0D0) the Hall angle is found to be H=82:3, which underlies the largely gyrotropic nature of the dynamics. If the full nonlinear damping is taken into account [Eq. (30)], we findH=68:3, which represents a significant reduction in the Hall e ect and a greater Newtonian response to an ap- plied force. Aside from a quantitative increase in the overalldamping, the presence of the nonlinear terms can therefore af- fect the dynamics qualitatively. Such considerations may be important for interpreting current-driven skyrmion dynamics in racetrack geometries, where the interplay between edge re- pulsion and spin torques is crucial for determining skyrmion trajectories [44, 45]. Finally, we conclude by commenting on the relevance of the chiral-dependent component of the damping term, 2. It has been shown theoretically that the Rashba spin-orbit cou- pling leading to Eq. (3) also gives rise to an e ective chiral interaction of the Dzyaloshinskii-Moriya form [34]. This in- teraction is equivalent to the interface-driven form considered earlier, which favors monochiral N ´eel wall structures in ul- trathin films with perpendicular magnetic anisotropy. Within this picture, a su ciently strong Rashba interaction should only favor domain wall or skyrmion spin textures with one given chirality as determined by the induced Dzyaloshinskii- Moriya interaction. 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2015-02-16

The consequences of nonlinear anisotropic damping, driven by the presence of Rashba spin-orbit coupling in thin ferromagnetic metals, are examined for the dynamics of topological magnetic solitons such as domain walls, vortices, and skyrmions. The damping is found to affect Bloch and N\'eel walls differently in the steady state regime below Walker breakdown and leads to a monotonic increase in the wall velocity above this transition for large values of the Rashba coefficient. For vortices and skyrmions, a generalization of the damping tensor within the Thiele formalism is presented. It is found that chiral components of the damping affect vortex- and hedgehog-like skyrmions in different ways, but the dominant effect is an overall increase in the viscous-like damping.

Role of nonlinear anisotropic damping in the magnetization dynamics of topological solitons

1502.04695v2

Brownian motion of magnetic domain walls and skyrmions, and their diusion constants Jacques Miltat,Stanislas Rohart, and Andr e Thiaville Laboratoire de Physique des Solides, Universit e Paris-Sud, Universit e Paris-Saclay, CNRS, UMR 8502, F-91405 Orsay Cedex, France (Dated: October 8, 2018) Extended numerical simulations enable to ascertain the diusive behavior at nite temperatures of chiral walls and skyrmions in ultra-thin model Co layers exhibiting symmetric - Heisenberg - as well as antisymmetric - Dzyaloshinskii-Moriya - exchange interactions. The Brownian motion of walls and skyrmions is shown to obey markedly dierent diusion laws as a function of the damping parameter. Topology related skyrmion diusion suppression with vanishing damping parameter, albeit already documented, is shown to be restricted to ultra-small skyrmion sizes or, equivalently, to ultra-low damping coecients, possibly hampering observation. I. INTRODUCTION The prospect of ultra-small stable information bits in magnetic layers in presence of the Dzyaloshinskii-Moriya (DM) interaction [1] combined to the expectation of their minute current propagation [2], notably under spin-orbit torques [3], builds up a new paradigm in information technology. In stacks associating a metal with strong spin-orbit interactions e.g. Pt and a ferromagnetic metal such as Co, that may host isolated skyrmions, large do- main wall velocities have also been forecast [4] and ob- served [5]. The DM interaction induces chiral magnetiza- tion textures, walls or skyrmions, that prove little prone to transformations of their internal structure, hence their extended stability and mobility. In order, however, to achieve low propagation cur- rents, steps will need to be taken towards a reduction of wall- or skyrmion-pinning. Recent experimental stud- ies indicate that skyrmions fail to propagate for cur- rents below a threshold roughly equal to 2 1011Am2for [Pt/Co/Ta]nand [Pt/CoFeB/MgO]nmultilayers [6], or 2:5 1011Am2for [Pt/(Ni/Co/Ni)/Au/(Ni/Co/Ni)/Pt] symmetrical bilayers [7]. Only in one seldom instance did the threshold current fall down to about 2 :5 1010Am2 for a [Ta/CoFeB/TaO] stack, still probably, however, one order of magnitude higher than currents referred to in simulation work applying to perfect samples [8]. In a wall within a Co stripe 50 nm wide, 3 nm thick, the number of spins remains large, typically 216for a 5 nm wide wall. A skyrmion within a Co monolayer (ML) over Pt or Ir, on the other hand, contains a mere 250 spins, say 28. Assuming that a sizeable reduction of pinning might somehow be achieved, then a tiny structure such as a skyrmion is anticipated to become sensitive, if not extremely sensitive, to thermal uctuations. In this work, we show, on the basis of extended nu- merical simulations, that both chiral walls and skyrmions within ferromagnets obey a diusion law in their Brow- nian motion at nite temperature [9, 10]. The diusion jacques.miltat@u-psud.fr x z q wS tS L a) b) q FIG. 1. a) Wall within a narrow stripe: wSis the stripe width, tSits thickness. The stripe element length Lis solely dened for computational purposes. qis the wall displacement; b) snapshot of the magnetization distribution: color coding after mx. The wall region mx1 appears red. Thermal uctua- tions are visible within domains: T= 25 K,wS= 100 nm, tS= 0:6 nm,= 0:5. law is shown to be valid over a broad range of damp- ing parameter values. The thermal diusion of domain walls seems to have attracted very little attention, ex- cept for walls in 1D, double potential, structurally un- stable, lattices [11], a source of direct inspiration for the title of this contribution. Chiral magnetic domain walls are found below to behave classically with a mobil- ity inversely proportional to the damping parameter. As shown earlier [12, 13], such is not the case for skyrmions, a behavior shared by magnetic vortices [14]. Vortices and skyrmions in ferromagnetic materials are both charac- terized by a denite topological signature. In contradis- tinction, skyrmions in antiferromagnetic compounds are characterized by opposite sign spin textures on each sub- lattice, with, as a result, a classical, wall-like, dependence of their diusion constant [15]. Lastly, ferrimagnets do display reduced skyrmion Hall angles [16], most likely conducive to modied diusion properties.arXiv:1806.03172v1 [cond-mat.mes-hall] 8 Jun 20182 II. DOMAIN WALL DIFFUSION We examine here, within the micromagnetic frame- work, the Langevin dynamics of an isolated domain wall within a ferromagnetic stripe with thickness tS, width wSand nite length L(see Fig. 1). The wall is located at mid-position along the stripe at time t= 0. Thermal noise is introduced via a stochastic eld ~HRduncorrelated in space, time and component-wise, with zero mean and varianceproportional to the Gilbert damping parame- terand temperature T[17] : h~HRdi=~0 hHi Rd(~ r;t)Hj Rd(~r0;t0)i=ij(~ r~r0)(tt0) =2kBT 00MS(1) where,kBis Boltzmann constant, 0and 0are the vac- uum permability and gyromagnetic ratio, respectively, MSthe saturation magnetization. Written as such, the functions(~ r~r0) and(tt0) have the dimension of reciprocal volume and time, respectively. Applied to nu- merical simulations, the variance of the stochastic eld becomes=2kBT 00MSVdt, whereVis the computation cell volume and dtthe integration time step. A. Simulation results The full set of numerical simulations has been per- formed by means of an in-house code ported to graph- ical processing units (GPU's). Double precision has been used throughout and the GPU-specic version of the "Mersenne twister" [18] served as a source of long- sequence pseudo-random numbers generator. Material parameters have been chosen such as to mimic a 3-ML Co layer (thickness tS= 0:6 nm) on top of Pt with an exchange constant equal to A= 1011J/m, a Ms= 1:09 106A/m saturation magnetization, a Ku= 1:25 106J/m3uniaxial anisotropy constant allowing for a perpendicular easy magnetization axis within domains, and a moderate-to-high DM interaction (DMI) constant DDM= 2 mJ/m2. In order to temper the neglect of short wavelength excitations [19], the cell size has been kept down toLx=Ly= 1 nm, whilst Lz=tS= 0:6 nm. The stripe length has been kept xed at L= 1m, a value compatible with wall excursions within the ex- plored temperature range. The latter has, for reasons to be made clear later, been restricted to 1=3 of the presumed Curie temperature for this model Co layer. Fi- nally, the integration time constant, also the uctuating eld refresh time constant, has been set to dt= 25 fs. As shown by the snapshot displayed in Fig. 1b, the wall may acquire some (moderate) curvature and/or slanting during its Brownian motion. Because wall diusion is treated here as a 1D problem, the wall position qis de- 0510152025 487488489Time [ns]Wall Position [nm]ΔtqFIG. 2. Excerpt of a wall trace displaying wall position uctu- ations vstime:T= 77 K,= 0:5,wS= 100 nm,tS= 0:6 nm. qis the wall displacement during time interval
t. ned as the average position owing to : q=L NxNyPNx i=1PNy j=1mz(i;j) [hmziLhmziR](2) where,iandjare the computation cell indices, Nxand Nythe number of cells along the length and the width of the stripe, respectively, hmziLis the uctuations aver- aged value of the zmagnetization component far left of the domain wall,hmziRthe average value of mzfar right. Regardless of sign, hmziRandhmziLare expected to be equal in the absence of any Hzeld. Fig. 2 displays the position as a function of time of a wall within a wS= 100 nm wide stripe immersed in a T= 77 K temperature bath. A 2 ns physical time win- dow has been extracted from a simulation set to run for 1:5s. The gure shows short term wall position uc- tuations superimposed onto longer time diusion. Ac- cording to Einstein's theory of Brownian motion [9], the probability P(x;t) of nding a particle at position xat timetobeys the classical diusion equation @tP(x;t) = D@2 x2P(x;t) with, as a solution, a normal (gaussian) dis- tributionP(x;t) = 1=p 4Dtexp(x2=4Dt), whereDis the diusion constant. So does the raw probability of nding a (sti) wall in a narrow stripe at position qafter a time interval
t, as shown in Fig. 3 (see Fig. 2 for variable denition). It ought to be mentioned that the average wall displace- menthq(
t)iis always equal to 0, with an excellent ac- curacy, provided the overall computation time is large enough. The t to a normal distribution proves rather satisfactory, with, however, as seen in Fig. 3, a slightly increasing skewness in the distributions as a function of increasing
t. Skewness, however, 1) remains moderate3 05 1031 1041.5 1042 104 -30-20-100102030Δt = 0.2 nsN q - <q> [nm]05 1031 1041.5 1042 104 -30-20-100102030Δt = 0.5 ns q - <q> [nm]N 05 1031 1041.5 1042 104 -30-20-100102030Δt = 1.0 ns q - <q> [nm]N 05 1031 1041.5 1042 104 -30-20-100102030q - <q> [nm]Δt = 2.0 nsN FIG. 3. Wall within stripe: event statistics with time interval
tas a parameter; = 0:5,wS= 100 nm, tS= 0:6 nm, T= 25K. The continuous blue lines are ts to a gaussian distribution, the variance of which increases with
t. up to
tvalues typically equal to 5 10 ns, 2) is seen to reverse sign with time interval (compare Fig. 3b and c), excluding intrinsic biasing. The distributions standard deviation is clearly seen to increase with increasing
t. Alternatively, one may represent the variance hq2i (hqi= 0) as a function of the time interval
t: if diu- sion applies, then a linear dependence is expected, with a 2Dslope for a one-dimensional diusion. Fig.4a shows, for various temperatures, that a linear law is indeed ob- served. Lastly, as shown in Fig.4b, the diusion constant increases linearly with increasing temperature. The er- ror bars measuring the departure from strict linearity in Fig.4a remain limited in extent. For the stripe width and damping parameter considered here ( wS= 100 nm, = 0:5), the ratio of diusion constant to temperature is found to amount to D=T= 0:187 nm2ns1K1. B. Wall diusion constant (analytical) Thiele's equation [20] states that a magnetic texture moves at constant velocity ~ vprovided the equilibrium of 3 forces be satised: ~G~ v+D~ v=~F (3) where,~Fis the applied force, ~FG=~G~ vis the gyrotropic force,~Gthe gyrovector, ~FD=D~ vthe dissipation force, Dthe dissipation dyadic. For the DMI hardened N eel wall considered here : ~G= 050100150200250300 012345Δt [ns]< q2 > [nm2] 25° K50° K77° K120° K150° K a)0102030 050100150T [K]D [nm2 ns-1] b)FIG. 4. a) Variance hq2i(nm2) of the wall displacement vs time interval
twith temperature Tas a parameter. Thick lines represent a linear t to data; b) Diusion constant D as a function of temperature (square full symbols). Dis pro- portional to the slope of the hq2ivs
tcurves in Fig.4a (see text for details). The error bars are deduced from the slopes of straight lines through the origin that encompass all data points in Fig.4a for a given temperature and the t time bracket, 15 ns. For the sake of legibility, the error bars have been moved-up by 2 :5 units. Continuous line: linear t through the origin. The dashed line is the analytical ex- pectation in the "low" noise limit. = 0:5,wS= 100 nm, tS= 0:6 nm. ~0. For a 1D wall, the Thiele equation simply reads : Dxxvx=Fx (4) where,Dxx=0MS 0R V(@~ m @x)2d3r. The calculation proceeds in two steps, rst evaluate the force, hence, according to Eqn.4, the velocity auto- correlation functions, then integrate vstime in order to derivehq2i. The force, per denition, is equal to minus the partial derivative of the energy Ew.r.t. the displace- mentq, namelyFx=@E @q=0MSR V@~ m @x
~H d3r. Formally, hFx(t)Fx(t0)i= (0MS)2 (5)*Z V@~ m(~ r;t) @x
~H(~ r;t)d3rZ V@~ m(~r0;t0) @x
~H(~r0;t0)d3r0+ As noticed earlier [14], since the random eld noise is "multiplicative" [17], moving the magnetization vector out of the average brackets is, strictly speaking, not al- lowed, unless considering the magnetization vector to only marginally dier from its orientation and modulus in the absence of uctuations (the so-called "low" noise limit [14]): hFx(t)Fx(t0)i= (0MS)2 (6) Z VX i;j" @mi(~ r;t) @x@mj(~r0;t0) @xD Hi(~ r;t)Hj(~r0;t0)E# d3r d3r0 If due account is being taken of the fully uncorrelated4 character of the thermal eld (Eqn.1), the force auto- correlation function becomes: hFx(t)Fx(t0)i= 2kBTDxx(tt0) (7) The velocity auto-correlation function follows from Eqn.4. Lastly, time integration ( q(t) =Rt 0vx(t0)dt0) yields : hq2(t)i= 2Dt;D=kBT Dxx(8) In order to relate the diusion constant to a more directly recognizable wall mobility, Dxxmay be expanded as : Dxx=0MS 02wStS
T(9) where,
Thas been called the Thiele wall width (implic- itly dened in [21]). Dmay thus be expressed as : D=kBT 20MS1 wStS 0
T (10) thus, proportional to the wall mobility 0
T=. A directly comparable result may be obtained after constructing a full Langevin equation from the ( q;) equations of domain wall motion (Slonczewski's equa- tions [22]), where is the azimuthal magnetization angle in the wall mid-plane. In this context, the wall mobility isW= 0
=, where
is the usual wall width, inci- dentally equal to the Thiele wall width in the case of a pure Bloch wall. The Langevin equation [10] here reads: mD 2wStSd2 q2 dt2+1 220Ms WwStSd q2 dt=kBT(11) where,mDis D oring's wall mass density (kg =m2): mD= 1 +2
 0 20Ms21 jDDMj(12) an expression valid in the limit jDDMjKE= Ku1 20M2 s. Note that the DMI constant DDMex- plicitly enters the expression of the wall mass, as a con- sequence of the wall structure stiening by DMI. In the stationary regime, hq2iis proportional to time tand the wall diusion constant exactly matches Eqn.10, after sub- stitution of
Tby
. Finally, the characteristic time for the establishment of stationary motion is: t0=mD1 20Ms 0
(13) For the parameters of our model 3-ML Co layer on top of Pt, D oring's mass density is equal to 3 108kg=m2 for= 0:5, and the characteristic time amounts to t0'25 ps. Still for = 0:5,wS= 100 nm and tS= 0:6 nm,D=Tamounts to 0 :153 nm2ns1K1for
T= 4:13 nm, i.e. the value computed from a properly converged wall prole at T= 0. The relative dierence 0255075 050100150T [K]a)wS = 25 nmwS = 50 nmwS = 100 nmD [nm2 ns-1] 00.250.50.751 00.010.020.030.040.051/wS [nm-1]D /T [nm2 ns-1 K-1] b)FIG. 5. a) Diusion constant Das a function of temperature with the stripe width wSas a parameter (full symbols); b) D=Tas a function of the inverse of the stripe width. = 0:5, tS= 0:6 nm, throughout. Solid blue lines: linear t through the origin, dashed line: analytical expectation. between simulation and theoretical values is found to be of the order of20%. Owing to Eqn.10, Dis expected to prove inversely pro- portional to both the stripe width wSand the Gilbert damping parameter , a behavior conrmed by simula- tions. Fig.5a displays the computed values of the dif- fusion coecient as a function of temperature with the stripe width as a parameter, whilst Fig.5b states the lin- ear behavior ofDvswS1. The slope proves, however, some 13:5% higher than anticipated from Eqn.10. Lastly, the 1=dependence is veried in Fig.6 showing the com- puted variation of Dvstemperature with as a param- eter for a narrow stripe ( wS= 25 nm) as well as the corresponding dependence ofD=T. The dotted line represents Eqn.10 without any adjusting parameter. The relative dierence between simulation data and theoret- ical expectation is beyond, say = 0:25, seen to grow with increasing but also appears to be smaller for a narrow stripe as compared to wider tracks. Altogether, simulation results only moderately depart from theoretical predictions. The Brownian motion of a DMI-stiened wall in a track clearly proves diusive. The diusion constant is classically proportional to the wall mobility and inversely proportional to the damping parameter. Unsurprisingly, the smaller the track width, the larger the diusion constant. In order to provide an order of magnitude, the diusion induced displacement expectation,p 2D
t, for a wall sitting in a 100 nm-wide, pinning-free, track for 25 ns at T= 300 K proves essen- tially equal tothe stripe width. III. SKYRMION DIFFUSION Outstanding observations, by means of Spin Polarized Scanning Tunneling Microscopy, have revealed the exis- tence of isolated, nanometer size, skyrmions in ultra-thin5 0255075100125150175200 020406080100α = 0.125α = 0.25α = 0.5α = 0.075α = 0.05 α = 0.8a)T [K]D [nm2 ns-1] 0246810 -20-1001020 00.20.40.60.81D /T [nm2 ns-1 K-1](%) αb)wS = 25 nmwS = 50 nmwS = 100 nm FIG. 6. a) Diusion constant Das a function of temperature with the damping constant as a parameter ( wS= 25 nm, tS= 0:6 nm). Solid blue lines: linear t through the ori- gin; b)D=T(large semi-open symbols) as a function of for wS= 25 nm and tS= 0:6 nm; dotted blue curve: analyt- ical expectation. Full symbols: relative dierence between computational and analytical results (%). FIG. 7. a) Snapshot of a skyrmion immersed in a 12.5 K temperature bath ( = 0:5), together with the underlying lattice. Red cells: sz+1, blue cells: sz1. The white cross indicates the barycenter of lattice site positions satisfy- ingsz0:5. lms such as a PdFe bilayer on an Ir(1111) single crystal substrate [23] [24]. We analyse below the thermal motion of skyrmions in a model system made of a Co ML on top of Pt(111). We deal with skyrmions with a diameter of about 2:5 nm containing at T= 0 about 250 spins. A. Simulation results In order to monitor the Brownian motion of an iso- lated skyrmion, rather than micromagnetics, it is pre- ferred to simulate the thermal agitation of classical spins,~ s(jsj= 1), on a triangular lattice. Lat- tice eects and frequency cutos in thermal excitations are thus avoided. Such simulations have already been used e.g. for the determination of the barrier to col- lapse of an isolated skyrmion [25, 26]. The parame- FIG. 8. Example of skyrmion trajectory. Distances in atomic units (1 at:u:= 2:51A). The trajectory started at the origin of coordinates at time t= 0 and stopped at the cross location at physical time t100 ns.T= 25 K,= 1. ters are: lattice constant a= 2:51A, magnetic mo- mentAt= 2:1B/atom, Heisenberg exchange nearest neighbor constant J= 29 meV/bond, Dzyaloshinskii- Moriya exchange d=1:5 meV/bond, magnetocrys- talline anisotropy 0 :4 meV/atom. The stochastic eld is still dened by Eqn.1 after substitution of the prod- uctMSVby the magnetic moment per atom. The code features full magnetostatic (dipole-dipole) interactions. Fast Fourier Transforms implementation ensues from the decomposition of the triangular lattice into two rectangu- lar sublattices, at the expense of a multiplication of the number of dipole-dipole interaction coecients. Lastly, the base time step, also the stochastic eld refresh time, has been given a low value in view of the small atomic volume, namely dt= 2:5 fs for0:1,dt= 1 fs below. Time steps that small may be deemed little compatible with the white thermal noise hypothesis [17]. They are in fact dictated by the requirement for numerical stability, primarily w.r.t. exchange interactions. Fig.7 is a snapshot of an isolated skyrmion in the model Co ML with a temperature raised to 12 :5 K. The skyrmion is at the center of a 200 at. u.- i.e. 50 nm-size square computation window, that contains 46400 spins and is allowed to move with the diusing skyrmion. Do- ing so alleviates the computation load without restricting the path followed by the skyrmion. Free boundary con- ditions (BC's) apply. The window, however, proves su- ciently large to render the conning potential created by BC's ineective. The skyrmion position as a function of time is dened simply as the (iso)barycenter of the con- tiguous lattice site positions x(k),y(k), wheresz0:5: qSk x=1 KKX k=1x(k) ;qSk y=1 KKX k=1y(k) (14)6 01 1042 1043 1044 1045 1046 1047 1048 104 01 1042 1043 1044 1045 1046 1047 104 -50050NΔt = 0.2 ns qx,y - <qx,y> [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104 01 1042 1043 1044 1045 1046 1047 104 -50050Δt = 0.5 nsN qx,y - <qx,y> [at. u.] 01 1042 1043 1044 1045 1046 1047 1048 104 01 1042 1043 1044 1045 1046 1047 104 -50050NΔt = 1.0 ns qx,y - <qx,y> [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104 01 1042 1043 1044 1045 1046 1047 104 -50050N qx,y - <qx,y> [at. u.]Δt = 2.0 ns FIG. 9. Skyrmion: event statistics with time interval
t as a parameter for the displacement components qx(black full symbols) and qy(red open symbols), labeled qx;yin the gures. In each panel, the curves have been oset vertically for legibility. Solid lines: t to a gaussian distribution. = 0:25,T= 25 K where,kis the lattice site index, Kthe number of lattice sites satisfying the above condition. Such a denition proves robust vsthermal disorder such as displayed in Fig. 7. Similarly to the case of wall diusion, we analyze rst the distributions of the displacement components qx;qy. The event statistics for each value of the time interval is clearly gaussian (see Fig.9). However, the noise in the distributions appears larger when compared to the wall case. It also increases faster with
t. On the other hand, the raw probabilities for hq2 xiandhq2 yibarely dier as anticipated from a random process. The behavior of hq2i(q2=q2 x+q2 y)vs
tis displayed in Fig.10a. The range of accessible temperatures is governed by the thermal stability of the tiny skyrmion within a Co ML: with a lifetime of '1s at 77 K [25{28], tem- peratures have been conned to a 50 K range. When compared to the wall case (Fig.4a), the linear dependence ofhq2iwith respect to
tappears less satisfactory, al- though, over all cases examined, the curves do not display a single curvature, but rather meander gently around a straight line. The slope is dened as the slope of the linear regression either for time intervals between 0 :25 and 2:5 ns (thick line segments in Fig.10a) or for the full range 0 to 5 ns (dashed lines). Then, the ratio of the diusion constant to temperature, D=T, for an isolated skyrmion within the model Co ML considered here is equal to 0:250 and 0:249 nm2ns1K1, respectively, for = 0:5 (see Fig.10b). The dierence proves marginal. Lastly, error bars appear even narrower than in the wall 01000200030004000 01234550° K25° K12.5° K4.2° K< q2 > [at.u.2] Δt [ns]a)01020 0255075T [K]b)D [nm2 ns-1]FIG. 10. a) Variance (at :u:2) of the skyrmion displacement hq2ivstime interval
twith temperature Tas a parameter. Thick and dashed lines represent a linear t to data with dierent time coverage, namely [0 :252:5 ns] and [05 ns]; b) Diusion constant Das a function of temperature for a [0:252:5 ns]- (open symbols) and [0 5 ns]- (full symbols) linear t. Solid blue line: linear t through the origin. Dashed line: analytical expectation in the "low" noise limit. In order to ensure legibility, the error bars as dened in the caption of Fig.4 and pertaining to the [0 :252:5 ns] t time bracket have been moved-up by one unit. = 0:5. case. B. Skyrmion diusion constant (analytical) The gyrovector ~Gin Thiele's equation (Eqn.3) has in the case of a skyrmion or a vortex, and in many other instances such as lines within walls, a single non-zero component, here Gz. Thiele's equation, in components form, reads: Gzvy+[Dxxvx+Dxyvy] =Fx +Gzvx+[Dyxvx+Dyyvy] =Fy(15) Because of the revolution symmetry of a skyrmion at rest, DxyorDyxmay safely be neglected and Dyy=Dxx. Accordingly, the velocities may be expressed as: vx=DFx+GFy G2+ (D)2;vy=DFyGFx G2+ (D)2(16) where,G=Gz,D=Dxx=Dyy. Similarly to the stochastic eld, the force components are necessarily uncorrelated. The velocity autocorrela- tion functions may now be obtained following the same lines as in the wall case, yielding, in the low noise ap- proximation: hvx(t)vx(t0)i=hvy(t)vy(t0)i= 2kBTD G2+ (D)2(tt0) (17)7 00.050.10.150.20.250.30.35 -20-1001020304050 0246810α(%)D /T [nm2 ns-1 K-1] FIG. 11. Computed values of D=Tvs(large open symbols); black line: guide to the eye; blue (resp. red) solid curves: an- alytical values with [ 0SAt=0At]D= 4(resp. 14:5). The blue curve thus corresponds to the Belavin-Polyakov prole limit. The relative dierence between simulation and theory is indicated by small full symbols (% : right scale). The average values of the displacements squared, hq2 xi andhq2 yifollow from time integration: q2 x(t) = q2 y(t) = 2kBTD G2+ (D)2t (18) As shown previously [12, 13], the diusion constant for a skyrmion thus reads: D=kBTD G2+ (D)2(19) The following relations do apply: q2 x(t) = q2 y(t) = 2Dt q2(t) = q2 x(t) +q2 y(t) = 4Dt(20) Relation (19) implies a peculiar damping constant de- pendence with, assuming for the time being DandGto have comparable values, a gradual drop to zero of the diusion constant with decreasing (1), termed "diusion suppression by G" by C. Sch utte et al. [12]. Diusion suppression is actually not a complete surprise since, for electrons in a magnetic eld, a similar eect is leading to the classical magnetoresistance. A similar de- pendenceD() is also expected for a vortex. Boundary conditions, however, add complexity to vortex diusion. What nevertheless remains, is a linear dependence of D vs[14], namely, diusion suppression. The classical expressions for GzandDxxvalid for a magnetization continuum need to be adapted when deal- ing with discrete spins. We obtain: Gz=0At 0X k[~ s(k)
[@x~ s(k)@y~ s(k)]] Dxx=0At 0X kh [@x~ s(k)]2i (21)where,Atis the moment per atom. The dimensionless product 0SAt 0AtGz(Eqn.21), where SAtis the surface per atom, amounts to 4 , irrespec- tive of the skyrmion size in a perfect material at T= 0. Stated otherwise, the skyrmion number is 1 [29]. In the Belavin-Polyakov prole limit [30], the dimention- less product 0SAt 0AtDxx(Eqn.21) also amounts to 4 . In this limit,Dis proportional to =(1+2).Dxxincreases with skyrmion radius beyond the Belavin-Polyakov pro- le limit (see supplementary material in [7]). For a skyrmion at rest in the model Co ML considered here, D=Dxx14:50At=( 0SAt). For that value of Dxx, and for the parameters used in the simulations, D=T, the ratio of the theoretical skyrmion diusion con- stant to temperature, is equal 0 :234 nm2ns1K1, for = 0:5 (SAt=a2p 3=2), to be compared to the 0 :250 value extracted from simulations. More generally, Fig.11 compares numerical D=Tvalues calculated for a broad spectrum of values with theoretical expectations for D= 14:50At=( 0SAt) and in the Belavin-Poliakov limit. The average dierence between analytical and sim- ulation results is, in the = (0;1) interval, seen to be of the order of'15%. IV. DISCUSSION In the present study of thermal diusion characteris- tics, satisfactory agreement between simulations and the- ory has been attained for DMI stiened magnetic tex- tures, be it walls in narrow tracks or skyrmions. The dependence of the diusion constants has been thor- oughly investigated, with, as a result, a conrmation of Brownian motion suppression in the presence of a non- zero gyrovector or, equivalently, a topological signature. The theory starts with the Thiele relation applying to a texture moving under rigid translation at constant ve- locity. Furthermore, the chosen values of the components of the dissipation dyadic, are those valid for textures at rest, atT= 0. Thedependence of the diusion con- stants clearly survives these approximations. And, yet, a wall within a narrow stripe or a skyrmion in an ultra-thin magnetic layer are deformable textures, as obvious from Figs.1,7. Simulations, on the other hand, rely on the pioneering analysis of Brownian motion, here meaning magnetization/spin orientation uctuations [17], within a particle small enough to prove uniformly magnetized and then extend the analysis to ultra-small computation cell volumes down to the single spin. Both approaches rely on the hypothesis of a white -uncorrelated- noise at nite temperature. The discussion of results is organized in two parts. In the rst, results are analyzed in terms of a sole action of structure plasticity on the diagonal elements of the dis- sipation dyadic. In the second, we envisage, without fur- ther justication, how the present results are amended if, in the diusion constants of walls and skyrmions (Eqns.8 and 19), the gyrotropic and dissipation terms are re-8 01 10-102 10-10 20406080100f (GHz)S 12.5 K 25 K T = 50 K a)0123 010203040506070T(K)< rEq > [nm] b) FIG. 12. a) Power spectrum Sof the time series rEq(t) for three temperatures. The hatched area corresponds to the fre- quency range where a signature of the fundamental skyrmion breathing mode is anticipated to be observed ( 39:3 GHz, in the present case); b) Equivalent skyrmion radius hrEqias a function of temperature. Error bars correspond to 1 of the gaussian distribution, itself a function of temperature. = 0:5, throughout. placed by their time average as deduced from simulations. A. Size eects The integral denition of wall position adopted in this work (Eqn.2) allows for a 1D treatment of wall diusion, thus ignoring any diusion characteristics potentially as- sociated with wall swelling, tilting, curving or meander- ing. Additional information is, however, available in the case of skyrmions. We concentrate here on the number, n, of spins within the skyrmion satisfying the condition sz0:5, and its uctuations as a function of time. The surface of the skyrmion is nSAtand its equivalent radius, rEq, is dened by r2 Eq=nSAt=. The skyrmion radius rEqis found to uctuate with time around its average value, according to a gaussian distribution that depends on temperature, but becomes independent of the autocor- relation time interval beyond 25 ps. The power spec- trum of the time series rEq(t), shown in Fig.12a, excludes the existence of a signicant power surge around the fundamental breathing mode frequency of the skyrmion (39:3 GHz for the present model Co ML) [31]. The skyrmion radius as dened from the discrete ndistribu- tion is thus subject to white noise. The average radius hrEqi, on the other hand, varies signicantly with tem- perature, increasing from 1:6 nm to 2:4 nm when the temperature is increased from 4 :2 K to 50 K (Fig.12b) and the diagonal element of the dissipation dyadic is ex- pected to increase with increasing skyrmion radius [3, 7]. Owing to relations (19,21), the maximum of D() is found for=Gz=Dxx=G=D . For < G=D , resp.  > G=D ,Dincreases, resp. decreases, with D, hence the relative positions of the blue and black continuous curves in Fig.11. At maximum, Dis independent of D and amounts to kBT 0SAt 0At1 2G=kBT 0SAt 0At1 8. It ensues z f α R/Δ a) b) 00.20.40.60.81 01020304050R /!"#D / #" < 0#D / #" > 0FIG. 13. Diusion suppression: a) general shape of function f(;R=
) with 0 <  < 1, 1< R=
<50; b) crest line separating the region of diusion suppression ( @D=@ > 0) from region @D=@< 0. that the discrepancy between numerical and analytical Dvalues around = 1 may not be relaxed by a sole variation of D. On the other hand, allowing Dto increase with skyrmion radius, itself a function of temperature, leads to an increase (decrease) of the diusion coecient for<G=D (>G=D ). Likely more important is the reduction, as a function of skyrmion size, of the window where diusion sup- pression is expected. If including the ( R=
+
=R) de- pendence of Dxx(see supplementary material in [7];
is the wall width and Rthe skyrmion radius), the skyrmion diusion constant may be expressed as: D=kBT 0SAt 0At1 8f ;R
 =R
;=1 21 +2  ;f(;) =2 1 + ()2(22) The general shape of function f(;R=
) is shown in Fig.13a. The maximum of f(;R=
) is equal to 1 for all values of andR=
. The crest line R=
is seen to divide the parameter space into two regions (see Fig.13b), a region close to the axes where @D=@ > 0, i.e. the region of diusion suppression, from the much wider region where @D=@ < 0, that is, the region of wall-like behavior for skyrmion diusion. Clearly, the  window for diusion suppression decreases dramatically with increasing skyrmion size R=
. A rst observation of skyrmion Brownian motion at a video recording time scale (25 ms) may be found in the Supplementary Ma- terial of Ref.[32]. Skyrmions are here unusually large and most likely escape the diusion suppression window (<0:02 forR=
= 50). Combining skyrmion thermal stability with general observability and damping parame- ter tailoring may, as a matter of fact, well prove extremely challenging for the observation of topology related diu- sion suppression.9 0.750.80.850.90.951 101214161820 020406080100120140160T (K)< DVF >< mz / mz(T=0) >< sz / sz(T=0) >1 ML3 ML : 0.6 nm FIG. 14. Average reduced zmagnetization or spin component as a function of temperature (left scale) and time averaged value of the sole vector function, hDVFi, within the diagonal element of the dissipation tensor in the skyrmion case (right scale). These results prove independent of the damping pa- rameter provided the time step in the integration of the LLG equation be suitably chosen. B. Time averaging One certainly expects from the simulation model a fair prediction of the average magnetization hMziorhSzivs temperature T, at least for temperatures substantially lower than the Curie temperature TC. Fig.14 shows the variation ofhMzi=Mz(T= 0) orhSzi=Sz(T= 0) with temperature for the two model magnetic layers of this work. Although simulation results do not compare unfa- vorably with published experimental data [33{35], where, typically, the Curie temperature amounts to 150Kfor 1 ML, and proves larger than 300 Kfor thicknesses above 2 ML, a more detailed analysis, potentially including dis- order, ought to be performed. hGzi=0Athszi 0hX k[~ s(k)
[@x~ s(k)@y~ s(k)]]i =0Athszi 0SAthGVF zi hDxxi=0Athszi 0hX k[@x~ s(k)]2i =0Athszi 0SAthDVF xxi(23) Let us now, without further justication, substitute in the expression of the skyrmion diusion coecient time averaged values of GandD, owing to relations (23). Keeping in mind the geometrical meaning of GVF z, the dimensionless vector function in G,hGziis anticipated to be a sole function of hszi. Inversely, DVF xx, the (di- mensionless) vector function in hDxxi, a denite posi- tive quantity, steadily increases with thermal disorder. It is even found to be proportional to temperature (notshown). Its time averaged value for the sole skyrmion may only be obtained by subtraction of values computed in the presence and absence of the skyrmion. For the skyrmion in our model Co monolayer, hDVF xxi is found to increase moderately with temperature (see Fig.14), a result also anticipated from an increase with temperature of the skyrmion radius. Besides, both hGzi andhDxxiare expected to decrease with temperature due to their proportionality to hszi.hDxxiis thus sub- ject to two competing eects of temperature T. Present evidence, however, points at a dominating in uence of hsz(T)i. V. SUMMARY AND OUTLOOK Summarizing, it has been shown that the Brownian motion of chiral walls and skyrmions in DMI materials obeys diusion equations with markedly dierent damp- ing parameter ( ) dependence. Although not a new re- sult, skyrmions Brownian motion suppression with de- creasing(<G=D ) is substantiated by a wide explo- ration of the damping parameter space. The observation of this astonishing topological property might, however, be hampered by the restriction to ultra-small skyrmion sizes or ultra-low values. The discrepancy (up to 20%) between simulation results and theoretical expectations could be reduced by the introduction of time averaged values for the gyrotropic and dissipation contributions to the analytical diusion coecients in the "low" noise limit, at the expense of a tiny upwards curvature in the D(T) curves. A strong theoretical justication for doing so remains, however, lacking at this stage. In this work, the sample has been assumed to be per- fect, i.e. devoid of spatial variations of the magnetic properties, even though the lifting of such a restriction is anticipated to prove mandatory for a proper description of experiments. Diusion in the presence of disorder has been theoretically studied for a number of disorder and random walk types [36, 37]. Generally, disorder changes the linear growth with time of the position variance into a power law, a behavior called superdiusion if the ex- ponent is larger than 1 and subdiusion if smaller. For instance, if the skyrmion motion in a disordered system may be mapped onto a 2D random walk with an onsite residence time , probability/(1+)(<1), then the diusion exponent will be , meaning subdiusion. Be- sides, choosing a physically realistic disorder model for a Co monolayer might well prove equally arduous [38]. Al- together, skyrmion diusion in the presence of disorder has been left out for future work. ACKNOWLEDGMENTS Support by the Agence Nationale de la Recherche (France) under Contracts No. ANR-14-CE26-0012 (Ul- trasky), No. ANR-17-CE24-0025 (TopSky) is gratefully10 acknowledged. [1] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Ku- betzka, R. Wiesendanger, G. Bihlmayer, and S. Bl ugel, Nat. Phys. 7, 713 (2011) [2] F. Jonietz, S. M uhlbauer, C. P eiderer, A. Neubauer, W. M unzer, A. Bauer, T. Adams, R. Georgii, P. B oni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Sci- ence330, 1648 (2010) [3] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotech. 8, 839 (2013) [4] A. Thiaville, S. Rohart, E. Ju e, V. Cros, and A. Fert, EPL100, 57002 (2012) [5]E. Ju e, A. Thiaville, S. Pizzini, J. Miltat, J. Sampaio, L. Buda-Prejbeanu, S. Rohart, J. Vogel, M. Bonm, O. Boulle, S. Auret, I. M. Miron, and G. Gaudin, Phys. Rev. B 93, 014403 (2016) [6] S. Woo, K. Litzius, B. Kr uger, M.-Y. Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. Reeve, M. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass, P. Fischer, M. Kl aui, and G. S. D. Beach, Nat. Mater. 15, 501 (2016) [7] A. Hrabec, J. Sampaio, M. Belmeguenai, I. Gross, R. Weil, S. M. Ch erif, A. Stashkevitch, V. Jacques, A. Thiaville, and S. Rohart, Nat. Commun. 8, 15765 (2017) [8] W. Jiang, X. Zhang, G. Yu, W. Zhang, M. B. Jung eisch, J. E. Pearson, X. Cheng, O. Heinonen, K. L. Wang, Y. Zhou, A. Homann, and S. G. E. te Velthuis, Nat. Phys. 13, 162 (2016) [9] A. Einstein, Ann. Phys. (Berlin) 17, 549 (1905) [10] P. Langevin, C. R. Acad. Sci. (Paris) 146, 530 (1908) [11] Y. Wada and J. R. 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2018-06-08

Extended numerical simulations enable to ascertain the diffusive behavior at finite temperatures of chiral walls and skyrmions in ultra-thin model Co layers exhibiting symmetric - Heisenberg - as well as antisymmetric - Dzyaloshinskii-Moriya - exchange interactions. The Brownian motion of walls and skyrmions is shown to obey markedly different diffusion laws as a function of the damping parameter. Topology related skyrmion diffusion suppression with vanishing damping parameter, albeit already documented, is shown to be restricted to ultra-small skyrmion sizes or, equivalently, to ultra-low damping coefficients, possibly hampering observation.

Brownian motion of magnetic domain walls and skyrmions, and their diffusion constants

1806.03172v1

Diffusion coefficients for multi-step persistent random walks on lattices Thomas Gilberty, David P. Sanders z yCenter for Nonlinear Phenomena and Complex Systems, Universit ´e Libre de Bruxelles, C. P. 231, Campus Plaine, B-1050 Brussels, Belgium zDepartamento de F ´ısica, Facultad de Ciencias, Universidad Nacional Aut ´onoma de M ´exico, 04510 M ´exico D.F., Mexico E-mail: thomas.gilbert@ulb.ac.be, dps@fciencias.unam.mx Abstract. We calculate the diffusion coefficients of persistent random walks on lattices, where the direction of a walker at a given step depends on the memory of a certain number of previous steps. In particular, we describe a simple method which enables us to obtain explicit expressions for the diffusion coefficients of walks with two-step memory on different classes of one-, two- and higher-dimensional lattices. Submitted to: J. Phys. A: Math. Theor. 1. Introduction Random walks are widely used throughout physics as a model for systems in which the state of the system can be viewed as evolving in a stochastic way from one time step to the next. Their properties have been extensively explored and the techniques to study them are well developed [1]. In particular, at a large scale, random walks behave diffusively. The most-studied case is that of random walkers which have no memory of their past history. Many physical applications, however, call for a model in which the choice of possible directions for the walker’s next step are given by probabilities which are influenced by the path it took prior to making that choice, so that its jumps are correlated; this is often called a persistent orcorrelated random walk [2]. A walk with zero memory corresponds to the Bernoulli process of the usual uncorrelated random walk. Walks with single-step memories are the most commonly studied cases of persistent random walks, where the random walker determines the direction it takes at a given step in terms of the direction taken on the immediately preceding step [1]. Such walks – not necessarily restricted to lattices, as will be the case here – were first discussed in the context of Brownian motion [3] and fluid dynamics [4], and have since found many applications in the physics literature, most prominently in polymer conformation theory [5] and tracer diffusion in metals [6], but also in relation to the telegrapher’s equation in the context of thermodynamics [7]. Previous works dealing with random walks on lattices with higher-order memory effects include that of Montroll [8], with applications to models of polymers, and Bender and Richmond [9]. The state of the walker is thus specified by two variables, its location and the direction it took at the preceding step. The statistical properties of such persistent walks can be describedarXiv:0908.1271v1 [cond-mat.stat-mech] 10 Aug 2009Diffusion coefficients for persistent random walks 2 by simple Markov chains and have already been thoroughly investigated in the literature; see in particular ref. [10]. We will only provide a short review of results relevant to our purposes, with specific emphasis on the diffusive properties. The statistics of random walks with multi-step memory can in principle be analysed in terms of Markov chains, in a similar way to their single-step memory counterpart. However, the number of states of these chains grows exponentially with the number of steps accounted for. This is the source of great technical difficulties, which are present already at the level of two-step processes. Of specific interest to us are random walks with two-step memory. Among the class of persistent walks under consideration, these are the simplest case beyond those with single- step memory, and are therefore relevant to problems dealing with the persistence of motion of tracer particles where the single-step-memory approximation breaks down. An example where this occurs is given in recent work by the present authors, on diffusion in a class of periodic billiard tables [11]. The paper is organised as follows. The general framework of walks on lattices is briefly reviewed in section 2, where we provide the expression of the diffusion coefficient of such walks in terms of the velocity auto-correlations. Successive approximation schemes for the computation of these auto-correlation functions are presented in sections 3, 4, and 5, pertaining to the number of steps of memory of the walkers, respectively 0, 1, and 2. Specific examples are discussed, namely the one-dimensional lattice and the two-dimensional square, honeycomb and triangular lattices, and their diffusion coefficients are computed. Some of the details of the computations presented in section 5 are deferred to appendices Appendix A and Appendix B. Section 6 provides an alternative derivation of the diffusion coefficients of two-step memory persistent walks with special left–right symmetries. Conclusions are drawn in section 7. 2. Diffusion on a lattice We consider the motion of independent tracer particles undergoing random walks on a regular latticeL. Their trajectories are specified by their initial position r0at time t=0, and the sequencefv0;:::;vngof the successive values vi2Vriof their direction vectors at positions ri, whereVridenotes the space of direction vectors allowed at site ri, which point to the lattice sites adjacent to ri. Here we consider dynamics in discrete time, so that the time sequences are simply assumed to be incremented by identical time steps tas the tracers move from site to site. In the sequel we will loosely refer to the direction vectors as velocity vectors; they are in fact dimensionless unit vectors. Examples of such motions are random walks on one- and two-dimensional lattices such as honeycomb, square and triangular lattices, but also include persistent random walks where memory effects must be accounted for, i.e. when the probability of occurrence of vndepends on the past history vn1;vn2;:::. The quantity we will be concerned with is the diffusion coefficient Dof such persistent processes, which measures the linear growth in time of the mean-squared displacement of walkers. This can be written in terms of velocity auto-correlations using the Taylor–Green– Kubo expression: D=`2 2dt" 1+2 lim K!¥K å n=1hv0
vni# ; (2.1) where ddenotes the dimensionality of the lattice L, and `is the lattice spacing. TheDiffusion coefficients for persistent random walks 3 (dimensionless) velocity auto-correlations are computed as averages h
iover the equilibrium distribution m, so that the problem reduces to computing hv0
vni=å v0;:::;vnv0
vnm(fv0;:::;vng): (2.2) As reviewed below, this can be easily carried out in the simple examples of random walks with zero- and single-step memories. The main achievement of this paper is to describe the computation of the velocity auto-correlations of random walks with two-step memory. All these cases involve factorisations of the measure m(fv0;:::;vng)by products of probability measures which depend on a number of velocity vectors, equal to the number of steps of memory of the walkers. These measures will be denoted by pthroughout the paper. The next three sections, sections 3, 4 and 5, are devoted to the computation of the diffusion coefficient (2.1) for random walks with zero-, one- and two-step memories, respectively. 3. No-Memory Approximation (NMA) In the simplest case, the walkers have no memory of their history as they proceed to their next position. This gives a Bernoulli process for the velocity trials, for which the probability measure factorises: m(fv0;:::;vng) =n Õ i=0p(vi): (3.1) Given that the lattice is isotropic and that pis uniform, the velocity auto-correlation (2.2) vanishes: hv0
vni=dn;0: (3.2) The diffusion coefficient of the random walk without memory is then given by DNMA=`2 2dt: (3.3) 4. One-Step Memory Approximation (1-SMA) We now assume that the velocity vectors obey a Markov process for which vntakes on different values according to the velocity at the previous step vn1. We may then write m(fv0;:::;vng) =n Õ i=1P(vijvi1)p(v0): (4.1) Here, P(bja)denotes the 1-step conditional probability that the walker moves in a direction b,given that it had direction aat the previous step. We denote by zthe coordination number of the lattice, i.e. the number of neighbouring sites accessible from each site, and we denote by Rthe rotation operation which takes a vector vthrough all the lattice directions v;Rv;:::;Rz1v. In general, the set of allowed orientations ofvdepends on the lattice site, such as in the two-dimensional honeycomb lattice. We denote byTthe symmetry operator that maps a cell to its neighbors, which corresponds simply to the identity for square lattices and to a reflection for the honeycomb lattice.Diffusion coefficients for persistent random walks 4 The idea of our calculation is to express each velocity vector vkin terms of the first one, v0, asvk=RikTkv0, where iklies between 0 and z1. Substituting this into the expression for the velocity auto-correlation hv0
vni, equation (2.2), we obtain å v0;:::;vnv0
vnn Õ i=1P(vijvi1)p(v0) =z å i0;:::;in=1v0
RinTnv0min;in1


mi1;i0pi0; (4.2) where min;in1P(RinTnv0jRin1Tn1v0) (4.3) are the elements of the stochastic matrix Mof the Markov chain associated to the persistent random walk, and pip(i)are the elements of its invariant (equilibrium) distribution, denoted P, evaluated with a velocity in the ith lattice direction. The invariance of Pis expressed as åjmi;jpj=pi. These notations will be used throughout this article. The terms involving Min (4.2) constitute the matrix product of ncopies of M. Furthermore, since the invariant distribution is uniform over the zpossible lattice directions, we can choose an arbitrary direction for v0, and hence write hv0
vni=v0
Tnv0m(n) 1;1+v0
RTnv0m(n) 2;1+


+v0
Rz1Tnv0m(n) z;1;(4.4) where m(n) i;jdenote the elements of Mn. Under special symmetry assumptions to be discussed in the examples below, one has hv0
vni=hcosqin; (4.5) wherehcosqidenotes the average angle between two successive velocity vectors. It is then a general, well-known, result for such symmetric persistent random walks with single-step memory [2] that their diffusion coefficients have the form D1SMA =DNMA1+hcosqi 1hcosqi(symmetricwalks ): (4.6) The actual value of the diffusion coefficient depends on the probabilities P(RjTvjv), which are parameters of the model. Specific applications of equation (4.6) are given in the examples below, such as shown in figure 1. To simplify the notation, we denote the conditional probabilities of these walks by Pj, where j=0;:::; z1 corresponds to the relative angle 2pj=zof the direction that the walker takes with respect to its previous step (up to a reflection in the case of the honeycomb lattice). These conventions are shown in figure 2. (a) (b) Figure 1. Examples of walks on (a) square and (b) honeycomb lattices. Note the inversion of the allowed directions at neighbouring sites on the honeycomb lattice.Diffusion coefficients for persistent random walks 5 (a) 0 1 (b) 01 2 (c) 0 321 (d) 0 5 432 1 Figure 2. The possible directions of motion at a given step for different lattices, relative to the incoming direction which is shown by the arrow, are labelled from 0 to z1, corresponding to the angle 2 pj=zthat the lattice direction jmakes with respect to the reference direction, up to a reflection inversion in the case of the honeycomb lattice. (a) One-dimensional lattice ( z=2); (b) honeycomb lattice ( z=3); (c) square lattice ( z=4); (d) triangular lattice ( z=6) . 4.1. One-dimensional lattice The simplest case is that of a regular one-dimensional lattice. In this case, each site is equivalent, and so Tis the identity. Each velocity vector vkhas only two possible values, vandv, so that Ris a reflection. We denote by P0P(vjv)the probability that the random walker continues in the same direction at the next step, and by P1P(vjv)the probability that it reverses direction. It is easy to check that the velocity auto-correlation (4.4) yields (4.5): hv0
vni= (P0P1)n=hcosqin; (4.7) and so the diffusion coefficient is given by D1SMA =DNMAP0 P1: (4.8) 4.2. Two-dimensional square lattice On a two-dimensional square lattice, vkcan take four possible values, and each lattice site is again equivalent. Thus Tis the identity and Rcan be taken as an anticlockwise rotation by angle p=2. We denote by P0P(vjv)the probability that the particle proceeds in the same direction as on its previous step, by P1P(Rvjv)the probability that the particle turns to the left relative to its previous direction, by P2P(R2vjv)the probability that it turns around, and by P3P(R3vjv)the probability that it turns right.Diffusion coefficients for persistent random walks 6 From (4.4), the velocity auto-correlation hv0
vniis given by hv0
vni=m(n) 1;1m(n) 1;3: (4.9) The transition matrix Mgiven by (4.3) is thus the following cyclic matrix: M=0 BB@P0P1P2P3 P3P0P1P2 P2P3P0P1 P1P2P3P01 CCA: (4.10) To calculate the elements m(n) i;jof the powers Mn, it is possible to compute the eigenvalues and eigenvectors of Mand then decompose it as M=Q
L
Q1, where Lis the diagonal matrix of the eigenvalues of MandQthe matrix of its eigenvectors. This procedure is, however, not necessary here, since we only require the combination of the m(n) i;jwhich appears in (4.9). To proceeed, we label the distinct entries of Mnasa(n) i, using the fact that Mnis also cyclic if M is: Mn=0 BBB@a(n) 1a(n) 2a(n) 3a(n) 4 a(n) 4a(n) 1a(n) 2a(n) 3 a(n) 3a(n) 4a(n) 1a(n) 2 a(n) 2a(n) 3a(n) 4a(n) 11 CCCA: (4.11) Writing Mn=MMn1, we can exploit the particular structure of the matrix to reduce it from a(44)-matrix to a (22)-matrix, by considering the following differences: a(n) 1a(n) 3 a(n) 2a(n) 4! = P0P2P3P1 P1P3P0P2 a(n1) 1a(n1) 3 a(n1) 2a(n1) 4! = P0P2P3P1 P1P3P0P2n1 P0P2 P1P3 : (4.12) The velocity auto-correlation (4.9) is thus given by hv0
vni=a(n) 1a(n) 3 = 1 0
 P0P2P3P1 P1P3P0P2n1 P0P2 P1P3 : (4.13) Summing the previous expression over all n, and using the fact that å¥ n=0An= (IA)1 for a matrix A, where Iis the identity matrix, we obtain the diffusion coefficient (2.1) as D1SMA =DNMA" 1+2 1 0 1P0+P2P1P3 P3P11P0+P21 P0P2 P1P3# : (4.14) For a symmetric process in which P1=P3, which is often imposed by a symmetry of the physical system, equation (4.5) holds, and the diffusion coefficient takes the form (4.6), viz. D1SMA =DNMA1+P0P2 1P0+P2: (4.15) If, however, P16=P3, then equation (4.6) is no longer valid. Instead, we have the more complicated expression D1SMA =DNMA1(P0P2)2(P1P3)2 (1P0+P2)2+(P1P3)2: (4.16) Such asymmetric walks are the lattice equivalent of the continuous-space models of persistent random walks with chirality considered in [12].Diffusion coefficients for persistent random walks 7 4.3. Two-dimensional honeycomb lattice On the two-dimensional honeycomb lattice, shown in figure 1(b), each site has z=3 neighbours. Here, Ris taken to be a clockwise rotation‡ by angle 2 p=3 and the arrangement of neighbours differs by a reflection T. We denote by P0P(vjv)the probability that the particle turns around, P1P(R2vjv)that it turns left relative to its previous direction, and P2P(Rvjv)that it turns right. The transition matrix Mis thus M=0 @P0P1P2 P2P0P1 P1P2P01 A: (4.17) Proceeding as with the square lattice, we let Mn0 B@a(n) 1a(n) 2a(n) 3 a(n) 3a(n) 1a(n) 2 a(n) 2a(n) 3a(n) 11 CA; (4.18) and obtain the matrix equation0 B@a(n) 1+1 2[a(n) 2+a(n) 3] a(n) 2+1 2[a(n) 1+a(n) 3] a(n) 3+1 2[a(n) 1+a(n) 2]1 CA=0 @P0P1P2 P2P0P1 P1P2P01 A0 B@a(n1) 1+1 2[a(n1) 2+a(n1) 3] a(n1) 2+1 2[a(n1) 1+a(n1) 3] a(n1) 3+1 2[a(n1) 1+a(n1) 2]1 CA; =1 20 @P0P1P2 P2P0P1 P1P2P01 An10 @13P0 13P2 13P11 A: (4.19) The velocity auto-correlation (4.4) is thus hv0
vni=a(n) 1+1 2[a(n) 2+a(n) 3]; =1 2 1 0 00 @P0P1P2 P2P0P1 P1P2P01 An10 @13P0 13P2 13P11 A: (4.20) Hence the diffusion coefficient (2.1) is D1SMA =DNMA2 641+ 1 0 00 @1P0P1P2 P21P0P1 P1P21P01 A10 @13P0 13P2 13P11 A3 75: (4.21) Again, in the case of an isotropic process for which P2=P1Ps(‘symmetric’), equation (4.5) holds, and substituting Ps= (1P0)=2 gives the following expression for the diffusion coefficient: D1SMA =DNMA1P0+Ps 1+P0Ps=DNMA3(1P0) 1+3P0: (4.22) For an asymmetric process for which P16=P2, defining the symmetric and antisymmetric parts Ps(P1+P2)=2 and Pa(P1P2)=2, we instead obtain D1SMA =DNMA13P2 a(P0Ps)2 3P2a+(1+P0Ps)2; =DNMA3(1P0)(1+3P0)12P2 a (1+3P0)2+12P2a: (4.23) ‡ We take a clockwise rotation as opposed to an anticlockwise one in the other examples so that the lattice directions are still labelled anticlockwise.Diffusion coefficients for persistent random walks 8 4.4. Two-dimensional triangular lattice Persistent walks on a triangular lattice were made popular by Fink and Mao [13] in connection to tie knots. Here, each site has z=6 neighbours, so that Ris an anti-clockwise rotation by angle p=3. Following our convention [see figure 2 (d)], we denote by P0P(vjv)the probability that the particle moves forward, P1P(Rvjv)that it moves in the forward left direction relative to its previous direction, and similarly for P2,P3,P4, and P5. Proceeding along the lines of the previous subsection, we let Mn0 BBBB@a(n) 1a(n) 2


a(n) 6 a(n) 6a(n) 1


a(n) 5............ a(n) 2a(n) 3


a(n) 11 CCCCA; (4.24) in terms of which the velocity auto-correlation (4.4) is given by hv0
vni=a(n) 1+1 2[a(n) 2+a(n) 6a(n) 3a(n) 5]a(n) 4: (4.25) A computation similar to that of equation (4.19) yields hv0
vni= 1 0


00 BBB@P0P1


P5 P5P0


P4 ............ P1P2


P01 CCCAn10 BBB@P0P3+1 2(P1P2P4+P5) P1P4+1 2(P2P3P5+P0) ... P5P2+1 2(P0P1P3+P4)1 CCCA: (4.26) The diffusion coefficient (2.1) is then D1SMA =DNMA2 66641+ 1 0


0
0 BBB@1P0P1


P5 P51P0


P4 ............ P1P2


1P01 CCCA1 0 BBB@P0P3+1 2(P1P2P4+P5) P1P4+1 2(P2P3P5+P0) ... P5P2+1 2(P0P1P3+P4)1 CCCA3 7775: (4.27) 4.5. d-dimensional hypercubic lattice The case of a hypercubic lattice in arbitrary dimension dwith coordination number z=2d can also be treated, provided that the same probability Psis assigned to scattering along all new directions which are perpendicular to the previous direction of motion. We then have P0+Pz=2+2(d1)Ps=1, and the invariant distribution of velocities p(v) =1=(2d)is uniform. We then recover the expression (4.15) for the diffusion coefficient in this case. 5. Two-Step Memory Approximation (2-SMA) We now turn to the main contribution of this paper, namely the calculation of the diffusion coefficient for persistent random walks with 2-step memory, for several two-dimensional lattices.Diffusion coefficients for persistent random walks 9 We thus assume that the velocity vectors obey a random process for which the probability ofvntakes on different values according to the velocities at the two previous steps, vn1and vn2, so that we may write m(fv0;:::;vng) =n Õ i=2P(vijvi1;vi2)p(v0;v1): (5.1) The velocity auto-correlation (2.2) function is then hv0
vni=å fvn;:::;v0gv0
vnn Õ i=2P(vijvi1;vi2)p(v0;v1): (5.2) The calculation of these correlations proceeds in the most straightforward way by transposing the calculation leading to equation (2.2) to the two-step probability transitions as characterizing the probability transitions of a two-dimensional Markov chain. Considering a lattice with coordination number z, the state of the Markov chain is a normalised vector of dimension z2. The time evolution is specified by the (z2z2)stochastic matrix Mwith entries mi=(i11)z+i2;j=(j11)z+j2di2;j1P(Ri1T2vjRj1Tv;Rj2v); (5.3) where i1,i2,i1, and i2take values between 1 and z. Denoting by Pthe invariant distribution of this Markov chain, i.e. the z2-dimensional vector with components pisuch that åz2 j=1mi;jpj= pi, equation (5.2) becomes hv0
vni=z2 å i0;in=1J(n) in;i0m(n1) in;i0pi0; (5.4) where J(n)is the (z2z2)matrix with elements J(n) i=(i11)z+i2;j=(j11)z+j2ei1
Tnej2; (5.5) andekdenotes the unit vector along the kth lattice direction. Using the symmetries of the problem and writing Pj k=P(RkTRjTvjRjTv;v)for the conditional probability of turning successively by angles 2 pj=zand 2 pk=zwith respect to the current direction (with reflection by Twhere needed), we define fexp(2ip=z), where i denotes the imaginary unit, i=p1, and show, through the examples below, that equation (5.4) reduces to the general expression hv0
vni=snz 2 1


12 66640 BBB@P00 P10


Pz1;0 fP01 fP11


fPz1;1 ............ fz1P0;z1fz1P1;z1


fz1Pz1;11 CCCAn1 Diag(1;f;:::;fz1) +0 BBB@P00 P10


Pz1;0 f1P01 f1P11


f1Pz1;1 ............ f1zP0;z1f1zP1;z1


f1zPz1;11 CCCAn1 Diag(1;f1;:::;f1z)3 77750 B@p1 ... pz1 CA; (5.6) where Diag (1;f;:::;fz1)denotes the matrix with elements listed on the main diagonal and 0 elsewhere. Here, sis a sign factor which is 1 for the honeycomb lattice and +1 for the other lattices, and reflects the action of T. Note that the second term in the summation is theDiffusion coefficients for persistent random walks 10 complex conjugate of the first, so that the result is real. The diffusion coefficient for persistent random walks with 2-step memory is therefore D2SMA DNMA=1+sz 1


1  (5.7) 2 66640 BBB@1P00P10


Pz1;0 fP01 1fP11


fPz1;1 ............ fz1P0;z1fz1P1;z1


1fz1Pz1;11 CCCA1 Diag(1;f;:::;fz1) +0 BBB@1P00P10


Pz1;0 f1P01f1P11


f1Pz1;1 ............ f1zP0;z1f1zP1;z1


1f1zPz1;11 CCCA1 Diag(1;f1;:::;f1z)3 77750 B@p1 ... pz1 CA: 5.1. One-dimensional lattice We first consider the simplest case, namely the one-dimensional lattice. The stochastic matrix Mfrom equation (5.3) is the 4 4 matrix M=0 BB@P(vjv;v)P(vjv;v) 0 0 0 0 P(vjv;v)P(vjv;v) P(vjv;v)P(vjv;v) 0 0 0 0 P(vjv;v)P(vjv;v)1 CCA; =0 BB@P00P100 0 0 0 P11P01 P01P110 0 0 0 P10P001 CCA; (5.8) where P00+P01=1 and P10+P11=1. Considering equation (5.4), we compute the invariant distribution of M, which is the vector Pwhose components correspond to the four states p(v;v);p(v;v);p(v;v)and p(v;v). Given that we must have p(v;v) = p(v;v)and p(v;v) = p(v;v), the equilibrium distribution is obtained as the solution of the system of equations p(v;v) =P00p(v;v)+P10p(v;v); p(v;v)+p(v;v) =1 2;(5.9) giving p1=p4=p(v;v) =P10 2[1P00+P10]; p2=p3=p(v;v) =1P00 2[1P00+P10]:(5.10) The matrix J(n), equation (5.5), is here the same for all n, and has the expression J=0 BB@11 11 11 11 1 11 1 1 11 11 CCA: (5.11)Diffusion coefficients for persistent random walks 11 The velocity auto-correlation (5.4) is thus hv0
vni=2h m(n1) 1;1m(n1) 1;4+m(n1) 3;4m(n1) 3;1 p1 + m(n1) 1;3m(n1) 1;2+m(n1) 3;2m(n1) 3;3 p2i ; =2 1 1 m(n1) 1;1m(n1) 1;4m(n1) 1;3m(n1) 1;2 m(n1) 3;4m(n1) 3;1m(n1) 3;2m(n1) 3;3! p1 p2 : (5.12) Since Mn, the nth power of M, has the symmetries of M, its entries m(n) i;jare such that m(n) 1;1=m(n) 4;4a(n) 1; m(n) 2;1=m(n) 3;4b(n) 4; m(n) 1;2=m(n) 4;3a(n) 2; m(n) 2;2=m(n) 3;3b(n) 3; m(n) 1;3=m(n) 4;2a(n) 3; m(n) 2;3=m(n) 3;2b(n) 2; m(n) 1;4=m(n) 4;1a(n) 4; m(n) 2;4=m(n) 3;1b(n) 1:(5.13) Writing Mn=MMn1, we obtain two separate sets of equations for 2 2 matrices, one involving a(n) 1;a(n) 4andb(n) 1;b(n) 4, and the other involving a(n) 2;a(n) 3andb(n) 2;b(n) 3: a(n) 1a(n) 4 b(n) 1b(n) 4! = P00P10 P01P11 a(n1) 1a(n1) 4 b(n1) 4b(n1) 1! ; a(n) 2a(n) 3 b(n) 2b(n) 3! = P00P10 P01P11 a(n1) 2a(n1) 3 b(n1) 3b(n1) 2! :(5.14) Note that these equations do not have a simple recursive form, since the elements of the matrices on the two sides do not appear in the same places. However, taking the differences a(n) 1a(n) 4,a(n) 3a(n) 2,b(n) 4b(n) 1, and b(n) 2b(n) 3, we obtain the recursive system a(n) 1a(n) 4a(n) 3a(n) 2 b(n) 4b(n) 1b(n) 2b(n) 3! = P00 P10 P01P11 a(n1) 1a(n1) 4a(n1) 3a(n1) 2 b(n1) 4b(n1) 1b(n1) 2b(n1) 3! ; = P00 P10 P01P11n1 a(1) 1a(1) 4a(1) 3a(1) 2 b(1) 4b(1) 1b(1) 2b(1) 3! ; = P00 P10 P01P11n1 P00P10 P01P11 ; = P00 P10 P01P11n 1 0 01 : (5.15) Plugging this equation into equation (5.12), we obtain hv0
vni=2 1 1 P00 P10 P01P11n1 p1 p2 : (5.16) This is equation (5.6), where f=exp(2ip=2) =1. The diffusion coefficient is therefore given by equation (5.7): D2SMA DNMA=1+4 1 1 1P00P10 P01 1+P111 p1 p2 ;Diffusion coefficients for persistent random walks 12 =DNMAP10 [1P00][1+P00P10] [1P00+P10]: (5.17) It is a function of the two parameters P00andP10; when these are equal, the process reduces to a walk with single-step memory, and the diffusion coefficient to that of the single-step memory approximation (4.8), as it should. 5.2. Two-dimensional honeycomb lattice For the two-dimensional honeycomb lattice, the stochastic matrix Mof equation (5.3) is a (99)matrix with the following non-zero entries: m1;1=m5;5=m9;9=P(vjv;v) =P00; m1;2=m5;6=m9;7=P(R2vjR2v;v) =P10; m1;3=m5;4=m9;8=P(RvjRv;v) =P20; m4;1=m8;5=m3;9=P(Rvjv;v) =P02; m4;2=m8;6=m3;7=P(vjR2v;v) =P12; m4;3=m8;4=m3;8=P(R2vjRv;v) =P22; m7;1=m2;5=m6;9=P(R2vjv;v) =P01; m7;2=m2;6=m6;7=P(RvjR2v;v) =P11; m7;3=m2;4=m6;8=P(vjRv;v) =P21:(5.18) Given the three constraints P00+P01+P02=1; P10+P11+P12=1; P20+P21+P22=1;(5.19) the actual number of independent parameters is six. The invariant distribution Pwith components pican be written in terms of the three probabilities p(v;v),p(v;Rv),p(v;R2v), p1=p5=p9=p(v;v); p2=p6=p7=p(v;Rv); p3=p4=p8=p(v;R2v);(5.20) which are solutions of the system of equations: p(v;v) =P00p(v;v)+P10p(v;R2v)+P20p(v;Rv); (5.21) p(v;R2v) =P01p(v;v)+P11p(v;R2v)+P21p(v;Rv); (5.22) p(v;v)+p(v;Rv)+p(v;R2v) =1 3: (5.23) The matrix (5.5) has the block structure J(n)= (1)n0 @B1B1B1 B2B2B2 B3B3B31 A; (5.24) where B1=0 @1 1=2 1=2 1 1=2 1=2 1 1=2 1=21 A;B2=0 @1=21 1=2 1=21 1=2 1=21 1=21 A;B3=0 @1=2 1=21 1=2 1=21 1=2 1=211 A: (5.25)Diffusion coefficients for persistent random walks 13 Substituting these expressions into equation (5.4), we find hvn+1
v0i=3(1)n+1(h m(n) 1;11 2m(n) 1;51 2m(n) 1;91 2m(n) 4;1+m(n) 4;51 2m(n) 4;91 2m(n) 7;1 1 2m(n) 7;5+m(n) 7;9i p1+h 1 2m(n) 1;21 2m(n) 1;6+m(n) 1;7+m(m;n) 4;21 2m(n) 4;6 1 2m(n) 4;71 2m(n) 7;2+m(n) 7;61 2m(n) 7;7i p2+h 1 2m(n) 1;3+m(n) 1;41 2m(n) 1;8 1 2m(n) 4;31 2m(n) 4;4+m(n) 4;8+m(n) 7;31 2m(n) 7;41 2m(n) 7;8i p3) ; =3(1)n+1 1 1 1  (5.26) 0 B@m(n) 1;11 2m(n) 1;51 2m(n) 1;91 2m(n) 1;21 2m(n) 1;6+m(n) 1;71 2m(n) 1;3+m(n) 1;41 2m(n) 1;8 1 2m(n) 7;11 2m(n) 7;5+m(n) 7;91 2m(n) 7;2+m(n) 7;61 2m(n) 7;7m(n) 7;31 2m(n) 7;41 2m(n) 7;8 1 2m(n) 4;1+m(n) 4;51 2m(n) 4;9m(n) 4;21 2m(n) 4;61 2m(n) 4;71 2m(n) 4;31 2m(n) 4;4+m(n) 4;81 CA0 @p1 p2 p31 A: Proceeding along the lines of the computation presented in subsection 5.1, we obtain a set of recursive matrix equations (A.9) involving the coefficients of Mn. We refer the reader to Appendix A for the details of this derivation. We note that the coefficients which appear in equation (5.26) satisfy identities such as, for instance, m(n) 1;11 2m(n) 1;51 2m(n) 1;9=1 2h m(n) 1;1+e2ip=3m(n) 1;5+e2ip=3m(n) 1;9i +1 2h m(n) 1;1+e2ip=3m(n) 1;5+e2ip=3m(n) 1;9i :(5.27) Thus, letting f=exp(2ip=3), we can combine the results of equation (A.9) with equation (5.26) to find: hvn
v0i= (1)n3 2 1 1 12 640 @P00 P10 P20 fP01fP11fP21 f2P02f2P12f2P221 An10 @1 0 0 0f0 0 0 f21 A +0 @P00 P10 P20 f2P01f2P11f2P21 fP02fP12fP221 An10 @1 0 0 0f20 0 0 f1 A3 750 @p1 p2 p31 A: (5.28) This is equation (5.6). The diffusion coefficient (2.1) is therefore given by (5.7), which is here D2SMA DNMA=13 1 1 12 640 @1+P00 P10 P20 fP011+fP11 fP21 f2P02 f2P121+f2P221 A10 @1 0 0 0f0 0 0 f21 A +0 @1+P00 P10 P20 f2P011+f2P11f2P21 fP02 fP12 1+fP221 A10 @1 0 0 0f20 0 0 f1 A3 750 @p1 p2 p31 A:(5.29) Given a symmetric process for which left and right probabilities are equal, but the probability of a left–left turn is different than that of a right–left turn, we let P02=P01Pbs=1P00 2; P10=P20Psb;P11=P22Pss;P12=P21=1PsbPss:(5.30)Diffusion coefficients for persistent random walks 14 Carrying out the matrix inversions in (5.29), we find the diffusion coefficient, D2SMA =DNMA3(1P00)(1+P00Psb)(2Psb2Pss) (1P00+Psb)[Psb(7+P008Pss)+2(1+P00)Pss4P2 sb]: (5.31) If we further assume complete left–right symmetry and identify the probabilities of left–left turns and left–right turns, thus letting P12=P21=P11=P22=1Psb 2; (5.32) equation (5.31) simplifies to D2SMA =DNMA3(1P00)(1+P00Psb) (1P00+Psb)(1+P00+2Psb): (5.33) 5.3. Two-dimensional square lattice For the two-dimensional square lattice, the matrix Mof equation (5.3) is the (1616)matrix with the following non-zero entries: m1;1=m6;6=m11;11=m16;16=P(vjv;v) =P00; m1;2=m6;7=m11;12=m16;13=P(R3vjR3v;v) =P10; m1;3=m6;8=m11;9=m16;14=P(R2vjR2v;v) =P20; m1;4=m6;5=m11;10=m16;15=P(RvjRv;v) =P30; m5;1=m10;6=m15;11=m4;16=P(Rvjv;v) =P03; m5;2=m10;7=m15;12=m4;13=P(vjR3v;v) =P13; m5;3=m10;8=m15;9=m4;14=P(R3vjR2v;v) =P23; m5;4=m10;5=m15;10=m4;15=P(R2vjRv;v) =P33; m9;1=m14;6=m3;11=m8;16=P(R2vjv;v) =P02; m9;2=m14;7=m3;12=m8;13=P(RvjR3v;v) =P12; m9;3=m14;8=m3;9=m8;14=P(vjR2v;v) =P22; m9;4=m14;5=m3;10=m8;15=P(R3vjRv;v) =P32; m13;1=m2;6=m7;11=m12;16=P(R3vjv;v) =P01; m13;2=m2;7=m7;12=m12;13=P(R2vjR3v;v) =P11; m13;3=m2;8=m7;9=m12;14=P(RvjR2v;v) =P21; m13;4=m2;5=m7;10=m12;15=P(vjRv;v) =P31: The matrix (5.5) is here J(n)J=0 BB@B1B1B1B1 B2B2B2B2 B1B1B1B1 B2B2B2B21 CCA; (5.34) where B1=0 BB@1 01 0 1 01 0 1 01 0 1 01 01 CCA;B2=0 BB@0 1 01 0 1 01 0 1 01 0 1 011 CCA; (5.35)Diffusion coefficients for persistent random walks 15 The invariant distribution Pwill not be written explicitly here. Due to symmetry, only 4 of the 16 components are distinct. These four components are most simply computed as the invariant vector of the following (44)matrix:0 BB@P00P10P20P30 P01P11P21P31 P02P12P22P32 P03P13P23P331 CCA0 BB@p1 p2 p3 p41 CCA=0 BB@p1 p2 p3 p41 CCA; (5.36) normalised so that p1+p2+p3+p4=1=4. In terms of these quantities, the velocity auto-correlations are found to be hv0
vn+1i=4h m(n) 1;1m(n) 1;11+m(n) 5;6m(n) 5;16m(n) 9;1+m(n) 9;11m(n) 13;6+m(n) 13;16i p1 +4h m(n) 1;7+m(n) 1;13+m(n) 5;2m(n) 5;12+m(n) 9;7m(n) 9;13m(n) 13;2+m(n) 13;12i p2 +4h m(n) 1;3+m(n) 1;9m(n) 5;8+m(n) 5;14+m(n) 9;3m(n) 9;9+m(n) 13;8m(n) 13;14i p3 +4h m(n) 1;5m(n) 1;15m(n) 5;4+m(n) 5;10m(n) 9;5+m(n) 9;15+m(n) 13;4m(n) 13;10i p4; =4 1 1 1 10 BBBB@m(n) 1;1m(n) 1;11m(n) 1;13m(n) 1;7m(n) 1;9m(n) 1;3m(n) 1;5m(n) 1;15 m(n) 13;16m(n) 13;6m(n) 13;12m(n) 13;2m(n) 13;8m(n) 13;14m(n) 13;4m(n) 13;10 m(n) 9;11m(n) 9;1m(n) 9;7m(n) 9;13m(n) 9;3m(n) 9;9m(n) 9;15m(n) 9;5 m(n) 5;6m(n) 5;16m(n) 5;2m(n) 5;12m(n) 5;14m(n) 5;8m(n) 5;10m(n) 5;41 CCCCA0 BB@p1 p2 p3 p41 CCA: Using the results of Appendix B, this expression reduces to hv0
vni=2 1 1 1 12 6640 BB@P00 P10 P20 P30 iP01 iP11 iP21 iP31 P02P12P22P32 iP03iP13iP23iP331 CCAn10 BB@1 0 0 0 0i0 0 0 01 0 0 0 0i1 CCA(5.37) +0 BB@P00 P10 P20 P30 iP01iP11iP21iP31 1P021P121P221P32 iP03 iP13 iP23 iP331 CCAn10 BB@1 0 0 0 0i0 0 0 01 0 0 0 0i1 CCA3 7750 BB@p1 p2 p3 p41 CCA; which is equation (5.6). Substituting this result into equation (2.1), we recover the diffuson coefficient given by equation (5.7) : D2SMA DNMA=1+4 1 1 1 1  (5.38) 2 6640 BB@1P00P10P20P30 iP011iP11iP21iP31 P02 P12 1+P22 P32 iP03 iP13 iP23 1+iP331 CCA10 BB@1 0 0 0 0i0 0 0 01 0 0 0 0i1 CCA +0 BB@1P00P10P20P30 iP01 1+iP11 iP21 iP31 P02 P12 1+P22P32 iP03iP13iP23iP331 CCA10 BB@1 0 0 0 0i0 0 0 01 0 0 0 0 i1 CCA3 7750 BB@p1 p2 p3 p41 CCA: For a symmetric walk, we substitute P03=P01=1P00P02 2;P23=P21=1P20P22 2; (5.39)Diffusion coefficients for persistent random walks 16 and write P33=P11Pss;P10=P30Psf; P12=P32Psb;P31=P13=1PsfPssPsb;(5.40) in terms of which the diffusion coefficient is found to be D2SMA =DNMA( 24P00+2P2 003Psb+6P00Psb3P2 00Psb+P2 sb2P00P2 sb+P2 00P2 sb +3Psf6P02Psf2P00Psf+2P02P00PsfP2 00Psf+2PsbPsf+3P02PsbPsf +3P00PsbPsfP02P00PsbPsf+P2 00PsbPsf2P2 sbPsf2P00P2 sbPsf+P2 sf +P02P2 sf+3P00P2 sfP02P00P2 sf4PsbP2 sf+2P02PsbP2 sf2P00PsbP2 sf2P3 sf +2P02P3 sf2h (1+P00)2(1+Psb)+[1+P02(3+P00)3P00 +2(1+P00)Psb]Psf2(1+P02)P2 sfi Pss+P2 20(P02Psb)[P02(2+Psb +Psf+2Pss)2Psb(1+Psb+Psf+2Pss)]+P20[(1+P00)PsbP02Psf] [3Psb+Psf+2Pss+P02(2+Psb+Psf+2Pss) 2Psb(Psb+Psf+2Pss)]P2 22h 2Psb+3Psf2Pss P2 00(2+Psb+Psf+2Pss)Psf[Psb+2PsbPsf+2Pss Psf(12Psf4Pss)]+P00[Psb(2+3Psf)4(1Pss) +Psf(2+3Psf+6Pss)]i P22n P2 sb[(1+P00)22(1+P00)Psf] +P02Psf[6+Psf+6PssP00(2+Psf+2Pss)+2Psf(Psf+2Pss)]+ Psbh Psf(3+3P022Psf+2P02Psf4Pss)+2(1+Pss) +P2 00(2+Psf+2Pss)P00[4+P02Psf+2P2 sf+4(1+Psf)Pss]i +P20h P02[42Psb+2Psf4Pss+2P00(2+Psb+Psf+2Pss) 3Psf(Psb+Psf+2Pss)]+Psb[4+Psb3Psf+2Pss +4Psf(Psb+Psf+2Pss)P00(4+3Psb+3Psf+6Pss)]io) ,( [1+PsbP00(1+Psb)+P20(P02+Psb)+P22(1+P00Psf) +Psf+P02Psf]h 2(1+P00)P2 sb2P02P2 sf+Psf[1+P22P00P22P00 +P02(2+P204Pss)]+2[1+P22+P20P02(1+P22)P00]Pss +Psb[3+P22+P20P02P22P002(1+P02)Psf4Pss +P00(3+2Psf+4Pss)]i) : (5.41) If we further assume complete left–right symmetry, so that P33=P11=P31=P13=1PsfPsb 2; (5.42)Diffusion coefficients for persistent random walks 17 then the diffusion coefficient becomes Ds 2SMA =DNMAh 1+P2 22+2P22P20P02+P2 20P2 02+2P002P2 22P002P22P20P02P00 P2 00+P2 22P2 00+PsbP22PsbP20Psb3P22P20Psb+P20P02Psb P2 20P02Psb2P00Psb+2P22P00PsbP20P00Psb+P22P20P00Psb +P20P02P00Psb+P2 00PsbP22P2 00Psb3Psf+3P2 22Psf+3P02Psf 3P02Psf+P20P02Psf+P22P20P02PsfP20P2 02Psf+P00PsfP2 22P00Psf P02P00Psf+P22P02P00Psfi =h 1+P2 22+2P22P20P02+P2 20P2 02+2P00 2P2 22P002P22P20P02P00P2 00+P2 22P2 00PsbP22PsbP20Psb P22P20PsbP20P02PsbP2 20P02Psb+2P00Psb+2P22P00Psb+P20P00Psb +P22P20P00Psb+P20P02P00PsbP2 00PsbP22P2 00PsbPsf+P2 22Psf P02PsfP22P02PsfP20P02Psf+P22P20P02PsfP20P2 02Psf+P00Psf P2 22P00Psf+P02P00Psf+P22P02P00Psfi : (5.43) It can be checked that this equation boils down to the expression (4.15) in the single-step memory approximation. The validity of equations (5.17), (5.29) and (5.38) has been checked by comparison with diffusion coefficients calculated from direct numerical simulations of the corresponding persistent random walk processes. 5.4. Triangular lattice Consider finally the triangular lattice with 6-fold symmetry. We denote the relative directions by numbers from 0 to 5, following the conventions shown in figure 2(d). The symmetry of Mis similar to the previous subsections, so that the structure of the problem is by now clear. Having identified the matrix Jand invariant measure Pin equation (5.4), the velocity auto-correlation is found to be hv0
vn+1i=3h 2m(n) 1;1+m(n) 1;8m(n) 1;152m(n) 1;22m(n) 1;29+m(n) 1;36+m(n) 7;1+2m(n) 7;8 +m(n) 7;15m(n) 7;222m(n) 7;29m(n) 7;36m(n) 13;1+m(n) 13;8+2m(n) 13;15+m(n) 13;22 m(n) 13;292m(n) 13;362m(n) 19;1m(n) 19;8+m(n) 19;15+2m(n) 19;22+m(n) 19;29m(n) 19;36 m(n) 25;12m(n) 25;8m(n) 25;15+m(n) 25;22+2m(n) 25;29+m(n) 25;36+m(n) 31;1m(n) 31;8 2m(n) 31;15m(n) 31;22+m(n) 31;29+2m(n) 31;36i p1 +3h m(n) 1;2m(n) 1;92m(n) 1;16m(n) 1;23+m(n) 1;30+2m(n) 1;31+2m(n) 7;2+m(n) 7;9 m(n) 7;162m(n) 7;23m(n) 7;30+m(n) 7;31+m(n) 13;2+2m(n) 13;9+m(n) 13;16m(n) 13;23 2m(n) 13;30m(n) 13;31m(n) 19;2+m(n) 19;9+2m(n) 19;16+m(n) 19;23m(n) 19;302m(n) 19;31 2m(n) 25;2m(n) 25;9+m(n) 25;16+2m(n) 25;23+m(n) 25;30m(n) 25;31m(n) 31;22m(n) 31;9 m(n) 31;16+m(n) 31;23+2m(n) 31;30+m(n) 31;31i p2 +3h m(n) 1;32m(n) 1;10m(n) 1;17+m(n) 1;24+2m(n) 1;25+m(n) 1;32+m(n) 7;3m(n) 7;10Diffusion coefficients for persistent random walks 18 2m(n) 7;17m(n) 7;24+m(n) 7;25+2m(n) 7;32+2m(n) 13;3+m(n) 13;10m(n) 13;172m(n) 13;24 m(n) 13;25+m(n) 13;32+m(n) 19;3+2m(n) 19;10+m(n) 19;17m(n) 19;242m(n) 19;25m(n) 19;32 m(n) 25;3+m(n) 25;10+2m(n) 25;17+m(n) 25;24m(n) 25;252m(n) 25;322m(n) 31;3m(n) 31;10 +m(n) 31;17+2m(n) 31;24+m(n) 31;25m(n) 31;32i p3 +3h 2m(n) 1;4m(n) 1;11+m(n) 1;18+2m(n) 1;19+m(n) 1;26m(n) 1;33m(n) 7;42m(n) 7;11 m(n) 7;18+m(n) 7;19+2m(n) 7;26+m(n) 7;33+m(n) 13;4m(n) 13;112m(n) 13;18m(n) 13;19 +m(n) 13;26+2m(n) 13;33+2m(n) 19;4+m(n) 19;11m(n) 19;182m(n) 19;19m(n) 19;26+m(n) 19;33 +m(n) 25;4+2m(n) 25;11+m(n) 25;18m(n) 25;192m(n) 25;26m(n) 25;33m(n) 31;4+m(n) 31;11 +2m(n) 31;18+m(n) 31;19m(n) 31;262m(n) 31;33i p4 +3h m(n) 1;5+m(n) 1;12+2m(n) 1;13+m(n) 1;20m(n) 1;272m(n) 1;342m(n) 7;5m(n) 7;12 +m(n) 7;13+2m(n) 7;20+m(n) 7;27m(n) 7;34m(n) 13;52m(n) 13;12m(n) 13;13+m(n) 13;20 +2m(n) 13;27+m(n) 13;34+m(n) 19;5m(n) 19;122m(n) 19;13m(n) 19;20+m(n) 19;27+2m(n) 19;34 +2m(n) 25;5+m(n) 25;12m(n) 25;132m(n) 25;20m(n) 25;27+m(n) 25;34+m(n) 31;5+2m(n) 31;12 +m(n) 31;13m(n) 31;202m(n) 31;27m(n) 31;34i p5 +3h m(n) 1;6+2m(n) 1;7+m(n) 1;14m(n) 1;212m(n) 1;28m(n) 1;35m(n) 7;6+m(n) 7;7 +2m(n) 7;14+m(n) 7;21m(n) 7;282m(n) 7;352m(n) 13;6m(n) 13;7+m(n) 13;14+2m(n) 13;21 +m(n) 13;28m(n) 13;35m(n) 19;62m(n) 19;7m(n) 19;14+m(n) 19;21+2m(n) 19;28+m(n) 19;35 +m(n) 25;6m(n) 25;72m(n) 25;14m(n) 25;21+m(n) 25;28+2m(n) 25;35+2m(n) 31;6+m(n) 31;7 m(n) 31;142m(n) 31;21m(n) 31;28+m(n) 31;35i p6 (5.44) Letting f=exp(2pi=6), we find hv0
vn+1i=3 1 1 1 1 1 1  (5.45) 2 66666640 BBBBBB@P00 P10 P20 P30 P40 P50 fP01 fP11 fP21 fP31 fP41 fP51 f2P02 f2P12 f2P22 f2P32 f2P42 f2P52 P03P13P23P33P33P53 f2P04f2P14f2P24f2P34f2P44f2P54 f1P05f1P15f1P25f1P35f1P45f1P551 CCCCCCAn0 BBBBBB@1 0 0 0 0 0 0f0 0 0 0 0 0 f20 0 0 0 0 01 0 0 0 0 0 0 f20 0 0 0 0 0 f11 CCCCCCA +0 BBBBBB@P00 P10 P20 P30 P40 P50 f1P01f1P11f1P21f1P31f1P41f1P51 f2P02f2P12f2P22f2P32f2P42f2P52 P03P13P23P33P33P53 f2P04 f2P14 f2P24f2P34 f2P44 f2P54 fP05 fP15 fP25fP35 fP45 fP551 CCCCCCAn0 BBBBBB@1 0 0 0 0 0 0f10 0 0 0 0 0 f20 0 0 0 0 01 0 0 0 0 0 0 f20 0 0 0 0 0 f1 CCCCCCA3 77777750 BBBBBB@p1 p2 p3 p4 p5 p61 CCCCCCA; which is again equation (5.6). The expression of the diffusion coefficient is thus given byDiffusion coefficients for persistent random walks 19 (5.7). 6. Two-Step Memory Approximation revisited As seen in the previous section, the symbolic computation of (5.4) quickly becomes tricky. However, an alternative to the above scheme can be found, provided that the walk has special symmetries. Returning to (5.2), we write hv0
vni=å v0å i1;:::;inv0
Si1;:::;inv0P(Si1;:::;inv0jSi1;:::;in1v0;Si1;:::;in2v0) 


 P(Si1;i2v0jSi1v0;v0)p(v0;Si1v0); (6.1) where we introduce the compact notation SiRiT, and sequences in the exponent denote multiple composition: Si1;:::;inSinSin1


 Si1, where each iktakes values between 1 andz. Note that in general SjSi6=Si+jwhen Tis non-trivial. The transition probabilities P(Sj2v0jSj1v0;Sj0v0)can be seen as matrix elements ˜Qj2j1;j1j0, so that (6.1) may be rewritten as hv0
vni=å v0å i1;:::;inv0
Si1;:::;inv0˜Qin;in1


˜Qi2;i1p(v0;Si1v0): (6.2) We would like to rewrite this expression as a matrix product. However, this is in general not possible, and further approximations are needed. Thus, assuming that the scalar product vn
v0factorises as Si1;:::;inv0
v0=n Õ k=1Sikv0
v0; (6.3) and defining Qi;j˜Qi;jSjv0
v0=P(Sj;iv0jSjv0;v0)Sjv0
v0; (6.4) equation (6.2) becomes hv0
vni=å v0å i1;:::;inv0
Sinv0Qin;in1


Qi2;i1p(v0;Si1v0); =zå i1;inV† inQn1 in;i1pi1; (6.5) where we have introduced the vectors Vinv0
Sinv0andpi1p(v0;Si1v0). As can be seen, equation (6.5) has an appropriate matrix form and can easily be resummed over nto compute the diffusion coefficient (2.1). Since Qis a zzmatrix, equation (6.5) is much easier to evaluate than (5.4). The trouble is that equation (6.3) is in general incorrect, and turns out to be strictly valid only for one-dimensional walks. Nonetheless, it may also be applied to higher-dimensional walks satisfying special symmetry conditions. We consider the different geometries separately in the following and discuss the conditions under which equation (6.5) can be applied. For higher- dimensional lattices, we recover by this simpler method the results obtained earlier under the relevant symmetry assumptions. 6.1. One-dimensional lattice The result (5.17) follows from equation (6.5). Indeed, Rin;:::;i1v0
v0=Rin+


+i1v0
v0=1 according to the parity of in+


+i1, and since this is also a property of the product Õn k=1Rikv0
v0, we see that equation (6.3) is valid.Diffusion coefficients for persistent random walks 20 The vector pion the right-hand side of equation (6.5) is p1=p(v;v) =1P00 2(1P00+P10); p2=p(v;v) =P10 2(1P00+P10):(6.6) The vector Vi, on the other hand, has components V1=1; V2=1:(6.7) The matrix elements Qi;jare defined according to equation (6.4), Q= P11P01 P10P00 = P101 1P00 P10 P00 : (6.8) Considering equation (2.1) and plugging the above expressions into equation (6.5), we have D2SMA =DNMAn 1+4V†[I2Q]1po ; (6.9) and we recover equation (5.17). 6.2. Two-dimensional honeycomb lattice Consider equation (6.3) in the case of a honeycomb lattice. The operation Sivis a clockwise rotation of vby anglep=3 ifi=1,p=3 ifi=2, or pifi=3. The operation Si1;:::;invis thus a rotation by angle [2(i1+:::+in)3n]p=3, and the scalar product Si1;:::;inv
v=cos[2(i1+:::+in)3n]p=3: (6.10) This expression is, however, in general different from the product Sinv
v


Si1v
v=n Õ k=1cos[2(ik)3]p=3: (6.11) This is so, for instance, when n=2 and i1=i2=1, for which (6.10) yields 1=2, whereas (6.11) yields 1 =4. There is however a special case under which the product structure that we seek can be retrieved, as follows. There are a priori nine transition probabilities P(Sj;ivjSjv;v). There are, however, a number of left–right symmetries in the system which reduce the number of independent transition probabilities to three: P(S1;1vjS1v;v);P(S3;3vjS3v;v);P(S1;2vjS1v;v): (6.12) In the event that the two probabilities P(S1;2vjS1v;v)andP(S1;1vjS1v;v)are equal, P(S1;2vjS1v;v) =P(S1;1vjS1v;v)Pss; (6.13) which is to say that forward–left and right scatterings are treated as identical events, and the number of independent parameters reduces to two, which we take to be P00andPss.Diffusion coefficients for persistent random walks 21 In this case, the expression of the diffusion coefficient can be obtained in a way similar to equation (5.17) for the one-dimensional lattice. This is so because Si1:::;in1;1v
v=cosn [2(i1+:::+in1+1)3n]p 3o =cosp 3cosn [2(i1+:::+in1)3(n1)]p 3o +sinp 3sinn [2(i1+:::+in1)3(n1)]p 3o ; (6.14) Si1:::;in1;2v
v=cosn [2(i1+:::+in1+2)3n]p 3o =cosp 3cosn [2(i1+:::+in1)3(n1)]p 3o sinp 3sinn [2(i1+:::+in1)3(n1)]p 3o ; (6.15) Si1:::;in1;3v
v=cosn [2(i1+:::+in1+3)3n]p 3o =cosn [2(i1+:::+in1)3(n1)]p 3o : (6.16) Thus, given the symmetry between forward–left and right scatterings, the two sine contributions in equations (6.14) and (6.15) cancel, whereas the cosines add up to 1: Si1:::;in1;1v
v+Si1:::;in1;2v
v=Si1:::;in1v
v; =Si1:::;in1v
v(S1v
v+S2v
v);(6.17) Si1:::;in1;3v
v=Si1:::;in1v
v=Si1:::;in1v
vS3v
v: (6.18) We therefore retrieve an effective product structure, as in equation (6.3), and can compute the diffusion coefficient using equation (6.5), with P=1 3(22PssP00)1 2(1P00) 12Pss ; (6.19) V= 1 1 ; (6.20) and Q= Pss 1=2P001=2 12PssP00 : (6.21) We obtain the expression of the diffusion coefficient for the symmetric [in the sense of equation (6.13)] two-step memory approximation on the honeycomb lattice: Ds 2SMA =DNMA3(1P00) (34Pss+P00)(2Pss+P00) (22PssP00): (6.22) This is equation (5.33). 6.3. Two-dimensional square lattice For the two-dimensional square lattice, recall that Tis the identity and the operation Siv=Riv is a anticlockwise rotation of vby angle ip=2, with i=0;:::; 3. The operation Si1;:::;invis thusDiffusion coefficients for persistent random walks 22 a rotation by angle (i1+


+in)p=2. Equations similar to (6.14)–(6.16) hold§: Ri1:::;inv
v=cosh (i1+


+in)p 2i ; =cosinp 2cosh (i1+


+in1)p 2i sininp 2sinh (i1+


+in1)p 2i ; =Ri1:::;in1v
vRinv
v (din;1din;3)sinh (i1+


+in1)p 2i : (6.23) The last term drops out provided P(Ri+1vjRiv;v) =P(Ri+3vjRiv;v): (6.24) Under the additional assumption that P(R1+ivjR1v;v) =P(R3+ivjR3v;v); (6.25) we retrieve an effective factorisation similar to that postulated in (6.3), and we can then use (6.5) to obtain the corresponding diffusion coefficient. We refer to equations (6.24) and (6.25) as defining a complete left–right symmetry. In this case, the invariant distribution is the solution of p(v;Riv) =å jP(Ri+jvjRjv;v)p(Riv;Rjv); (6.26) å ip(v;Riv) =1 4: (6.27) We solve these equations for p(v;v)andp(v;v), identifying p(v;R1v)andp(v;R3v), and define V= 1 1 ;P= p(v;v) p(v;v) ; (6.28) with the transition matrix Q= P(vjv;v)P(vjv;v) P(vjv;v)P(vjv;v) = P22P02 P20P00 : (6.29) The resulting expression of the diffusion coefficient is identical to (5.43). The validity of this expression extends to d-dimensional orthogonal lattices under the symmetry assumptions (6.24)–(6.25). 7. Conclusions We have shown that it is possible to find exact results for the diffusion coefficient of persistent random walks with two-step memory on regular lattices, by finding the matrix elements which give the velocity auto-correlation function and then resumming then. § Note that, in general, we have the decomposition Ri1:::;inv
v=å w1;:::;wn2f0;1gsgn(w1;:::;wn)Ri1v
vw1


Rinv
vwn; where we introduced the notation vw=vifw=0 and vw=v?ifw=1, and the function sgn (w1;:::;wn) =1, depending on the sequence w1;:::;wn. Equation (6.5) would then be replaced by a more complicated expression involving the mixed products of two matrices P(Rj+ivjRjv;v)Rjv
vandP(Rj+ivjRjv;v)Rjv
v?.Diffusion coefficients for persistent random walks 23 We have applied the results obtained here to approximate the diffusion coefficients of certain periodic billiard tables in [11]. The extension to lattice random walks with longer memory is possible, albeit difficult for obvious technical reasons. Finally, we remark that the extension to lattices in three dimensions is not direct, since in that case, additional information must be specified in order to uniquely define relative directions [12]. Acknowledgments The authors thank Hern ´an Larralde for helpful discussions. This research benefitted from the joint support of FNRS (Belgium) and CONACYT (Mexico) through a bilateral collaboration project. The work of TG is financially supported by the Belgian Federal Government under the Inter-university Attraction Pole project NOSY P06/02. TG is financially supported by the Fonds de la Recherche Scientifique F.R.S.-FNRS. DPS acknowledges financial support from DGAPA-UNAM project IN105209, and the hospitality of the Universit ´e Libre de Bruxelles, where most of this work was carried out. TG acknowledges the hospitality of the Weizmann Institute of Science, where part of this work was completed. Appendix A. 2SMA on a honeycomb lattice In analogy to equation (5.13), we may write Mn0 BBBBBBBBBBBBBBB@a(n) 1a(n) 2a(n) 3a(n) 4a(n) 5a(n) 6a(n) 7a(n) 8a(n) 9 c(n) 9c(n) 7c(n) 8c(n) 3c(n) 1c(n) 2c(n) 6c(n) 4c(n) 5 b(n) 5b(n) 6b(n) 4b(n) 8b(n) 9b(n) 7b(n) 2b(n) 3b(n) 1 b(n) 1b(n) 2b(n) 3b(n) 4b(n) 5b(n) 6b(n) 7b(n) 8b(n) 9 a(n) 9a(n) 7a(n) 8a(n) 3a(n) 1a(n) 2a(n) 6a(n) 4a(n) 5 c(n) 5c(n) 6c(n) 4c(n) 8c(n) 9c(n) 7c(n) 2c(n) 3c(n) 1 c(n) 1c(n) 2c(n) 3c(n) 4c(n) 5c(n) 6c(n) 7c(n) 8c(n) 9 b(n) 9b(n) 7b(n) 8b(n) 3b(n) 1b(n) 2b(n) 6b(n) 4b(n) 5 a(n) 5a(n) 6a(n) 4a(n) 8a(n) 9a(n) 7a(n) 2a(n) 3a(n) 11 CCCCCCCCCCCCCCCA: (A.1) We have the three sets of equations 0 B@a(n) 1a(n) 5a(n) 9 c(n) 1c(n) 5c(n) 9 b(n) 1b(n) 5b(n) 91 CA=0 @P00P10P20 P01P11P21 P02P12P221 A0 B@a(n1) 1a(n1) 5a(n1) 9 c(n1) 9c(n1) 1c(n1) 5 b(n1) 5b(n1) 9b(n1) 11 CA; (A.2) 0 B@a(n) 2a(n) 6a(n) 7 c(n) 2c(n) 6c(n) 7 b(n) 2b(n) 6b(n) 71 CA=0 @P00P10P20 P01P11P21 P02P12P221 A0 B@a(n1) 2a(n1) 6a(n1) 7 c(n1) 7c(n1) 2c(n1) 6 b(n1) 6b(n1) 7b(n1) 21 CA; (A.3) 0 B@a(n) 3a(n) 4a(n) 8 c(n) 3c(n) 4c(n) 8 b(n) 3b(n) 4b(n) 81 CA=0 @P00P10P20 P01P11P21 P02P12P221 A0 B@a(n1) 3a(n1) 4a(n1) 8 c(n1) 8c(n1) 3c(n1) 4 b(n1) 4b(n1) 8b(n1) 31 CA: (A.4) Proceeding with our analogy, we seek linear combinations of the elements in the rows of the matrices on the left-hand side of the above equations, so as to obtain a single matrix equationDiffusion coefficients for persistent random walks 24 similar to equation (5.15). Considering the elements in equation (A.2), we write 0 B@a(n) 1+fa(n) 5+f2a(n) 9 fc(n) 1+f2c(n) 5+c(n) 9 f2b(n) 1+b(n) 5+fb(n) 91 CA= 0 @P00 P10 P20 fP01fP11fP21 f2P02f2P12f2P221 A0 B@a(n1) 1+fa(n1) 5+f2a(n1) 9 fc(n1) 1+f2c(n1) 5+c(n1) 9 f2b(n1) 1+b(n1) 5+fb(n1) 91 CA: (A.5) Comparing with equation (A.2), we infer f3=1,8 >< >:f=1; f=exp(2ip=3) =1ip 3 2; f=exp(2ip=3) =1+ip 3 2:(A.6) Applying the same procedure to equations (A.3)–(A.4), we obtain 0 B@fa(n) 2+f2a(n) 6+a(n) 7 f2c(n) 2+c(n) 6+fc(n) 7 b(n) 2+fb(n) 6+f2b(n) 71 CA= 0 @P00 P10 P20 fP01fP11fP21 f2P02f2P12f2P221 A0 B@fa(n1) 2+f2a(n1) 6+a(n1) 7 f2c(n1) 2+c(n1) 6+fc(n1) 7 b(n1) 2+fb(n1) 6+f2b(n1) 71 CA: (A.7) 0 B@f2a(n) 3+a(n) 4+fa(n) 8 c(n) 3+fc(n) 4+f2c(n) 8 fb(n) 3+f2b(n) 4+b(n) 81 CA= 0 @P00 P10 P20 fP01fP11fP21 f2P02f2P12f2P221 A0 B@f2a(n1) 3+a(n1) 4+fa(n1) 8 c(n1) 3+fc(n1) 4+f2c(n1) 8 fb(n1) 3+f2b(n1) 4+b(n1) 81 CA: (A.8) The system of equations (A.5), (A.7), (A.8) reduces to the single recursive matrix equation 0 B@a(n) 1+fa(n) 5+f2a(n) 9fa(n) 2+f2a(n) 6+a(n) 7f2a(n) 3+a(n) 4+fa(n) 8 fc(n) 1+f2c(n) 5+c(n) 9f2c(n) 2+c(n) 6+fc(n) 7c(n) 3+fc(n) 4+f2c(n) 8 f2b(n) 1+b(n) 5+fb(n) 9b(n) 2+fb(n) 6+f2b(n) 7fb(n) 3+f2b(n) 4+b(n) 81 CA =0 @P00 P10 P20 fP01fP11fP21 f2P02f2P12f2P221 A 0 B@a(n1) 1+fa(n1) 5+f2a(n1) 9:::f2a(n1) 3+a(n1) 4+fa(n1) 8 fc(n1) 1+f2c(n1) 5+c(n1) 9:::c(n1) 3+fc(n1) 4+f2c(n1) 8 f2b(n1) 1+b(n1) 5+fb(n1) 9:::fb(n1) 3+f2b(n1) 4+b(n1) 81 CA; =0 @P00 P10 P20 fP01fP11fP21 f2P02f2P12f2P221 An10 @P00 fP10f2P20 fP01f2P11P21 f2P02P12 fP221 A;Diffusion coefficients for persistent random walks 25 =0 @P00 P10 P20 fP01fP11fP21 f2P02f2P12f2P221 An0 @1 0 0 0f0 0 0 f21 A: (A.9) Appendix B. 2SMA on a square lattice In analogy to equations (A.2)–(A.4), there are 64 different entries among the 256 elements of Mn, which can be obtained through the set of equations 0 BBB@a(n) 1a(n) 6a(n) 11a(n) 16 d(n) 1d(n) 6d(n) 11d(n) 16 c(n) 1c(n) 6c(n) 11c(n) 16 b(n) 1b(n) 6b(n) 11b(n) 161 CCCA=0 BB@P00P10P20P30 P01P11P21P31 P02P12P22P32 P03P13P23P331 CCA0 BBB@a(n1) 1a(n1) 6a(n1) 11a(n1) 16 d(n1) 16d(n1) 1d(n1) 6d(n1) 11 c(n1) 11c(n1) 16c(n1) 1c(n1) 6 b(n1) 6b(n1) 11b(n1) 16b(n1) 11 CCCA;(B.1) 0 BBB@a(n) 2a(n) 7a(n) 12a(n) 13 d(n) 2d(n) 7d(n) 12d(n) 13 c(n) 2c(n) 7c(n) 12c(n) 13 b(n) 2b(n) 7b(n) 12b(n) 131 CCCA=0 BB@P00P10P20P30 P01P11P21P31 P02P12P22P32 P03P13P23P331 CCA0 BBB@a(n1) 2a(n1) 7a(n1) 12a(n1) 13 d(n1) 13d(n1) 2d(n1) 7d(n1) 12 c(n1) 12c(n1) 13c(n1) 2c(n1) 7 b(n1) 7b(n1) 12b(n1) 13b(n1) 21 CCCA;(B.2) 0 BBB@a(n) 3a(n) 8a(n) 9a(n) 14 d(n) 3d(n) 8d(n) 9d(n) 14 c(n) 3c(n) 8c(n) 9c(n) 14 b(n) 3b(n) 8b(n) 9b(n) 141 CCCA=0 BB@P00P10P20P30 P01P11P21P31 P02P12P22P32 P03P13P23P331 CCA0 BBB@a(n1) 3a(n1) 8a(n1) 9a(n1) 14 d(n1) 14d(n1) 3d(n1) 8d(n1) 9 c(n1) 9c(n1) 14c(n1) 3c(n1) 8 b(n1) 8b(n1) 9b(n1) 14b(n1) 31 CCCA;(B.3) 0 BBB@a(n) 4a(n) 5a(n) 10a(n) 15 d(n) 4d(n) 5d(n) 10d(n) 15 c(n) 4c(n) 5c(n) 10c(n) 15 b(n) 4b(n) 5b(n) 10b(n) 151 CCCA=0 BB@P00P10P20P30 P01P11P21P31 P02P12P22P32 P03P13P23P331 CCA0 BBB@a(n1) 4a(n1) 5a(n1) 10a(n1) 15 d(n1) 15d(n1) 4d(n1) 5d(n1) 10 c(n1) 10c(n1) 15c(n1) 4c(n1) 5 b(n1) 5b(n1) 10b(n1) 15b(n1) 41 CCCA:(B.4) Combining these quantities, we let A(n) 1(k)a(n) 1+fka(n) 6+f2 ka(n) 11+f3 ka(n) 16; A(n) 5(k)f3 ka(n) 4+a(n) 5+fka(n) 10+f2 ka(n) 15; A(n) 9(k)f2 ka(n) 3+f3 ka(n) 8+a(n) 9+fka(n) 14; A(n) 13(k)fka(n) 2+f2 ka(n) 7+f3 ka(n) 12+a(n) 13; B(n) 2(k)b(n) 2+fkb(n) 7+f2 kb(n) 12+f3 kb(n) 13; B(n) 6(k)f3 kb(n) 1+b(n) 6+fkb(n) 11+f2 kb(n) 16; B(n) 10(k)f2 kb(n) 4+f3 kb(n) 5+b(n) 10+fkb(n) 15; B(n) 14(k)fkb(n) 3+f2 kb(n) 8+f3 kb(n) 9+b(n) 14; C(n) 3(k)c(n) 3+fkc(n) 8+f2 kc(n) 9+f3 kc(n) 14; C(n) 7(k)f3 kc(n) 2+c(n) 7+fkc(n) 12+f2 kc(n) 13; C(n) 11(k)f2 kc(n) 1+f3 kc(n) 6+c(n) 11+fkc(n) 16; C(n) 15(k)fkc(n) 4+f2 kc(n) 5+f3 kc(n) 10+c(n) 15;Diffusion coefficients for persistent random walks 26 D(n) 4(k)d(n) 4+fkd(n) 5+f2 kd(n) 10+f3 kd(n) 15; D(n) 8(k)f3 kd(n) 3+d(n) 8+fkd(n) 9+f2 kd(n) 14; D(n) 12(k)f2 kd(n) 2+f3 kd(n) 7+d(n) 12+fkd(n) 13; D(n) 16(k)fkd(n) 1+f2 kd(n) 6+f3 kd(n) 11+d(n) 16; in terms of which we have 0 BBB@A(n) 1(k)A(n) 13(k)A(n) 9(k)A(n) 5(k) D(n) 16(k)D(n) 12(k)D(n) 8(k)D(n) 4(k) C(n) 11(k)C(n) 7(k)C(n) 3(k)C(n) 15(k) B(n) 6(k)B(n) 2(k)B(n) 14(k)B(n) 10(k)1 CCCA =0 BB@P00 P10 P20 P30 fkP01fkP11fkP21fkP31 f2 kP02f2 kP12f2 kP22f2 kP32 f3 kP03f3 kP13f3 kP23f3 kP331 CCA0 BBB@A(n1) 1(k)A(n1) 13(k)A(n1) 9(k)A(n1) 5(k) D(n1) 16(k)D(n1) 12(k)D(n1) 8(k)D(n1) 4(k) C(n1) 11(k)C(n1) 7(k)C(n1) 3(k)C(n1) 15(k) B(n1) 6(k)B(n1) 2(k)B(n1) 14(k)B(n1) 10(k)1 CCCA; =0 BB@P00 P10 P20 P30 fkP01fkP11fkP21fkP31 f2 kP02f2 kP12f2 kP22f2 kP32 f3 kP03f3 kP13f3 kP23f3 kP331 CCAn10 BBB@A(1) 1(k)A(1) 13(k)A(1) 9(k)A(1) 5(k) D(1) 16(k)D(1) 12(k)D(1) 8(k)D(1) 4(k) C(1) 11(k)C(1) 7(k)C(1) 3(k)C(1) 15(k) B(1) 6(k)B(1) 2(k)B(1) 14(k)B(1) 10(k)1 CCCA; =0 BB@P00 P10 P20 P30 fkP01fkP11fkP21fkP31 f2 kP02f2 kP12f2 kP22f2 kP32 f3 kP03f3 kP13f3 kP23f3 kP331 CCAn10 BB@P00fkP10f2 kP20f3 kP30 fkP01f2 kP11f3 kP21P31 f2 kP02f3 kP12P22fkP32 f3 kP03P13fkP23f2 kP331 CCA; =0 BB@P00 P10 P20 P30 fkP01fkP11fkP21fkP31 f2 kP02f2 kP12f2 kP22f2 kP32 f3 kP03f3 kP13f3 kP23f3 kP331 CCAn0 BB@1 0 0 0 0fk0 0 0 0 f2 k0 0 0 0 f3 k1 CCA; (B.5) provided f4 k=1,fk=exp(ikp=2);k=0;1;2;3; (B.6) We have the following identities, 2[a(n) 1a(n) 11] =A(n) 1(1)+A(n) 1(3); 2[a(n) 5a(n) 15] =A(n) 5(1)+A(n) 5(3); 2[a(n) 9a(n) 3] =A(n) 9(1)+A(n) 9(3); 2[a(n) 13a(n) 7] =A(n) 13(1)+A(n) 13(3); 2[b(n) 2b(n) 12] =B(n) 2(1)+B(n) 2(3); 2[b(n) 6b(n) 16] =B(n) 6(1)+B(n) 6(3); 2[b(n) 10b(n) 4] =B(n) 10(1)+B(n) 10(3); 2[b(n) 14b(n) 8] =B(n) 14(1)+B(n) 14(3); 2[c(n) 3c(n) 9] =C(n) 3(1)+C(n) 3(3);Diffusion coefficients for persistent random walks 27 2[c(n) 7c(n) 13] =C(n) 7(1)+C(n) 7(3); 2[c(n) 11c(n) 1] =C(n) 11(1)+C(n) 11(3); 2[c(n) 15c(n) 5] =C(n) 15(1)+C(n) 15(3); 2[d(n) 4d(n) 10] =D(n) 4(1)+D(n) 4(3); 2[d(n) 8d(n) 14] =D(n) 8(1)+D(n) 8(3); 2[d(n) 12d(n) 2] =D(n) 12(1)+D(n) 12(3); 2[d(n) 16d(n) 6] =D(n) 16(1)+D(n) 16(3); which, using equation (B.5) yield the velocity auto-correlation (5.37). References [1] Weiss G H 1994, Aspects and Applications of the Random Walk (North-Holland Publishing Co., Amsterdam). [2] Haus J W and Kehr K W 1987, Diffusion in regular and disordered lattices, Phys. Rep. 150, 263–406. [3] F ¨urth R 1920, Die Brownsche Bewegung bei Ber ¨ucksichtigung einer Persistenz der Bewegungsrichtung. Mit Anwendungen auf die Bewegung lebender Infusorien, Zeit. f. Physik 2, 244. [4] Taylor G I 1922, Diffusion by continuous movements, Proc. London Math. Soc. 20, 196. [5] Kuhn W 1934, Kolloid Z. 68, 2; 1936 76, 258. [6] Manning J R 1959, Correlation effects in impurity diffusion, Phys. Rev. 116, 819. [7] Weiss G H 2002, Some applications of persistent random walks and the telegrapher’s equation, Physica A 311, 381. [8] Montroll EW 1950, Markoff chains and excluded volume effect in polymer chains J. Chem. Phys. 18, 734. [9] Bender E A and Richmond L B, 1984 Correlated random walks Ann. Prob. 12, 274. [10] Renshaw E and Henderson R 1981, The correlated random walk J. Appl. Probab. 18, 403; 1994, The general correlated random walk, J. Appl. Probab. 31, 869. [11] Gilbert T and Sanders DP 2009, Persistence effects in deterministic diffusion. Preprint, arXiv:0908.0600v1. [12] Larralde H 1997, Transport properties of a two-dimensional “chiral” persistent random walk Phys. Rev. E 56, 5004. [13] Fink T M and Mao Y 1999, Designing tie knots using random walks, Nature 398, 31.

2009-08-10

We calculate the diffusion coefficients of persistent random walks on lattices, where the direction of a walker at a given step depends on the memory of a certain number of previous steps. In particular, we describe a simple method which enables us to obtain explicit expressions for the diffusion coefficients of walks with two-step memory on different classes of one-, two- and higher-dimensional lattices.

Diffusion coefficients for multi-step persistent random walks on lattices

0908.1271v1

arXiv:1812.07244v2 [cond-mat.mes-hall] 26 May 2019Thermal gradient driven domain wall dynamics M. T. Islam,1,2X. S. Wang,3,4and X. R. Wang1,5,∗ 1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2Physics Discipline, Khulna University, Khulna, Banglades h 3School of Electronic Science and Engineering and State Key L aboratory of Electronic Thin Film and Integrated Devices, University of Electronic Science and Technology of China, C hengdu 610054, China 4Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Tr ondheim, Norway 5HKUST Shenzhen Research Institute, Shenzhen 518057, China The issue of whether a thermal gradient acts like a magnetic fi eld or an electric current in the domain wall (DW) dynamics is investigated. Broadly speakin g, magnetization control knobs can be classified as energy-driving or angular-momentum drivin g forces. DW propagation driven by a static magnetic field is the best known example of the former i n which the DW speed is proportional to the energy dissipation rate, and the current-driven DW mo tion is an example of the latter. Here we show that DW propagation speed driven by a thermal gradien t can be fully explained as the angular momentum transfer between thermally generated spi n current and DW. We found DW- plane rotation speed increases as DW width decreases. Both D W propagation speed along the wire and DW-plane rotation speed around the wire decrease with th e Gilbert damping. These facts are consistent with the angular momentum transfer mechanis m, but are distinct from the energy dissipation mechanism. We further show that magnonic spin- transfer torque (STT) generated by a thermal gradient has both damping-like and field-like compo nents. By analyzing DW propagation speed and DW-plane rotational speed, the coefficient ( β) of the field-like STT arising from the non- adiabatic process, is obtained. It is found that βdoes not depend on the thermal gradient; increases with uniaxial anisotropy K/bardbl(thinner DW); and decreases with the damping, in agreement w ith the physical picture that a larger damping or a thicker DW leads t o a better alignment between the spin-current polarization and the local magnetization, or a better adiabaticity. I. INTRODUCTION Manipulating domain walls (DW) in magnetic nanos- tructures has attracted much attention because of its po- tential applications in data storage technology [ 1] and logic gates [ 2]. The traditional DW control knobs, namely magnetic fields and spin-polarized currents, have certain drawbacks in applications. In the magnetic-field- driven DW motion, energy dissipation is the main cause ofDWpropagationwhosespeedisproportionaltotheen- ergy dissipation rate [ 3,4], and the magnetic field tends todestroyunfavorabledomainsandDWs, insteadofdriv- ing a series of DWs synchronously [ 5–7]. An electrical current drives a DW to move mainly through the angu- lar momentum transfer so that it pushes multiple DWs [8–11] in the same direction. To achieve a useful DW speed, it requires high electrical current densities that result in a Joule heating problem [ 12–14]. To avoid these problems, spin-wave spin current has been proposed as a moreenergy-efficientcontrolparameter[ 15–18]. Thermal gradient, a way to generate spin-wave spin current, is an alternative control knob of the DW motion. The inves- tigation on thermal-gradient-driven domain wall motion is meaningful not only for conventional applications, but also for the understanding of spin wave and domain wall dynamics [ 16,17,20–23], as well as for possible recycling of waste heat [ 19,24]. ∗[Corresponding author:]phxwan@ust.hkTo understand the mechanism behind thermal- gradient-drivenDWdynamics, therearemicroscopicthe- ories [15–17,25,26] and macroscopic thermodynamic theories [ 21,22]. Briefly speaking, the microscopic theo- ries suggest that magnons populated in the hotter region diffuses to the colder region to form a magnon spin cur- rent. The magnon spin currentpassesthrough a DWand exerts a torque on the DW by transferring spin angular momentum to the DW. Thus, magnons drive the DW propagating toward the hotter region of the nanowire, opposite to the magnon current direction [ 15,16,18]. The thermodynamic theories anticipate that a thermal gradient generates an entropy force which always drives the DW towards the hotter region in order to minimize the system free energy. The macroscopic theories do not provide any microscopic picture about DW dynamics al- though a thermal gradient is often considered as an effec- tive magnetic field to estimate DW speed [ 21,22] from field-driven DW theories. Thus, one interesting issue is whether a thermal gradient in DW dynamics acts like a magnetic field or an electric current. DW propagation speed should be sensitive to both DW width and types of a DW (transverse DW) under an energy-driving force while the speed should be insensitive to the DW and DW structure in the angular-momentum-driving force. This is the focus of the current work. In this paper, we investigate DW motion along a uni- axial wire with the easy axis along the wire direction under a thermal gradient. We found that the DW al- ways propagates to the hotter region with an accom- panied DW-plane rotation. DW propagation speed and2 z xy FIG. 1. Schematic diagram of a uniaxial magnetic nanowire with a head-to-head DW at the center under a thermal gra- dient∇T. Black (white) color represents colder (hotter) end of the sample. DW-plane rotation speed increases as the magnetic easy- axis anisotropy and damping decreases. We show that DW motion can be attributed to the angular momen- tum transfer between magnonic spin current and the DW. Thus, we conclude that a thermal gradient in- teracts with DW through angular-momentum transfer rather than through energy dissipation. Similar to an electric current [ 27], a thermal gradient can generate both damping-like (or adiabatic) STT and field-like (or non-adiabatic) STT. From the damping-dependence and anisotropy-dependence of the average DW velocity and DW-plane rotation angular velocity, we extract field-like STT coefficient ( β). It is found that βis independent of thermal gradient; is bigger for a thinner DW; and de- creases with the damping coefficient. We also show that in the presence of a weak hard-axis anisotropy perpen- dicular to the wire, the DW still undergoes a rotating motion. The DW propagation speed increases slightly while the DW-plane rotation speed decreases with the strength of the hard-axis anisotropy. II. MODEL AND METHOD We consider a uniaxial nanowire of length Lxand cross-section Ly×Lzalong the x-axis (easy axis) with a head-to-head DW at the center, as shown in Fig. 1. Ly,Lzis much smaller than the DW width ∆, and ∆ is much smaller than Lx. A thermal gradient is applied along the wire. The highest temperature is far below the Curie temperature Tc. The magnetization dynam- ics is governed by the stochastic Landau-Lifshitz-Gilbert (LLG) equation [ 28,29], dm dt=−γm×(Heff+hth)+αm×∂m ∂t,(1) wherem=M/MsandMsare respectively the magne- tization direction and the saturation magnetization. α is the Gilbert damping constant and γis the gyromag-netic ratio. Heff=2A µ0Ms/summationtext σ∂2m ∂x2σ+2K/bardbl µ0Msmxˆx+hdipoleis the effective field, where Ais the exchange constant, xσ (σ= 1,2,3) denote Cartesian coordinates x,y,z,K/bardblis the easy-axis anisotropy, and hdipoleis the dipolar field. hthis the stochastic thermal field. The stochastic LLG equation is solved numerically by MUMAX3 package [ 30] in which we use adaptive Heun solver. To balance stability and efficiency, we choose the time step 10−14s with the cell size (2 ×2×2) nm3. Mag- netic charges at the two ends of the wire are removed to avoid their attraction to the DW. The saturation mag- netization Ms= 8×105A/m and exchange constant A= 13×10−12J/m are used to mimic permalloy in our simulations. The thermal field follows the Gaussian process characterized by following statistics [ 31] /angb∇acketlefthth,ip(t)/angb∇acket∇ight= 0, /angb∇acketlefthth,ip(t)hth,jq(t+∆t)/angb∇acket∇ight=2kBTiαi γµ0Msa3δijδpqδ(∆t),(2) whereiandjdenote the micromagnetic cells, and p,q represent the Cartesian components of the thermal field. Tiandαiare respectively temperature and the Gilbert damping at cell i, andais the cell size. kBis the Boltz- mann constant [ 28]. The numerical results presented in this study are averaged over 15 random configurations (for DW velocity) and 4000-5000 random configurations (for spin current). Underthethermalgradient ∇xT,magnetizationatdif- ferent positions deviate from their equilibrium directions differently and small transverse components myandmz are generated. The transverse components vary spatial- temporally and depend on the local temperature. This variation generates a magnonic spin current [ 16]. This magnonic spin current can interact with spin textures such as DWs. In the absence of damping (the thermal field also vanishes), the spin currentalong the xdirection can be defined from the spin continuity equation derived from Eq. ( 1) as follows [ 15], ∂m ∂t=−1 1+α2m׈xmxK/bardbl−∂J ∂x,(3) where J(x) =2γA µ0Msm×∂m ∂x, (4) is the spin current density along x-direction due to the exchangeinteraction. J(x) can be numerically calculated [15,23]. In the presence of damping as well as the ther- mal field, the contribution of the damping term and the thermal term is proportional to α, which is relatively small. More importantly, according to the fluctuation- dissipation theorem [ 28], the damping term and the ther- mal term should cancel each other after average over a long time. Since the time scale of DW dynamics is much longer than the thermal fluctuation, the combined con- tribution of damping and thermal terms should be very small.3 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s52/s56/s49/s50/s49/s54/s50/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s50/s52/s54/s56/s49/s48/s49/s50/s45/s56/s48/s48 /s45/s52/s48/s48 /s48 /s52/s48/s48 /s56/s48/s48/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50 /s32/s32 /s32/s118 /s115/s105/s109/s117 /s32/s118 /s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41 /s120/s84 /s32/s40/s75/s47/s110/s109/s41/s32/s32/s32 /s120/s84 /s32/s40/s75/s47/s110/s109/s41 /s32/s32 /s48/s46/s48/s55/s32 /s75/s47/s110/s109 /s48/s46/s49 /s75/s47/s110/s109 /s48/s46/s49/s53/s32 /s75/s47/s110/s109 /s48/s46/s50/s32 /s75/s47/s110/s109 /s48/s46/s50/s53/s32 /s75/s47/s110/s109 /s48/s46/s51/s32 /s75/s47/s110/s109/s74 /s116/s111/s116/s40/s120/s41/s40 /s115/s41/s41 /s120 /s32/s40/s110/s109/s41 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41/s40/s97/s41 /s32 /s75 /s32/s40/s49/s48/s52 /s32/s74/s47/s109/s51 /s41/s118 /s32/s40/s109/s47/s115/s41/s40/s99/s41/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41 /s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41 /s51/s54/s51/s57/s52/s50 /s32 FIG. 2. (a) The spatial dependence of spin current densities Jtot(x) for various ∇xT. The DW center is chosen as x= 0. (b) Thermal gradient dependence of DW velocity vsimu from micromagnetic simulations (open squares) and vcurrent computedfrom total spin current(solid squares). (c)Therm al gradient dependence of DW-plane rotation angular velocity (squares). In (a)(b)(c) model parameters are Lx= 2048 nm, Ly=Lz= 4 nm, α= 0.004 and K/bardbl= 5×105J/m3. (d) vsimu(solid squares) and dφ/dt(open squares) as a function ofK⊥forLx= 1024 nm and ∇xT= 0.5 K/nm. Integrating the x−component of Eq. ( 3) over a space enclosed the DW in the center and noticing the absence of the first term on the right, we have vcurrent=1 2/integraldisplayLx/2 −Lx/2∂mx ∂tdx =−2γA µ0Ms/bracketleftbig1 2(Jx|left−Jx|right)].(5) where we have assumed the fluctuations in the domains are small and the DW is not far from a symmetric one. Jx|left,Jx|rightmean the x-components of the total spin current on the left and right sides of the DW. The equa- tion clearly shows that the DW propagates opposite to the spin current. This is the theoretical DW velocity un- der the assumption of angular momentum conservation, and it will be compared with the directly simulated DW velocity below. III. RESULTS A. Average spin current and DW velocity TosubstantiateourassertionthatDWpropagationun- der a thermal gradient is through angular-momentum ef- fect instead of energy effect, we would like to compare the DW velocity obtained from micromagnetic simula- tions and that obtained from total spin current based on/s49 /s50 /s51 /s52 /s53 /s54 /s55/s56/s49/s50/s49/s54/s50/s48 /s52/s56/s49/s50/s49/s54/s50/s48 /s32/s118 /s115/s105/s109/s117 /s32/s118 /s99/s117/s114/s114/s101/s110/s116 /s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41/s32/s118 /s32/s40/s109/s47/s115/s41 /s32/s76 /s120/s61/s50/s48/s52/s56/s32/s110/s109 FIG. 3. Damping αdependence of the DW dynamics: vsimu (Open squares); vcurrent(solid squares ); and dφ/dt(solid circles). Model parameters are ∇xT= 0.2 K/nm, K/bardbl= 5× 105J/m3,Lx= 2048 nm and Ly=Lz= 4 nm. Eq. (5). Eq. (4) is used to calculate Jx(x). Fig.2(a) is spatial distribution of the ensemble averaged Jx(x) with DW atx= 0 for various thermal gradients. The sud- den sign change of Jx(x) at the DW center is a clear evidence of strong angular-momentum transfer from spin current to the DW. Technically, magnetizationof the two domains separated by the DW point to the opposite di- rections, thus the spin current polarization changes its sign. In calculating DW velocity vcurrentfrom Eq. ( 5), the spin currents before entering DW and after passing DW are the averages of Jx(x) overx∈[−2∆,−∆] and x∈[∆,2∆], where ∆ is the DW width which is 16 nm in the current case. The thermal gradient dependence ofvcurrentis shown in Fig. 2(b) (solid squares). vcurrent compares well with the velocity vsimu(open squares) ob- tained directly from simulations by extracting the speed of the DW center along x-direction. The DW veloc- ity is linearly proportional to the temperature gradient v=C∇xT, with the thermal mobility C= 6.66×10−8 m2s−1K−1forvsimuorC= 6.59×10−8m2s−1K−1for vcurrent. It is noted that vcurrentalmost coincides with vsimuexcept a small discrepancy at very high thermal gradient when the nonlinear effects is strong. The small discrepancy may be attributed to the large fluctuations as well as the contribution from the damping, the dipo- lar and stochastic fields. These observations are consis- tent with magnonic STT [ 15,16,25,26]. It is observed that the DW-plane rotates around the x-axis counter- clockwise for head-to-head DW and clockwise for tail-to- tail DW during DW propagation. DW rotation speed dφ/dt(squares) is shown in Fig. 2(c)) as a function of ∇xT.4 /s48 /s50 /s52 /s54 /s56 /s49/s48/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54 /s52/s54/s56/s49/s48/s49/s50/s49/s52/s32/s32 /s75 /s124/s124/s32/s40/s49/s48/s53 /s32/s74/s47/s109/s51 /s41/s32/s118 /s115/s105/s109/s117 /s32/s118 /s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41 /s100 /s47/s100/s116/s40/s100/s101/s103/s47/s110/s115/s41/s32 FIG. 4. Anisotropy K/bardbldependence of the DW dynamics: vsimu(open squares); vcurrent(solid squares); and dφ/dt(solid circles). Model parameters are Lx= 2048 nm, Ly=Lz=4 nm, α= 0.004 and ∇xT= 0.2 K/nm. B. Damping and anisotropy dependence of DW dynamics An energy-effect and angular-momentum-effect have different damping-dependence and anisotropy- dependence of DW dynamics. To distinguish the roles of energy and the angular-momentum in thermal-gradient driven DW dynamics, it would be useful to probe how the DW dynamics depends on αandK/bardbl. Damping have two effects on the spin currents: one is the decay of spin current during its propagation so that the amount of spin angular momentum deposited on a DW should decrease with the increase of the damping coefficient. As a result, the DW propagation speed and DW-plane rotation speed should also be smaller for a larger α. Indeed, this is what we observed in our simulations as shown in Fig. 3(a) for DW speed and DW-plane rotation speed (open squares for vsimu, solid circles for vcurrent, and stars for dφ/dt). The model parameters are Lx= 2048, Ly=Lz= 4 nm, ∇xT= 0.2 K/nm and K/bardbl= 5×105J/m3. The second damping effect is that the larger αhelps the spin current polarization to align with the local spin. This second effect enhances the adiabatic process that is important for non-adiabatic STT or field-like torque discussed in the next subsection. Therefore, α−dependence of DW dynamics supports the origin of thermal driven DW dynamics to be the angular-momentum effect, not the energy effect that would lead to a larger vsimuanddφ/dtfor a larger α [3,4,33–35] instead of a decrease observed here. Here we would like to see how the DW dynamics de- pendsonuniaxialanisotropy K/bardbl. Fig.4showsboth vsimu (open squares), vcurrent(filled squares) and dφ/dt(cir- cles) for Lx= 2048 nm, α= 0.004 and ∇xT= 0.2. The DW propagation speed, vsimudecreases with K/bardblwhileDW-plane rotational speed increases with K/bardbl. These re- sults seem follow partially the behavior of magnetic-field induced DW motion, in which DW propagation speed is proportional to DW width (∆ ∼/radicalBigg A K/bardbl) or decrease withK/bardbl, and partially electric current driven DW mo- tion, in which DW-plane rotational speed increases with K/bardbl. Thus, one may tend to conclude that a thermal gra- dient behaves more like a magnetic field rather than an electric current from the DW width dependence of DW propagation speed, opposite to our claim of the angular- momentum effects of the thermal gradient. It turns out, this is not true. The reason is that magnon spectrum, ωk=2γ µ0Ms/parenleftbig Ak2+K/bardbl/parenrightbig , has a gap in a system with mag- netic anisotropy. The larger K/bardblis, the bigger the energy gap will be. Thus, it becomes harder to thermally excite magnon. As a result, the spin current decreasesas K/bardblin- creases. To see whether the thermal-gradient driven DW motion is due to the angular-momentum transfer or not, one should compare whether vsimuandvcurrentmaintain a good agreement with each other as K/bardblvaries. Indeed, a good agreement between vsimuandvcurrentis shown in Fig.4. This conclusion is also consistent with existing magnonic STT theories [ 33–35]. C. Separation of adiabatic and non-adiabatic torques We have already demonstrated that a thermal gradi- ent interacts with DW through magnonic STT rather than through energy dissipation. It is then interesting to know what kind of STTs a thermal gradient can gen- erate. Specifically, whether a magnonic spin current generates damping-like (adiabatic), or field-like (Non- adiabatic) torques, or both just like an electric current /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52 /s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s73/s32/s40/s49/s48/s49/s48 /s32/s65/s47/s109/s50 /s41 /s32 /s120/s84 /s32/s40/s75/s47/s110/s109/s41 /s32/s32 FIG. 5. Model parameters are K/bardbl= 5×105J/m3,α= 0.004,Lx= 1024 and Ly=Lz=4 nm. Effective electric current densityI(open squares) and β(solid squares) are plotted as functions of ∇xT.5 /s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50 /s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s46/s49/s52/s48/s46/s50/s49/s48/s46/s50/s56 /s40/s49/s48/s45/s51 /s41 /s32/s32 /s75 /s124/s124/s32/s40/s49/s48/s53 /s32/s74/s47/s109/s51 /s41/s40/s97/s41 /s40/s98/s41/s32/s32 FIG. 6. Model parameters are ∇xT=0.5 K/nm, Lx= 1024 nm and Ly=Lz=4 nm. (a) α-dependence of βforK/bardbl= 106 and J/m3. (b)K/bardbl-dependence of βforα= 0.004. [27] does. To extract the STT generated from a thermal gradient, we approximate DW dynamics by the motion of its collective modes of DW center Xand the titled angleφof DW-plane. Subject to both damping-like and field-like torques, using the travelling-wave ansatz [ 33– 35], tan(θ/2) = exp[( x−X)/∆] where ∆ ∼/radicalbig A/K/bardbl, one can derive the equations for X and φ, α ∆dX dt+dφ dt=β αu,1 ∆dX dt−αdφ dt=u α.(6) From the above two equations, one can straightfor- wardly find DW propagating speed and DW-plane ro- tation speed, v=(1+αβ) (1+α2)u,˙φ=(β−α) (1+α2)u. (7) One can extract βand equivalent electric current den- sityI= (2eMsu)/gµBPfromvanddφ/dtobtained in simulations. For α= 0.004,K/bardbl= 106J/m3, theIandβ are obtained and plotted in Fig. 5as a function of ∇xT. It is evident that Ilinearly increases with ∇xTandβ is independent of ∇xTas it should be. We then fixed ∇xT= 0.5 K/nm, and repeat simulations and analysis mentioned above for various αandK/bardbl. Fig.6(a) and (b) shows βas a function of αandK/bardbl. From the figure, it is evident that βdecreases with α. This is because the larger damping favors the alignment of spin current polarization with the local spin so that the non-adiabatic effect,β, becomes smaller. βincreases with K/bardblfor the similar reason: Larger K/bardblmeans a thinner DW so that it is much harder for the spin current polarization to re- verse its direction after passing through the thinner DW, i.e. a stronger non-adiabatic effect. In some experiments, the temperature gradient is gen- erated by a laser spot[ 36]. The laser spot will induce a Gaussian distribution of the temperature over the space [36,37]. In Fig. 7, weshowtheDWmotionin aGaussian temperature profile T(x) =T0exp/parenleftBig −(x−xL)2 2σ2/parenrightBig by plot- ting the DW position against the time. Here we use the/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48 /s32/s32/s68/s87/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s40/s110/s109/s41 /s116/s105/s109/s101/s32/s40/s110/s115/s41 FIG. 7. Domain wall position versus time in a Gaussian tem- perature profile. The gray lines are raw data for different random seeds and the red line is the averaged result. The green dashed line is theoretical result using the thermal mo - bilityC= 6.66×10−8m2s−1K−1obtained from Fig. 2(b). same parameters as those in Fig. 2(b), except a longer wireLx= 2048 nm, and T0= 400 K, σ= 200 nm, xL= 200 nm. Theoretically, if the instantaneous DW speed under a Gaussian temperature is the same as that in the constant thermal-gradient case, we should expect dx dt=CdT dx, where the thermal mobility Cis the same as that in Fig. 2(b). Using C= 6.66×10−8m2s−1K−1, the above differential equation for x(t) can be numeri- cally solved with initial condition x(0) = 0. The result is plotted in Fig. 7in green dashed line. The simu- lated speed is smaller than this theoretical result. This is probably because, for the constant thermal-gradient, we focus on the steady-state DW motion speed. In a Gaussian temperature, the DW cannot immediately fol- low the local temperature gradient. Before the DW can reach the steady-state speed corresponding to the local temperature, it already moves to a position of smaller temperature gradient. More details about DW motion in Gaussian temperature profile may be an issue of future studies. IV. DISCUSSION AND SUMMARY We have studied the thermal gradient-driven DW dy- namicsinanuniaxialnanowire. Inreality, thereisalways certain hard anisotropy in a wire whose cross-section is not a perfect ellipse. Thus, it is interesting to see how the above results will change in a weak biaxial nanowire with a small hard anisotropy K⊥= 1/2µ0M2 s(Nz−Ny), say along y-direction. Our simulations show that a DW still propagatestowardsthe higher temperature region in a similar way as that in a uniaxial wire. Interestingly, as shown in Fig. 2(d) for the K⊥-dependence of vsimu(solid squares) and dφ/dt(open squares), DW speed increases6 slightly with K⊥. This may be due to the increase of torque along θ-direction [ 33] since Γ θis proportional to (Nz−Ny). This is also consistent with the early results forthe uniaxialwire that vsimu(which includes stochastic thermal field and demagnetisation fields) is always larger thanvcurrent(where the transverse fields are neglected). At the meanwhile, dφ/dtdecreases with K⊥. The main purpose of this paper is to study the magnonic effects in thermal-gradient-driven domain wall dynamics. We consider the spin waves explicitly and all the material parameters (exchange constant A, crys- talline anisotropy K, saturation magnetization Ms, and Gilbert damping α) are assumed to be constant. Indeed, the atomistic magnetic moments are independent of tem- perature. At the atomistic level, the exchange constant Aoriginating from the Pauli exclusion principle and the crystalline anisotropy Koriginating from the spin-orbit coupling onlyweaklydepend on the temperature because of the vibration of atoms [ 39]. In micromagnetic models, because finite volumes that contains many magnetic mo- ments are considered as unit cells, the parameters A,K, andMsdepend on the temperature. This is because the thermally excited spin waves with wavelengths shorter than the length scale of the unit cells are included in the effective A,K, andMsby doing an average [ 16,38]. Since we use small mesh size 2 ×2×2 nm3, only spin waves of very short wavelength affect the parameters A, K, andMsin our model. Those short-wavelength spin wavespossess high energyaswell as low density ofstates, so their contributions to the effective A,K, andMsare not significant. The Gilbert damping αdepends on the temperature non-monotonically [ 40–43]. The underlying mechanism is still under debate, but for many cases the dependence is not significant in a wide range of temper- ature. In summary, our results show that the uniform ther- mal gradient always drives a DW propagating towards the hotter region and the DW-plane rotates around the easy axis. The DW velocity and DW-plane rotational speed decrease with the damping coefficient. The DW velocity obtained from simulation agrees with the veloc- ity obtained from angular momentum conservation when the magnon current density ( J(x)) from the simulation is usedtoestimatetheamountofangularmomentumtrans- ferred from magnon current to the DW. All the above findings lead to the conclusion that the thermal gradient interacts with DW through angular-momentum transfer rather than through energy dissipation. Furthermore, we demonstrated that the magnonic STT generated by a thermal gradient has both damping-like and field-like components. The field-like STT coefficient βis deter- mined from DW speed and DW-plane rotation speed. β does not depend on the thermal gradient as expected, but increases with a decrease of DW width. This behav- ior can be understood from the expected strongmisalign- ment of magnon spin polarization and the local spin so that non-adiabatic torque (also called field-like torque) is larger. For the same reason, a larger Gilbert dampingresults in a better alignment between spin current polar- ization and the local spin, thus βshould decrease with α. The thermal gradientcan be a veryinteresting control knob for nano spintronics devices, especially those made from magnetic insulators. This work was supported by the National Natural Sci- ence Foundation of China (Grant No. 11774296) as well as Hong Kong RGC Grants Nos. 16300117, 16301518 and 16301816. X.S.W acknowledges support from NSFC (GrantNo. 11804045),ChinaPostdoctoralScienceFoun- dation (Grant No. 2017M612932and 2018T110957),and the Research Council of Norway through its Centres of Excellence funding scheme, Project No. 262633, “QuS- pin.” M. T. I acknowledges the Hong Kong PhD fellow- ship.7 [1] Parkin S S P, Hayashi M and Thomas L 2008 Science 320 190 [2] Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit D and Cowburn R P 2005 Science309 1688 [3] Wang X R, P Yan, Lu J and He C 2009 Ann. Phys. (N. Y.)324 1815 [4] Wang X R, Yan P and Lu J 2009 Europhys. Lett. 86 67001 [5] Atkinson D, Allwood D A, Xiong G, Cooke M D, Faulkner C C, and Cowburn R P 2003 Nat. Mater. 2 85 [6] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine J L 2005 Nat. 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Mater.11 391 [20] Jiang W, Upadhyaya P, Fan Y B, Zhao J, Wang M S, Chang L T, Lang M R, Wong K L, Lewis M, Lin Y T, Tang J S, Cherepov S, Zhou X Z, Tserkovnyak Y, Schwartz R N and Wang K L 2013 Phys. Rev. Lett. 110 177202 [21] Schlickeiser F, Ritzmann U, Hinzke D and Nowak U 2014 Phys. Rev. Lett. 113 097201 [22] Wang X S and Wang X R 2014 Phys. Rev. B 90 014414[23] Wang X-G, Chotorlishvili L, Guo G-H, Sukhov A, Dugaev V, Barnas J and Berakdar J 2016 Phys. Rev. B94 104410 [24] Safranski C, Barsukov I, Lee H K, Schneider T, Jara A, Smith A, Chang H, Lenz K, Lindner J,Tserkovnyak Y, Wu M and Krivorotov I 2017 Nat. Com. 8 117 [25] Wang X G, Guo G H, Nie Y Z, Zhang G F and Li Z X 2012Phys. Rev. B 86 054445 [26] Yan P, Cao Y, and Sinova J 2015 Phys. Rev. B 92 100408 [27] Kishine Jun-ichiro and Ovchinnikov A S 2010 Phys. Rev. B81 134405 [28] Brown W F 1963 Phys. Rev. 130 1677 [29] Gilbert T L 2004 IEEE Trans. Magn. 40 3443 [30] Vansteenkiste A, Leliaert J, Dvornik M, Helsen M, Garcia-Sanchez F and Van Waeyenberge B 2014 AIP Ad- vances4 107133 [31] HinzkeD,KazantsevaN,NowakU,MryasovON,Asselin P and Chantrell R W 2008 Phys. Rev. B 77 094407 [32] Etesami S R, Chotorlishvili L, Sukhov A and Berakdar J 2014Phys. Rev. B 90 014410 [33] Mougin A, Cormier M, Adam J P, Metaxas P J and Ferre J 2007Europhys. Lett. 78 57007 [34] Wieser R, Vedmedenko E Y, Weinberger P and Wiesen- danger R 2010 Phys. Rev. B 82 144430 [35] Sun Z Z, Schliemann J, Yan P and Wang X R 2011 Eur. Phys. J. B 79 449453 [36] Ramsay A J, Roy P E, Haigh J A, Otxoa R M, Irvine A C, JandaT,CampionRP,Gallagher BLandWunderlich J 2015Phys. Rev. Lett .114 067202 [37] Moretti S, Raposo V, Martinez E and Lopez-Diaz L 2017 Phys. Rev. B 95 064419 [38] Staunton J B, Szunyogh L, Buruzs A, Gyorffy B L, Os- tanin S, and L. Udvardi 2006 Phys. Rev. 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2018-12-18

The issue of whether a thermal gradient acts like a magnetic field or an electric current in the domain wall (DW) dynamics is investigated. Broadly speaking, magnetization control knobs can be classified as energy-driving or angular-momentum driving forces. DW propagation driven by a static magnetic field is the best-known example of the former in which the DW speed is proportional to the energy dissipation rate, and the current-driven DW motion is an example of the latter. Here we show that DW propagation speed driven by a thermal gradient can be fully explained as the angular momentum transfer between thermally generated spin current and DW. We found DW-plane rotation speed increases as DW width decreases. Both DW propagation speed along the wire and DW-plane rotation speed around the wire decrease with the Gilbert damping. These facts are consistent with the angular momentum transfer mechanism, but are distinct from the energy dissipation mechanism. We further show that magnonic spin-transfer torque (STT) generated by a thermal gradient has both damping-like and field-like components. By analyzing DW propagation speed and DW-plane rotation speed, the coefficient ( \b{eta}) of the field-like STT arising from the non-adiabatic process, is obtained. It is found that \b{eta} does not depend on the thermal gradient; increases with uniaxial anisotropy K_(||) (thinner DW); and decreases with the damping, in agreement with the physical picture that a larger damping or a thicker DW leads to a better alignment between the spin-current polarization and the local magnetization, or a better adiabaticity.

Thermal gradient driven domain wall dynamics

1812.07244v2

arXiv:2105.07376v1 [cond-mat.mes-hall] 16 May 2021Anatomy of inertial magnons in ferromagnetic nanostructur es Alexey M. Lomonosov1,∗Vasily V. Temnov2,3,†and Jean-Eric Wegrowe3‡ 1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991, Moscow, Russia 2Institut des Mol´ ecules et Mat´ eriaux du Mans, UMR CNRS 6283 , Le Mans Universit´ e, 72085 Le Mans, France and 3LSI, Ecole Polytechnique, CEA/DRF/IRAMIS, CNRS, Institut Polytechnique de Paris, F-91128, Palaiseau, Fran ce (Dated: May 18, 2021) We analyze dispersion relations of magnons in ferromagneti c nanostructures with uniaxial anisotropy taking into account inertial terms, i.e. magnet ic nutation. Inertial effects are parametrized by damping-independent parameter β, which allows for an unambiguous discrimi- nation of inertial effects from Gilbert damping parameter α. The analysis of magnon dispersion relation shows its two branches are modified by the inertial e ffect, albeit in different ways. The up- per nutation branch starts at ω= 1/β, the lower branch coincides with FMR in the long-wavelength limit and deviates from the zero-inertia parabolic depende nce≃ωFMR+Dk2of the exchange magnon. Taking a realistic experimental geometry of magnet ic thin films, nanowires and nanodiscs, magnon eigenfrequencies, eigenvectors and Q-factors are found to depend on the shape anisotropy. The possibility of phase-matched magneto-elastic excitat ion of nutation magnons is discussed and the condition was found to depend on β, exchange stiffness Dand the acoustic velocity. PACS numbers: Valid PACS appear here I. INTRODUCTION After the first description of the dynamics of the mag- netization by Landau and Lifshitz [1], Gilbert proposed an equation that contains a correction due to the preces- sional damping [2, 3]. Since then, the so-called Landau- Lifshitz-Gilbert (LLG) equation is known to give an ex- cellent description of the dynamics of the magnetization, including ferromagnetic resonance (FMR) and magneto- static waves [4, 5], as well as the magnetization reversal [6, 7]. Ferromagnetic resonance and time-resolved mag- netization measurementsallow its spatially homogeneous precession ( k= 0) but also non-uniform modes of the magnetizationprecession( k∝ne}ationslash= 0, where kisthe wavevec- tor of spin waves) to be measured [8–10]. During the last decades, these techniques have been advanced in the con- text of ultrafast demagnetization dynamics [11, 12] that paved the way for the description of new physics at the sub-picosecond regime. High-frequency resonant modes of exchange magnons have been measured with ultrafast time-resolved optical techniques [8, 10, 13]. Therefore, the validity of the LLG equations has been confirmed down to the picosecond time scale and below. However, limitations of LLG equations has been es- tablished in the stochastic derivation performed by W. F. Brown in a famous paper published in 1963 [14]. This limit is due to the hypothesis that the typical time scales of magnetization dynamics are much longer than those of other degrees of freedom forming the dissipative envi- ronment. In analogy to the common description of the diffusion process of a Brownianparticle, the inertial (mo- mentum) degrees of freedom are supposed to relax much ∗lom@kapella.gpi.ru †vasily.temnov@univ-lemans.fr ‡jean-eric.wegrowe@polytechnique.edufaster than its spatial coordinate. This means that the degrees of freedom related to the linear momentum (in the case of the usual diffusion equation), or to the an- gular momentum (in the case of the magnetization) are included into the heat bath. As a consequence, the iner- tial terms do not explicitly appear in the equations, but are considered to be part of the damping term [15]. The possibility of measuring the contribution to iner- tial degrees of freedom led to a generalizationof the LLG equation with an additional term, incorporating the sec- ond time-derivative of magnetization: ˙m=−γm×Heff+αm×˙m+βmרm,(1) wherem=M/Msis the unit magnetization vector thatgivesthedirectionofthemagnetizationateachpoint (andMsis the modulus of the magnetization, which is constant), γ=γ0µ0is the gyromagnetic ratio, αstands for the Gilbert damping. Inertial effects are character- ized by the parameter β, which is introduced in a phe- nomenological way, i.e. independent on αandγ[16]. This generalized LLG equation has been derived in the framework of different and independent theoretical con- texts [15, 17–32]; its solutions have been studied in a series of publications [33–36]. The main consequence of inertia for the uniform magnetization (magnon with the wavevector k= 0) is the existenceofnutation oscillations that are superimposed to the precession. This leads to an appearance of the second resonance peak at a higher frequency in FMR spectra. The direct measurement of nutation has been reported recently [37, 38]. The goal of the present report is to study the conse- quences of these inertial effects on the exchange magnons (i.e.k∝ne}ationslash= 0 modes), in the perspective of experimental studies. Magnons are defined as linear magnetic exci- tations propagating in ferromagnets at the micromag- netic limit. This work completes the first description2 FMR magnon ys(t) s(t) Nutation magnon (a) (b) xΨz k FIG. 1. (a) Inertial magnons propagating in ferromagnetic nanostructures with wavevector kalong the zdirection under an external magnetic field Hresult in complex magnetization dynamics. (b) They can be decomposed in FMR magnon and nutation magnon precessing in opposite directions on el - liptical trajectories at different frequencies, giving ris e to a characteristic flower-shaped trajectory. published in 2015, Section IV of the remarkable work of Toru Kikuchi and Gen Tatara [22], and independently reconsidered by Makhfudz et al. in 2020 [39]. The paper is organized as follows. Section II presents the derivation of the linear magnetic excitations deduced from (1). Section III describes the dispersion relation in a simple case of zero dipolar field (spherical symme- try). The first consequence of the inertia is that the dis- persion relation splits in two branches: the lower one s1exp(ikz−iω1t) (FMR magnons ) and the upper one fors2exp(ikz−iω2t) (nutation magnons ). The second consequence is that the quality factor Qincreases with thek-vector. SectionIVgeneralizesthedescriptiontothe caseofauniaxialanisotropyquantifiedbythe dimension- less (shape) anisotropy parameter ξ. In the anisotropic case the trajectories of both FMR magnons and nuta- tion magnons become elliptical and rotating in opposite directions at each point in space. For a given k-vector the magnetization vector corresponding to a superposi- tion of both magnons draws a typical trochoidal trajec- tory (see Fig. 1). Section V discusses the conditions for phase-matched excitation the nutation magnons by co-propagatinglongitudinalacoustic phonons, illustrated by the material parameters for Gd-doped Permalloy thin films [13]. II. EXCHANGE MAGNONS IN FERROMAGNETIC THIN FILMS WITH MAGNETIC INERTIA We start with the LLG equation for unit magnetiza- tion vector mwith aneffective field Heff, which includes exchange interactions with stiffness D, an external field H= (Hx,0,Hz) and a demagnetizing field induced by the shape anisotropy Hd=−MS/hatwideNm. The demagne- tization tensor /hatwideNdepends on the specific shape of the ferromagnetic sample. Hereafter we assume the diagonal formof/hatwideNwithdiagonalelements Nx,NyandNz. Damp-ing of the magnetization dynamics is described by the conventional Gilbert term with parameter α. In addition to the conventional LLG equation we take into account the inertial effect characterized by the independent pa- rameterβ. Then the inertial LLG equation (ILLG) takes the form of Eq.(1) with Heff=H+D∆m+Hd. The coordinate system was chosen such that the ex- ternal field lies in the y= 0 plane, as is shown in Fig. 1. The material is assumed to be magnetically isotropic, so that the unperturbed magnetization vector also lies in they= 0 plane. We seek for time- and space-dependent solutions in the form m=m0+s(z,t) with spin-wave solutions s(z,t) = (sx,sy,sz)exp(ikz−iωt) (2) propagating as plane waves with a real wave vector k along the z-axis, see Fig. 1. Substitution m(z,t) into equation (1) and its linearization with respect to small perturbations sx,sy,szresults in a homogeneous system of three linear equations: /hatwideA sx sy sz = 0 (3) where the matrix Ais given by −iω A 12(ω,k) 0 A21(ω,k)−iω A 23(ω,k) 0A32(ω,k)−iω (4) with coefficients Aij(ω,k) defined as: A12=mz(γDk2+γMSξyz−iαω−βω2)+γHz A21=−mz(γDk2+γMSξxz−iαω−βω2)+γHz A23=mx(γDk2+γMSξzx−iαω−βω2)+γHx(5) A32=−mx(γDk2+γMSξyx−iαω−βω2)−γHx where coefficients ξij=Ni−Njcharacterize the shape anisotropy. The condition for the nontrivial solution of thehomogeneoussystem(3) toexist, i.e. det A= 0, gives rise to the secular equation ω2+A12(ω,k)A21(ω,k)+A23(ω,k)A32(ω,k) = 0 (6) which is used to calculate the spin wave dispersion re- lationω(k) for different shapes/symmetries, i.e.charac- terized by different types of the /hatwideNtensor. III. INERTIAL EXCHANGE MAGNONS IN SAMPLES WITH SPHERICAL SYMMETRY Examples of such symmetry are infinite homogeneous isotropic ferromagnetic media, or any spherical body. In3 thesecasesthedemagnetizationtensor /hatwideNisdiagonalwith all nonzero elements equal 1 /3, so that its contribution to the magnetization dynamics (1) and correspondingly to the wave matrix components(5) vanishes. The secular equation (6) takes a concise form: /parenleftbig γH+γDk2−βω2−iαω+ω/parenrightbig ×(7) ×/parenleftbig γH+γDk2−βω2−iαω−ω/parenrightbig = 0. Due to the symmetry of /hatwideN, equation (7), and hence all its roots, remains independent on the direction of Hand the equilibrium magnetization m0with respect to the wave propagation direction along the z-axis. For each positive wavenumber k, the determinant (7) is solved forω. Given that the presumed solution has a form ∼exp(ikz−iωt), positive ωdesignates the waves trav- elling in the positive direction. The two positive roots corresponding to the first parenthesis in (7) have the fol- lowing forms: ω1=1 2β/parenleftig −1−iα+/radicalbig 4γβ(Dk2+H)+(1+iα)2/parenrightig (8) ω2=1 2β/parenleftbigg 1−iα+/radicalig 4γβ(Dk2+H)+(1−iα)2/parenrightbigg (9) Thefirstrootisthelowermagnonbranchorprecession, slightly modified by the inertial term and the second one exhibits the inertial magnon branch or nutation. It is convenient to split these roots into real and imaginary parts:ω1,2=ω′ 1,2+iω′′ 1,2. Taylor series approximation of those roots assuming the smallness of γβDk2,γβH,α ≪ 1 results in the following expressions for their real parts: ω′ 1≈γ[Dk2+H−2βγHDk2+...] (10) ω′ 2≈1 β+ω′ 1 (11) In this approximation the nutation magnon branch is simply shifted by 1 /βwith respect to FMR magnon branch. The validity of this approximation is illustrated in Fig. 2, where the exact roots given by (8) and (9) are depicted with solid lines, whereas dashed lines represent the power series approximation of (10) and (11). Lower branch emerges from the Larmor’s frequency γHand grows parabolically with k. Effect of inertia re- duces the coefficient at the term quadratic in k. Upper branch is simply displaced by +1 /βand has the simi- lar shape. Imaginary parts of the roots ω′′ 1,2represent attenuation of the corresponding magnetization dynam- ics in time as ∝exp/parenleftbig ω′′ 1,2t/parenrightbig , and therefore they must be negative. In the frequency domain they characterize the width ∆f=|ω′′|/π(FWHM) of the Lorentzian spectral line.0.3 0.5 0.8 1.0 1.3 1.5 0.25 0.50 0.75 1.00 1.25 β= 0.17ps β= 0.27ps β= 0.17ps Frequency, THz wavenumber k, rad/nm β= 0.27ps 0.25 0.50 0.75 1.00 1.25 1.50 10 20 30 β= 0β= 0 α= 0.023 α= 0.023 α= 0.01 α= 0.01 Line Width, GHz wavenumber k, rad/nm β= 0.27ps FIG. 2. (a) Dispersion of the inertial magnon (red, magenta) and conventional magnon (blue,green). Dashed curves shows the approximations by (10) and (11). (b) corresponding line widths. In order to indicate the effect of inertia on the pre- cession, dashed curves show the line width without inertia β= 0. ω′′ 1≈ −αγ/bracketleftbig Dk2+H−6βγ2HDk2+.../bracketrightbig (12) ω′′ 2≈ −α β−ω′′ 1 (13) Note that in the limiting case of (12), (13) field and exchange stiffness have opposite effects on the attenua- tion of the two magnon branches: they increase the at- tenuation in the lower branch ω1and decrease it for the inertial branch ω1. In the other limiting case of large field and large kattenuation of both branches tends to exp/parenleftig −α 2βt/parenrightig . The damping for both branches appears to be naturally proportional to the Gilbert damping pa- rameterα. A conspicuous decrease of nutation linewidth ω′′ 2(k= 0) with growing αreported by Cherkasski et al. [36] roots back to the parametrization of the nu- tation phenomenon in terms of a product ατ, whereτ denotes the characteristic nutation lifetime. Within this parametrization, a variation of α, while keeping τcon- stant, leads to the simultaneous decrease of the nutation4 0.2 0.4 0.6 0.810203040Q k, nm-11/2H=0 2 T5 T 0.2 0.4102030in a thin film FIG. 3. Q-factor dependencies on kfor different values of the external field. Dashed line shows the quadratic in kapprox- imation for H= 0. Inset shows Q-factors for the ordinary magnon (blue curve) and inertial magnon (red curve) in a thin film, H= 0 frequency 1 /β= 1/(ατ) rendering the analysis of damp- ing extremely difficult. An alternative notation in terms ofαandβ, introduced in this paper, resolves this prob- lem and allows for an independent investigation of iner- tial and damping effects. Another parameter, which characterizes the resonant spectral line centered at frequency f0, is its quality factor defined as Q=f0/∆f=ω′/(2ω′′). As can be seen from equations (8) and (9), Q-factors for both branches coin- cide within the accuracy of ∼(ωα)2. Dependence of the Q-factor on the wavenumber klooks counterintuitive in that it essentially grows with k. Assuming for simplicity H= 0, for small kthe Q-factor can be approximated by expansion of ω′andω′′in the power series in k, which results in: Q(k)∼1 2α1+γβDk2 1−γβDk2+...∼1 2α/parenleftbig 1+2γβDk2/parenrightbig (14) Exact values for the Q-factor in comparison to the estimate of (14) are shown in Fig. 3 for the external field ranging from 0 to 5 T. Field effect for small kcan be approximated as Q∼ 1/(2α)(1+2γβH). IV. INERTIAL EXCHANGE MAGNONS IN SAMPLES WITH CYLINDRICAL SYMMETRY Examplesofsuchbodies aredisks, wires, infinite plates and films. Axial symmetry about the z-axis retains the diagonal form of /hatwideNwith the diagonal elements satisfying thefollowingconditions: Nx=NyandNx+Ny+Nz= 1. As a result, components of the matrix /hatwideAgiven by (5)acquire terms proportional to γMS. Lack of symmetry makes the magnon propagation dependent on the orien- tation of vectors m0andHwith respect to the z-axis. We consider two limiting cases: collinear arrangement withm0parallel to the axis of symmetry (Θ = Ψ = 0 in Fig. 1); and orthogonal arrangement with m0paral- lel to the x-axis and Θ = Ψ = π/2. In the collinear case the demagnetizing field acts simply against the ex- ternal field, hence the secularequation remains similarto (7), but with field Hsubstituted with the reduced field H′=H−ξMS: /parenleftbig γH′+γDk2−βω2−iαω+ω/parenrightbig ×(15) ×/parenleftbig γH′+γDk2−βω2−iαω−ω/parenrightbig = 0 whereξ=Nz−Nx=Nz−Nycharacterizes the shape effect on demagnetizing, so that in an infinite wire ξ= −1/2, in the spherical symmetric (or unbounded) body ξ= 0 and in the infinite film ξ= 1. Correspondingly the roots to (15) are similar to ones given in (8) and (9) with modified field: ω1=1 2β/parenleftbigg −1−iα+/radicalig 4γβ(Dk2+H′)+(1+iα)2/parenrightbigg (16) ω2=1 2β/parenleftbigg 1−iα+/radicalig 4γβ(Dk2+H′)+(1−iα)2/parenrightbigg (17) At the low- klimit the lower branch roughly tends to the Larmor’s frequency γ(H−ξMS) and the upper branch limit is 1 /β+γ(H−ξMS), which is similar to the case of spherical symmetry, but with modified field. In the orthogonal configuration with m0andHperpendic- ular to the axis of symmetry and to the magnon propa- gation direction, roots of the determinant (4) generally cannot be found in an analytical form. Therefore we firstconsideranapproximatesolutions,andthendescribe brieflythe numericalgorithmforobtainingthedispersion curves. By neglecting the Gilbert attenuation ( α= 0), the approximate solutions to (4) for the in-plane mag- netization and field can be found in a concise analytical form: ω1,2=1 β√ 2{2γβ(Dk2+H)+γβξM s+1 ∓/radicalbig 4γβ(Dk2+H)+(γβξM s+1)2}1/2(18) Here indices 1 and 2 denote the frequencies of the conventional and inertial magnons respectively, sign ‘- ‘ prior to the square root in (18) corresponds to the lower branch ω1; sign ‘+’ denotes the inertial branch ω2. Numerical procedure for building the dispersion relations of the magnonic modes for nonzero αor ar- bitrary orientation of the external field H starts with calculation of the stationary equilibrium magnetization5 m0= (mx,my,mz). This can be done by solving (1) in its stationary form, i.e.with all time derivatives set zero. In a thin film, for example, quantities Hiandmj are related by MSmxmz+Hxmz−Hzmx= 0. Thus obtained stationary magnetization components are then substituted into (5) and (4). At some fixed small kthe determinant (4) as a function of complex-valued ωpos- sesses two minima, which correspond to the FMR and nutational branches. Their exact locations can be evalu- atedbyanumericalroutinewhichminimizestheabsolute value of the determinant (4) in the vicinity of the guess values for those branches, for example given by equa- tions (19) and (20). Then we give ka small increment and repeat the extremum search using the ωs obtained at the previous step as guess values, and so on. As a result, calculated values for ω1andω2follow the disper- sion curves of both branches. Note that for nonzero α rootsω1,2possess imaginary parts, which determine the line width and Q-factor for each mode. Let us consider the magnetization behavior in a thin film in more detail. For this geometry /hatwideNpossesses the only nonzero compo- nentNz= 1, and correspondingly ξ= 1. In the small- k limit, the lower branch approaches the Kittel’s frequency ωFMR=γ/radicalbig H(H+MS) from below as β,k→0: ω1≈ωFMR−1 2γωFMR(2H+MS)β+...(19) Effect of the demagnetizing field on the inertial branch is exhibited by an upward shift by1 2γMS; whereas effect of inertia is opposite: ω2≈1 β+γH+1 2γMS−/parenleftbigg ω2 FMR+1 8γ2M2 S/parenrightbigg β+...(20) For the orthogonal configuration, when both m0and Hlie in the film plane, we can estimate the trajectories of the magnetization dynamics of both modes for small kand small α. For each root given by (18) we solve the homogeneous equation for perturbations s(3). Nor- malization of the solutions can be chosen in an arbitrary way,here for simplicity we define sz= 1. In orthogonal geometry sxcomponent is obviously negligible or equals zero, so the system reduces to two equations in syand sz. Results shown as a power series expansion for small inertiaβω≪1 for the precession: sp= 0 −i/radicalig 1+ξMS Dk2+H/parenleftig 1+βγξM S 2/parenrightig 1 exp(−iω1t) (21) and nutation: sn= 0 i(1−βγξM S/2) 1 exp(−iω2t) (22)0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.0 1 T0.2 Tellipticity sy’/sx’ k, nm-1sx’ sy’0.1 T FMR nutation FIG. 4. Ratio of the polarization axes for the external field of 0.1T, 0.2T, and 1T. Handm0are parallel to the inward normal to the figure plane. with the parameter of anisotropy ξ= 1 for a thin film normal to the z−axis and ξ= 1/2 for a thin wire spread along the z−axis. Both perturbations exhibit elliptical polarization within the ( x′,y′) plane. Precession trajec- tory is deformed by the demagnetizing effect so that the y-axis of the ellipse is stretched with the/radicalbig 1+MS/H factor due to demagnetizing effect, and in addition on account of inertial effect. On the contrary, the nutation ellipse is squeezed along the y-axis proportionally to the inertial parameter β. Ellipticity of the lower branch de- pends on the external field (21), whereas that of the up- per branch in this approximation shows no dependence on the field. Signs of sycomponents are opposite for nutation and precession, this indicates that they are ro- tating in the opposite directions. Exact polarizationscan be found numerically for a reasonable set of material pa- rameters and fields, as is shown in Fig. 4.6 V. EXCITATION MECHANISMS OF INERTIAL EXCHANGE MAGNONS The only experimental evidence of inertial effects in ferromagnets has been reported for k= 0 nutation man- gons in Py-thin films resonantly excited with a mag- netic field of an intense quasi-monochromatic THz pulse [37]. In order to excite k∝ne}ationslash= 0 exchange magnon modes one would need to have either spatially localized and instantaneous stimuli [10] or any other source of effec- tive magnetic field characterized by spectral and spatial overlap with investigated magnon modes. The letter can be provided through ultrashort large-amplitude acoustic pulses [40, 41] producing effective magneto-elastic fields rapidly varying in time and space [42]. Acoustic pulses propagating through a thin ferromagnetic sample at an acoustic velocity vare quantified by a linearized disper- sion relation ωac=vk. Crossing between acoustic and magnon brunches, i.e. satisfying the phonon-magnon phase-matchingcondition, usually facilitatesthe acoustic excitation of magnetization dynamics [43, 44]. A ques- tion arises under which conditions the crossing between dispersion curves for longitudinal phonons and inertial magnons can occur. Whereas for realistic magnetic fields the acoustic dispersion always intersects the lower FMR- branch at a frequency close to FMR frequency [42], the crossing of the upper nutation brunch is less obvious. FIG. 5. The magneto-acoustic phase matching condition for nutation magnons can be tuned vie the reduction of exchange stiffness in Gd-doped Py samples. Gd concentration x varies from 0 to 13%. The dashed line displays the acoustic disper- sion relation ωac/(2π). Magneto-elastic coupling with inertial mangon is efficient when the dashed line lies within the pink tinted area. Material parameters are taken from Ref. [13] an d β=0.276 ps. It is possible to quantify the criterion for magneto- elastic crossing with nutation magnons analytically. To do that we note that for larger wavenumbers ksatisfying Dk2≫H,MStheexchangetermplaysthedominantrole and the asymptotic behaviourfor both branchesbecomeslinear in k: ω1,2≈ ∓1 2β+k/radicaligg γD β. (23) It follows from (23) that the condition for the nutation magnon branch to intersect the acoustical dispersion re- lationωac(k), requires the asymptotic slope of ω2(k) to be smaller than the acoustic velocity v: /radicaligg γD β< v. (24) This expression shows that for a given βthe magneto- elastic crossing is facilitated by small exchange stiffness Dand small acousticvelocity. This approximateanalysis breaks down for acoustic frequencies in above-THz spec- tral range, where the acoustic dispersion starts deviating from its linear approximation. Figure 5 highlights the remarkable role of exchange stiffness to achieve the dispersion crossing between nuta- tionmagnonsandlongitudinalacousticphonons. Doping Py thin films with Gadolinium has been shown to gradu- ally reduce the exchange stiffness upon Gd-concentration from 300 to 100 [meV ·˚A2] [13]. For a fixed value of in- ertial parameter β=0.276 ps, nutation magnons for pure Py samples do not display any crossing with acoustic phonons within the displayed range of k-vectors but the Gd-doped Py with 13% Gd concentration does. The nu- tation magnon-phonon crossing point occurs at 0.75 THz frequency and k= 0.85 nm−1(magnon wavelength of approximately 5 nm), i.e. magnon parameters readily accessible in ultrafast magneto-optical experiments [10]. VI. CONCLUSIONS In this paper we have theoretically studied exchange inertial magnons in ferromagnetic samples of different shapes under the action of an external magnetic field. The parametrizationof magnetization dynamics in terms of two independent parameters, the Gilbert damping α and the inertial time β, allows for unambiguous discrim- ination between the inertial and damping effects as well as their impact on both branches of magnon dispersion. Inertial effects are found to strongly effect not only the frequencies (magnon eigenvalues) of both branches but also result in a monotonous increase of the Q-factor as a function of the external magnetic field and magnon k- vector. The two magnon branchesare found to precessin opposite directions along the elliptical trajectories with perpendicularlyorientedlongaxisoftheellipses(magnon eigenvectors). Their ellipticity is found to depend on the components of the demagnetizing tensor. An analyti- cal criterion for the existence of phase-matched magneto- elastic excitation of nutation magnons has been derived andillustratedforGd-dopedpermalloysampleswithtun- able exchange stiffness.7 ACKNOWLEDGMENTS Financial support by Russian Basic Research Founda- tion (Grant No. 19-02-00682)is gratefully acknowledged. [1] L. D. Landau and L. M. Lifshitz, Physik. Zeits. Sowjetu- nion8, 153 (1935). [2] T. L. Gilbert, Ph. D. Thesis (1956). [3] T. L. Gilbert, IEEE transactions on magnetics 40, 3443 (2004). [4] R. W. Damon and J. R. Eshbach, Journal of Physics and Chemistry of Solids 19, 308 (1961). [5] M. Farle, Reports on progress in physics 61, 755 (1998). [6] L. Thevenard, J.-Y. Duquesne, E. Peronne, H. J. Von Bardeleben, H. Jaffres, S. Ruttala, J.-M. George, A. Lemaitre, and C. Gourdon, Physical Review B 87, 144402 (2013). [7] V. S. Vlasov, A. M. Lomonosov, A. V. Golov, L. N. Ko- tov, V. Besse, A. Alekhin, D. A. Kuzmin, I. V. Bychkov, and V. V. Temnov, Phys. Rev. B 101, 024425 (2020). [8] M. Van Kampen, C. Jozsa, J. Kohlhepp, P. LeClair, L. Lagae, W. deJonge, and B. Koopmans, Physical re- view letters 88, 227201 (2002). [9] V. Kruglyak, S. Demokritov, and D. Grundler, Journal of Physics D: Applied Physics 43, 264001 (2010). [10] I. Razdolski, A. Alekhin, N. Ilin, J. P. Meyburg, V. Rod- datis, D. Diesing, U. Bovensiepen, and A. Melnikov, Na- ture communications 8, 1 (2017). [11] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Physical review letters 76, 4250 (1996). [12] A. Kirilyuk, A. V. Kimel, and T. Rasing, Reviews of Modern Physics 82, 2731 (2010). [13] R. Salikhov, A. Alekhin, T. Parpiiev, T. Pezeril, D. Makarov, R. Abrudan, R. Meckenstock, F. Radu, M. Farle, H. Zabel, et al., Physical Review B 99, 104412 (2019). [14] W. F. Brown Jr, Physical review 130, 1677 (1963). [15] J.-E. Wegrowe and M.-C. Ciornei, American Journal of Physics80, 607 (2012). [16] In a majority of the published reports on the magnetic inertia, the coefficient used is the relaxation time τof the inertial degrees of freedom τ=β/α, which has a clear physical meaning. However, only the parameter β is intrinsic, i.e. proper to the material. Indeed, in the framework of the mechanical analogy, βis defined by the first and second inertial moment I1=I2see Ref. [15]. [17] J.-E. Wegrowe, Physical Review B 62, 1067 (2000). [18] M. F ¨ahnle, D. Steiauf, and C. Illg, Physical Review B 84, 172403 (2011). [19] M. F ¨ahnle, D. Steiauf, and C. Illg, Physical Review B 88, 219905 (2013). [20] M.-C. Ciornei, J. Rub´ ı, and J.-E. Wegrowe, Physical Review B 83, 020410(R) (2011). [21] S. Bhattacharjee, L. Nordstr ¨om, and J. Fransson, Phys- ical Review Letters 108, 057204 (2012). [22] T. Kikuchi and G. Tatara, Physical Review B 92, 184410 (2015). [23] J.-E. Wegrowe and E. Olive, Journal of Physics: Con- densed Matter 28, 106001 (2016).[24] M. Sayad, R. Rausch, and M. Potthoff, EPL (Euro- physics Letters) 116, 17001 (2016). [25] D. Thonig, O. Eriksson, and M. Pereiro, Scientific re- ports7, 1 (2017). [26] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppe- neer, Physical Review B 96, 024425 (2017). [27] U. Bajpai and B. K. Nikoli´ c, Physical Review B 99, 134409 (2019). [28] M. F ¨ahnle, Journal of Magnetism and Magnetic Materi- als469, 28 (2019). [29] R. Mondal and P. M. Oppeneer, Journal of Physics: Con- densed Matter 32, 455802 (2020). [30] S. Giordano and P.-M. D´ ejardin, Physical Review B 102, 214406 (2020). [31] R. Mondal, S. Großenbach, L. R´ ozsa, and U. Nowak, Physical Review B 103, 104404 (2021). [32] S. Titov, W. Coffey, Y. P. Kalmykov, M. Zarifakis, and A. Titov, Physical Review B 103, 144433 (2021). [33] E. Olive, Y. Lansac, and J.-E. Wegrowe, AppliedPhysics Letters100, 192407 (2012). [34] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E. Wegrowe, Journal of Applied Physics 117, 213904 (2015). [35] D. Bottcher and J. Henk, Physical Review B 86, 020404(R) (2012). [36] M. Cherkasskii, M. Farle, and A. Semisalova, Physical Review B 102, 184432 (2020). [37] K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Z. Hagstr¨om, S. S. P. K. Arekapudi, A. Semisalova, K. Lenz, B. Green, J.-C. Deinert, et al., Nature Physics 17, 245 (2021). [38] Y. Li, A.-L. Barra, S. Auffret, U. Ebels, and W. E. Bailey, Physical Review B 92, 140413(R) (2015). [39] I. Makhfudz, E. Olive, and S. Nicolis, Applied Physics Letters117, 132403 (2020). [40] V. V. Temnov, C. Klieber, K. A. Nelson, T. Thomay, V. Knittel, A. Leitenstorfer, D. Makarov, M. Albrecht, and R. Bratschitsch, Nature Communications 4, 1468 (2013). [41] V. V. Temnov, I. Razdolski, T. Pezeril, D. Makarov, D. Seletskiy, A. Melnikov, and K. A. Nelson, Journal of Optics 18, 93002 (2016). [42] V. Besse, A. V. Golov, V. S. Vlasov, A. Alekhin, D. Kuzmin, I. V. Bychkov, L. N. Kotov, and V. V. Tem- nov, J. Magn. Magn. Mater. 502, 166320 (2020). [43] J. Januˇ sonis, C. L. Chang, T. Jansma, A. Gatilova, V. S. Vlasov, A. M. Lomonosov, V. V. Temnov, and R. I. Tobey, Phys. Rev. B 94, 024415 (2016). [44] C. L. Chang, A. M. Lomonosov, J. Janusonis, V. S. Vlasov, V. V. Temnov, and R. I. Tobey, Phys. Rev. B 95, 060409(R) (2017).

2021-05-16

We analyze dispersion relations of magnons in ferromagnetic nanostructures with uniaxial anisotropy taking into account inertial terms, i.e. magnetic nutation. Inertial effects are parametrized by damping-independent parameter $\beta$, which allows for an unambiguous discrimination of inertial effects from Gilbert damping parameter $\alpha$. The analysis of magnon dispersion relation shows its two branches are modified by the inertial effect, albeit in different ways. The upper nutation branch starts at $\omega=1/ \beta$, the lower branch coincides with FMR in the long-wavelength limit and deviates from the zero-inertia parabolic dependence $\simeq\omega_{FMR}+Dk^2$ of the exchange magnon. Taking a realistic experimental geometry of magnetic thin films, nanowires and nanodiscs, magnon eigenfrequencies, eigenvectors and $Q$-factors are found to depend on the shape anisotropy. The possibility of phase-matched magneto-elastic excitation of nutation magnons is discussed and the condition was found to depend on $\beta$, exchange stiffness $D$ and the acoustic velocity.

Anatomy of inertial magnons in ferromagnets

2105.07376v1

Spin Waves in Ferromagnetic Insulators Coupled via a Normal Metal Hans Skarsv ag,Andr e Kapelrud, and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Dated: May 27, 2022) Herein, we study the spin-wave dispersion and dissipation in a ferromagnetic insulator{normal metal{ferromagnetic insulator system. Long-range dynamic coupling because of spin pumping and spin transfer lead to collective magnetic excitations in the two thin-lm ferromagnets. In addition, the dynamic dipolar eld contributes to the interlayer coupling. By solving the Landau-Lifshitz- Gilbert-Slonczewski equation for macrospin excitations and the exchange-dipole volume as well as surface spin waves, we compute the eect of the dynamic coupling on the resonance frequencies and linewidths of the various modes. The long-wavelength modes may couple acoustically or optically. In the absence of spin-memory loss in the normal metal, the spin-pumping-induced Gilbert damp- ing enhancement of the acoustic mode vanishes, whereas the optical mode acquires a signicant Gilbert damping enhancement, comparable to that of a system attached to a perfect spin sink. The dynamic coupling is reduced for short-wavelength spin waves, and there is no synchronization. For intermediate wavelengths, the coupling can be increased by the dipolar eld such that the modes in the two ferromagnetic insulators can couple despite possible small frequency asymmetries. The surface waves induced by an easy-axis surface anisotropy exhibit much greater Gilbert damping enhancement. These modes also may acoustically or optically couple, but they are unaected by thickness asymmetries. PACS numbers: 76.50.+g,75.30.Ds,75.70.-i,75.76.+j I. INTRODUCTION The dynamic magnetic properties of thin-lm fer- romagnets have been extensively studied for several decades.1,2Thin-lm ferromagnets exhibit a rich vari- ety of spin-wave modes because of the intricate inter- play among the exchange and dipole interactions and the material anisotropies. In ferromagnetic insulators (FIs), these modes are especially visible; the absence of disturb- ing electric currents leads to a clear separation of the magnetic behavior. Furthermore, the dissipation rates in insulators are orders of magnitude lower than those in their metallic counterparts; these low dissipation rates enable superior control of travelling spin waves and facil- itate the design of magnonic devices.3 In spintronics, there has long been considerable in- terest in giant magnetoresistance, spin-transfer torques, and spin pumping in hybrid systems of normal met- als and metallic ferromagnets (MFs).4{7The experimen- tal demonstration that spin transfer and spin pumping are also active in normal metals in contact with insu- lating ferromagnets has generated a renewed interest in and refocused attention on insulating ferromagnets, of which yttrium iron garnet (YIG) continues to be the prime example.8{19In ferromagnetic insulators, current- induced spin-transfer torques from a neighboring normal metal (NM) that exhibits out-of-equilibrium spin accu- mulation may manipulate the magnetization of the insu- lator and excite spin waves.8,20,21The out-of-equilibrium spin accumulation of the normal metal may be induced via the spin Hall eect or by currents passing through other adjacent conducting ferromagnets. Conversely, ex- cited spin waves pump spins into adjacent NMs, and this spin current may be measured in terms of the inverse spinHall voltages or by other conducting ferromagnets.8{14 The magnetic state may also be measured via the spin Hall magnetoresistance.16{19,23,24Because of these devel- opments, magnetic information in ferromagnetic insula- tors may be electrically injected, manipulated, and de- tected. Importantly, an FI-based spintronic device may eciently transport electric information carried by spin waves over long distances15without any excessive heat- ing. The spin-wave decay length can be as long as cen- timeters in YIG lms.22These properties make FI{NM systems ideal devices for the exploration of novel spin- tronic phenomena and possibly also important for future spintronic applications. Magnonic devices also oer ad- vantages such as rapid spin-wave propagation, frequen- cies ranging from GHz to THz, and the feasibility of cre- ating spin-wave logic devices and magnonic crystals with tailored spin-wave dispersions.25 To utilize the desirable properties of FI{NM systems, such as the exceptionally low magnetization-damping rate of FIs, it is necessary to understand how the mag- netization dynamics couple to spin transport in adjacent normal metals. The eective damping of the uniform magnetic mode of a thin-lm FI is known to signi- cantly increase when the FI is placed in contact with an NM. This damping enhancement is caused by the loss of angular momentum through spin pumping.26{30Re- cent theoretical work has also predicted the manner in which the Gilbert damping for other spin-wave modes should become renormalized.31For long-wavelength spin waves, the Gilbert damping enhancement is twice as large for transverse volume waves as for the macrospin mode, and for surface modes, the enhancement can be ten times stronger or more. Spin pumping has been demon- strated, both experimentally9and theoretically,31to be suppressed for short-wavelength exchange spin waves.arXiv:1407.0635v1 [cond-mat.mes-hall] 2 Jul 20142 A natural next step is to investigate the magnetization dynamics of more complicated FI{NM heterostructures. In ferromagnetic metals, it is known that spin pumping and spin-transfer torques generate a long-range dynamic interaction between magnetic lms separated by normal metal layers.32The eect of this long-range dynamic in- teraction on homogeneous macrospin excitations can be measured by ferromagnetic resonance. The combined ef- fects of spin pumping and spin-transfer torque lead to an appreciable increase in the resonant linewidth when the resonance elds of the two lms are far apart and to a dramatic narrowing of the linewidth when the reso- nant elds approach each other.32This behavior occurs because the excitations in the two lms couple acous- tically (in phase) or optically (out of phase). We will demonstrate that similar, though richer because of the complex magnetic modes, phenomena exist in magnetic insulators. In the present paper, we investigate the magnetization dynamics in a thin-lm stack consisting of two FIs that are in contact via an NM. The macrospin dynamics in a similar system with metallic ferromagnets have been studied both theoretically and experimentally.32We ex- pand on that work by focusing on inhomogeneous mag- netization excitations in FIs. For long-wavelength spin waves travelling in-plane in a ferromagnetic thin lm, the frequency as a function of the in-plane wave number Qstrongly depends on the direction of the external magnetic eld with respect to the propagation direction. If the external eld is in- plane and the spin waves are travelling parallel to this direction, the waves have a negative group velocity. Be- cause the magnetization precession amplitudes are usu- ally evenly distributed across the lm in this geometry, these modes are known as backward volume magneto- static spin waves (BVMSW). Similarly, spin waves that correspond to out-of-plane external elds are known as forward volume magnetostatic spin waves (FVMSW), i.e., the group velocity is positive, and the precession amplitudes are evenly distributed across the lm. When the external eld is in-plane and perpendicular to the propagation direction, the precession amplitudes of the spin waves become inhomogeneous across the lm, ex- periencing localization to one of the interfaces. These spin waves are thus known as magnetostatic surface spin waves (MSSW).33,34 When two ferromagnetic lms are coupled via a normal metal, the spin waves in the two lms become coupled through two dierent mechanisms. First, the dynamic, nonlocal dipole-dipole interaction causes an interlayer coupling to arise that is independent of the properties of the normal metal. This coupling is weaker for larger thicknesses of the normal metal. Second, spin pumping from one ferromagnetic insulator induces a spin accu- mulation in the normal metal, which in turn gives rise to a spin-transfer torque on the other ferromagnetic in- sulator, and vice versa. This dynamic coupling, is in contrast to the static exchange coupling35rather long-ranged and is limited only by the spin-diusion length. This type of coupling is known to strongly couple the macrospin modes. When two ferromagnetic lms become coupled, the characterization of the spin waves in terms of FVMSW, BVMSW, and MSSW still holds, but the dispersion relations are modied. It is also clear that the damping renormalization caused by spin pumping into the NM may dier greatly from that in a simpler FI jN bilayer system. To understand this phenomenon, we per- form a detailed analytical and numerical analysis of a trilayer system, with the hope that our ndings may be used as a guide for experimentalists. This paper is organized as follows. Section II intro- duces the model. The details of the dynamic dipolar eld are discussed, and the boundary conditions associ- ated with spin pumping and spin transfer at the FI jN interfaces are calculated. Sec. III provides the analyti- cal solutions of these equations in the long-wavelength regime dominated by the dynamic coupling attributable to spin pumping and spin transfer. To create a more complete picture of the dynamic behavior of this system, we perform a numerical analysis for the entire spin-wave spectrum of this system, which is presented in Sec. IV. We conclude our work in Sec. V. II. EQUATIONS OF MOTION Consider a thin-lm heterostructure composed of two ferromagnetic insulators (FI1 and FI2) that are in elec- trical contact via an NM layer. The ferromagnetic in- sulators FI1 and FI2 may have dierent thicknesses and material properties. We denote the thicknesses by L1, dN, andL2for the FI1, NM, and FI2 layers, respectively (see Fig. 1(a)). The in-plane coordinates are ;, and the transverse coordinate is (see Fig. 1(b)). We will rst discuss the magnetization dynamics in isolated FIs and will then incorporate the spin-memory losses and the cou- pling between the FIs via spin currents passing through the NM. A. Magnetization Dynamics in Isolated FIs The magnetization dynamics in the ferromagnetic in- sulators can be described by using the Landau-Lifshitz- Gilbert (LLG) equation, _Mi= MiHe+Mi_Mi; (1) where Miis the unit vector in the direction of the mag- netization in layer i= 1;2, is the gyromagnetic ratio, is the dimensionless damping parameter, and Heis the space-time-dependent eective magnetic eld. The eective magnetic eld is He=Hint+hex+hd+hsurface; (2) where Hintis the internal eld attributable to an external magnetic eld and the static demagnetization eld, hex=3 dN2+L2 dN2 -dN2 -dN2-L1NFI2 FI1 SUBx (a) (b) FIG. 1: (Color online) a) A cross section of the FI1 jNjFI2 het- erostructure. The ferromagnetic insulators FI1 and FI2 are in contact via the normal metal N. The transverse coordinate is indicated along with the thicknesses L1,dN, andL2of FI1, N, and FI2, respectively. b) The coordinate system of the internal eld (blue) with respect to the coordinate system of the FI1jNjFI2 structure (red). denotes the angle between the lm normal and the internal eld, and is the angle be- tween the in-plane component of the magnetic eld and the in-plane wave vector. 2Ar2M=MSis the exchange eld ( Ais the exchange constant), hdis the dynamic demagnetization eld, and hsurface =2KS M2 S(Mi
^n)(i)^n (3) is the surface anisotropy eld located at the FI jN in- terfaces. In this work, hsurface is assumed to exist only at the FIjN interfaces and not at the interfaces between the FIs and the substrate or vacuum. It is straightfor- ward to generalize the discussion to include these surface anisotropies as well. We consider two scenarios: one with an easy-axis surface anisotropy ( KS>0) and one with no surface anisotropy ( KS= 0). Note that a negative value ofKS 0:03 erg=cm2, which implies an easy-plane surface anisotropy, has also been observed for sputtered YIGjAu bilayers.36In general, the eective eld Hemay dier in the two FIs. We assume the two FIs consist of the same material and consider external elds that are either in-plane or out-of-plane. Furthermore, we consider devices in which the internal magnetic elds in the two FI layers are aligned and of equal magnitude. In equilibrium, the magnetization inside the FIs is ori- ented along the internal magnetic eld, Mi=M0. In the linear response regime, Mi=M0+mi, where the rst- order correction miis small and perpendicular to M0.The magnetization vanishes outside of the FIs. Because the system is translationally invariant in the anddi- rections, we may, without loss of generality, assume that mconsists of plane waves travelling in the direction, mi(;; ) =miQ()ei(!tQ): (4) Linearizing Maxwell's equations in miimplies that the dynamic dipolar eld must be of the same form, hd(;; ) =hdQ()ei(!tQ): (5) Furthermore, the total dipolar eld (the sum of the static and the dynamic dipolar elds) must satisfy Maxwell's equations, which, in the magnetostatic limit, are r
(hd+ 4MSm) = 0; (6a) rhd= 0; (6b) with the boundary equations (hd+ 4MSm)?;in= (hd)?;out; (7a) (hd)k;in= (hd)k;out; (7b) where the subscript in (out) denotes the value on the FI (NM, vacuum or substrate) side of the FI interface and ? (k) denotes the component(s) perpendicular (parallel) to the FI{NM interfaces. Solving Maxwell's equations (6) with the boundary conditions of Eq. (7) yields33 hdQ() =Z d0^G(0)mQ(0); (8) where ^G(rr0) is a 33 matrix acting on min the (;; ) basis, ^G() =0 @GP()() 0iGQ() 0 0 0 iGQ() 0GP()1 A: (9) Here,GP() =QeQjj=2, andGQ() =sign()GP. Note that the dynamic dipolar eld of Eq. (8) accounts for both the interlayer and intralayer dipole-dipole cou- plings because the magnetization varies across the two magnetic insulator bilayers and vanishes outside these materials. It is now convenient to perform a transformation from the--coordinate system dened by the sample geome- try to thex-y-zcoordinate system dened by the internal eld (see Fig. 1(b)). In the linear response regime, the dynamic magnetization milies in thex-yplane, and the linearized equations of motion become33  i! 1 1 +11 !H+2A MS Q2d2 d2 miQxy() =2X i=1Z d0^Gxy(0)miQxy(0): (10)4 N m1,QFI1m2,QFI2 ee FIG. 2: (Color online) Two coupled spin waves with ampli- tudem1Qin ferromagnet FI1 and amplitude m2Qin ferro- magnet FI2. The spin-waves inject a spin current into the nor- mal metal (NM) via spin pumping. In the NM, the spins dif- fuse and partially relax, inducing a spin accumulation therein. In turn, the spin accumulation causes spin-transfer torques to arise on FI1 and FI2. The combined eect of spin transfer and spin pumping leads to a dynamic exchange coupling that, to- gether with the dynamic demagnetization eld, couples the spin waves in the two FIs. Here, miQxy = (miQx;miQy) is the Fourier transform of the dynamic component of the magnetization in the x- yplane and ^Gxy() is the 22 matrix that results from rotating ^G() into thex-y-zcoordinate system (see Ap- pendix A), and considering only the xx,xy,yxandyy- components. B. Boundary Conditions and Spin Accumulation The linearized equations of motion (10) must be sup- plemented with boundary conditions for the dynamic magnetization at the FI jN interfaces. A precessing mag- netization at the FI jN boundaries injects a spin-polarized current, jSP, into the NM, an eect known as spin pumping .8,28{30The emitted spin currents at the lower and upper interfaces ( i= 1;2) are jSP i=~ eg?Mi_Mi =i; (11) wherei=dN=2 at the lower and upper interfaces, respectively, and g?is the real part of the transverse spin- mixing conductance per unit area.37We disregard the imaginary part of the spin-mixing conductance because it has been found to be small at FI jN interfaces.38The reciprocal eect of spin pumping is spin transfer into the FIs because of a spin accumulation Sin the NM. In the normal metal at the lower and upper interfaces ( i=1,2),the associated spin-accumulation-induced spin current is jST i=1 eg?Mi(MiS) =i: (12) The signs of the pumped and spin-accumulation-induced spin currents in Eqs. (11) and (12) were chosen such that they are positive when there is a ow of spins from the NM toward the FIs. The pumped and spin-accumulation-induced spin cur- rents of Eqs. (11) and (12) lead to magnetic torques act- ing on the FI interfaces. The torques that correspond to the spin pumping and spin transfer localized at the FI jN interfaces are SP i= ~2 2e2g?(i)Mi_Mi; (13a) ST i= ~ 2e2g?Mi(MiS)(i);(13b) respectively. In the presence of spin currents to and from the normal metal, the magnetization dynamics in the FIs is then governed by the modied Landau-Lifshitz- Gilbert-Slonczewski (LLGS) equation, _M= MiHe+Mi_Mi+X i=1;2SP i+ST i:(14) By integrating Eq. (14) over the FI jN interfaces and the interfaces between the FI and vacuum/substrate, we nd5 thatmimust satisfy the boundary conditions21,31  Lidmi d+i _mi1 ~M0 +LiKS Acos (2)mi x =dN=2= 0;(15a)  Lidmi d+i _mi1 ~M0 +LiKs Acos2()mi y =dN=2= 0;(15b) dm1 d =dN=2L1= 0;dm2 d =dN=2+L2= 0:(15c) Here, we have introduced the timescale i= Li~2g?=4Ae2. The subscripts xandyin Eqs. (15a) and (15b) denote the xandycomponents, respectively. In our expressions for the boundary conditions (15), we have also accounted for the possibility of a surface anisotropy arising from the eective eld described by Eq. (3), whereKS>0 indicates an easy-axis surface anisotropy (EASA). The boundary conditions of Eq. (15), in combi- nation with the transport equations in the NM , which we will discuss next, determine the spin accumulation in the NM and the subsequent torques caused by spin transfer. In the normal metal, the spins diuse, creating a spa- tially dependent spin-accumulation potential Q, and they relax on the spin-diusion length scale lsf. The spin accumulation for an FI jNjFI system has been cal- culated in the macrospin model.39The result of this calculation can be directly generalized to the present situation of spatially inhomogeneous spin waves by re- placing the macrospin magnetization in each layer with the interface magnetization and substituting the spin- diusion length with a wave-vector-dependent eective spin-diusion length lsf!~lsf(Q) such that Q=~ 2M0[(_mQ(1) +_mQ(2))1() (_mQ(1)_mQ(2))2()]:(16) See Appendix B for the details of the functions 1and 2. The eective spin-diusion length is found by Fouriertransforming the spin-diusion equation (see Appendix C), resulting in ~lsf=lsf=p 1 + (Qlsf)2: (17) We thus have all the necessary equations to de- scribe the linear response dynamics of spin waves in the FI1jNjFI2 system. We now provide analytical solutions of the spin-wave modes in the long-wavelength limit and then complement these solutions with an extensive nu- merical analysis that is valid for any wavelength. III. ANALYTIC SOLUTIONS FOR THE SPIN WAVE SPECTRUM The eect that the exchange and dipolar elds have on the spin-wave spectrum depends on the in-plane wave numberQ. WhenQLi1, the dipolar eld dominates over the exchange eld. In the opposite regime, when QLi1, the exchange eld dominates over the dipo- lar eld. The intermediate regime is the dipole-exchange regime. Another length scale is set by the spin-diusion length. When Qlsf1, the eective spin-relaxation length ~lsfof Eq. (17) becomes small, and the NM acts as a perfect spin sink. In this case, only the relatively short-ranged dipolar eld couples the FIs. We therefore focus our attention on the dipole-dominated regime, in which the interchange of spin information between the two FIs remains active. In the limit QLi1, the magnetization is homoge- neous in the in-plane direction. We may then use the ansatz that the deviation from equilibrium is a sum of transverse travelling waves. Using the boundary condi- tions on the outer boundaries of the stack, Eq. (15c), we nd miQxy() = Xi Yi cos ki (Li+dN 2) ;(18) wherei= 1 whenis inside FI1 and i= 2 whenis inside FI2.k1andk2are the out-of-plane wave vectors of the lower and upper lms, respectively. The eigenfrequencies of Eq. (10) depend on ki. To rst order in the damping parameter, we have !(ki) =!M" s!H !M+A 2M2 Sk2 i!H !M+A 2M2 Sk2 i+ sin2 +i!H !M+A 2M2 Sk2 i+1 2sin2# : (19) We can, without loss of generality, consider only those frequencies that have a positive real part. The eigen-6 frequency!is a characteristic feature of the entire sys- tem, so we must require !(k1) =!(k2), which implies thatk1=k2. We will discuss the cases of symmetric (L1=L2) and asymmetric ( L16=L2) geometries sepa- rately. A. Symmetric FI lms without EASA Consider a symmetric system in which the FIs are of identical thickness and material properties. We assume that the eect of the EASA is negligible, which is the case for thin lms and/or weak surface anisotropy ener- gies such that KSL=A1, whereL=L1=L2. The other two boundary conditions, (15a) and (15b), cou- ple the amplitude vectorsX1Y1
TandX2Y2
Tof Eq. (18). A non-trivial solution implies that the deter- minant that contains the coecients of the resulting 4 4 matrix equation vanishes. Solving the secular equation, we nd the following constraints on k, iA!A=kLtan(kL); (20a) iO!O=kLtan(kL); (20b) where A= 1 1 +2g?lsf tanh(dN=2lsf)1! ;(21a) O= 1 1 +2g?lsf coth(dN=2lsf)1! ;(21b) and=L~2g?=4Ae2. The two solutions correspond to a symmetric mode (acoustic) and an antisymmetric mode (optical). This result can be understood in terms of the eigenvectors that correspond to the eigenvalues of Eqs. (20), which are m1= +m2andm1=m2for the acoustic and optical modes, respectively. Typically, because spin pumping only weakly aects the magne- tization dynamics, the timescale that is proportional to the mixing conductance g?is much smaller than the FMR precession period. In this limit, kLtan(kL)1. This result allows us to expand the secular equations (20) aroundkL=n, wherenis an integral number, which yields i!;n(kL+n)kL; (22) where= A;O. This result can be reinserted into the bulk dispersion relation of Eq. (19), from which we can determine the renormalization of the Gilbert damping coecient attributable to spin pumping,
. We dene
= Im[!(SP)]Im[!(0)] =Im[!(0)] (23) as a measure of the spin-pumping-enhanced Gilbert damping, where !(0)and!(SP)are the frequencies of the same system without and with spin pumping, respec- tively.Similar to the case of a single-layer ferromagnetic insulator,31we nd that all higher transverse volume modes exhibit an enhanced magnetization dissipation that is twice that of the macrospin mode. The enhance- ment of the Gilbert damping for the macrospin mode (n= 0) is
;macro = ~2g? 2LMSe2 ; (24) and for the other modes, we obtain
;n6=0= 2
;macro: (25) Compared with single-FI systems, the additional fea- ture of systems with two FIs is that the spin-pumping- enhanced Gilbert damping diers signicantly between the acoustic and optical modes via the mode-dependent ratio=. This phenomenon has been explored both experimentally and theoretically in Ref. 32 for the macrospin modes n= 0 when there is no loss of spin transfer between the FIs, lsf!1 . Our results repre- sented by Eqs. (24) and (25) are generalizations of these results for the case of other transverse volume modes and account for spin-memory loss. Furthermore, in Sec. IV, we present the numerical results for the various spin-wave modes when the in-plane momentum Qis nite. When the NM is a perfect spin sink, there is no transfer of spins between the two FIs, and we recover the result for a sin- gle FIjN system with vanishing back ow, !.31 Naturally, in this case, the FI jNjFI system acts as two independent FIjN systems with respect to magnetiza- tion dissipation. The dynamical interlayer dipole cou- pling is negligible in the considered limit of this section (QL1). In the opposite regime, when the NM lm is much thin- ner than the spin-diusion length and the spin conductiv- ity of the NM is suciently large such that g?dN=1, thenA!0 andO!. This result implies that for the optical mode, the damping is the same as for a sin- gle FI in contact with a perfect spin sink, even though the spin-diusion length is very large. The reason for this phenomenon is that when the optical mode is ex- cited, the magnetizations of the two lms oscillate out of phase such that one layer acts as a perfect spin sink for the other layer. By contrast, there is no enhance- ment of the Gilbert damping coecient for the acoustic mode; when the lm is very thin and the magnetizations of the two layers are in phase, there is no net spin ow or loss in the NM lm and no spin-transfer-induced losses in the ferromagnets. Finally, when the NM is a poor con- ductor despite exhibiting low spin-memory loss such that g?dN=(lsf=dN)1, then!0 because there is no exchange of spin information. For the macrospin modes in the absence of spin-memory loss, these results are in exact agreement with Ref. 32. Beyond these results, we nd that regardless of how much spin memory is lost, it is also the case that in trilayer systems, all higher trans- verse modes experience a doubling of the spin-pumping- induced damping. Furthermore, these modes can still7 be classied as optical and acoustic modes with dierent damping coecients. B. Symmetric Films with EASA Magnetic surface anisotropy is important when the spin-orbit interaction at the interfaces is strong. In this case, the excited mode with the lowest energy becomes inhomogeneous in the transverse direction. For a nite KS, the equations for the xandycomponents of the magnetization in the boundary condition (15) dier, re- sulting in dierent transverse wave vectors for the two components, kxandky, respectively. Taking this situa- tion into account, we construct the ansatz miQxy() = Xicos (kx;ikx;i(L+dN=2)) Yicos (ky;iky;i(L+dN=2)) ;(26) which, when inserted into the boundary conditions of Eqs. (15a) and (15b), yields i!+LKS Acos (2) =kxdtan (kxd);(27a) i!+LKS Acos2() =kydtan (kyd);(27b) wherecontinues to denote an acoustic (A) or optical (O) mode, = A;O. Depending on the sign of KSand the angle, the resulting solutions kxandkycan be- come complex numbers, which implies that the modes are evanescent. Let us consider the case of KS>0 and an in-plane magnetization ( ==2). Although kyis unchanged by the EASA, with LKS=A> 1!,kx is almost purely imaginary, =ik=KS=Ai!, so that miQx() =Xcosh((d+dN=2)): (28) The magnetization along the xdirection is exponentially localized at the FI jN surfaces. Following the same proce- dure as in Sec. III A for the KS= 0 case, we insert this solution into the dispersion relation (19) and extract the renormalization of the eective Gilbert damping:
EASA  = ~2g? 2LMSe2 1 +!H !M 1 +2LKS A K2 S 2M2 SA 1 + 2!H !MK2s 2M2 SA: (29) In the presence of EASA, the damping coecient is a ten- sor; thus, the eective damping of Eq. (29) is an average, as dened in Eq. (23). This Gilbert damping enhance- ment may become orders of magnitude larger than the
macro of Eq. (24). For thick lms,
macroL1, whereas
EASA  reaches a constant value that is in- versely proportional to the localization length at the FI jN interface. Note that for large EASA, the equilibrium magnetization is no longer oriented along the external eld, and Eq. (29) for
EASA  becomes invalid.C. Asymmetric FI Films Let us now consider an asymmetric system in which L16=L2. In this conguration, we will rst consider KS= 0, but we will also comment on the case of a - niteKSat the end of the section. Because the analytical expressions for the eigenfrequencies and damping coe- cients are lengthy, we focus on the most interesting case: that in which the spin-relaxation rate is slow. As in the case of the symmetric lms, the dispersion relation of Eq. (10) dictates that the wave numbers in the two layers must be the same. To satisfy the boundary equations (15), we construct the ansatz miQxy() = Xicos (kk(L+dN=2)) Yicos (kk(L+dN=2)) : (30) The dierence between this ansatz and the one for the symmetric case represented by Eq. (26) is that the mag- nitudes of the amplitudes, XiandYi, of the two layers, i= 1;2, that appear in Eq. (30) is no longer expected to be equal. When the two ferromagnets FI( L1) and FI(L2) are completely disconnected, the transverse wave vectors must be equivalent to standing waves, qn;1=n=L 1and qm;2=m=L 2in the two lms, respectively, where nand mmay be any integral numbers. Because spin pumping is weak, the eigenfrequencies of the coupled system are close to the eigenfrequencies of the isolated FIs. This nding implies that the wave vector kof the coupled sys- tem is close to either qn;1orqm;2. The solutions of the linearized equations of motion are then k=kn;1=qn;1+kn;1or (31a) k=km;2=qm;2+km;2; (31b) wherekn;1andkm;2are small corrections attributable to spin pumping and spin transfer, respectively. Here, the indices 1 and 2 represent the dierent modes rather than the layers. However, one should still expect that mode 1(2) is predominantly localized in lm 1(2). In this manner, we map the solutions of the wave vectors in the coupled system to the solutions of the wave vectors in the isolated FIs. Next, we will present solutions that correspond to the qn;1of Eq. (31a). The other family of solutions, corresponding to qm;2, is determined by inter- changingL1$L2and making the replacement n!m. Inserting Eq. (31a) into the boundary conditions of Eq. (15) and linearizing the resulting expression in the weak spin-pumping-induced coupling, we nd, for the macrospin modes, i!~A,O 1;macro = (L1k0;1)2; (32) where ~A 1;macro1 2dN lsf g?lsfL1 L1+L21; (33a) ~O 1;macro1 2L1+L2 L21: (33b)8 Here,1=L1~2g?=4Ae2. Inserting this parameter into the dispersion relation of Eq. (19), we obtain the follow- ing damping renormalizations:
A macro = ~2g? 2MSe21 2dN lsf g?lsf1 L1+L2;(34a)
O macro = ~2g? 2MSe21 21 L1+1 L2 : (34b) These two solutions correspond to an acoustic mode and an optical mode, respectively. The corresponding eigenvectors are m1=m2for the acoustic mode and L1m1=L2m2for the optical mode. As in the sym- metric case, the damping enhancement of the acoustic mode vanishes in the thin-NM limit. In this limit, the behavior of the acoustic mode resembles that of a single FI of thickness L1+L2. It is the total thickness that determines the leading-order contribution of the damp- ing renormalization. The optical mode, however, experi- ences substantial damping enhancement. For this mode, the damping renormalization is the average of two sepa- rate FIs that are in contact with a perfect spin sink. The cause of this result is as follows. When there is no spin- memory loss in the NM, half of the spins that are pumped out from one side return and rectify half of the angular- momentum loss attributable to spin pumping. Because the magnetization precessions of the two lms are com- pletely out of phase, the other half of the spin current causes a dissipative torque on the opposite layer. In ef- fect, spin pumping leads to a loss of angular momentum, and the net sum of the spin pumping across the NM and the back ow is zero. The total dissipation is not aected by spin transfer, and thus, the result resembles a system in which the NM is a perfect spin sink. For the higher excited transverse modes, there are two scenarios, which we treat separately. I. The allowed wave number for one layer matches a wave number for the other layer. Then, for some integer n > 0,qn;1=qm;2 for some integer m. In this case, we expect a coupling of the two layers. II. The allowed wave number for one layer does not match any of the wave numbers for the other layer, and thus, for some integer n > 0, we have qn;16=qm;2for all integers m. We then expect that the two layers will not couple. I. In this case, we nd two solutions that correspond to acoustic and optical modes. These modes behave very much like the macrospin modes; however, as in the sym- metric case, the damping renormalization is greater by a factor of 2:
A,O n6=0= 2
A,O macro;Case I: (35) The eigenvectors of these coupled modes have the same form as for the macrospin modes, such that m1=m2 andL1m1=L2m2for the acoustic and optical modes, respectively. II. In this case, the two layers are completely decou-pled. To the leading order in dN=lsf, we nd
n6=0= ~2g? 2L1MSe2;Case II; (36) for all modes that correspond to excitations in FI1. The damping renormalization is thus half that of the FI(L1)jN(lsf= 0) system.31This result can be explained by the zero loss of spin memory in the NM. Although half of the spins are lost to the static FI2, half of the spins return and rectify half of the dissipation attributable to spin pumping. The amplitudes of these modes are strongly suppressed in FI2 (or FI1, upon the interchange of FI1$FI2), such thatjm2j=jm1j!2. Finally, let us discuss the case in which EASA is present. In the limit KSLi=A1, the excitation en- ergies of the surface modes are independent of the FI thicknesses. However, the surface modes do not behave like the macrospin modes for the asymmetric stack. The excitation volume of these modes is determined by the decay length A=KSin accordance with Eq. (28). This nding is in contrast to the result for the macrospin modes, where the excitation volume spans the entire FI. Thus, the surface modes couple in the same manner as in the symmetric case. With a good experimental control of surface anisotropy, the coupling of the surface modes is thus robust to thickness variations. The higher ex- cited transverse modes, in the presence of EASA, have thickness-dependent frequencies, which means that these modes behave similarly to the n>0 modes in the KS= 0 case. IV. NUMERICAL RESULTS When the spin-wave wavelength becomes comparable to the lm thickness, the dipolar eld becomes a compli- cated function of the wavelength. We study the proper- ties of the system in this regime by numerically solving the linearized equations of motion (10) with the bound- ary conditions (15). We use the method presented in Ref. 31, which solves the spin-wave excitation spectrum for an FIjN system, and extend this approach to the present trilayer system. The physical parameters used in the numerical calculations are listed in Table I. We investigate two geometries: I. the BWMSW geometry, in which the spin wave propagates parallel to the external eld, and II. the MSSW geometry, in which the spin wave propagates perpendicular to the external eld. To calculate the renormalization of the Gilbert damp- ing, we perform one computation without spin pumping and one computation with spin pumping, in which the intrinsic Gilbert damping is excluded. Numerically, the renormalization can then be determined by calculating
=Im[!(SP)]=0=Im[!(0)], where!(0)is the eigenfre- quency obtained for the computation without spin pump- ing and!(SP)is the frequency obtained for the compu- tation with spin pumping.319 TABLE I: Physical parameters used in the numerical calcu- lations Constant Value Units g?a3:4
1015cm2e2=h b5:4
1017s1 4MSc1750 G Ac3:7
107erg=cm Hint 0:58
4MS c3
104 KS 0;d0:05 erg=cm2 a) Ref. [47], b) Ref. [48], c) Ref. [34] d) Reported to be in the range of 0 :10:01 erg=cm2in Ref. [21] A. BVMSW FIG. 3: (Color online) FI(100nm) jN(50nm)jFI(101nm): a) Spin-pumping-enhanced Gilbert damping
as a function ofQL1of the uniform modes and the n= 1 modes. The inset presents the corresponding dispersion relation. b) Relative phase and c) amplitude between the out-of-plane magnetiza- tions along xat the edges of FI1 jN and FI2jN. The apparent discontinuity in the green line in c) appears because the phase is dened on the interval to. Let us rst discuss the BVMSW geometry. The cou- pling of the uniform modes in the two lms is robust;it is not sensitive to possible thickness asymmetries. In contrast, at Q= 0, the sensitivity to the ratio between the thickness and the rather weak dynamic coupling at- tributable to spin pumping implies that the coupling of the higher transverse modes in the two bilayers is fragile. Small asymmetries in the thicknesses destroy the cou- pling. This eect can best be observed through the renor- malization of the damping. However, we will demon- strate that a nite wave number Qcan compensate for this eect such that the higher transverse modes also become coupled. To explicitly demonstrate this result, we numerically compute the real and imaginary parts of the eigenfrequencies of a slightly asymmetric system, FI(100nm)jN(50nm)jFI(101nm) with lsf= 350 nm. The asymmetry between the thicknesses of the ferromagnetic insulators is only 1%. The surface anisotropy is consid- ered to be small compared with the ratio Li=A, and we setKS= 0. In Fig. 3, the numerical results for the eective Gilbert damping, the dispersion of the modes, and the relative phase and amplitude between the magnetizations in the two FIs are presented. As observed in the relative phase results depicted in Fig. 3(c), the two uniform modes in widely separated FIs split into an acoustic mode and an optical mode when the bilayers are coupled via spin pumping and spin transfer. Figure 3(a) also demon- strates that the acoustic mode has a very low renor- malization of the Gilbert damping compared with the optical mode. Furthermore, there is no phase dierence between the two modes with a transverse node ( n= 1) in Fig. 3(a), which indicates that the modes are decoupled. Thesen= 1 modes are strongly localized in one of the two lms; see Fig. 3(b). For small QL1, Fig. 3(a) demon- strates that these modes have approximately the same renormalization as the optical mode, which is in agree- ment with the analytical results. Because the magnetiza- tion in the layer with the smallest amplitude is only a re- sponse to the spin current from the other layer, the phase dierence is =2 (Fig. 3(b)). When Qincreases, the dipo- lar and exchange interactions become more signicant. The interlayer coupling is then no longer attributable only to spin pumping but is also caused by the long-range dipole-dipole interaction. This additional contribution to the coupling is sucient to synchronize the n= 1 modes. The relative amplitude between the two layers then be- comes closer to 1 (see Fig. 3(b)). Again, we obtain an acoustic mode and an optical n= 1 mode, which can be observed from the phase dierence between the two lay- ers in Fig. 3(c). The spin-pumping-induced coupling only occurs as long as the eective spin-diusion length ~lsfis large or on the order of dN. Once this is no longer the case, the modes rapidly decouple, and the system reduces to two separate FI jN systems with a relatively weak in- terlayer dipole coupling. In the limit of large QL1, the exchange interaction becomes dominant. The energy of the wave is then predominantly attributable to the mo- mentum in the longitudinal direction, and the dynamic part of the magnetization goes to zero at the FI jN inter-10 faces, causing the renormalization attributable to spin pumping to vanish.31 We also note that the dispersion relation depicted in the inset of Fig. 3(a) reveals that the acoustic mode (blue line) exhibits a dip in energy at lower QL1than does the optical mode (red line). We suggest that this feature can be understood as follows: The shift in the position of the energy dip can be interpreted as an increase in the eective FI thickness for the acoustic mode with re- spect to that for the optical mode. When ~lsfis larger than the NM thickness, the uniform mode behaves as if the NM were absent and the two lms were joined. This result indicates that the dispersion relation for the acoustic mode exhibits frequency behavior as a function ofQ~L=2, where the eective total thickness of the lm is ~L=L1+L2. The optical mode, however, \sees" the NM and thus behaves as if ~L=L1. Consequently, the dip in the dispersion occurs at lower QL1for the acoustic mode than for the optical mode. B. MSSW Finally, let us study the dynamic coupling of mag- netostatic surface spin waves (MSSWs). We now con- sider a perfectly symmetric system, FI(1000 nm) jN(200 nm)jFI(1000 nm), with lsf= 350 nm. For such thick lms, surface anisotropies may play an important role. We therefore discuss a case in which we include a surface anisotropy of KS= 0:05 erg=cm2. According to the an- alytical result presented in Eq. (28), the lowest-energy modes with QL11 are exponentially localized at the FIjN surfaces, with a decay length of A=KS200 nm. We now compute the eigenfrequencies, !, as a function of the wave vector in the range 104< QL 1<103. In Fig. 4(a), we present the real part of the frequency for the six lowest-energy modes with a positive real part, and in Fig. 4(b), we present the corresponding renormaliza- tions of the Gilbert damping for the four lowest-energy modes. The dispersion relations indicate that the mode pairs that are degenerate at QL11 rapidly split in energy when QL1approaches 102. Strong anticrossings can be observed between the n= 1 andn= 2 modes. Such anticrossings are also present between the surface mode and the n= 1 mode; they are almost too strong to be recognized as anticrossings. The enhanced damping renormalizations exhibit very dierent behavior for the dierent modes. We recognize the large-
mode of one pair as the surface optical mode and the low-
mode as the volume n= 1 acoustic mode. Without EASA, the anticrossings in Fig. 4(a) would become crossings. The lowest-energy modes at QL11 would then cut straight through the other modes. In the case considered here, this behavior is now observed only as steep lines at QL10:05 and atQL10:5. WhenQis increased, the eective spin-diusion length decreases (see Eq. (17)), which reduces the spin- pumping-induced coupling between the modes at largeQ. WhenQL1100, the coupling becomes so weak that the two FIs decouple. This phenomenon can be ob- served from the behavior of
in Fig. 4(b), where the damping of the acoustic modes become the same as for the optical modes. FIG. 4: (Color online) FI(1000nm) jN(200nm)jFI(1000nm) lsf= 350 nm, KS= 0:05 erg=cm2: a) The dispersion rela- tion as a function of QL1for the six lowest positive-real-part modes. b) The renormalization of the damping attributable to spin pumping for the four lowest modes with frequencies with positive real parts as a function of QL1. At largeQL1, the computation becomes increasingly demanding, and the point density of the plot becomes sparse. We have therefore individually marked the plotted points in this region. In the MSSW geometry, an isolated FI has magneto- static waves that are localized near one of the two sur- faces, depending on the direction of propagation with respect to the internal eld.34Asymmetries in the exci- tation volume are therefore also expected for the trilayer in this geometry. In Fig. 5, we present the eigenvectors of the surface modes as functions of the transverse co- ordinatefor increasing values of the wave vector Q. AtQL1= 0:5, the modes have already begun to ex- hibit some asymmetry. Note that the renormalization of the damping observed in Fig. 4(b) is approximately one order of magnitude larger than the intrinsic Gilbert damping for the optical mode and that the damping of any one mode may vary by several orders of magnitude as a function of QL1.31Therefore, these eects should be experimentally observable. The greatest damping oc- curs when the two layers are completely decoupled; see Figs. Fig. 4(b) and 5. Because the damping of the opti- cal mode is equivalent to that of a system with a perfect spin sink, one might expect that the greatest damping11 FIG. 5: (Color online)FI(1000nm) jN(200nm)jFI(1000nm), lsf= 350nm, KS= 0:05 erg=cm2: a) and b) present the real parts of the xcomponents of the out-of-equilibrium mag- netization vectors for the acoustic and optical surface modes, respectively, for several values of QL1. For values of QL1&1, the modes decouple and become localized in one of the two layers. For large values of QL1100, the two modes are strongly localized at one of the two FI jN interfaces, which correspond to the peaks in the damping that are apparent in Fig. 4(b). should occur for this mode. However, the large localiza- tion, which is achieved only at large QL1, in combination with the vanishing of the eective spin-diusion length leads to damping that is much greater than that of the synchronized optical mode. V. CONCLUSIONS We investigated the dynamic coupling of spin-wave ex- citations, which are present in single FI thin lms, pri- marily through spin pumping and spin transfer but also through the dynamic demagnetization eld created when two FI thin lms are in contact via an NM layer. Because of this coupling, the modes are split into acoustical and optical excitations. When the NM is thin compared with lsf, the renormalization of the Gilbert damping vanishes for the acoustic modes, whereas for the optical modes, the renormalization is equally as large as for a single- FIjN system in which the NM is a perfect spin sink. A spin current pumped by a travelling magnetic wave has a wavelength of equal magnitude, which leads to traversal paths across the NM that are longer than the thickness of the NM. Consequently, the spin-memory loss is greater for short-wavelength spin currents. This phenomenonleads to an eective spin-diusion length in the NM that decreases for increasing values of Q. As a result, the dy- namic coupling strength is reduced for short-wavelength spin waves. At some critical value of Q, the coupling be- comes so weak that the acoustic- and optical-mode con- gurations are lost in favor of modes that are localized in one of the two FIs. At these values of Q, the inter- layer dipole coupling is also dominated by the intralayer exchange coupling. For these high-wave-number modes, the system behaves similar to two separate FI jN(lsf= 0) systems. When the two lms are of dierent thicknesses, the exchange energies of the higher-order transverse n > 1 modes dier between the two layers. Because of the rel- atively small coupling attributable to spin pumping, the synchronization of these modes at QL11 requires that the FI thicknesses be very similar. A small asymmetry breaks the synchronization; however, for larger QL11, the modes can again become coupled through interlayer dipole interaction. This coupling arises in addition to the spin-pumping- induced coupling. For even larger Q, the eective spin-diusion length becomes small, and the coupling attributable to spin pumping vanishes. The rel- atively small dipole coupling alone is not sucient to couple the modes when there is a nite dierence in lm thickness , and the synchronization breaks down. Depending on the quality of the interface between the FIs and the strength of the spin-orbit coupling in the NM , additional eective surface elds may be present because of surface anisotropy energies. For the EASA case, the lowest-energy modes are localized at the FI jN surfaces. These modes couple in the same manner as the macrospin modes. For lms that are much thicker than the decay length A=KS, the energies of the surface modes do not depend on the lm thickness. Consequently, the coupling of these modes is independent of the thickness of the two FIs. Similar to the simpler FI jN system, the damping enhancement may attain values as high as an or- der of magnitude larger than the intrinsic Gilbert damp- ing. However, in the trilayer system, the presence of both acoustic and optical modes results in large variations in the eective damping within the same physical sample. Because of this wide range of eective damping, which spans a dierence in
of several orders of magnitude as a function of Q, we suggest that trilayer modes should be measurable in an experimental setting. With more complicated FI structures in mind, we be- lieve that this work may serve as a guide for experimen- talists. The large variations in eective damping for dif- ferent modes make the magnetic properties of the system detectable both with and without EASA. For spin waves, dipole-dipole interactions assist spin pumping in inter- layer synchronization, which may facilitate the design of future spintronic devices.12 Acknowledgments We acknowledge support from InSpin 612759 and the Research Council of Norway, project number 216700. Electronic address: hans.skarsvag@ntnu.no 1C. Kittel, Phys. Rev. 73, 155 (1948). 2R.W. Damon, J.R. Eshbach, J. Phys. Chem. Solids 19, 308 (1961). 3V. Cherepanov, I. Kolokolov and V. L'vov, Phys. Rep. 229, 81 (1993). 4A. Brataas, A. D. Kent, and H. Ohno, Nature Mat., 373 (2012). 5G. Binasch, P. Gr unberg, F. Saurenbach, and W. 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B 82, 054403 (2010). 42J. Xingtao, L. Kai, K. Xia, and G. E. W. Bauer, EPL 96, 17005 (2011). 43A. Kapelrud and A. Brataas, (unpublished). 44Y. Zhou, H.J. Jiao, Y.T. Chen, G. E. W. Bauer, and J. Xiao, Phys. Rev. B 88, 184403 (2013). 45Z. Q. Qiu , J. Pearson and S. D. Bader Phys. Rev. B 46, 8659 (1992). 46J.-M.L. Beaujour, W. Chen, A. D. Kent and J. Z. Sun, J. Appl. Phys. 99, 08N503 (2006). 47M. B. Jung eisch, V. Lauer, R. Neb, A. V. Chumak and B. Hillebrands, Appl. Phys. Lett. 103, 022411 (2013). 48D. Giancoli, "25. Electric Currents and Resistance". In Jo- celyn Phillips. Physics for Scientists and Engineers with Modern Physics (4th ed.), (2009) [1984].13 Appendix A: Dipole Tensor The dipole tensor in the coordinate system, ^G() from Eq. (9) can be rotated by the xyzcoordinate system with the rotation matrix R=0 @scscc 0cs csssc1 A; (A1)where we have introduced the shorthand notation s sin,ccosand so on. We then get that (* ^Gxyz=R^GRT =0 @s2 Gcs2G+c2 c2 GssG+sccGscGscc2 G+c(s2 c2 )G ssG+sccG s2 G scG+sscG scGscc2 G+c(s2 c2 )GscG+sscGc2 G+s2cG+c2 s2 G1 A: (A2) Because we work in the linear respons regime the equilibrium magnetization should be orthogonal to the dynamic deviation, mi
^z= 0, it is therefor sucient to only keep the xypart of ^Gxyz. We then nd ^Gxy=s2 Gcs2G+c2 c2 GssG+sccG ssG+sccG s2 G : (A3) Appendix B: Spin Accumulation The functions 1() and 2() are taken directly from Ref.39, and modied to cover the more complicated mag- netic texture model. We then have 1()cosh =~lsf cosh =~lsf +sinh =~lsf =2g?~lsf; 2()sinh =~lsf sinh =~lsf +cosh =~lsf =2g?~lsf:(B1) ForQlsf1 the eective spin diusion length becomes short, 1!1 and 2!0 at the FIjN interfaces. Appendix C: Eective spin diusion length The diusion in the NM reads @tS=Dr2S1 sfS; (C1)whereDis the diusion constant and sfis the spin ip relaxation time. We assume that the FMR frequency is much smaller than the electron traversal time, D=d2 N, and the spin- ip relaxation rate, 1 =sf.39This means the LHS of Eq. (C1) can be disregarded. In linear response the spin accumulation, which is a direct consequence of spin pumping, must be proportional to the rate of change of magnetization at the FI jN interfaces. We do the same Fourier transform, as for the magnetization, so that  expfi(!tQ)g. The spin diusion equation then takes the form @2 S= Q2+1 Dsf S: (C2) The spin diusion length is then lsf=pDsf, and by introducing the eective spin diusion length ~lsf= lsf=q 1 + (Qlsf)2one gets @2 S=1 ~l2 sfS: (C3)

2014-07-02

Herein, we study the spin-wave dispersion and dissipation in a ferromagnetic insulator--normal metal--ferromagnetic insulator system. Long-range dynamic coupling because of spin pumping and spin transfer lead to collective magnetic excitations in the two thin-film ferromagnets. In addition, the dynamic dipolar field contributes to the interlayer coupling. By solving the Landau-Lifshitz-Gilbert-Slonczewski equation for macrospin excitations and the exchange-dipole volume as well as surface spin waves, we compute the effect of the dynamic coupling on the resonance frequencies and linewidths of the various modes. The long-wavelength modes may couple acoustically or optically. In the absence of spin-memory loss in the normal metal, the spin-pumping-induced Gilbert damping enhancement of the acoustic mode vanishes, whereas the optical mode acquires a significant Gilbert damping enhancement, comparable to that of a system attached to a perfect spin sink. The dynamic coupling is reduced for short-wavelength spin waves, and there is no synchronization. For intermediate wavelengths, the coupling can be increased by the dipolar field such that the modes in the two ferromagnetic insulators can couple despite possible small frequency asymmetries. The surface waves induced by an easy-axis surface anisotropy exhibit much greater Gilbert damping enhancement. These modes also may acoustically or optically couple, but they are unaffected by thickness asymmetries.

Spin Waves in Ferromagnetic Insulators Coupled via a Normal Metal

1407.0635v1

Prepared for submission to JHEP An effective field theory of damped ferromagnetic systems Jingping Li Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213 E-mail: jingpinl@andrew.cmu.edu Abstract: Using the in-in formalism, we generalize the recently constructed magnetoelastic EFT [1] to describe the damping dynamics of ferromagnetic systems at long wavelengths. We find that the standard Gilbert damping term naturally arises as the simplest leading-order symmetry- consistentnon-conservativecontributionwithinthein-inframework. TheEFTiseasilygeneralized to scenarios with anisotropy and inhomogeneity. In particular, we find the classic Landau-Lifshitz damping term emerges when isotropy is broken by a constant external background field. This provides a first principle explanation for distinguishing the two types of damping dynamics that were originally constructed phenomenologically. Furthermore, the EFT framework could also in- corporate intrinsic anisotropy of the material in a straightforward way using the spurion method. For systems with inhomogeneity such as nontrivial spin textures, we find that the leading order derivative correction yields the generalized Gilbert damping equations that were found in con- densed matter literature. This shows that the EFT approach enables us to derive the form of higher-derivative-order corrections in a systematic way. Lastly, using the phonon-magnon cou- pling deduced in the magnetoelastic EFT, we are able to make a prediction for the generic form of the phononic contribution to the damping equation.arXiv:2312.13093v1 [hep-th] 20 Dec 2023Contents 1 Introduction 1 2 Review of the magnon EFT and Schwinger-Keldysh formalism 2 2.1 Symmetry breaking in magnetoelastic systems 2 2.2 The magnon EFT and the conservative equation of motion 4 2.3 The Schwinger-Keldysh formalism 4 3 Gilbert damping term from EFT 6 3.1 Coupling at the leading order 6 3.2 Gilbert damping 6 4 More general materials 9 4.1 Anisotropic materials 9 4.2 Inhomogeneous materials 10 5 Magnon damping term from phonons 11 6 Conclusion and discussions 12 1 Introduction It has long been established that the methodology of coset construction serves as a powerful tool of relativistic effective field theories (EFTs) of Goldstone bosons (e.g. pions from spontaneously broken approximate chiral symmetry) [2–5]. In recent years, many works have demonstrated that its versatility is extendable to condensed matter systems where we are interested in the macroscopic behavior which are usually the massless low energy excitations [6, 7]. Using this approach, a recent paper [1] constructed an EFT of magnetoelastic systems where the phonons and magnons are considered Goldstones associated with translations spontaneously broken by the ground state location of the material (the lattice) and an SO(3)symmetry of the magnetic moments by their ground state orientations. The EFT approach provides a systematic way to understand phonon-magnon interactions from first principles and predict the forms of higher-order corrections which has not been done previously. While the paper was focused on conservative dynamics, there has also been extensive study on the theoretical description of non-conservative dynamics of damped magnetic systems since the seminal work of Landau, Lifshitz, and later Gilbert [8, 9]. However, to our knowledge, the prior works were mostly model-dependent phenomenological descriptions. It is therefore desirable to have a first principle derivation from a similar many-body EFT perspective. On the other hand, the Schwinger-Keldysh formalism [10, 11] (and the related in-in formalism [12, 13]) has been known to describe the quantum field theory of open systems and hence fully – 1 –capable of describing dissipative effects. In recent years, its power has been successfully extended to the EFT framework of dissipative dynamics in astrophysics and black holes [14, 15, 17] that systematically derives dissipative equations of motion. Therefore, it is natural to consider its utility in describing deriving damping equations in condensed matter EFTs. In this paper, combining the power of the two techniques, we apply the Schwinger-Keldysh formalism to incorporate dissipative effects into the EFT of magnons to reproduce the known results of magnetic damping. In section 2, we review the coset construction of the magnon EFT as well as the techniques in Schwinger-Keldysh formalism to be applied in this paper. Section 3 derives the original Gilbert damping equation for homogeneous and isotropic materials. In section 4, we move on to more general materials and recover the Landau-Lifshitz damping equation for anisotropic systems and generalized Gilbert damping for spatially inhomogeneous materials. In section 5, we derive the damping terms originating from the magnon-phonon interaction. Conventions: we use natural units where ℏ= 1. Unless specified, the uppercase Latin indices A, B, C . . . denote the full internal spin symmetry space which runs over 1,2,3while the lower case ones a, b, c . . . in the begining alphabet index the broken subspace 1,2. Those in the middle alphabet i, j, k . . . run over the three spatial dimensions (to generalize to higher dimensions, the internal spin symmetry will have to be modified accordingly). 2 Review of the magnon EFT and Schwinger-Keldysh formalism In this section, we first provide a self-contained review on the symmetries and the corresponding coset construction of magnon-phonon EFT proposed by [1]. Furthermore, we summarize the Schwinger-Keldysh formalism which is the central tool for deriving the dissipative equations of motion. 2.1 Symmetry breaking in magnetoelastic systems We follow the derivations in [1]. The symmetries under consideration are the spatial Galilean group (generated by translations Pi, rotations, Li, boost Ki) and internal symmetries (internal translations Ti, internal rotations Qi, spin rotations SA). The algebra of the generators is given by [Li, Kj] =iϵijkKk,[Li, Pj] =iϵijkPk, (2.1) [Ki, H] =−iPk,[Ki, Pj] =−iMδ ij, (2.2) [Qi, Tj] =iϵijkTk,[Qi, Qj] =iϵijkQk, (2.3) [SA, SB] =iϵijkKk,[Li, Lj] =iϵijkLk. (2.4) In particular, the TiandQigenerators generate translations and rotations on the “comoving” coordinates ϕI(x)(or the Lagrangian coordinates in continuum mechanics which simply gives an initial labeling to the continuum) ϕI(x)→ϕI(x) +aI, ϕI(x)→RI JϕJ(x), (2.5) andSAgenerates the internal rotation on the orientation of the spin NA→OA BNB, (2.6) – 2 –where O=eiχaSaand the Néel vector ⃗Nis the order parameter for the spin orientation. In the ground state, the order parameters gain vacuum expectation values (VEVs) which we choose to be D ⃗ϕ(x)E =⃗ x,D ⃗NE = ˆx3. (2.7) The VEVs are not invariant under the transformations and hence spontaneously break some of the symmetries Unbroken = H Pi+Ti≡¯Pi Li+Qi S3 M, Broken = Ki Ti Qi S1, S2≡Sa. (2.8) The parametrization of the ground state manifold which is simply the broken symmetry transfor- mations plus the unbroken translations is given by Ω =e−itHeixi¯PieiηiKieiπiTieiθiQieiχaSa, (2.9) where ηi,θi,χa, and πi=ϕi−xiare the corresponding Goldstone fields. For any Goldstone fields ψicorresponding to the broken generators Xi, their covariant deriva- tives are the basic building blocks of the low energy EFT. They are systematically computed by the coset construction using the Mauer-Cartan forms of the broken group Ω−1∂µΩ⊃ ∇µψi
Xi, (2.10) by computing the coefficients of the broken generator Xi. In addition, in the case that one broken generator X′appears in the commutation algebra of the other X’ with the unbroken translations ¯P, X ⊃X′, (2.11) it means that the two Goldstones are not independent and one of them can be eliminated. This is known as the inverse Higgs phenomenon. The result of this exercise is that ηi,θiare eliminated and the only independent degrees of freedom are the magnons χaand phonons πi, and at leading order in derivatives, they appear in the following combinations: ∇(iπj)= (D√ DTDD−1)ij−δij, (2.12) ∇tχa=1 2ϵaBCn O−1h ∂t−∂tπk(D−1)j k∂ji Oo BC, (2.13) ∇iχa=1 2ϵaBC O−1∂iO
BC, (2.14) where Dij=δij+∂iπj. [1] found the most general action for ferromagnetic material in the form L=c1 2det (D)ϵabh O−1∂tO
ab−∂tπk D−1
j k O−1∂jO
abi (2.15) −1 2Fij 2(∇(iπj))∇iχa∇jχa−1 2F3(∇(iπj))∇tχa∇tχa, (2.16) where the first term is similar to a Wess-Zumino-Witten (WZW) term that differs by a total derivative under the symmetry transformation. – 3 –2.2 The magnon EFT and the conservative equation of motion To derive the equations of motion for magnons, it is more convenient to express the magnon fields in the nonlinear form ˆn=R(χ)ˆx3= (sin θcosϕ,sinθsinϕ,cosθ), (2.17) where the two magnon fields are related to the angular fields by χ1=θsinϕ, χ 2=θcosϕ. (2.18) Physically, this unit vector represents the direction of the magnetic moment. Under this repre- sentation, the pure magnon Lagrangian (in the absence of phonon excitations) becomes L →c2 2ϵab O−1∂tO
ab+c6 2(∂tˆn)2−c7 2(∂iˆn)2, (2.19) where F3(0) = c6andFij 2(0) = c7δij. The dispersion relation for the quadratic Lagrangian has two solutions [1] ω2 +=c2 c62 +O(k2), ω2 −=c7 c22 k4+O(k6). (2.20) For ferromagnetic materials, where c2(c6c7)3/4, the first mode is gapped around the EFT cutoff scale, while the second has ω∼k2scaling and exits the EFT. In the long wavelength limit, we may assign the scaling ∂1/2 t∼∂ito the derivatives for ferromagnets. To derive the equation of motion, we notice that an action of this form has a symmetry under the infinitesimal spin rotation δˆn=⃗ ω׈n, where ⃗ ωis the constant infinitesimal parameter. It can be shown that the Wess-Zumino-Witten term contributes to a total derivative ∂µ⃗Fµunder this transformation. Using the equation for Noether current in Lagrangian mechanics ⃗Jµ= ˆn×∂L ∂∂µˆn−⃗Fµ, (2.21) we find the conserved current ⃗J0=−c2ˆn−c6∂tˆn׈n,⃗Ji=c7∂in׈n. (2.22) The continuity equation ∂µ⃗Jµ= 0is then explicitly c2∂tˆn=− c6∂2 tˆn−c7∇2ˆn
׈n, (2.23) which is the equation of motion for Landau-Lifshitz model of magnetism. 2.3 The Schwinger-Keldysh formalism The appropriate formalism for non-conservative system is the so-called in-in or Schwinger-Keldysh formalism [10–13]. The basic idea is that there is an external sector Xthat the energy is dissipated into since the total energy needs to be conserved. The external sector could evolve into any final state which we do not observe, so all the dynamics are inclusive of the final states in the Hilbert space of XX Xout⟨Xin|. . .|Xout⟩⟨Xout|. . .|Xin⟩ ≡ ⟨. . .⟩in, (2.24) – 4 –and depends only on the initial state (hence the name in-in). We can generate an effective action for the in-in observables via the Schwinger-Keldysh closed time path integral exp iΓ[q; ˜q] =Z initialDXD˜Xexp iS[q, X]−iS[˜q,˜X] , (2.25) where we are integrating over an additional copy of variables ˜Xwhich corresponds to evolving back to the boundary conditions fixed at the initial time. The equation of motion for the degrees of freedom in the observed sector qcan be derived from the action functional Γ[q; ˜q]by δ δqΓ[q; ˜q] q=˜q= 0. (2.26) Any external sector operator O(X)coupled to some operator in the observable sector F(q)by the interaction termR dxO(X(x))F(q(x′))(where xis the corresponding spacetime coordinates) would enter the equations of motion in terms of ⟨O(X(x))⟩in=Z initialDXD˜Xexp iS[q, X]−iS[˜q,˜X] O(X), (2.27) where we have abbreviated ⟨O(X)⟩in≡ ⟨Xin|O(X)|Xin⟩. Just as in the perturbative quantum field theory correlation functions calculated by Feynman propagators, this can be similarly calcu- lated using the Schwinger-Keldysh propagators ⟨Oa(x)Ob(x′)⟩= ⟨TO(x)O(x′)⟩ ⟨O (x′)O(x)⟩ ⟨O(x)O(x′)⟩ ⟨˜TO(x)O(x′)⟩! , (2.28) where Tand ˜Trepresent time and anti-time orderings. The sub-indices label the first and the second copy, which determines the relative time-ordering of the operators. Explicitly, the linear response gives ⟨O(x)⟩=iZ dx′{⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩}F q(x′)
+O(F2). (2.29) or equivalently ⟨O(x)⟩=Z dx′GR(x, x′)F x′
, (2.30) with the retarded Green’s function given by GR(x, x′) =iθ(t−t′)⟨[O(x),O(x′)]⟩ =i(⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩). (2.31) Therefore, the exact form of the damping term in the equation of motion would depend on the detailed structure of these retarded response functions. – 5 –3 Gilbert damping term from EFT 3.1 Coupling at the leading order The composite operators Or(X)that encapsulate the external sector transform under arbitrary representations (labeled by r), provided they form invariants of the unbroken SO(2)with the magnon χaand derivatives. In order to achieve this, the operators have to be dressed with the broken SO(3)/SO(2)subgroup parametrized by the Goldstones TR(χ)in the corresponding representation ˜Or(X)≡ Rr(χ)Or(X), (3.1) such that they transform covariantly under the unbroken subgroup [16]. In the long wavelength regime, the theory is organized by a spatial derivative expansion. In fact, the simplest invariant operator at zeroth-order in the derivative expansion is the singlet aligned along the ground state orientation ˆx3 Sint=Z d4x˜O3(X)≡Z d4xˆx3·˜⃗O(X), (3.2) (note that we are adopting manifestly relativistic notations for spacetime and energy-momentum for convenience, albeit the system may or may not be relativistic). Equivalently, we may write Sint=Z d4xˆn·⃗O(X) (3.3) where ˆn=O(χ)ˆx3as defined previously. At the same order in this expansion, there could be more operators that can be added, such as when the operator is a two-index tensor operator O(2)and we may have combinations of the form ˆn· O(2)·ˆn. However, as will be explained in the next subsection, due to the constraint ˆn2= 1, these will not lead to any new contributions, and from the perspective of the EFT, they are redundant operators. Hence, Eq. (3.3) is the only non-trivial operator at this order. Furthermore, we observe that the form of the operator restricts the possible external sector that can couple in this way. For example, one can form such operators from fermions, where ⃗O(ψ) =ψ†⃗ σψ+ψ⃗ σψ†, (3.4) where ⃗ σare the Pauli matrices acting as the intertwiner between the spinor and SO(3)spin space. On the other hand, phonons cannot form operators in this form, and hence will not contribute in this way. 3.2 Gilbert damping The undamped equation of motion is derived from the continuity equation ∂µ⃗Jµ= 0 (3.5) corresponding to the spin rotation transformation δˆn=⃗ ω׈n. To derive the non-conservative equation of motion, we need to find out how the additional term Eq. (3.3) affects the continuity equation. – 6 –The spin rotation transformation on ˆnalone is itself a valid symmetry for the pure magnon action Eq. (2.19), but the interaction term Eq. (3.3) is not invariant if we keep the external sector fixed. A standard trick of Noether theorem is that for an arbitrary symmetry transformation ϕ(x)7→ϕ(x) +f(x)ϵ, if we promote the global symmetry variation parameter to be an arbitrary local variation ϵ(x), the total variation takes the form δS=−Z d4xJµ∂µϵ, (3.6) such that when ϵis a constant, invariance under the symmetry transformation is guaranteed δS= 0even off-shell (equation of motion is not satisfied). Integrate by parts, we find that on-shell δS=Z d4x(∂µJµ)ϵ. (3.7) However, for an arbitrary ϵ(x), this is also the equation of motion, since δS=Z d4xδS δϕf(x)ϵ(x). (3.8) Therefore, the effect of an additional term in the action ∆Sis adding a term to the current divergence ∂µJµ= 0→∂µJµ+δ∆S δϕf(x) = 0 . (3.9) For the spin rotations, the variation of the pure magnon EFT with local ω(x)is given by δS=Z d4x ∂µ⃗Jµ ·⃗ ω. (3.10) Correspondingly, the addition of Eq. (3.3) leads to the modification δS=Z d4x ∂µ⃗Jµ+ ˆn×⃗O(X) ·⃗ ω. (3.11) Thus, the (non-)continuity equation becomes ∂µ⃗Jµ=−ˆn×⃗O(X). (3.12) When we focus on the measurements of the magnons, the effect of the external sector enters as an in-in expectation valueD ⃗OE in. This may be evaluated via the in-in formalism, and the leading order contribution is given by ⃗ ω·Z d4x∂µ⃗Jµ=−⃗ ω·Z d4xˆn×D ⃗OE in=−⃗ ω·Z d4xˆn×Z d4x′ GR·ˆn′
, (3.13) where GR(t′, ⃗ x′;t, ⃗ x)is the retarded response function of the operator ⃗O. In frequency space, we have GR(t, ⃗ x) =Rd3⃗kdω (2π)4e−iωt+⃗k·⃗ x˜GR(ω,⃗k)and furthermore, using the spectral representation (making the spin space indices explicit temporarily) ˜GAB R(ω,⃗k) =Z∞ −∞dω0 πi ω−ω0+iϵρAB(ω0,⃗k), (3.14) – 7 –we can separate the prefactor using the identity i ω−ω0+iϵ=πδ(ω−ω0) +Pi ω−ω0, (3.15) into a δ-function and a principal part. The dissipative part is captured by the former ˜GAB R,diss (ω,⃗k) =Z∞ −∞dω0δ(ω−ω0)ρAB(ω0,⃗k) =ρAB(ω,⃗k). (3.16) Theindicesofthisspectralfunctionlivesinthespin SO(3)spaceand, forisotropicsystems, should be built from invariant tensors δAB,ϵABC. However, the latter could not neither form a two-index object nor respect parity invariance by itself, so the only symmetry-consistent possibility is ρAB(ω,⃗k) =f(ω,|⃗k|2)δAB, (3.17) with fassumed to be an analytic function of its arguments such that it has a smooth limit as ωgoes to zero. Dissipative dynamics is antisymmetric under time reversal, so it should be odd under the simultaneous transformation ω↔ −ωand(A, B)↔(B, A), meaning that the leading order contribution is given by ρAB(ω,⃗k) =−iCωδAB(3.18) or in real spacetime GAB R,diss (t, ⃗ x) =C∂ ∂tδ(t)δ3(⃗ x)δAB. (3.19) Ccould be understood as a Wilson coefficient in this non-conservative sector. From this, we arrive at the equation ∂µ⃗Jµ=−Cˆn×∂ ∂tˆn. (3.20) Combining with the conservative part of the continuity equation Eq. (2.23), we find the Gilbert damping equation ∂ ∂t⃗ m=−γ ⃗ m×∂ ∂t⃗ m+. . . , (3.21) where γ=C/(c2ms),msˆn=⃗ m(for uniform materials), and the higher-order terms on the right- hand side of the original equation of motion Eq. (2.23) are contained in . . .which will be omitted in the following. When we have another singlet of the form ˆn·O(2)·ˆn, the effect of the extra ˆns is a replacement ofthespectraldensity ρAB→ρACBDˆnCˆnD. ThetensorbasisstillconsistsofKroneckerdeltasince the only structure that Levi-Civita tensors could contract to ˆnand would vanish automatically. Therefore, any additional structures will appear in the form of the inner product ˆn·ˆndue to contractions with the Kronecker delta. Consequently, they do not lead to anything new due to the normalization condition ˆn2= 1. We see that using the in-in formalism, the form of the damping equation and the coeffi- cients are completely fixed by the principles of EFT: the symmetries, power counting, and Wilson coefficients. – 8 –4 More general materials By choosing the spectral function to depend only on invariant tensors and the frequency, we naturally arrive at the Gilbert damping Eq. (3.21) which applies to isotropic and homogeneous systems. However, in generic materials, we may be interested in situations with more general materials which for instance have non-trivial spin textures or highly anisotropic lattices and thus inhomogeneous or anisotropic. The advantage of the EFT framework is that these generalizations can be systematically incorporated by including additional couplings. In this section, we explore several possibilities along these lines. 4.1 Anisotropic materials For homogeneous systems, one can still have anisotropy due to a background field. The retarded response function can then depend on the background. Given a homogeneous background vector field⃗heff, the Levi-Civita tensor can now be incorporated into the response ρAB=DϵAB ChC eff. Since dissipative effects need to be antisymmetric under the simultaneous transformation A↔B, ω↔ − ωandA↔Bantisymmetry is already included in the Levi-Civita tensor structure, the response function has to be symmetric under ω↔ −ω. This means that the leading order contribution is now independent of ω.1The corresponding response function is GA,B R,diss(t, ⃗ x) =DϵABChC effδ(t)δ3(⃗ x) (4.1) and gives rise to the conservation equation ∂ ∂t⃗ m=−λ⃗ m×(⃗ m×⃗heff). (4.2) For systems with conserved parity, the external field ⃗heffis an effective magnetic field in the sense of being parity odd to ensure an even-parity response function. This damping equation involving an effective external background magnetic field is known as the Landau-Lifshitz damping equation [8]. We observe that from the EFT point of view, the Landau-Lifshitz and Gilbert damping terms are distinguished by symmetries. For completeness, we note that in the literature there are generalizations to Gilbert damping by introducing anisotropic damping tensors. In field theoretic language, anisotropy corresponds to explicitly breaking of SO(3)and is thus straightforwardly realized in the EFT by a spurion condensation. Instead of introducing a symmetry-breaking VEV to an explicit spurion operator in the action, one may assume the 2-point functions acquire a VEV and the most general resulting dissipative response function at LO is given by GA,B R,diss(ω) =SABω+AAB, (4.3) where SABandAABare general symmetric and antisymmetric tensors which are not SO(3) invariant. Thegeneralizeddampingequationforananisotropicbuthomogeneousmagneticsystem then become∂ ∂t⃗ m=−⃗ m×S·∂ ∂t⃗ m+⃗ m×A·⃗ m. (4.4) 1This is analogous to the dissipative EFT of a spinning black hole in which case the role of this background is played by the direction of the spin vector [17]. – 9 –The exact form of these anisotropy tensors can then be extracted from the microscopic details of the given full theory. 4.2 Inhomogeneous materials For inhomogenous materials, e.g. configurations with background spin textures, we may have contributions from higher orders in (spatial) derivative expansion. The simplest possible operator is given by the coupling Sint=Z d4x∂iχa˜Oai(X). (4.5) In terms of the orientation vector, this is equivalent to Sint≈Z d4x(ˆn×∂iˆn)·⃗Oi(X), (4.6) at leading order in χ-field. Again promoting the spin rotation transformation δˆn=⃗ ω׈nto a local parameter ⃗ ω(x), we find an additional contribution to the current divergence δSint=Z d4x⃗ ω·(2∂iˆnˆn+ (ˆnˆn−δ)·∂i)·⃗Oi, (4.7) where we have used the fact that ˆn·∂iˆn=1 2∂iˆn2= 0to simplify the expression. For the spectral function of the form ρiA,jB(ω) =EωδijδAB, (4.8) this leads to the damping term c2∂tˆn=E 2∂iˆnˆn·(∂tˆn×∂iˆn) + (ˆnˆn−δ)·∂t ˆn×⃗∇2ˆn =E 2ˆn×(ˆn×∂iˆn) (ˆn×∂iˆn)·∂tˆn+ (ˆnˆn−δ)·∂t ˆn×⃗∇2ˆn , (4.9) where we have used that ˆn×(ˆn×∂iˆn) =−∂iˆnon the first term. More compactly, this is ∂ ∂t⃗ m=⃗ m×A·∂ ∂t⃗ m+E(ˆnˆn−δ) c2ms·∂t ⃗ m×⃗∇2⃗ m , (4.10) where the first term contains the generalized damping tensor A=2E c2ms(⃗ m×∂i⃗ m) (⃗ m×∂i⃗ m). (4.11) This corresponds to the generalized Gilbert damping in the presence of non-trivial spin textures (i.e. when ∇⃗ m̸= 0) [18]. We note that Eq. (4.6) is only the leading order derivative correction to the damping dynam- ics. The EFT framework is capable of systematically generating higher derivative corrections. For example, other types of inhomogeneity may be attributed to interactions in the lattice model [19] by ⃗ mi×X ijGij·⃗ mj, (4.12) – 10 –where the i, jindices label the lattice sites associated with the magnetic moments. In the contin- uum field theory, the lattice variables become ⃗ mi7→⃗ m(⃗ xi)and the tensorial structure becomes the response function Gij7→GR,diss (t, ⃗ xi−⃗ xj), except that unlike the one in Eq. (3.19), it is non-local (no longer proportional to δ(⃗ x)). In the simpler case that the long-range coupling falls off sufficiently quickly, these terms are traded for a series expansion ⃗ m×X nAi1...in∂n i1...in∂t⃗ m, (4.13) for some coefficient tensors Ai1...in. In terms of the action, this means the spectral functions are now dependent on the wave vectors ρAB(ω,⃗k) =X n˜Ai1...inki1. . . k inωδAB. (4.14) 5 Magnon damping term from phonons From the magnetoelastic EFT, the generic magnon-phonon couplings are given by [1] Lph=−1 2Fij 2(∇(iπj))∂iˆn·∂jˆn+1 2ρ˜F3(∇(iπj))Dtˆn·Dtˆn, (5.1) where Dt≡∂t+vi∂iwith the velocity of the material given by vi=−∂tϕ(D−1)i j. The "full theory" (technically the EFT at the next level of the hierarchy) action constrains the form of the couplings between magnons the external sector to be Lph=1 2∂iˆn·∂jˆnOij 2(π) +1 2∂tˆn·∂tˆnO3(π) +1 2∂tˆn·∂iˆnOi 4(π), (5.2) where the last term arises from the linear-in- vcontribution in the expansion Dtˆn·Dtˆn. For ferromagnets, the dispersion relation dictates the first term to be dominant. After inte- grating out the external sector using the Schwinger-Keldysh methodR DπD˜π, we find its contri- bution to the in-in equation of motion is given by c2∂tˆn= ˆn×∂i ∂jˆnD Oij 2E , (5.3) where the in-in expectation value is given by D Oij 2(x)E =Z d4x′Gij,kl R,2(x−x′)∂kˆn(x′)·∂lˆn(x′). (5.4) For 4-index tensor structures under SO(3), there are two invariant tensors corresponding to the symmetric-traceless and trace irreps. Therefore, one may write the leading-order dissipative con- tribution to the retarded response function as Gij,kl R,2(x)≃1 2δijδklC2+δi(kδl)jD2 δ3(⃗ x)∂tδ(t), (5.5) where C2andD2are the independent (Wilson) coefficients. – 11 –Substituting the results, we find the damping equation in a similar form c2∂tˆn=1 2C2ˆn×∂i ∂iˆn∂t ∂jˆn·∂jˆn

+D2ˆn×∂i ∂jˆn∂t ∂iˆn·∂jˆn

=C2ˆn×∂i ∂iˆn∂jˆn·∂j∂tˆn
+D2ˆn×∂i ∂jˆn∂iˆn·∂j∂tˆn
+D2ˆn×∂j ∂iˆn∂iˆn·∂j∂tˆn
,(5.6) or more compactly ∂ ∂t⃗ m=⃗ m×D·∂ ∂t⃗ m, (5.7) where the "damping tensor" Dis given by D=1 c2msh ∂i C2∂iˆn∂jˆn+D2∂jˆn∂iˆn
+D2∂j ∂iˆn∂iˆn
+ C2∂iˆn∂jˆn+D2∂jˆn∂iˆn
∂i+D2∂iˆn∂iˆn∂ji ∂j. (5.8) We notice that the form of the couplings restricts the damping tensor to appear at higher- orders in derivative expansions and hence they are expected to be small compared with contribu- tions from fermions (e.g. electrons) in the long wavelength limit. However, for insulating materials that have electron-magnon coupling suppressed, we expect their effects to be more significant. 6 Conclusion and discussions In this paper, we used the in-in (Schwinger-Keldysh) formalism to generalize the recently con- structed EFT of magnetoelasticity [1] to describe damped magnetic dynamics. We discover that the Gilbert damping term naturally arises as the simplest symmetry consistent dissipative cor- rection within the in-in formalism. Systematic generalizations to anisotropic and inhomogeneous setups also yield desired results such as the Landau-Lifshitz magnetic damping equation. More- over, we are able to predict the form of phononic contribution to the damping dynamics. Thus we have shown that this is a useful framework to derive dissipative dynamics from first principles and to predict the forms of higher-order corrections in a systematic way. It would be interesting to investigate the explicit full theory model of the “external sector” such as the fermionic fields in Eq. (3.4) and extract the relevant Wilson coefficients by matching the response functions. In this way, we may gain better insights into what controls the damping parameter and give more predictive power to the EFT approach. It would also be interesting to match the relevant coefficients in Eq. (4.14) to obtain an EFT framework for a generalized class of models. Furthermore, various applications of the magnetoelastic EFT [20, 21] have appeared more recently. It would be interesting to investigate the effects of adding dissipative terms into these problem. There are also further developments in the technical aspects of such EFTs [22, 23]. It is natural to consider their implications on the non-conservative sector. We leave these problems for future works. Acknowledgement The author thanks Ira Rothstein for advising throughout the project and a careful reading of the manuscript. The author also thanks Riccardo Penco for important discussions, Shashin Pavaskar – 12 –for other useful discussions, and Witold Skiba for comments on the draft. This work is partially supported by the grants DE- FG02-04ER41338 and FG02- 06ER41449. References [1] S. Pavaskar, R. Penco, I. Z. Rothstein, An Effective Field Theory of Magneto-Elasticity , SciPost Phys.12.5.155 (2022), arXiv:2112.13873 [hep-th]. [2] S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological Lagrangians. 1. , Phys.Rev. 177 2239 (1969). [3] J. Callan, Curtis G., S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological Lagrangians. 2. , Phys.Rev. 177 2247 (1969). [4] D. V. Volkov, Phenomenological Lagrangians , Fiz. Elem. Chast. Atom. Yadra 4 3 (1973). [5] V. I. Ogievetsky, Nonlinear Realizations of Internal and Space-time Symmetries , Proc. of. X-th Winter. School of Theoretical Physics in Karpacz, Vol. 1, Wroclaw 227 (1974) . [6] M. Baumgart et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter Systems, in2021 Snowmass Summer Study. 10, 2022, arXiv:2210.03199[hep-ph]. [7] T. Brauner et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter Systems , in2022 Snowmass Summer Study. 3, 2022„ arXiv:2203.10110[hep-th]. [8] L. D. Landau and E. M. Lifshitz, Theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowjetunion. 8, 153 (1935). [9] T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials , IEEE Transactions on Magnetics, vol. 40, no. 6, (2004). [10] J. Schwinger, Brownian Motion of a Quantum Oscillator , J. Math. Phys. 2 407 (1961). [11] L. V. Keldysh, Diagram technique for nonequilibrium processes , Zh. Eksp. Teor. Fiz. 47 1515 (1964). [12] C. R. Galley, The classical mechanics of non-conservative systems , Phys. Rev. Lett. 110, 174301 (2013), arXiv:1210.2745 [gr-qc]. [13] C. R. Galley, D. Tsang, and L. C. Stein, The principle of stationary nonconservative action for classical mechanics and field theories (2014), arXiv:1412.3082 [math-ph]. [14] S. Endlich, R. Penco, An effective field theory approach to tidal dynamics of spinning astrophysical systems, Phys. Rev. D. 93.064021 (2016), arXiv:1510.08889 [gr-qc]. [15] W. D. Goldberger and I. Z. Rothstein, Horizon radiation reaction forces , JHEP 10 026 (2020), arXiv:2007.00731[hep-th]. [16] L. V. Delacrétaz, S. Endlich, A. Monin, R. Penco, F. Riva, (Re-)Inventing the Relativistic Wheel: Gravity, Cosets, and Spinning Objects , JHEP 11 (2014) 008, arXiv:1405.7384 [hep-th]. [17] W. D. Goldberger, J. Li, and I. Z. Rothstein, Non-conservative effects on spinning black holes from world-line effective field theory , JHEP 06 053 (2021) arXiv:2012.14869[hep-th]. [18] S. Zhang, S. S.-L. Zhang, Generalization of the Landau-Lifshitz-Gilbert Equation for Conducting Ferromagnets , Phys. Rev. Lett. 102, 086601 (2009). [19] S. Brinker, M. dos Santos Dias, S. Lounis, Generalization of the Landau-Lifshitz-Gilbert equation by multi-body contributions to Gilbert damping for non-collinear magnets , J. Phys.: Condens. Matter 34 285802 (2022), arXiv:2202.06154 [cond-mat.mtrl-sci]. – 13 –[20] A. Esposito, S. Pavaskar, Optimal anti-ferromagnets for light dark matter detection (2022), arXiv:2210.13516 [hep-ph]. [21] S. Pavaskar, I. Z. Rothstein, The Dynamics of Line Defects and Their Sensitivity to the Lattice Structure (2022), arXiv:2212.10587 [hep-th]. [22] A. Nicolis, I. Z. Rothstein, Apparent Fine Tunings for Field Theories with Broken Space-Time Symmetries (2022), arXiv:2212.08976 [hep-th]. [23] C. O. Akyuz, G. 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2023-12-20

Using the in-in formalism, we generalize the recently constructed magnetoelastic EFT arXiv:2112.13873 [hep-th] to describe the damping dynamics of ferromagnetic systems at long wavelengths. We find that the standard Gilbert damping term naturally arises as the simplest leading-order symmetry-consistent non-conservative contribution within the in-in framework. The EFT is easily generalized to scenarios with anisotropy and inhomogeneity. In particular, we find the classic Landau-Lifshitz damping term emerges when isotropy is broken by a constant external background field. This provides a first principle explanation for distinguishing the two types of damping dynamics that were originally constructed phenomenologically. Furthermore, the EFT framework could also incorporate intrinsic anisotropy of the material in a straightforward way using the spurion method. For systems with inhomogeneity such as nontrivial spin textures, we find that the leading order derivative correction yields the generalized Gilbert damping equations that were found in condensed matter literature. This shows that the EFT approach enables us to derive the form of higher-derivative-order corrections in a systematic way. Lastly, using the phonon-magnon coupling deduced in the magnetoelastic EFT, we are able to make a prediction for the generic form of the phononic contribution to the damping equation.

An effective field theory of damped ferromagnetic systems

2312.13093v1

Magnetic lump motion in saturated ferromagnetic films Xin-Wei Jin,1, 2Shi-Jie Shen,2Zhan-Ying Yang,1, 3and Ji Lin2,∗ 1School of Physics, Northwest University, Xi’an 710127, China 2Department of Physics, Zhejiang Normal University, Jinhua 321004, China 3Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China (Dated: July 4, 2023) In this paper, we study in detail the nonlinear propagation of magnetic soliton in a ferromagnetic film. The sample is magnetized to saturation by an external field perpendicular to film plane. A new generalized (2+1)- dimensional short-wave asymptotic model is derived. The bilinear-like forms of this equation are constructed and exact magnetic line soliton solutions are exhibited. It is observed that a series of stable lumps can be generated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable and can maintain their shapes and velocities during evolution or collision. The interaction between lump and magnetic soliton, as well as interaction between two lumps, are numerically investigated. We further discuss the nonlinear motion of lumps in ferrites with Gilbert-damping and inhomogeneous exchange effects. The results show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay exponentially during propagation. And the shock waves are generated from a lump when quenching the strength of inhomogeneous exchange. I. INTRODUCTION The propagation of electromagnetic wave in ordered magnetic materials, especially in a ferromagnetic medium, plays a vital role in faster and higher density storage fields [1– 3]. In particular, magnetic soliton(MS), which exists in both ferro- and antiferro-magnets, is becoming a very promising information carrier because of its particle-like behavior and maneuverability [4–9]. In the past few decades, a wide range of soliton-type propagation phenomena has been theoretically predicted [10–13], and some of them have been confirmed experimentally [14, 15]. Indeed, wave propagation in ferromagnetic media is well- known as a highly nonlinear problem. A complete description of all types of nonlinear excitations is governed by the Maxwell equations coupled with Landau-Lifschitz equation. For this moment, let us notice that a fully nonlinear theory has not been developed. But the linear theory for sufficiently small amplitudes was established and validated experimentally [16]. In order to obtain results valid in nonlinear regimes, or at least weakly nonlinear, one has to resort to intermediate models (by introducing a small perturbative parameter related to the soliton wavelength) [17]. These models include long- wave model [18–20], modulational asymptotic model [21], and short-wave model [22–25]. Both long-wave model and modulational asymptotic model are mainly used to explain and predict the behavior of large-scale phenomena owing to their long-wave-type approximate condition [26]. However, this condition is not always applicable because the scale of magnetic materials and devices are getting more refined and more sophisticated. Moreover, the main practical interest of ferrites is that they propagate microwaves [27, 28]. On the contrary, from the viewpoint of applied physics, the short- wave-type approximation is much more relevant to available experiments than the former one. Since Kraenkel et al. first proposed the short-wave model [29], quite a few related nonlinear evolution equations have been derived, which belong to the Kraenkel-Manna-Merle (KMM) system [22, 23, 30–32]. Some significant works ∗Corresponding author: linji@zjnu.edu.cnhave been devoted to searching and explaining different excitation patterns of ferromagnetic insulators. As for (1+1)- dimensional KMM system, the existence of multi-valued waveguide channel solutions has been verified, and the nonlinear interaction properties were investigated between the localized waves alongside the depiction of their energy densities [22]. By applying the Hirota bilinear transformation method, the one- and two-soliton solutions were constructed while studying in details the solitons scattering properties [23]. This system is also solvable using the inverse scattering method [25]. It is noteworthy that this system possesses the loop-soliton and spike-like soliton [33, 34], and the magnetic loop-soliton dynamics have been extensively studied [35–37]. The propagation of electromagnetic waves in higher-dimensional ideal ferromagnets has also been studied, corresponding to the (2+1)-dimensional KMM system [26, 31, 38, 39]. The analytical one-line-soliton solution as well as its transverse stability have been reported [26]. It has been shown that these structures were stable under certain conditions. On the other hand, most previous studies have only focused on the propagation of MS in ideal ferrites, which means some important properties of the magnetic material were neglected. The main reason is that the nonlinear wave equation describing the propagation of electromagnetic waves in non-ideal ferromagnetic materials is no longer integrable. However, the Gilbert-damping and inhomogeneous exchange effects are essential features in a real ferromagnetic film, and their connection with MS motion is an important issue that has not been explored so far. In this paper, we aim to investigate theoretically and numerically the dynamics of the MS in a ferromagnetic film including damping and the inhomogeneous exchange effect. The rest of this paper is organized as follows. In Section 2, we review the physical background and derive a new (2+1)-dimensional short-wave asymptotic model in ferromagnetic media. In Section 3, the bilinear-like form of the reduced system is constructed and the analytical MS solutions are acquired. In Section 4, the transmission stability of the magnetic soliton is numerically explored. The results show that an unstable MS will split to some magnetic lumps by a small perturbation. The motions of these lumps under the influence of damping and inhomogeneous exchange are analysed in detail. We end this work in Section 5 with a brief conclusion and perspectives.arXiv:2307.00903v1 [nlin.PS] 3 Jul 20232 II. PHYSICAL BACKGROUND A. Basic equations The physical system under consideration is a saturated magnetized ferrite film lying in the x−yplane, as shown in Fig. 1. Different from Ref. [32], we consider the external field H∞ 0perpendicular to the film, i.e., M0= (0,0,m). So the transverse drift is avoided. The typical thickness of the film is about 0.5mm, and the width is approximately 10mm. We assume the propagation distance is large enough with regard to the wavelength, say more than 50cm. The evolution of the magnetic field Hand the magnetization density Mis governed by the Maxwell equations coupled with Landau-Lifschitz- Gilbert equation, which read as −∇(∇·H)+∆H=1 c2∂2 ∂t2(H+M), (1a) ∂ ∂tM=−γµ0M×Heff+σ MsM×∂ ∂tM, (1b) where c=1/p µ0˜εis the speed of light with the scalar permittivity ˜εof the medium, γis the gyromagnetic ratio, µ0being the magnetic permeability of the vacuum, σis the damping constant, and Msis the saturation magnetization. The effective field Heffis given by [30] Heff=H−βn(n·M)+α∆M. (2) Here αandβare the constants of the inhomogeneous Figure 1. Ferrite film under consideration. The sample is magnetized to saturation by long strong magnetic field H∞ 0applied in the z-direction. The x-direction of the short wave propagation is perpendicular to the direction of static magnetization. exchange and the magnet anisotropy ( β>0 corresponds to the easy-plane case), respectively. For a simple tractability, the unit vector nof the anisotropy axis is assumed to be along the zaxis (i.e., n≡ez). In order to transform the above systems to dimensionless equation, we rescale the quantities M,H, and t intoµ0γM/c,µ0γH/c, and ct. Thus, the constants µ0γ/cand cin Eqs.(2) and (3) are replaced by 1, Msbym=µ0γMs/c, andσby˜σ=σ/µ0γ[30]. B. Linear analysis To study the linear regime we look at a small perturbation of a given solution. Equations (1) are linearized about the steady state: M0= (0,0,m),H0=µM0. (3)where µis the strength of the internal magnetic field. Before proceeding further we assume that the ferromagnetic materials have weak damping ¯σ∼ε˜σ. The exchange interaction parameter αand anisotropy parameter βare of order ε2and ε3, respectively (i.e. ¯α=ε2α,¯β=ε3β). Let us seek for the plane wave perturbation solution propagating along the x- direction such as M=M0+εmexp[i(kx+ly−ωt)], H=H0+εhexp[i(kx+ly−ωt)],(4) where kandlare the wave numbers in the xandydirections, ω is the frequency. Vectors m= (mx,my,mz)andh= (hx,hy,hz) are arbitrary real scalar quantities. Substituting Eq. (4) into (1) and (2) in the linear limit, it is reduced to ω20 0 ω2−l2kl 0 0 ω20 kl ω2−k20 0 0 ω20 0 ω2−k2−l2 −iωmµ 0 0 −m 0 −mµ−iω 0 m 0 0 0 0 −iω 0 0 0 · mx my mz hx hy hz =0 Then we obtain the following dispersion relation m2(µ+1) µ(k2+l2−ω2)−ω2 −ω2(k2+l2−ω2) =0 (5) Note that we focus on studying the short-wave approximation k→∞[2]. It comes k0∼ε−1through a small parameter ε≪1 linked to the magnitude of the wavelength. Consequently, the frequency expands accordingly as ω=ω−1ε−1+ω1ε+ω3ε3+.... (6) This assumption guarantees the phase velocity ω(k)/kand the group velocity ∂ω/∂kare always bounded [3]. Now, replacing Eq. (6) into the dispersion relation above, we obtain a set of equations: •At order of ε−4:ω−1=±k0 •At order of ε−2:ω1= (µ+1)m2+l2 /2k0 •higher order equations which determines ω3,ω5,... The direction of the wave propagation is assumed to be close to the xaxis, thus yvariable gives only account of a slow transverse deviation[40, 41]. Therefore lis assumed to be very small with respect to kand we write l=l0of order 0 with respect to ε. The phase up to order εis thus (x−t)/ε+l0y−εω1t,which motivates the introduction of new variables: ζ=1 ε(x−Vt),y=y,τ=εt. (7) The variable ζdescribes the shape of the wave propagating at speed V; it assumes a short wavelength about 1 /ε. The slow time variable τaccounts for the propagation during very long time on very large distances with regard to the wavelength. The transverse variable yhas an intermediate scale, as in KP- type expansions [26, 41] C. Multiple scale approach In order to derive the nonlinear model, fields MandHare expanded in power series of εas M=M0+εM1+ε2M2+ε3M3+..., H=H0+εH1+ε2H2+ε3H3+....(8)3 where M0,H0,M1,H1,...are functions of (ζ,y,τ). We consider the boundary conditions: lim ζ→−∞M0= (0,0,m),lim ζ→−∞Mj=lim ζ→−∞Hj=0,(j̸=0). We derive the following expressions by substituting Expansions (8) into equation (1): •At order ε−2: M0is a constant vector M0=(0,0,m), •At order ε−1: Hx 0=0,My 1=0,Mz 1=0, •At order ε0: Mx 1ζ=mHy 0, Mx 2ζζ=−Hx 2ζζ−Hy 1ζτ My 2ζζ=−Hx 1ζy+Hx 0ζy Mz 2ζζ=Hz 2ζτ+Hz yy •At order ε1: Mx 2ζ=−mHy 1 My 2ζ=m¯αMx 1ζζ+¯σM1ζx−Mx 1Hz 0+mHx 1 Mz 2ζ=Mx 1Hy 0 let us introduce some independent variables XandTdefined asX=−mζ/2,Y=my,T=mτ. After eliminating H2andM2, we finally obtain the (2+1)- dimensional KMM equation: CXT=−BBX+CYY, BXT=BCX+BYY−sBX+ρBXX,(9) where observables B,Cand constants s,ρare defined by C=−X−ZX (Hz 0/m)dX,B=Mx 1/2m, s=−¯σ/2,ρ=¯αm2/4.(10) This equation is new, which describes the evolution of magnetization field Mand magnetic field Hwithin a ferrite film in presence of Gilbert-damping and inhomogeneous exchange. The quantities H0andM1refer to the zeroth and first-order expansion coefficients of the external magnetic field and the magnetization, respectively. For some simplicity, in the next, the independent variables X,YandTwill be rewritten as their lower cases x,yandt, respectively. III. HIROTA’S BILINEARIZATION AND SOLITON SOLUTIONS OF THE (2+1)-DIMENSIONAL KMM EQUATION To explore soliton solutions for the (2+1)-dimensional KMM equation (9), we consider a specific dependent variable transformation B=G F,C=δx−2(lnF)t−2(lnF)y, (11) where δis an arbitrary constant. Consequently, the bilinear- like forms of the (2+1)-dimensional KMM equation can be derived as follow F·(DxDt+sDx−D2 y)G·F+G·(DxDy+D2 y)F·F=δF2G (12a) ∂xG2 2F2−(DyDt+D2 t)F·F F2 +∂y(DyDt+D2 t)F·F F2 =0 (12b)where G,Fare all differential functions of (x,y,t)to be determined. The symbols Dx,Dtrefer to the Hirota’s operators with respect to the variable x,t, respectively. In order to construct the solitary wave solutions of Eq.(6), we expand GandFwith respect to a formal expansion parameter as G=εG1+ε3G3+ε5G5+...,F=1+ε2F2+ε4F4+ε6F6+..., in which εis a perturbation parameter and functions Gi,Fi,(i= 1,2,3,...)are expansion coefficients of the above series. The one-soliton solution could be constructed by truncating the perturbation expansion of GandFas follow G=eη1,F=1+k2A2 16δ2e2η1. (13) Substituting these expressions into Eq.(9) and solving the bilinear system recursively, in the absence of damping, the analytical one-soliton solution of the (2+1)-dimensional KMM equation can be transformed into B=2δ ksech(η1+η0),C=δx−2δ k[tanh(η1+η0)+1], (14) where η1=kx+ly+ [(l2−kl)/2k]t,η0=ln(k/4δ),kandl are arbitrary real constants. It should be noted that this soliton solution exists only when the damping is neglected (s=0). Similar to the procedure for constructing one-soliton solution, the two-soliton solution can be given by treating the truncated perturbation expansions of GandFas G=A1eξ1+A2eξ2+C12eξ1+2ξ2+C21e2ξ1+ξ2, (15a) F=1+B11e2ξ1+B22e2ξ2+B12eξ1+ξ2+E12e2ξ1+2ξ2,(15b) where A1,A2,k1,k2are real constants, ξi=kix+liy+ (l2 i+δ)/ki t,(i=1,2), and the remaining parameters have the following forms: Bii=A2 ik2 i 16δ2,B12=A1A2 2δ2k2 1k2 2 k2+,k1l2=k2l1, Ci j=AiA2 j 16δ2k2 jk2 − k2+,E12=A2 1A2 2 256δ4k2 1k2 2k4 − k4+,(16) where k+=k1+k2,k−=k1−k2. Parameters Ai,Aj,ki,kj andli,(i=1,2,j=3−i)are arbitrary real constants. IV . NUMERICAL INVESTIGATION OF LINE-SOLITON AND MAGNETIC LUMPS A. Unstable MS splits into lumps We now turn to the stability and interactions between MSs in a ferromagnetic film. The initial data is a MS perturbed by some position-dependent Gaussian wave packets with the following expression: f=bexp" −x−x0 xr2 −y yr2# , (17) where b,xrandyrcorrespond to the shape of the wave packet andx0is related to the perturbation position. The time evolution results clearly show the instability of the MS. For small bi, the MS will break up and eventually4 (a) (b) (c) (d) Figure 2. Propagation of MS perturbed by a Gaussian disturbance. (a) Component Hz, (b) Component Hy, (c) and (d) are enlarged views of the indicated areas circled in red and black, respectively. The parameters are chosen as A1=A2=1,δ=−1,l1=l2=0,k1= 1,k2=2,x0=−29,b=0.1,xr=1.5,yr=2.5 in (16) and (17). evolve into some stable two-dimensionally localized lumps , as displayed in Figs. 2(a) and 2(b). We observe that most of the energy is always propagated as a lump, even if its speed may differ from the input. Such a magnetic lump is a solitary wave packet that maintains its shape and speed during propagation or collision. A complete single lump of magnetic field component Hz (component Hy) is circled in red (black) in Fig.2. The enlarged views (see Figs.2(c) and 2(d)) provide a clear picture of the shape and contour map of the lump. It can be found that component Hzis a dipole-mode lump, whereas component Hyis a standard KP-lump. We also show the vector field of the magnetic lump in Fig.3(a). Note that magnetic field component Hxis zero, the magnetic field is always in the y−z plane, hence the lump can be regarded as a 360◦domain wall localized in xandydirections. Fig.3(b) presents the magnetic field along y=0. The blue and red arrows correspond to the magnetic field intensity of component Hz,Hy, respectively. The rest of this work is concerned with the propagation and interaction behavior of these lumps in ferrite medium. (a) (b) Figure 3. (a) The vector field of the magnetic lump. (b) The magnetic lump along y=0. The blue and red arrows correspond to the magnetic field intensity of components Hz,Hy, respectively.B. Lump motion in ferromagnets with damping or inhomogeneous exchange effects Figure 4. Three dimensional projections of lump at t=0,HandW represent the definitions of lump height and width, respectively. The evolution behavior of the magnetic lump in the ideal ferrite is quite simple and imaginable. Each lump maintains its shape while it travels at a constant speed. However, in most of real ferromagnetic materials, we have to take the Gilbert- damping into account . For instance, the dimensionless damping constant sranges from 0.048 to about 0.385 in garnet ferrite films. Here we are going to study the dynamics of magnetic lump in a damped ferrite film. The typical ferromagnetic film under consideration is a garnet ferrite film with the dimensionless damping constant s=0.1. For a clearer view of the change in shape of the lump, we define HandWas the height and width of the lump, which are the vertical distance between the highest point and the lowest point and the horizontal distance along the propagation direction, respectively. All of these are summarized in Fig.4. The propagation of a lump on the garnet ferrite film is presented in Fig.5. As shown in Fig.5(a), the lump travels forward a visible distance in the damped ferrite. Beyond that, comparing the profiles of lump between t=0 and t=10, we evidently observe that the lump becomes smaller and narrower. Fig.5(b) shows the lump height and width exhibit a tendency of exponential decay. The solid blue line is the exponential fitting curve to H(t), with the function expression being H(t) = A0e−st. We confirm the above-mentioned amplitude attenuation law is universal by simulating the motion of lump in ferrites with virous damping factors. Moreover, a definite relationship between the amplitude and the localization region of solitons is important for the soliton excitations. We analyze different sizes of numerical lumps and mark the width and height of lumps in the phase diagram (see Fig. 5(c)). The results show that for a magnetic lump excitation, its width and height meet a linear relationship within the error range ( W/H∼0.305). So the lump excitation, upon decay, retains a soliton form. Therefore, in this system, the Gilbert-damping plays a role of dissipating energy during the motion of magnetic lumps and it is characterized by decreasing the amplitude and width of lump. The inhomogeneities otherwise referred to as deformities is inevitable in real magnetic materials, and it can be caused by either external fields or the presence of defects, voids and gaps in the material. It has already been reported that the MS may be deformed by the presence of inhomogeneities, in particular5 (a) (b) (c) Figure 5. Evolution of a magnetic lump in a damped ferrite film with dimensionless damping constant s=0.1. (a) Comparison picture of lump wave at t=0 and t=10. (b) The variation of lump height H, lump width Wand velocity V . (c) Numerical relationship between the width and height of magnetic lump. its structure and speed [35, 42]. In this present system, the inhomogeneous exchange process is unignorable when the wavelength of lump is comparable to the characteristic exchange length. (a) (b) (c) (d) Figure 6. Propagation of lump with and without the inhomogeneous interaction, respectively. We now move to study the lump motion in the presence of inhomogeneous exchange effect. The initial data is the stable magnetic lump shown in Fig.5. As can be observed from Fig. 6(a) and 6(b), in ferrite without exchange interaction, the lump solution propagates at a constant speed and along the previous path. We then consider the non-equilibrium dynamics of lump by performing a sudden interaction quench. The pictures of component Hyat dimensionless times t=2 and t=4.5 are shown in Fig. 6(c) and 6(d). As we see, for a quench from the non-interacting to strong inhomogeneous exchange ferrite film, the lump oscillates rapidly and diffracts alongthe propagation direction. A two-dimensional shock wave is generated and propagates forward. The shock wave front continues to propagate in the negative direction along x-axis. Finally, the energy of lump will be dissipated into numberless tiny waves. Accordingly, considering that the lump would be destroyed by the inhomogeneous exchange process, one has to consider keeping its wavelength away from the characteristic exchange length in the lump-based microwave applications. C. Some examples of excitations and interactions The evolution pattern given in Fig.2 reveals that the lump moves at a larger velocity than the broken MS in the propagation. The reason is that the velocity of soliton solution is proportional to the soliton amplitude. During the formation of the lump, the original MS will be destroyed, and most of the energy is concentrated in some certain centers, which causes the amplitude (and velocity) of the lump to be greater than that of MS. These lumps with various speeds enable us to explore the interaction between lump and soliton, as well as between two lumps. A typical example of lump-MS collision is shown in Fig.7(a). The MS begins to break up around at t=4. Subsequently, the splitting lump is going to catch up and collide with the front-MS. After the collision, the front- MS is destroyed and broken into several lumps with various sizes. It is remarkable that the lump keep its localized form before and after the collision almost unchanged. This phenomenon implies such two-component lumps are natural results from this nonlinear propagation equations. Further simulation shows these lump structures could be generated by a MS with random disturbance. Fig.7(b) depicts a characteristic inelastic collision between two lumps. We initially generate two adjoining lumps. They are emitted by MS at dimensionless time t=6.5. The merging process can be performed as follows. From t=7.5 tot=9.5, two lumps merge simultaneously together and give birth to a new lump whose amplitude is significantly greater than the amplitude of previous lumps. Obviously there is a weak attraction between two lumps which results in their fusion. In addition to the fusion of the two lumps, we also observed an extraordinary peak at a specific moment (about t=9.5), which looks like a6 (a) (b) Figure 7. (a) Collision between lump and MS. (b) Mergence of two lumps and the formation of a second-order rogue wave-like structure. second-order rogue wave. It appears to be the result of the interaction between the ripples surrounding the two lumps. After the fusion, the rouge wave-like structure disappears and the dynamics of the output is determined mainly by a single high-amplitude lump. V . CONCLUSION As a conclusion, the nonlinear propagation of MS in a saturation magnetized ferromagnetic thick film is studied in detail. In the starting point, we derive the (2+1)-dimensional KMM system that governs the evolution of short MS waves in a saturated ferromagnetic film. The bilinear form of the KMM system is constructed and the MS solutions are obtained analytically. After that, numerical simulations are performed to analyse the evolution behaviours of MS. A significant observation is that the unstable MS can be destroyed by Gaussian perturbation and broken into some stable magnetic lumps. These lumps exhibit high stability during the propagation. Furthermore, some examples are given to analyse the collision behaviours between lump and MS, and the interaction between two lumps. It is found the lump keeps its shape and speed in the collision with MS. The results confirm that the lump is astable propagation mode in this system and, more to the point, the velocity of lump can be adjusted by its amplitude. Their robustness and controllability provide the possibility for future information memory and logic devices. We also study the propagation of such a lump in ferrites subjected to influence of damping and inhomogeneous exchange effects. When the Gilbert-damping of ferrite is considered, the lumps undergo the following changes: the amplitude and the speed of lump are decreased, and the width of lump along the propagation direction is getting narrow. It would cause a strong diffraction of the lump if we quench the interaction strength. We hope our work will invoke follow-up experimental studies of lump-based microwave applications. Addition- ally, since only one- and two-line-soliton are obtained, the integrability of the (2+1)-dimensional system Kraenkel- Manna-Merle (KMM) remains an open issue. The existence of the higher-dimensional evolution system as well as the bulk polariton solution is an intriguing avenue for future exploration. ACKNOWLEDGMENT This work was supported by the National Natural Science Foundation of China under Great Nos. 11835011; 11675146; 11875220;. [1] M Daniel, V Veerakumar, and R Amuda. Soliton and electromagnetic wave propagation in a ferromagnetic medium. 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[38] Aly R Seadawy, Naila Nasreen, Dianchen Lu, and Muhammad Arshad. Arising wave propagation in nonlinear media for the (2+ 1)-dimensional heisenberg ferromagnetic spin chain dynamical model. Physica A: Statistical Mechanics and its Applications , 538:122846, 2020. [39] Victor K Kuetche, Thomas B Bouetou, and Timoleon C Kofane. Fractal structure of ferromagnets: The singularity structure analysis. Journal of mathematical physics , 52(9):092903, 2011. [40] Herv ´e Leblond. The reductive perturbation method and some of its applications. Journal of Physics B: Atomic, Molecular and Optical Physics , 41(4):043001, 2008. [41] Herv ´e Leblond and M Manna. Two-dimensional electromag- netic solitons in a perpendicularly magnetized ferromagnetic slab. Physical Review B , 80(6):064424, 2009. [42] M Saravanan and Russell L Herman. Perturbed soliton solutions for an integral modified kdv equation. Communications in Nonlinear Science and Numerical Simulation , 91:105437, 2020.

2023-07-03

In this paper, we study in detail the nonlinear propagation of magnetic soliton in a ferromagnetic film. The sample is magnetized to saturation by an external field perpendicular to film plane. A new generalized (2+1)-dimensional short-wave asymptotic model is derived. The bilinear-like forms of this equation are constructed, and exact magnetic line soliton solutions are exhibited. It is observed that a series of stable lumps can be generated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable and can maintain their shapes and velocities during evolution or collision. The interaction between lump and magnetic soliton, as well as interaction between two lumps, are numerically investigated. We further discuss the nonlinear motion of lumps in ferrites with Gilbert-damping and inhomogeneous exchange effects. The results show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay exponentially during propagation. And the shock waves are generated from a lump when quenching the strength of inhomogeneous exchange.

Magnetic lump motion in saturated ferromagnetic films

2307.00903v1

Gilbert and Landau-Lifshitz damping in the presense of spin-torque Neil Sm ith San J ose Reserch Center, Hitachi Global Strage Tec hnologies, Sa n Jose, CA 95135 (Dated 6/12/07) A recent arti cle by Stiles et al. (cond-mat/0702020) argued in favor of th e Land au-Lifshitz dampin g term in the micromagnetic equations of motion over that of the more commo nly accepted Gilbert d amping for m. Much of their argument r evolved around spin-torque dr iven domai n wall motion in n arrow magnetic wires, since th e presence of spin -torques can mor e acut ely draw a distinct ion b etween the two form s of dam ping. In this ar ticle, the author uses simple argum ents and exam ples to offer an alterna tive po int of view favoring Gilb ert. I. PREL IMINARIES The Gilbert1 (G) or La ndau-Li fshitz2 (LL) equations of motion for unit magnetization vect or are formally descri bed by the gene ric form sMt t /),( ),(ˆ rM rm≡ ) derivativeal (variationˆ/ /1)] (ˆ[ /ˆ effeff totdamp tot NC m HH H HH Hm m ∂∂−≡+≡+×γ−= E Mdt d s (1) where the satu ration magnetizatio n, and sM γ is th e gyromagnetic ratio (tak en here to be a po sitive constant). The total (p hysical) field by h as con tributions fro m the usual "effecti ve field" term , pl us t hat of a "nonconservative-field" that is supp osed not to be derivable from the -gradient of the (internal) free-energy density functional . Although also non conservative by defi nition, the "dam ping-field" is p rimarily a math ematica l vehicle for describing a physical d amping torque , and i s properly treated separat ely. For most of the rem ainder of th is article, an y sp atial depe ndence of will be im plicitly unde rstood. effH NCH mˆ )ˆ(mE dampH dampˆHm×sM ),(ˆtrm As was described by Brown,3 the Gilb ert eq uations of motion m ay be de rived using st andard techni ques of Lagra ngian mechanics .4 In particular, a phenomenological damping of the motion in included via the use of a R ayleigh dissipation function : )/ˆ( dtdmℜ dt d dt d M/dtdM GG ss /ˆ)/ ( )/ˆ(/ /1ˆ )2/ ( G damp2 m m Hm γα−= ∂∂ℜ−≡γα=ℜ (2) where dimensionless is th e Gilb ert damping parameter. By d efinition,Gα 4 2 dampˆ / /ˆ 2G/dtd M dtds G m m H γα=⋅−=ℜ is th e in stantaneous rate o f energy lost fro m the magnetizatio n syste m to its thermal environment (e.g ., to the lattice) d ue to the viscous "friction" re present ed by the damping field . dt d/ˆG dampm H−∝ The Lag rangian method is well su ited to include nonconservative fields , which can be generally defined using the principles of virtual work:0NC≠H 3,4 )ˆ ( /1 ) ˆ() ˆ( ˆ NC NCNC NC NC mN H HmHm m H × =⇔×=⇒δ⋅×=δ⋅ =δ s ss s M MNM M W θ (3) The latter exp ressio n is useful in cases (e.g., spin-torques) where the torque density funct ional is specified. Treatin g as fi xed, the (virtual) dis placem ent )ˆ(mN sM mˆδ is of the fo rm m m ˆ ˆ×δ=δθ , and only the orthogonal compone nts of the torque mNmN ˆ ˆ××↔ are physically signi ficant. Combining (1) an d (2) gives the Gilbert equations: )/ˆ ˆ( ) ˆ( /ˆtot dtd dt dG mm Hm m ×α+×γ−= (4) As is well known, the G eq uations o f (4) may be rearra nged into their equivalent (and perhaps m ore common) f orm: )] ˆ(ˆ ˆ[ 1/ˆtot tot2 GHmm Hm m ××α+× α+γ−=G dt d (5) With re gard to the LL e quations, the form of is not uniquely defined in problems where LL dampH 0NC≠H , whic h have only c ome to the forefront with the recent interest in spin-torque phen omena. Two d efinitions conside red are ) ˆ(eff damp LLLLHm H ×α≡ , (6a) (6b) ) ˆ(tot damp LLLLHm H ×α≡ The fi rst de finition of (6a) is the historical/conventional form of LL, and is that em ployed by Stiles et al.5 Howe ver, in this a uthor's view, the re is no a-priori reason, other than historical, to not replace as in (6b). Doing so yields a form of LL that reta ins it "usual" e quivalence (i.e., to first o rder in tot eff H H→ α) to G w hether or not 0NC≠H , as is seen by com paring (5) and (6b). The form o f (6b) treats both and on an equal footin g. effHNCH Nonet heless, to facilita te a com parative discussi on with the analysis of Stiles et al. ,5 (6a) will he ncefort h be used to define what will be re ferred to below as the LL eq uatio ns of motion: )] ˆ(ˆ ˆ[ /ˆeff tot LL Hmm Hm m ××α+×γ−=dt d (7) In cases of pre sent interest where , the difference betwee n G in (4) (or (5)) and the f orm of LL give n in ( 7) are first orde r in the dam ping param eter, and thus o f a more fundam ental nature. T hese differences a re the subj ect of the remainder of t his article. 0NC≠H II. SPIN-TOR QUE EXAM PLES Two distinct situations where spin-torque effects have garnere d substantial intere st are those of CPP-GMR nanopillars, and spin-torque driven dom ain wall motion i n nanowires as was conside red in R ef. 5. The spi n-torque functio n is taken t o have a predominant "adiabatic" c omponent , alon g with a small "nona diabatic" com pone nt described phenomenolog ically by the relation )ˆ( )ˆ( )ˆ(nad ad ST m NmNmN + = )ˆ(admN )ˆ(nadm N ad nadˆNm N ×β−≡ , with . In t he case of a narrow nanowire along the - axis, with m agnetization and electron curre nt density , the torque function and associate d field (see ( 2)) are descri bed by1<<β xˆ )(ˆxm x J ˆe eJ= )ˆ(STmN )ˆ(STmH5 )/ˆ /ˆ ˆ() 2/ ()/ˆ()2/ ()ˆ( STad dxd dxd eM PJdxde PJ s ee m mm Hm mN β+× −== hh (8) where P is the spin-polarization of t he electron curre nt. To check if is conse rvative, one ca n "discretize" the spatial derivatives app earing in (8) in the form STH x dxdi iixx∆ −→−+=2/)ˆ ˆ( /ˆ1 1m m m , whe re )(ˆ ˆi i xmm≡ and , not unlik e the com mon m icrom agnetics approxim ation. For a c onservative H-field where , the set of Cartesian tensorsi i x xx−≡∆+1 i i Em H ∂∂∝ / 33×6 j i j iuv ji E H mm mH ∂∂∂∝∂∂≡ /2/t will be sym metric, i.e., vu ijuv ji H Htt = , under sim ultaneo us reversal of s patial indices and vect or in dices ji, z yxvu or,, ,= . For the adiabatic term in (6), it can be readily shown that the uv jiHt are in gene ral asymmetric , i.e., always antisym metric in vect or indices (du e to cross pr oduct) , but asy mmetr ic in spatial indices , being antisym metric he re only for locally uniform magnetization . The nonadiabatic term yields an -inde pendent 1 ,±=iji i im m ˆ ˆ1=± mˆuv jiHt that is always antisym metric, i.e., symmetric in ve ctor i ndices, but antisym metric in spatial indic es . The concl usion here t hat is in ge neral nonconservative a grees with that f ound in Ref. 5, by way of a rathe r diffe rent argument. 1 ,±=iji STH Anot her well known example is a nanopillar stack wit h only two fe rrom agnetic (FM ) layers, the " refere nce" layer having a m agnetization rigidly fixe d in time, and a dynamically varia ble "free " lay er refˆm )(ˆ)( ˆfree t tm m= . As descri bed by Sloncze wski,7 the (adiaba tic) spin-t orque density function a nd field is given by: )ˆ(STmH ]ˆ )ˆ ˆ[() 4/ ()ˆ ˆ(]ˆ ˆˆ[) 4/ ()ˆ ˆ( ref reffree refref free ref ad ST m m mm m Hm mm m m N β+××⋅−=×× ⋅−= tMe PJ gte PJ g s ee hh (9) where is the free layer thickness, and freet )ˆ ˆ(refm m⋅ g is a functio n of order unity, the de tails of which are not relevant to the present discu ssion. From the the -tenso r, or by simple inspection, t he adiabatic term in (7) is manifestly nonc onservative . However, app roxim ating uvHt m m ˆ ˆref× )ˆ ˆ(refm m⋅ g ~ consta nt, the conse rvative nonadiabatic ter m resem bles a magnetic field d escribe d by the -gradient of an Zeem an-like ene rgy function mˆ m m ˆ ˆref nad ⋅∝ E . The remaining discussion will restrict attention to nonconservative contrib utions. III. STATIO NARY SOLU TIONS OF G AND LL With , stationary (i.e, ST NC H H→ 0 /ˆ=dt dm ) solutions of G-equatio ns (4) satisfy the c onditions that 0ˆm ST STST 0 eff 0 0G damp eff 0 ˆ ˆ 0 ˆ0 /ˆ ;0) ( ˆ Hm Hm Hmm H H H m ×−=×⇒≠×=∝ =+× dt d (10) The clear and physically intuitiv e interpreta tion of (10) is that stationary state satisfies a condition of zero physical tor que, 0ˆm 0 ˆtot 0=×Hm , includin g bot h conservative ( ) an d nonconservative s pin-torque ( ) fields. Being visco us in nature, the G dam ping torque inde pendently vanishes.. effH STH 0 /ˆ ˆG damp 0 ≡∝× dt dm Hm Previous measurem ents6 of the angular depe ndence of spin-torque critical curre nts in CPP-GMR nanopillar syste ms by this author and colleagues demonstrated t he existence of such stationary states with non-collinear )ˆ ˆ(refcritm m⋅eJ 0 ˆ ˆ0 ref≠×m m and crit0e eJ J<< . In t his situation, it follows from (9) an d (10) that the stationa ry state satisfies 0ˆm 0 ˆ ˆeff 0 0 ST ≠×=×− Hm Hm . It is noted that the last result i mplies that is not a (therm al) 0ˆmequilibrium state which m inimizes the free energy , i.e., )ˆ(mE 0 ) ˆ()ˆ ()ˆ/(ˆ/eff 0 0 ≠δ⋅×∝×δ⋅∂∂=δδ θ θ Hm m m m E E for arbitra ry . θδ In the present described circum stance of stationary with , the LL equatio ns of (7) differ from G in a fundam ental respect. Setting in (7) yield s 0ˆm 0 ˆST 0≠×Hm 0 /ˆ=dt dm ) ˆ( ˆ ) ( ˆeff 0 0 eff 0 LL ST Hm m H H m ××α−=+× (11) Like (10), (11) im plies that 0 ˆeff 0≠×Hm whe n . However, (11) also imply a static , nonzero physical tor que , alon g with a static, nonzero damping tor que (see (6a) ) to cancel it out . In sim ple mechanical term s, the latte r amounts to non-visco us "static-frictio n". It has n o anal ogue with G in a ny circum stance, or with LL in conventional situations with and equilibrium for which LL dam ping wa s origi nally develo ped as a phenomenolog ical dam ping f orm. It furth er contra dicts th e viscous (o r -depe ndent) nature o f the damping mechanism s desc ribed by physical (rathe r than phenomenolog ical) base d theoretical m odels0 ˆST 0≠×Hm 0 ˆtot 0≠×Hm 0 ˆLL damp 0 ≠×Hm 0ST NC =↔H H ↔0ˆm dtd/ˆm ,8,9. The above arguments ignored therm al fluctuatio ns of . However, thermal fluctuations mˆ10 scale approxim ately a s , while ( 10) or (11) are scale-inva riant with 2 eff 0 ) /( Hm⋅ kT H. In t he simple CPP nanopillar exam ple of (9), one can (conce ptually at least) continually increase both eJ and a n applied field contri bution to to scale up appHeffH ST 0ˆHm× and eff 0ˆHm× while approxim ately keeping a fixe d statio nary state (satis fying 0ˆm 0 ˆ ˆref 0≠×mm with fixed ). However, unique to LL eq uatio ns (11) based on (6a) is t he additional requi rement that the static dam ping mechanism be able to produce an refˆm eff dampˆLLHm H ×∝ which sim ilarly scales (without li mit). This author finds thi s a physically unreasonable proposition. IV. ENERGY A CCOUN TING If one ignores/forgets t he Lagrangian formulation3 of the Gilbert e quatio ns (4), one may derive t he followin g energy relationships, substitutin g the right side of (4) for evaluating vecto r products of form : dtd/ˆmH⋅ )/ˆ ˆ( ) ˆ()/ˆ ˆ( ) ˆ(/ˆ )/ˆˆ/ /( /1 eff effeff effeff NCNC dtddtddtd dtd E dtdE Ms mm H Hm Hmm H Hm Hm H mm ×⋅α−×⋅γ−=×⋅α−×⋅γ=⋅−≡⋅∂∂= (12a) )/ˆ ˆ( ) ˆ(/ˆ / /1 NC NCNC NC eff dtddtd dt dWMs mm H Hm Hm H ×⋅α+×⋅γ−=⋅≡ (12b) )/ˆ ˆ() () ˆ(/ˆ /ˆ NC efftot2 dt ddt d dt d mm H HHm m m ×⋅+γ=×γ−⋅= (12c) Subtractin g (12b) from (12a), and usin g (12c) one finds dt d M dt dWdt d M dt dW dtdE ss /ˆ //ˆ / / / :G G NCG NC damp2 m Hm ⋅ + =γα− = (13) The re sult o f (13) is essentially a state ment of energy conservation. Nam ely, that the rate of change of the internal free e nergy (density) of the magnetic sy stem is give by the work done on the system by the (exte rnal) no nconservative forces/fields , minus the loss of energy (t o the lattice) due t o dam ping. The G damping term in ( 13) is ( not surprisi ngly) t he sam e as expected from (2). It is a strictly lossy, negative-definite contributio n to . NCH dtdE/ Over a finite interval of motion from time to , the change 1t2t )ˆ( )ˆ(1 2 m m E EE −=∆ is, from (12b ) and (13): ∫⋅γα− =∆2 1G NCˆ)/ˆ / )ˆ( (t tsdtddt d dt MEmm m H (14) Since is nonc onservative, t he work NCHNCW∆ is pat h- depe ndent, and so use of (14) requires indepe ndent knowledg e of the solution of (4). Sin ce itself depe nds on ) (ˆ2 1 ttt≤≤m )(ˆtmGα, the term's contribution to (14) also can vary with . Regardless, NCH Gα 0>∆E can only result in the case of a positive amount of work done by . ∫⋅ =∆2 1NC NC )/ˆ (t ts dtdt d M W m HNCH Working out the results analogous to (12 a,b) for the LL equatio ns of (6a) and (7), one finds ) ˆ() ˆ( /ˆ/ˆ / / :LL tot eff dampdamp LLLL NC Hm Hm m Hm H ×⋅×αγ−=⋅⋅ + = dtddtd M dt dW dtdEs (15) The form of (15) is the sam e as the latter result in (13). However, unlike G, the LL damping term in (15) is not manifestly negative-definite, except when 0NC=H . The results of (13)-(15) apply equally to situations where one inte grates over the spatial distribution of to evaluate the total syste m free energy, rat her tha n (local ) free e nergy density . Total time derivatives may be replace d by partial deri vative s whe re appropriate. ),(ˆtrm dtd/ t∂∂/ Dropping terms of order (and sim plifying notation ), (7) is easily transfor med to a Gilbert-like form : 2 LLα α→αLL )ˆˆ( )] ˆ( [ˆˆ:LLNC totdtd dtd mm Hm Hmm×α+×α−×γ−= (16) whic h differs from G in (4) by the term ) ˆ(NCHm×α which is first order in both and . For the "wire problem" described by (8), the equation s of m otion bec ome αNCH )ˆ ˆ(ˆ ˆˆ ˆ:LL)ˆ ˆ(ˆ ˆˆ ˆ effeff :G dxdv dtd dxdv dtddxdv dtd dxdv dtd m mm Hmm mm mm Hmm m αβ+α+×α+×γ−=+αβ+×α+×γ−=+ (17) where , and terms of or der eM PJ vs e2/γ=h βα are dropped for LL. A s noted previously,5,9,11 (17) permits "translational" solutions ) (ˆ)(ˆeq vtx x,t −=m m whe n α=β (G) or (LL), with the static , equilibrium (minimum E) solution of . Evaluatin g by takin g from (8), and with , one finds that 0=β )(ˆeqxm 0) ˆ(eff eq =×H m dtd M dt dWs /ˆ /ST ST m H⋅ =STH ) (ˆ /ˆeq vtx v dtd −′−→m m dqd q /ˆ )(ˆ m m≡′ 2 eq2) (ˆ)/ ( /ST vtx Mv dt dWs −′γβ= m . In transl ational cases where is exactly collinear to , only the nonadiabatic term does work on t he -system. dtd/ˆm dx d/ˆm mˆ Interestingly, the energy interpretation of these translational solutions is very diffe rent fo r G or LL. For G, the positive rate of work when dt dW /ST α=β exactly balances t he negative damping l oss as given in (13), the latter alway s nonzero and scaling as . For LL by contrast, the wo rk done by vanishes when 2v STH 0=β , matching t he damping l oss whic h, from (6a) or (15), is always zero since regardle ss of 0) ˆ(eff eq =×H m v. If is a sharp domain wall, )(ˆeqxm ) (ˆ /ˆeq vtx v dtd −′−=m m represents, from a spatially local perspective at a fixed point x, an abrupt, irreversi ble, non-equilibrium reor ientation of at/near tim e whe n the wall core passes by. The prediction of LL/(6a) that t his magnetization re versal c ould take place locally (at arbitrarily large v), with the com plete absence of the spin-orbit couple d, ele ctron scatteri ng processesmˆ vxt /≈ 8 that lead to spin-latti ce dam ping/r elaxation in all other known circum stances (e. g., external field-driven domain wall motion) is, in the vi ew of this author, a rather dubious, nonphysical aspect of (6a) when 0ST≠H . Stiles et al.5 repo rt that micromagnetic com putation s using G in the case sho w (non-translational) time/distance l imited dom ain wall displacement, resulting in a net positive increase 0=βE∆. They claim that 1) "spi n trans fer to rques do not cha nge the ene rgy of the sy stem", and that 2) "Gilbert dam ping to rque is the only torque th at changes the e nergy". Acce pting as accurate, it is this aut hor's view that t he elementary physics/ mathematics leading t o (13) and (14) demonstrably prove t hat bot h of these claim s must be incorrect (err or in the first per haps leading t o the misinterpretation of the second). On a related point, the res ults of (1 3) and (1 5) shows that excludi ng work or 0>∆E dt dW /ST STW∆ , only LL- damping may possibly lead to a positive contri bution to or dtdE/ E∆ when 0ST≠H , in a pparent contradictio n to the claim in R ef. 5 that LL damping "uniquely and irre versibly reduces magnetic free energy whe n spin-transfer torque is prese nt". ACK NOWLEDGM ENTS The aut hor would like to ackno wledge em ail discussions on these or related topics with W. Sa slow and R. Duine, as well as an extende d series of friendly discussio ns with Mark Stiles. Obviously, the latter have not (as of yet) achieve d a mutually agree d viewpoint on thi s subj ect. REFERENCES 1 T. L. Gilber t, Armour Research Report, M ay 1956; IEEE Tran s. Magn., 40, 3343 (2004). 2 L. Landau and E. Lifshit z, Phys. Z. Sow jet 8, 153 (1935). 3 W. F. Brown, Micromagnetics (Krieger , New Y ork 1978). 4 H. Gol dstein, Classical Me chanics , (Addison Wesley, Reading Massachusetts, 1 950). 5M. D. Stiles, W. M. Saslow, M. J. Donahue, and A . Zangwill, arXiv:cond-m at/0702020. 6 N. Smith, J. A. Katine, J. R. Childress, and M. J. Carey, IEEE Trans. M agn. 41, 2935 (2005) ; N. Sm ith, J Appl. Ph ys. 99, 08Q703 (2006). 7 J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996); J. Magn. Magn . Mater. 247, 324 (2 002) 8 V. Kambersky, Can. J. Phys. 48, 2906 (1970) ; V. Kam bersky and C. E. Patton , Phys. Rev . B 11, 2668 (1975). 9 R. Duine, A. S. Nunez, J. Sinova, and A. H. MacDonald, arXiv:cond-m at/0703414. 10 N. Sm ith, J. A ppl. Ph ys. 90, 57 68 (2001). 11 S. E Barnes and S . Maekaw a, Phys. R ev. Lett. 95, 10720 4 (2005).

2007-06-12

A recent article by Stiles et al. (cond-mat/0702020) argued in favor of the Landau-Lifshitz damping term in the micromagnetic equations of motion over that of the more commonly accepted Gilbert damping form. Much of their argument revolved around spin-torque driven domain wall motion in narrow magnetic wires, since the presence of spin-torques can more acutely draw a distinction between the two forms of damping. In this article, the author uses simple arguments and examples to offer an alternative point of view favoring Gilbert.

Gilbert and Landau-Lifshitz damping in the presense of spin-torque

0706.1736v1

CoFeB/MgO/CoFeB structures with orthogonal easy axes: perpendicular anisotropy and damping H. G lowi nskia, A. _Zywczakb, J. Wronac, A. Krysztoka, I. Go scia nskaa, T. Stobieckid,e, J. Dubowika, aInstitute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17, 60-179 Poznan, Poland bAGH University of Science and Technology, Academic Centre of Materials and Nanotechnology, Al. Mickiewicza 30, 30-059 Krakow, Poland cSingulus Technologies AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany dAGH University of Science and Technology, Department of Electronics, Al. Mickiewicza 30, 30-059 Krakow, Poland eAGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Al. Mickiewicza 30, 30-059, Krakw Poland Abstract We report on the Gilbert damping parameter , the eective magnetization 4Meff, and the asymmetry of the g-factor in bottom-CoFeB(0.93 nm)/MgO(0.90{ 1.25 nm)/CoFeB(1.31 nm)-top as-deposited systems. Magnetization of CoFeB layers exhibits a specic noncollinear conguration with orthogonal easy axes and with 4Meffvalues of +2 :2 kG and2:3 kG for the bottom and top layers, respectively. We show that 4 Meffdepends on the asymmetry g?gk of theg-factor measured in the perpendicular and the in-plane directions re- vealing a highly nonlinear relationship. In contrast, the Gilbert damping is practically the same for both layers. Annealing of the lms results in collinear easy axes perpendicular to the plane for both layers. However, the linewidth Corresponding author Email address: dubowik@ifmpan.poznan.pl (J. Dubowik) Preprint submitted to Journal of Physics: Condensed Matter February 21, 2019arXiv:1902.07563v1 [cond-mat.mes-hall] 20 Feb 2019is strongly increased due to enhanced inhomogeneous broadening. Keywords: ferromagnetic resonance, perpendicular magnetic anisotropy, magnetization precession damping PACS: 75.30.Gw, 75.70.Tj, 75.78.-n, 76.50.+g 1. Introduction CoFeB/MgO/CoFeB systems are extensively employed in magnetic tun- nel junctions (MTJs), which are important for modern spintronic devices such as read-heads and magnetic random-access memory [1]. In these ap- plications the two key features are the perpendicular magnetic anisotropy (PMA) with PMA constant K?and magnetization damping with inhomoge- neous (extrinsic) and Gilbert (intrinsic) contributions to the ferromagnetic resonance (FMR) linewidth. The FMR linewidth is usually enhanced in Ta/CoFeB/MgO stacks for which the values of PMA and the Gilbert damping parameter are scattered [2, 3, 4]. Recent experimental results [4, 5] indicate that there is no correlation betweenK?andin these systems. Specically, is approximately constant while the PMA tends to improve on annealing. However, systems with a high PMA have often an increased linewidth due to an inhomogeneous broadening [6, 7] so that an extrinsic contribution to the linewidth may be as high as 400{500 Oe [8] despite is of 0.01 { 0.02 in these systems. An increase in linewidth is attributed to an angular dispersion of the easy PMA axis, which results in a high inhomogeneous broadening attributed to the zero-frequency linewidth
H0[6]. It has been shown that PMA in CoFe/Ni multilayers is linearly propor- 2tional to the orbital-moment asymmetry [7, 9] in accordance with the Bruno's model [see Ref. [7] for discussion]. On the other hand, substantial PMA in Ta/CoFeB/MgO systems [2] has been considered as related to an inhomoge- neous concentration of the anisotropy at the interface [10] so that the Bruno's model may be not valid in this case. Based on our experimental results, we aim to shed some light on possible correlation between asymmetry of the g-factor and the eective magnetization 4 Meff, which are the magnetic parameters measured directly in a broadband FMR experiment. According to well known Kittel's formula, a departure from the free electron g-factor is proportional to L=S[11] so that we can discuss the asymmetry of the g-factor as well as on the asymmetry of the orbital moment on equal footing. Here, we prefer to use asymmetry in g-factor for evaluating the relationship between orbital moment and PMA. As far as we know, FMR has not yet been thoroughly investigated in "full" Ta/CoFeB/MgO/CoFeB/Ta MTJ structures. In particular, a depen- dence of PMA on the asymmetry in the g-factor has not yet been proved in CoFeB/MgO/CoFeB systems. In this paper, we aim to independently characterize each CoFeB layer separated by a MgO tunnel barrier in terms of theparameter and 4 Meff. By analyzing FMR measurements in the in-plane and out-of-plane congurations, we nd that PMA correlates with theg-factor asymmetry in a highly nonlinear relationship. 2. Experimental methods The samples were sputtered in an Ar atmosphere using a Singulus Timaris PVD Cluster Tool. The CoFeB magnetic lms were deposited by dc-sputtering 3from a single Co 40Fe40B20target, whereas the MgO barriers were deposited by rf-sputtering directly from a sintered MgO target. The samples were de- posited on an oxidized silicon wafer with 5 Ta/ 20 Ru /Ta 3 buer layers and capped with 5 Ta/ 5 Ru (numbers indicate the nominal thickness in nanometres). The studied structures consist of two ferromagnetic CoFeB (0.93 nm { bottom and 1.31 nm { top) lms separated by a MgO barrier of dierent thicknesses (0.90, 1.1, and 1.25 nm). It is important to note that we investigated the as-deposited samples so that the CoFeB layers were amor- phous [3, 12]. The eect of annealing treatment (330oC for 1 hr) on magnetic properties of the system will be discussed at the end of the paper. Hysteresis loops of the samples were measured by vibrating sample mag- netometer (VSM) with the perpendicular and in-plane magnetic elds. The saturation magnetization Msof 1200 G in the as-deposited state was deter- mined from magnetic moment per unit area vs. CoFeB thickness dependen- cies [13]. To investigate anisotropy and damping in studied samples, vector network analyzer ferromagnetic resonance (VNA-FMR) spectra of the S21 parameter were analyzed [14]. VNA-FMR was performed at a constant fre- quency (up to 40 GHz) by sweeping an external magnetic eld, which was applied either in-plane or perpendicular to the sample plane. These two con- gurations will be referred to as the in-plane and out-of-plane congurations. Experimental data were tted using the Kittel formula ! k=q (Hr+Ha) (Hr+Ha+ 4Meff) (1) for the in-plane conguration and ! ?= (Hr4Meff) (2) 4for the out-of-plane conguration, where != 2fis the angular microwave frequency,Hrthe resonance eld, k;?=gk;?B=~the gyromagnetic ratio, gkandg?are the spectroscopic g-factors for the in-plane and out-of-plane congurations, respectively, ~the reduced Planck constant, Bthe Bohr magneton, and Hathe in-plane uniaxial anisotropy eld. 4 Meff= 4Ms H?is the eective magnetization , where Msis the saturation magnetization, andH?= 2K?=Msis the perpendicular anisotropy eld and K?is the perpendicular anisotropy constant. For the in-plane easy axis 4 Meff>0 whereas for the perpendicular to the plane easy axis 4 Meff<0. According to Eqs. (1) and (2), 4 Meff=2Keff=Ms, whereKeffis the eective anisotropy constant dened as K?2M2 s[15]. 3. Results and discussion Figure 1 (e) presents hysteresis loops of the sample with a 1.25 nm thick MgO barrier measured in the out-of-plane (red line) and in-plane congu- ration (black line). The shape of the loops in both directions is nearly the same for each conguration as the saturation elds (of Hs2 kOe) for both layers have nearly the same magnitude with the opposite signs in 4 Meff. Each hysteresis loop is a sum of the loops typical for the easy and hard axis and, as explained below, we can infer from magnetization reversals which layer possesses PMA. Let us assume that the bottom CoFeB layer (B) has an in-plane easy axis and the top layer (T) has a perpendicular to the plane easy axis so that their magnetization directions are orthogonal at remanence. Three congurations of a magnetic eld Happlied for the magnetization measurements are shown 5-10 -5 0 5 10-101 Normalized moment H (kOe)HTT BH e.a.e.a. B He.a.e.a. T Be.a. e.a.a) b) c) B T B+T d) 10 51 -1 -10 -5 0e) H (kOe)Figure 1: (a)-(c) Congurations used for the magnetic measurements with a magnetic eld applied perpendicular or parallel to the lm plane. (d) Example of schematic pictures of the magnetization reversals of a CoFeB/MgO/CoFeB structure for conguration (a). (e) Hysteresis loops of a CoFeB/MgO/CoFeB structure measured in congurations (a) - black line and (b) - red line. The inset shows schematically the model reversals for congurations (a)-black and (b)-red . in Figs. 1 (a) - (c). These congurations enable magnetization reversals to be observed with Horiented parallel- (a) (perpendicular- (b)) to the easy axis of B (T) layer, respectively, or perpendicular to both easy axes (c). Further, we will refer to these congurations as (a), (b), and (c) congurations. As it is schematically shown in Fig. 1 (d), an apparent magnetization reversal of B+T for the conguration (a) is a sum of independent magnetization reversals of B and T. For the perfectly asymmetric structure with 4 MB eff=4MT eff with the same thickness (i.e. with the same magnetic moments MSVT;B) the 6apparent magnetization reversals taken in congurations (a) and (b) would overlay. However, as it is seen in Fig. 1 (e) they do not completely overlay so that the curve taken in the conguration (b) lies a bit higher than that taken in (a). As it is shown in the inset of (e), a simple model explains that the T layer (i.e. the with nominal thickness tof 1.3 nm) possesses an easy axis perpendicular to the plane, while the B layer with t= 0:93 nm has an in-plane easy axis. In the model, the magnetization reversals in each layer can be approxi- mated with a normalized relation [16] M(H;S) = arctan[H=H stan(S=2)]= arctan[H=H maxtan(S=2)], where Hsof 2 kOe is a saturation eld for the hard direction and Sis dened as a ratio of remanence to the satura- tion moment. For Hkparallel to the easy axis, S= 1 (B layer in Fig. 1 (d)) and for H?perpendicular to the easy axis (T layer in Fig. 1 (d)), S= 0:66 as well as Hmax= 10 kOe are arbitrary chosen for the sake of simplicity. The apparent magnetization curve for conguration (a) is a sum [tBM(H;S = 1) +tTM(H;S = 0:66)]=(tB+tT). For the conguration (b),tTandtBare reversed in the sum. In order to satisfy the experimental data shown in (e), a ratio tB=tT= 0:79. It is easily seen that if the B layer had an in-plane easy axis and the T layer had an easy axis perpendicular to the plane, a curve taken in conguration (b) would lie lower than that taken in conguration (a). Hence, the thin B layer is that with the in-plane easy axis. Figures 2 (a) and (b) show typical VNA-FMR spectra of the CoFeB/MgO(1.25 nm)/CoFeB system measured (see Figs. 1) in conguration (a) and (b) , re- spectively. Two FMR peaks associated with the bottom and top CoFeB lay- 76 8ImS21(a.u.) H(kOe)topbottom(a) 20GHz in-planeconfiguration 4 6 8 10(b)ImS21(a.u.) H(kOe)top bottom20GHz out-of-planeconfiguration ( ) ()Figure 2: Typical VNA-FMR spectrum of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB structure with resonance peaks from bottom (B) and top (T) layers measured in the in- plane (a) and out-of-plane (b) congurations. Solid red lines represent the Lorentzian ts to the experimental data. (c) Dependence of the FMR eld on the polar angle  of applied eld in X band (9.1 GHz). The easy axis of magnetization of the B is in the in-plane orientation. For the T layer, the out-of-plane direction becomes the easy axis. ers are clearly visible. To determine the resonance eld Hrand the linewidth
Hat constant frequency with a high precision, the spectra were tted with Lorentzians (marked by solid lines in Fig. 2 (a) and (b)). Figure 2 (c) shows dependencies of the X-band (9.1 GHz) resonance elds of the B and T layers on the polar angle between the lm normal and the direction of an applied eld. It is clearly seen that the T layer has 4 Meff<0 (i.e., a perpendicular easy axis) and the B layer with 4 Meff>0 has an in-plane easy axis. From Figs. 2 (a) and (b), we can clearly see that the intensity (area under the FMR 8peak) of the T layer is higher than that of the B layer. This additionally conrms that the bottom layer has the lower magnetic moment than that of the top layer. A typicalHrvs.fdependence, observed for the CoFeB/MgO(1.25 nm)/CoFeB system is shown in Fig. 3 (a) and (b) for the in-plane (a) and out-of-plane (b) conguration, respectively. The observed data points are tted using Eqs. (1) and (2). The values of 4 Meff, obtained from the tting are found to be of +2 :2 kG and2:3 kG for the bottom and top layers, respectively. ThefversusHrdata for the B layer were tted assuming Haof 30 Oe as conrmed by VSM measurements (not shown) in the conguration presented in Fig. 1(c). The values of gkof the top and bottom layers are equal to 2.04 and 2.08, respectively, in contrast, the values of g?for these layers are 2.06 and 2.22. One can notice the dierences in values of g?resulting from clear dierences in the slopes of the f(Hr) dependencies (see, Fig. 3 (b)) for the bottom ( ?= 2:88 MHz/Oe) and top ( ?= 3:11 MHz/Oe) layer, respec- tively. To sum up, VSM and FMR measurements conrmed the presence of or- thogonal easy axes in our CoFeB/MgO/CoFeB systems and showed that the thickness ratio tB=tT= 0:79 is slightly higher than the ratio of nominal thick- ness (tB nom=tT nom= 0:71). The thinner B layer has an in-plane easy axis while the T layer has a perpendicular easy axis. However, keeping in mind our for- mer studies of a dead magnetic layer (DML) in the Ta/CoFeB/MgO (B) and MgO/CoFeB/Ta (T) structures [13] deposited in the same Timaris system, we estimated DMLB'0:23 nm and DMLT'0:4. With such asymmetric DMLs the eective thickness tB eff'0:7 nm andtT eff'0:9 nm which satises 90 5 1005101520 0 5 10010203040 f (GHz) Hr(kOe) (b)(a) in-plane configuration out-of-plane configuration f (GHz) Hr(kOe)1.25 nm MgO bottom top0 2 46 8 10048121620 top 1.25 nm MgO 1.25 nm MgO 0.96 nm MgO 0.96 nm MgO 0.85 nm MgO 0.85 nm MgOf (GHz) H (kOe)bottomFigure 3: FMR dispersion relations of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB structure measured in the in-plane conguration (a) and out-of-plane conguration (b). The solid lines show the ts given in accordance with Eqs. (1) and (2). Inset in (a) shows that the tting parameter practically do not depend on the MgO thickness. tB=tT= 0:78. VNA-FMR measurements, which oer a greater precision than VSM measurements, give 4 Meff=2:3 kG (K?= 10:4106erg/cm3) and 4Meff= +2:2 kG (K?= 7:7106erg/cm3) for the T and B layers, respec- tively. All tting parameters for a CoFeB/MgO(1.25 nm)/CoFeB structure are juxtaposed in Table 1. As it is shown in the inset of Fig. 3 (a), the thick- ness of MgO spacer within a range of 0.9 { 1.25 nm had almost no in uence on the tting parameters, therefore, the values of tting parameters 4 Meff, g,, and
H0are typical for all samples with various MgO thickness. 10Table 1: Parameters determined from VNA-FMR spectra for the as-deposited CoFeB(0.93 nm)/MgO (1.25 nm)/CoFeB(1.31 nm) for the in-plane and out-of-plane con- gurations: the in-plane anisotropy eld ( Ha), the eective magnetization (4 M eff), spec- troscopicg-factors for in-plane and out-of-plane conguration, Gilbert damping ( ), the frequency-independent FMR linewidth (
H0). The values of the tting parameters do not depend on the MgO thickness. The values of g?are marked by asterisks. In-plane conguration Ha(Oe) 4Meff(kG)gk,g? 
H0(Oe) top 0 -2.290.05 2.040.02 0.0180.002 10222 bottom 30 2.220.15 2.080.03 0.0170.002 6923 Out-of-plane conguration top { -2.30.01 2.220.01?0.0180.001 9513 bottom { 2.190.04 2.060.02?0.0170.003 16030 Although it is counter-intuitive that the thinner B layer possesses an in- plane easy axis, the same feature has been reported for other Ta/CoFeB(1 nm)/MgO systems deposited in the same Timaris equipment [17]. Similar ef- fect has been recently observed in a substrate/MgO/CoFeB/Ta/CoFeB/MgO structure, where the thicker CoFeB layer exhibits a strong PMA in con- trast to the relatively weak PMA in the thinner CoFeB layer [18, 19]. It is possible that the growth mode of the MgO layer in contact with an amor- phous CoFeB layer might be responsible. The perpendicular anisotropy in these systems originates from the CoFe/MgO interface [20]. The structure of the unannealed CoFeB layers is amorphous regardless of underlying lay- ers, whereas the MgO barrier deposited on the amorphous CoFeB has an amorphous structure of up to four monolayers (that is about 0.9 nm) [21]. 11Hence, there are subtle dierences between the CoFeB/MgO (bottom) and MgO/CoFeB (top) interfaces; the interface of the bottom CoFeB layer is mainly amorphous whereas the interface of the top layer is crystalline, be- cause the barrier thickness of the investigated samples is above the transition from amorphous to crystalline phase. Therefore, dierent structures for the CoFeB/MgO interfaces may result in dierent values of anisotropy constant. Another explanation is that the measured dependence Keffteffvs.teff in lms with PMA is often strongly nonlinear due to either intermixing at interfaces [22] or magnetoelastic eects [15], with Keffteffexhibiting a maximum as a function of decreasing teffand with the PMA eventually being lost for small teffof, for example, 0.7 nm. The values of gfactor yield the ratio of the orbital Land spinSmag- netic moments in accordance with equation [9, 11] L S=g2 2; (3) whereS=B. Hence, the dierence between orbital moments
Lalong the easy and hard direction in the in-plane [Fig. 1 (a)] and out-of-plane [Fig. 1 (b)] congurations is proportional to ( g?gk) and reads
L=B(g? gk)=2.
Lis of 0.09Band0:01Bfor the T and B layer, respectively. In CoFe/Ni multilayers [7], the PMA has been shown to be proportional to the orbital moment anisotropy in accordance to Bruno model [23]. How- ever, in the case of the CoFeB/MgO systems this direct relationship between the orbital moment asymmetry and the perpendicular anisotropy is not ful- lled. As can be seen in Table 1, ( g?gk)0 for the B layer corresponds to 4Meff= 2:2 kG. Hence, while ( g?gk) is negligible, a decrease in 4 Meff due to PMA from 4 MS= 15 kG to 2.2 kG is substantial. In contrast, 12(g?gk)0:18 is exceptionally large for the T layer, while 4 Meffmerely decreases to - 2.3 kG. In accordance with the earlier report [24], this conrms that any relationship between the orbital moment asymmetry and the per- pendicular anisotropy in CoFeB/MgO systems is highly nonlinear. Of course, other factors controlled by annealing such as disorder at interfaces and over- or underoxidized interfaces would also play a signicant role in PMA [20]. Future work conrming such a nonlinear relationship for a broad range of tCoFeB might resolve this issue. At present, there is no doubt that PMA in MgO/CoFeB structures is an interface eect and it is correlated with the presence of oxygen atoms at the interface despite the weak spin-orbit coupling [20, 25]. The origin of PMA is attributed to hybridization of the O-p with Co(Fe)-d orbitals at the interface [20] and/or to a signicant contribution of thickness dependent magnetoelastic coupling [15]. A deviation of the g-factor from the 2.0 value is expressed by g'24=
, where < 0 is the spin-orbit constant for Fe(Co) and
is the energy levels splitting in the ligand eld [11]. While the deviation of the g-factor is inversely proportional to
, PMA (and hence 4Meff) is proportional to the enhanced spin-orbit-induced splitting around the Fermi level [20]. This may result in a complex relationship between PMA andg-factor anisotropy. The Gilbert damping parameter is evaluated from the dependence of the linewidth
Hon the resonance frequency as shown in Fig. 4 for the in-plane (a) and the out-of-plane (b) congurations. The lines are linear ts to
H=4f k;?+
H0; (4) 13/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s50/s48/s48/s52/s48/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48 /s40/s98/s41/s40/s97/s41 /s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32 /s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56/s32/s32/s72/s32/s40/s79/s101/s41 /s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110 /s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32 /s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56 /s32/s32/s72/s32/s40/s79/s101/s41 /s102/s32/s40/s71/s72/s122/s41Figure 4: Linewidth as a function of frequency measured in the in-plane conguration (a) and out-of-plane conguration (b). The damping parameter is obtained using Eq. (4). The thickness of MgO was 1.25 nm. where
H0is the inhomogeneous broadening related to CoFeB layer quality. The values of and
H0are shown in Table 1. The top and the bottom layers show almost the same of 0.017 - 0.018. This suggests that the damping has no relation to PMA. While
H0for the top layer is almost the same for both congurations,
H0for the bottom layer at the (b) conguration is nearly twice as large as that for the (a) conguration. Such a behavior suggests that the layer B is rather inhomogeneous with a large angular dispersion of magnetization across the layer [26, 27]. Spin pumping to Ta layers (which are a part of the buer and cap- 14ping layers, as shown in Fig. 1 (e)) may also in uence the damping in CoFeB/MgO/CoFeB systems since magnetization precession induces a spin current to the adjacent nonmagnetic Ta layers that result in an enhanced damping [8]. This is an interface eect and hence scales inversely propor- tional to the CoFeB layer thickness. Because the bottom layer with an in- plane easy axis is thinner than the top layer with a perpendicular easy axis, the spin pumping eect aects it more. To estimate spin pumping eect the standard equation [28] without back ow is used
=gBg#" 4Msteff; (5) whereteffis the eective thickness of CoFeB and g#"is the mixing con- ductance. The measured damping of both layers is of 0.017 - 0.018, while damping of a bulk CoFeB is around 0.004 [12]. Therefore, an increase of
 due to spin pumping is of 0.014 which gives the mixing conductance g#"= 0:8 and 11015cm2for the eective thickness 0.7 nm and 0.9 nm of B and T layer, respectively. The value of mixing conductance g#"for Ta/CoFeB interface found in the literature lies in a broad range from 1 :671014to 1:41015cm2[29, 30, 31, 32]. Taking into account our simplication (the lack of back ow), this estimation gives the maximal values of mixing conduc- tance. Hence, we can conclude that spin pumping substantially in uences the damping in our structures. It is worth mentioning that the measured  of 0.017 - 0.018 for CoFeB/MgO/CoFeB systems agrees with = 0:015 for the Ta/CoFeB(1)/MgO structure reported in [3]. Finally, we would like to make a further comment on postdeposition an- nealing of our CoFeB/MgO/CoFeB systems. We found that annealing at 330oC for 1 hr, beside increasing Msto 1500 G, enhances also PMA so that 15both layers possess easy axes perpendicular to the plane. 4 Meffattains -1 kG and -4 kG for the B and T layers, respectively. We found that an increase in K?of 7:7106erg/cm3equally contributes to both layers and, for example, K?= 17106erg/cm3for the T layer. On the other hand, the linewidth
Hstrongly broadens to 400 Oe and700 Oe for the B layer and the T layer, respectively. These values are in agreement with recently reported values for a similar systems [17]. Moreover, as it is shown in Fig. 5,
Hdoes not follow the linear dependence described by Eq. (4). Therefore, it is impossible to determine precisely for the annealed systems. Such a behavior of
Hand the decreased remanence with respect to the saturation magnetization (see, [17]) both conrm a strong angular dispersion of the easy PMA axis in both layers. It has been observed that with increasing PMA the dispersion of anisotropy also increases [6, 7, 27]. As a result, dispersion in PMA leads to a large two magnon scattering contribution to the linewidth for in-plane magnetization and to an enhanced Gilbert damping [6]. While the magnetic parameters practically do not depend on the MgO thickness in as-deposited structures, the annealed structures show a substantial spread in 4Meffas it is shown in Fig. 6, which may imply some dierent CoFeB/MgO interfaces due to, for example, boron diusion [30, 33]. 4. Conclusion We investigated the CoFeB/MgO/CoFeB as-deposited systems with the in-plane and out-of-plane orthogonal easy axes due to the substantial dif- ference in PMA for the bottom (B) and the top (T) CoFeB layers, respec- tively. The T and the B layer had comparable Gilbert damping suggesting 16/s53 /s49/s48 /s49/s53 /s50/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48 /s32/s32 /s32/s116/s111/s112/s32/s67/s111/s70/s101/s66 /s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s72/s32/s40/s79/s101/s41 /s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 5: Linewidth as a function of frequency measured in the in-plane conguration for the annealed structure. The thickness of MgO was 1.25 nm. that there is no correlation between the Gilbert damping and PMA. We also showed that 4 Meffcorrelates with the asymmetry in the g-factor (and hence with
L) and this correlation is highly nonlinear. Annealing enhances PMA in both layers but it has detrimental eect on the linewidth, however. Therefore, despite the Gilbert parameter shows no correlation with PMA, it seems that there is some correlation between the linewidth (see Eq. 4) and PMA in the annealed systems through a combined eect between dispersion of local anisotropy easy axes in crystallites with a high PMA. 17/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48 /s32/s32 /s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32 /s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32 /s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79 /s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79 /s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32 /s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32 /s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79 /s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79/s102/s32/s40/s71/s72/s122/s41 /s72/s32/s40/s107/s79/s101/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 6: FMR dispersion relations of CoFeB/MgO(0.9 { 1.25 nm)/CoFeB annealed struc- ture measured in the in-plane conguration. Acknowledgments We acknowledge support from the the project \Marie Sk lodowska-Curie Research and Innovation Sta Exchange (RISE)" Contract No. 644348 with the European Commission, as part of the Horizon2020 Programme, and partially by the project NANOSPIN PSPB-045/2010 under a grant from Switzerland through the Swiss Contribution to the enlarged European Union. 18References [1] B. Dieny and M. Chshiev, \Perpendicular magnetic anisotropy at transi- tion metal/oxide interfaces and applications," Rev. Mod. Phys. , vol. 89, p. 025008, Jun 2017. [2] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, \A perpendicular- anisotropy CoFeB/MgO magnetic tunnel junction," Nature Materials , vol. 9, p. 721, 2010. [3] T. Devolder, P.-H. Ducrot, J.-P. Adam, I. Barisic, N. Vernier, J.-V. Kim, B. Ockert, and D. 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Silva, \Roughness induced mag- netic inhomogeneity in Co/Ni multilayers: Ferromagnetic resonance and switching properties in nanostructures," Journal of Applied Physics , vol. 108, no. 9, p. 093922, 2010. [27] M. Frankowski, J. Chciski, W. Skowroski, and T. Stobiecki, \Perpendic- ular magnetic anisotropy in uence on voltage-driven spin-diode eect in magnetic tunnel junctions: A micromagnetic study," Journal of Mag- netism and Magnetic Materials , vol. 429, pp. 11 { 15, 2017. [28] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, \Enhanced gilbert damping in thin ferromagnetic lms," Phys. Rev. Lett. , vol. 88, p. 117601, Feb 2002. [29] G. Allen, S. Manipatruni, D. E. Nikonov, M. Doczy, and I. A. Young, \Experimental demonstration of the coexistence of spin Hall and Rashba eects intantalum/ferromagnet bilayers," Phys. Rev. B , vol. 91, p. 144412, Apr 2015. [30] M. Cecot, L. Karwacki, W. Skowro nski, J. Kanak, J. Wrona, A._Zywczak, L. Yao, S. Dijken, J. Barna s, and T. 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2019-02-20

We report on the Gilbert damping parameter $\alpha$, the effective magnetization $4\pi M_{eff}$, and the asymmetry of the $g$-factor in bottom-CoFeB(0.93~nm)/MgO(0.90--1.25~nm)/CoFeB(1.31~nm)-top as-deposited systems. Magnetization of CoFeB layers exhibits a specific noncollinear configuration with orthogonal easy axes and with $4\pi M_{eff}$ values of $+2.2$ kG and $-2.3$ kG for the bottom and top layers, respectively. We show that $4\pi M_{eff}$ depends on the asymmetry $g_\perp - g_\parallel$ of the $g$-factor measured in the perpendicular and the in-plane directions revealing a highly nonlinear relationship. In contrast, the Gilbert damping is practically the same for both layers. Annealing of the films results in collinear easy axes perpendicular to the plane for both layers. However, the linewidth is strongly increased due to enhanced inhomogeneous broadening.

CoFeB/MgO/CoFeB structures with orthogonal easy axes: perpendicular anisotropy and damping

1902.07563v1

Intrinsic and non-local Gilbert damping in polycrystalline nickel studied by Ti:Sapphire laser fs spectroscopy J Walowski1, M Djordjevic Kaufmann1, B Lenk1, C Hamann2 and J McCord2, M M unzenberg1 1Universit at G ottingen, Friedirch-Hund-Platz 1, 37077 G ottingen, Germany 2IFW Dresden, Helmholtzstrae 20, 01069 Dresden E-mail: walowski@ph4.physik.uni-goettingen.de Abstract. The use of femtosecond laser pulses generated by a Ti:Sapphire laser system allows us to gain an insight into the magnetization dynamics on time scales from sub-picosecond up to 1 ns directly in the time domain. This experimental technique is used to excite a polycrystalline nickel (Ni) lm optically and probe the dynamics afterwards. Dierent spin wave modes (the Kittel mode, perpendicular standing spin-wave modes (PSSW) and dipolar spin-wave modes (Damon-Eshbach modes)) are identied as the Ni thickness is increased. The Kittel mode allows determination of the Gilbert damping parameter extracted from the magnetization relaxation time . The non-local damping by spin currents emitted into a non-magnetic metallic layer of vanadium (V), palladium (Pd) and the rare earth dysprosium (Dy) are studied for wedge-shaped Ni lms 1 nm 30 nm. The damping parameter increases from = 0:045 intrinsic for nickel to  > 0:10 for the heavy materials, such as Pd and Dy, for the thinnest Ni lms below 10 nm thickness. Also, for the thinnest reference Ni lm thickness, an increased magnetic damping below 4 nm is observed. The origin of this increase is discussed within the framework of line broadening by locally dierent precessional frequencies within the laser spot region.arXiv:0805.3495v1 [cond-mat.other] 22 May 2008Gilbert damping in Nickel thin lms 2 1. Introduction The understanding of picosecond-pulsed excitation of spin packets, spin wave modes and spin currents is of importance in developing a controlled magnetic switching concept beyond the hundred picosecond timescale and to test the speed of magnetic data storage media heading to the physical limits. Over the last years profound progress has been made within that eld by using femtosecond laser spectroscopy. The recent discoveries in ultrafast magnetization dynamics are heading to a new understanding [1{5] and new all-optical switching concepts have been discovered [6]. In addition, the all-optical method has developed into a valuable tool to study the magnetization dynamics of the magnetic precession and thereby access magnetocrystalline anisotropies and the magnetic damping [7{11] or the dynamics of magnetic modes in nanometer sized arrays of magnetic structures [12, 13] and single magnetic nanostructures [14, 15]. Naturally, one nds similarities and dierences as compared to magnetic resonance techniques in frequency space (FMR) [16], optical techniques such as Brillouin light scattering (BLS) [17,18] and time-resolved techniques, for example pulsed inductive magnetometry (PIMM) [19]. Advantages and disadvantages of the dierent techniques have already been compared in previous work [20{22]. The same concepts can be applied to the femtosecond-laser-based all-optical spectroscopy techniques. Here we discuss their abilities, highlighting some aspects and peculiarities [11,23{27]: i. After excitation within the intense laser pulse, the nature of the magnetic relaxation mechanisms determine the magnetic modes observed on the larger time scale [5]. For a Ni wedge dierent modes are found as the thickness is increased: coherent precession (Kittel mode), standing spin waves (already found in [28]) and dipolar surface spin waves (Damon-Eshbach modes) appear and can be identied. ii. Magnetic damping has been extracted by the use of fs spectroscopy experiments already in various materials, epitaxial lms, as a function of the applied eld strength, eld orientation and laser excitation power [7{11]. Using the Kittel mode, we study the energy dissipation process caused by non-local damping by spin currents [29] in Ni by attaching a transition metal lm (vanadium (V), palladium (Pd) and a rare earth lm (dysprosium (Dy)) as a spin sink material and compare them to a Ni reference sample. The present advantages and disadvantages of the method are discussed. iii. A modication of the magnetic damping is found for the thinnest magnetic layers below 4 nm. The understanding of this eect is of high interest because of the increase in methods used to study magnetic damping processes in the low eld region in the current literature. We present a simple model of line broadening known from FMR [30{32] and adapted to the all-optical geometry that pictures the eect of the increased intrinsic apparent damping observed. Therein a spread local magnetic property within the probe spot region is used to mimic the increased apparent damping for the low eld region.Gilbert damping in Nickel thin lms 3 a) b)Side view: Figure 1. a) Schematics of the pump probe experiment to determine the change in Kerr rotation as a function of the delay time . b) Experimental data on short and long time scales. On top a schematic on the processes involved is given. 2. Experimental Technique The all-optical approach to measuring magnetization dynamics uses femtosecond laser pulses in a pump-probe geometry. In our experimental setup a Ti:Sapphire oscillator generates the fs laser pulses which are then amplied by a regenerative amplier (RegA 9050). This systems laser pulse characteristics are 815 nm central wave length, a repetition rate of 250 kHz, a temporal length of 50 80 fs and an energy of 1J per pulse. The beam is split into a strong pump beam (95% of the incoming power), which triggers the magnetization dynamics by depositing energy within the spot region,Gilbert damping in Nickel thin lms 4 and a weaker probe pulse (5% of the incoming power) to probe the magnetization dynamics via the magneto-optical Kerr eect delayed by the time , in the following abbreviated as time-resolved magneto-optic Kerr eect (TRMOKE). The schematic setup and sample geometry is given in gure 1a). The spot diameters of the pump and probe beam are 60 m and 30m respectively. A double-modulation technique is applied to detect the measured signal adapted from [33]: the probe beam is modulated with a photo-elastic modulator (PEM) at a frequency f1= 250 kHz and the pump beam by a mechanic chopper at a frequency f2= 800 Hz. The sample is situated in a variable magnetic eld (0 150 mT), which can be rotated from 0(in-plane) to 90 (out of plane) direction. The degree of demagnetization can be varied by the pump uence (10 mJ =cm260 mJ=cm2) to up to 20% for layer thicknesses around 30 nm and up to over 80% for layers thinner than 5 nm. The samples studied were all grown on Si(100) substrates by e-beam evaporation in a UHV chamber at a base pressure of 51010mbar. For a variation of the thickness, the layers are grown as wedges with a constant gradient on a total wedge length of 15 mm. 3. Results and discussion 3.1. Kittel mode, standing spin waves and Damon-Eshbach surface modes To give an introduction to the TRMOKE signals
Kerr() measured on the timescale from picoseconds to nanoseconds rst, the ultrafast demagnetization on a characteristic time scaleMand the magnetic precessional motion damped on a time scale is shown for a Ni lm in gure 1b); the schematics of the processes involved on the dierent time scales are given on the top. The change in Kerr rotation
Kerr() shows a sudden drop at= 0 ps. This mirrors the demagnetization within a timescale of 200 fs [34{36]. For the short time scale the dynamics are dominated by electronic relaxation processes, as described phenomenologically in the three temperature model [34] or by connecting the electron-spin scattering channel with Elliot-Yafet processes, as done by Koopmans [36] and Chantrell [4] later. At that time scale the collective precessional motion lasting up to the nanosecond scale is initiated [28, 37]: the energy deposited by the pump pulse leads to a change in the magnetic anisotropy and magnetization, and thus the total eective eld. Within 10 ps the total eective eld has recovered to the old value and direction again. However, the magnetization, which followed the eective eld, is still out of equilibrium and starts to relax by precessing around the eective eld. This mechanism can be imagined as a magnetic eld pulse a few picoseconds long, and is therefore sometimes called an anisotropy eld pulse. The resulting anisotropy eld pulse is signicantly shorter than standard eld pulses [38]. This makes the TRMOKE experiment dierent to other magnetization dynamics experiments. The fact that the situation is not fully described by the model can be seen in the following. Already van Kampen et al. [28] not only observed the coherent precessional mode, they also identied another mode at a higher frequency than the coherentGilbert damping in Nickel thin lms 5 precession mode, shifted by !k;n2Ak2= 2An=t Ni2, the standing spin wave (PSSW) mode. It originates from the connement of the nite layer thickness, where Ais the exchange coupling constant and nis a given order. Here we also present the nding of dipolar propagating spin waves. For all three, the frequency dependence as a function of the applied magnetic eld will be discussed, a necessity for identifying them in the experiments later on. For the coherent precession the frequency dependence is described by the Kittel equation. It is derived by expressing the eective eld in the Landau-Lifshitz-Gilbert (LLG equation) as a partial derivative of the free magnetic energy [39, 40]. Assuming negligible in-plane anisotropy in case of the polycrystalline nickel (Ni) lm and small tilting angles of the magnetization out of the sample plane (eld is applied 35out of plane gure 1a)), it is solved as derived in [41]: != 0s 0Hx 0Hx+0Ms2Kz Ms ; (1) For the standing spin waves (PSSW) a similar equation is given. For the geometry with the eld applied 35out of plane (gure 1a) the frequencies !and!k;ndo not simply add as in the eld applied in plane geometry [41]: != 0s (0Hx+2Ak2 Ms) 0Hx+0Ms2Kz Ms+2Ak2 Ms ; (2) While the exchange energy dominates in the limit of small length scales, the magnetic dipolar interaction becomes important at larger length scales. Damon and Eshbach [42] derived by taking into account the dipolar interactions in the limit of negligible exchange energy, the solution of the Damon-Eshbach (DE) surface waves propagating with a wave vector qalong the surface, decaying within the magnetic layer. The wavelengths are found to be above the >m range for Ni [27]. != 0s 0Hx 0Hx+0Ms2Kz Ms+M2 S 4[1exp(2qtNi)] ; (3) The depth of the demagnetization by the femtosecond laser pulse is given by the optical penetration length opt15 nm (= 800 nm). From the nature of the excitation process in the TRMOKE experiment one can derive that for dierent thicknesses tNiit will change from an excitation of the full lm for a 10 nm lm to a thin excitation layer only for a few 100 nm thick lm; thus the excitation will be highly asymmetric. The model of the magnetic anisotropy eld pulse fails to explain these eects since it is based on a macrospin picture. Another way to look at the excitation mechanism has been discussed by Djordjevic et al. [5]. When the magnetic system is excited, on a length scale of the optical penetration depth short wavelength (high kvector) spin-wave excitations appear. As time evolves, two processes appear: the modes with high frequency owning a fastGilbert damping in Nickel thin lms 6 oscillation in space are damped very fast by giving part of the deposited energy to the lattice. In addition, through multiple magnon interaction lower k-vector states are populated, resulting in the highest occupation of the lowest energy modes at the end (e.g. the PSSW and DE modes here). As the Ni thickness is increased, the excitation prole becomes increasingly asymmetric, favoring inhomogeneous magnetic excitations, as the PSSW mode. The DE modes, due to their nature based on a dipolar interaction, are expected to be found only for higher thicknesses. Figure 2. Change in Kerr rotation after excitation on the long time scale for Cu 2nm = Ni tNinm=Si(100) with tNi= 20 and b) their Fourier transform for dierent applied elds 0150 mT, (35out of plane (blue)). In c) the Fourier power spectra as color maps for three Ni thicknesses tNi= 20, 40 and 220 nm are given. The data overlaid is determined form the peak positions. The straight lines are the analysis of the dierent modes and are identied in the graph (Kittel model), perpendicular standing spin wave (PSSW) and dipolar surface spin wave (Damon Eshbach mode).Gilbert damping in Nickel thin lms 7 The identication of the mode is important in determining a value for the magnetic damping. Figure 2 pictures the identication of the dierent modes and their appearance for dierent Ni thicknesses. The data are handled as follows: for a tNi= 20 nm lm on Si(100), covered with a 2 nm Cu protection layer, in a) the original data after background subtraction and in b) its corresponding Fourier transform, shown for increasing applied magnetic eld. The evolution of the mode frequency and its amplitude increase can be followed. An exponentially decaying incoherent background is subtracted from the data. This has to be done very carefully, to avoid a step- like background which will be evident after Fourier transform as a sum of odd higher harmonics. The frequency resolution is limited by the scan range of 1 ns corresponding to
!=2= 1 GHz. However, since the oscillation is damped within the scan range, the datasets have been extended before Fourier transform to increase their grid points. A color map of the power spectrum is shown in gure 2c), where the peak positions are marked by the data points overlaid. For the 20 nm thick lm with tNi= 20 nmopt only a single mode is observed. The mode is analyzed by 1 indicating the Kittel mode being present (data points and line in gure 2c), top) using Kz= 3:03
104J=m3. With increasing nickel thickness tNi= 40 nm>  opt, the perpendicular standing spin waves (PSSW) of rst order are additionally excited and start to appear in the spectra (gure 2c), middle). An exchange constant A= 9:5
1012J=m is extracted. In the limit of tNi= 220 nmopt(gure 1c)) the excitation involves the surface only. Hence, modes with comparable amplitude prole, e.g. with their amplitude decaying into the Ni layer, are preferred. Consequently DE surface waves are identied as described by 3 and dominate the spectra up to critical elds as high as 0Hcrit= 100 mT. For tNi= 220 nm the wave factor is k= 2m (data points and line in gure 2c), bottom). For larger elds than 100 mT the DE mode frequency branch merges into the Kittel mode [27]. To resume the previous ndings for the rst subsection, we have shown that in fact the DE modes, though they are propagating spin-wave modes, can be identied in the spectra and play a very important role for Ni thicknesses above tNi= 80 nm. They appear for thicknesses much thinner than the wavelength of the propagating mode. Perpendicular standing spin waves (PSSW) give an important contribution to the spectra for Ni thicknesses above tNi= 20 nm. For thicknesses below tNi= 20 nm we observe the homogeneously precessing Kittel mode only. This thickness range should be used to determine the magnetic damping in TRMOKE experiments. 3.2. Data analysis: determination of the magnetic damping For the experiments carried out in the following with tNi<25 nm the observed dynamics can be ascribed to the coherent precession of the magnetization (Kittel mode). The analysis procedure is illustrated in the following using the data given in gure 3a). A Pd layer is attached to a Ni lm with the thicknesses (Ni 10 nm =Pd 5 nm=Si(100)) to study the non-local damping by spin currents absorbed by the Pd. The dierent spectra with varying the magnetic eld strength from 0 mT 150 mT are plotted from bottomGilbert damping in Nickel thin lms 8 to top (with the magnetic eld tilted 35out of the sample plane). 0 .0 4 5 0 .0 5 0 0 .0 5 5 α 0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 02468 ν [GHz] µ0 He x t [m T ]0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 8 - 4 048 ∆θk [a.u.] τ [p s ]µ0He x t = 1 5 0 m T 1 4 0 m T 1 3 0 m T 1 2 0 m T 1 1 0 m T 1 0 0 m T 9 0 m T 8 0 m T 7 0 m T 6 0 m T 5 0 m T 4 0 m T 3 0 m T 0 m T a) b)5 nm Pd/ 10 n m Ni 5 nm Pd/ 10 n m Ni Figure 3. a) Kerr rotation spectra for a Cu 2nm =Ni 10 nm=Pd 5 nm=Si(100) layer, measured for elds applied from 0 150 mT (35out of plane, (blue)) and the tted functions (white, dashed). b) The magnetic damping and precession frequencies extracted from the ts to the measured spectra. The line is given by the Kittel mode (gray). The data can be analyzed using the harmonic function with an exponential decay within:
kexp  
sin(2(0)) +B(); (4) The precession frequency =!=2and the exponential decay time of the precession amplitude is extracted, where the function B() stands for the background arising from the uncorrelated magnetic and phonon excitations. To determine the Gilbert damping parameter as given in the ansatz by Gilbert, the exponential decay timehas to be related with . The LLG equation is solved under the same preconditions as for equation 1 using an exponential decay of the harmonic precession withinfrom 4. Then the damping parameter and can be expressed by the followingGilbert damping in Nickel thin lms 9 equation [41]: =1   HxKz 0Ms+Ms 2: (5) It is evident from 5 that in order to determine the Gilbert damping from the decay of the Kittel mode , the variables ,MsandKzhave to be inserted, and therefore Kz has to be determined beforehand. In gure 3a), the background B() is already subtracted. The ts using 4 are plotted with the dashed lines on top of the measured spectra. The results are presented in b). The frequencies range from 3 GHz for 30 mT to 7 :5 GHz for the 150 mT applied magnetic eld. They increase linearly with the strength of the applied magnetic eld for high eld values. The extrapolated intersection with the ordinate is related to the square root of the dipolar and anisotropy eld. Using the Kittel equation (1), one determines the out-of-plane anisotropy constant KzofKz= 6:8
104J=m3. The calculated magnetic dampingas a function of the applied eld is given in the graph below: this is mostly constant but increases below 60 mT. Within the ansatz given by Gilbert, the damping constantis assumed to be eld-independent. We nd that this is fullled for most of the values: the average value of = 0:0453(4), consistent with earlier ndings by Bhagat and Lubitz from FMR experiments [43], is indicated by the line in the plot. The given in the following will always be averaged over a eld region where the damping is Gilbert-like. A deviation from this value occurs for the small external eld strengths. It originates for two reasons: tting 5 with a few periods only does not determine a reliable value of the exponential precession decay time and leads to a larger error. Second, magnetic inhomogeneities mapping a spread in anisotropy energies within the probe spot region can also be a source, and this becomes generally more important for even thicker lms below 4 nm [31]. This will be discussed in more detail in the last section of the manuscript. 3.3. Intrinsic damping: nickel wedge For our experiments Ni was chosen instead of Fe or Py as a ferromagnetic layer. The latter would be preferable because of their lower intrinsic damping int, which make the lms more sensitive for detecting the non-local contribution to the damping. The reason for using Ni for our experiments is the larger signal excited in the TRMOKE experiments. The magnetic damping intis used as a reference later on. The dierent spectra with varying the Ni thickness tNiNixnm=Si(100) from 2 nm x22 nm are plotted from bottom to top (with the constant magnetic eld 150 mT and tilted 30 out of plane) in gure 4a). The measurements were performed immediately after the sample preparation, in order to prevent oxidation on the nickel surface caused by the lack of a protection layer (omitted on purpose). The spectra show similar precession frequency and initial excitation amplitude. However, the layers with tNi<10 nm show a frequency shift visually recognized in the TRMOKE data. Furthermore, the precessionGilbert damping in Nickel thin lms 10 amplitude decreases faster for the thinner layers. Figure 5 shows the frequencies and the damping parameter extracted from the measured data in the intrinsic case for the nickel wedge sample (black squares). While the precession frequency given for 150 mT is almost constant above 8 nm Ni thickness, it starts to drop by about 25% for the thinnest layer. The magnetic damping (black squares) is found to increase to up to = 0:1, an indication that in addition to the intrinsic there are also extrinsic processes contributing. It has to be noted that the change in is not correlated with the decrease of the precession frequency. The magnetic damping is found to increase below a thickness of 4 nm, while the frequency decrease is observed below a thickness of 10 nm. A priori ,MsandKzcan be involved in the observed frequency shift, but they can not be disentangled within a t of our eld-dependent experiments. However, from our magnetic characterization no evidence of a change of andMsis found. A saturation magnetization 0Ms= 0:659 T and g-factor of 2.21 for Ni are used throughout the manuscriptzandKzis determined as a function of the Ni thickness, which shows a 1=tNibehavior, as expected for a magnetic interface anisotropy term [44]. The knowledge of the intrinsic Gilbert damping intof the Ni lm of a constant value for up to 3 nm thickness allows us to make a comparative study of the non-local damping0, introduced by an adjacent layer of vanadium (V) and palladium (Pd) as representatives for transition metals, and dysprosium (Dy) as a representative of the rare earths. Both damping contributions due to intrinsic intand non-local spin current damping0are superimposed by: =int+0: (6) They have to be disentangled by a study of the thickness dependence and compared to the theory of spin-current pumping, plus a careful comparison to the intrinsic value inthas to be made. 3.4. Non-local spin current damping: theory Dynamic spin currents excited by a precessing moment in an adjacent nonmagnetic layer (NM) are the consequence of the fact that static spin polarization at the interface follows a dynamic movement of a collective magnetic excitation. The eect has already been proposed in the seventies [45,46] and later calculated within a spin reservoir model with the spins pumped through the interfaces of the material by Tserkovnyak [29, 47]. For each precession, pumping of the spin current results in a corresponding loss in magnetization, and thus in a loss of angular momentum. The spin information is lost and the backward diusion damps the precession of the magnetic moment. In addition to the rst experiments using ferromagnetic resonance (FMR) [48{54] it has been observed in time- resolved experiments using magnetic eld pulses for excitation [55,56]. In fact zAn altered g-factor by interface intermixing can not decrease its value below 2. Also, there is no evidence for a reduced Msfor lower thicknesses found in the Kerr rotation versus Ni thickness data. More expected is a change in the magnetic anisotropy Kz. For the calculation of later on, the in both cases (assuming a variation of Kzor an altered ) the dierences are negligible.Gilbert damping in Nickel thin lms 11 a) b) 0 250 500 750 10 00- 10 - 8 - 6 - 4 - 2 0 2 2 n m 1 8 n m 1 0 n m 1 4 n m 8 n m 7 n m 6 n m 5 n m 4 n m 3 n m 2 n m ∆θk [a.u.] τ [ p s ] 0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 1 0 -8-6-4-20 1 7 n m 2 3 n m 1 4 n m 1 0 n m 8 n m 7 n m 6 n m 5 n m 4 n m 2 n m 3 n m ∆θk [a.u.] τ [p s]Ni r efer enc e x Ni/ 5 nm DydNi = dNi = Figure 4. a) Kerr rotation spectra for nickel layers from tNi= 2 nm22 nm, measured on the nickel wedge tNi= nm Ni=Si(100) and opposed in b) by a nickel wedge Al 2nm = Dy 5nm= tNi= nm Ni=Si(100) with a 5 nm Dy spin-sink layer. the non-local spin current damping is very closely related to the damping by spin- ip scattering described within the s-d current model [57, 58] that uses the approximation of strongly localized d-states and delocalized s-states [59]. A review describes the underlying circuit theory and dynamics of the spin currents at interfaces in detail [60]. The outcome of the theoretical understanding is that the additional Gilbert damping is proportional to the angular momentum Ar;ltransmitted through the interface. Since each interface owns a characteristic re ection and transmission, the size of Ar;ldepends on the matching of the Fermi surfaces. The absolute value is given by the total balance between transmitted angular momentum and the back ow. For the non-local damping 0one nds: 0= ~G"# 4MstFM1 1 +q sf eltanh tNM sd1: (7) The tanh function stems thereby from the diusion prole of the spin currents determined by the spin diusion length sdwithin the non-magnetic material withGilbert damping in Nickel thin lms 12 thicknesstNM. Also, one nds from the analysis the ratio of the electron scattering rateelversus the spin ip rate sf. The total amount of spin current through the interfaces is determined by the interface spin mixing conductance G"#. It is related to the magnetic volume. It is therefore that scales with the thickness of the magnetic layertFM. The eective gyromagnetic ratio altered by the spin-current implies that in addition to an increased damping a small frequency shift will be observed. The non-local Gilbert damping becomes important when it exceeds the intrinsic damping int. 3.5. Non-local damping: vanadium, palladium and dysprosium Diering from other techniques, TRMOKE experiments require optical access for excitation and detection, setting some restrictions to the layer stack assembly that can be investigated with this method: a thick metallic layer on top of the magnetic layer is not practical. Placing the damping layer below the magnetic layer is also unfavorable: by increasing the spin sink thickness the roughness of the metal lm will increase with the metals layer thickness and introduce a dierent defects density, altering int. In the following the nickel thickness will be varied and the spin sink thickness will be kept xed at 5 nm. To warrant that the nickel lms magnetic properties are always comparable to the reference experiment ( Kz,int), they are always grown rst on the Si(100). For the Pd case the damping layer is below the Ni layer. Here the excitation mechanism did not work and the oscillations were too weak in amplitude to analyze the damping , probably due to the high re ectivity of Pd. The results are presented in gure 4b) for the nickel wedge sample Ni xnm=Si(100) with a 5 nm dysprosium (Dy) as a spin sink layer, covered by an aluminum protection layer, as opposed to the nickel wedge sample data without this in a). The nickel layer thickness is varied from 2 nm x22 nm. All spectra were measured in an external magnetic eld set to 150 mT and tilted 30out of plane. For the thinnest Ni thickness, the amplitude of the precession is found to be smaller due to the absorption of the Dy layer on the top. While the precession is equally damped for the Ni thicknesses ranging from 7 to 23 nm, an increased damping is found for smaller thicknesses below this. The dierence in damping of the oscillations is most evident for tNi= 4 and 5 nm. The result of the analysis as described before is summarized in gure 5. In this graph the data are shown for the samples with the 5 nm V, Pd, Dy spin-sink layer and the Ni reference. While for the Ni reference, and Ni with adjacent V and Dy layer, the frequency dependence is almost equal, indicating similar magnetic properties for the dierent wedge-like shaped samples, the frequency for Pd is found to be somewhat higher and starts to drop faster than for the others. The most probable explanation is that this dierence is due to a slightly dierent anisotropy for the Ni grown on top of Pd in this case. Nevertheless, the magnetic damping found for larger thicknesses tNiis comparable with the Ni reference. In the upper graph of gure 5 the Gilbert damping as a function of the Ni layer thickness is shown. While for the Pd and Dy as a spin sink material a additional increase below 10 nm contributing to the damping can beGilbert damping in Nickel thin lms 13 identied, for V no additional damping contribution is found. 0 . 05 0 . 10 0 . 15 0 5 1 0 1 5 20 685 1 0 1 5 0 . 0 0 0 0 . 0 2 5 0 . 0 5 0 dN i [ nm] α ν [GHz] x N i w i th : 5 n m V 5 n m P d 5 n m D y α−αint dN i [n m ] Figure 5. Gilbert damping parameters and frequency as a function of the nickel layer thickness for the intrinsic case and for dierent damping materials of 5 nm V, Pd, and Dy adjacent to the ferromagnet. is extracted from experiments over a large eld region. The ts are made using equation 5 and equation 7. In the inset the data is shown on a reciprocal scale. Below, the frequency is given (150 mT). The lines are guides for the eye. For the adjacent V layer, since it is a transition metal with a low spin orbit- scattering (light material with low atomic number Z), with a low spin- ip scattering rate and thus a spin diusion length larger than the thickness tNM(dsd), no additional damping will occur. For Pd and Dy the situation is dierent: whereas the heavier Pd belongs to the transition metals with a strong orbit-scattering (heavy material with high atomic number Z), Dy belongs to the rare earth materials. It owns a localized 4fGilbert damping in Nickel thin lms 14 magnetic moment: therefore, both own a high spin- ip scattering rate and we expect the latter two to be in the region where ( tsd). In their cases the thickness of 5 nm of the spin-sink layer is chosen to be larger than the spin diusion length ( tNMsd). In this limit the spin current emitted from the magnetic layer through the interface is totally absorbed within the non-magnetic layer. One can simplify 6 to: 0(1) = ~G"# 4Mst1 FM: (8) This is called the limit of a perfect spin sink. The additional non-local spin current damping is expected to behave inversely proportional with the nickel layer thickness t1 FM. The inset gives the analysis and the data point on a reciprocal scale. The slope shows a linear increase for thinner nickel layers, as expected for an inverse proportionality for both the Pd and the Dy. Since the value for the intrinsic damping of the nickel lm increases below 4 nm this contribution has to be subtracted to reveal the spin-current contribution. The value for 0is then found to be 0 :07 for the 2 nm Ni =5 nm Pd lm, which is in the order found by Mizukami by FMR for sputtered Permalloy lms with a Pd spin sink ( 0= 0:04 for 2 nm Py =5 nm Pd) [49, 50]. A further analysis of the thickness dependence of yields values for the prefactor in 7 for Pd (0 :33(3) nm) and Dy (0:32(3) nm) with the t given in the graph. From that value the real part of the interface spin mixing conductance in 7 can be calculated. It is found to be G"#= 4:5(5)
1015 cm1 for the Ni/Pd and Ni/Dy interface. The increase of the intrinsic damping inthas been analyzed using an inverse thickness dependence (prefactor of 0 :1 nm). While it describes the data in the lower thickness range, it can be seen that it does not describe the thickness dependence for the thicker range and thus, probably the increase does not originate from an interface eect. 3.6. Increased damping caused by anisotropy uctuations: consequences for the all-optical approach In this last part we want to focus on the deviation from the intrinsic damping intfor the thin nickel layers itself ( tNi<4 nm). In the low eld range (10 50 mT) small magnetization inhomogeneities can build up even when the magnetization appears to be still saturated from the hysteresis curve (the saturation elds are a few mT). For these thin layers the magnetization does not align parallel in an externally applied eld any more, but forms ripples. The in uence of the ripples on the damping is discussed in reference [32]. In the following we adopt this ansatz to the experimental situation of the TRMOKE experiment. We deduce a length scale on which the magnetization reversal appears for two dierent Ni thicknesses and relate it to the diameter of our probe spot. Lateral magnetic inhomogeneities were studied using Kerr microscopy at dierent applied magnetic elds [44]. Magnetization reversal takes place at low elds of a -0.5 to 2 mT. The resolution of the Kerr microscopy for this thin layer thickness does not allow us to see the extent of the ripple eect in the external eld where the increase of and its strong eld dependence is observed. However, the domains in theGilbert damping in Nickel thin lms 15 demagnetized state also mirror local inhomogeneities. For our Ni xnm=Si(100) sample this is shown in gure 6a) and b). The domains imaged using Kerr microscopy are shown for a 3 nm and a 15 nm nickel layer in the demagnetized state. The domains of the 15 nm layer are larger than the probe spot diameter of 30 m, whereas the domains of the 3 nm layer are much smaller. d =15nmNi d =3nmNi demagnetized demagnetizeda) b)c) d)20µm 20µm Figure 6. a) and b) Kerr microscopy images for the demagnetized state for 15 nm and 3 nm. c) and d) corresponding model representing the areas with slightly varying anisotropy From that observation, the model of local anisotropy uctuations known from FMR [30, 31] is schematically depicted in gure 6c) and d). A similar idea was also given by McMichael [61] and studied using micromagnetic simulations. While for the thick lm the laser spot probes a region of almost homogeneous magnetization state, for the thin layer case the spot averages over many dierent regions with slightly dierent magnetic properties and their magnetization slightly tilted from the main direction averaging over it. The TRMOKE signal determined mirrors an average over the probed region. It shows an increased apparent damping and a smaller resulting from the line broadening and dierent phase in frequency space. While for the thick layer the typical scale of the magnetic inhomogeneity is as large as the probe laser spot given and only 1-2 regions are averaged, for the thinner lm of dNi= 3 nm many regions are averaged within a laser spot, as can be seen in 6b) and d). Because the magnetic inhomogeneity mapping local varying anisotropies becomes more important for smaller elds, it also explains the strong eld dependence of observed within that region. Figure 7 shows data calculated based on the model, in which the upper curve (i)Gilbert damping in Nickel thin lms 16 is calculated from the values extracted from the experimental data for the 10 nm nickel layer, curve (ii) is calculated by a superposition of spectra with up to 5% deviation from the central frequency at maximum and curve (iii) is calculated by a superposition of spectra of 7% deviation from the central frequency at maximum to mimic the line broadening. The corresponding amplitudes of the superposed spectra related to dierent Kzvalues is plotted in the inset of the graph to the given frequencies. The apparent damping is increased by 0.01 (for 5%) and reaches the value given in gure 3b) for the 10 nm lm determined for the lowest eld values of 30 mT. These eects generally become more important for thinner lms, since the anisotropy uctuations arising from thickness variations are larger, as shown by the Kerr images varying on a smaller length scale. These uctuations can be vice versa determined by the analysis. /s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s50/s52 /s55/s46/s48 /s55/s46/s53 /s56/s46/s48 /s32/s32/s77/s32/s91/s97/s46/s117/s46/s93 /s32/s91/s112/s115/s93/s40/s105/s105/s105/s41/s40/s105/s41 /s40/s105/s105/s41/s40/s105/s105/s105/s41/s40/s105/s41/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s65 /s32/s91/s71/s72/s122/s93/s40/s105/s105/s41 Figure 7. a) Datasets generated by superposing the spectra with the frequency spread according to the inset: (i) is calculated from the values extracted from the experimental data for the 10 nm Ni layer, (ii) by a superposition of spectra with up to 5% and (iii) is calculated by a superposition of spectra owing 7% variation from the central frequency at maximum. The average precession amplitude declines faster if a higher spread of frequencies (i.e. dierent anisotropies) are involved.Gilbert damping in Nickel thin lms 17 4. Conclusion To conclude, we have shown that all-optical pump-probe experiments are a powerful tool to explore magnetization dynamics. Although the optical access to the magnetic layer allows an access to the surface only, magnetization dynamics can be explored directly in the time domain, resolving dierent types of spin-wave modes (Kittel mode, perpendicular standing spin waves and Damon-Eshbach dipolar surface waves). This is in contrast to FMR experiments, where the measured data is a response of the whole sample. The obtained data can be similar to the eld-pulsed magnetic excitations and the Gilbert damping parameter , needed for the analysis of magnetization dynamics and the understanding of microscopic energy dissipation, can be determined from these experiments. We have evaluated the contributions of non-local spin current damping for V, Pd and Dy. 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2008-05-22

The use of femtosecond laser pulses generated by a Ti:Sapphire laser system allows us to gain an insight into the magnetization dynamics on time scales from sub-picosecond up to 1 ns directly in the time domain. This experimental technique is used to excite a polycrystalline nickel (Ni) film optically and probe the dynamics afterwards. Different spin wave modes (the Kittel mode, perpendicular standing spin-wave modes (PSSW) and dipolar spin-wave modes (Damon-Eshbach modes)) are identified as the Ni thickness is increased. The Kittel mode allows determination of the Gilbert damping parameter alpha extracted from the magnetization relaxation time tau_alpha. The non-local damping by spin currents emitted into a non-magnetic metallic layer of vanadium (V), palladium (Pd) and the rare earth dysprosium (Dy) are studied for wedge-shaped Ni films 1 nm-30 nm. The damping parameter increases from alpha=0.045 intrinsic for nickel to alpha>0.10 for the heavy materials, such as Pd and Dy, for the thinnest Ni films below 10 nm thickness. Also, for the thinnest reference Ni film thickness, an increased magnetic damping below 4 nm is observed. The origin of this increase is discussed within the framework of line broadening by locally different precessional frequencies within the laser spot region.

Intrinsic and non-local Gilbert damping in polycrystalline nickel studied by Ti:Sapphire laser fs spectroscopy

0805.3495v1

1 Current -induced motion of twisted skyrmions Chendong Jin1, Chunlei Zhang1, Chengkun Song1, Jinshuai Wang1, Haiyan Xia1, Yunxu Ma1, Jianing Wang1, Yurui Wei1, Jianbo Wang1,2 and Q ingfang Liu1,* 1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, People’ s Republic of China . 2Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education , Lanzhou University, Lanzhou 730000, People ’s Republic of China. Abstract Twisted skyrmions, whose helicity angles are different from that of Bloch skyrmion s and Né el skyrmion s, have already been demonstrated in experiments recently. In this work, we first contrast the magnetic structure and origin of the twisted skyrmion with other three types of skyrmion including Bloch skyrmion , Né el skyrmion and antiskyrmion. Following, we investigate the dynamics of twisted skyrmions driven by the spin transfer toque (STT) and the spin Hall effect (SHE) by using micromagnetic simulations. It is found that the spin Hall angle of the twisted skyrmion is related to the dissipative force tensor and the Gilbert damping both for the motions induced by the STT and the SHE, especially for the SHE induced motion, the skyrmion Hall angle depends s ubstantially on the skyrmion helicity. At last, we demonstrate that the trajectory of the twisted skyrmion can be controlled in a t wo dimensional plane with a Gilbert damping gradient . Our results provide the understanding of current -induced motion of twisted skyrmions, which may contribute to the applications of skyrmion -based racetrack memories. Keywords: Twisted skyrmion, spin transfer torque, spin Hall effect _____________________________ *Corresponding author: Qingfang Liu , liuqf @lzu.edu.cn 2 Introduction It has been recognized that the spin -polarized current -induced the motion and reversal of magnetic structures arises as a result of the spin transfer torque (STT) effect [1-4], which has attracted large interests due to the fundamental physics and potential applications in spintronic devices, such as magnetic random access memorie s (MRAM s)[5, 6] , racetrack memori es[7, 8] , nano -oscillator s[9-11] and logic device s[12-14]. Recently, it has been reported that the spin Hall effect (SHE )[15, 16] , generated by the pure spin currents flowed from the heavy metal substrate due to t he strong spin -orbit coupling at the interface of ferromagnet/heavy -metal , is an alternative efficient method to manipulate the magnetization dynamics in magnetic materials [17-20]. Compared with the STT, the SHE does not require currents flow through the magnetic layer, and then reducing the Joule h eat and electromigration, i.e., avoiding the restricted effect of large current density in traditional STT devices [21]. Magne tic skyrmions are chiral spin magnetization structures with topological properties and can be divided into the following types according to different types of Dzyaloshinskii -Moriya interaction (DMI) [22-26]: (i) Bloch skyrmmions are first discovered in bulk non -centrosymmetric B20 -type lattice structures such as MnSi [27], FeCoS i[28-30], and FeGe [31, 32] due to the presence of bulk DMI; (ii) Né el skyrmions are observed in multilayered ultrathin films lacking inversion symmetry with str ong spin -orbit coupling like Ir (111)/F e[33], Ta/CoFeB [34] and Pt/Co [35] due to the presence of interfacial DMI ; (iii) Antiskyrmions are reported in Heu sler compound s such as MnPtSn [36] due to the presence of anisotropic DMI [37, 38] . Recently, at the interface of chiral bulk Cu2OSeO 3 below a certain thickness, the so -called twisted skyrmions are demonstrated directly by the circularly polarized resonant e lastic x -ray scattering, due to the breaking of translational symmetry at the surface of bulk ferromagnet [39, 40] . Up to now, the dynamics of twisted skyrmions driven by current have not been reported. Therefore, in this paper, on the basis of comparing the magnetic structure , origin and topological properties of the above four types of skyrmion, we focus on the dynamics of twisted skyrmions driven by the STT and the SHE and also analysis the simulation results by using Thiele ’s equations [41]. 3 Micromagnetic simulation details Our magnetic simulation results are performed by using t he Object Oriented MicroMagnetic Framework (OOMMF) public code [42], which includ es the additional modules for bulk DMI, interfacial DMI, anisotropic DMI and twisted DMI. The magnetization dynamics is described by numerical ly solving the Landau -Lifshitz -Gilbert (LLG) equation containing terms of the STT and the SHE [17, 20] , as follow : eff STT SHEd+ + ,m dmm H mdt dt (1) where m is the unit vector of the local magnet ization, is the gyromagnetic ratio, is the Gilbert damping, effH is the effective field including the exchange field, anisotropy field, demagnetization field and DMI effective field . The STT term is expressed as STT s s ( ) ( ),mmv m m v mxx (2) where is the non -adiabatic factor , and vs is the velocity of the conduction electrons with the form s 0s2PvJeM , where J is the current density, e is the electron char ge, P is the spin polarizatio n, is the reduced Planck c onstant , 0 is the permeability of free space , and Ms is the saturation magnetization . The electrons flowing toward + x direction when vs > 0. The SHE term is given by SHE SH HM 0s( ),2m m z jeM L (3) where L is the thickness of the magnetic layer with the value of 1 nm, SH is the spin-Hall angle of Pt substrate with the value of 0.07 , z is the unit vectors o f the surface normal direction, and HMj is the current density injected into the heavy metal . In order to eliminate the influence of the boundary effect on the size and dynamics of skyrmions, t he 2D plane is assumed to 500 × 500 × 1 nm3 (length × width × thickness) with the mesh size of 1 × 1 × 1 nm3, and the initial position of the skyrmion is set in the center of the 2D plane. The material parameters are chosen similar to Ref. [8]: saturation magnetization Ms = 580 × 103 A/m, exchange constant A = 1.5 × 10-11 J/m, perpendicular magnetic anisotropy constant 4 Ku = 8 × 105 J/m3, and DMI strength DDMI = 2.5 ~ 3.5 × 10-3 J/m2. Four types of skyrmions According to the different helicity of skyrmions, there are four types of skyrmions: Bloch skyrmion, Né el skyrmion , antiskyrmion and twisted skyrmion as shown in Figs. 1(a)–(d), respectively. Figure s 1(e)–(h) display the corresponding spatial profiles of the local magnetization across the skyrmions. It can be seen that the mz of the four types of skyrmions are consistent, while the mx and my of the four types of skyrmions are different, which again proves the different distribution of the in-plane magnetic moments of the four types of skyrmions. We emphasize that the distribut ion of the in-plane magnetic moments in the skyrmion structure is determined by the direction of the DMI vector , that is to say, the existence of the twisted skyrmion in this work is achieved by changing the DMI vector, which is much different as the reaso n that observed in the experiments. Figure s 1(i)–(l) show the four types of DMIs: bulk DMI, interfacial DMI, anisotropic DMI and twisted DMI that promise the existence of the Bloch skyrmion, Né el skyrmion , antiskyrmion and twisted skyrmion, respectively. The four types of DMI considered in C4 symmetry can be written as: ˆ ˆ ˆ ˆ BulkDMI ˆ ˆ ˆ ˆ InterDMI ˆ ˆ ˆ ˆ AnisoDMI TwisDMIˆ ˆ ˆ ˆ ( ),2 ˆ ˆ ˆ ˆ ( ),2 ˆ ˆ ˆ ˆ ( ),2 (2i i x i x i y i y i i i x i x i y i y i i i x i x i y i y i iiDE S S x S x S y S y DE S S y S y S x S x DE S S y S y S x S x DE S S ˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆˆ ˆˆ ( ) ( ) ( ) ( ),x i x i y i y ix y S x y S x y S x y (4) where D is the DMI constant representing the DMI strength , iS is the atomic moment unit vector, ˆx and ˆy are the unit vectors in the model. Topological properties of four types of skyrmions A. Helicity, winding number and topological number In order to better understand the helicity and winding number of skyrmions , we use the two-dimensional polar coordinates to describe a general magnetic skyrmion structure, as shown in Fig . 2 which display s a Bloch skyrmion, as 5 example, in the polar coordinates with azimuthal angle ( ) and radial coordinate ( ). Therefore , the unit vector of the local magnetization mx, my and mz in the C artesian coordinates can be written as [26, 43, 44] : sin ( )cos ( ), sin ( )sin ( ), cos ( ),x y zm m m (5) where () is the radial profile of the perpendicular component of the magnetization, and fro m the center to the boundary, its value chan ges from 0.5 to 0.5; () is the angle between the magnetic moment and the radial coordinate . The vorticity of skyrmions is obtained by calculat ing the full turns of the transverse magnetic moments on the perimeter and is defined by the winding num ber[45] 2 01d ( )2W . Therefore, the winding number W = 1 for twisted skyrmion , Bloch skyrmion and Né el skyrmion as shown in Fig. 3(a), and W = 1 for antiskyrmion as shown in Fig. 3(b). The helicity of a skyrmion is given by ( ) ( 0) W with the value ranging form to, that is, for the Bloch skyrmion, 0.5; for Né el skyrmion , 0 or ; for twisted skyrmion , 0.5, 0 and , and the helicity of the twi sted skyrm ion shown in Fig. 1 (d) equals to 0.25; for antiskyrmion as shown in Fig. 1(c), . The topological number Q relates to the winding number and counts how many times the unit vector along the magnetic moment wraps the unit sphere with the form [26] 1,,4mmQ qdxdy q mxy (6) where q is the topological density. Figure s 3(c) and (d) show the topological densities corresponding to the magnet ic skyrmions shown in Figs. 3(a) and (b), respectivel y. It can be seen that Q = 1 in Fig. 3(c) and Q = 1 in Fig. 3(d), i.e., Q = W when the spins point down in the central region and point up in the boundary region. B. Skyrmion size and d issipative force tensor The diameter of twisted skyrmion size ( d) is usually defined as the distance from in -plane to in -plane magnetization, i.e., the distance between the region mz = 0, as shown in the inset of Fig. 4. The dissipative force tensor D is used to describe the effect of the dissipative forces on the moving skyrmion [46-48]. For a single twisted skyrmion, D is given by 6 0 14 , ,0 4mmdxdyxx DDDD (7) wher e D is the diagonal element of the dissipative tensor and also called dissipative parameter. The dissipative parameter D is determined by t he diameter and domain wall width of the twisted skyrmion. Therefore, both d and D are affected by DMI strength as shown in Fig. 4. With the increase in DDMI from 2.5 to 3.5 mJ/m2, d increases from 7.9 to 34.8 nm and D increases from 1.0577 to 1.961, respectively, for the twisted skyrmion. Dynamics of twisted skyrmion driven by the STT To understand the STT-induced motion of the twisted skyrmion s, we first use the Thiele equation [41] to describe the dynamics of the four kinds of skyrmions mentioned above by casting the L LG Eqs. (1) and (2) to the following equation [46, 47] : s d s d( - ) ( ) 0,v v v v D G (8) where G is the gyrovector with the form G = (0 0 G) = (0 0 4Q), and vd is the drift velocity of the skyrmion. When the velocity of the conduction electrons vs applied along the x direction, vd = (vx, vy) is derived from Eq. (8) as 2 xs 2 2 2 ys 2 2 2()+,() ().QvvQ v Q vQ D D D (9) It can be seen that the direction of the skyrmion deviates from the direction of the conduction elect rons when, and this phenomenon is called the skyrmion Hall effect and can be further defined by the skyrmion Hall angle x Sky y22 xy= sign( ) arccos( ),vv vv (10) which defines the angle in the range from 180o to 180o. For the situation of STT -induced skyrmion motion, the sign of the vx is always the same with vs, i.e., the skyrmion Hall angle is in the range of (-90o, 90o), and therefore the Eq. (10) can be reduced to Sky 22()= arctan( )Q Q D D . The trajectories of the f our types of skyrmion driven by the in -plane STT with vs = 100 m/s, = 0.4, = 0.2 and 7 DDMI = 3 mJ/m2 is shown in Fig. 5. The positions of the skyrmions are obtain ed by solving the guiding center ( Rx, Ry) with the form [49, 50] xy ,xqdxdy yqdxdy R = , R = qdxdy qdxdy (11) where q is the topological density. One can see that the antiskyrmion deflects to the y direction, while for Bloc h skyrmion, Né el skyrmion and twisted skyrmion deflect to the y direction, i.e., θSky of the skyrmions with Q =1 (antiskyrmion) and Q =1 (Bloch, Né el and twisted skyrmion ) equal to 12.89o and 12.89o, respectively. Following we focus on the STT -induced motion of twisted skyrmion with differen t conditions, as shown in Fig. 6. Figure s 6 (a) and (b) show the vx and vy as a function of vs for different with = 0.2 and DDMI = 3 mJ/m2, respectively. It can be seen that vx and vy both increase linearly with the increase in vs for different α, it should be also note that vy is a negative value for < , a positive value for > , and zero for = . Then we chose the situation of vs = 100 m/s to investigate the skyrmion Hall angle of the twisted skyrmion as a function of vs, as shown in Fig. 6(c), the skyrmion Hall angel Sky remains almost unchanged with the increase in vs. Figure 6(d) shows the simulation and calculation of Sky as a function of with = 0.2 , the skyrmion Hall angle θSky decreases from 13.7 o to 12.89 o with the increasing from 0. 01 to 0.4. According to Eq s. (9) and (10), both the velocity and the skyrmion Hall angle Sky are affected by the dissipative parameter D, and the dissipative parameter D is determined by the DMI strength DDMI. Therefore, it is necessary to investigate the dynamics of the twisted skyrmion under different DDMI, as shown in Figs. 6 (e) and (f) wit h vs = 100 m/s, = 0.4 and = 0.2 . vx increases at first and then decreases w ith DDMI increasing from 2.5 to 3.5 mJ/m2, while vy keeps decreasing (the a bsolute value of vy is continuously increasing ), and both simulation and calculation results support that the corresponding skyrmion Hall angle Sky decreases from 11.4 o to 16.8 o ( the absolute value of Sky is proportional to the DDMI). We have known that the STT -induced twisted skyrmion motion is affected by the damping in the previous paragraph . Following, we investigate the dynamics of twisted skyrmion induced by the STT under a damp ing gradient, as shown in Fig. 7. Figure 7(a) shows t he position along the y axis of the twisted skyrmion as a functio n of distance along the x axis 8 with vs = 100 m/s, = 0.4 and DDMI = 3 mJ/m2. The damping decreases from 0.5 to 0.25 linearly from 0 to 50 nm along the x axis, as indicated by the color code. Figure 7 (b) sho ws the skyrmion Hall angle Sky of the twisted skyrmion as a function of its position along the x axis. In the region > , the twisted skyrmion moves along the x axis direction from 0 nm and deflects in the –y direction until moving to the x axis of 20 nm , where = = 0.4 ; from the region of 20 to 50 nm along x axis, the twisted skyrmion begins to deflect in the + y direction because of < . Therefore , the trajectory of twisted skyrmion induced by the ST T can be controlled under a damping gradient. Dynamics of twisted skyrmion driven by the SHE SHE -induced motion of antiskyrmion has already been studied in Ref. [38], which demonstrates that the antiskyrmion Hall angle depends on the direction of the current strongly. In thi s section, we focus on the SHE -induced motion s of the skyrmio ns whose winding number W = 1(Bloch, Né el and twisted skyrmion ). The LLG Eqs. (1) and (3) can be cast into the following form : d d HM 4 ( ) 0 v v B J D GR (12) where G = (0 0 4) due to Q = 1, B is linked to the SHE , and the sign of B is determined by th e SHE angle; R(χ) is the in-plane rotation matrix with the form cos sin()sin cosR [49, 51] . When the current JHM injected into the heavy metal along the x direction, vd = (vx, vy) is derived from Eq. (12) as x HM 22 y HM 22cos sin,1 sin cos.1v B J v B J D D D D (13) The skyrmion Hall angle Sky can be obtained by the Eq. (10), which is in the range of 180o to 180o. The Eq. (13) suggests that the direction of motion of the skyrmions depends on their helicities. Therefore, we first investigate the trajectories of skyrmions driven by the SHE with JHM = 10 × 1010 A/m2, = 0.2 and DDMI = 3 mJ/m2 for different helicities of skyrmions, as shown in Fig. 8. These skyrmions with different helicities are achieved by changing the direction of DMI vector. The simulation results in Fig. 8(a) show that the skyrmion Hall angles Sky are 150.4o, 9 165.4o, 121.4o, 75.6o, 29.6o, 14.6o, 58.6o and 104.4o for the helicities χ = 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75 and , respective ly. Figure 8(b) shows the skyrmion Hall angle as a function of the helicity both supported by simulations and calculations . Following we take the case of χ = 5 (the twisted skyrmion shown in Fig. 1(d) ) and investigate the motion induced by the SHE , as shown in Fig. 9. Figure 9(a) shows the simulation results of vx and vy of the twisted skyrmion as a function of JHM with = 0.2 and DDMI = 3 mJ/m2. It can be seen that vx and vy both increase linearly with the increase in JHM, and the corresponding skyrmion Hall angle Sky is shown in Fig. 9(b). The skyrmion Hall angle Sky almost remains at 29.6o when JHM is no more than 200 × 1010 A/m2, while for the case JHM =500 × 1010 A/m2, the skyrmion Hall angle Sky decreases to 28.9o. This is because the size of the twisted skyrmion, i.e., the dissipative parameter D, increases slightl y with JHM increasing to 500 × 1010 A/m2, the skyrmion Hall angle Eq. (10) can be reduced to Sky1= arctan( ).1+D D (14) For χ = 5, which indicates that the skyrmion Hall angle Sky decreases with the increase in D. Figure 9(c) shows the simulation results of vx and vy of the twisted skyrmion as a function of with JHM =100 × 1010 A/m2 and DDMI = 3 mJ/m2, vx first increases and then decreases w ith increasing from 0.01 to 1, while vy keeps decreasing (the a bsolute value of vy decreases at first and then increases), and therefore the corresponding skyrmion Hall angle Sky decreases from 44.3o to 8.2o (the trend of Sky is consistent with vy), which also supported by calculation, as shown in Fig. 9(d). Figure 9(e) shows that vx and vy both increases with DDMI increasing from 2.5 to 3.5 mJ/m2 when JHM = 100 × 1010 A/m2 and = 0.2. Figure 9(f) shows that the corresponding skyrmion Hall angle Sky decreases with the increase in DDMI, which is similar to the res ults by calculating the Eq. (14) with the increase in D. In contrast to the STT -induced twisted skyrmion motion under a damping gradient , we investigate th e dynamics of twisted skyrmion driven by the S HE under a damping gradient , as shown in Fig. 10. Figure 10 (a) shows the trajectory of the twisted skyrmion as a functio n of its position along the x axis with JHM = 100 × 1010 A/m2 and DDMI = 3 mJ/m2. The damping increases from 0.2 to 1.2 linearly from 0 to 20 0 nm along the x axis, as indicated by the color code. Figure 10 (b) 10 shows the corresponding skyrmion Hall angle Sky as a function of its position along the x axis. The Eq. (14) implies that: in the region 1D > 0, the twisted skyrmion moves along the x axis direction from 0 nm and deflects in the y direction until moving to the x axis of 114 nm where 1D = 0; in the region 1D < 0, i.e., from 114 to 200 nm along x axis, the twisted skyrmion deflects in the y direction . Therefore , the trajectory of the SHE -induced motion of twisted skyrmion can also be controlled by a damping gradient. Conclusions In summary, we first introduce the magnetic structure and the corresponding DMI of the twisted skyrmion in contr ast to that of Bloch skyrmion , Né el skyrmion and antiskyrmion. Furthermore, we discuss and calculate the helicity, winding number, topological number, size and dissipative force tensor of the twisted skyrmion, which pave the way for the following study of the dynamics of twisted skyrmion driven by the STT and the SHE . For the STT -induced motion of twisted skyrmion , it is found that the skyrmion Hall angle is determined by the topological number, the dissipative force tensor and the difference between the Gilbert damping and the non -adiabatic factor . For the SHE -induced motion of twisted skyrmion, apart from the dissipative force tensor and the Gilbert damping , the skyrmion angle depends on the helicity significantly. At last, we demonstrate that the trajectories of both the STT -induced and the SHE -induced motion of twisted skyrmion can be controlled by a Gilbert damping gradient . These results may present guidance for the design of twisted skyrmion -based racetrack memories . Acknowledgments This work is supported by National Science Fund of China (11574121 and 51771086 ). C. J. acknowledges the funding by the China Scholarship Council. References [1] J.C. Slonczewski, Current -driven excitation of magnetic multilayers, J. Magn. Magn. Mater. 159, L1 (1996) . [2] L. 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Porter, OOMMF User’ s Guide, v. 1.0 (National Institute of Standards and Technology, Gaithersburg, MD, 1999); NIST Interagency Report No. NISTIR6376, 1999. [43] S.L. Zhang, G. van der Laan, T. Hesjedal, Direct experimental determination of the topological winding number of skyrmions in C u2OSeO3, Nat. Commun. 8, 14619 (2017) . [44] X. Yu, K. Shibata, W. Koshibae, Y . Tokunaga, Y . Kaneko, T. Nagai, K. Kimoto, Y . Taguchi, N. Nagaosa, Y . Tokura, Thermally activated helicity reversals of skyrmions, Phys. Rev. B 93, 134417 (2016) . [45] Y . Yamane, J. Sinova, Skyrmion -number dependence of spin -transfer torque on magnetic bubbles, J. Appl. Phys. 120, 233901 (2016) . [46] K. Everschor, M. Garst, R. Duine, A. Rosch, Current -induced rotational torques in the skyrmion lattice phase of chira l magnets, Phys. Rev. B 84, 064401 (2011) . [47] K. Everschor, M. Garst, B. Binz, F. Jonietz, S. Mü hlbauer, C. Pfleiderer, A. Rosch, Rotating skyrmion lattices by spin torques and field or temperature gradients, Phys. Rev. B , 86, 054432 (2012) . [48] J. Iwasaki, M. Mochizuki, N. Nagaosa, Universal current -velocity relation of skyrmion motion in chiral magnets, Nat. Commun. 4, 1463 (2013) . [49] N. Papanicolaou, T. Tomaras, Dynamics of magnetic vortices, Nucl. Phys. B 360, 425 (1991) . [50] C. Moutafis, S. Komineas, J. Bland, Dynamics and switching processes for magnetic bubbles in nanoelements, Phys. Rev. B 79, 224429 (2009) . [51] M. Knoester, J. Sinova, R. Duine, Phenomenology of current -skyrmion interactions in thin films with perpendicular magnetic anisotropy, Phys. Rev. B 89, 064425 (2014) . 13 Figures FIG. 1. Four types of skyrmions. (a) –(d) display the magnetization dist ribution of Bloch skyrmion, Né el skyrmion , antiskyrmion and twisted skyrmion , respectively. We only intercept the central region of the 2D plane with the size of 50 nm × 50 nm. The red, white and blue represent where the z comp onent of the magnetization is positive, zero and negative, respectively. The black arrows denote the distribution of the in -plane magnetization. (e)–(h) are the spatial profiles of the local magnetization corresponding to the yellow dotted line which marked in the Fig. 1(a). (i)–(l) are the configuration s of bulk DMI, interfacial DMI ，anisotropic DMI and twisted DMI , respectively. The orange arrows denote the directions of the DMI vector. 14 FIG. 2. Schematic of a general skyrmion in two-dimensional polar co ordinates ., , and () indica te the radial coordinate, azimuthal angle, skyrmion helicity and the angle between the magnetic moment and the radial coordinate, respectively. 15 FIG. 3. (a) display s the magnetization distribution s of twisted skyrmion , Bloch skyrmion and Né el skyrmion with W = 1. (b) display s the magnetization distribution of anti skyrmion with W = 1. (c) and (d) show the distribution s of topological density corresponding to the magnetizations shown in (a) and (b) with Q = 1 and Q = 1, respectively. 16 FIG. 4. Skyrmion diameter (d) and the diagonal element of the dissipative tensor (D) as a function of DMI stre ngth. The inset is the spatial profile of mz across the twisted skyrmion. It should be note that the twisted skyrmion exist s stably in region of 250 nm × 250 nm, the diagram only show the central part of 50 nm × 50 nm . 17 FIG. 5. The trajectories of four types of skyrmion driven by the STT. The initial position of the skyrmions is at the center of the 2D magnetic film, the size of 2D plane is 250 nm × 250 nm, vs = 100 m/s in x direction , = 0.4, = 0.2 and DDMI = 3 mJ/m2. The big yellow solid arrow and white dotted arrow s represent the direction of conduction electrons and the trajectories of skyrmions, respectively. It should be note that the four types of skyrmions are enlarged with the purpose to see their helicities clearly . The actual sizes of the four skyrmions are almost the same as the skyrmion at the center position. 18 FIG. 6. The STT-induced motion of the twisted skyrm ion (0.25). (a) and (b) display the vx and vy as a function of vs for = 0.01, 0.1, 0.2, 0.3 and 0.4 with = 0.2 and DDMI = 3 mJ/m2, respectively. (c) T he skyrmion Hall an gle Sky as a function of vs corresponding to the situation of = 0.4 shown in Figs. (a) and (b). (d) The skyrmion Hall a ngle Sky as a function of α corresponding to the situation of vs = 100 m/s shown in Figs. 6 (a) and (b). (e) and (f) display the skyrmon velocity and the skyrmion Hall angle as a function of DDMI with vs = 100 m/s, = 0.4 and = 0.2 , respectively. 19 FIG. 7. The STT -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance (y axis) of the skyrmion and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively. The initial position of the skyrmions is defined as 0 nm both in x and y axis, vs = 100 m/s, = 0.4 and DDMI = 3 mJ/m2. The c olor code represents that the damping decreases from 0.5 to 0.25 linearly in the region from 0 to 50 nm along the x direction. The red dotted line represents the position w here = . 20 FIG. 8. The SHE -induced motion of skyrm ions with different (a) The trajectories of eight types of skyrmions with χ = 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75 and driven by the SHE. The initial position of the eight skyrmions is in the center of the 2D magnetic film whose size is 250 nm × 250 nm, = 0.2 and DDMI = 3 mJ/m2. The big yellow solid arrow denote s the direction of current JHM = 10 × 1010 A/m2. The wh ite dotted arrow s represent the trajectories of skyrmions. It also should be note here that the eight types of skyrmi ons are enlarged to see their helicities clearly . The actual sizes of the eight skyrmions are almost the same as them at the center position. (b) The skyrmion Hall angle Sky as a function of the helicitiy . The black solid squares c orrespond to the e ight types of skyrmion in Fig. 8(a), and the black hollow squares are calculated by the equation. 21 FIG. 9. The SHE -induced motion of the twisted skyrm ion (0.25). (a) and (b) display the skyrmion velocity and skyrmion Hall angle Sky as a function of JHM with = 0.2 and DDMI = 3 mJ/m2, respectively. (c) and (d) denote the skyrmion velocity and skyrmion Hall angle Sky as a function of with JHM = 100 × 1010 A/m2 and DDMI = 3 mJ/m2, respectively. (e ) and (f) represent the skyrmion velocity and skyrmion Hall angle Sky as a function of DDMI with JHM = 100 × 1010 A/m2 and = 0.2, respectively. 22 FIG. 10. The SHE -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance (y axis) of the skyrmion and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively. The initial position of the skyrmions is defined as 0 nm both in x and y axis, JHM = 100 × 1010 A/m2 and DDMI = 3 mJ/m2. The c olor code represents that the damping increases from 0.2 to 1.2 linearly in the region from 0 to 200 nm along the x direction. The red dotted line represents the position w here 1D = 0, i.e., the skyrmion Hall angle Sky = 0o.

2019-03-07

Twisted skyrmions, whose helicity angles are different from that of Bloch skyrmions and N\'eel skyrmions, have already been demonstrated in experiments recently. In this work, we first contrast the magnetic structure and origin of the twisted skyrmion with other three types of skyrmion including Bloch skyrmion, N\'eel skyrmion and antiskyrmion. Following, we investigate the dynamics of twisted skyrmions driven by the spin transfer toque (STT) and the spin Hall effect (SHE) by using micromagnetic simulations. It is found that the spin Hall angle of the twisted skyrmion is related to the dissipative force tensor and the Gilbert damping both for the motions induced by the STT and the SHE, especially for the SHE induced motion, the skyrmion Hall angle depends substantially on the skyrmion helicity. At last, we demonstrate that the trajectory of the twisted skyrmion can be controlled in a two dimensional plane with a Gilbert damping gradient. Our results provide the understanding of current-induced motion of twisted skyrmions, which may contribute to the applications of skyrmion-based racetrack memories.

Current-induced motion of twisted skyrmions

1903.02812v1

arXiv:2106.13702v1 [math.OC] 25 Jun 2021Perturbed inertial primal-dual dynamics with damping and s caling terms for linearly constrained convex optimization problems⋆ Xin Hea, Rong Hub, Ya-Ping Fanga,∗ aDepartment of Mathematics, Sichuan University, Chengdu, S ichuan, P.R. China bDepartment of Applied Mathematics, Chengdu University of I nformation Technology, Chengdu, Sichuan, P.R. China Abstract Weproposeaperturbedinertialprimal-dualdynamicwithdampingan dscalingcoefficients, whichinvolvesinertial terms both for primal and dual variables, for a linearly constrained convex optimization problem in a Hilbert setting. With different choices of damping and scaling coefficients, by a Lyapunov analysis approach we discuss the asymptotic properties of the dynamic and prove its fast conve rgence properties. Our results can be viewed extensions of the existing ones on inertial dynamical systems for t he unconstrained convex optimization problem to the linearly constrained convex optimization problem. Keywords: Perturbed inertial primal-dual dynamic, linearly constrained conve x optimization problem, damping and scaling, Lyapunov analysis approach, convergence rate 1. Introduction 1.1. Problem statement LetH1andH2be two real Hilbert spaces with inner /angb∇acketleft·,·/angb∇acket∇ightand norm /ba∇dbl·/ba∇dbl. Letf:H1→Rbe a differentiable convex function and A:H1→ H2be a continuous linear operator with its adjoint operator AT. Consider the perturbed inertial primal-dual dynamical system ¨x(t)+α(t)˙x(t) =−β(t)(∇f(x(t))+AT(λ(t)+δ(t)˙λ(t))+σAT(Ax(t)−b))+ǫ(t), ¨λ(t)+α(t)˙λ(t) =β(t)(A(x(t) +δ(t)˙x(t))−b)(1) wheret∈[t0,+∞) witht0≥0,σ≥0,α: [t0,+∞)→(0,+∞) is a viscous damping coefficient, β: [t0,+∞)→ (0,+∞) is a scaling coefficient, δ: [t0,+∞)→(0,+∞) is an extrapolation coefficient, and ǫ: [t0,+∞)→ H1 is an integrable source term that can be interpreted as a small exte rnal perturbation. In terms of the dynamic (1), in this paper, we shall develop a fast primal-dual dynamic appro ach to solve the linearly constrained convex optimization problem min xf(x), s.t. Ax =b. (2) ⋆This work was supported by the National Science Foundation o f China (11471230) and the Scientific Research Foundation of the Education Department of Sichuan Province (16ZA0213). ∗Corresponding author Email addresses: hexinuser@163.com (Xin He), ronghumath@aliyun.com (Rong Hu), ypfang@aliyun.com (Ya-Ping Fang) Preprint submitted to Elsevier June 28, 2021The primal-dualdynamic(1) involvesthree important parameters: the dampingcoefficient α(t), the extrapolation coefficient δ(t), and the scalingcoefficient β(t), which play crucialrolesin derivingthe fast convergencepropert ies. The importance of the damping coefficient and the scaling coefficient h as been widely recognized in inertial dynamical approaches[4, 19, 40] as well as fast algorithms [35, 40, 9, 39, 44] for unstrained optimization problems. Recently, the damping technique and the scaling technique were also used to develop inertial primal-dual dynamic approachesand inertial primal-dual algorithms for linearly constra ined optimization problems, see [47, 27, 28, 26]. Extrapolation coefficients were also considered in [47, 27]. LetL(x,λ) andLσ(x,λ) be the Lagrangian function and the augmented Lagrangian funct ion of the problem (2) respectively, i.e., L(x,λ) =f(x)+/angb∇acketleftλ,Ax−b/angb∇acket∇ight and Lσ(x,λ) =L(x,λ)+σ 2/ba∇dblAx−b/ba∇dbl2=f(x)+/angb∇acketleftλ,Ax−b/angb∇acket∇ight+σ 2/ba∇dblAx−b/ba∇dbl2, (3) whereσ≥0 is the penalty parameter and λis the Lagrangian multiplier. Let Ω ⊂ H1×H2be the saddle point set ofL(Lσ). It is known that ( x∗,λ∗)∈Ω if and only if −ATλ∗=∇f(x∗), Ax∗−b= 0.(4) Throughout this paper, we always assume that fis a convex continuously differentiable function and Ω /negationslash=∅. We will investigate the asymptotical behavior of the dynamic (1) with the damping coefficient α(t) =α trand the extrapolation coefficient δ(t) =δts, whereα >0,δ >0, and 0 ≤r≤s≤1. 1.2. Related works 1.2.1. Inertial dynamical systems with damping coefficients Let’s recall some important inertial dynamical systems with damping coefficients for the unstrained optimiza- tion problem minΦ(x), (5) where Φ( x) is a smooth convex function. The following inertial gradient system : (IGSα) ¨x(t)+α(t)˙x(t)+∇Φ(x(t)) = 0, and its perturbed version (IGSα,ǫ) ¨x(t)+α(t)˙x(t)+∇Φ(x(t)) =ǫ(t), have been intensively studied in the literature. When damping coefficie ntα(t) =αwithα >0: (IGS α) becomes the heavy ball with friction system, which was introduced by Polyak [ 36], and the asymptotic behavior has been investigated in [1, 15]; under the assumption/integraltext+∞ t0/ba∇dblǫ(t)/ba∇dbldt <+∞, Haraux and Jendoubi [25] studied the 2asymptotic behavior of solutions of (IGS α,ǫ). When α(t) =α trwithα >0, r∈(0,1): Cabot and Frankel [20] and May [33] investigated the asymptotic behavior of (IGS α) astgoes to infinity; Jendoubi and May [29] generalized the results of [20] to (IGS α,ǫ) with/integraltext+∞ t0/ba∇dblǫ(t)/ba∇dbldt <+∞and/integraltext+∞ t0t/ba∇dblǫ(t)/ba∇dbldt <+∞respectively; Balti and May [13] obtained the O(1/t2r) convergence rate with/integraltext+∞ t0tr/ba∇dblǫ(t)/ba∇dbldt <+∞and theo(1/t1+r) convergence rate with /integraltext+∞ t0t(1+r)/2/ba∇dblǫ(t)/ba∇dbldt <+∞for (IGS α,ǫ); Sebbouh et al. [37] investigated the convergence rate of the va lues along the trajectory of (IGS α,ǫ) under some additional geometrical conditions on Φ( x). When α(t) =α t: Su et al. [40] pointed out that (IGS α) withα= 3 can be viewed as a continuous version of the Nesterov’s accelera ted gradient algorithm ([14, 34]), and obtained the convergence rate Φ( x(t))−minΦ = O(1/t2) asα≥3; Attouch et al. [6] investigated the asymptotic behavior of (IGS α,ǫ) asα≥3 under the assumption/integraltext+∞ t0t/ba∇dblǫ(t)/ba∇dbldt <+∞; May [32] proved an improved convergence rate Φ( x(t))−minΦ = o(1/t2) withα >3; in the case α≤3 of (IGS α) and (IGSα,ǫ), theO(1/t2α/3) rate of convergence can be found in [7, 43]; the optimal converge nce rates under some additional geometrical conditions was studied by [11] for (IGS α) withα >0. For general damping coefficient α(t), it has been investigated by [4, 8, 19]. 1.2.2. Inertial dynamical systems with scaling coefficients Balhag el al. [12] considered following inertial gradient system with t ime scaling and constant damping coefficient: ¨x(t)+α˙x(t)+β(t)∇Φ(x(t)) = 0, (6) for solving problem (5), under the assumption β(t) =eβtwithβ≤α, they can obtain the linear convergence without strong convexity of Φ. From the calculus of variations, Wibis ono et al. [44] proposed the following dynamic ¨x(t)+α t˙x(t)+C(α−1)2tα−3∇Φ(x(t)) = 0, (7) with time scaling β(t) =C(α−1)2tα−3for problem (5) where α >1 andC >0, and obtained the O(1/tα−1) rate of convergence. Fazlyab et al. [23] extended the dynamic (7) to fo llowing dual dynamic for solving problem (2) : ¨λ(t)+α t˙λ(t)+C(α−1)2tα−3∇G(λ(t)) = 0, whereG(λ) = min xL(x,λ),α >1C >0, the convergence rate G(λ∗)−G(λ(t)) =O(1/tα−1) also obtained. In [9], they consider following dynamic: ¨x(t)+α t˙x(t)+β(t)∇Φ(x(t)) = 0 for problem (5), and showed O(1/t2β(t)) rate of convergence under assumption t˙β(t)≤(α−3)β(t). The general damped inertial gradient system with time scaling can be found in [3, 10 , 17]. 1.2.3. Inertial primal-dual dynamics Fortheaffineconstrainedconvexoptimizationproblem(2), themos tpopularnumericalmethodsanddynamics are based on the primal-dual framework. In recent years, many fi rst-order dynamical systems were proposed for 3a better understanding of iterative schemes of the numerical algo rithms, (see [5, 16, 31, 38]). How to extend the dynamics (IGS α) and (IGS α,ǫ) to second-order primal-dual dynamics for solving problem (2) is a p roblem worth studying. Recently, Zeng et al. [47] proposed the following damped p rimal-dual dynamical system for solving the problem (2): ¨x(t)+α t˙x(t) =−∇f(x(t))−AT(λ(t)+δt˙λ(t))−σAT(Ax(t)−b), ¨λ(t)+α t˙λ(t) =A(x(t)+δt˙x(t))−b,(8) In this dynamic, the damping coefficients α(t) =α t,δ(t) =δt. When α >3 andδ=1 2, they showed that the trajectory satisfies the following asymptotic convergence rate L(x(t),λ∗)−L(x∗,λ∗) =O(1/t2),/ba∇dblAx(t)−b/ba∇dbl=O(1/t), (9) they also obtained L(x(t),λ∗)− L(x∗,λ∗) =O(1/t2α/3) withα≤3, δ=3 2α. He et al. [27] and Attouch et al. [3] extended dynamic (8) to solve separable convex optimization p roblems with general conditions. The “second-order” + “first-order” primal-dual dynamics with time sc aling was investigated by [26, 28]. In the next, by the substitution of variables in dynamic (8), let’s illust rate the role of time scaling β(t) in dynamic (1). Suppose that α >3 andδ=1 2in (8), (x∗,λ∗)∈Ω. Let’s make the change of time variable t=υ(p), whereυ:R→Rand lim p→+∞υ(p) = +∞. Set ¯x(p) =x(υ(p)) and¯λ(p) =λ(υ(p)). By the chain rule, we have ˙¯x(p) = ˙x(υ(p))˙υ(p),¨¯x(p) = ˙x(υ(p))¨υ(p)+ ¨x(υ(p))˙υ(p)2 and ˙¯λ(p) =˙λ(υ(p))˙υ(p),¨¯λ(p) =˙λ(υ(p))¨υ(p)+¨λ(˙υ(p))υ(p)2. Then rewritten (8) in terms of ¯ x(·),¯λ(·) and its derivatives, we obtain ¨¯x(p)+/parenleftBig α˙υ(p) υ(p)−¨υ(p) ˙υ(p)/parenrightBig ˙¯x(p) =−˙υ(p)2(∇f(¯x(p))+AT(¯λ(p)+υ(p) 2˙υ(p)˙¯λ(p))+σAT(A¯x(p)−b), ¨¯λ(p)+/parenleftBig α˙υ(p) υ(p)−¨υ(p) ˙υ(p)/parenrightBig˙¯λ(p) = ˙υ(p)2(A(¯x(p)+υ(p) 2˙υ(p)˙¯x(p))−b).(10) This leads to the time scaling coefficient β(p) = ˙υ(p)2and the damping coefficients α(p) =α˙υ(p) υ(p)−¨υ(p) ˙υ(p), δ(p) = υ(p) 2˙υ(p).The convergence rate (9) becomes L(¯x(p),λ∗)−L(x∗,λ∗) =O(1 υ(p)2),/ba∇dblA¯x(p)−b/ba∇dbl=O(1 υ(p)). In the next, we investigate two model examples. First, taking υ(p) =ep, then (10) reads ¨¯x(p)+(α−1)˙¯x(p) =−e2p(∇f(¯x(p))+AT(¯λ(p)+1 2˙¯λ(p))+σAT(A¯x(p)−b)), ¨¯λ(p)+(α−1)˙¯λ(p) =e2p(A(¯x(p)+1 2˙¯x(p))−b).(11) Inthiscase, thedampingcoefficients α(p) =α−1,δ(p) =1 2areconstants,thetimescalingcoefficientis β(p) =e2p, and the convergence rate becomes L(¯x(p),λ∗)−L(x∗,λ∗) =O(1 e2p),/ba∇dblA¯x(p)−b/ba∇dbl=O(1 ep). 4Takingυ(p) =pκwithκ >0, then (10) reads ¨¯x(p)+1+(α−1)κ p˙¯x(p) =−κ2p2(κ−1)(∇f(¯x(p))+AT(¯λ(p)+p 2κ˙¯λ(p))+σAT(A¯x(p)−b)), ¨¯λ(p)+1+(α−1)κ s˙¯λ(p) =κ2p2(κ−1)(A(¯x(p)+p 2κ˙¯x(p))−b).(12) the convergence rate becomes L(¯x(p),λ∗)−L(x∗,λ∗) =O(1 p2κ),/ba∇dblA¯x(p)−b/ba∇dbl=O(1 tκ), the damping coefficient α(p) =1+(α−1)κ p. Forκ≥1, we have 1+( α−1)κ≥α, so damping coefficient similar to (8), where α(t) =α t. 1.3. Organisation In Section 2, we present the rate of convergence in the different c hoice of damping coefficient and extrapola- tion coefficient under the suitable assumptions on time scaling coefficie nt and external perturbation. Section 3 concludes the paper. Some technical proofs and lemmas are postp oned to Appendix . 2. Main results In this paper, we will investigate the dynamic (1) with damping coefficie ntα(t) =α trand extrapolation coefficient δ(t) =δts, whereα >0,δ >0, 0≤r≤s≤1. The the dynamic (1) becomes: ¨x(t)+α tr˙x(t) =−β(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+ǫ(t), ¨λ(t)+α tr˙λ(t) =β(t)(A(x(t) +δts˙x(t))−b).(13) Beforeinvestigatingtherateofconvergence,wefirstdiscussth eexistenceanduniquenessofsolutionsfordynamical system (13). When∇f(x) is Lipschitz continuous on H1, from [3, Theorem 4.2], for any ( x0,λ0,u0,v0), the dynamic (13) has a unique strong global solution ( x(t),λ(t)), in which (i): x(t)∈ C2([t0,+∞),H1),λ(t)∈ C2([t0,∞),H2); (2): (x(t),λ(t)) and (˙x(t),˙λ(t)) are locally absolutely continuous; (3): for almost every t∈[0,+∞), (13) holds, and (x(t0),λ(t0)) = (x0,λ0) and (˙x(t0),˙λ(t0)) = (u0,v0). When∇f(x) is locally Lipschitz continuous on H1, following from the Picard-Lindelof Theorem (see [42, Theorem 2.2]), we can establish the local existence and uniqueness s olution of dynamic (13) as follows: Proposition 2.1. Letfbe continuously differentiable function such that ∇fis locally Lipschitz continuous, β: [t0,+∞)→(0,+∞)be a continuous function, ǫ: [t0,+∞)→ H1be locally integrable. Then for any (x0,λ0,u0,v0), there exists a unique solution (x(t),λ(t))withx(t)∈ C2([t0,T),H1),λ(t)∈ C2([t0,T),H2)of the dynamic (13)satisfying (x(t0),λ(t0)) = (x0,λ0)and(˙x(t0),˙λ(t0)) = (u0,v0)on a maximal interval [t0,T)⊆ [t0,+∞). 5So under the assumptions in Proposition 2.1, we obtain that there ex ists a unique solution ( x(t),λ(t)) defined on maximal interval [ t0,T)⊆[t0,+∞). If we can prove that the derivative of trajectory (˙ x(t),˙λ(t)) is bounded on [t0,T), it follows from assumptions that (¨ x(t),¨λ(t)) is also bounded on [ t0,T). This implies that ( x(t),λ(t)) and its derivative (˙ x(t),˙λ(t)) have a limit at t=T, and therefore can be continued, a contradiction. Thus T= +∞, we obtain the existence and uniqueness of global solution of dynamic (13). To simplify the proof process, we assume that the global solution of dynamic (1) exists. We will discuss the existence and uniqueness of global solution of dynamics (13) in the case r= 0,s∈[0,1] later, and it can be proved similarly for other cases. In order to investigate the convergence rates of dynamic (13) un der different choices of r,s. We construct the different energy functions, fixed ( x∗,λ∗)∈Ω, for any λ∈ H2, define the energy function Eλ,ρ ǫ: [t0,+∞)→Ras Eλ,ρ ǫ(t) =Eλ,ρ(t)−/integraldisplayt t0/angb∇acketleftθ(w)(x(w)−x∗)+wρ˙x(w),wρǫ(w)/angb∇acket∇ightdw, (14) where Eλ,ρ(t) =E0(t)+E1(t)+E2(t), (15) with E0(t) =t2ρβ(t)(Lσ(x(t),λ)−Lσ(x∗,λ)), E1(t) =1 2/ba∇dblθ(t)(x(t)−x∗)+tρ˙x(t)/ba∇dbl2+η(t) 2/ba∇dblx(t)−x∗/ba∇dbl2, E2(t) =1 2/ba∇dblθ(t)(λ(t)−λ)+tρ˙λ(t)/ba∇dbl2+η(t) 2/ba∇dblλ(t)−λ/ba∇dbl2, θ,η: [t0,+∞)→Rare two smooth functions, and ρ≥0. The key point of our proof is to find the appropriate θ(t),η(t) to ensure that the energy function Eλ,ρ ǫ(t) is decreasing. To avoid repeated calculations, we list the main calculatio n procedures in Appendix A.1. 2.1. Case r= 0, s∈[0,1] Let us first consider the case when r= 0,s∈[0,1], i.e., the dynamic (13): ¨x(t)+α˙x(t) =−β(t)(∇f(x(t))+AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+ǫ(t), ¨λ(t)+α˙λ(t) =β(t)(A(x(t) +δts˙x(t))−b),(16) withα >0, δ >0, σ≥0, t≥t0>0. Theorem 2.1. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with ts˙β(t)≤(1 δ−sts−1)β(t), (17) andǫ: [t0,+∞)→ H1is a integrable function with /integraldisplay+∞ t0ts 2/ba∇dblǫ(t)/ba∇dbldt <+∞. Suppose αδ >1whens= 0;δ≤1whens= 1, andσ >0. Let(x(t),λ(t))be a global solution of the dynamic (16)and(x∗,λ∗)∈Ω. Then(x(t),λ(t))is bounded, and the following conclusions hold: 6(i)/integraltext+∞ t0((1 δ−sts−1)β(t)−ts˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞. (ii)/integraltext+∞ t0ts(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)dt <+∞,/integraltext+∞ t0β(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞. (iii)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1 ts/2). (iv) When limt→+∞tsβ(t) = +∞: L(x(t),λ∗)−L(x∗,λ∗) =O(1 tsβ(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1 ts/2/radicalbig β(t)). Proof.Givenλ∈ H2, define energy functions Eλ,ρ(t) andEλ,ρ ǫ(t) same as (15), (14) with r= 0,s∈[0,1],ρ=s 2, and θ(t) =1 δt−s/2, η(t) =1 δ(α−1 δt−s). (18) By computations, we can verify that (A.2) and (A.4) hold. Cases= 0:θ(t) =1 δandη(t) =αδ−1 2δ2. Sinceαδ >1, we obtain that (A.1), (A.3) hold, and then (A.5) holds, θ(t)+ρtρ−1−αtρ−r=1 δ−α <0. (19) It follows from (17) that tρ˙β(t)+(2ρtρ−1−θ(t))β(t) =˙β(t)−1 δβ(t)≤0 (20) for allt≥t0. Taking λ=λ∗, thenLσ(x(t),λ∗)−Lσ(x∗,λ∗)≥0, it follows from (19), (20) and (A.5) that ˙Eλ∗,ρ ǫ(t)≤(1 δ−α)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t) 2δ/ba∇dblAx(t)−b)/ba∇dbl2 +(˙β(t)−1 δβ(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗)) (21) ≤0. SoEλ∗,ρ ǫ(·) is nonincreasing on [ t0,+∞), and then Eλ∗,ρ ǫ(t)≤ Eλ∗,ρ ǫ(t0),∀t≥t0. (22) By the definition of Eλ∗,ρ(·) andEλ∗,ρ ǫ(·), we have 1 2/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl2≤ Eλ∗,ρ ǫ(t0)+/integraldisplayt t0/angb∇acketleft1 δ(x(w)−x∗)+ ˙x(w),ǫ(w)/angb∇acket∇ightdw. By Cauchy-Schwarz inequality, we get 1 2/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl2≤ |Eλ∗,ρ ǫ(t0)|+/integraldisplayt t0/ba∇dbl1 δ(x(w)−x∗)+ ˙x(w)/ba∇dbl/ba∇dblǫ(w)/ba∇dbldw, then applying Lemma Appendix A.1 with µ(t) =/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl, we obtain sup t≥t0/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl ≤/radicalBig 2|Eλ∗,ρ ǫ(t0)|+/integraldisplay+∞ t0/ba∇dblǫ(t)/ba∇dbldt <+∞. (23) 7It is easy to verify Eλ∗,ρ(t)≥0 for all t≥t0, then we have inf t≥t0Eλ∗,ρ ǫ(t)≥ −sup t≥t0/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl×/integraldisplay+∞ t0/ba∇dblǫ(s)/ba∇dblds >−∞ and sup t≥t0Eλ∗,ρ(t)≤ Eλ∗,ρ ǫ(t0)+ sup t≥t0/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl×/integraldisplay+∞ t0/ba∇dblǫ(s)/ba∇dblds <+∞. Thistogetherwith(22)andthedefinitionof Eλ∗,ρ(·)yieldstheboundednessof Eλ∗,ρ(·)andEλ∗,ρ ǫ(·). Byintegrating inequality (21) on [ t0.+∞), it follows the boundedness of Eλ∗,ρ ǫ(·) that (α−1 δ)/integraldisplay+∞ t0/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2dt+/integraldisplay+∞ t0(1 δβ(t)−˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt +σ 2δ/integraldisplay+∞ t0β(t)/ba∇dblAx(t)−b)/ba∇dbl2 <+∞. This together with1 δ< αyields (i)−(ii). Sinceη(t) =αδ−1 2δ2>0, By the boundedness of Eλ∗,ρ(·), we obtain that /ba∇dblx(t)−x∗/ba∇dbl2,/ba∇dblλ(t)−λ∗/ba∇dbl2,/ba∇dbl1 δ(x(t)− x∗)+ ˙x(t)/ba∇dbland/ba∇dbl1 δ(λ(t)−λ∗)+˙λ(t)/ba∇dblare bounded, and then the trajectory ( x(t),λ(t)) is bounded, sup t0∈[t0,+∞)/ba∇dbl˙x(t)/ba∇dbl ≤1 δsup t∈[t0,+∞)/ba∇dblx(t)−x∗/ba∇dbl+ sup t∈[t0,+∞)/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl<+∞, similarly, supt0∈[t0,+∞)/ba∇dbl˙λ(t)/ba∇dbl<+∞, this is ( iii). When lim t→+∞β(t) = +∞, following from the boundedness ofEλ∗,ρ(·), we get Lσ(x(t),λ∗)−Lσ(x∗,λ∗) =O(1 β(t)). SinceLσ(x(t),λ∗)−Lσ(x∗,λ∗) =L(x(t),λ∗)−L(x∗,λ∗)+σ 2/ba∇dblAx(t)−b/ba∇dbl2, then we obtain ( iv). Cases∈(0,1]:There exists t1≥t0such that 1 δt−s+s 2t−1≤α 2,∀t≥t1, (24) this together with (18) yields η(t)≥α 2δ>0,∀t≥t1. (25) We can compute that θ(t)˙θ(t)+˙η(t) 2= 0. Then (A.1)-(A.4) are satisfied for any t≥t1. It follows from (24) that θ(t)+ρtρ−1−αtρ−r=ts/2(1 δt−s+s 2t−1−α)≤ −α 2ts/2,∀t≥t1. (26) By computation, tρ(tρ˙β(t)+(2ρtρ−1−θ(t))β(t)) =ts˙β(t)−(1 δ−sts−1)β(t)≤0,∀t≥t0. (27) 8Letλ=λ∗,Lσ(x(t),λ∗)−Lσ(x∗,λ∗)≥0. Combining (26), (27) and (A.5), we get ˙Eλ∗,ρ ǫ(t)≤ −α 2ts(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t) 2δ/ba∇dblAx(t)−b)/ba∇dbl2 +(ts˙β(t)−(1 δ−sts−1)β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗)) (28) ≤0 for allt≥t1.Eλ∗,ρ ǫ(·) is nonincreasing on [ t1,+∞), Eλ∗,ρ ǫ(t)≤ Eλ∗,ρ ǫ(t1),∀t≥t1. By the definition of Eλ∗,ρ(·) andEλ∗,ρ ǫ(·), for all t≥t1we have 1 2/ba∇dbl1 δt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl2≤ Eλ∗,ρ ǫ(t1)+/integraldisplayt t1/angb∇acketleft1 δw−s/2(x(w)−x∗)+ws/2˙x(w),ws/2ǫ(w)/angb∇acket∇ightdw. By similar arguments in Case s=0 , we obtain the boundedness of Eλ∗,ρ(·) andEλ∗,ρ ǫ(·). Integrating inequality (28) on [ t1,+∞), we get the results ( i)−(ii). SinceEλ∗,ρ(·) is bounded, following from the definition of Eλ∗,ρ(·), we obtain ( iv), sup t≥t0η(t)/ba∇dblx(t)−x∗/ba∇dbl2<+∞ and sup t≥t0/ba∇dbl1 δt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl<+∞. This together with (25) and s∈(0,1] implies sup t≥t0/ba∇dblx(t)−x∗/ba∇dbl<+∞ and sup t≥t0ts/2/ba∇dbl˙x(t)/ba∇dbl ≤1 δsup t≥t0t−s/2/ba∇dblx(t)−x∗/ba∇dbl+ sup t≥t0/ba∇dbl1 δt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl ≤1 δts/2 0sup t≥t0/ba∇dblx(t)−x∗/ba∇dbl+ sup t≥t0/ba∇dbl1 δt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl <+∞. Similarly we have supt≥t0/ba∇dblλ(t)−λ∗/ba∇dbl<+∞, supt≥t0ts/2/ba∇dbl˙λ(t)/ba∇dbl<+∞. Then we obtain the boundedness of (x(t),λ(t)) and (iii). Remark 2.1. From Proposition 2.1, there exists a unique local solution (x(t), λ(t))of the dynamic (16)defined on a maximal interval [t0,T)withT≤+∞. If we pick a appropriate t0>0, following from the proof process in Theorem 2.1 and (iii), we can obtain supt∈[t0,T)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl<+∞, and then T= +∞, the existence and uniqueness of global solution of the dynamic (16)is established. 9Remark 2.2. From Theorem 2.1, we can see that for same damping α(t) =α, choosing another damping δ(t) =δ tsdifferent, the different rates of convergence can be obtained . Taking A= 0,b= 0, we can obtain the O(1/tsβ(t))convergence rate for dynamic (6)under the assumption ts˙β(t)≤(1 δ−sts−1)β(t)withδ >0, so Theorem 2.1 complements the results in [12]. The assumption/integraltext+∞ t0/ba∇dblǫ(t)/ba∇dbldt <+∞for perturbation ǫ(t)has been used in [25] for asymptotic analysis of heavy ball dynamic. Remark 2.3. Whens= 0, choosing β(t)≡1, then(17)is automatically satisfied. Then from (i), we have /integraltext+∞ t0Lσ(x(t),λ∗)−Lσ(x∗,λ∗)dt <+∞. SinceLσ(·,λ∗)is a convex function with respect to first variable, taking ¯x(t) =/integraltextt t0x(s)ds t−t0, we have Lσ(¯x(t),λ∗)−Lσ(x∗,λ∗)≤1 t−t0/integraldisplayt t0Lσ(x(s),λ∗)−Lσ(x∗,λ∗)ds ≤1 t−t0/integraldisplay+∞ t0Lσ(x(s),λ∗)−Lσ(x∗,λ∗)ds. Following from the definition of Lσ(¯x(t),λ∗), we obtain L(¯x(t),λ∗)− L(x∗,λ∗) =O(1/t)and/ba∇dblA¯x(t)−b/ba∇dbl= O(1/√ t), theO(1/t)ergodic convergence rate corresponds to the convergence ra te of the discrete heavy ball algorithm in [24]; for general β(t)withs= 0, the similar convergence rate results can be found in [28]. W hen s= 1, choosing β(t)≡1andδ≤1, theO(1/t)rate of convergence also was investigated in [27, Theorem 4. 4] with r= 0for problem (2), and it is consistent with results of heavy ball dynamic and a lgorithm in [41] for problem (5). In Theorem 2.1, when lim t→+∞tsβ(t) = +∞, we show the O(1/tsβ(t)) convergence rate of Lagrangian function and O(1/ts/2/radicalbig β(t)) convergence rate of constraint, then |f(x(t))−f(x∗)| ≤ L(x(t),λ∗)−Lσ(x∗,λ∗)+/ba∇dblλ∗/ba∇dbl/ba∇dblAx(t)−b/ba∇dbl=O/parenleftBigg 1 ts/2/radicalbig β(t)/parenrightBigg . We only can obtain the O(1/ts/2/radicalbig β(t)) convergence rate of objection function. In the next, we will investigate the best convergence rates of obj ection function and constrain for suitable β(t). When s= 0, let ˙β(t) =1 δβ(t). then˙β(t) β(t)=1 δ, integrating it on [ t0,t], we have β(t) =β(t0) et0/δet δ. In this case, from Theorem 2.1, we can obtain the O(1 et 2δ) convergence rate of objective function and constraint. Letβ(t) =µet/δwithµ >0, we list the following improved convergence rate results, which also can be found in [3, Proposition 6.2] with ǫ(t) = 0. Theorem 2.2. Letβ(t) =µet/δwithµ >0,αδ >1,s= 0,σ≥0. Assume/integraltext+∞ t0/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t)) be a solution of dynamic (16)and(x∗,λ∗)∈Ω. Then: |f(x(t))−f(x∗)|=O(1 et/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1 et/δ). 10Proof.Givenλ∈ H2, recall the energy functions Eλ,ρ(t) andEλ,ρ ǫ(t) from Theorem 2.1 with β(t) =µet/δ, s= ρ= 0. Then tρ˙β(t)+(2ρtρ−1−θ(t))β(t) = 0, this together with (19) and (A.5) yields ˙Eλ,ρ ǫ(t)≤(1 δ−α)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t) 2δ/ba∇dblAx(t)−b)/ba∇dbl2≤0,∀t≥t0,λ∈ H2. (29) So for any λ∈ H2,Eλ,ρ ǫ(·) is nonincreasing on [ t0,+∞) such that, Eλ,ρ ǫ(t)≤ Eλ,ρ ǫ(t0),∀t≥t0. By the definition of Eλ,ρ ǫ(·) andσ≥0, we have f(x(t))−f(x∗)+/angb∇acketleftλ,Ax(t)−b/angb∇acket∇ight ≤1 µet/δ/parenleftbigg Eλ,ǫ(t0)+ sup t≥t0/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl/integraldisplay+∞ t0/ba∇dblǫ(t)/ba∇dbldt/parenrightbigg for anyλ∈ H1andt≥t0. Taking ̺ >/ba∇dblλ∗/ba∇dbl, it follows from Lemma Appendix A.2 that f(x(t))−f(x∗)+̺/ba∇dblAx(t)−b/ba∇dbl ≤1 µet/δ/parenleftBigg sup /bardblλ/bardbl≤̺Eλ,ǫ(t0)+ sup t≥t0/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl/integraldisplay+∞ t0/ba∇dblǫ(t)/ba∇dbldt/parenrightBigg .(30) DenoteC= sup/bardblλ/bardbl≤̺Eλ,ǫ(t0)+supt≥t0/ba∇dbl1 δ(x(t)−x∗)+ ˙x(t)/ba∇dbl/integraltext+∞ t0/ba∇dblǫ(t)/ba∇dbldt. Since̺ >/ba∇dblλ∗/ba∇dbl, sup/bardblλ/bardbl≤̺Eλ,ǫ(t0)≥ Eλ∗,ǫ(t0)≥0, this together with (23) yields 0 ≤C <+∞. Following from (4), we have f(x(t))−f(x∗)≥ −/ba∇dblλ∗/ba∇dbl/ba∇dblAx(t)−b/ba∇dbl, this together with (30) implies /ba∇dblAx(t)−b/ba∇dbl ≤C µ(̺−λ∗)et/δ and then −/ba∇dblλ∗/ba∇dblC µ(̺−λ∗)et/δ≤f(x(t))−f(x∗)≤C µet/δ. We obtain results from above inequalities. Remark 2.4. Whens= 0andβ(t) =µet/δ, Theorem 2.1 obtains O(1 et/2δ)convergence rate of objective function and constraint, it is consistent with convergence rates of d ynamic(11), which is derived from dynamic (8). Theorem 2.2 shows that the rate of convergence is actually O(1 et/δ). Then we can obtain the linear convergence rate of dynamic (16)merely under the convexity assumption of f, and in this case we also allow the penalty parameter σof augmented Lagrangian function to be zero, which is differe nt in Theorem 2.1. Whens∈(0,1), letts˙β(t) = (1 δ−sts−1)β(t). It leads β(t) =ts 0β(t0) e1 δ(1−s)t1−s 0e1 δ(1−s)t1−s ts. Takeβ(t) =µe1 δ(1−s)t1−s tswithµ >0. We investigate the following optimal results. 11Theorem 2.3. Letβ(t) =µe1 δ(1−s)t1−s tswithµ >0, s∈(0,1), σ≥0. Suppose/integraltext+∞ t0ts/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let (x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω. Then |f(x(t))−f(x∗)|=O/parenleftbigg1 e1 δ(1−s)t1−s/parenrightbigg ,/ba∇dblAx(t)−b/ba∇dbl=O/parenleftbigg1 e1 δ(1−s)t1−s/parenrightbigg . Proof.Givenλ∈ H2,recalltheenergyfunctions Eλ,ρ(t)andEλ,ρ ǫ(t)fromTheorem2.1with β(t) =µe1 δ(1−s)t1−s ts, s∈ (0,1), ρ=s 2. Then tρ˙β(t)+(2ρtρ−1−θ(t))β(t) = 0, this together with (19) and (A.5) yields ˙Eλ,ρ ǫ(t)≤ −α 2ts(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t) 2δ/ba∇dblAx(t)−b)/ba∇dbl2≤0,∀t≥t1,λ∈ H2, (31) for some t1≥t0. By similar arguments in Theorem 2.1, we obtain the results. Whens= 1, lett˙β(t) = (1 δ−1)β(t). It leads β(t) =β(t0) t1 δ−1 0t1 δ−1. Takingβ(t) =µt1 δ−1withµ >0. By similar arguments in Theorem 2.2 and Theorem 2.3, we obtain the f ollowing results. Theorem 2.4. Letβ(t) =µt1 δ−1withµ >0,δ≤1,s= 1,σ≥0. Suppose/integraltext+∞ t0t1/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let (x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω. We have |f(x(t))−f(x∗)|=O(1 t1/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1 t1/δ). Remark 2.5. Whens= 1, takingδ= 1andβ(t)≡1, from Theorem 2.4, we obtain O(1 t)convergence rates of objective function and constraint, which improves results in Theorem 2.1 with time scaling β(t)≡1. Remark 2.6. For damping α(t) =α,δ=δ tswiths∈[0,1], theO(1/tsβ(t))convergence rate in Theorem 2.1 shows that convergence results is better as slarger in [0,1]. Conversely, following from Theorem 2.2-Theorem 2.4, when sis smaller in [0,1], we can obtain better optimal convergence rates with suitab leβ(t). 2.2. Case r∈(0,1), s∈[r,1] In the case r∈(0,1),s∈[r,1], the dynamic (1) reads: ¨x(t)+α tr˙x(t) =−β(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+ǫ(t), ¨λ(t)+α tr˙λ(t) =β(t)(A(x(t) +δts˙x(t))−b).(32) withα >0, δ >0, σ≥0, t≥t0>0. We will investigate the convergence properties of dynamic (32). 12Theorem 2.5. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with ts˙β(t)≤(1 δ−τts−1)β(t) (33) andǫ: [t0,+∞)→ H1satisfies/integraldisplay+∞ t0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞, whereτ∈(0,r+s). Assume αδ >1whens=r;τδ≤1whens= 1. Let(x(t),λ(t))be a global solution of the dynamic (32)and(x∗,λ∗)∈Ω. The following results hold: (i)/integraltext+∞ t0((1 δtτ−s−τtτ−1)β(t)−tτ˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞. (ii)/integraltext+∞ t0tτ−sβ(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞,/integraltext+∞ t0tτ−r(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)dt <+∞. (iii)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1 tτ/2). (iv) When limt→+∞tτβ(t) = +∞, L(x(t),λ∗)−L(x∗,λ∗) =O(1 tτβ(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1 tτ/2/radicalbig β(t)). Proof.Givenλ∈ H2, recall energy functions Eλ,ρ(t) andEλ,ρ ǫ(t) from (15), (14) with r∈(0,1),s∈[r,1],ρ=τ 2 and θ(t) =1 δtτ/2−s, η(t) =−1 δtτ−s−r(1 δtr−s+(τ−s)tr−1−α). (34) Then the equations (A.2) and (A.4) are automatically satisfied. We claim that there exists C1<0 andt1≥t0such that 1 δtr−s+τ 2tr−1−α≤C1,∀t≥t1. (35) Indeed, when s=r, sinceαδ >1 andr∈(0,1), there exists t1≥t0such that1 δtr−s+τ 2tr−1−α=1 δ−α+τ 2tr−1≤ 1 2(1 δ−α)<0; whens∈(r,1], since r∈(0,1), there exists t1≥t0such that1 δtr−s+τ 2tr−1−α≤ −α 2<0. Since τ 2<r+s 2≤s, it follows from (35) that 1 δtr−s+(τ−s)tr−1−α≤C1,∀t≥t1, and it yields η(t)≥−C1 δtτ−s−r≥0,∀t≥t1. (36) Sinceτ∈(0,s+r), then there exist t2≥t1such that α(τ−s−r)−(τ−s−1)(τ−s)tr−1<0,∀t≥t2, so we can compute θ(t)˙θ(t)+˙η(t) 2=1 2δtτ−s−r−1(α(τ−s−r)−(τ−s−1)(τ−s)tr−1)<0 13for allt≥t2. Then (A.1) and (A.3) hold for any t≥t2. It follows from (35) that θ(t)+ρtτ 2−1−αtτ 2−r=tτ 2−r(1 δtr−s+τ 2tr−1−α)≤C1tτ 2−r<0,∀t≥t1. (37) By computation, and from (33), we have tρ˙β(t)+(2ρtρ−1−θ(t))β(t) =tτ 2−s(ts˙β(t)−(1 δ−τts−1)β(t))≤0. for allt≥t0. Letλ=λ∗, thenLσ(x(t),λ∗)−Lσ(x∗,λ∗)≥0, this together with (34), (37) and (A.5) yields ˙Eλ∗,ρ ǫ(t)≤C1tτ−r(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σtτ−sβ(t) 2δ/ba∇dblAx(t)−b)/ba∇dbl2(38) +(tτ˙β(t)−(1 δtτ−s−τtτ−1)β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗)) ≤0 for allt≥t2. ThenEλ∗,ρ ǫ(·) is nonincreasing on [ t2,+∞), Eλ∗,ρ ǫ(t)≤ Eλ∗,ρ ǫ(t2),∀t≥t2. Since/integraltext+∞ t0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞, by similar arguments in proof of Theorem 2.1 and using the fact C1<0, we obtain that Eλ∗,ρ(·) andEλ∗,ρ ǫ(·) are bounded on [ t0,+∞), and then ( i),(ii),(iv) hold. It follows from (34), (36) and the definition of Eλ∗,ρ(·) that sup t≥t0t(τ−s−r)/2/ba∇dblx(t)−x∗/ba∇dbl<+∞,sup t≥t0/ba∇dbl1 δtτ/2−s(x(t)−x∗)+tτ/2˙x(t)/ba∇dbl<+∞. Sinces∈[r,1], then sup t≥t0tτ/2/ba∇dbl˙x(t)/ba∇dbl ≤1 δsup t≥t0tτ/2−s/ba∇dblx(t)−x∗/ba∇dbl+ sup t≥t0/ba∇dbl1 δtτ/2−s(x(t)−x∗)+tτ/2˙x(t)/ba∇dbl ≤1 δsup t≥t0t(τ−s−r)/2/ba∇dblx(t)−x∗/ba∇dbl+ sup t≥t0/ba∇dbl1 δtτ/2−s(x(t)−x∗)+tτ/2˙x(t)/ba∇dbl <+∞. Similarly, supt≥t0tτ/2/ba∇dbl˙λ(t)/ba∇dbl<+∞, the result ( iii) holds. If we take β(t) satisfying ts˙β(t)≤(1 δ−(r+s)ts−1)β(t), then for any τ∈(0,r+s), (33) is satisfied, and then we obtain the following results from The orem 2.5. Corollary 2.1. Assume that ts˙β(t)≤(1 δ−(r+s)ts−1)β(t),/integraldisplay+∞ t0t(r+s)/2/ba∇dblǫ(t)/ba∇dbldt <+∞. (39) Suppose αδ >1whens=r;δ(r+s)≥1whens= 1. Let(x(t),λ(t))be a global solution of the dynamic (32). Then for any (x∗,λ∗)∈Ωandτ∈(0,r+s): 14(i)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1 tτ/2). (ii) When limt→+∞tτβ(t) = +∞, L(x(t),λ∗)−L(x∗,λ∗) =O(1 tτβ(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1 tτ/2/radicalbig β(t)). Remark 2.7. In proof process of Theorem 2.5, we can note that the boundedn ess of trajectory (x(t),λ(t))is not guaranteed. If (39)holds, we can obtain sup t≥t0t(τ−s−r)/2(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)<+∞ is satisfied for any (x∗,λ∗)∈Ωandτ∈(0,r+s), then we get that tp(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)is bounded for anyp <0. When objective function fsatisfying the following coercive condition: lim /bardblx/bardbl→+∞f(x) = +∞, (40) we also can obtain the boundedness of x(t)of dynamic (32)from(iv)of Theorem 2.5. Remark 2.8. Takingβ(t)≡1,s= 1, and letting/integraltext+∞ t0t(r+1)/2/ba∇dblǫ(t)/ba∇dbldt <+∞andδ(r+1)≥1. We obtain the O(1/tτ)rate of convergence for any τ∈(0,r+1), sincer∈(0,1),r+ 1>2r, so the results in Corollary 2.1 improve the corresponding results in [27, Theorem 3.4] whic h only obtain the O(1/t2r)convergence rate. In the caseα(t) =α trwithα >0, r∈(0,1), theo(1/tr+1)convergence of (IGSα)and(IGSα,ǫ)for problem (5)have been obtained in [4, Corollary 4.5] and [13, Theorem 1.2] res pectively, which have subtle differences of dynamic (32)for problem (2). The assumption/integraltext+∞ t0t(r+1)/2/ba∇dblǫ(t)/ba∇dbldt <+∞also can find in [13]. By similar discussions in Section 2.1, we obtain the following optimal conv ergence rates of Theorem 2.5, and the proof is similar to Theorem 2.2, so we omit it. Theorem 2.6. Letβ(t) =µe1 δ(1−s)t1−s tτwithµ >0, τ∈(0,r+s), r∈(0,1), s∈[r,1), σ≥0. Assume αδ >1 whens=r. Suppose/integraltext+∞ t0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω. Then |f(x(t))−f(x∗)|=O/parenleftbigg1 e1 δ(1−s)t1−s/parenrightbigg ,/ba∇dblAx(t)−b/ba∇dbl=O/parenleftbigg1 e1 δ(1−s)t1−s/parenrightbigg . Theorem 2.7. Letβ(t) =µt1 δ−τwithµ >0,τ∈(0,r+ 1),r∈(0,1), s= 1, δτ≤1, σ≥0. Suppose /integraltext+∞ t0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω. Then |f(x(t))−f(x∗)|=O(1 t1/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1 t1/δ). Remark 2.9. Whens= 1, taking τ=1 δ, thenβ(t) =µ >0is a positive constant time scaling. For any 1 δ< r+1, we can obtain the O(1 t1/δ)convergence rates of objective function and constraint. 152.3. Case r= 1, s= 1 Consider the case when r= 1,s= 1, i.e., the dynamic (1) becomes: ¨x(t)+α t˙x(t) =−β(t)(∇f(x(t))+AT(λ(t)+δt˙λ(t))+σAT(Ax(t)−b))+ǫ(t), ¨λ(t)+α t˙λ(t) =β(t)(A(x(t) +δt˙x(t))−b).(41) We will discuss dynamic (41) with α≤3 andα >3 respectively. Theorem 2.8. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with t˙β(t)≤τβ(t), (42) andǫ: [t0,+∞)→ H1satisfies/integraldisplay+∞ t0tα−τ 3/ba∇dblǫ(t)/ba∇dbldt <+∞. Let0≤τ≤α≤3,δ=3 2α+τand(x(t),λ(t))be a global solution of the dynamic (41). Then for any (x∗,λ∗)∈Ω, the following conclusions hold: (i)/integraltext+∞ t0t2(α−τ) 3−1β(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞. (ii) When τ∈[0,α): for any ρ∈[0,α−τ 3), /integraldisplay+∞ t0t2ρ−1β(t)(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞,/integraldisplay+∞ t0t2ρ−1/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2dt <+∞. (iii) When limt→+∞t2(α−τ) 3β(t) = +∞: L(x(t),λ∗)−L(x∗,λ∗) =O(1 t2(α−τ) 3β(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1 tα−τ 3/radicalbig β(t)). (iv) When τ= 0andα= 3: /ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1 tρ),∀ρ∈(0,1). Otherwise: /ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1 tα−τ 3). Proof.Givenλ∈ H2, defineEλ,ρ(t) andEλ,ρ ǫ(t) as (15), (14) with r=s= 1,ρ∈[0,α−τ 3] and θ(t) =2α+τ 3tρ−1, η(t) =2α+τ 3(1+α−τ 3−2ρ)t2ρ−2. (43) By computation, we have η(t)≥2α+τ 3(1−α−τ 3)t2ρ−2≥0, (44) and (A.2), (A.4) are satisfied. Since 0 ≤τ≤α≤3 andρ∈[0,α−τ 3], we also can verify that θ(t)˙θ(t)+˙η(t) 2=2α+τ 3(α+1−2ρ)(ρ−1)t2ρ−3≤0. 16Then (A.1)-(A.4) hold for any t≥t0. It is easy to verify that θ(t)+ρtρ−1−αtρ−r= (ρ−α−τ 3)tρ−1≤0 (45) and tρ˙β(t)+(2ρtρ−1−θ(t))β(t) =tρ−1(t˙β(t)−τβ(t)+(τ+2ρ−2α+τ 3)β(t)) ≤2(ρ−α−τ 3)tρ−1β(t) (46) ≤0 for allt≥t0. This together with (A.5) in case λ=λ∗implies ˙Eλ∗,ρ ǫ(t)≤(ρ−α−τ 3)t2ρ−1(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)+2(ρ−α−τ 3)t2ρ−1β(t)(Lσ(x(t),λ∗)−Lσ(x∗,λ∗)) −σt2ρ−1β(t) 2δ/ba∇dblAx(t)−b)/ba∇dbl2(47) ≤0. ThenEλ∗,ρ ǫ(·) is nonincreasing on [ t0,+∞), Eλ∗,ρ ǫ(t)≤ Eλ∗,ρ ǫ(t0),∀t≥t0. Since/integraltext+∞ t0tα−τ 3/ba∇dblǫ(t)/ba∇dbldt <+∞, for any ρ∈[0,α−τ 3], we have /integraldisplay+∞ t0tρ/ba∇dblǫ(t)/ba∇dbldt <+∞. By similar arguments in proof of Theorem 2.1, we obtain the boundedn ess ofEλ∗,ρ(·) andEλ∗,ρ ǫ(·). Since (47) holds for any ρ∈[0,α−τ 3], integrating it on [ t0,+∞), and following from the boundedness of Eλ∗,ρ ǫ(·), we get the results (i)−(ii). SinceEλ∗,ρ(·) is bounded for any ρ∈[0,α−τ 3], by the definition of Eλ∗,ρ(·) and (3), (43), we obtain ( iii), sup t≥t0/radicalbigg 1+α−τ 3−2ρ×tρ−1/ba∇dblx(t)−x∗/ba∇dbl<+∞ (48) and sup t≥t0/ba∇dbl2α+τ 3tρ−1(x(t)−x∗)+tρ˙x(t)/ba∇dbl<+∞ (49) for anyρ∈[0,α−τ 3]. Whenτ= 0 and α= 3: for any ρ∈(0,1), 1+α−τ 3−2ρ= 2(1−ρ)>0, it follows from (48) and (49) that sup t≥t0tρ−1/ba∇dblx(t)−x∗/ba∇dbl<+∞ and then sup t≥t0tρ/ba∇dbl˙x(t)/ba∇dbl ≤2sup t≥t0tρ−1/ba∇dblx(t)−x∗/ba∇dbl+ sup t≥t0/ba∇dbl2tρ−1(x(t)−x∗)+tρ˙x(t)/ba∇dbl)<+∞, 17for anyρ∈(0,1). Similarly supt≥t0tρ/ba∇dbl˙λ(t)/ba∇dbl<+∞. Otherwise α−τ <3, taking ρ=α−τ 3, then 1+α−τ 3−2ρ= 1−α−τ 3>0, by similar discussions in above, we get (iv). Remark 2.10. Following from above proof process, when τ= 0andα= 3: sup t≥t0tρ−1(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)<+∞,∀ρ∈(0,1), Otherwise: sup t≥t0tα−τ 3−1(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)<+∞. When the coercive condition (40)satisfied , we also can obtain the boundedness of x(t)of dynamic (42)with α≤3. Remark 2.11. Theorem 2.8 extends the results in [27, Corollary 2.9] and [4 7, Theorem 3.2] to general case. TakingA= 0,b= 0, the dynamic (41)reduces to ¨x(t)+α t˙x(t)+β(t)∇f(x(t)) =ǫ(t), withα≤3for solving unconstrained optimization problem, then Theo rem 2.8 also complements the results in [9, Theorem A.1], which considered the case α≥3. Takingt˙β(t) =τβ(t), in which β(t) =µtτwithµ >0, we investigate the improved rate of convergence. Theorem 2.9. Letβ(t) =µtτwithµ >0,0≤τ≤α≤3,δ=3 2α+τ, σ≥0. Suppose/integraltext+∞ t0t(α−τ)/3/ba∇dblǫ(t)/ba∇dbldt < +∞. Let(x(t),λ(t))be a solution of dynamic (41). For any (x∗,λ∗)∈Ω: |f(x(t))−f(x∗)|=O(1 t(2α+τ)/3),/ba∇dblAx(t)−b/ba∇dbl=O(1 t(2α+τ)/3). Proof.Recall the energy functions Eλ,ρ(t) andEλ,ρ ǫ(t) from Theorem 2.8 with β(t) =µtτ,ρ=α−τ 3. Then tρ˙β(t)+(2ρtρ−1−θ(t))β(t) = 0, this together with (45) and (A.5) yields ˙Eλ,ρ ǫ(t)≤ −σβ(t) 2δ/ba∇dblAx(t)−b)/ba∇dbl2≤0,∀t≥t0,λ∈ H2. (50) By similar arguments in Theorem 2.2, we obtain the results. From Theorem 2.9, we obtain the following results in the case τ= 0 and τ=α, respectively. Corollary 2.2. Letβ(t) =β >0,α≤3,δ=3 2α, σ≥0. Suppose/integraltext+∞ t0tα/3/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a solution of dynamic (41). For any (x∗,λ∗)∈Ω: |f(x(t))−f(x∗)|=O(1 t2α/3),/ba∇dblAx(t)−b/ba∇dbl=O(1 t2α/3). 18Corollary 2.3. Letβ(t) =µtαwithµ >0,α≤3,δ=1 α, σ≥0. Suppose/integraltext+∞ t0/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t)) be a solution of dynamic (41). For any (x∗,λ∗)∈Ω: |f(x(t))−f(x∗)|=O(1 tα),/ba∇dblAx(t)−b/ba∇dbl=O(1 tα). Remark 2.12. Takingβ= 1, the dynamic (41)has been investigate in [27] and [47] for α≤3. Corollary 2.2 improves the convergence rates of [27, Corollary 2.9] an d [47, Theorem 3.2], which only obtain O(1 tα/3) convergence rate of |f(x(t))−f(x∗)|and/ba∇dblAx(t)−b/ba∇dbl, and it also can be viewed as analogs of the results in [7, 43], where the convergence rate analysis of (IGSα,ǫ)withα(t) =α t, α≤3for unconstrained optimization problem (5). Corollary 2.3 shows the optimal convergence rate we can exp ect of dynamic (41)withα≤3. Next, we investigate the convergence rate of dynamic (41) with α >3. The similar results can be found in [26]. Theorem 2.10. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with t˙β(t)≤(1 δ−2)β(t), and2≤1 δ< α−1. Letǫ: [t0,+∞)→ H1with /integraldisplay+∞ t0t/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a global solution of the dynamic (41)and y(x∗,λ∗)∈Ω. Then(x(t),λ(t))is bounded and the following conclusions hold: (i)/integraltext+∞ t0t((1 δ−2)β(t)−t˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞. (ii)/integraltext+∞ t0tβ(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞,/integraltext+∞ t0t/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2dt <+∞. (iii)/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2=O(1 t). (iv) When limt→+∞t2β(t) = +∞: L(x(t),λ∗)−L(x∗,λ∗) =O(1 t2β(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1 t/radicalbig β(t)). Proof.Givenλ∈ H2, defineEλ,ρ(t) andEλ,ρ ǫ(t) as (15), (14) with r=s=ρ= 1 and θ(t) =1 δ, η(t) =αδ−δ−1 δ2. Sinceα−1>1 δ≥2, by simple computations we can verify (A.1)-(A.4). It follows from a ssumptions that tρ˙β(t)+(2ρtρ−1−θ(t))β(t) =t˙β(t)+(2−1 δ)β(t)≤0. Takingλ=λ∗, this together with (A.5) implies ˙Eλ∗,ρ ǫ(t)≤(1 δ+1−α)t(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)+t(t˙β(t)+(2−1 δ)β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗)) 19−σtβ(t) 2δ/ba∇dblAx(t)−b/ba∇dbl2 ≤0. By similarly arguments in proof of Theorem 2.1, we obtain the bounded ness ofEλ∗,ρ(·) andEλ∗,ρ ǫ(·). This yields (i),(ii),(iv). Sinceη(t) =αδ−δ−1 δ2>0, we get that ( x(t),λ(t)) is bounded and sup t≥t0t(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dblλ(t)/ba∇dbl2)<+∞. This implies ( iii). Remark 2.13. Theorem 2.10 extends the results in [9, Theorem A.1] and [10, Section 3.2] from (IGSα,ǫ)with α(t) =α t, α >3for problem (5)to primal-dual dynamic for problem (2). Taking β(t)≡1, we recover the convergence rate of [27, Corollary 2.9] and [47, Theorem 3.1 ], moreover when A= 0, b= 0, we get the classical results for (IGSα)and(IGSα,ǫ)withα(t) =α twithα >3, which can be seen as a continuous version of the Nesterov method, see [6, 11, 33, 40]. Lett˙β(t) = (1 δ−2)β(t). We have β(t) =µt1 δ−2withµ >0. By similar proof of Theorem 2.2, we obtained following results, and the corresponding results of unperturbed c ase can be found in [3, Proposition 6.3]. Theorem 2.11. Letβ(t) =µt1/δ−2withµ >0,2≤1 δ< α−1,σ≥0. Suppose/integraltext+∞ t0t/ba∇dblǫ(t)/ba∇dbldt <+∞. Let (x(t),λ(t))be a solution of dynamic (41). For any (x∗,λ∗)∈Ω: |f(x(t))−f(x∗)|=O(1 t1/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1 t1/δ). From Theorem (2.11), we have following result. Corollary 2.4. Letβ(t) =β >0,δ=1 2,α >3,σ≥0. Suppose/integraltext+∞ t0t/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a solution of dynamic (41). For any (x∗,λ∗)∈Ω: |f(x(t))−f(x∗)|=O(1 t2),/ba∇dblAx(t)−b/ba∇dbl=O(1 t2). Remark 2.14. Theorem 2.11 shows the optimal convergence rates of dynamic (41)in the case α >3. The O(1/tp+2)convergence rate results associated with the time scaling β(t) =µtpfor unconstrained optimization problem (5)can be found in [9, 44], it also can be found in [23] with Euclid ean setting of Bregman distance for problem (2). Corollary 2.4 showst the convergence rate of objective fun ction and constraint of dynamical system (8)isO(1 t2)instead of O(1 t). 2.4. Summary of results In the subsection, we complete the tables giving a synthetic view of c onvergence results in before. For dynamic (13) with different rands, chose suitable parameters α, δ. Table 1 lists the convergences rates forL(x(t),λ∗)− L(x∗,λ∗) of dynamic (1) under different assumptions of β(t) andǫ(t). Table 2 summarizes 20the properties of trajectory ( x(t),λ(t)) and its derivates (˙ x(t),˙λ(t)). (See Theorem 2.1, Corollary 2.1, Theorem 2.8, Theorem 2.10, Remark 2.7, Remark 2.10). The results extend th e inertial dynamic with time scaling in [9, 10, 12, 44] for problem (5) to primal-dual dynamic (1) for proble m (2). Taking A= 0,b= 0, our results also can complement the existing results the inertial dynamic with time sca ling. Table 1: Convergence rates for L(x(t),λ∗)−L(x∗,λ∗) of dynamic (1) r,s β(t) ǫ(t) L(x(t),λ∗)−L(x∗,λ∗) r= 0,s∈[0,1] ts˙β(t)≤(1 δ−sts−1)β(t)/integraltext+∞ t0ts 2/ba∇dblǫ(t)/ba∇dbldt <+∞ O(1 tsβ(t)) r∈(0,1),s∈[r,1]ts˙β(t)≤(1 δ−(r+s)ts−1)β(t)/integraltext+∞ t0tr+s 2/ba∇dblǫ(t)/ba∇dbldt <+∞O(1 tρβ(t)),∀ρ∈(0,r+s) r=s= 1α≤3t˙β(t)≤τβ(t),τ∈[0,α]/integraltext+∞ t0tα−τ 3/ba∇dblǫ(t)/ba∇dbldt <+∞ O/parenleftBig 1 t2(α−τ)/3β(t)/parenrightBig α >3 t˙β(t)≤(1 δ−2)β(t)/integraltext+∞ t0t/ba∇dblǫ(t)/ba∇dbldt <+∞ O(1 t2β(t)) Table 2: Summary of trajectory properties r,s /ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl I=/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl r=s= 0 bounded Ibounded r= 0, s∈(0,1] O(1 ts/2) Ibounded r∈(0,1), s∈[r,1] O(1 tρ),∀ρ∈(0,r+s 2) tρIbounded, ∀ρ∈(−r+s 2,0) r=s= 1α= 3,τ= 0 O(1 tρ),∀ρ∈(0,1) tρIbounded, ∀ρ∈(−1,0) α≤3,0≤α−τ <3 O(1 t(α−τ)/3) tα+τ 3−1Ibounded α >3 O(1 t) Ibounded Select a specific time scaling β(t) with suitable parameters α, δ. Table 3 shows optimal convergence rates we can expect for different choices of coefficients. (See Theorem 2 .2, Theorem 2.3, Theorem 2.4, Theorem 2.6, Theorem 2.7, Corollary 2.3, Theorem 2.11) Taking time scaling β(t)≡1, Table 4 lists the corresponding convergence rates (See Remark 2.3, Theorem 2.1, Theorem 2.4, Theorem 2.5, Theorem 2.7, Corollary 2.2, Corollary 2 .4), it extends the convergence rates of (IGSα) and (IGSα,ǫ) in [6, 7, 13, 40, 41, 43] for unconstrained optimization problems to primal-dual dynamic (1) for linear equality constrained optimization problems. It also ext end and complements the existing results of inertial primal-dual dynamic in [5, 26, 27, 28, 47]. 21Table 3: Optimal convergence rates of |f(x(t)−f(x∗))|and/bardblAx(t)−b/bardbl r,s β(t) |f(x(t)−f(x∗))|and/ba∇dblAx(t)−b/ba∇dbl r= 0,s∈[0,1) µe1 δ(1−s)t1−s ts O/parenleftbigg 1 e1 δ(1−s)t1−s/parenrightbigg r= 0,s= 1 µt1 δ−1O(1 t1/δ) r∈(0,1),s∈[r,1) µe1 δ(1−s)t1−s tτ,∀τ∈(0,r+s) O/parenleftbigg 1 e1 δ(1−s)t1−s/parenrightbigg r∈(0,1),s= 1 µt1 δ−τ,∀τ∈(0,r+1) O(1 t1/δ) r=s= 1α≤3 µtαO(1 tα) α >3 µt1/δ−2O(1 t1/δ) Table 4: Convergence rates of dynamic (1) with β(t)≡1 r,s |f(x(t))−f(x∗)|and/ba∇dblAx(t)−b/ba∇dbl L(x(t),λ∗)−L(x∗,λ∗) r= 0,s= 0 O(1√ t) ergodic sence O(1 t) ergodic sence r= 0,s∈(0,1) O(1 ts/2) O(1 ts) r= 0,s= 1 O(1 t) r∈(0,1),s∈[r,1) O(1 tτ/2),∀τ∈(0,r+s) O(1 tτ),∀τ∈(0,r+s) r∈(0,1),s= 1 O(1 tτ),∀τ∈(0,r+1) r=s= 1α≤3 O(1 t2α/3) α >3 O(1 t2) 3. Conclusion In this paper, we propose a family of damped inertial primal-dual dyn amical systems with time scaling for solving problem (2) in Hilbert space. We extend the inertial dynamic in [6 , 7, 9, 12, 13, 40, 41, 43, 44] for solving unconstrained optimization problems to primal-dual dynamic ( 1) for solving linear equality constrained convex optimization problems. Our results also extend and compleme nt the existing results of inertial primal- dual dynamics in [5, 26, 27, 28, 47]. Taking A= 0,b= 0, our results also complement the convergence rate results of existing inertial dynamic for solving unconstrained conve x optimization problems. By discretization of primal-dual dynamic (13), it may lead to new primal-dual algorithms for solving problem (2), how to chose suitable discretization scheme of (13) to get rate-matchingalgorit hms is an interesting direction of research. From 22references [26, 28], it seems achievable, and we will consider it in the f uture works. Appendix A. Some auxiliary results Appendix A.1. Differentiating the energy function In this part, we list the main calculation procedures for differentiatin g the energy function Eλ,ρ ǫ(t). Multiplying the first equation of (13) by tρ, we have tρ¨x(t) =−αtρ−r˙x(t)−tρβ(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+tρǫ(t). This yields ˙E1(t) =/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),˙θ(t)(x(t)−x∗)+θ(t)˙x(t)+ρtρ−1˙x(t)+tρ¨x(t)/angb∇acket∇ight +˙η(t) 2/ba∇dblx(t)−x∗/ba∇dbl2+η(t)/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight =/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),˙θ(t)(x(t)−x∗)+(θ(t)+ρtρ−1−αtρ−r)˙x(t) −tρβ(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+tρǫ(t)/angb∇acket∇ight +˙η(t) 2/ba∇dblx(t)−x∗/ba∇dbl2+η(t)/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight = (θ(t)˙θ(t)+˙η(t) 2)/ba∇dblx(t)−x∗/ba∇dbl2+tρ(θ(t)+ρtρ−1−αtρ−r)/ba∇dbl˙x(t)/ba∇dbl2 +(θ(t)(θ(t)+ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t))/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight −δθ(t)tρ+sβ(t)/angb∇acketleftx(t)−x∗,AT˙λ(t)/angb∇acket∇ight−δt2ρ+sβ(t)/angb∇acketleftA˙x(t),˙λ(t)/angb∇acket∇ight −θ(t)tρβ(t)(/angb∇acketleftx(t)−x∗,∇f(x(t))+ATλ(t)+σAT(Ax(t)−b)/angb∇acket∇ight −t2ρβ(t)/angb∇acketleft˙x(t),∇f(x(t))+ATλ(t)+σAT(Ax(t)−b)/angb∇acket∇ight +/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),tρǫ(t)/angb∇acket∇ight. Similarly, we have ˙E2(t) = (θ(t)˙θ(t)+˙η(t) 2)/ba∇dblλ(t)−λ/ba∇dbl2+tρ(θ(t)+ρtρ−1−αtρ−r)/ba∇dbl˙λ(t)/ba∇dbl2 +(θ(t)(θ(t) +ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t))/angb∇acketleftλ(t)−λ,˙λ(t)/angb∇acket∇ight +θ(t)tρβ(t)/angb∇acketleftλ(t)−λ,Ax(t)−b/angb∇acket∇ight+δθ(t)tρ+sβ(t)/angb∇acketleftλ(t)−λ,A˙x(t)/angb∇acket∇ight +t2ρβ(t)/angb∇acketleft˙λ(t),Ax(t)−b/angb∇acket∇ight+δt2ρ+sβ(t)/angb∇acketleft˙λ(t),A˙x(t)/angb∇acket∇ight. Differentiating of E0(t) to get ˙E0(t) =t2ρβ(t)/angb∇acketleft∇f(x(t)) +ATλ+σAT(Ax(t)−b),˙x(t)/angb∇acket∇ight +(2ρt2ρ−1β(t)+t2ρ˙β(t))(Lσ(x(t),λ)−Lσ(x∗,λ)). Letθ(t) satisfy t2ρβ(t) =δθ(t)tρ+sβ(t). Adding ˙E0(t),˙E1(t),˙E2(t) together, using Ax∗=band rearranging the terms, we get ˙Eλ,ρ(t) =˙E0(t)+˙E1(t)+˙E2(t) =5/summationdisplay i=1Vi(t), 23where V1(t) =/parenleftbigg θ(t)˙θ(t)+˙η(t) 2/parenrightbigg (/ba∇dblx(t)−x∗/ba∇dbl2+/ba∇dblλ(t)−λ/ba∇dbl2), V2(t) = (θ(t)(θ(t) +ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t))(/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight+/angb∇acketleftλ(t)−λ,˙λ(t)/angb∇acket∇ight), V3(t) =tρ(θ(t)+ρtρ−1−αtρ−r)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2), V4(t) =tρ(tρ˙β(t)+(2ρtρ−1−θ(t))β(t))(Lσ(x(t),λ)−Lσ(x∗,λ)) +θ(t)tρβ(t)(f(x(t))−f(x∗)−/angb∇acketleftx(t)−x∗,∇f(x(t))/angb∇acket∇ight)−σθ(t)tρβ(t) 2/ba∇dblAx(t)−b/ba∇dbl2, V5(t) =/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),tρǫ(t)/angb∇acket∇ight. To investigate the rates of convergence of dynamical system (13 ), we need to find the appropriate θ(t) andη(t) to satisfy the following conditions: θ(t)≥0, η(t)≥0, (A.1) t2ρβ(t)−δθ(t)tρ+sβ(t) = 0, (A.2) θ(t)˙θ(t)+˙η(t) 2≤0, (A.3) θ(t)(θ(t) +ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t) = 0, (A.4) ThenV1≤0,V2= 0, this together with the convexity of fyields ˙Eλ,ρ ǫ(t) =˙Eλ,ρ(t)−/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),tρǫ(t)/angb∇acket∇ight ≤tρ(θ(t)+ρtρ−1−αtρ−r)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σθ(t)tρβ(t) 2/ba∇dblAx(t)−b)/ba∇dbl2 +tρ(tρ˙β(t)+(2ρtρ−1−θ(t))β(t))(Lσ(x(t),λ)−Lσ(x∗,λ)) (A.5) for anyλ∈ H2. Appendix A.2. Technical lemmas: In convergence analysis for the dynamical system, we shall recall the following lemmas. Lemma Appendix A.1. [18, Lemma A.5] Let ν: [t0,T]→[0,+∞)be integrable, and M≥0. Suppose µ: [t0,T]→Ris continuous and 1 2µ(t)2≤1 2M2+/integraldisplayt t0ν(s)µ(s)ds for allt∈[t0,T]. Then|µ(t)| ≤M+/integraltextt t0ν(s)dsfor allt∈[t0,T]. Lemma Appendix A.2. 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2021-06-25

In Hilbert space, we propose a family of primal-dual dynamical system for affine constrained convex optimization problem. Several damping coefficients, time scaling coefficients, and perturbation terms are thus considered. By constructing the energy functions, we investigate the convergence rates with different choices of the damping coefficients and time scaling coefficients. Our results extend the inertial dynamical approaches for unconstrained convex optimization problems to affine constrained convex optimization problems.

Perturbed primal-dual dynamics with damping and time scaling coefficients for affine constrained convex optimization problems

2106.13702v1

arXiv:1910.10977v2 [cond-mat.mes-hall] 11 Feb 2020Topological damping Rashba spin orbit torque in ballistic magnetic domain walls D. Wang1,∗and Yan Zhou2,† 1College of Engineering Physics, Shenzhen Technology University, Guangdong 518118, P. R. China 2School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, P. R. China (Dated: February 12, 2020) Abstract Rashba spin orbit torque derived from the broken inversion s ymmetry at ferromagnet/heavy metal interfaces has potential application in spintronic d evices. In conventional description of the precessional and damping components of the Rashba spin orbi t torque in magnetization textures, the decomposition coefficients are assumed to be independent of the topology of the underlying structure. Contrary to this common wisdom, for Schr¨ odinge r electrons trespassing ballistically across a magnetic domain wall, we found that the decompositi on coefficient of the damping component is determined by the topology of the domain wall. T he resultant damping Rashba spin orbit torque is protected by the topology of the underly ing magnetic domain wall and robust against small deviations from the ideal domain wall profile. Our identification of a topological damping Rashba spin orbit torque component in magnetic doma in walls will help to understand experiments on current driven domain wall motion in ferroma gnet/heavy metal systems with broken inversion symmetry and to facilitate its utilizatio n in innovative device designs. 1One main theme in the field of nanomagnetism is to search for new appr oaches to realize fast and energy efficient manipulation of magnetic state, rat her than using the conventional magnetic field. In the past three decades, several promising candidates, such as electric field1, laser pulses2and spin current through the spin transfer torque (STT)3–6, were proposed. A recent development along this line is the emergenc e of the Rashba spin orbittorque(RSOT)inmagneticsystemswithoutinversionsymmetr y. Inasimplepicture7, the electric field along the symmetry breaking direction is equivalent t o a magnetic field, dubbed the Rashba field, in the rest reference frame of an electro n in motion. Due to the s-dexchange between the local and itinerant spin degrees of freedom , the Rashba field is transformed into the RSOT acting on the local magnetization. When it was first proposed, only the precessional component8–11of the RSOT, corre- sponding to the torque caused by an effective Rashba field acting on the local magnetiza- tion, was considered. Upon considering the impurity and spin-flip sca ttering, an additional damping torque in accordance with the effective Rashba field can aris e12. Subsequent the- oretical investigations were devoted to exposition of the physics o f the RSOT, adopting different approachesandconsideringsamplegeometrieswithfinitee xtension13–20. However, most of the previous theoretical investigations focus on the case of uniform magnetization distribution or slowly varying magnetization textures, the moreimpo rtant case of magnetic domain walls (DWs), which will be the focus of the current work, is almo st not touched upon. Thetopologicaldescriptionofelectrontransportinperiodicpoten tialsappearsnaturally by considering the geometric Berry phase21of itinerant electrons. In the simplest case of one dimensional (1D) motion of electrons, it leads to the Zak phase22, and the Thouless- Kohmoto-Nightingale-den Nijs (TKNN) invariant23for two dimensional (2D) motion. The Berry phase is generically caused by the existence of a gauge field24, which is given by the spatial variationof the periodic modulationwave function inthe case of Bloch electrons. In the presence of spin orbit interaction and a background magnetic fi eld, which is generated byamagneticDW, theitinerant electronswill alsoexperience aspatia llyvarying, emergent gauge field. By analogy with the TKNN invariant and the Zak phase, we speculate that topological phase factors should arise for electrons traversing c ross a DW. Actually, the effect of spatially varying magnetization on the motion of electrons w as already discussed theoretically by Bruno et al.25. Whether a similar topological effect will emerge in RSOT remains a question. For a simple demonstration of the physics, we will use the following minim al model Hamiltonian to study the magnetization dynamics of itinerant electro ns confined to the 2interface between a ferromagnet and a heavy metal8–11, H=p2 2me+µBσ·M+αR ¯hσ·(p׈z). (1) p=−i¯h∇is the momentum operator, meis the electron mass, ¯ his the Planck constant divided by 2 π, andµBis the Bohr magneton. αRis the Rashba constant, which measures the degree of the inversion symmetry breaking26. We consider only the motion of the electrons in the interface, which is a 2D xyplane in our coordinate system, since previous density functional theory investigation found that the RSOT is prim arily an interface effect13. The third term in the Hamiltonian (1) is the Rashba spin orbit interact ion term, showing that the main effect of the broken inversion symmetry is to in troduce an effective in-plane magnetic field, which is everywhere tangential to the in-plan e linear momentum p.σ= ˆxσx+ ˆyσy+ ˆzσzis a vector in the spinor space where σx,σyandσzare the Pauli matrices, and ˆ x, ˆyand ˆzare unit vectors along the x,yandzdirections, respectively. The Hamiltonian (1) gives the energy of conduction electrons intera cting through the s-dexchange interaction with the local magnetization M. In our model treatment, we consider only the itinerant Hamiltonian as given in Eq. (1), while the loca l magnetic moments are assumed to be static, as described by M. The variation of the vector M insidemagnetizationtexturesisusedtoprovideaneffective’exchan ge’ fieldfortheitinerant magnetization. The Walker DW profile27considered for the study of the RSOT is characterized by an angleθthrough the expression M=M(ˆxsinθ+ ˆzcosθ) with cosθ=−qtanh(x/λ) and sinθ=χsech(x/λ), whereλ=/radicalBig A/KistheDWwidth. Aistheexchange constant and K theanisotropyconstantoftheferromagnet. Forageneraldes cription, weconsider explicitly the charge qand chirality χof a DW28. Using the time dependent Pauli-Schr¨ odinger equationi¯h∂ψ/∂t=Hψfor the spinor wave function ψ, the equation of motion for the spin density s=ψ†σψof conduction electrons is given by 2me ¯h∂s ∂t=∇·Q+2k2 BˆM×s+τ, (2) where the spin current density is defined as Q=i(ψ†∇σψ−∇ψ†σψ)+kαǫij3ˆiˆjψ†ψ. (3) ǫijkis the antisymmetric Levi-Civita symbol and a summation over repeat ed indices is implied in the expression for Q. A substitution of x,yandzby numbers 1, 2 and 3 is made to compactify the expression. The parameter kBis related to the Zeeman energy splitting ¯h2k2 B/2me=µBM, and the constant kα= 2meαR/¯h2is an effective wave number 3characterizing the strength of the Rashba interaction. ˆMis a unit direction vector for the local magnetization, ˆM=M/M. The precessional term τfollows directly from the Rashba term in Eq. (1), and is given by τ(k,ρ) = 2kαℑ(ˆzψ†σ·∇ψ−ψ†σz∇ψ). (4) For later convenience, the momentum and position dependence of τis explicitly written out in Eq. (4). Our equation of motion for the spin density is identical in form to a pre- vious result29, if the angular momentum operator is replaced by the Rashba field op erator considered here. However, this connection is superficial, as the dy namics for the angular momentum are not considered here. With Eq. (2), it is obvious that the itinerant magnetization dynamics is governed by three torques. The first term on the right hand side of Eq. (2) cor responds to the spin current torque acting on the itinerant magnetization, which is just the divergence of the spin current density. In the ground state, the spin current torq ue reduces to the exchange torqueformagnetizationtextures, whichisproportionalto ˆ m×∇2ˆm, with ˆmanunit vector for the itinerant magnetization. The second term describes the to rque originating from the static local magnetization, whose net effect can be viewed as an effe ctives-dexchange field acting on the itinerant magnetization. The Rashba term in the Hamilto nianHgives rise to the last torque on the right hand side of Eq. (2). In equilibrium, th is Rashba torque has a form identical to the Dzyaloshinskii-Moriya torque30–34. If a steady state electronic current is allowed to flow, the spin current torque and the Rashba t orque transform into the conventional STT and RSOT, respectively. In the current car rying steady state, there is no time variation of the itinerant magnetization. Hence the various torques on the right hand side must sum to zero. Due to this torque balance, the torque induced by the spin accumulation, which corresponds to the second term on the right h and side of Eq. (2), contains both the STT and RSOT contributions. Eq. (4) gives only the RSOT for a single Bloch state in the momentum sp ace. Using the relaxation time approximation35, the physical RSOT induced in the presence of an electric fieldEalong thexdirection can be obtained through an integration in the momentum space as τ(ρ) =−eEτ0 (2π)2¯h/contintegraldisplay dϕkxτ(k,ρ), (5) whereτ0is the relaxation time constant, ethe electron charge, and ϕthe angle of the wave vector relative to the x-axis. As the temperature is assumed to be absolute zero, the integration is confined to the 2D Fermi surface, which is a circle. 4-20 -10 0 10 20-2-1012Spin orbit torque (a. u.)x y z (a) -700 -350 0 350 700-4-202Spin orbit torque (a. u.)x y z (b) FIG. 1. RSOT with q= 1 and χ= 1. The DW widths correspond to λkF= 2 (a) and λkF= 70 (b). For the small DW width λkF= 2 (a), both the precessional ( xandz) and the damping (y) components are comparable in magnitude. As the DW width inc reases to λkF= 70 (b), the damping component decreases in comparison to the precessio nal one. For the long DW width λkF= 70, although the damping component is negligibly small at t he DW center, its magnitude is sizable far away from the DW center. We adopt a scattering matrix method36,37to numerically solve the eigenvalue problem Hψ=ǫkψ (6) for the Pauli-Schr¨ odinger equation with energy ǫk. The idea behind this scattering matrix method is intuitively simple. In order to construct the eigenfunction s of Eq. (6), we first solve it at infinity to obtain the asymptotic wave functions with specifi c momentum and spin. Then we evolve the obtained asymptotic wave functions towar ds the DW center, according to Eq. (6). Generally, the evolved wave functions are no t continuous at the DW center, and are thus not eigenfunctions in the whole space. This problem can be overcome by forming linear combinations of the evolved wave functio ns with the same energy but different momenta and spin projections along the zdirection, requiring that the continuity condition is satisfied at the DW center. The resultant wave functions are eigenfunctions over the whole space. Previously, the same method was successfully applied to the discussion of STT in DWs38. In the actual numerical implementation, we can employ a particle-hole or charge-parity-time-reversal symmetry of the Hamiltonian (1), H=σxPTHTPσx, to reduce the number of the wave functions to be computed. Wav e functions related to each other by the particle-hole symmetry, ψandσxPψ, are conjugate pairs with opposite momenta but identical spin projections along the zdirection, injecting fromoppositeendsoftheDW.Itisinteresting tonotethatasimilarp article-holesymmetry was found formagnons inside DWs39. Further numerical details of thecalculation are given 5-20 -10 0 10 201.21.62.02.4 & (a. u.) FIG. 2. Precessional ( α) and damping ( β) RSOT coefficients with q= 1 and χ= 1. The DW width is λkF= 2. The quantum confinement induced oscillation around the D W center ( xkF= 0) and far away from the DW center ( xkF=±20), which is obvious for the displayed DW width, is smoothed out as the DW width is increased to λkF= 70, as shown in Fig. 3. Unspecified parameters are the same as those used to generate Fig. 1. in Ref. 40. With the numerical wave functions thus obtained, the RSOT can be c omputed using Eqs. (4) and (5). The resultant RSOT for the DW width λkF= 2 andλkF= 70 with kB/kF= 0.4 andkα/kF= 0.1 is shown in Fig. 1, where we have measured the DW width in terms of the inverse Fermi wave vector k−1 Ffor the free electron system that is described by only the kinetic energy term in the Hamiltonian (1). For the shorte r DW width λkF = 2, the RSOT has both sizable precessional and damping component s. The precessional component is caused by the effective Rashba field, which has the for m ˆm׈y, while the corresponding damping component is ˆ m×(ˆm׈y)9. The total RSOT is a sum of both components, τ=αˆm׈y+βˆm×(ˆm׈y). (7) The corresponding decomposition coefficients αandβare displayed in Fig. 2. Due to the confinement of electrons caused by such short DWs, quantum interference of wave functions shows up as the observable spatial variation of the RSOT and decomposition coefficients far away from the DW center. This spatial variation dec ays out as the DW width is increased (cf. Figs. 1 (a) and (b), 2 and 3). AstheDWwidthincreases, themagnitudeoftheprecessionalcomp onentincreaseswhile the magnitude of the damping component decreases, as can be exp ected from a previous investigation on STT38. However, the scaling of the non-adibaticity for the RSOT, which is defined as β/α, is algebraic instead of exponential40. At the DW center, the residue 6-700 -350 0 350 7002.63.03.4 (a. u.)(a) -700 -350 0 350 700-0.8-0.400.40.8 (a. u.)(b) FIG. 3. Topological behaviour of the precessional ( α) and damping ( β) RSOT coefficients for all four combinations of qandχ. The DW width is λkF= 70. Other parameters are the same as those used to generate Fig. 1. damping component is negligible, but it is sizable far away from the DW ce nter, as evident from Fig. 3 (b) for the longer DW width λkF= 70. This finite residue damping component of the RSOT will demonstrate itself in the current driven magnetizat ion dynamics of mag- netization textures, and warrants further attention in consider ing its effects in spintronic devices. Furthermore, our numerical result shows that the coeffi cientβdepends on the topology of the underlying DW. As shown in Fig. 3, for the four possib le combinations of the DW charge and chirality, we have only two traces for β, reversed to each other, for the longer DW width λkF= 70: the product of the DW charge and chirality, qχ, determines the sign of β. The physical origin of the damping RSOT can be determined through a perturbation analysis of the same Pauli-Schr¨ odinger equation (6) which is used fo r our numerical sim- ulation. Using the first order wave function, the damping RSOT comp onent atx=±∞ can be calculated. It has the form as given in Eq. (7) with the coefficie nt40 β∝qχk2 α/parenleftbigg c+a λ2+be−γλ/parenrightbigg (8) to the lowest order in kα, wherea,b,candγare all constants. The constant cis of the order of unity, hence as the DW width increases to a very large value ,λ≫λc, the damping RSOT will approach to a constant value cat±∞. The critical length λc=kF/k2 B, which isλckF= 6.25 using our parameters, determines the DW width where transition from non- adiabatic to adiabatic behaviour occurs for STT in DWs without spin or bit interaction38. Theappearanceofthefactor qχintheexpression of βindicates thatthedampingRSOT is a topological quantity. The factor k2 αsignifies that the damping RSOT is a higher order effect, askαis proportional to αR. In the perturbation calculation, the adiabatic or zeroth 7order wave functions give rise to only the precessional RSOT. Due t o this origin from the zeroth order wave functions, the adiabatic coefficient αis almost independent of the topological features of the underlying DW, whether in the adiabatic limit or not: For α, the dominant contribution does not sense the topology of the DW, a nd the topological contribution only enters as a higher order correction (cf. Fig. 3 (a )). Inclusion of the first order wave functions brings about the damping RSOT. The firs t order wave functions at infinity are determined by the scattering of the incident, zeroth order waves under the influence of the perturbation potential. To the first order of kα, the explicit form of the perturbation potential in momentum space V(kf,ki) for incident and scattered momenta kiandkfis give by V(kf,ki) =pcschp 4πλ−χkα 4ky k2 Bπ2+4p2 2π2λsechp+qχkα 4sechp −qχks 4/parenleftBigg sechp−2χkαky πk2 Bpcschp/parenrightBigg σy+χλkα 2kyσzsechp, (9) withp=Qλπ/2 andks=kf+ki.Q=kf−kiis the momentum transfer. In comparison to the original potential in (1), the potential (9) corresponds to a m agnetic field with only y andzcomponents and a scalar electric potential, while the Rashba interac tion is absorbed into the magnetic field and electric potential. When the momentum tra nsfer is zero, the scaling ofV(ki,ki) with respect to the DW width λis algebraic. For finite momentum transferQ=kf−ki,V(kf,ki) brings about theexponential decay ofthe physical quantities on the DW width through the hyperbolic secant and cosecant funct ions41. Not all of the topological terms in potential (9) contribute to the e xpression for the damping RSOT. In the case of zero momentum transfer, Q= 0, theycomponent of the effective magnetic field in(9), which is the coefficient of σy, does not contribute at all; while thezeffective field, which is the coefficient of σz, andthe scalar potential contribute partly: The product of the first and second terms in the scalar potential g ives rise to the term proportional to λ−2in (8), while the product of the zcomponent of the effective magnetic field with the first term of the scalar potential contributes the con stant term in β. Both those two contributions areproportionalto thechirality χ. The dependence of qinthe final expression for βis derived from its dependence on the zcomponent of the magnetization, mz. Hence the topological feature of βis characterized by the relation β∝χmz, far away from the DW center. This behaviour is similar to that of Bloch wave fun ctions in periodic potentials, as demonstrated by the Zak phase22. Themzis a dynamical contribution, and χis a manifestation of the existence of a topological phase with the va lue of 0 or π. The topological dependence of βobtained using the potential (9), Eq. (8), is actually borne out by the numerical results, as shown in Fig. 3. 8The topological nature of βexplains mathematically why the damping RSOT remains finite even when the DW width is large, λ≫λc. Due to the different topologies of the DW and a uniformly magnetized state, a continuous transition betwe en the two states is prohibited. Thus βcannot be reduced to zero, which is the value for βin a uniform state. Physically, the topological protection of the damping RSOT c an be traced back to the nonlocal character of quantum particles, which means that the wave functions are not determined locally by the potential. In particular, the damping RS OT atx=±∞ is determined by V(ki,ki) in the adiabatic limit ( λ≫λc), which is an integration of the perturbation potential over the whole space and gives rise to the t opological characteristics of the damping RSOT. Therefore, the damping RSOT at x=±∞is finite due to the pure existence of the DW, even though the magnetization variation there is infinitesimal, approaching to the value for a uniform magnetization distribution. ToseehowthenewlyidentifieddampingRSOTinfluencesthecurrentd rivenDWmotion (CDWM), we consider the expression for the normalized DW velocity v nαGv=qhzcosθh−nξu+qχβ∆θ0, (10) obtained using a simple 1D model description of CDWM42. A detailed derivation of the velocity (10) is given in the Supplemental Material [43], and referenc es [44, 45] therein. αGis the Gilbert damping constant, and n,θhand ∆θ0are constants related to the equi- librium DW configuration. ξis the non-adiabaticity and uis an equivalent speed for the STT.hzis a perpendicularly applied magnetic field, normalized to the anisotrop y field. In obtaining Eq. (10), we have assumed that the current density is small, and the RSOT only causes infinitesimal deviation from the equilibrium DW configuratio n. Even with this rather simple assumption, Eq. (10) shows that the CDWM can exhibit very complicated behaviour: For short DWs, βis not completely determined by the product qχ, then the RSOT contribution to the DW velocity has both qχdependent and independent compo- nents. In the adiabatic limit, the qχdependent component of βfades out, and the RSOT contribution to the DW velocity is qχindependent, resembling the behaviour of the STT contribution. In systems with a sizable Rashba interaction, the sign of the produc tqχfor a stable N´ eel wall is determined by the sign of the Dzyaloshinskii-Moriya inter action constant D, as qχD<0 gives a lower energy. Additional control over the DW chirality can b e realized by applying an in-plane magnetic field hx46, with the DW charge fixed. With this freedom in manipulating the DW chirality, the velocity of CDWM can be tuned by the application of an in-plane magnetic field, for DW width in the non-adiabatic limit. Furth er complication 9can arise from the chirality dependence of the Gilbert damping const ant and gyromagnetic factor47–49, as well as the STT non-adiabaticity50. Before turning to our conclusion, it is appropriate to mention that our above discussion is based on a simple 1D treatment of the CDWM, which is a very rough approximation based on the assumption t hat the ground state of the DW is a N´ eel configuration. The applicability of this assu mption is dubious in the presence of an electric current, since the current induced eff ective Rashba field tends to stabilize a Bloch wall. Our discussion is only to illustrate the complication o f the CDWM in the presence of the RSOT. Further detailed investigation is neede d for a thorough understanding of the CDWM in systems with sizable Rashba spin orbit in teraction. In conclusion, we have studied the RSOT in magnetic DWs, which is deriv ed from the broken inversion symmetry at ferromagnet/heavy metal interfa ces. By numerically solving the Pauli-Schr¨ odinger equation for 2D electrons moving inside a N´ e el DW, a topological damping RSOT component is identified. Even in the adiabatic limit, the ma gnitude of the topological damping component is sizable, in stark contrast to t he negligible non- adiabatic STT in the same limit. This finite damping RSOT is a manifestation of the nontrivial topology of the underlying DW. The identification of a topo logical damping RSOT component in magnetization textures will promote the applicat ion of RSOT in spintronic devices and facilitate a thorough understanding of the e xperimental data in current driven motion of magnetic DWs in ferromagnet/heavy meta l bilayer systems. ACKNOWLEDGEMENTS WewouldliketoexpressourgratitudetoProf. JiangXiaoforhisvalua blecommentsand discussions, especially for bringing us to the topic of RSOT in magnetic DWs and sharing his code on STT simulation. Y. Z. acknowledges the support by the Pr esident’s Fund of CUHKSZ, Longgang Key Laboratory of Applied Spintronics, Natio nal Natural Science Foundation of China (Grant No. 11974298, 61961136006), and Sh enzhen Fundamental Research Fund (Grant No. JCYJ20170410171958839). ∗wangdaowei@sztu.edu.cn †zhouyan@cuhk.edu.cn 1H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature (London) 408, 944 (2000). 2A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rev. Mod. 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Barna´ s, A. /suppress Lusakowski, and /suppress L. A. Turski, P hys. Rev. B 65, 224419 (2002). 42D. Wang, Y. Dong, Y. Zhou, X. Wang, J. He, and G. Guo, J. Phys.: C ondens. Matter 28, 206005 (2016). 43See Supplemental Material at for a detailed derivation of th e velocity for the CDWM. 44M. C. Hickey, Phys. Rev. B 78, 180412(R) (2008). 45V. V. Slastikov, C. B. Muratov, J. M. Robbins, and O. A. Tretia kov, Phys. Rev. B 99, 100403(R) (2019). 46P. P. J. Haazen, E. Mur` e, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Nat. Mater. 12, 299 (2013). 47E. Ju´ e, C. K. Safeer, M. Drouard, A. Lopez, P. Balint, L. Buda -Prejbeanu, O. Boulle, S. Auffret, A. Schuhl, A. Manchon, I. M. Miron, and G. Gaudin, Nat. Mater. 15, 272 (2016). 48C. A. Akosa, I. M. Miron, G. Gaudin, and A. Manchon, Phys. Rev. B 93, 214429 (2016). 49F. Freimuth, S. Bl¨ ugel, and Y. Mokrousov, Phys. Rev. B 96, 10 4418 (2017). 50D. Wang and Y. Zhou, arXiv: 1910.11031. 12

2019-10-24

Rashba spin orbit torque derived from the broken inversion symmetry at ferromagnet/heavy metal interfaces has potential application in spintronic devices. In conventional description of the precessional and damping components of the Rashba spin orbit torque in magnetization textures, the decomposition coefficients are assumed to be independent of the topology of the underlying structure. Contrary to this common wisdom, for Schr\"{o}dinger electrons trespassing ballistically across a magnetic domain wall, we found that the decomposition coefficient of the damping component is determined by the topology of the domain wall. The resultant damping Rashba spin orbit torque is protected by the topology of the underlying magnetic domain wall and robust against small deviations from the ideal domain wall profile. Our identification of a topological damping Rashba spin orbit torque component in magnetic domain walls will help to understand experiments on current driven domain wall motion in ferromagnet/heavy metal systems with broken inversion symmetry and to facilitate its utilization in innovative device designs.

Topological damping Rashba spin orbit torque in ballistic magnetic domain walls

1910.10977v2

Origin of Intrinsic Gilbert Damping M. C. Hickeyand J. S. Moodera Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, 150 Albany Street, Cambridge, Massachusetts 02139 USA. The damping of magnetization, represented by the rate at which it relaxes to equilibrium, is successfully modeled as a phenomenological extension in the Landau-Lifschitz-Gilbert equation. This is the damping torque term known as Gilbert damping and its direction is given by the vector product of the magnetization and its time derivative. Here we derive the Gilbert term from rst principles by a non-relativistic expansion of the Dirac equation. We nd that this term arises when one calculates the time evolution of the spin observable in the presence of the full spin-orbital coupling terms, while recognizing the relationship between the curl of the electric eld and the time varying magnetic induction. PACS numbers: 76.20.-m, 75.30.-m and 75.45.+j The Gilbert damping torque in magnetic systems de- scribes the relaxation of magnetization and it was intro- duced into the Laudau-Lifschitz equation [1, 2] for de- scribing spin dynamics. Gilbert damping is understood to be a non-linear spin relaxation phenomenon and it con- trols the rate at which magnetization spins reach equilib- rium. The introduction of this term is phenomenological in nature [3] and the question of whether it has an in- trinsic physical origin has not been fully addressed, in the face of rather successful modeling of the relaxation dynamics of measured systems. Correlating ferromag- netic resonance spectral line-widths [4, 5] in magnetic thin lms with the change in damping has been success- ful for conrming the form of the damping term in the underlying dynamical equations. The intrinsic origin of the damping itself is still an open question. The damping constant,is often reformulated in terms of a relaxation time, and the dominant relaxation processes are invoked to calculate this, but this approach presupposes preces- sional damping torque. It has been long thought that intrinsic Gilbert damp- ing had its origin in spin-orbital coupling because this mechanism does not conserve spin, but it has never been derived from a coherent framework. Non-local spin re- laxation processes [6] and disorder broadening couple to the spin dynamics and can enhance the Gilbert damp- ing extrinsically in thin lms and heterostructures. This type of spin relaxation, which is equivalent to ensemble dephasing [7], is modeled as the (S-S 0)/T 2decay term in the dynamical Bloch equation, where T 2is the decay time of the ensemble of spins. Crudely speaking, during spin relaxation, some spins lag behind the mean mag- netization vector and the exchange and magnetostatic elds then exert a time dependent torque. Calculations on relaxation driven damping of this kind presuppose the Gilbert damping term itself which begs the question. The inhomogeneous damping term can be written as Mdr2M=dtwhich gives rise to non-local eects such Electronic mail : hickey@mit.eduas spin wave dissipation [6, 8]. These non-local theo- ries are successful in quantifying the enhancement of the Gilbert damping, but do not derive the intrinsic Gilbert term itself. There are models [9, 10] which deal with the scattering of electron spins from thermal equilibrium in the presence of phonon and spin-orbital interactions which is a dynamic interaction and this allows us to de- termine the strength of the Gilbert damping for itiner- ant ferromagnetic metals, generalizing the Gilbert damp- ing response to a tensorial description. Both the s-d exchange relaxation models [11, 12] and the Fermi sur- face breathing models of Kambersky [9, 13] either pre- suppose a Gilbert damping term in the dynamical equa- tion or specify a phenomenological Hamiltonian H = - 1/( Ms)^.dM/dt. While this method is ab initio from the point of view of electronic structure, it already as- sumes the Gilbert term ansatz. Hankiewicz et al. [14] construct the inhomogeneous Gilbert damping by con- necting the spin density-spin current conservation law with the imaginary part of magnetic susceptibility ten- sor and show that both electron-electron and impurity scattering can enhance the damping through the trans- verse spin conductivity for nite wavelength excitations (q6= 0). In previous work [15], there are derivations of the Gilbert constant by comparing the macroscopic damping term with the torque-torque correlations in ho- mogeneously magnetized electron gases possessing spin orbital coupling. For the case of intrinsic, homogeneous Gilbert damping, it is thought that in the absence of spin-orbital scattering, the damping vanishes. We aim to focus on intrinsic, homogeneous damping and its physical origin in a rst-principles framework and the question as to whether spin in a homogeneous time-varying magne- tization can undergo Gilbert damping is addressed. In this work, we show that Gilbert damping does indeed arise from spin-orbital coupling, in the sense that it is due to relativistic corrections to the Hamiltonian which couple the spin to the electric eld and we arrive at the Gilbert damping term by rst writing down the Dirac equation for electrons in magnetic and electric potentials. We transform the Hamiltonian in such a way as to write it in a basis in which the canonical momentum terms arearXiv:0812.3184v2 [cond-mat.other] 1 Apr 20092 even powers. This is a standard approach in relativistic quantum mechanics and we do this in order to calculate the terms which couple the linear momentum to the spin in a basis which is diagonal in spin space. This is often referred to as a non-relativistic expansion of the Dirac equation. This allows us to formulate the contributions as a perturbation to an otherwise non-relativistic parti- cle. We then wish to calculate the rate equation for the spin observable with all of the spin-orbital corrections in mind. Now, we start with a purely relativistic particle, a Dirac particle and we write the Dirac-Pauli Hamiltonian, as follows : H=c:(peA c) +m 0c2+e (1) =O+m 0c2+" (2) where Aandare the magnetic vector potential and the electrostatic potential, respectively and = 0i i0 while = 1 0 01 : We observe immediately that O=O.Ois the Dirac canonical momentum , c and e are the speed of light in a vacuum and the electronic charge, respectively. We now need to rewrite the Hamiltonian in a basis where the odd operators (whose generators are o diagonal in the Pauli-Dirac basis : i, i, 5..) and even operators (whose generators are diagonal in the Pauli-Dirac basis : (1,, ,.. ) are decoupled from one another. If we are to nd S so that H0does not contain odd powers of spin operators, we must chose the operator S, in such a way as to satisfy the following constraint : [S;] =O im0c2(3) In order to satisfy cancelation of the odd terms of O to rst order, we require S=iO 2m0c2and this is known as the Foldy-Wouthuysen transformation in relativistic quantum mechanics and it is treated in some detail in, for example, reference [16]. We now would like to collect all of the terms into the transformed Hamiltonian, and this is written as H0= m0c2+O2 2m0c2O4 8m3 0c6 +"1 8m2 0c4[O;[O;"]] + 2m0c2[O;"]O3 3m2 0c4 The expression above contains odd powers of the canon- ical momentumO, so we redene the canonical momen- tum to encapsulate all of these odd power terms. So wenow apply the procedure of eliminating odd powers once again : S0=i 2m0c2O0=i 2m0c2 2m0c2[O;"]O3 3m2 0c4 (4) H00=eiS0 H0eiS0 =m 0c2+"0+O00; (5) whereO00is now O(1 m2 0c4), which can be further elimi- nated by applying a third transformation (S00=iO00 2m0c4), we arrive at the following Hamiltonian : H000=eiS00 H00 eiS00 =m 0c2+"0 = m0c2+O2 2m0c2O4 8m3 0c6 + "1 8m2 0c4[O;[O;"]] Thus we have the fully Foldy-Wouthuysen transformed Hamiltonian : H000= m0c2+(peA=c)2 2m0p4 8m3 0c6 +e e~ 2m0c2:Bie~2 8m2 0c2:(rE) e~ 4m2 0c2:Epe~2 8m2 0c2(r:E) The terms which are present in the above Hamiltonian, show us that we have a p4kinetic part which is the rela- tivistic expansion of the mass of the particle. The terms which couple to the spin  are of importance and we see that these terms correspond to the Zeeman, spin-orbital (comprising momentum and electric eld curl terms) and the Darwin term, respectively. Strictly speaking, the presence of the scalar potential breaks the gauge invari- ance in the Pauli-Dirac Hamiltonian and a fully gauge in- variant theory would require that this contain the gauge- free electromagnetic eld energy. We omit the term e2~ 4m2c3:(AE) (which establishes gauge invariance in the momentum terms) in this rotated Hamiltonian, as it is O(1=m2c3) and we are only interested in calculating semiclassical rate equations for elds, which are mani- festly gauge-invariant, and not wavefunctions or energy eigenvalues. We can now dene the spin dependent cor- rections to a non-relativistic Hamiltonian : H=e~ 2m0c2:Be~ 4m2 0c2:Epie~2 8m2 0c2:(rE): (6) where = i0 0i ^Si:3 andiare the Pauli matrices. Note that the last two terms in Equation 6 encapsulate the entire spin orbital coupling in the sense that these terms couple the particle's linear momentum to the spin ^Si. The rst spin-orbital term in the Hamiltonian is well known and give rise to momentum dependent magnetic elds. When the ensuing dynamics are calculated for this case, it gives rise to spin relaxation terms which are linear in spin [17]. Note that, while neither spin-orbital term is Hermitian, the two terms taken together are Hermitian and so the particles angular momentum is a conserved quantity and the total energy lost in going from collective spin excitations (spin waves) to single particles states via spin-orbital coupling is gained by the electromagnetic eld. Recognizing the curl of the electric eld in the last term, we now rewrite this the time varying magnetic eld as given by Maxwells equations asrE=@B @t. We now have an explicitly time-dependent perturbation on the non-relativistic Hamiltonian. We can write the time-varying magnetic eld seen by the spin (in, for example a magnetic material) as@B @t=@B @M
@M @t=0(1 +1 m)@M @t. We now have the spin dependent Hamiltonian : HS=e~ 2m0c2S:Be~ 4m2 0c2S:Ep +ie~20 8m2 0c2S: 1 +1 m
:dM dt= HS=HS 0+HS(t): We focus our attention on the explicitly time-dependent part of the Hamiltonian HS(t) ; HS(t) =ie~20 8m2 0c2S: 1 +1 m
:dM dt: (7) In this perturbation scheme, we allow the Hermitian components of the Hamiltonian to dene the ground sate of the system and we treat the explicitly time-dependent Hamiltonian (containing the spin orbital terms) as a time dependent perturbation. In this way, the rate equation is established from a time dependent perturbation expan- sion in the quantum Liouville description. We now dene the magnetization observable as ^M=X gB VTr^S(t) where the summation is taken over the site of the magne- tization spin . We now examine the time dependence of this observable by calculating the rate equation according to the quantum-Liouville rate equation ; d(t) dt+1 i~[^;H] = 0 (8) This rate equation governs the time-evolution of the magnetization observable as dened above, in the non- equilibrium regime. We can write the time derivative ofthe magnetization [18], as follows ; dM dt=X n;gb Vh n(t)j1 i~[S;H] +@ @tS+@S @tj n(t)i; and we can use the quantum Liouville rate equation as dened by Equation 8 to simplify this expression and we arrive at the following rate equation : dM dt=X gb V1 i~Trf[S;HS(t)]g (9) In the case of the time dependent Hamiltonian derived in equation 7, we can assume a rst order dynamical equation of motion given bydM dt= MHand calculate the time evolution for the magnetization observable : dM dt=X ;gB V1 i~Tr[Si ;ie~20 8m2c2Sj ]:(1 +1 m) !@M dt =X gB Vie~20 8m2c21 i~Tri ~ijkSk (1 +1 m)jl !@ Ml dt =ie~0 8m2c2(1 +1 m)M !@M dt; where, in the last two steps, we have used the fol- lowing commutation relations for magnetization spins : [Si ;Sj ] =i~ijkSk which implies that the theory pre- sented here is that which relates to local dynamics and that the origin of the damping is intrinsic. We now rec- ognize the last equation as the which describes Gilbert damping, as follows : dM dt= Ms:M !@M @t(10) whereby the constant is dened as follows : =ie~0Ms 8m2 0c2 1 +1 m
(11) Thedened above corresponds with the Gilbert damping found in the phenomenological term in the Landau-Lifschitz-Gilbert equation and mis the mag- netic susceptibility. In general, the inverse of the suscep- tibility can be written in the form [19], 1 ij(q;!) = ~1 ?(q;!)!ex 0M0ij; (12) where the equilibrium magnetization points along the z- axis and!exis the excitation frequency associated with the internal exchange eld. The ijterm in the in- verse susceptibility does not contribute to damping mech- anisms as it corresponds to the equilibrium response.4 In the basis (M xiMy,Mz), we have the dimensionless transverse magnetic susceptibility, as follows : ~m?(q;!) = 0M0i ?q2 !0!i ?q2!0=M0 The rst term in the dimensionless Gilbert coecient (Equation 11) is small ( 1011) and the higher damp- ing rate is controlled by the the inverse of the suscep- tibility tensor. For uniformly saturated magnetization, the damping is critical and so the system is already at equilibrium as far as the Gilbert mechanism is concerned (dM/dt = 0 in this scenario). The expression for the dimensionless damping constant in the dc limit ( !=0 ) is : =e~0Ms 8m2 0c2Im0 @!0 0M0i?q2!0 0M2 0 1i ?q2=M01 A; (13) and we have the transverse spin conductivity from the following relation (in units whereby ~=1) : ?=n 4m!2 01 dis ?+1 ee ? ; wheredis ?andee ?are the impurity disorder and electron electron-electron scattering times as dened and param- eterized in Reference [14]. We calculate the extrinsically enhanced Gilbert damping using the following set of pa- rameters as dened in the same reference ; number den- sity of the electron gas, n=1.4 1027m3, polarization p, equilibrium magnetization M 0= pn/2, equilibrium ex- citation frequency !0=EF[(1 +p)2=3(1p)2=3] and wave-number dened as q = 0.1 k F, where E Fand kF are the Fermi energy and Fermi wave number, respec- tively. mis taken to be the electronic mass. Using these quantities, we evaluate values and these are plotted as a function of both polarization and disorder scattering rate in Figure 1. In general, the inverse susceptibility 1 mwill deter- mine the strength of the damping in real inhomogeneous magnetic systems where spin relaxation takes place, sub- bands are populated by spin orbit scattering and spin waves and spin currents are emitted. The susceptibil- ity term gives the Gilbert damping a tensorial quality, agreeing with the analysis in Reference [10]. Further, the connection between the magnetization dynamics and the electric eld curl provides the mechanism for the energy loss to the electromagnetic eld. The generation of radi- ation is caused by the rotational spin motion analog of electric charge acceleration and the radiation spin inter- action term has the form : HS(t) =ie~20 8m2 0c2X  1 +1 m
S:dM dt: (14)In conclusion, we have shown that the Gilbert term, heretofore phenomenologically used to describe damping FIG. 1: (Color Online) Plot of the dimensionless Gilbert damping constant in the dc limit ( !=0), as a function of electron spin polarization and disorder scattering rate. in magnetization dynamics, is derivable from rst prin- ciples and its origin lies in spin-orbital coupling. By a non-relativistic expansion of the Dirac equation, we show that there is a term which contains the curl of the elec- tric eld. By connecting this term with Maxwells equa- tion to give the total time-varying magnetic induction, we have found that this damping term can be deduced from the rate equation for the spin observable, giving the correct vector product form and sign of Gilberts' origi- nal phenomenological model. Crucially, the connection of the time-varying magnetic induction and the curl of the electric eld via the Maxwell relation shows that the damping of magnetization dynamics is commensu- rate with the emission of electromagnetic radiation and the radiation-spin interaction is specied from rst prin- ciples arguments. Acknowledgments M. C. Hickey is grateful to the Trinity and the uni- formity of nature. We thank the U.S.-U.K. Fulbright Commission for nancial support. The work was sup- ported by the ONR (grant no. N00014-09-1-0177), the NSF (grant no. DMR 0504158) and the KIST-MIT pro- gram. The authors thank David Cory, Marius Costache and Carlos Egues for helpful discussions.5 [1] L. Landau and E. Lifshitz, Phys. Z. Sowiet. Un. 8, 153 (1935). [2] E. M. Lifschitz and L. P. Pitaevskii, Statistical Physics Part 2 (Pergamon Press, Oxford, United Kingdom, 1980). [3] T. Gilbert, Magnetics, IEEE Transactions on 40, 3443 (2004). [4] C. E. Patton, C. H. Wilts, and F. B. Humphrey, J. Appl. Phys. 38, 1358 (1967). [5] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki, Japanese Journal of Ap- plied Physics 45, 3889 (2006). [6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [7] Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phys. Lett. 84, 5234 (2004). [8] G. Eilers, M. L uttich, and M. M unzenberg, Phys. Rev. B74, 054411 (2006). [9] V. Kambersk y, Canadian Journal of Physics 48, 2906 (1970).[10] D. Steiauf and M. F ahnle, Phys. Rev. B 72064450 (2005). [11] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). [12] M. F ahnle, R. Singer, D. Steiauf, and V. P. Antropov, Phys. Rev. B 73, 172408 (2006). [13] J. Kune s and V. Kambersk y, Phys. Rev. B 65, 212411 (2002). [14] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B 78, 020404 (2008). [15] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys. Rev. B 75, 174434 (2007). [16] W. Greiner, Relativistic Quantum Mechanics (Springer- Verlag, Berlin, Germany, 1987). [17] H.-A. Engel, E. I. Rashba, and B. I. Halperin, Phys. Rev. Lett. 98, 036602 (2007). [18] J. Ho, F. C. Khanna, and B. C. Choi, Phys. Rev. Lett. 92, 097601 (2004). [19] Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404 (2002).

2008-12-17

The damping of magnetization, represented by the rate at which it relaxes to equilibrium, is successfully modeled as a phenomenological extension in the Landau-Lifschitz-Gilbert equation. This is the damping torque term known as Gilbert damping and its direction is given by the vector product of the magnetization and its time derivative. Here we derive the Gilbert term from first principles by a non-relativistic expansion of the Dirac equation. We find that the Gilbert term arises when one calculates the time evolution of the spin observable in the presence of the full spin-orbital coupling terms, while recognizing the relationship between the curl of the electric field and the time varying magnetic induction.

Origin of intrinsic Gilbert damping

0812.3184v2

arXiv:2103.03407v3 [math.NA] 6 Oct 2022Multilevel quasi-Monte Carlo for random elliptic eigenval ue problems II: Efficient algorithms and numerical results Alexander D. Gilbert1Robert Scheichl2 October 7, 2022 Abstract Stochastic PDE eigenvalue problems often arise in the field of uncert ainty quan- tification, whereby one seeks to quantify the uncertainty in an eige nvalue, or its eigen- function. Inthis paperwepresentanefficient multilevelquasi-Mont eCarlo(MLQMC) algorithm for computing the expectation of the smallest eigenvalue o f an elliptic eigen- value problem with stochastic coefficients. Each sample evaluation re quires the solu- tion of a PDE eigenvalue problem, and so tackling this problem in practic e is noto- riously computationally difficult. We speed up the approximation of this expectation in four ways: we use a multilevel variance reduction scheme to sprea d the work over a hierarchyofFEmeshes andtruncationdimensions; weuse QMCmeth ods to efficiently compute the expectations on each level; we exploit the smoothness in parameter space and reuse the eigenvector from a nearby QMC point to reduce the n umber of itera- tions of the eigensolver; and we utilise a two-grid discretisation sche me to obtain the eigenvalue on the fine mesh with a single linear solve. The full error ana lysis of a basic MLQMC algorithm is given in the companion paper [Gilbert and Scheichl, 20 22], and so in this paper we focus on how to further improve the efficiency and provide theo- retical justification for using nearby QMC points and two-grid meth ods. Numerical results are presented that show the efficiency of our algorithm, an d also show that the four strategies we employ are complementary. 1 Introduction In this paper we develop efficient methods for computing the ex pectation of an eigenvalue of the stochastic eigenvalue problem (EVP) −∇·/parenleftbig a(x,y)∇u(x,y)/parenrightbig +b(x,y)u(x,y) =λ(y)c(x,y)u(x,y),forx∈D, u(x,y) = 0 for x∈∂D,(1.1) where the differential operator ∇is with respect to x, which belongs to the physical domainD⊂Rdford= 1,2,3. Randomness is incorporated into the PDE (1.1) through the dependence of the coefficients a,bon the stochastic parameter y= (yj)j∈N, which is a countably infinite-dimensional vector with i.i.d. unif ormly distributed entries: yj∼ U[−1 2,1 2] forj∈N. The whole stochastic parameter domain is denoted by Ω := [−1 2,1 2]N. 1School of Mathematics and Statistics, University of New Sou th Wales, Sydney NSW 2052, Australia. alexander.gilbert@unsw.edu.au 2Institute for Applied Mathematics & Interdisciplinary Cen tre for Scientific Computing, Universit¨ at Heidelberg, 69120 Heidelberg, Germany and Department of Ma thematical Sciences, University of Bath, Bath BA2 7AY UK. r.scheichl@uni-heidelberg.de 1The study of stochastic PDE problems is motivated by applica tions in uncertainty quantification—where one is interested in quantifying how u ncertain input data affect modeloutputs. Inthecaseof (1.1)theuncertaininputdataa rethecoefficients aandb,and the outputs of interest are the eigenvalue λ(y) and its correspondingeigenfunction u(x,y), which are now also stochastic objects. As such, to quantify u ncertainty we would like to compute statistics of the eigenvalue (or eigenfunction), a nd in particular, in this paper we compute the expectation of the smallest eigenvalue λwith respect to the countable product of uniform densities. This is formulated as the infin ite-dimensional integral Ey[λ] =/integraldisplay Ωλ(y) dy:= lim s→∞/integraldisplay [−1 2,1 2]sλ(y1,y2,...,ys,0,0...) dy1dy2···dys. EVPs corresponding to differential operators appear in many a pplications from engi- neering and the physical sciences, e.g., structural vibrat ion analysis [43], nuclear reactor criticality [14, 27] or photonic crystal structures [13, 29 , 34]. In addition, stochastic EVPs, such as (1.1), have recently garnered more interest due to th e desire to quantify the un- certainty present in such applications [41, 2, 44, 3, 38]. Th us, significant development has recently also gone into efficient numerical methods for tackl ing such stochastic EVPs in practice, the most common being Monte Carlo [41], stochasti c collocation [1] and stochas- tic Galerkin/polynomial chaos methods [16, 44]. The latter two classes perform poorly for high-dimensional problems, so in order to handle the high-d imensionality of the param- eter space, sparse and low-rank versions of those methods ha ve been developed, see e.g., [1, 22, 15]. Furthermore, to improve upon classical Monte Ca rlo, while still performing well in high dimensions, the present authors with their coll eagues have analysed the use of quasi-Monte Carlo methods [17, 18]. In practice, for each parameter value y∈Ω the elliptic EVP (1.1) must be solved numerically, which we do here by the finite element (FE) metho d, see, e.g., [4]. First, the spatial domain is discretised by a family of triangulati ons{Th}h>0indexed by the meshsize h >0, and then (1.1) is solved on the finite-dimensional FE space corresponding toTh. This leads to a large, sparse, symmetric matrix EVP, which i s typically solved by an iterative method (such as Rayleigh quotient iteration or the Lanczos algorithm, see, e.g., [37, 40]), requiring several solves of a linear system . To speed up the solution of each EVP we use the accelerated two-grid method developed independently in [25, 26] and [47]. In particular, to obtain an eigenvalue approximation corre sponding to a “fine” mesh Th, one first solves the FE EVP on a “coarse” mesh TH, withH≫h, to obtain a coarse eigenpair ( λH,uH). An eigenvalue approximation λhon the fine grid This then obtained by performing a single step of shifted inverse iteration wit h shiftλHand start vector uH. Typically, the fast convergence rates of FE methods and of sh ifted inverse iteration for eigenvalue problems allow for a very large difference between the coarse and fine meshsize, e.g., for piecewise polynomial FE spaces it is sufficient to ta keH/equalorsimilarh1/4, so that the cost of the two-grid method essentially reduces to the cost of a si ngle linear solve on the fine mesh. Multilevel sampling schemes also exploit a hierarchy of FE meshes, and so the two-grid method very naturally fits into this framework. The multilevel Monte Carlo (MLMC) method [20, 23] is a varian ce reduction scheme that has achieved great success when applied to problems inc luding path integration [23], stochastic differential equations [20] and also stochastic P DEs [6, 8]. For stochastic PDE problems, it is based on a hierarchy of L+1 increasingly fine FE meshes {Thℓ}L ℓ=0(i,.e., the meshwidths are decreasing h0> h1>···> hL>0), and an increasing sequence of truncation dimensions s0< s1<···< sL<∞of the infinite-dimensional parameter domain Ω. For the eigenvalue problem (1.1), letting the trun cation-FE approximation on levelℓbe denoted by λℓ:=λhℓ,sℓ, the key idea is to write the expectation on the desired 2finest level Las a telescoping sum of differences: Ey[λL] =Ey[λ0]+L/summationdisplay ℓ=1Ey[λℓ−λℓ−1], (1.2) and then compute each expectation Ey[λℓ−λℓ−1] by an independent MC approximation. Asℓ→∞, provided hℓ→0 andsℓ→∞, we have λℓ→λand hence also λℓ−λℓ−1→0. Thus the variance on each level decreases, and so less sample s will be needed on the finer levels. In this way, the MLMC method achieves a significa nt cost reduction by spreading the work across the hierarchy of levels, instead o f performing all evaluations on the finest level L. For any linear functional G, we can write a similar telescoping sum for Ey[G(uL)], where we define uℓ:=uhℓ,sℓ. The smallest eigenvalue is simple, therefore we can ensure that the corresponding eigenfunction is unique b y normalising it and choosing the sign consistently. Similarly, we normalise each approx imationuℓand choose the sign to match the eigenfunction, which ensures they are also well defined. Our method can also be applied to approximate the expectation of other simp le eigenvalues higher up the spectrum, without any essential modifications. If the ei genvalue in question is well- separated from the rest of the spectrum (uniformly in y) then our analysis can also be extended in a straightforward way. However, for simplicity and clarity of presentation in this paper we focus on the smallest eigenvalue. Quasi-Monte Carlo methods are equal-weight quadrature rul es that are tailored to efficiently approximate high-dimensional integrals, see, e .g., [11, 12]. In particular, by deterministically choosing well-distributed quadrature points, giving preference to more important dimensions, QMC rules can be constructed such tha t the error converges faster than for MC methods, whilst still being independent of dimen sion. Using a QMC rule to approximate the expectation on each level in (1.2) instea d of Monte Carlo gives a Multilevel quasi-Monte Carlo (MLQMC) method. MLQMC method s were first developed in [21] for option pricing, and since then have also had great success for UQ in stochastic PDE problems [19, 32, 31]. The gains are complementary, so th at for several problems MLQMCmethodscanbeshowntoresultinfasterconvergenceth aneitherMLMCmethods or single level QMC approximations. In this paper, we present an efficient MLQMC method for computi ng the expectation of the smallest eigenvalue of stochastic eigenvalue proble ms of the form (1.1). We employ four complementary strategies: 1) we use the ML strategy to r educe the variance and spread the work across a hierarchy of FE meshes and truncatio n dimensions; 2) we use QMC methods to compute the expectation on each level more effic iently; 3) we use the two-grid method for eigenvalue problems [25, 26, 47] to comp ute the eigenpair on finer grids using an eigensolve on a very coarse grid followed by a s ingle linear solve; and 4) we reuse the eigenvector corresponding to a nearby QMC point as the starting vector for the Rayleigh quotient algorithm to solve each eigenvalue probl em. Thefocusofthispaperisondevelopingtheabovepracticals trategies togiveanefficient MLQMC method. A rigorous analysis of MLQMC methods for the st ochastic EVP (1.1) is the focus of a separate paper [19]. However, we do give a the oretical justification of the enhancement strategies 3) and 4). First, we extend the two-g rid method and its analysis to stochastic EVPs, allowing also for a reduced truncation d imension on the “coarse grid”. Second, we analyse the benefit of using an eigenvector corres ponding to a nearby QMC point as the starting vector for the iterative eigensolve. The structure of the paper is as follows. In Section 2 we prese nt the necessary back- ground material. Thenin Section 3we extend the two-grid met hodfor deterministic EVPs to stochastic EVPs and analyse the error. In Section 4 we desc ribe a basic MLQMC algo- rithm, and thenoutline how onecan reducethecost by usingtw o-grid methods andnearby 3QMC points. Finally, in Section 5 we present numerical resul ts for two test problems. 2 Mathematical background In this section we briefly summarise the relevant material on variational EVPs, two-grid FE methods and QMC methods. For further details we refer the r eader to the references indicated throughout or to [17]. We make the following assumptions on the physical domain and on the boundedness of the coefficients (from above and below). These ensure the we ll-posedness of (1.1) and admit a fast convergence rate of our MLQMC algorithm. Assumption A1. 1.D⊂Rd, ford= 1,2,3, is bounded and convex. 2.aandbare of the form a(x,y) =a0(x)+∞/summationdisplay j=1yjaj(x)andb(x,y) =b0(x)+∞/summationdisplay j=1yjbj(x),(2.1) whereaj, bj∈L∞(D), for allj≥0, andc∈L∞(D)depend on xbut noty. 3. There exists amin>0such that a(x,y)≥amin,b(x,y)≥0andc(x)≥amin, for all x∈D,y∈Ω. 4. There exists p∈(0,1)andq∈(0,1)such that ∞/summationdisplay j=1max/parenleftbig /ba∇dblaj/ba∇dblL∞,/ba∇dblbj/ba∇dblL∞/parenrightbigp<∞and∞/summationdisplay j=1/ba∇dbl∇aj/ba∇dblq L∞<∞. For convenience, we then let amax<∞be such that max/braceleftbig /ba∇dbla(y)/ba∇dblL∞,/ba∇dbl∇a(y)/ba∇dblL∞,/ba∇dblb(y)/ba∇dblL∞,/ba∇dblc/ba∇dblL∞/bracerightbig ≤amax,for ally∈Ω.(2.2) 2.1 Variational eigenvalue problems For the variational form of the EVP (1.1), we introduce the us ual function space setting for second-order elliptic PDEs: the first-order Sobolev spa ce of functions with zero trace is denoted by V:=H1 0(D) and equipped with the norm /ba∇dblv/ba∇dblV:=/ba∇dbl∇v/ba∇dblL2. Its dual space isV∗:=H−1(D). We will also use the Lebesgue space L2(D), equipped with the usual inner product/a\}b∇acketle{t·,·/a\}b∇acket∇i}htL2, and the induced norm /ba∇dbl·/ba∇dblL2. Next, for each y∈Ω define the bilinear form A(y) :V×V→Rby A(y;w,v):=/integraldisplay Da(x,y)∇w(x)·∇v(x)dx+/integraldisplay Db(x,y)w(x)v(x)dx, which is also an inner product on Vand admits the induced norm /ba∇dblv/ba∇dblA(y):=/radicalbig A(y;v,v). We define also the inner product M:V×V→Rby M(w,v):=/integraldisplay Dc(x)w(x)v(x)dx, again with induced norm given by /ba∇dblv/ba∇dblM:=/radicalbig M(v,v). Further, letM(·,·) also denote the duality paring on V×V∗. 4The variational form of the EVP (1.1) is: Find λ(y)∈R,u(y)∈Vsuch that A(y;u(y),v) =λ(y)M(u(y),v) for all v∈V , (2.3) /ba∇dblu(y)/ba∇dblM= 1. Thevariational EVP(2.3)is symmetricandsoit is well-know n that(2.3)admits countably many, strictly positive eigenvalues, see, e.g., [4]. The ei genvalues – labelled in ascending order, counting multiplicities – and the corresponding eig enfunctions are denoted by 0< λ1(y)≤λ2(y)≤···,andu1(y), u2(y), ... . Fory∈Ω, we define the solution operator T=T(y) :V∗→Vby A(y;Tf,v) =M(f,v) for all v∈V. Clearly, if λ(y) is an eigenvalue of (2.3) then µ(y) = 1/λ(y) is an eigenvalue of Tand the corresponding eigenspaces are the same. The Krein–Rutmann Theorem ensures the smallest eigenvalue is simple, and then in [17, Prop. 2.4] it was shown that the spectral gap can be bound ed away from 0 indepen- dently of y. That is, there exists ρ >0, independent of y, such that λ2(y)−λ1(y)≥ρfor ally∈Ω. (2.4) The eigenfunctions {u(y)}k∈Ncan be chosen to form a basis for Vthat is orthonormal with respect toM(·,·), and hence, by (2.3), also orthogonal with respect to A(y;·,·). Fory∈Ω, let the eigenspace E(λk(y)) be the subspace spanned by all eigenfunctions corresponding to λk(y), and let/hatwideE(λk(y)):={v:v∈E(λk(y)),/ba∇dblv/ba∇dblV= 1}. Since the coefficients are uniformly bounded away from 0 and fr om above, theA(y)- andM-norms are equivalent to the V- andL2-norms, respectively, with cA/ba∇dblv/ba∇dblV≤/ba∇dblv/ba∇dblA(y)≤CA/ba∇dblv/ba∇dblV, (2.5) cM/ba∇dblv/ba∇dblL2≤/ba∇dblv/ba∇dblM≤CM/ba∇dblv/ba∇dblL2, (2.6) wherethe constants are independentof y, see [19, eqs. (2.7), (2.8)] for their explicit values. ByCPoin>0 we denote the Poincar´ e constant, which is independent of yand such that /ba∇dblv/ba∇dblL2(D)≤CPoin/ba∇dblv/ba∇dblV,for allv∈V. (2.7) For the remainder of the paperwe denote the smallest eigenva lue and its corresponding eigenfunction by λ=λ1andu=u1, respectively. 2.2 Stochastic dimension truncation In order to evaluate the stochastic coefficients a(y) andb(y) in practice, we must first truncate the infinite-dimensional stochastic domain Ω. Thi s is done by choosing a finite truncation dimension s∈Nand by setting yj= 0 for all j > s. We define the following notation: ys= (y1,y2,...,ys), as(x,y):=a0(x)+s/summationdisplay j=1yjaj(x) and bs(x,y):=b0(x)+s/summationdisplay j=1yjbj(x). In this way, the truncated coefficients as(y) andbs(y) can be evaluated in practice, since they only depend on finitely many terms. 5Similarly, the truncated approximations of the eigenvalue and eigenfunction are de- noted by λs(y),us(y), respectively. Defining the bilinear form As(y) :V×V→R corresponding to the truncated coefficients by As(y;w,v):=/integraldisplay Das(x,y)∇w(x)·∇v(x)dx+/integraldisplay Dbs(x,y)w(x)v(x)dx,(2.8) we have that λs(y),us(y) satisfy As(y;us(y),v) =λs(y)M(us(y),v),for allv∈V . (2.9) 2.3 Finite element methods for eigenvalue problems The eigenvalue problem (2.3) will be discretised in the spat ial domain using piecewise linear finiteelements (FE). First, wepartitionthespatial domainDusingafamilyofshape regular triangulations {Th}h>0, indexed by the meshwidth h= max{diam(τ) :τ∈Th}. Then, for h >0 letVhbe the conforming FE space of continuous functions that are piecewise linear on the elements of the triangulation Th, and let Mh:= dim(Vh)<∞ denote the dimension of this space. Additionally, we assume that each mesh This such that the dimension of the corresponding FE space Vhis Mh/equalorsimilarh−d, (2.10) which will be satisfied by quasi-uniform meshes, but also all ows for locally refined meshes. For each y∈Ω, the FE eigenvalue problem is: Find λh(y)∈R,uh(y)∈Vhsuch that A(y;uh(y),vh) =λh(y)M(uh(y),vh) for all vh∈Vh, (2.11) /ba∇dbluh(y)/ba∇dblM= 1. The FE eigenvalue problem (2.11) admits Mheigenvalues and corresponding eigenvectors 0< λ1,h(y)≤λ2,h(y)≤···≤ λMh,h(y),andu1,h(y), u2,h(y), ..., u Mh,h(y), which converge from above to the first Mheigenvalues and eigenfunctions of (2.3) as h→0, see, e.g., [4] or [17] for the stochastic case. Asbefore,let E(λk,h(y))betheeigenspacecorrespondingto λk,h(y)anddefine/hatwideE(λk,h(y)):= {v∈E(λk,h(y)) :/ba∇dblv/ba∇dblV= 1}. If the exact eigenvalue λk(y) has multiplicity m(and we assume without loss of generality that λk(y) =λk+1(y) =···=λk+m−1(y)), then there existmFE eigenvalues, λk,h(y), λk+1,h(y),...,λ k+m−1,h(y), that converge to λk(y), but are not necessarily equal. As such, we also define Eh(λk(y)) to be the direct sum of all the eigenspaces E(λℓ,h(y)) such that λℓ,h(y)→λk(y). Finally, we define /hatwideEh(λk(y):={v∈ Eh(λk(y)) :/ba∇dblv/ba∇dblV= 1}. In Assumption A1 we have only assumed that the physical domai nDis convex and thata∈W1,∞(D). Hence, piecewise linear FEs are sufficient to achieve the op timal rates of convergence with respect to hin general. In particular, in [17, Thm. 2.6] it was shown that the FE error for the minimal eigenpair can be bounded ind ependently of ywith the usual rates in terms of h. Explicitly, if h >0 is sufficiently small, then for all y∈Ω /ba∇dblu(y)−uh(y)/ba∇dblV≤Cuh,|λ(y)−λh(y)|≤Cλh2, (2.12) and forG∈H−1+t(D) witht∈[0,1] /vextendsingle/vextendsingleG(u(y))−G(uh(y))/vextendsingle/vextendsingle≤CGh1+t, (2.13) 6where 0< Cλ, Cu, CG<∞are independent of yandh. In the companion paper [19], it is shown that for hsufficiently small1the spectral gap of the FE eigenvalue problem (2.11) satisfies the uniform low er bound λ2,h(y)−λ1,h(y)≥ρ 2>0, (2.14) and that the eigenvalues and eigenfunctions of both (2.3) an d (2.11) satisfy the bounds λk≤λk(y)≤λk,h(y)≤λk,and (2.15) max/braceleftbig /ba∇dbluk(y)/ba∇dblV,/ba∇dbluk,h(y)/ba∇dblV/bracerightbig ≤uk, (2.16) whereλk,λk,ukare also independent of both yandh. Note that the use of piecewise linear FEs is not a restriction on our MLQMC methods. The algorithms presented in Section 4 are very general, and w ill work with higher order FE methods as well, without any modification of the overall al gorithm structure. 2.4 Iterative solvers for eigenvalue problems The discrete EVP (2.11) from the previous section leads to a g eneralised matrix EVP of the form Ahuh=λhBhuh, where, in general, the matrices AhandBhare large, sparse and symmetric positive definite. Since we are only interested in computing a single eigenpair , we will use Rayleigh quotient (RQ) iteration to compute it. It is well-known that for symmetric matrices RQ iteration converges cubically for almost all starting vect ors, see, e.g., [37]. 2.5 Quasi-Monte Carlo integration A quasi-Monte Carlo (QMC) method is an equal weight quadratu re rule Qs,Nf=1 NN−1/summationdisplay k=0f(tk) (2.17) withN∈Ndeterministically-chosen quadrature points {tk}N−1 k=0, as opposed to random quadrature points as in Monte Carlo. The key feature of QMC me thods is that the points are cleverly constructed to be well-distributed within hig h-dimensional domains, which allows for efficient approximation of high-dimensional inte grals such as Isf:=/integraldisplay [−1 2,1 2]sf(y)dy. There are many different types of QMC point sets, and for furthe r details we refer the reader to, e.g., [11]. In this paper, we use a simple to construct, yet powerful, cla ss of QMC methods called randomly shifted rank-1 lattice rules . A randomly shifted lattice rule approximation to IsfusingNpoints is given by Qs,N(∆)f:=1 NN−1/summationdisplay k=0f({tk+∆}−1 2) (2.18) wherez∈Nsis thegenerating vector and the points tkare given by tk=/braceleftbiggkz N/bracerightbigg fork= 0,1,...,N−1, 1The explicit condition is that h≤hwithh:=/radicalbig ρ/(2Cλ). 7∆∈[0,1)sis a uniformly distributed random shift , and{·}denotes the fractional part of each component of a vector. Note that we have subtracted 1 /2 in each dimension to shift the quadrature points from [0 ,1]sto [−1 2,1 2]s. Goodgeneratingvectorscanbeconstructedinpracticeusin gthecomponent-by-component (CBC) algorithm, or the more efficient Fast CBC construction [35, 36]. In fact, it can be shown that for functions in certain first-order weighted S obolev spaces such as those introduced in [42], the root-mean-square (RMS) error of a ra ndomly shifted lattice rule using a CBC-constructed generating vector achieves almost the optimal rate of O(N−1). To state the CBC error bound, we briefly introduce the followi ng specific class of weighted Sobolev spaces, which are useful for the analysis o f lattice rules. Given a col- lection of weightsγ:={γu>0 :u⊆{1,2,...,s}}, which represent the importance of different subsetsof variables, let Ws,γbethes-dimensionalweighted (unanchored)Sobolev space of functions with square-integrable mixed first deriv atives, equipped with the norm /ba∇dblf/ba∇dbl2 Ws,γ=/summationdisplay u⊆{1:s}1 γu/integraldisplay [−1 2,1 2]|u|/parenleftbigg/integraldisplay [−1 2,1 2]s−|u|∂|u| ∂yuf(y) dy−u/parenrightbigg2 dyu,(2.19) where we use the notation {1 :s}={1,2,...,s},yu= (yj)j∈uandy−u= (yj)j∈{1:s}\u. Then, for f∈Ws,γandNa power of 2, the RMS error of a CBC-constructed randomly shifted lattice rule approximation satisfies /radicalBig E∆/bracketleftbig |Isf−Qs,Nf|2/bracketrightbig /lessorsimilarN−1+δ/ba∇dblf/ba∇dblWs,γ, δ > 0, (2.20) where under certain conditions on the decay of the weights γthe constant is independent of the dimension. Note that similar results also hold for gen eralN, but with Non the RHS of (2.20) replaced by the Euler Totient function, which c ounts the number of integers less than and coprime to N, see, e.g., [11, Theorem 5.10]. For more details on the gener al theory of lattice rules see [11], and for a theoretical analy sis of randomly shifted lattice rules for MLQMC applied to (1.1) see [19]. The generating vectors given by the CBC algorithm are extens ible in dimension, how- ever, they are constructed for a fixed value of N. By modifying the error criterion that is minimised in each step of the CBC algorithm, one can construc t a generating vector that works well for a range of values of N, where now Nis given as some power of a prime base, e.g.,Nis a power of 2. The resulting quadrature ruleis called an embedded lattice rule and was developed in [9]. Not only do embedded lattice rules work well for a range of values ofN, but the resulting point sets are nested. Hence, one can impr ove the accuracy of a previously computed embedded lattice rule approximation b y simply adding the function evaluations corresponding to the new points to the sum from t he previous approximation. As will be clear later, the extensibilty in both sandNof embedded lattice rules makes them extremely convenient for use in MLQMC methods in practi ce. Currently there is not any theory for the error of embedded la ttice rules, however, a series of comprehensive numerical tests conducted in [9] sh ow empirically that the optimal rateofN−1isstillobserved, andthattheworst-caseerrorforanembed dedlattice increases at most by a factor of 1.6 as compared to the normal CBC algorit hm with Nfixed. Finally, instead of using a single random shift, in practice it is better to average over several randomlyshiftedapproximationsthatcorrespondt oasmallnumberofindependent random shifts. The practical benefits are (i) that averaging gives more consistent results, by reducing the chance of using a single “bad” shift, and (ii) that the sample variance of the shifted approximations provides a practical error esti mate. Let ∆(1),∆(2),...,∆(R) beRindependent uniform random shifts, then the average of the Q MC approximations 8corresponding to the random shifts is /hatwideQs,N,Rf:=1 RR/summationdisplay r=1Qs,N(∆(r))f, and the mean-square error of /hatwideQs,N,Rfcan be estimated by the sample variance /hatwideV[/hatwideQs,N,R]:=1 R(R−1)R/summationdisplay r=1/bracketleftbig/hatwideQs,N,Rf−Qs,N(∆(r))f/bracketrightbig2. (2.21) 2.6 Discrepancy theory Much of the modern theory for QMC rules is based on weighted fu nction spaces as dis- cussed in Section 2.5, however, the traditional analysis of QMC rules is based on the discrepancy of the quadrature points. Loosely speaking, for a given poin t set the discrep- ancy measures the difference between the number of points that actually lie within some subset of the unit cube and the number of points that are expec ted to lie in that subset if the point set were perfectly uniformly distributed. This more geometric notion of the quality of a QMC point set will be useful later when we analyse the use of an eigenvector corresponding to a nearby QMC point as the starting vector fo r the RQ iteration. We now recall some basic notation and definitions from the fiel d of discrepancy theory for a point setPN={t0,t1,...,tN−1}⊂[0,1]son the unit cube. Note that by a simple translation the results from this section are also applicab le on [−1 2,1 2]s. The axis-parallel box with corners a,b∈[0,1]swithaj< bjis denoted by [ a,b):= [a1,b1)×[a2,b2)×···× [as,bs). The number of points from PNthat lie in [ a,b) is denoted by|{PN∩[a,b)}|and the Lebesgue measure on [0 ,1]sbyLs. Definition 2.1. Thestar discrepancy of a point setPNis defined by D∗ N(PN):= sup b∈[0,1]s/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[0,b)}| N−Ls/parenleftbig [0,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.22) PNis called a low discrepancy point set if there exists CPN<∞, independent of s, such that D∗ N(PN)≤CPNlog(N)s−1 N. (2.23) There exist several well-known points sets that have low-di screpancy, such as Ham- mersley point sets, see [12] for more details. The connection between star discrepancy and quadrature is g iven by the Koksma– Hlawka inequality, which for a function fwith bounded Hardy–Krause variation states that the quadrature error of a QMC approximation (2.17) sati sfies the bound /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay [0,1]sf(y) dy−Qs,Nf/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftBigg/summationdisplay ∅/\e}atio\slash=u⊆{1:s}/integraldisplay [0,1]|u|/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u| ∂yuf(yu;1)/vextendsingle/vextendsingle/vextendsingle/vextendsingledyu/parenrightBigg D∗ N(PN),(2.24) see, e.g., [12]. Here, ( yu;1) denotes the anchored point with jth component yjifj∈uand 1 otherwise. Hence, low-discrepancy point sets lead to QMC a pproximations for which the error converges like O(log(N)s−1/N). Lattice rules can also be constructed such that their discre pancy is log( N)s/N(see [12, Corollary 3.52]). By the Koksma–Hlawka inequality (2. 24), they then admit error bounds similar to (2.20), but with an extra log( N)sfactor. By instead considering the 9weighted discrepancy, one can construct lattice rules that have a weighted discrepancy (and similarly error bounds) without this log factor, see [2 8]. Finally, we also define the extreme discrepancy of a point set , which removes the restriction that the boxes are anchored to the origin. Definition 2.2. Theextreme discrepancy of a point setPNis defined by /hatwideDN(PN):= sup [a,b)⊂[0,1]s/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[a,b)}| N−Ls/parenleftbig [a,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.25) 3 Two-grid-truncation methods for stochastic EVPs Two-grid FEdiscretisation methodsforEVPswerefirstintro ducedin[46]andlater refined independentlyin[25,26]and[47]. Theideabehindthemissi mple: tocombineFEmethods with iterative solvers for matrix EVPs. Letting H > h > 0 be the meshwidths of a coarse and a fine FE mesh, THandTh, respectively, one first solves the EVP (2.11) on the coarse FE space VHto giveλH,uH. This coarse eigenpair ( λH,uH) is then used as the starting guess for an iterative eigensolver for the EVP on th e fine FE space Vh. Since FE methods for PDE EVPs and iterative methods for matrix EVPs bo th converge very fast, Handhcan be chosen such that a single linear solve is all that is req uired to obtain the same order of accuracy as can be expected from solving the ori ginal FE EVP on the fine mesh. This strategy can be adapted to a full multigrid method for EVPs as in, e.g., [45]. However, it was shown in [25, 26, 47] that the maximal ratio H/hbetween the coarse and fine meshwidth in a two grid method is so large that, in general , two grids are sufficient. Here, we present a new algorithm that extends the two-grid me thod to stochastic (or parametric) EVPs, by also using a reduced (i.e., cheaper and less accurate) truncation of the parameter space when solving the parametric EVP on the in itial coarse mesh. Since our new algorithm combines this truncation and FE approxima tions, we first introduce some notation. For some h >0 ands∈N, the FE EVP that approximates the truncated problem (2.9) is: Find λh,s(y)∈Randuh,s(y)∈Vhsuch that As(y;uh,s(y),vh) =λh,s(y)M(uh,s(y),vh) for all vh∈Vh, (3.1) /ba∇dbluh,s(y)/ba∇dblM= 1. We also define the solution operator Th,s=Th,s(y) :V∗→Vhfor (3.1), which for f∈V∗ satisfies As(y;Th,sf,vh) =M(f,vh) for all vh∈Vh, (3.2) and theAs(y)-orthogonal projection operator Ph,s=Ph,s(y) :V→Vh, which for u∈V satisfies As(y;u−Ph,su,vh) = 0 for all vh∈Vh. (3.3) Although both operators depend on ywe will not specify this dependence. In Algorithm 1 below we detail our new two-grid and truncatio n method for paramet- ric EVPs. The algorithm is based on the accelerated version o f the two-grid algorithm (see [25, 26] and also [47]), which uses the shifted-inverse power method for the update step. In addition, we add a normalisation step so that /ba∇dbluh(y)/ba∇dblM= 1, which simplifies the RQ update but does not affect the theoretical results. As in the papers above, we will consistently use the notation that two-grid approxima tions use superscripts whereas ordinary approximations (i.e., eigenpairs of truncated/F E problems) use subscripts. To perform step 2 in practice, one must interpolate the uH,S(y) at the nodes of the fine mesh Thto obtain the corresponding start vector. Since we use piece wise linear FE methods, linear interpolation is sufficient. 10Algorithm 1 Two-grid-truncation method for parametric EVPs GivenH > h > 0, 0< S < s andy∈Ω: 1:FindλH,S(y)∈RanduH,S(y)∈VHsuch that AS(y;uH,S(y),vH) =λH,S(y)M(uH,S(y),vH) for all vH∈VH, /ba∇dbluH,S(y)/ba∇dblM= 1. 2:Finduh,s∈Vhsuch that As(y;uh,s(y),vh)−λH,S(y)M(uh,s(y),vh) =M(uH,S(y),vh) for all vh∈Vh.(3.4) 3:uh,s(y)←uh,s(y)//ba∇dbluh,s(y)/ba∇dblM ⊲normalise the eigenfunction approximation 4: λh,s(y) =As(y;uh,s(y),uh,s(y)). (3.5) The following lemmas will help us to extend the error analysi s of two-grid methods to include a component that corresponds to truncating the para meter dimension. Lemma 3.1. LetB,/tildewideB:V×V→R, be two bounded, coercive, symmetric bilinear forms. Suppose that (λ,u)is an eigenpair of B(u,v) =λM(u,v)for allv∈V , and letw∈V. Then /tildewideB(w,w) M(w,w)−λ=/ba∇dblu−w/ba∇dbl2 /tildewideB /ba∇dblw/ba∇dbl2 M−λ/ba∇dblu−w/ba∇dbl2 M /ba∇dblw/ba∇dbl2 M+1 /ba∇dblw/ba∇dbl2 M/parenleftbig B(u,u−2w)−/tildewideB(u,u−2w)/parenrightbig .(3.6) Proof.Expanding, then using the fact that ( λ,u) is an eigenpair gives /ba∇dblu−w/ba∇dbl2 /tildewideB−λ/ba∇dblu−w/ba∇dbl2 M =/tildewideB(u,u)+/tildewideB(w,w)−2/tildewideB(u,w)−λM(u,u)−λM(w,w)+2λM(u,w) =/tildewideB(u,u)+/tildewideB(w,w)−2/tildewideB(u,w)−B(u,u)−λM(w,w) +2B(u,w) =/tildewideB(w,w)−λM(w,w)−B(u,u−2w)+/tildewideB(u,u−2w). Dividing by/ba∇dblw/ba∇dbl2 Mand rearranging leads to the desired result. Lemma 3.2. Let Assumption A1 hold, then /ba∇dblT−Th,s/ba∇dbl≤CT(s−1/p+1+h), (3.7) whereCTis independent of y,sandh. Proof.The differential operator A(y)v=−∇·(a(y)∇v)+b(y)vfrom the EVP (1.1) fits into the general framework of [10]. Defining Ts:=T(ys) to be the solution operator for the truncated EVP (2.9), it follows from the triangle inequa lity that /ba∇dblT−Th,s/ba∇dbl≤/ba∇dblT−Ts/ba∇dbl+/ba∇dblTs−Th,s/ba∇dbl≤C1s−1/p+1+C2h, where we have used [10, Theorem 2.6 and eq. (2.17)] in the last step. 11The error of the outputs of Algorithm 1 are given in the theore m below. The proof follows a similar proof technique as used in [47], and also re lies on an abstract approxi- mation result for operators from that paper. Note that the FE component of the error is the same as the results in [25, 26, 47], but here we have extra t erms corresponding to the truncation error. The proof is deferred to the appendix. Theorem 3.1. Suppose that Assumption A1 holds, let S∈Nbe sufficiently large and let H >0be sufficiently small. Then, for s > Sand0< h < H , /ba∇dblu(y)−uh,s(y)/ba∇dblV/lessorsimilarH4+h+S−2(1/p−1)+s−(1/p−1)+H2S−(1/p−1),and(3.8) |λ(y)−λh,s(y)|/lessorsimilarH8+h2+S−4(1/p−1)+s−(1/p−1)+H4S−2(1/p−1), (3.9) where both constants are independent of s,S,h,H andy. It follows that in our two-grid-truncation method, to maint ain the optimal order h convergence for the eigenfunction we should take H/equalorsimilarh1/4,s/equalorsimilarh−p/(1−p)andS/equalorsimilars1/2, whereas for the eigenvalue error we should take a higher trun cation dimension, namely s/equalorsimilarh−2p/(1−p)andS/equalorsimilars1/4. The difference in conditions comes from the fact that for EVPs, the truncation error for the eigenvalue and eigenfunc tion are of the same order, whereas the FE error for the eigenvalue is double the order of the eigenfunction FE error. Itis similar to howahigher precision numerical quadrature ruleshouldbeusedtocompute the elements of the stiffness matrix for eigenvalue approxima tion, see, e.g., [5]. 4 MLQMC algorithms for random eigenvalue problems In this section, we present two MLQMC algorithms for approxi mating the expectation of a random eigenvalue. First, we briefly give a straightforward MLQMC algorithm, for which a rigorous theoretical analysis of the error was presented i n [19]. After analysing the cost of this algorithm we then present a second, more efficient MLQM C algorithm, where we focus on reducing the overall cost by reducing the cost of eva luating each sample. 4.1 A basic MLQMC algorithm for eigenvalue problems The starting point of our basic MLQMC algorithm is the telesc oping sum (1.2), along with a collection of L+1 FE meshes corresponding to meshwidths, h0> h1>···> hL>0, andL+ 1 truncation dimensions, 0 < s0≤s1≤···≤ sL<∞. Recall that we denote the eigenvalue approximation on level ℓbyλℓ:=λhℓ,sℓwithλ−1≡0. The expectation on each level ℓin the sum (1.2) can be approximated by a QMC rule using Nℓpoints, which we denote by Qℓ:=Qsℓ,Nℓas in (2.18), so that our MLQMC approximation of E[λ] is QML L(∆)λ:=L/summationdisplay ℓ=0Qℓ(∆ℓ)/parenleftbig λℓ−λℓ−1/parenrightbig . (4.1) Here, each ∆ℓ∈[0,1)sℓis an independent random shift, so that the QMC approximatio ns on different levels are independent. To simplify the notation , we also concatenate the L+1 shifts into a single random shift ∆= (∆0;∆1;...;∆L) (where “;” denotes concatenation of column vectors). For a linear functional G ∈V∗, the MLQMC approximation to Ey[G(u)] can be defined analogously. As described in Section 2.5, in practice it is beneficial to us eRindependent random shifts∆(1),∆(2),...,∆(R). Then the shift-averaged MLQMC approximation is /hatwideQML L,Rλ:=L/summationdisplay ℓ=01 RR/summationdisplay r=1Qℓ(∆(r) ℓ)/parenleftbig λℓ−λℓ−1/parenrightbig . (4.2) 12In this case, the variance on each level can be estimated by th e sample variance as given in (2.21) and denoted by Vℓ. Due to the independence of the QMC approximations across the levels, the total variance of the MLQMC estimator is /hatwideV/bracketleftbig/hatwideQML L,Rλ/bracketrightbig =L/summationdisplay ℓ=0Vℓ. (4.3) 4.2 Cost & error analysis The cost of the MLQMC estimator (4.2) for the expected value o fλis given by cost(/hatwideQML L,Rλ) =RL/summationdisplay ℓ=0Nℓcost(λℓ−λℓ−1), where cost( λℓ−λℓ−1) denotes the cost of evaluating the difference at a single para meter value. Since cost( λℓ−λℓ−1)≤2cost(λℓ), the cost of evaluating λℓat a single parameter value, cost(/hatwideQML L,Rλ)/lessorsimilarRL/summationdisplay ℓ=0Nℓcost(λℓ). The cost of evaluating the eigenvalue approximation λℓconsists of two parts: cost(λℓ) =Csetup ℓ+Csolve ℓ, whereCsetup ℓdenotes the setup cost of constructing the stiffness and mass m atrices, and Csolvedenotes the cost of solving the eigenvalue problem. Since th e coefficient cis inde- pendent of yso too is the mass matrix, and as such we only compute it once pe r level. Thus,Csetup ℓis dominated by constructing the stiffness matrix for each qua drature point. Constructing the stiffness matrix at each parameter value inv olves evaluating the co- efficients, which are sℓ-dimensional sums, at the quadrature points for each elemen t in the mesh. Under the assumption (2.10) on the number of FE degrees of freedom, the number of elements in the mesh is also O(h−d), which implies the setup cost is Csetup ℓ/lessorsimilarsℓh−d ℓ. At each each step of an iterative eigensolver a linear system must be solved, and this forms the dominant component of the cost for that step. Essen tially, the cost of each eigenproblem solve is of the order of a source problem solve ( on the mesh Thℓ) multiplied by the number of iterations. As in the case of the source probl em (see e.g., [32, 31]), we assume that the linear systems occurring in each iteratio n of the eigensolver can be solved inO(h−γ) operations, with d < γ < d +1. Assuming that the number of iterations required is independent of y, the cost of each eigensolve is then Csolve/lessorsimilarh−γ ℓ. (4.4) We discuss how to bound the number of iterations of the eigens olver in Section 4.4. It then follows that the cost of evaluating λℓat a single parameter value satisfies cost(λℓ)/lessorsimilarsℓh−d ℓ+h−γ ℓ, and hence the total cost of the MLQMC estimator (4.2) satisfi es cost(/hatwideQML L,Rλ)/lessorsimilarRL/summationdisplay ℓ=0Nℓ(sℓh−d ℓ+h−γ ℓ). (4.5) 13Since the eigenfunction approximation uℓis computed at the same time as λℓ, and we assume that the cost of applying a linear functional Gis constant, the cost of the MLQMC estimator/hatwideQML L,RG(u) is of the same order as the cost of the eigenvalue estimator i n (4.5). The error of the approximation (4.1) is analysed rigourousl y in [19], and so here we only give a brief summary of one of the key results. First, sup pose that Assumption A1 holds with 0 < p < q < 1, and let each Qℓuse a generating vector given by the CBC construction. Next, choose hℓ/equalorsimilar2−ℓandsℓ=sL/equalorsimilarh2p/(2−p) L. Then, it was shown in [19, Corollary 3.1] that for 0 ≤ε <exp(−1), we can choose Land the number of points on each level, Nℓ, such that the mean-square error of the estimator (4.1) is bo unded by E∆/bracketleftbig |Ey[λ]−QML L(∆)λ|2/bracketrightbig ≤ε2, and forδ >0 the cost is bounded by cost/parenleftbig QML L(λ)/parenrightbig /lessorsimilar ε−2/η−p/(2−p)ifη <4/d, ε−2/η−p/(2−p)log2(ε−1)1+1/ηifη= 4/d, ε−d/2−p/(2−p)ifη >4/d,(4.6) whereηis the convergence rate of the variance of Vℓwith respect to Nℓ, which by [19, Theorem 5.3] is given by η= 2−δifq∈(0,2 3] 2 q−1 ifq∈(2 3,1). In practice, we typically set hℓ/equalorsimilar2−ℓ,sℓ/equalorsimilar2ℓand use the adaptive algorithm from [21] to choose{Nℓ}andL. Although this is a greedy algorithm, it was shown in [31, Sec tion 3.3] that the resulting choice of Nℓleads to the same asymptotic order for the overall cost as the choice of Nℓin the theoretical complexity estimate from [19, Corollary 3.1]. 4.3 An efficient MLQMC method with reduced cost per sample To reduce the cost of computing each sample in the MLQMC appro ximation (4.1) in practice, we employ the following two strategies for each ev aluation of the difference λℓ− λℓ−1on a given level: 1) we use the two-grid-truncation method (c f. Algorithm 1) to evaluate the eigenpairs in the difference; and 2) we use the eig envector from a nearby quadrature point as the starting vector for the eigensolve o n the coarse mesh. Two-grid-truncation methods Our strategy for how to use the two-grid-truncation method from Section 3 for a given sample yis as follows. First, we solve the EVP (3.4) corresponding to a coarse discretisation, with meshwidth a nd truncation dimension given by Hℓ= min/parenleftbig h1/4 ℓ,h0/parenrightbig andSℓ= max/parenleftbig/ceilingleftbig s1/2 ℓ/ceilingrightbig ,s0/parenrightbig , to get the coarseeigenpair ( λHℓ,Sℓ(y),uHℓ,Sℓ(y)). Then, we let uℓ(y):=uhℓ,sℓ(y)∈Vℓbe the solution to the following source problem Asℓ(y;uℓ(y),v)−λHℓ,sℓ(y)M(uℓ(y),v) =M(uHℓ,sℓ(y),v) for all v∈Vℓ,(4.7) and define the eigenvalue approximations for ℓ= 1,2,...,Lby the Rayleigh quotient λℓ(y):=λhℓsℓ(y):=Asℓ(y;uℓ(y),uℓ(y)) M(uℓ(y),uℓ(y)). (4.8) 14The eigenpair on level ℓ−1 for the same sample is computed in the same way and we setλ−1(y) = 0 and λ0(y) =λh0,s0(y). In this way, the MLQMC approximation with two-grid update, a ndRrandom shifts, is given by /hatwideQTG L,Rλ:=1 RR/summationdisplay r=1L/summationdisplay ℓ=0Qℓ(∆(r) ℓ)/parenleftbig λℓ−λℓ−1/parenrightbig . (4.9) Note that for a given sampleon level ℓwe use(λHℓ,Sℓ(y),uHℓ,Sℓ(y)) to compute λℓ−1as well. Technically, this violates the telescoping property , sinceλℓ−1from the previous level (ℓ−1) will use ( λHℓ−1,Sℓ−1(y),uHℓ−1,Sℓ−1(y)), but in practice this difference is negligible and does not justify an extra coarse solve. Furthermore, sin ce the two-grid method allows for such a large difference in parameters of the coarse grid and the fine grid ( H/equalorsimilarh1/4and S/equalorsimilars1/2), often we will have the case where Hℓ−1=Hℓ=h0andSℓ−1=Sℓ=s0. So that reusing ( λHℓ,Sℓ(y),uHℓ,Sℓ(y)) to compute λℓ−1in the difference on level ℓdoes not violate the telescoping property. As an example, if we take h0= 1/8, then we can use Hℓ=h0 as the coarse meshwidth for all levels ℓup tohℓ≤2−12= 1/4096. In all of our numerical results, 1 /4096 was below the finest grid size hLrequired. Using the two-grid method still involves solving a source pr oblem on the fine mesh, so that the cost of a two-grid solve is of the same order as Csolvebut with an improved constant. The reduction in cost is proportional to the numbe r of RQ iterations that are required to solve the eigenproblem on the fine mesh without tw o-grid acceleration, so that the highest gains will be achieved for problems where the RQ i teration converges slowly. Reusing samples from nearby QMC points Now, foreach samplewemuststillsolve the EVP (4.7) corresponding to a coarse mesh and a reduced tru ncation dimension, which we do using the RQ algorithm (see, e.g., [37]). To reduce the n umber of RQ iterations to compute this coarse eigenpair at some QMC point tk, we use the eigenvector from a nearby QMC point (say t′) as the starting vector: v0=uHℓ,Sℓ(t′). For the initial shift in the RQ algorithm we use the Rayleigh quotient of this nearby v ector with respect to the bilinear form at the currentQMC point: σ0=ASℓ(tk;v0,v0). In practice, we have found that a good choice of the nearby QMC point is simply the previo us point tk−1. Explicit details on how these two strategies areimplemente d to construct theestimator (4.9) in practice are given in Algorithm 2. First we introduc e some notation to simplify the presentation. Denote the kth randomly shifted rank-1 lattice point on level ℓby tℓ,k:=/braceleftbiggkzℓ Nℓ+∆ℓ/bracerightbigg , (4.10) wherezℓis ansℓ-dimensional generating vector and ∆ℓ∼U[0,1)sℓ. Finally, by relaxing the restriction that approximations o n different levels are inde- pendent from one another we can use the same set of random shif ts for all levels. In this case, the variance decomposition (4.3) becomes an inequali ty with a factor Lin front of the sum. Following the arguments in [7, Section 3.1 and Remar k 2] it can be shown that this does not significantly change the overall complexity, a t worst the cost increases by a factor of|log(ǫ)|. Then, if we also use nested QMC rules we can reuse approximati ons from lower levels on the higher levels. In particular, for ℓ≥1 we will have tℓ,k=t0,kand can set Hℓ=h0so that we can omit the coarse eigenvalue solves (steps 6 and 8) i n Algorithm 2. Furthermore, thereis noneed tocalculate λℓ−1,uℓ−1again either, sosteps 12 and14 can also beskipped. In this case, because the optimal choice for the parameters i n the two-grid-truncation methods are H/equalorsimilarh1/4andS/equalorsimilars1/2, the range of possible meshwidths and truncation 15Algorithm 2 Two-grid MLQMC for eigenvalue problems Givenv0,L,R,{sℓ}L ℓ=0,{hℓ}L ℓ=0and{Nℓ}L ℓ=0: 1:forℓ= 0,1,2,...,Ldo 2:Hℓ←min(h1/4 ℓ,h0) andSℓ←max(s1/2 ℓ,s0) 3:forr= 1,2,...,Rdo 4: generate ∆ℓ∼U[0,1)sℓ 5: fork= 0,1,...,N ℓdo 6: generate tℓ,kusing the shift ∆ℓas in (4.10) ⊲shifted QMC point 7: compute ( λHℓ,Sℓ(tℓ,k),uHℓ,Sℓ(tℓ,k)) usingv0as start value 8: v0←uHℓ,Sℓ(tℓ,k) ⊲update starting value 9: ifℓ >0then 10: solve the source problem (4.7) for uℓ(tℓ,k)∈Vℓ 11: setλHℓ−1,Sℓ−1←λHℓ,SℓanduHℓ−1,Sℓ−1←uHℓ,Sℓ 12: solve the source problem (4.7) for uℓ−1(tℓ,k)∈Vℓ−1 13: λℓ(tℓ,k)←Asℓ(tℓ,k;uℓ(tℓ,k),uℓ(tℓ,k)) M(uℓ(tℓ,k),uℓ(tℓ,k))⊲two-grid updates 14: λℓ−1(tℓ,k)←Asℓ−1(tℓ,k;uℓ−1(tℓ,k),uℓ−1(tℓ,k)) M(uℓ−1(tℓ,k),uℓ−1(tℓ,k)) 15: end if 16: Q(r) ℓλ←Q(r) ℓλ+(λℓ(tℓ,k)−λℓ−1(tℓ−1,k)) ⊲update QMC sum 17: end for 18: Q(r) ℓλ←1 NℓQ(r) ℓλ 19:/hatwideQℓ,Rλ←/hatwideQℓ,R+Q(r) ℓλ 20:end for 21:/hatwideQℓ,Rλ←1 R/hatwideQℓ,Rλ ⊲average over shifts 22:/hatwideQML R,Lλ←/hatwideQML L,Rλ+/hatwideQℓ,Rλ ⊲ update ML estimator 23:end for dimensions are restricted. With this in mind, we let meshwid th of the finest triangulation be denoted by hand let the coarsest possible triangulation have h0≤h1/4, then we define the maximum number of levels Land the meshwidth on each level so that h=hL< hL−1<···< h1< h0≤h1/4. For example, if h= 2−8, thenh0= 2−2and we could take L= 6 with hℓ= 2−ℓ−2. Again, this does not affect the asymptotic complexity bounds p roved in [19]. To overcome this restriction on the coarse and fine meshwidths, one could instead use a full multigrid method as in [39]. Such an extension would be an interesting t opic for future work. Note that for a given problem, it may be possible that a meshwi dth ofh1/4is not sufficiently fine to resolve the coefficients. For this reason, w e only demand that h0≤h1/4 and not equality. Note also that, asymptotically, the coars est meshwidth h0must decrease with the finest meshwidth hL, but only at the rate h0/equalorsimilarh1/4 L. Similarly, defining sLto be the highest truncation dimension, the lowest truncation di mension increases like s0/equalorsimilars1/2 L. 4.4 Analysis of using nearby QMC samples The argument for why starting from the eigenvector of a nearb y QMC point reduces the number of RQ iterations is very intuitive: As the number of po intsNin a QMC rule increases the points necessarily become closer, and since t he eigenvectors are Lipschitz in 16the parameter (see [17, Proposition 2.3]) this implies that the eigenvectors corresponding to nearby QMC samples become closer as Nincreases. Hence the starting guess for the RQ algorithm becomes closer to the eigenvector that is to be f ound, and so for a fixed tolerance the number of RQ iterations also decreases. In this section we provide some basic analysis to justify our intuition above. Through- out it will be convenient to use the more geometric notions fr om the classical discrepancy theory of QMC point sets on the unit cube [0 ,1]s, which were discussed in Section 2.6. From the definition of the star discrepancy (see Definition 2. 1) follows a simple upper bound on how close nearby points are in a low-discrepancy poi nt set. The result is given in terms of dist(·,·), the distance function with respect to the ℓ∞norm. Proposition 4.1. LetPNbe a low-discrepancy point set for N >1, then max t∈PNdist(t,PN\{t})≤3C1/s PNlog(N)1−1/sN−1/s, (4.11) whereCPNis the constant from the discrepancy bound (2.23)onPN. Proof.Lett∈PN. Clearly dist( t,PN\{t})≤1 holds trivially because supx,y∈[0,1]s/ba∇dblx− y/ba∇dblℓ∞≤1. Hence, we can assume, without loss of generality, that the upper bound in (4.11) satisfies 3C1/s PNlog(N)1−1/sN−1/s<1, (4.12) which will be satisfied for Nsufficiently large. For any box [ a,b)⊂[0,1]s, it follows from the definition of the extreme discrepancy /hatwideDNin Definition 2.2 that /vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[a,b)}| N−Ls/parenleftbig [a,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup a≤b∈[0,1]s/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[a,b)}| N−Ls/parenleftbig [a,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle=:/hatwideDN(PN). By the reverse triangle inequality it then follows that |{PN∩[a,b)}|≥N/parenleftbig Ls/parenleftbig [a,b)/parenrightbig −/hatwideDN(PN)/parenrightbig . (4.13) Now, define τ=/parenleftbig 2/N+/hatwideDN(PN)/parenrightbig1/sand consider the box [ a,b) given by [aj,bj) =/braceleftBigg [tj,tj+τ) iftj+τ <1, [1−τ,1) otherwise.(4.14) From [12, Proposition 3.14], the extreme discrepancy can be bounded by the star dis- crepancy:/hatwideDN(PN)≤2sD∗ N(PN), and then due to (2.23) and (4.12) we have the upper bound τ≤/parenleftbigg2 N+2sCPNlog(N)s−1 N/parenrightbigg1/s ≤3C1/s PNlog(N)1−1/sN−1/s<1, (4.15) where we have also used the fact that CPNlog(N)s−1>1 forNsufficiently large and 2+2s≤3s. As such, we have [ a,b)⊂[0,1]s, witht∈[a,b), andLs([a,b)) =τs<1. Applying the lower bound (4.13) to the box [ a,b) defined in (4.14) gives |{PN∩[a,b)}|≥N/parenleftbig τs−/hatwideDN(PN)/parenrightbig =N/parenleftbig 2/N+/hatwideDN(PN)−/hatwideDN(PN)/parenrightbig = 2, which implies that there are at least 2 points in the box [ a,b). By the construction of the box [a,b) it then follows from (4.15) that there exists a t′∈PNsuch that t′/\e}atio\slash=tand /ba∇dblt−t′/ba∇dblℓ∞≤τ≤3C1/s PNlog(N)1−1/sN−1/s. 17Since the eigenvalue and eigenfunction are analytic and thu s Lipschitz in y, we can now bound how close eigenpairs corresponding to nearby QMC p oints are, explicit in N. Proposition 4.2. LetPNbe a low-discrepancy point set, let s∈N, leth >0be sufficiently small and suppose that Assumption A1holds. Then for any t∈PNthere exists t/\e}atio\slash=t′∈PN such that the eigenvalue and eigenfunction satisfy |λh,s(t)−λh,s(t′)|/lessorsimilarlog(N)1−1/sN−1/s,and (4.16) /ba∇dbluh,s(t)−uh,s(t′)/ba∇dblV/lessorsimilarlog(N)1−1/sN−1/s, (4.17) where the constants are independent of t,t′,sandh. Proof.We only prove the result for the eigenfunction. The eigenval ue result follows the same argument. For hsufficiently small, the eigenfunction uhis analytic. In particular, uhadmits a Taylor series that converges in Vfor ally∈Ω. Hence, for any y,y′∈Ω the zeroth order Taylor expansion of uh(y) abouty′(see [24]) gives uh(y) =uh(y′)+∞/summationdisplay j=1(yj−y′ j)/integraldisplay1 0∂j yuh(τy)dτ. Rearranging and taking the V-norm, this can be bounded by /ba∇dbluh(y)−uh(y′)/ba∇dblV≤/ba∇dbly−y′/ba∇dblℓ∞∞/summationdisplay j=1sup τ∈[0,1]/ba∇dbl∂j yuh(τy)/ba∇dblV ≤/ba∇dbly−y′/ba∇dblℓ∞∞/summationdisplay j=1uCβmax/parenleftbig /ba∇dblaj/ba∇dblL∞,/ba∇dblbj/ba∇dblL∞/parenrightbig , where in the last inequality we have used the upper bound [19, eq. (4.4)] on the stochastic derivatives of uh, andCβis independent of handy. From Assumption A1.4 the sum is finite, and hence uhis globally Lipschitz in ywith a constant that is independent of h. Since this bound holds for all y, it also holds for all ywithyj= 0 forj > s, and thus clearlyuh,sis also Lipschitz with a constant that is independent of sandh. The Lipschitz continuity of uh,stogether with Proposition 4.1 then imply (4.17). Since C1/s PN≤max(1,CPN), the result holds with a constant independent of s. Suppose now that for t∈PNwe wish to compute the eigenpair ( λh,s(t),uh,s(t)) using the RQ algorithm with the initial vector v0=uh,s(t′) and initial shift σ0=As(t;v0,v0), wheret′∈PNis the nearby QMC point from Proposition 4.2. Then, there exi sts anN sufficiently large, such that these starting values satisfy /ba∇dbluh,s(t)−v0/ba∇dblV<1,and|λ2,h,s(t)−σ0|>ρ 2, i.e., distance between the initial vector and the eigenvect or to be found is less than one, and the initial shift is closer to λh,s(t) than to λ2,h,s(t). In particular, for any t∈PNwe can choose the starting values such that this holds. Since the RQ algorithm converges cubically (see, e.g., [37] ) for all sufficiently close starting vectors, for a fixed tolerance ε >0 it follows that the number of iterations will be bounded independently of the current QMC point t, if the starting vector is sufficiently close. For Nsufficiently large, Proposition 4.1 implies that for each QMC point there is a starting vector (taken to be the eigenvector correspond ing to a nearby QMC point) that is sufficiently close to the target eigenvector, with a un iform upper bound on the 18distance (4.11). This uniform upper bound implies that for a ll QMC points the target eigenvector and the starting vector will be sufficiently clos e, and hence that the number of RQ iterations is bounded independently of the QMC point. Fur thermore, as Nincreases the starting vector becomes closer to the eigenvector to be f ound due to (4.17), and so the number of iterations decreases with increasing N. 5 Numerical results In this section we present numerical results for two different test problems, which demon- strate the efficiency of MLQMC and also show the computational gains achieved by our efficient MLQMC algorithm using two-grid methods and nearby Q MC points as described in Section 4.3. The superiority of MLQMC for the two test prob lems is also clearly demonstrated by a comparison with single level Monte Carlo ( MC), multilevel Monte Carlo (MLMC) and single level QMC. All tests were performed o n a single node of the computational cluster Katana at UNSW Sydney. Note also that we use “e” notation for powers of 10, e.g., 5e −3 = 5×10−3. Thenumberof quadraturepoints for all methods ( NorNℓ), includingthe MC/MLMC tests, are chosen to be powers of 2, and for the QMC methods we u se a randomly shifted embedded lattice rule [9] in base 2 given by the generating ve ctorlattice-39102-1024- 1048576.3600 from [30] with R= 8 random shifts. For base-2 embedded lattice rules, the points are enumerated in blocks of powers of 2, where each subsequent block fills in the gaps between the previous points and retains a lattice st ructure, see [9] for further details. The FE triangulations are uniform, with geometric ally decreasing meshwidths given by hℓ= 2−(ℓ+3),ℓ≥0. For the two-grid method, we take as the coarse meshwidth Hℓ=h0= 2−3= 1/8, which satisfies Hℓ≤h1/4 ℓfor allℓ≤10. Note that none of our tests required a FE mesh as fine as h=h10= 1/4096. To choose NℓandL, we use the adaptive MLQMC algorithm from [21], with error tolerances ranging fr omε= 0.625,...,6.1e−5. For the eigensolver we use the RQ algorithm with an absolute e rror tolerance of 5e −8, which is below the smallest error tolerance εgiven as input to our MLQMC algorithm. Numerical tests in [17] for almost the same EVPs, show that th e error corresponding to dimension truncation with s= 64 is less than 1e −5. The smallest error tolerance we use below is bigger than 5e −5. Thus, for simplicity we take a single truncation dimensio n sℓ=s= 64 for all ℓbelow. Consequently, the “coarse” truncation dimension fo r the two-grid method is then Sℓ=S=s1/2= 8 for all ℓ. 5.1 Problem 1 First let D= (0,1)2and consider the eigenvalue problem (1.1) with b≡0,c≡1 andaas in (2.1) with a0= 1 orπ/√ 2 and aj(x) =1 j/tildewidepsin(jπx1)sin((j+1)πx2), (5.1) for several different values of the decay parameter /tildewidep >1. Taking the L∞(D) norm of the basis functions we get /ba∇dblaj/ba∇dblL∞=j−/tildewidep, so that for all /tildewidep the bounds on the coefficient are given by amin=a0−ζ(/tildewidep) 2andamax=a0+ζ(/tildewidep) 2, whereζis the Riemann Zeta function and thus, for /tildewidep <2, a choice of a0=π/√ 2 ensures 1910-410-310-210-1 eps10-1100101102103104105time (s)1MC SLQMC MLMC MLQMC TG-MLQMC 10-410-310-210-1 eps10-1100101102103104105time (s) 1MC SLQMC MLMC MLQMC TG-MLQMC Figure 1: Problem 1: Complexity (measured as time in seconds ) of MC, QMC, MLMC, plain-vanilla MLQMC and the enhanced MLQMC method using the two-grid method and nearby QMC points for /tildewidep= 4/3 (left) and/tildewidep= 2 (right). amin>0. For/tildewidep≥2 we choose a0= 1. Furthermore ∇aj(x) = jπ j/tildewidepcos(jπx1)sin((j+1)πx2) (j+1)π j/tildewidepsin(jπx1)cos((j+1)πx2) , (5.2) so that/ba∇dblaj/ba∇dblW1,∞= (j+1)π/j/tildewidep≤2πj−(/tildewidep−1). Thus, Assumption A1 holds for p >1//tildewidepand q >1/(/tildewidep−1). In Figure 1, we plot the cost, measured as computational time in seconds, against the tolerance εfor our two MLQMC algorithms, benchmarked against single le vel MC and QMC, and against MLMC. To ensure an identical bias error for t he single- and multilevel methods, the FE meshwidth for the single-level methods is ta ken to be equal to hL, the meshwidth on the finest level of the multilevel methods. The d ecreasing sequence of tolerances εcorresponds to a reduction in the finest meshwidth hLby a factor 2 at each step. The axes are in log-log scale, and as a guide the black tr iangle in the bottom left corner of each plot indicates a slope of −1. As expected, the MLQMC algorithms are clearly superior in all cases, and for /tildewidep= 2, the MLMC and single level QMC methods seem to converge at the same rate of approximately −2. Also, as we expect the cost of the two MLQMC algorithms grow at the same rate of roughly −1, but the enhancements introduced in Algorithm 2 yields a reduction in cost by a (rou ghly) constant factor of about 2. Note that for this problem the RQ algorithm requires only 3 iterations for almost all cases tested, and so at best we can expect a speedup factor of 3. In almost all of our numerical tests using the eigenvector of a nearby QMC p oint as the starting vector reduced the number of RQ iterations to 2. A similar speedup by a factor of 2 was also observed in [39], which recycled samples from the multigrid hierarchy within a MLQMC algorithm for the elliptic source problem. From [19, Corollary 3.1], for our MLQMC algorithms we expect a rate of−1 (with a log factor) when q≤2/3, or equivalently /tildewidep≥5/2. However, we observe for our MLQMC algorithms are close to −1, regardless of the decay /tildewidep. A possible explanation of this is that we use an off-the-shelf lattice rule that hasn’t been tail ored to this problem, and so we observe nearly the optimal rate but the constant may still depend on the dimension (which is fixed for these experiments). For the other methods we observe the expected rates, with the exception of QMC, which appears to not yet be i n the asymptotic regime. Results for/tildewidep= 3 are very similar to those for /tildewidep= 2, and so have been omitted. 205.2 Problem 2: Domain with interior islands Consider again the domain D= (0,1)2, and the subdomainconsisting of four islandsgiven byDf:= [1 8,3 8]2∪[5 8,7 8]2∪[1 8,3 8]×[5 8,7 8]∪[5 8,7 8]×[1 8,3 8],see Figure 2 for a depiction. Since we use uniform triangular FE meshes with hℓ= 2−ℓ+3the FE triangulation aligns with the boundaries of the components Dfon all levels ℓ= 0,1,2,.... 1 1 01 81 8 3 83 8 7 87 8 5 85 8 Figure 2: Domain Dwith four islands forming Df(in grey). The coefficients are now given by a0(x) =/braceleftBigg σdiff:= 0.01 ifx∈Df, σ′ diff:= 0.011 ifx∈D\Df,aj(x) =/braceleftBigg σdiffw(j+1)/2(/tildewidepa;x) forjodd, σ′ diffw′ j/2(/tildewidep′ a;x) for jeven, b0(x) =/braceleftBigg σabs:= 2 if x∈Df, σ′ abs:= 0.3 ifx∈D\Df,bj(x) =/braceleftBigg σabsw(j+1)/2(/tildewidepb;x) forjodd, σ′ absw′ j/2(/tildewidep′ b;x) for jeven, where wk(q;x) =/braceleftBigg 1 kqsin/parenleftbig 8kπx1/parenrightbig sin/parenleftbig 8(k+1)πx2/parenrightbig forx∈Df, 0 for x∈D\Df,and w′ k(q;x) =/braceleftBigg 0 for x∈Df, 1 kqsin/parenleftbig 8kπx1/parenrightbig sin/parenleftbig 8(k+1)πx2/parenrightbig forx∈D\Df, and where the parameters /tildewidepa,/tildewidep′ a,/tildewidepb,/tildewidep′ b≥4/3 give the different decays of the coefficients on the different areas of the domain. As for Problem 1, if any of /tildewidepa,/tildewidep′ a,/tildewidepb,/tildewidep′ bare less than 2, then we scale the corresponding zeroth term in the coefficie nt byπ/√ 2. The complexity of MLMC, MLQMC and the enhanced MLQMC using tw o-gird meth- ods and nearby QMC points for this problem is given in Figure 3 . As expected for both MLQMC algorithms we observe a convergence rate of −1, and the MLMC results ap- proach the expected convergence rate of −2. Also, since this problem is more difficult for eigensolvers to handle, we now observe that the two-grid MLQ MC gives a speedup by a factor of more than 3. Other tests using different values of /tildewidepa,/tildewidep′ a,/tildewidepb,/tildewidep′ byielded similar results. Note also that numerical results for single level Q MC methods applied to this problem (with slightly different aj,bj) were given previously in [17]. 6 Conclusion We have developed an efficient MLQMC algorithm for random elli ptic EVPs, which uses two-grid methods and nearby QMC points to obtain a speedup co mpared to an ordinary MLQMC implementation. We provided theoretical justificati on for the use of both strate- gies. Finally, we presented numerical results for two test p roblems, which validate the 2110-510-410-3 eps100101102103104time (s) 1MLMC MLQMC TG-MLQMC 10-510-410-3 eps10-1100101102103104time (s) 1MLMC MLQMC TG-MLQMC Figure 3: Problem 2: Complexity (measured as time in seconds ) of MLMC, plain-vanilla MLQMC and the enhanced MLQMC method using the two-grid metho d and nearby QMC points for/tildewidepa=/tildewidepb= 4/3,/tildewidep′ a=/tildewidep′ b= 2 (left), and /tildewidepa=/tildewidep′ a=/tildewidepb=/tildewidep′ b= 2 (right). theoretical results from the accompanying paper [19] and al so demonstrate the speedup obtained by our new MLQMC algorithm. Acknowledgements. This work is supported by the Deutsche Forschungsgemeinsch aft (GermanResearchFoundation)underGermany’sExcellenceS trategyEXC2181/1-390900948 (theHeidelbergSTRUCTURESExcellenceCluster). Also, thi sresearchincludescomputa- tions using the computational cluster Katana supported by R esearch Technology Services at UNSW Sydney. 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A Proof of Theorem 3.1 Proof.First of all, we can use the triangle inequality to split the eigenfunction error into /ba∇dblu(y)−uh,s(y)/ba∇dblV≤/ba∇dblu(y)−us(y)/ba∇dblV+/ba∇dblus(y)−uh,s(y)/ba∇dblV+/ba∇dbluh,s(y)−uh,s(y)/ba∇dblV /lessorsimilarh+s−(1/p+1)+/ba∇dbluh,s(y)−uh,s(y)/ba∇dblV, (A.1) where we have used [17, Theorem 4.1] and (2.12), and the constant is independent of h,sandy. All that remains for the eigenfunction result is to bound the third te rm above. To this end, we can rewrite Step 2 of Algorithm 1 using (3.2) as As(y;uh,s(y)−λH,S(y)Th,suh,s(y),vh) =As(y;Th,suH,S(y),vh) for all vh∈V , which is equivalent to the operator equation: Find uh,s(y)∈Vhsuch that /parenleftbigg1 λH,S(y)−Th,s/parenrightbigg uh,s(y) =1 λH,S(y)Th,suH,S(y). 24This is in turn equivalent (up to a constant scaling factor) to the pro blem: find/tildewideu∈Vhsuch that /parenleftbigg1 λH,S(y)−Th,s/parenrightbigg /tildewideu=λH,S(y)Th,suH,S(y) /ba∇dblλH,S(y)Th,suH,S(y)/ba∇dblV=:u0. (A.2) Explicitly, /tildewideu=λH,S(y)uh,s(y) /ba∇dblTh,suH,S(y)/ba∇dblV, but after normalisation (Step 3) uh,s(y) =/tildewideu//ba∇dbl/tildewideu/ba∇dblM. We now apply Theorem 3.2 from [47] to (A.2), using the space X=Vand with µ0=1 λH,S(y)andu0=λH,S(y)Th,suH,S(y) /ba∇dblλH,S(y)Th,suH,S(y)/ba∇dblV. To do so, we must first verify that µ0andu0satisfy the required assumptions of [47, Theorem 3.2], namely,/ba∇dblu0/ba∇dblV= 1,µ0is not an eigenvalue of Th,s, and for all y∈Ω dist(u0,Eh(λs(y))≤1 2and (A.3) |µ0−µ2,h,s(y)|≥µ1,s(y)−µ2,s(y) 2=:/tildewideρs(y) 2. (A.4) Recall that µk(y) = 1/λk(y) is an eigenvalue of T, and similarly, subscripts handsdenote their FE and dimension-truncated counterparts, respectively. Clearly , the first two assumptions hold, and so it remains to verify (A.3) and (A.4). To show (A.3), since λs(y) is simple we have dist(u0,Eh(λs(y)) = inf α∈R/ba∇dblu0−αuh,s(y)/ba∇dblV =1 λH,S(y)/ba∇dblTh,suH,S(y)/ba∇dblVinf α∈R/ba∇dblλH,S(y)Th,suH,S(y)−αuh,s(y)/ba∇dblV.(A.5) Toshowthatthefirstfactorcanbeboundedbyaconstant,weus ethereversetriangleinequality along with the lower bound (2.15), which since H >0 was assumed to be sufficiently small gives λH,S(y)/ba∇dblTh,suH,S(y)/ba∇dblV≥λ/vextendsingle/vextendsingle/ba∇dblTuH,S(y)/ba∇dblV−/ba∇dbl(T−Th,s)uH,S(y)/ba∇dblV/vextendsingle/vextendsingle. (A.6) Now, by the equivalence of norms (2.5) we have /ba∇dblTuH,S(y)/ba∇dblV≥1 CA/radicalBig A(y;TuH,S(y),TuH,S(y)). Then using the definition of T, along with the facts that uH,S(y) is an eigenfunction and A(y) is symmetric, we can simplify this as A(y;TuH,S(y),TuH,S(y)) =M(uH,S(y),TuH,S(y)) =1 λH,S(y)A(y;uH,S(y),TuH,S(y)) =1 λH,S(y)M(uH,S(y),uH,S(y))≥1 λ, where for the last inequality we have used (2.15) and /ba∇dbluH,S(y)/ba∇dblM= 1. Hence, we have the constant lower bound /ba∇dblTuH,S(y)/ba∇dblV≥C−1 Aλ−1/2, (A.7) which is independent of y,SandH. For the second term in (A.6), by (3.7) we have the upper bound /ba∇dbl(T−Th,s)uH,S(y)/ba∇dbl=/ba∇dblT−Th,s/ba∇dbl/ba∇dbluH,S(y)/ba∇dblV ≤CT(s−1/p+1+h)/radicalBig λH,S(y)/ba∇dbluH,S(y)/ba∇dblM ≤λ1/2CT(s−1/p+1+h), (A.8) 25where we have used that uH,S(y) is an eigenfunction, normalised in M, and also (2.15). Returning to (A.6), since /ba∇dblTuH,S(y)/ba∇dblVis bounded from below by a constant, by (A.8) there existsS0∈Nsufficiently large and H0>0 sufficiently small such that for all s≥S0andh≤H0 we have/ba∇dblTuH,S(y)/ba∇dblV>/ba∇dbl(T−Th,s)uH,S(y)/ba∇dblV. Thus, substituting (A.7) and (A.8) into (A.6), we have the lower bound λH,S(y)/ba∇dblTh,suH,S(y)/ba∇dblV≥C−1 Aλλ−1/2−λλ1/2CT(s−1/p+1+h) ≥C−1 Aλλ−1/2−λλ1/2CT(S−1/p+1 0+H0) =:1 Cu0>0, where 0< Cu0<∞is independent of s,S,h,H andy. It follows that dist/parenleftbig u0,Eh(λs(y))/parenrightbig ≤Cu0/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblV. For the second factor in (A.5), using (3.2), for all vh∈Vhwe have the identity As(y;uh,s(y)−λH,STh,suH,S(y),vh) =/bracketleftbig λh,s(y)−λH,S(y)/bracketrightbig M(uH,S(y),vh) +λh,s(y)M(uh,s(y)−uH,S(y),vh). Lettingvh=uh,s(y)−λH,S(y)Th,suH,S(y), then using that Asis coercive, as well as applying the triangle and Cauchy–Schwarz inequalities, we have /ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dbl2 V /lessorsimilar/vextendsingle/vextendsingleλh,s(y)−λH,S(y)/vextendsingle/vextendsingle/ba∇dbluH,S/ba∇dblM/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblM +λh,s(y)/ba∇dbluh,s(y)−uH,S(y)/ba∇dblM/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblM. Dividing through by /ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblVand applying the Poincar´ e inequality (2.7) gives /ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dblV/lessorsimilar/vextendsingle/vextendsingleλh,s(y)−λH,S(y)/vextendsingle/vextendsingle+λ/ba∇dbluh,s(y)−uH,S(y)/ba∇dblM, where we have also used that /ba∇dbluH,S(y)/ba∇dblM= 1 and (2.15). We can incorporate λinto the constant, and then split the right hand side again using the triangle inequality to g ive /ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dblV/lessorsimilar/vextendsingle/vextendsingleλ(y)−λs(y)|+|λs(y)−λh,s(y)| +|λ(y)−λS(y)|+|λS(y)−λH,S(y)/vextendsingle/vextendsingle+/ba∇dblu(y)−us(y)/ba∇dblV +/ba∇dblus(y)−uh,s(y)/ba∇dblM+/ba∇dblu(y)−uS(y)/ba∇dblV+/ba∇dbluS(y)−uH,S(y)/ba∇dblM, where we have also applied the Poincar´ e inequality (2.7) again to switc h to the V-norms for the eigenfunction truncation errors. Now, each of the terms in (A.5) can be bounded by [17, Theorems 2.6 & 4.1] to give dist(u0,Eh(λs(y))/lessorsimilar/ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dblV /lessorsimilars−1/p+1+S−1/p+1+h2+H2, (A.9) where to bound the FE error in the M-norm we have used [17, eqn. (2.35)] with the functional G=M(·,us(y)−uh,s(y))//ba∇dblus(y)−uh,s(y)/ba∇dblM∈L2(D) (and similarly for uS(y)−uH,S(y)). It followsfrom(A.9)thatthereexists Ssufficientlylargeand Hsufficientlysmall—bothindependent ofy— such that (A.3) holds. Next, to verify (A.4), since µ0= 1/λH,S(y) =:µH,S(y) and since the FE eigenvalues converge from above and thus µ2,h,s(y)≤µ2,s(y), |µ0−µ2,h,s(y)|=µH,S(y)−µ2,h,s(y)≥µH,S(y)−µ2,s(y) =/tildewideρs(y)−/parenleftbig µs(y)−µH,S(y)/parenrightbig . (A.10) Now, suppose that/parenleftbig µs(y)−µH,S(y)/parenrightbig ≤0, then (A.10) simplifies to |µ0−µ2,h,s(y)|≥/tildewideρs(y)≥/tildewideρs(y) 2, 26as required. Alternatively, if/parenleftbig µs(y)−µH,S(y)/parenrightbig >0 then (A.10) becomes |µ0−µ2,h,s(y)|≥/tildewideρs(y)−/vextendsingle/vextendsingleµs(y)−µH,S(y)/vextendsingle/vextendsingle. By the triangle inequality we can bound the second term on the right, again using the bounds from [17, Theorems 2.6 & 4.1], as well as (2.15), to give /vextendsingle/vextendsingleµs(y)−µH,S(y)/vextendsingle/vextendsingle≤|λ(y)−λs(y)| λ(y)λs(y)+|λ(y)−λS(y)| λ(y)λS(y)+|λS(y)−λH,S(y)| λS(y)λH,S(y) ≤C λ2(s−1/p+1+S−1/p+1+H2). The upper bound is independent of y, thus we can take Ssufficiently large and Hsufficiently small, such that, using the bound on the spectral gap in (2.4) toget her with (2.15), /vextendsingle/vextendsingleµs(y)−µH,S(y)/vextendsingle/vextendsingle≤1 2ρ λ1λ2≤1 2λ2,s(y)−λs(y) λs(y)λ2,s(y)=/tildewideρs(y) 2. (A.11) Then, to show (A.4) we can substitute the bound above into (A.10). Hence, we have verified the assumptions for [47, Theorem 3.2] for ally. Sinceλs(y),λh,s(y) are simple, dist( uh,s(y),/hatwideEh(λs(y)) =/ba∇dbluh,s(y)−uh,s(y)/ba∇dblVand hence, it now follows from [47, Theorem 3.2] that /ba∇dbluh,s(y)−uh,s(y)/ba∇dblV≤16 /tildewideρs(y)|λh,s(y)−λH,S(y)| λh,s(y)λH,S(y)/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblV.(A.12) We handle the three factors in turn. For the first factor, by the a rgument used in (A.11) we have 1//tildewideρs(y)≤λ1λ2/ρ, independently of y. For the second factor, we can use the uniform lower bound (2.15), and then the triangle inequality to give the upper boun d |λh,s(y)−λH,S(y)| λh,s(y)λH,S(y)≤1 λ2/parenleftBig |λ(y)−λs(y)|+|λs(y)−λh,s(y)| +|λ(y)−λS(y)|+|λS(y)−λH,S(y)|/parenrightBig . Each term above can be bounded by using one of Theorems 2.6 or 4.1 f rom [17] to give |λh,s(y)−λH,S(y)| λh,s(y)λH,S(y)/lessorsimilars−(1/p−1)+h2+S−(1/p−1)+H2, (A.13) where the constant is again independent of s,S,h,H andy. Finally, the third factor in (A.12) can be bounded using (A.9). Hence, substituting (A.13) and (A.9) into (A.12) we obtain the upper bound /ba∇dbluh,s(y)−uh,s(y)/ba∇dblV/lessorsimilars−2(1/p−1)+h4+S−2(1/p−1)+H4 +2/parenleftbig s−(1/p−1)h2+s−(1/p−1)S−(1/p−1)+s−(1/p−1)H2+h2S−(1/p−1)+h2H2+S−(1/p−1)H2/parenrightbig /lessorsimilarH4+S−2(1/p−1)+H2S−(1/p−1), (A.14) where we have used the fact that s≥Sandh≤Hto obtain the last inequality. Then to give the error bound (3.8), we simply substitute (A.14) into (A.1). The second result (3.9) follows from Lemma 3.1, by choosing B=A(y;·,·),/tildewideB=As(y;·,·) = A(ys;·,·),u=u(y) andw=uh,s(y). Noting that/ba∇dbluh,s/ba∇dblM= 1 and using the definition of λh,s(y) in (3.5), this gives λh,s(y)−λ(y) =/ba∇dblu(y)−uh,s(y)/ba∇dbl2 A(ys)−λ(y)/ba∇dblu(y)−uh,s(y)/ba∇dbl2 A(y) +A/parenleftbig y−ys;u(y),u(y)−2uh,s(y)/parenrightbig /lessorsimilar/ba∇dblu(y)−uh,s(y)/ba∇dbl2 V+A/parenleftbig y−ys;u(y),u(y)−2uh,s(y)/parenrightbig , (A.15) 27where we simplified using the linearity of A(y) inyand used the equivalence of norms in (2.5) and (2.15), which both hold for all y. The last term from (A.15) is bounded as follows A(y−ys;u(y),u(y)−2uh,s(y)) =/integraldisplay D/summationdisplay j>s/parenleftbig yjaj(x)∇u(y)·∇[u(y)−2uh,s(y)] +yjbj(x)u(y)[u(y)−2uh,s(y)]/parenrightbig dx ≤1 2/summationdisplay j>s/bracketleftbig /ba∇dblaj/ba∇dblL∞/ba∇dblu(y)/ba∇dblV(/ba∇dblu(y)/ba∇dblV+2/ba∇dbluh,s(y)/ba∇dblV) +/ba∇dblbj/ba∇dblL∞/ba∇dblu(y)/ba∇dblL2(/ba∇dblu(y)/ba∇dblL2+2/ba∇dbluh,s(y)/ba∇dblL2)/bracketrightbig /lessorsimilar/summationdisplay j>smax/parenleftbig /ba∇dblaj/ba∇dblL∞,/ba∇dblbj/ba∇dblL∞/parenrightbig /lessorsimilars−(1/p−1), (A.16) where in the second last inequality we have bounded the V-norms using (2.16) and the L2-norms using the equivalence to the M-norm (2.6), and then used that u(y) anduh,s(y) are normalised. The tail sum in the last inequality is bounded using [33, Theorem 5.1]. Finally, the result (3.9) is obtained by substituting (3.8) and (A.16) int o (A.15). 28

2021-03-05

Stochastic PDE eigenvalue problems often arise in the field of uncertainty quantification, whereby one seeks to quantify the uncertainty in an eigenvalue, or its eigenfunction. In this paper we present an efficient multilevel quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the smallest eigenvalue of an elliptic eigenvalue problem with stochastic coefficients. Each sample evaluation requires the solution of a PDE eigenvalue problem, and so tackling this problem in practice is notoriously computationally difficult. We speed up the approximation of this expectation in four ways: we use a multilevel variance reduction scheme to spread the work over a hierarchy of FE meshes and truncation dimensions; we use QMC methods to efficiently compute the expectations on each level; we exploit the smoothness in parameter space and reuse the eigenvector from a nearby QMC point to reduce the number of iterations of the eigensolver; and we utilise a two-grid discretisation scheme to obtain the eigenvalue on the fine mesh with a single linear solve. The full error analysis of a basic MLQMC algorithm is given in the companion paper [Gilbert and Scheichl, 2022], and so in this paper we focus on how to further improve the efficiency and provide theoretical justification for using nearby QMC points and two-grid methods. Numerical results are presented that show the efficiency of our algorithm, and also show that the four strategies we employ are complementary.

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

2103.03407v3

arXiv:1803.10064v2 [physics.gen-ph] 19 Oct 2018Dynamics of a Magnetic Needle Magnetometer: Sensitivity to Landau–Lifshitz–Gilbert Damping Y. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6 1Department of Chemistry, Department of Physics, Department of Electro-Optics, and the Ilse Katz Center for N ano-Science, Ben-Gurion University, Beer-Sheva 84105, Israel 2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China 3Department of Physics, and the Ilse Katz Center for Nano-Sci ence, Ben-Gurion University, Beer-Sheva 84105, Israel 4Yukawa Institute for Theoretical Physics, Kyoto, Japan 5Institut f¨ ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many 6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany An analysis of a single-domain magnetic needle (MN) in the pr esence of an external magnetic fieldBis carried out with the aim of achieving a high precision magn etometer. We determine the uncertainty ∆ Bof such a device due to Gilbert dissipation and the associate d internal magnetic field fluctuations that give rise to diffusion of the MN axis dir ectionnand the needle orbital angular momentum. The levitation of the MN in a magnetic trap and its s tability are also analyzed. A rigid single-domain magnet with large total spin, e.g.,S≃1012/planckover2pi1, can be used as a magnetic needle magne- tometer (MNM). Recently Kimball, Sushkov and Budker [1] predicted that the sensitivity of a precessing MNM can surpass that of present state-of-the-art magnetome- ters by orders of magnitude. This prediction motivates our present study of MNM dynamics in the presence of an external magnetic field B. Such analysis requires in- clusion of dissipation of spin components perpendicular to the easy magnetization axis (Gilbert damping). It is due to interactions of the spin with internal degrees of freedom such as lattice vibrations (phonons), spin waves (magnons), thermal electric currents, etc. [2, 3]. Once there is dissipation, fluctuations are also present [6], and result in a source of uncertainty that can affect the ac- curacy of the magnetometer. Here we determine the un- certainty in the measurement of the magnetic field by a MNM. We also analyze a related problem concerning the dynamics of the needle’s levitation in an inhomogeneous magnetic field, e.g., a Ioffe-Pritchard trap [8]. The Hamiltonian for a MN, treated asa symmetric top with body-fixed moments of inertia IX=IY≡ I ∝negationslash=IZ, subject to a uniform magnetic field Bis, H=1 2IˆL2+(1 2IZ−1 2I)ˆL2 Z /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright HR−(ω0//planckover2pi1)(ˆS·ˆn)2 /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright HA−ˆµ·B/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright HB, (1) where a hat denotes quantum operator. In the rotational Hamiltonain HR,ˆLis the orbital angular momentum op- erator and ˆLZ=ˆL·ˆZis its component along the body- fixed symmetry axis. ˆSis the needle spin angular mo- mentum operator, and ˆnis the operator for nthat is the unit vector in the direction of the easy magnetization axis. The frequency appearing in the anisotropy Hamil- tonianHA[4] isω0= 2γ2KS/V, whereKis the strengthofthe anisotropy, Vis the needle volume, and γ=gµB//planckover2pi1 is the gyromagnetic ratio, in which µBis the Bohr mag- netron, and gis theg-factor (taken to be a scalar for simplicity). In the expression for the Zeeman Hamilto- nianHB,ˆµ=gµBˆSis the magnetic moment operator. The Heisenberg equations of motion are ˙ˆS=−gµBB׈S+2ω0 /planckover2pi1(ˆS׈n)(ˆS·ˆn),(2) ˙ˆL=-2ω0 /planckover2pi1(ˆS׈n)(S·ˆn), (3) ˙ˆJ=−gµBB׈S, (4) ˙ˆn=I−1 /planckover2pi1[ˆL׈n+i/planckover2pi1ˆn], (5) whereˆJ=ˆL+ˆSis the total angularmomentum operator andIis the moment of inertia tensor. The dynamics of a MN can be treated semiclassically because Sis very large. A mean–field approximation [9–11] is obtained by taking quantum expectation values of the operator equations and assuming that for a given operator ˆA, the inequality/radicalBig ∝angbracketleftˆA2∝angbracketright−∝angbracketleftˆA∝angbracketright2≪ |∝angbracketleftˆA∝angbracketright|holds, (an assumption warranted for large S). Hence, the ex- pectation values of a product of operators on the RHS of Eqs. (2)-(5) can be replaced by a product of expecta- tion values. The semiclassical equations are equivalent to those obtained in a classical Lagrangian formulation. Dissipation is accounted for by adding the Gilbert term [2, 4]−αS×(˙S//planckover2pi1−Ω×S//planckover2pi1) to the RHS of the expecta- tion value of Eq. (2) and subtracting it from the RHS of Eq. (3). Here αis the dimensionless friction parameter, and the term Ω×Stransforms from body fixed to space fixed frames. Note that Gilbert damping is due to inter- nalforces, hence Jis not affected and Eq. (4) remains intact.2 It is useful to recast the semiclassical dynamical equa- tions of motion in reduced units by defining dimension- less vectors: the unit spin m≡S/S, the orbital angu- lar momentum ℓ≡L/S, the total angular momentum, j=m+ℓand the unit vector in the direction of the magnetic field b=B/B: ˙m=ωBm×b+ω0(m×n)(m·n)−αm×(˙m−Ω×m),(6) ˙ℓ=−ω0(m×n)(m·n)+αm×(˙m−Ω×m),(7) ˙n=Ω×n, (8) ˙j=ωBm×b, (9) where the angular velocity vector Ωis given by Ω= (ω3−ω1)(ℓ·n)n+ω1ℓ = (ω3−ω1)[(j−m)·n]n+ω1(j−m).(10) HereωB=γ|B|is the Larmor frequency, ω1=S/IX, andω3=S/IZ. Similar equations were obtained in Ref. [5], albeit assuming that the deviations of n(t) and m(t) frombare small. We show below that the dynam- ics can be more complicated than simply precession of the needle about the magnetic field, particularly at high magnetic fields where nutation can be significant. For the numerical solutions presented below we are guided by Ref. 1, which uses parameters for bulk cobalt, and take ω1= 100 s−1,ω3= 7000 s−1, anisotropy fre- quencyω0= 108s−1, Gilbert constant α= 0.01, tem- perature T= 300 K, and N=S//planckover2pi1= 1012. First, we elu- cidate the effects of Gilbert dissipation, and consider the shorttimebehaviorin aweakmagneticfield, ωB= 1s−1. The initial spin direction is intentionally chosen notto be along the easy magnetic axis; n(0) = (1 /2,1/√ 2,1/2), m(0) = (1 /√ 2,1/√ 2,0),ℓ(0) = (0 ,0,0). Figure 1(a) shows the fast spin dissipation as it aligns with the easy axis of the needle, i.e., m(t)→n(t) after a short time, and Fig. 1(b) shows relaxation of the oscillations in ℓ(t), whileℓx(t) andℓy(t) approach finite values. Figure 1(c) showsthe innerproduct m·n, which clearlytendstounity onthe timescaleofthe figure. Increasing αleadsto faster dissipation of m(t), but the short-time saturation values of bothm(t) andℓ(t) are almost independent of α. We consider now the long time dynamics (still in the weak field regime) and take the initial value of the spin to coincide with the easy magnetization axis, e.g., m(0) =n(0) = (1 /√ 2,1/√ 2,0), with all other param- eters unchanged. The spin versus time is plotted in Fig. 2(a). The unit vectors m(t) andn(t) are almost identical, andsincetheir z-componentisnearlyzero,they move together in the x-yplane. In this weak field case, the nutation is small, and the fast small-oscillations due to nutation are barely visible. The orbital angular mo- mentum dynamics is plotted in Fig. 2(b) [note the differ- enttimescalein (a)and(b)] andshowsthat ℓ(t) oscillates with a frequency equal to that of the fast tiny-oscillation
















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{} FIG. 1: (color online) (a) The normalized spin vector mver- sus time for the low-field case at short times (5 orders of magnitude shorter than in Fig. 2) when the initial spin is not along the fast axis. (b) The reduced orbital angular mo- mentum vector ℓ(t). (c) The inner product m(t)·n(t) (the projection of the spin on the fast magnetic axis of the needle . ofm(t) [the oscillation amplitude is 0 .02|m(t)|]. Fig- ure 2(c) shows a parametric plot of m(t) versus time. The nutation is clearly very small; the dynamics of m(t) consists almost entirely of precession at frequency ωB. Figure 3 shows the dynamics at high magnetic field (ωB= 105s−1) with all the other parametersunchanged. Figure 3(a) shows mversus time, and now the nutation is clearly significant. For the high magnetic field case, m(t) is also almost numerically equal to n(t).ℓ(t) is plotted in Fig. 3(b). Its amplitude is very large, ℓ(t)≈ 40m(t). However, its oscillation frequency is comparable with that of m(t). In contrast with the results in Fig. 2, here, in addition to precession of the needle, significant nutation is present, as shown clearly in the parametric plot of the needle spin vector m(t) in Fig. 3(c). We now determine the uncertainty of the MNM due to internal magnetic field fluctuations related to the Gilbert damping. A stochastic force ξ(t), whose strength is de- termined by the fluctuation–dissipation theorem [6], is3





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} FIG. 2: (color online) Dynamics for thelow-field case ( ωB= 1 s−1), over relatively long timescales relative to those in Fig. 1. (a)mversus time in units of seconds (note that nis indistin- guishable from mon the scale of the figure). (b) ℓ(t) (note that it stays small compared to S). (c) Parametric plot of the needle spin vector m(t) showing that nutation is almost im- perceptible for small fields [contrast this with the large fie ld result in Fig. 3(c)]; only precession is important. added to Eq. (6), in direct analogy with the treatment of Brownianmotionwherebothdissipationandastochastic force are included [12]: ˙m=m×(ωBb+ξ)+ω0(m×n)(m·n) −αm×(˙m−Ω×m). (11) ξ(t) is internal to the needle and therefore it does not affect the total angular momentum jdirectly, i.e., ξ(t) does not appear in Eq. (9) [since the term −m×ξis also added to the RHS of (7)]. However, as shown below, ξ(t) affectsℓas well as m, causing them to wobble stochas- tically. This, in turn, makes jstochastic as well via the Zeeman torque [see Eq. (9)]. The fluctuation-dissipation theorem [6] implies ∝angbracketleftξαξβ∝angbracketrightω≡/integraldisplay dt∝angbracketleftξα(t)ξβ(0)∝angbracketrighteiωt =δαβαωcoth(/planckover2pi1ω/2kBT) N≈δαβ2αkBT /planckover2pi1N,(12)













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} FIG. 3: (color online) High-field case ( ωB= 105s−1). (a) m(t) [which is almost numerically equal to n(t)]. (b)ℓ(t) (note the ordinate axis scale is [ −40,40]). (c) Parametric plot of the needle spin vector m(t) showing that strong nutation occurs for large fields in addition to precession. whereN=S//planckover2pi1, and the last approximation is ob- tained under the assumption that /planckover2pi1ω≪kBT. Note that Eq. (11) should be solved together with Eqs. (8) and (9). The presence of the anisotropy term in Eq. (11) makes numerical solution difficult for large ω0. Hence, we con- sider a perturbative expansion in powers of λ≡ω1/ω0: m(t) =n0(t)+λδm(t)+...,n(t) =n0(t)+λδn(t)+..., j(t) =j0(t) +λδj(t) +.... Sinceω0is the largest fre- quency in the problem, the inequalities αω0≫ωB,ω1,ω3 hold. Moreover, the Gilbert constant αis large enough to effectively pin m(t) ton(t) [hencej(t) =ℓ(t)+m(t)≈ ℓ(t)+n(t)]. Therefore, anadiabaticapproximationtothe set of dynamical stochastic equations can be obtained. The zero order term in λreads: ˙j0=ωBn0×b,˙n0=ω1j0×n0,(13)4 whereΩwas approximated by Ω0= (ω3−ω1)(j0·n0− 1)n0+ω1(j0−n0) in Eqs. (8) and (10) in obtaining (13) [7]. The solution to Eqs. (13) [for times beyond which Gilbert dissipation is significant so m(t)≈n(t)] is very close to that obtained from Eqs. (6)-(8). Expanding Eq. (11) in powers of λand keeping only the first order terms (the zeroth order term on the LHS vanishes since m0=n0), we get: ω1(δm−δn)×n0= ˙n0−ωBn0×b+αn0×(˙n0−Ω0×n0)−n0×ξ. Taking Eq. (13) into account and introducing the notation δη≡ δm−δn, we obtain δη×n0=j0×n0−(ωB/ω1)n0×b−(1/ω1)n0×ξ,(14) and from Eqs. (8) and (9) we find d dtδj=ωB(δn+δη)×b, (15) d dtδn=ω1(j0−n0)×δn+ω1(δj−δn−δη)×n0 =ω1j0×δn+ω1(δj−δη)×n0. (16) To first order in λ,δn⊥n0(sincenmust be a unit vector), and δm⊥n0, henceδη⊥n0. Therefore, δη× b= [j0−(j0·n0)n0]×b+(ωB/ω1)[b−(b·n0)n0]×b+ ω−1 1[ξ−(ξ·n0)n0]×bon the RHS of Eq. (15) and d dtδj=ωBδn×b+ωB[j0−(j0·n0)n0]×b −ω2 B ω1(b·n0)n0×b+ωB ω1[ξ−(ξ·n0)n0]×b.(17) Equations (13), (16) and (17) form a closed system of stochastic differential equations [upon using Eq. (14) to substitute for δη×n0on the RHS of Eq. (16)]. With the largest frequency ω0eliminated, a stable numerical solution is obtained. Moreover, for small magnetic field (whereωBis the smallest frequency in the system), an analytic solution of these equations is achievable. To ob- tain an analytic solution to Eqs. (13), let us transform to the frame rotating around Bwith frequency ωBto get equations of the formd dτv=d dtv+ωBb×v(which definesτ): d dτn0=−ω1n0×/parenleftbigg n0−j0+ωB ω1b/parenrightbigg ,(18) d dτj0=ωBb×/parenleftbigg n0−j0+ωB ω1b/parenrightbigg .(19) If the initial condition is n0(0)−j0(0)+(ωB/ω1)b= 0, then, in the rotating frame j0(τ) andn0(τ) are constant vectors. Note that this initial condition is only slightly different from the “ordinary” initial condition n0(0) = j0(0)since( ωB/ω1)≪1forsmallmagneticfields. Hence, in the rotating frame, d dτδn=ω1n0×(δn−δj+δη),(20)d dτδj=−ωBb×(δn−δj+δη).(21) With the special initial conditionbeing satisfied, Eq. (14) becomes δη×n0=−(1/ω1)n0×ξ, and Eqs. (20)-(21) become a set of first order differential equations with time-independent coefficients. Their solution for initial conditions, δn(t= 0) = 0, δj(t= 0) = 0 is, /parenleftbiggδn(t) δj(t)/parenrightbigg =t/integraldisplay 0dt1exp[C(t−t1)]C/parenleftbiggδη(t1) 0/parenrightbigg ,(22) where the constant matrix C=/parenleftbiggA−A −B B/parenrightbigg has di- mension 6 ×6 and the 3 ×3 matrices AandBare given by Aij=−ω1ǫijknk 0,Bij=−ωBǫijkbk. Without loss of gen- eralitywecanchoose n0=ˆzandb=ωB(cosθˆz+sinθˆx), whereθis the angle between the easy magnetization axis and the magnetic field. In this basis, ∝angbracketleftδηxδηx∝angbracketrightω= ∝angbracketleftδηyδηy∝angbracketrightω≈ω−2 0∝angbracketleftξxξx∝angbracketrightω=ω−2 0∝angbracketleftξyξy∝angbracketrightω=Sa(ω), and ∝angbracketleftδηzδηz∝angbracketrightω= 0. Here ∝angbracketleftxx∝angbracketrightω≡/integraltext dteiωt∝angbracketleftx(t)x(0)∝angbracketrightand [see Eq. (12)] Sa(ω) =αωcoth(/planckover2pi1ω/2kBT) ω2 0N≈2αkBT N/planckover2pi1ω2 0. We are particularly interested in the quantities ∝angbracketleftδn2 y(t)∝angbracketright ≡ ∝angbracketleftδny(t)δny(t)∝angbracketrightand∝angbracketleftδj2 y(t)∝angbracketright ≡ ∝angbracketleftδjy(t)δjy(t)∝angbracketright because, in the basis chosen above, the y-axis is the di- rection of precession of n0aroundb. Using Eq. (22) we obtain∝angbracketleftδn2 y(t)∝angbracketright ≈tω2 1Sa(ω∼ω1). Assuming the pre- cession of nis measured, [or equivalently, the precession ofm, since they differ only for short timescales of or- der (αω0)−1], the uncertainty in the precession angle is ∝angbracketleft(∆ϕ)2∝angbracketright ≈tω2 1Sa(ω∼ω1). We thus arrive at our central result: the precision with which the precession frequency can be measured is, ∆ ωB=√ /angbracketleft(∆ϕ)2/angbracketright t≈ω1 ω0/radicalBig 2αkBT /planckover2pi1N1√ t. Equivalently, the magnetic field precision is, ∆B=∆ωB γ≈/planckover2pi1 gµBω1 ω0/radicalbigg 2αkBT /planckover2pi1N1√ t.(23) For the parameters used in this paper we find ∆ B≈ 5×10−18√ t[s]Tesla (independent of ωB). This result should be compared with the scaling ∆ B∝t−3/2obtained in Ref. 1. Therein, the initial uncertainty of the spin di- rection relative to the needle axis was estimated from the fluctuation-dissipation relation and the deterministic precession resulted in the t−3/2scaling of the precession angle uncertainty (in addition this angle was assumed to be small). In contrast, we consider the uncertainty ac- quired due to Gilbert dissipation duringthe precession, allowing the precession angle to be large. Thus, the stan- dard1/√ tdiffusion scalingis obtained and dominates for times that are even much longer than those considered in Ref. 1. IntheSupplementalMaterial[13]wediscussthreerele- vant related issues. (a) The time at which diffusion stops because equipartition is reached (we estimate the time5 when the energy stored in stochastic orbital motion be- comes of order kBT). (b) The uncertainty of the mag- netic field for experiments in which the fast precession of naroundjis averaged out in the measurement, and the diffusion of jdetermines ∆ B. (c) We consider the related problem of the dynamics and stability of a rotating MN in an inhomogeneous field (e.g., levitron dynamics in a Ioffe-Pritchard trap [14, 15]). In conclusion, we show that ∆ Bdue to Gilbert damp- ing is very small; external noise sources, as discussed in Ref. [1], will dominate over the Gilbert noise for weak magnetic fields. A closed system of stochastic differen- tial equations, (13), (16) and (17), can be used to model the dynamics and estimate ∆ Bfor large magnetic fields. A rotating MN in a magnetic trap can experience levi- tation, although the motion does not converge to a fixed point or a limit cycle; an adiabatic–invariant stability analysis confirms stability [13]. This work was supported in part by grants from the DFG through the DIP program (FO703/2-1). Useful discussions with Professor Dmitry Budker are gratefully acknowledged. A. S. was supported by DFG Research Grant No. SH 81/3-1. [1] D. F. J. Kimball, A. O. Sushkov, and D. Budker, Phys. Rev. Lett. 116, 190801 (2016). [2] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443 (2004) [3] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8153 (1935). In L. D. Landau, Collected Papers. Ed. by D. ter Haar, (Gordon and Breach, New York, 1967), p. 101. [4] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963). [5] H. Keshtgar, et al., Phys. Rev. B 95, 134447 (2017). [6] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951). [7] We note in passing that Eqs. (13) are equivalent to the equations of motion of a symmetric top in a gravita- tional field when the top is anchored at a point a=an on its axis a distance afrom the center of mass. The equations of motion are: dL/dt=T, where Land T=an×(−mgz) are taken with respect to the fixedpoint, and dn/dt=Ω×n. The angular velocity is given byΩ=I−1 1[L−(L·n)n] +I−1 3(L·n)n, where the mo- ments of inertia ( I1,I1,I3) are calculated relative to the fixed point. Introducing a characteristic scale L0so that L=L0j(jis not a unit vector and its length is not conserved) we obtain Eqs. (13) with ωB=mga/L 0and ω1=L0/I1. Here, the analog of the magnetic field is the gravitational field and the analog of bisz. [8] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein. [9] O. Zobay and B. M. Garraway, Phys. Rev. A 61, 033603 (2000); J. Liu, L. Fu, B.-Y. Ou, S.-G. Chen, D.-I. Choi, B. Wu, and Q. Niu, Phys. Rev. A 66, 023404 (2002). [10] Y. B. Band, I. Tikhonenkov, E. Pazyy, M. Fleischhauer, and A. Vardi, J. of Modern Optics 54, 697-706 (2007). [11] Y. B. Band, Phys. Rev. E 88, 022127 (2013); Y. B. Band and Y. Ben-Shimol, Phys. Rev. E 88, 042149 (2013). [12] H. P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, Cambridge, 2002); M. Schlosshauer, Decoherence and the Quantum- to-Classical Transition (Springer, Berlin, 2007). [13] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLet t.121.160801 which contains a discussion of the three issues enumer- ated in the text, and which includes Refs. 16-20. [14] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996). [15] A movie showing the dynamics of a Levitron can be seen athttps://www.youtube.com/watch?v=wyTAPW_dMfo . [16] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a Magnetic Needle Magnetometer: Sensitivity to Landau– Lifshitz–Gilbert Damping”, Phys. Rev. Lett. (to be pub- lished). [17] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein; D. E. Pritchard, Phys. Rev. Lett. 51, 15 (1983). [18] C. C.Rusconi, V.P¨ ochhacker, K.Kustura, J.I.Ciracan d O. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017); C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romero-Isart, Phys. Rev B 96, 134419 (2017); C. C. Rusconi and O. Romero-Isart, Phys. Rev B 93, 054427 (2016). [19] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842). [20] D. R. Merkin, Introduction to the Theory of Stability , (Springer–Verlag, New York, 1997); F. Verhulst, Non- linear Differential Equations and Dynamical Systems , (Springer–Verlag, Berlin, 1990).arXiv:1803.10064v2 [physics.gen-ph] 19 Oct 2018Supplemental Material for “Dynamics of a Magnetic Needle Ma gnetometer: Sensitivity to Landau–Lifshitz–Gilbert Damping” Y. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6 1Department of Chemistry, Department of Physics, Department of Electro-Optics, and the Ilse Katz Center for N ano-Science, Ben-Gurion University, Beer-Sheva 84105, Israel 2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China 3Department of Physics, and the Ilse Katz Center for Nano-Sci ence, Ben-Gurion University, Beer-Sheva 84105, Israel 4Yukawa Institute for Theoretical Physics, Kyoto, Japan 5Institut f¨ ur Theorie der Kondensierten Materie, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many 6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany In this supplemental material we expand the discussion of the main t ext [1] and address the following three issues. (a) The time τeat which the diffusion of the magnetic needle axis direction nand the magnetic needle orbital angular momentum ℓstops because equipartition is reached, i.e., we estimate the time req uired for the energy stored in stochastic orbital motion to become of order kBT. (b) The uncertainty ∆ Bof the magnetic field for experiments in which the fast precession of naroundjis averaged out in the measurement process and the uncertainty ∆ Bis determined by the diffusion of j. (c) The dynamics of a magnetic needle in an inhomogeneous field, e.g., levitron dynamics of a rotating magnetic needle in a Ioffe-Pritchard trap [2], s ee Refs. [3–5]. (a):τecan be estimated by noting that the diffusion determined in [1] stops once equipartition is reached. The energy ∆ Estored in stochastic orbital motion is given by ∆E∼/planckover2pi1ω1N/angbracketleftδℓ2/angbracketright, (1) where where N=S//planckover2pi1(note that δj−δn=δℓ). By requiring ∆ E∼kBTwe can estimate that the diffusion given by Eqs. (20-21) of [1] stops when τe∼ω2 0/(αω3 1) (this result can also be obtained by expanding Eq. (11) further in powers of λ≡ω1/ω0). For the parameters used in [1] this is an extremely long time ( τe∼1012s∼5 years). Hence, we conclude that the diffusion of Eqs. (20-21) and the error estima tes given for ∆ Bin Ref. [1] are relevant for all reasonable times. (b): In [1] we calculate ∆ Bassuming the experimental measurement follows the temporal dyn amics of nandj. An alternative assumption is that the precession of naroundjis averaged out by the measurement process and one measures the diffusion of j. For the latter we obtain the leading term /angbracketleftδj2 y(t)/angbracketright ≈tω2 Bcos2θSa(ω∼ω1), (2) whereSa(ω) is given in Eq. (23) of [1]. At θ=π/2 the leading contribution obtained in Eq. (2) vanishes and the remaining sub-leading term is /angbracketleftδn2 y(t)/angbracketright ≈t2ω4 B ω2 1Sa(ω∼ω1), (3) hence for θ/negationslash=π/2 we obtain ∆B=∆ωB γ≈/planckover2pi1 gµBωB ω0cosθ/radicalbigg 2αkBT /planckover2pi1N1√ t, (4) whereas at θ=π/2, ∆B=∆ωB γ≈/planckover2pi1 gµBω2 B ω0ω1/radicalbigg 4αkBT /planckover2pi1N1√ t. (5) TakingωB= 1s−1we obtain ∆ B≈cosθ×5×10−23√ t[s]Tesla for θ/negationslash=π/2, and ∆ B≈7×10−25√ t[s]Tesla for θ=π/2.2 (c): A rotating magnet can be levitated in an inhomogeneous magnet ic field [3–5]. This is possible despite Earn- shaw’s theorem [6] from which one can conclude that levitation of a non-rotating ferromagnetin a static magnetic field is not possible. Two important factors regarding magnetic levitation are the forces on the magnet and its stability (ensuring that it does not spontaneously slide or flip into a configura tion without lift). The dynamics of a magnetic needle in an inhomogeneous magnetic field can be modelled using Eqs. (6 ), (7) and (8) of [1] augmented by the equations of motion for the center of mass (CM) degrees of freed om of the needle, ˙p=∇(µ·B(r)), (6) ˙r=p/m , (7) whererandpare the needle CM position and momentum vectors. Our numerical re sults show levitation of the magnetic needle when the initial rotational angular momentum vecto r of the needle is sufficiently large and points in the direction of magnetic field at the center of the trap. We shall s ee that the dynamical variables do not evolve to a fixed point or a simple cyclic orbit. Moreover, a linear stability analy sis yields a 15 ×15 Jacobian matrix with eigenvalues having a positive real part, so the system is unstable. However, a stability analysis of the system using the adiabatic invariant |µ||B|[3] does yield a stable fixed point (contrary to the full numerical re sults which show a more complicated levitation dynamics). Figure 1 shows the dynamics of the system over time in the trap. We u se the same magnetic needle parameters used in Fig. 2 of [1] and a Ioffe-Prichard magnetic field [2] B(r) =ex/parenleftbigg B′x−B′′ 2xz/parenrightbigg +ey/parenleftbigg B′y−B′′ 2zy/parenrightbigg +ez/parenleftbigg B0+B′′ 2(z2−x2+y2 2)/parenrightbigg , (8) with field bias B0, gradient B′, and curvature B′′parameters chosen so that the Zeeman energy and its variation ov er the trajectory of the needle in the trap are substantial (as is clea r from the results shown in the figure). We start the dynamics with initial conditions: r(0) = (0,0,0),p(0) = (0,0,0),m(0) = (0,0.0011/2,−(1−0.001)1/2) (almost along the −zdirection), n(0) =m(0),ℓ(0) = (0 ,0,0.001) [this is large orbital angular momentum since ℓis the orbital angular momentum divided by S]. Figure 1(a) shows the needle CM position r(t) versus time. Fast and slow oscillations are seen in the xandymotion, whereas z(t) remains very close to zero. Figure 1(b) shows oscillations of the CM momentum p(t) with time. px(t) andpy(t) oscillate with time, and pz(t) remains zero. Figure 1(c) plots the spinm(t) versus time. Initially, m(0) points almost in the −zdirection, and the tip of the needle n(t) =m(t) carries out nearly circular motion in the nx-nyplane. Figure 1(d) plots the orbital angular momentum ℓ(t). The components ℓx(t) andℓy(t) undergo a complicated oscillatory motion in the ℓx(t)-ℓy(t) plane but ℓz(t)≈ℓz(0). Figure 1(e) is a parametric plot of m(t); the motion consists of almost concentric rings that are slightly dis placed one from the other. The full dynamics show levitation but they do not converge to a fixed point or a limit cycle. Quite generally, for a system of dynamical equations, ˙ yi(t) =fi(y1,...,y n),i= 1,...n, a linear stability analysis requires calculating the eigenvalues of the Jacobian matrix evaluate d at the equilibrium point y∗wheref(y∗) =0, Jij=/parenleftBig ∂fi ∂yj/parenrightBig y∗[7]. The system is unstable against fluctuations if any of the eigenvalu es ofJijhave a positive real part. Equations (6), (7) and (8) of [1] together with Eqs. (6) and (7) above have a Jacobian matrix with eigenvalues whose real part are positive, so the linear stability test fails. Howev er, if the Zeeman force −∇HZin Eq. (6) is replaced by the gradient of the adiabatic invariant, µ·∇|B(r)|, none of the eigenvalues of the Jacobian matrix have a positive real part and the system is linearly stable, i.e., the stability a nalysis using the adiabatic-invariant predicts stability. Note that substituting the adiabatic invariant for the Zee man energy in the full equations of motion yields r(t) andp(t) vectors that are constant with time and n(t),m(t) andℓ(t) are similar to the results obtained with the full equations of motion (but the parametric plot of m(t) is a perfectly circular orbit). Thus, adiabatic–invariant stability analysis of a rotating magnetic needle in a magnetic trap confi rms stability of its levitation as obtained in the numerical solution of the dynamical equations.3






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 FIG. 1: (color online) Dynamics of a needle in a Ioffe-Pritcha rd magnetic field. (a) rversus time, (b) pversus time, (c) m versus time (note that n(t) is indistinguishable from m(t) on the scale of the figure). (d) ℓversus time (note that |ℓ(t)|is small compared to Sbut rotational angular momentum L(t) =Sℓ(t) is large since S= 1012). (e) Parametric plot of the needle spin vectorm(t) (nutation is very small for this case of small magnetic field ).4 [1] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a Magnet ic Needle Magnetometer: Sensitivity to Landau–Lifshitz– Gilbert Damping”, Phys. Rev. Lett. (to be published). [2] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein; D. E. Pritchard, Phys. Rev. Lett. 51, 15 (1983). [3] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996). [4] A movie a a Levitron can be seen at https://www.youtube.com/watch?v=wyTAPW_dMfo . [5] C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017); C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romer o-Isart, Phys. Rev B 96, 134419 (2017); C. C. Rusconi and O. Romero-Isart, Phys. Rev B 93, 054427 (2016). [6] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842). [7] D. R. Merkin, Introduction to the Theory of Stability , (Springer–Verlag, New York, 1997); F. Verhulst, Nonlinear Differential Equations and Dynamical Systems , (Springer–Verlag, Berlin, 1990).

2018-03-19

An analysis of a single-domain magnetic needle in the presence of an external magnetic field ${\bf B}$ is carried out with the aim of achieving a high precision magnetometer. We determine the uncertainty $\Delta B$ of such a device due to Gilbert dissipation and the associated internal magnetic field fluctuations that gives rise to diffusion of the magnetic needle axis direction ${\bf n}$ and the needle orbital angular momentum. The levitation of the magnetic needle in a magnetic trap and its stability are also analyzed.

Dynamics of a Magnetic Needle Magnetometer: Sensitivity to Landau-Lifshitz-Gilbert Damping

1803.10064v2

arXiv:1303.4922v1 [cond-mat.mes-hall] 20 Mar 2013Spin-pumping and Enhanced Gilbert Damping in Thin Magnetic Insulator Films Andr´ e Kapelrud and Arne Brataas Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Precessing magnetization in a thin film magnetic insulator p umps spins into adjacent metals; however, this phenomenon is not quantitatively understood . We present a theory for the dependence of spin-pumpingon the transverse mode number and in-plane w ave vector. For long-wavelength spin waves, the enhanced Gilbert damping for the transverse mode volume waves is twice that of the macrospin mode, and for surface modes, the enhancement can b e ten or more times stronger. Spin- pumping is negligible for short-wavelength exchange spin w aves. We corroborate our analytical theory with numerical calculations in agreement with recen t experimental results. PACS numbers: 76.50.+g, 75.30.Ds, 75.70.-i, 75.76.+j, 75. 78.-n Metallic spintronics have been tremendously success- ful in creating devices that both fulfill significant market needs and challenge our understanding of spin transport in materials. Topics that are currently of great interest are spin transfer and spin-pumping [1–3], spin Hall ef- fects [4], and combinations thereof for use in non-volatile memory, oscillator circuits, and spin wave logic devices. A recent experimental demonstration that spin transfer and spin-pumping can be as effective in magnetic insula- tors as in metallic ferromagnetic systems was surprising and has initiated a new field of inquiry [5]. In magnetic insulators, no moving charges are present, and in some cases, the dissipative losses associated with the magnetization dynamics are exceptionally low. Nev- ertheless, when a magnetic insulator is placed in con- tact with a normal metal, magnetization dynamics in- ducespin-pumping,whichinturncausesangularmomen- tum to be dumped to the metal’s itinerant electron sys- tem. Duetothisnon-localinteraction, themagnetization losses become enhanced. Careful experimental investiga- tions of spin-pumping and the associated enhanced mag- netization dissipation were recently performed, demon- strating that the dynamic coupling between the magne- tization dynamics in magnetic insulators and spin cur- rentsinadjacentnormalmetalsisstrong. Importantly,in magnetic insulators, an exceptionally low intrinsic damp- ing combined with good material control has enabled the study of spin-pumping for a much larger range of wave vectorsthan has previously been obtained in metallic fer- romagnets [5–14]. In thin film ferromagnets, the magnetization dynamics are strongly affected by the long-range dipolar interac- tion, which has both static and spatiotemporal contribu- tions. This yields different types of spin waves. When the in-plane wavelength is comparable to the film thick- ness or greater, the long-range dipolar interaction causes the separation of the spin-wave modes into three classes depending on the relative orientation of the applied ex- ternal field, in relation to the film normal, and the spin- wave propagation direction [15–20]. These spin waves are classified according to their dispersion and transverse magnetization distribution as forward volume magneto-static spin waves (FVMSWs) when the external field is out-of-plane, backward volume magnetostatic spin waves (BVMSWs) when the external field is in-plane and along the direction of propagation, and magnetostatic surface waves (MSSWs) when the external field is in-plane but perpendicular to the direction of propagation. In volume waves, the magnetic excitation is spatially distributed across the entire film, while surface modes are localized near one of the surfaces. “Backward” waves have a fre- quencydispersionwithanegativegroupvelocityforsome wavelengths. While these spin waves have been studied in great detail overthe last decades, the effect of an adja- cent normal metal on these waves has only recently been investigated. Experimentally, it has been observed that spin- pumping differs for FVMSWs, BVMSWs and MSSWs and that it depends on the spin-wave wavelength[6, 8, 9, 12–14]. Recent experiments [8] have demonstrated that the magnetization dissipation is larger for surface spin waves in which the excitation amplitude is localized at the magnetic insulator-normal metal interface. To uti- lize spin-pumping from thin film magnetic insulatorsinto adjacent normal metals, a coherent theoretical picture of these experimental findings must be developed and un- derstood, which is the aim of our work. In this Letter, we present a theory for energy dissipa- tionfromspin-waveexcitationsinaferromagneticinsula- tor (FI) thin film via spin-pumping when the ferromag- netic insulator layer is in contact with a normal metal (NM). To this end, consider a thin film magnetic insu- lator of thickness Lon an insulating substrate with a normal metal capping (see Figure 1). We consider a nor- mal metal such as Pt at equilibrium, where there is rapid spin relaxation and no back-flow of spin currents to the magnetic insulator. The normal metal is then a perfect spin sink and remains in equilibrium even though spins are pumped into it. The magnetization dynamics are described by the Landau-Lifshitz-Gilbert (LLG) equation [21] with a torque originating from the FI/NM interfacial spin-2 pumping [22] ˙M=−γM×Heff+α MSM×˙M+τsp,(1) whereαis the Gilbert damping coefficient, MSis the saturation magnetization, γis the gyromagnetic ratio, Heffis the effective field including the external field, ex- change energy, surface anisotropy energy, and static and dynamic demagnetization fields. Spin-pumping throughinterfaces between magneticin- sulators and normal metals gives rise to a spin-pumping induced torque that is described as [2] τsp=γ/planckover2pi12 2e2M2 Sg⊥δ/parenleftBig ξ−L 2/parenrightBig M×˙M,(2) whereg⊥is the transverse spin (“mixing”) conductance per unit area at the FI/NM interface. We disregard the imaginary part of the mixing conductance because this part has been found to be small at FI/NM interfaces [12]. In addition, the imaginary part is qualitatively less important and only renormalizes the gyromagnetic ratio. Assuming only uniform magnetic excitations, “macrospin” excitations, the effect of spin-pumping on the magnetization dissipation is well known [2, 3]. Spin-pumping leads to enhanced Gilbert damping, α→α+∆αmacro, which is proportional to the FI/NM cross section because more spin current is then pumpedout, but inversely proportional to the volume of the ferromagnet that controls the total magnetic moment: ∆αmacro=γ/planckover2pi1 4πLMSh e2g⊥. (3) Thus, the enhanced Gilbert damping due to spin- pumping is inversely proportional to the film thickness Land is important for thin film ferromagnets. However, a “macrospin” excitation, or the FMR mode, is only one out of many types of magnetic excitations in thin films. The effect of spin-pumping on the other modes is not known, and we provide the first analytical results for this important question, which is further supported and com- plemented by numerical calculations. We consider weak magnetic excitations around a ho- mogenous magnetic ground state pointing along the di- rection of the internal field Hi=Hiˆz, which is the com- bination of the external applied field and the static de- magnetizing field [19]. We may then expand M=MSˆz+ mQ,xy(ξ)ei(ωt−Qζ), wheremQ,xy·ˆz= 0,|mQ,xy| ≪MS, andQis the in-plane wave number in the ζ-direction. Following the linearization approach of the LLG equa- tion (1) as in Ref. [19], we arrive at a two-dimensional integro-differential equation of the dynamic magnetiza- tion (in the xy-plane) in the film’s transverse coordinate ξ: /bracketleftbigg iω ωM/parenleftbigg α−1 1α/parenrightbigg +11/parenleftbiggωH ωM+8πγ2A ω2 M/bracketleftbigg Q2−d2 dξ2/bracketrightbigg +iαω ωM/parenrightbigg/bracketrightbigg mQ,xy(ξ) =/integraldisplayL 2 −L 2dξ′/hatwideGxy(ξ−ξ′)mQ,xy(ξ′),(4) whereωis the spin-wave eigenfrequency, Ais the ex- change stiffness, ωH=γHi,ωM= 4πγMS, and/hatwideGxyis the dipole-dipole field interaction tensor, which fulfills the boundary conditions resulting from Maxwell’s equa- tions (see [23]). The eigensystem must be supplemented by boundary conditions that account for spin-pumping and surface anisotropy. These boundary conditions are obtained by integrating Eq. (1) over the interface [24] and expanding to the lowest order in the dynamic magnetization. When an out-of-plane easy axis surface anisotropy is present, the boundary conditions are /parenleftbigg L∂ ∂ξ+iωχ+LKs Acos(2θ)/parenrightbigg mQ,x(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ξ=L 2= 0,(5a) /parenleftbigg L∂ ∂ξ+iωχ+LKs Acos2(θ)/parenrightbigg mQ,y(ξ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle ξ=L 2= 0,(5b) whereKsis the surface anisotropy energy with units erg·cm−2andχ=L/planckover2pi12g⊥ 4Ae2is a parameter relating the ex-change stiffness and the spin mixing conductance ([ χ] = s). The boundary condition at the magnetic insulator– substrate interface at ξ=−L/2 is similar to Eq. (5), but simpler because χ→0 andKs→0 at that interface. A mathematical challenge induced by spin-pumping arisesbecausethesecondterminthe linearizedboundary condition(5)isproportionaltotheeigenvalue ωsuchthat the eigenfunctions cannot simply be expanded in the set ofeigenfunctionsobtainedwhenthereisnospin-pumping or dipolar interaction. Instead, we follow an alternative analytical route for small and large wave vectors. Fur- thermore, we numericallydetermine the eigenmodeswith a custom-tailored technique, where we discretize the dif- ferential equation (4), include the spin-pumping bound- ary conditions (5), and transform the resulting equations into an eigenvalue problem in ω[25]. Let us now outline how we obtain analytical results for smallQL≪1 and large QL≫1 wave vectors. First, we consider the case of vanishing surface anisotropy and compute the renormalization of the Gilbert damping for3 (a)L/Slash12 /MinusL/Slash12NM FI SUBΞ (b) FIG. 1. a) A thin film magnetic insulator of thickness L in its coordinate system; ξis the normal axis, the infinite ηζ-plane is coplanar with the interfaces, and the spin waves propagate along the ζ-axis. The internal field and saturation magnetization are along the z-axis. The y-axis is always kept in-plane, and the x-axis is selected such that the x-,y- and z-axes form a right-handed coordinate system. b) A cross- section showing the material stack. the resulting modes. Next, we demonstrate that the sur- face anisotropycreates a surface wavewith a comparably large enhancement of the Gilbert component. WhenQL≪1, the convolution integral on the right- hand side of Eq. (4) only contains the homogeneous de- magnetization field. The magnetization is then a trans- verse standing wave mQ,xy/parenleftbig eikξ+e−ikξ+φ/parenrightbig ,wherekis a transverse wave number, φis a phase determined by the BC at the lower interface, and the two-dimensional coefficient vector mQ,xyallows for elliptical polarization in thexy-plane. By employing exchange-only boundary conditions [24] at the lower interface and using Eq. (5) with Ks= 0 on the upper interface, the transverse wave number kis de- termined by kLtankL=iωχ. Together with the bulk dispersion relation ω=ω(k), calculated from Eq. (4), thisexpressionallowsustocalculatethe magneticexcita- tion dispersion relation parameterized by the film thick- ness, the Gilbert damping α, and the transverse conduc- tanceg⊥. When spin-pumping is weak, ωχis small, and the so- lutions of the transcendental equation can be expanded around the solutions obtained when there is no spin- pumping, kL=nπ, where nis an integer. When n∝negationslash= 0, we expand to first order in kLand obtain kL≈nπ+iωχ/(nπ). When n= 0, we must perform a second-order expansion in terms of kLaround 0, which results in ( kL)2≈iωχ. Using these relations in turn to eliminate kfrom the bulk dispersion relation while maintaining our linear approximation in small terms and solving for ω, we obtain complex eigenvalues, where the imaginary part is proportional to a renormalized Gilbert damping parameter, α∗=α+ ∆α. When n= 0, our results agree with the spin-pumping-enhanced Gilbert damping of the macrospin (FMR) mode derived in [2](see Eq. (3)), ∆ α0= ∆αmacro. Whenn∝negationslash= 0, we compute ∆αn= 2∆αmacro. (6) These new results indicate that allhigher transverse vol- ume modes have an enhanced magnetization dissipation that is twice that of the macrospin mode. Thus, coun- terintuitively, with the exception of the macrospin mode, increasingly higher-order standing-wave transverse spin- wave modes have precisely the same enhanced Gilbert damping. Next, let us discuss spin-pumping for surface waves induced by the presence of surface anisotropy. When Ks∝negationslash= 0, the lowest volume excitation mode develops into a spatially localized surface wave. Expanding the ex- pression for the localized wave to the highest order in LKs/A, we determine after some algebra that the result- ing enhancement of the Gilbert damping is ∆αn=0=γ/planckover2pi1Ks 4πMsAh e2g⊥ωH ωM/bracketleftbiggωH ωM+1 2−K2 s 4πM2sA/bracketrightbigg−1 . (7) Comparing Eqs. (7) and (6), we see that for large sur- face anisotropy LKs/A≫1, the spin-pumping-induced enhanced Gilbert damping is independent of L. This re- sult occurs because a large surface anisotropy induces a surface wave with a decay length A/Ks, which re- places the actual physical thickness Las the effective thickness of the magnetic excitations, i.e., for surface wavesL→A/Ksin the expression for the enhanced Gilbert damping of Eq. (3). This replacement implies that the enhanced Gilbert damping is much larger for surface waves because the effective magnetic volume de- creases. For typical values of AandKs, we obtain an effective length A/Ks∼10nm. Compared with the film thicknesses used in recent experiments, this value corre- sponds to a tenfold or greater increase in the enhance- ment of the Gilbert damping. In contrast, for the volume modes (n∝negationslash= 0), we note from Eq. (5) that the dynamic magnetization will decrease at the FI/NM interface due to the surface anisotropy; hence, ∆ αdecreases compared with the results of Eq. (6). Finally, we can also demonstrate that for large wave vectorsQL≫1, the excitation energymostly arisesfrom the in-plane (longitudinal) magnetization texture gradi- ent. Consequently, spin-pumping, which pumps energy out of the magnetic system due to the transverse gradi- ent of the magnetization texture, is much less effective and decays as 1 /(QL)2with respect to Eq. (3). To complement our analytical study, we numerically computed the eigenfrequencies ωn(Q). The energy is de- termined by the real part of ωn(Q), whileImωn(Q) de- terminesthe dissipationrateandhencethespin-pumping contribution. Recent experiments [6, 11, 13, 14] on controlling and optimizing the ferrimagnetic insulator yttrium-iron-garnet (YIG) have estimated that the mix- ingconductancesofbothYIG—AuandYIG—Ptbilayers4 10/Minus610/Minus40.01 1 100QL12345/CΑpDeltΑΑ/LParen110/Minus4/RParen1 /MinusL/Slash120 L/Slash12 Ξ/LBracketBar1m/RArroΩ /LParen1Ξ/RParen1/RBracketBar1 FIG. 2. ∆ αversus wave vector for the MSSW geometry ( θ= φ=π/2) for thefour lowest eigenvalues. Inset: Magnitudesof eigenvectors (in arbitrary units) across the film at QL= 1.5. are in the range of g⊥h/e2∼0.02–3.43·1015cm−2. We useg⊥h/e2= 1.2·1014cm−2from Ref. [6] in this work. All of our results can be linearly re-scaled with other val- ues of the mixing conductance. In the following section, we also use A= 2.9·10−8erg/cm,Ks= 0.05erg/cm2, L= 100nm, 4 πMS= 1750G, and α= 3·10−4. To distinguish the spin-pumping contribution∆ αfrom the magnetization dissipation due to intrinsic Gilbert damping α, we first compute the eigenvalues, ωd, with intrinsic Gilbert damping, α∝negationslash= 0, and no spin-pumping, g⊥= 0. Second, we compute the eigenvalues ωspwith dissipation arising from spin-pumping only, α= 0 and g⊥∝negationslash= 0. Because Imωd∝α, we define a measure of the spin-pumping-induced effective Gilbert damping as ∆α=αImωsp/Imωd. Wefirstconsiderthe caseofnosurfaceanisotropy. Fig- ure 2showsthe spin-pumping-enhancedGilbert damping ∆αas a function of the product of the in-plane wave vec- tor and the film thickness QLin the MSSW geometry. In the long-wavelength limit, QL≪1, the numerical re- sult agreespreciselywithouranalyticalresultsofEq.(6). The enhanced Gilbert damping of all higher transverse modes is exactly twice that of the macrospin mode. In the dipole-exchange regime, for intermediate values of QL, the dipolar interaction causes a small asymmetry in the eigenvectors for positive and negative eigenfrequen- cies because modes traveling in opposite directions have different magnitudes of precession near the FI/NM in- terface [26], and spin-pumping from these modes there- fore differ. This phenomenon also explains why the en- hanced damping, ∆ α, splits into different branches in this regime, as shown in Fig. 2. For exchange spin waves, QL≫1, the exchange interaction dominates the dipo- lar interaction and removes mode asymmetries. We also see that ∆ α→0 for large QL, in accordance with our analytical theory. Figure 3 shows ∆ αfor the BVMSW geometry. The eight first modes are presented; however, as no substan- tial asymmetry exists between eigenmodes traveling in10/Minus610/Minus40.01 1 100QL123456/CΑpDeltΑΑ/LParen110/Minus4/RParen1 0.1 1 100.00.51.01.52.02.5Re/LBrace1Ω/Slash1ΩM/RBrace1 /MinusL/Slash12 0 L/Slash12 Ξ/LBracketBar1m/RArroΩ /LParen1Ξ/RParen1/RBracketBar1 FIG. 3. ∆ αversus wave vector for the BVMSW geometry (θ=π/2 andφ= 0). Left inset: Magnitude of eigenvectors (in arbitrary units) across the film when QL= 1.5. Right inset: The real part of the dispersion relation for the same modes. 10/Minus40.001 0.01 0.1 1 10 100QL510152025/CΑpDeltΑΑ/LParen110/Minus4/RParen1 /MinusL/Slash12 0 L/Slash12 Ξ/LBracketBar1m/RArroΩ /LParen1Ξ/RParen1/RBracketBar1 FIG. 4. ∆ αversus wave vector for the MSSW geometry (θ=φ=π/2) with surface anisotropy added at the inter- face. Inset: Magnitudes of eigenvectors (in arbitrary unit s) across the film. different directions, the modes have the same pairwise renormalization of α. This symmetry occurs because the direction of the internal field coincides with the direction of propagation. As in the previous case, the dipolar in- teraction causes a slight shift in the eigenvectors in the intermediate QLregime, thereby altering ∆ αfrom that of Eq. (6). Figure 4 shows ∆ αfor the MSSW geometry but with surface anisotropy at the FI/NM interface. As expected from our analytical results, surface anisotropy induces two localized surface modes with a ten-fold larger en- hancement of∆ αcomparedwith the volume modes. The horizontal dashed line in Figure 4 indicates the analyti- cal result for the enhanced Gilbert damping of the n∝negationslash= 0 modes when Ks= 0. For the volume modes, it is clear thattheeigenvectorshavealowermagnitudeclosertothe FI/NM interfaceandthat ∆ αis lowercomparedwith the case ofKs= 0, which is consistent with our analytical analysis. Our results also agree with recent experiments. Sandweg et al.[8] found that spin-pumping is signifi-5 cantly higher for surface spin waves compared with vol- ume spin-wave modes. In addition, in Ref. [9], exchange waves were observed to be less efficient at pumping spins than dipolar spin waves, which is consistent with our re- sults. Furthermore, our results are consistent with the theoretical finding that spin-transfer torques preferen- tially excite surface spin waves with a critical current inversely proportional to the penetration depth [27]. In conclusion, we have analyzed how spin-pumping causes a wave-vector-dependent enhancement of the Gilbert damping in thin magnetic insulators in con- tact with normal metals. In the long-wavelength limit, our analytical results demonstrate that the enhancement of the Gilbert damping for all higher-order volumetric modes is twice as large as that of a macrospin excita- tion. Importantly, surface anisotropy-pinnedmodes have a Gilbert renormalization that is significantly and lin- early enhanced by the ratio LKs/A. A. Kapelrud would like to thank G. E. W. Bauer for his hospitality at TU Delft. This work was supported by EU-ICT-7 contract No. 257159 “MACALO”. [1] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012). [2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [3] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005). [4] T. Jungwirth, J. Wunderlich, and K. Olejnik, Nat. Mater.11, 382 (2012). [5] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanasahi, S. Maekawa, and E. Saitoh, Nature 464, 262 (2010). [6] B. Heinrich, C. Burrowes, E. Montoya, B. Kar- dasz, E. Girt, Y.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107, 066604 (2011). [7] C. Burrowes, B. Heinrich, B. Kardasz, E. A. Mon- toya, E. Girt, Y. Sun, Y.-Y. Song, and M. Wu, Appl. Phys. Lett. 100, 092403 (2012). [8] C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and B. Hillebrands, Appl. Phys. Lett. 97(2010). [9] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jungfleisch, E. Saitoh, and B. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011). [10] L. H. Vilela-Leao, A. A. C. Salvador, and S. M. Rezende, Appl. Phys. Lett. 99, 102505 (2011). [11] S. M. Rezende, R. L. Rodriguez-Suarez, M. M. Soares, L. H. Vilela-Leao, D. L. Dominguez, and A. Azevedo, Appl. Phys. Lett. 102, 012402 (2013). [12] X. Jia, K. Liu, X. K, and G. E. W. B. Bauer, EPL 96, 17005 (2011). [13] M. B. Jungfleisch, V. Lauer, R. Neb, A. V. Chu- mak, and B. Hillebrands, ArXiv e-prints (2013), arXiv:1302.6697 [cond-mat.mes-hall]. [14] Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Taka- hashi, T. An, Y. Fujikawa, and E. Saitoh, ArXiv e-prints(2013), arXiv:1302.7091 [cond-mat.mes-hall]. [15] J. R. Eshbach and R. W. Damon, Phys. 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2013-03-20

Precessing magnetization in a thin film magnetic insulator pumps spins into adjacent metals; however, this phenomenon is not quantitatively understood. We present a theory for the dependence of spin-pumping on the transverse mode number and in-plane wave vector. For long-wavelength spin waves, the enhanced Gilbert damping for the transverse mode volume waves is twice that of the macrospin mode, and for surface modes, the enhancement can be ten or more times stronger. Spin-pumping is negligible for short-wavelength exchange spin waves. We corroborate our analytical theory with numerical calculations in agreement with recent experimental results.

Spin-pumping and Enhanced Gilbert Damping in Thin Magnetic Insulator Films

1303.4922v1

Experimental investigation of the effect of topological insulator on the magnetization dynamics of ferromagnetic metal: BiSbTe 1.5Se1.5andNi80Fe20heterostructure Sayani Pal, Soumik Aon, Subhadip Manna, Sambhu G Nath, Kanav Sharma & Chiranjib Mitra∗ Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India (Dated: November 27, 2023) We have studied the spin-pumping phenomenon in ferromagnetic metal( Ni80Fe20)/topological insulator( BiSbTe 1.5Se1.5) bilayer system to understand magnetization dynamics of ferromagnetic metal (FM) in contact with a topological insulator (TI). TIs embody a spin-momentum-locked surface state that spans the bulk band gap. Due to this special spin texture of the topological surface state, the spin-charge interconversion efficiency of TI is even higher than that of heavy metals. We evaluated the parameters like effective damping coefficient ( αeff), spin-mixing conductance ( g↑↓ eff) and spin current density ( j0 S) to demonstrate an efficient spin transfer in Ni80Fe20/BiSbTe 1.5Se1.5 heterostructure. To probe the effect of the topological surface state, a systematic low-temperature study is crucial as the surface state of TI dominates at lower temperatures. The exponential increase of ∆Hfor all different thickness combinations of FM/TI bilayers and the enhancement of effective damping coefficient ( αeff) with lowering temperature confirms that the spin chemical potential bias generated from spin-pumping induces spin current into the TI surface state. Furthermore, low- temperature measurements of effective magnetization (4 πMeff) and magnetic anisotropy field ( Hk) showed anomaly around the same temperature region where the resistivity of TI starts showing metallic behavior due to the dominance of conducting TI surface state. The anomaly in Hkcan result from the emerging exchange coupling between the TI surface state and the local moments of the FM layer at the interface without any long-range ferromagnetic order in TI at the interface. INTRODUCTION Spintronics is one of the emerging fields that has witnessed remarkable progress on both fundamen- tal and technological fronts over the past couple of decades. Phenomena like spin-orbit torque [1], spin Hall effect[2], giant magnetoresistance [3], tunnelling magnetoresistance [4], domain wall motion [5] provide basics for applications in memory devices[6], storage technology[7], logic gates [8] and magnetic sensors [9]. These devices utilize the spin degrees of freedom of electrons and their interaction with orbital moments through spin-orbit coupling. Complete knowledge of the process of generation, manipulation, and detection of spin degrees of freedom or the spin current is essential for widespread applications in this field. If one focuses on the currently available spin current generation processes, spin-pumping [10, 11] is one of the most efficient methods where the precessing magnetization in the ferromagnet (FM) injects spin current into the adjacent layer by transferring spin angular momentum. This raises a need to study the effect of spin pumping with special emphasis on exploring new materials which can give rise to significant spin-charge interconversion efficiency. Topological insulators (TI) are a new class of materials that have an interesting spin texture of the surface state, owing to spin-momentum locking[14–17]. The momentum direction of the electron in the surface state of TI is perpendicularly locked to its spin polarization ∗Corresponding author:chiranjib@iiserkol.ac.indirection. Thus the spin-charge interconversion for TI is even higher than the heavy metals which makes TIs suitable for spintronics application[12]. As surface states are robust against deposition of FM layers on top of TI [18], the topological surface states remain intact and gapless [19]. TI/FM bilayers have been successfully used for the spin current generation in spin-pumping experiments[20, 34–37]. The effect of spin pumping can be witnessed in the enhanced damping coefficient ( αeff) value of the ferromagnet upon excitations of ferromag- netic resonance (FMR) because, in the spin pumping process, the net transfer of spin angular momentum into TI layer brings about an additional damping torque on the precessing magnetization in the FM. It is difficult to fabricate a perfect TI thin film where the bulk state of TI is completely insulating. Thus for a complete understanding of the effect of TI surface state on FM magnetization dynamics, the low-temperature study is necessary where the surface states of TI dominate. In this paper, we present the study of the spin-pumping phenomenon in ferromagnetic metal (FM)/ topological insulator (TI) bilayer system. We chose Ni80Fe20 as the FM layer and BiSbTe 1.5Se1.5as the TI layer. Currently, BiSbTe 1.5Se1.5is one of the best 3D TI materials in which bulk conduction in thin films is negligible even at room temperature and the dominance of surface state is very prominent at lower temperatures [21–23]. In our low-temperature measurements, we have witnessed exponential enhancement of FMR linewidth (∆H) and effective damping coefficient ( αeff) at lower temperatures. It supports the proposal of the spin chemical potential bias induced spin current injectionarXiv:2303.07025v2 [cond-mat.mes-hall] 24 Nov 20232 into the surface state of TI given by Abdulahad et al. [50]. For further investigation of the effect of the TI surface state on the FM magnetization, we have also studied low-temperature variations of effective magnetization and anisotropy field. We calculated the interfacial magnetic anisotropy of the bilayer to be in-plane of the interface. At low temperatures, this magnetic anisotropy field shows a hump-like feature concomitant with the resistivity behavior of BiSbTe 1.5Se1.5with temperature. It predicts the existence of exchange coupling between the surface states of TI and the local moments of the FM layer which acts perpendicular to the TI/FM interface. We have also evaluated the values of spin-transport parameters like spin-mixing conductance, g↑↓ effand spin current density, j0 sat room temperature to ensure a suc- cessful spin injection into the TI layer from the FM layer. SAMPLE PREPARATION AND CHARACTERIZATION For this particular work, we have prepared dif- ferent thickness combinations of topological insu- lator(TI)/ferromagnet(FM) bilayer heterostructure. BiSbTe 1.5Se1.5(BSTS ) has been taken as the TI material and Permalloy( Ni80Fe20) has been used as the ferromagnetic material. BSTS thin films were grown on silicon (Si 111) substrate using pulsed laser deposition(PLD) technique [24, 25]. The target material was prepared using 99 .999% pure Bi, Sb, Te, and Se in a 1:1:1.5:1.5 stoichiometric ratio. The films were deposited through ablation of the target by a KrF excimer laser (248 nm, 25 ns pulse width) at a low repetition rate of 1Hz and 1 .2Jcm−2laser fluence keeping the substrate temperature fixed at 2500Cand the chamber partial pressure at 0.5 mbar (base pressure 2 ×10−5mbar) with a continuous flow of Ar gas. After deposition, TI films were immediately transferred into the thermal evaporation chamber for the deposition of the FM layer. Commercially available 99 .995% pure permalloy (Ni80Fe20) pallets were used for deposition. The Py film was deposited [26] on top of TI film at a rate of 1.2˚A(crystal monitor: Inficon SQM 160) keeping the chamber pressure fixed at 1 ×10−6torr (base pressure 1×10−7torr). For the characterization of the films X-ray diffraction analysis (XRD), field emission scanning electron microscope (FE-SEM) imaging, and atomic force microscopy (AFM) facilities have been used. X-ray reflectometry technique has been used for thickness measurements here. For convenience we are defining the BSTS of different thicknesses as follows: 10nm BSTS as BSTS1, 21nm BSTS as BSTS2, 28nm BSTS as BSTS3, and 37nm BSTS as BSTS4.RESULTS AND DISCUSSION For a systematic study of the FM/TI bilayer system, we have done in-plane FMR measurements in reflection mode geometry using a short-circuited CPW as shown in fig.1a. We obtained typical FMR signal at different microwave frequencies for Py(15nm)/BSTS2 sample in fig.1b. From the Lorentz formula fitting [53] of the FMR signal we extracted the frequency dependence of the field linewidth (∆ Hvs.f) and the resonance frequency vs. resonance field ( fvsH) data as shown in fig.2a and fig.2b respectively. These give us valuable information about the magnetization dynamics in ferromagnet which can be described within the framework proposed by Landau, Lifshitz, and Gilbert [30], d⃗M dt=−γ⃗M×⃗Heff+αeff MS⃗M×d⃗M dt(1) where, γis the gyromagnetic ratio, ⃗Mis the magneti- zation vector, MSis the saturation magnetization, Heff is the effective magnetic field which includes the exter- nal field, demagnetization and crystalline anisotropy field andαeffis the effective damping coefficient of the sys- tem. For a given magnetic material at ferromagnetic res- onance, the resonance field and frequency follow Kittel equation[27] given by, f=γ 2πq (H+Hk)(H+Hk+ 4πMeff) (2) where H,Hk, and 4 πMeffare the externally ap- plied field, magnetic anisotropy field, and effective mag- netization respectively. We have obtained Hkand 4πMefffor different FM/TI bilayer systems by fitting the Kittel equation to the fvs. Hcurve as shown in fig.2b. The obtained 4 πMeffvalue contains satura- tion magnetization(4 πMs) and other anisotropic contri- butions. We can evaluate 4 πMsvalue by analyzing the thickness dependent measurement of 4 πMeffof the FM layer. In the lower thickness region of the ferromagnetic thin films, 4 πMeffis inversely proportional to the film thickness and follows the equation[28], 4πMeff= 4πMs−2Ks Msd(3) where Ksis the surface anisotropy constant and dis the thickness of the FM film. The slope of the linear fit gives the anisotropy field contribution to 4 πMeffand the intercept gives the 4 πMsvalue as shown in fig.2c. The 4 πMeffdoes not depend on the thickness varia- tion of BSTS at room temperature but 4 πMefffor Py(t) monolayer samples and for Py(t)/BSTS2 bilayer sam- ples vary linearly with the inverse Py thickness as shown in Fig.2c. From the linear fitting (Eq.3) of 4 πMeff3 (a) (b) FIG. 1. (a) In the left diagram, a schematic illustration of the experimental set-up has shown where the FM/TI bilayer is placed upside down on top of a CPW, and in the right diagram, net injected spin current ( Ipump S ) due to spin-pumping into the TI layer (BSTS) from the FM layer (Py) has shown, it results faster magnetization relaxation in FM; (b) Ferromagnetic Resonance spectra of absorption at different frequencies for Py/BSTS bilayer system at room temperature after background subtraction. (a) (b) (c) FIG. 2. (a) Field linewidth (∆ H) variation with resonance frequencies ( f) at 300K for Py/BSTS bilayer samples with different Py thicknesses. Eq.4 has been used for fitting the curve and to determine the damping coefficient ;(b) Resonance field ( H) vs. resonance frequency ( f) for Py(20nm)/BSTS2 system at different temperatures . Eq.2 has been used for fitting the curve and to determine the effective magnetization; (c) Effective magnetization (4 πMeff) variation with thickness of Py(t), Py(t)/BSTS2 and Py(15nm/BSTS(t) at room temperature. Eq3 has been used for fitting the curve and to evaluate saturation magnetization (4πMS) and magnetic anisotropy field( Hk). (a) (b) FIG. 3. (a) αeffvariation with Py thickness for Py(t)/BSTS2 heterostructure at room temperature which fits in Eq.5; (b) αeffas a function of BSTS thickness for Py(15nm)/BSTS(t) heterostructure at room temperature.4 (a) FIG. 4. Temperature dependence of resistivity of the BSTS sample of thickness 21nm deposited on Si(111) substrate. vs. 1 /tPydata for the Py(t) and Py(t)/BSTS2 sam- ples, we evaluated the saturation magnetization, Msof the Py/BSTS bilayer that has been decreased from that of the bare Py sample by an amount of 183 emu/cc3. It is a result of the loss of ferromagnetic order in the Permalloy layer. Due to the intermixing of the Py and BSTS at the interface, a magnetic dead layer could have formed at the interface which resulted in the reduction of saturation magnetization value as suggested by some previous studies [42–44] also. The Ksvalue has de- creased from 0 .092±0.008erg/cm2in bare Py film to 0.091±0.015erg/cm2in Py/BSTS2 bilayer. So interfa- cial anisotropy constant, Ki(=KPy/TI s −KPy s) for the Py/BSTS2 sample is −0.001erg/cm2. From the nega- tive value of Ki, we can ensure an in-plane magnetic anisotropy in the Py/BSTS interface at room temper- ature. A detailed discussion of magnetic anisotropy has been provided in the last section where the temperature variation of Hkis discussed. αeffcan be determined by analysing ∆ Hat different frequencies. ∆ Hcontains both the intrinsic and extrin- sic contributions to the damping. Linewidth due to in- trinsic damping is directly proportional to the resonance frequency( f) and follows the equation[29], ∆H= ∆H0+ (2παeff γ)f (4) where ∆ H0describes inhomogeneous linewidth broad- ening [38, 39] due to different extrinsic contributions like magnetic inhomogeneities [40, 41], surface roughness, and defects in the sample. We have evaluated the αeff values by fitting the ∆ Hvsfcurve for FM/TI bilayers as shown in fig.2a. This αeffconsists of Gilbert damp- ing in the bulk ferromagnet( αFM) and the enhanced damping( αSP) resulting from spin pumping into the ad- jacent TI layer [31–33], αeff=αFM+αSP. The αFM value for bare Py film of thickness 15nm was calculated to be 0.0074 and for the FM/TI bilayer system there has been significant enhancement in the αeffvalue over thebare Py value due to spin pumping, αSP. In this het- erostructure, αeffincreases gradually as the thickness of Py decreases both for Py(t) and Py(t)/BSTS2 samples as shown in fig.3a. From the linear fit of αeffvs. 1/tPydata we have obtained the spin-mixing coefficient, g↑↓ efffor the BSTS/Py interface to be 5 .26×1018±0.71×1018m−2 by using the equation[34, 37], αeff−αFM=gµB 4πMstFMg↑↓ eff(5) where, gandµBare the g-factor and Bohr magneton respectively. We have also calculated the spin current density( j0 s) for the FM/TI heterostructure using the g↑↓ eff value in the following equation[20, 36], j0 s=g↑↓ effγ2h2 mℏ[4πMsγ+p (4πMs)2γ2+ 4ω2] 8πα2[(4πMs)2γ2+ 4γ2](6) where γ,hm,ℏ,ω, and αare the gyromagnetic ratio, microwave magnetic field, Planck’s constant, Larmour precession frequency, and effective damping parameter respectively. Using Eq.6 the j0 svalue for Py/BSTS2 was obtained to be 0 .901×10−10±0.122×10−10Jm−2 in our experiment. The g↑↓ effandj0 svalues obtained from Py thickness-dependent study of αeffare in a comparable range of the previously reported values for other combinations of ferromagnet and TI bilayer structures [34, 35, 37]. This gives evidence of successful spin injection into the BSTS layer from the Py layer due to spin pumping [31–33, 50]. We also report the TI thickness-dependent study of αeffas shown in fig.3b. For bilayer structures of Py(15nm)/BSTS2(t) there is a sudden jump in the αeffvalue from that of the bare FM film ( αFM = 0.0074) because of spin pumping. Then with the thickness variation of TI layer in the range of 10nm to 37nm, αeffincreases slowly from 0.015 to 0.02. The TI thickness dependence of αeff for Py(15nm)/BSTS(t) bilayer is almost linear which certainly can not be described by the conventional spin diffusion theory [48] for FM/NM proposed by Tserkovnyak et al. [47]. For Py/BSTS heterostructure, αeffvs. tBSTS study suggests an efficient spin-sink nature of the TI bulk with increasing thickness at room temperature [49]. From the room temperature study we certainly can not distinguish the TI surface state contribution from the TI bulk state contribution because growing a BSTS thin film with a perfectly insulating bulk state is still very challenging. Thus it was imperative to study the effect of topological surface state at low-temperature where bulk states of TI get suppressed and surface states of TI starts to dominate. In this section, we have focused on low-temperature measurements specifically to understand the effect of topological surface states (TSS) on the magnetization relaxation of FM. At higher temperatures, a significant amount of bulk carriers are available to participate in the transport but with the reduction of phonon5 (a) (b) FIG. 5. (a)Temperature dependence of the field linewidth (∆ H) for different thickness combinations of Py/BSTS bilayer systems and for a bare Py thin film. The solid lines are the fits in the expression exp(−T/T 0); (b)Temperature dependence of effective damping coefficient, αeffof Py(20nm)/BSTS2 and bare Py(20nm) film. (a) (b) FIG. 6. (a)Temperature dependence of effective magnetization of Py(20nm/BSTS2); (b)Temperature dependence of the anisotropy field of Py(20nm)/BSTS2. scattering, surface carriers dominate at a lower tem- perature. From the resistivity vs. temperature data of BSTS2 in fig.4, we can see an insulating behavior of resistivity due to the enhanced insulating nature of the bulk state of TI at higher temperatures and a metallic behavior of resistivity below a certain temperature where the topological surface states dominate. We measured temperature variation of FMR linewidth (∆H), enhanced damping coefficient ( αeff), anisotropy field ( Hk) and effective magnetization (4 πMeff). For different thickness combinations of Py/BSTS bilayer, we obtained the ∆ Hvariation with temperature. It increases exponentially with decreasing temperature that fits the expression, exp(−T/T 0) as shown in fig.5a. For bare Py(15nm) film, we can note that there is no significant variation in ∆ Hat low temperatures as can be seen from the curve at the bottom of fig.5a. To gain further understanding, the temperature variation ofαeffhas also been studied for Py(20nm)/BSTS2 as shown in fig.5b and compared with αefffor barePy film. From the enhancement of αeffvalue for Py(20nm)/ BSTS2 at room temperature we can ensure a successful spin injection due to the spin pumping effect. But the exponential increase of αeffwith decreasing temperature for the bilayer implies a huge increment in the amount of spin angular momentum transfer into the TI layer at lower temperatures. We attribute the origin of the exponential increase of αeffand ∆ Hat lower temperatures to the spin chemical potential bias induced spin current into the surface state of TI as proposed by Abdulahad et al. [50]. The induced spin current into the TI surface state at lower temperatures corresponds to the rapid relaxation of magnetization precession of FM which is reflected in the exponential increase of ∆ H andαeffof the ferromagnet. To further investigate the effect of TI surface state on the magnetization of FM, we studied the temperature variation of 4 πMeffandHkfor Py(20nm)/BSTS2. In our previous study [26] with bare Py thin films, we have6 seen that 4 πMeffincreases monotonically as saturation magnetization increases with lowering the temperature. But from fig.6a, we can see that the low-temperature dependence of 4 πMefffor Py/BSTS2 bilayer deviates from the single layer Py film [Supplementary fig.S11(a)]. This anomaly in 4 πMeffis related to the change of mag- netic anisotropy energy of the system as well as the other effects like spin chemical potential induced current and exchange coupling between TSS and FM as mentioned by Abdulahad et al. [50]. In a previous section, we evaluated the interfacial magnetic anisotropy coefficient (Ki=−0.001erg/cm2) to be in-plane of the interface of the Py/BSTS2 bilayer. The anisotropy field associated with the system anisotropy energy shows an interesting nature as we lower the temperature. We can see from fig.6b that Hkincreases initially with decreasing tem- perature until a certain value is reached and then the anisotropy field weakens against a further decrease in temperature. Thus we get a hump-like feature of HK for the same temperature region where 4 πMeffshows the anomaly and it is concomitant with the resistivity vs temperature behavior of the BSTS2 sample. The low- temperature behavior of Hkand 4 πMeffcan be justified by the argument proposed by Abdulahad et al. [50]. In their phenomenological model, they propose an existence of exchange interaction between the surface states of TI and local moments of the ferromagnetic layer. Several theoretical as well as experimental predictions confirm the existence of gapless topological surface states even after transition metal deposition on TI [51, 52]. These surface states can couple with the local moments of the FM through exchange interaction without any long-range ferromagnetic order. This exchange coupling acts per- pendicular to the TI surface and weakens the in-plane anisotropy at lower temperatures where the surface states of TI dominate. CONCLUSIONS In summary, we have carried out spin-pumping ex- periment in BiSbTe 1.5Se1.5(TI)/ Ni80Fe20(FM) bilayer system. From the thickness-dependent measurements of FM/TI bilayers, we obtained the spin-transport param- eters like damping coefficient due to spin-pumping, spin mixing conductance, and spin current density at room temperature. These results demonstrate a successful spin transfer from the FM layer to the TI layer due to spin- pumping. We have performed low-temperature measure- ments to specifically understand the surface state con- tribution of TI on the FM magnetization because the surface states of TI are more pronounced at lower tem- peratures. We have confirmed the suppression of the insulating bulk state of TI at lower temperatures from the resistivity vs. temperature data of TI. In our low- temperature measurements of FMR linewidth and ef-fective damping coefficient, we have witnessed an expo- nential increase in both parameters with the decrease in temperature. It suggests a spin chemical potential bias- induced spin current injection into the surface states of TI that gets enhanced at low temperatures [50]. We have also studied temperature variations of the effective mag- netization of the system. It showed a deviation from the bare Py film [26] in the temperature regime where TI surface states dominate. This deviation of effective magnetization results from the change in the anisotropy energy of the system. At room temperature, we eval- uated the magnetic anisotropy energy coefficient which is found to be in-plane of the interface. This in-plane anisotropy weakens when conducting surface state of TI starts to dominate. It reflects from the hump-like feature in the magnetic anisotropy field vs. temperature data of the bilayer system. The decrease in in-plane magnetic anisotropy below a certain temperature can result from the exchange coupling between the surface states of TI and the local moments of the FM layer which act per- pendicular to the interface [50]. Combining the results of our low-temperature measurements we can conclude that there exists an exchange coupling between the TI surface state and FM which does not create any long-range ferro- magnetic order in the TI and is unable to alter the overall spin texture of the TI surface state at the interface[18]. However, it affects the magnetization dynamics of the ferromagnetic metal quite significantly. These added fea- tures of enhancing the damping coefficients enables an- other fast control of magnetization dynamics in the FM layer. ACKNOWLEDGEMENTS The authors sincerely acknowledge the Ministry of Education, Government of India and Science and Engineering Research Board (SERB) (grant no: EMR/2016/007950), and Department of Science and Technology (grant no. DST/ICPS/Quest/2019/22) for financial support. S.P. acknowledges the Department of Science and Technology(DST)-INSPIRE fellowship In- dia, S. A. acknowledges the Ministry of Education of the Government of India, S.M. acknowledges the Council Of Scientific and Industrial Research(CSIR), India, S.G.N and K.S acknowledges the University Grant Commis- sion, India for research fellowship. 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[49] Baker, A.A., Figueroa, A.I., Collins-McIntyre, L.J., Van Der Laan, G. and Hesjedal, T., 2015. Spin pump- ing in ferromagnet-topological insulator-ferromagnet het- erostructures. Scientific reports, 5(1), p.7907. [50] Abdulahad, F.B., Lin, J.H., Liou, Y., Chiu, W.K., Chang, L.J., Kao, M.Y., Liang, J.Z., Hung, D.S. and Lee, S.F., 2015. Spin chemical potential bias induced sur- face current evidenced by spin pumping into the topo- logical insulator B i 2 T e 3. Physical Review B, 92(24), p.241304. [51] Ye, M., Eremeev, S.V., Kuroda, K., Krasovskii, E.E., Chulkov, E.V., Takeda, Y., Saitoh, Y., Okamoto, K., Zhu, S.Y., Miyamoto, K. and Arita, M., 2012. Quasipar- ticle interference on the surface of Bi 2 Se 3 induced by cobalt adatom in the absence of ferromagnetic ordering. Physical Review B, 85(20), p.205317. [52] Scholz, M.R., S´ anchez-Barriga, J., Marchenko, D., Varykhalov, A., Volykhov, A., Yashina, L.V. and Rader, O., 2012. Tolerance of topological surface states towards magnetic moments: Fe on Bi 2 Se 3. Physical Review Letters, 108(25), p.256810. [53] Celinski, Z., Urquhart, K.B. and Heinrich, B., 1997. Us- ing ferromagnetic resonance to measure the magnetic mo- ments of ultrathin films. Journal of Magnetism and Mag- netic Materials, 166(1-2), pp.6-26.

2023-03-13

We have studied ferromagnetic metal/topological insulator bilayer system to understand magnetization dynamics of ferromagnetic metal (FM) in contact with a topological insulator (TI). At magnetic resonance condition, the precessing magnetization in the metallic ferromagnet ($Ni_{80}Fe_{20}$) injects spin current into the topological insulator ($BiSbTe_{1.5}Se_{1.5}$), a phenomenon known as spin-pumping. Due to the spin pumping effect, fast relaxation in the ferromagnet results in the broadening of ferromagnetic resonance linewidth ($\Delta H$). We evaluated the parameters like effective Gilbert damping coefficient ($\alpha_{eff}$), spin-mixing conductance ($g_{eff}^{\uparrow \downarrow}$) and spin current density ($j_S^0$) to confirm a successful spin injection due to spin-pumping into the $BiSbTe_{1.5}Se_{1.5}$ layer. TIs embody a spin-momentum locked surface state that span the bulk band-gap. It can act differently to the FM magnetization than the other normal metals. To probe the effect of topological surface state, a systematic low temperature study is crucial as surface state of TI dominates at lower temperatures. The exponential growth of $\Delta H$ for all different thickness combination of FM/TI bilayers and effective Gilbert damping coefficient ($\alpha_{eff}$) with lowering temperature confirms the prediction that spin chemical bias generated from spin-pumping induces surface current in TI due to spin-momentum locking. The hump-like feature of magnetic anisotropy field ($H_K$)of the bilayer around 60K suggests that the decrease of interfacial in-plane magnetic anisotropy can result from exchange coupling between the TI surface state and the local moments of FM layer.

Experimental investigation of the effect of topological insulator on the magnetization dynamics of ferromagnetic metal: $BiSbTe_{1.5}Se_{1.5}$ and $Ni_{80}Fe_{20}$ heterostructure

2303.07025v2

Temperature dependent magnetic damping of yttrium iron garnet spheres H. Maier-Flaig,1, 2,S. Klingler,1, 2C. Dubs,3O. Surzhenko,3R. Gross,1, 2, 4M. Weiler,1, 2H. Huebl,1, 2, 4and S. T. B. Goennenwein1, 2, 4, 5, 6 1Walther-Meiner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany 2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany 3INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany 4Nanosystems Initiative Munich, 80799 M unchen, Germany 5Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden, Germany 6Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden, Germany (Dated: June 5, 2017) We investigate the temperature dependent microwave absorption spectrum of an yttrium iron garnet sphere as a function of temperature (5 K to 300 K) and frequency (3 GHz to 43 :5 GHz). At temperatures above 100 K, the magnetic resonance linewidth increases linearly with temperature and shows a Gilbert-like linear frequency dependence. At lower temperatures, the temperature dependence of the resonance linewidth at constant external magnetic elds exhibits a characteristic peak which coincides with a non-Gilbert-like frequency dependence. The complete temperature and frequency evolution of the linewidth can be modeled by the phenomenology of slowly relaxing rare- earth impurities and either the Kasuya-LeCraw mechanism or the scattering with optical magnons. Furthermore, we extract the temperature dependence of the saturation magnetization, the magnetic anisotropy and the g-factor. I. INTRODUCTION The magnetization dynamics of the ferrimagnetic insu- lator yttrium iron garnet (YIG) recently gained renewed interest as YIG is considered an ideal candidate for spin- tronic applications as well as spin-based quantum infor- mation storage and processing1{4due to the exception- ally low damping of magnetic excitations as well as its magneto-optical properties5{7. In particular, consider- able progress has been made in implementing schemes such as coupling the magnetic moments of multiple YIG spheres2,8or interfacing superconducting quantum bits with the magnetic moment of a YIG sphere3,9. Magnetization dynamics in YIG have been investi- gated in a large number of studies in the 1960s.10{12 However, a detailed broadband study of the magnetiza- tion dynamics in particular for low temperatures is still missing for bulk YIG. Nevertheless, these parameters are essential for the design and optimization of spintronic and quantum devices. Two recent studies13,14consider the temperature dependent damping of YIG thin lms. Haidar et al.13report a large Gilbert-like damping of unknown origin, while the low damping thin lms inves- tigated by Jermain et al.14show a similar behavior as reported here. Our systematic experiments thus provide an impor- tant link between the more recent broadband studies on YIG thin lms and the mostly single-frequency studies from the 1960s: We investigate the magnetostatic spin wave modes measured in a YIG sphere using broadband magnetic resonance up to 43.5 GHz in the temperature range from 5 to 300K. We extract the temperature de- pendent magnetization, the g-factor and the magnetic anisotropy of YIG. Additionally, we focus our analysis on the temperature dependent damping properties of YIGand identify the phenomenology of slowly relaxing rare- earth impurities and either the Kasuya-LeCraw mecha- nism or the scattering with optical magnons as the mi- croscopic damping mechanism. The paper is organized as follows. We rst give a short introduction into the experimental techniques followed by a brief review of the magnetization damping mechanisms reported for YIG. Finally, we present the measured data and compare the evolution of the linewidth with temper- ature and frequency with the discussed damping models. The complete set of raw data and the evaluation routines are publicly available.15 II. EXPERIMENTAL DETAILS AND FERROMAGNETIC RESONANCE THEORY The experimental setup for the investigation of the temperature dependent broadband ferromagnetic reso- nance (bbFMR) is shown schematically in Fig. 1. It consists of a coplanar wave guide (CPW) onto which a 300 µm diameter YIG sphere is mounted above the 300µm wide center conductor. The [111] direction of the single crystalline sphere is aligned along the CPW surface normal as conrmed using Laue diraction (not shown). We mount a pressed diphenylpicrylhydrazyl (DPPH) powder sample in a distance of approximately 5 mm from the sphere. The identical sample with the same alignment has been used in Ref. 16. This assembly is mounted on a dip stick in order to place the YIG sphere in the center of a superconducting magnet (Helmholtz conguration) in a Helium gas- ow cryostat. End-launch connectors are attached to the CPW and connected to the two ports of a vector network analyzer (VNA) mea- suring the phase sensitive transmission of the setup uparXiv:1703.09444v2 [cond-mat.mtrl-sci] 2 Jun 20172 liquid Helium side view CPWP1 P2H0hMW hMWVNAP1P2 side viewside view5mm DPPH YIG com ±1‰ ±1‰ FIG. 1. The coplanar waveguide (CPW), on which the YIG sphere and a DPPH marker are mounted (right), is inserted into a magnet cryostat (left). The microwave transmission through the setup is measured phase sensitively using a vec- tor network analyzer (VNA). As the Oersted eld hMW(red) around the center conductor of the CPW extends into the YIG sphere and the DPPH, we can measure the microwave response spectra of both samples. A superconducting magnet provides the static external magnetic eld H0(orange) at the location of the sample. Also shown are the lines correspond- ing to the specied 1 ‰homogeneity of the eld for a on-axis deviation from the center of magnet (com). to 43:5 GHz. The sphere is placed within the microwave Oersted eldhMWof the CPW's center conductor which is excited with a continuous wave microwave of variable frequency. We apply a static external magnetic eld H0perpen- dicular to the CPW surface and thus hMWis oriented primarily perpendicular to H0. The microwave Oersted eld can therefore excite magnetization precession at fre- quencies that allow a resonant drive. The magnetization precession is detected by electromagnetic induction via the same center conductor.17This induction voltage in combination with the purely transmitted microwave sig- nal is measured phase sensitively as the complex scatter- ing parameter Sraw 21(!) at port 2 of the VNA. The frequency-dependent background is eliminated as follows: A static external magnetic eld suciently large that no resonances are expected in the given microwave frequency range is applied and the transmission at this eld is recorded as the background reference SBG 21. Then, the external eld is set to the value at which we ex- pect resonances of YIG in the given frequency range and record the transmission Sraw 21. We nally divide Sraw 21by SBG 21. This corrects for the frequency dependent attenu- ation and the electrical length of the setup. We choose this background removal method over a microwave cal- ibration because it additionally eliminates the eld andtemperature dependence of S21that arises from the ther- mal contraction and movement of the setup and magnetic materials in the microwave connectors. In the following we always display S21=Sraw 21=SBG 21. For the evaluation of the magnetization dynamics, we t the transmission data to S21=ifZ +A1+A2f for eachH0. Here,A1;2describe a complex-valued back- ground and (f;H 0) =0Ms 2 20H0i
f
f2resf2if
f(1) is the ferromagnetic high-frequency susceptibility.17,18 The free parameters of the t are the resonance frequency fres, the full width at half maximum (FWHM) linewidth
fas well as the complex scaling parameter Z, which is proportional to the strength of the inductive coupling between the specic magnetic resonance mode and the CPW. For a given xed magnetic eld the t parame- ters 2(gyromagnetic ratio) and Ms(saturation magne- tization) are completely correlated with Zand are thus xed. They are later determined from tting the disper- sion curves.19 In spheres various so-called magnetostatic modes (MSM) arise due to the electromagnetic boundary conditions.20These modes can be derived from the Landau-Lifshitz equation in the magnetostatic limit ( ~r ~H= 0) for insulators.21The lineshape of all modes is given by Eq. 1. Due to the dierent spatial mode proles and the inhomogeneous microwave eld, the inductive coupling and thus Zis mode dependent.22A detailed re- view of possible modes, their distribution and dispersion is given in R oschmann and D otsch20. We will only dis- cuss the modes (110) and (440) in detail in the following as all the relevant characteristics of all other modes can be related to these two modes. Their linear dispersions are given by20 f110 res= 20(H0+Hani) (2) f440 res= 20 H0+Hani+Ms 9 (3) whereHaniis the magnetic anisotropy eld and 2is the gyromagnetic ratio which relates to the g-factor by 2=B hg. It is thus generally assumed that gis the same for all modes. We note that the apparent g-factor may still vary in between modes if the modes experience a dierent anisotropy.23,24Such an anisotropy contribu- tion can be caused by surface pit scattering as it aects modes that are localized at the surface stronger than bulk like modes25. In our experiment, no such variation in g coinciding with a change in anisotropy was observed and we use a mode number independent gin the following. Knowledge of the dispersion relations of the two modes allows to determine the saturation magnetization from 0Ms(T) = 92
fM= 92 f440 resf110 res
:(4)3 The anisotropy eld is extracted by extrapolating the dispersion relations in Eqs. (2) and (3) to H0= 0. The temperature dependent linewidth
fof the modes is the central result of this work. For a short review of the relevant relaxation processes we refer to the dedicated Sec. III. In this work, we investigate the T-dependence of Ms, Hani,gand
f. Accurate determination of the g-factor and the anisotropy Hanirequires accurate knowledge of H0. In order to control the temperature of the YIG sphere and CPW, they are placed in a gas- ow cryo- stat as displayed schematically in Fig. 1. The challenge in this type of setup is the exact and independent deter- mination of the static magnetic eld and its spatial in- homogeneity. Lacking an independent measure of H0,26 we only report the relative change of gandHanifrom their respective room temperature values which were de- termined separately using the same YIG sphere.16Note that we determine the resonance frequencies directly in frequency space. Our results on linewidth and magneti- zation are hence independent of a potential uncertainty in the absolute magnitude of H0and its inhomogeneity. III. RELAXATION THEORY When relaxation properties of ferromagnets are dis- cussed today, the most widely applied model is the so- called Gilbert type damping. This purely phenomeno- logical model is expressed in a damping term of the form MdM dtin the Landau-Lifshitz equation. It describes a viscous damping, i.e. a resonance linewidth that de- pends linearly on the frequency. A linear frequency de- pendence is often found in experiments and the Gilbert damping parameter serves as a gure of merit of the ferromagnetic damping that allows to compare samples and materials. It contains, however, no insight into the underlying physical mechanisms. In order to understand the underlying microscopic re- laxation processes of YIG, extensive work has been car- ried out. Improvements on both the experimental side (low temperatures27, temperature dependence10,28,29, separate measurements of MzandMxy11) and on the sample preparation (varying the surface pit size25, pu- rifying Yttrium10, doping YIG with silicon30and rare- earth elements30{33) led to a better understanding of these mechanisms. However, despite these eorts the microscopic origin of the dominant relaxation mechanism for bulk YIG at room temperature is still under debate. It has been described by a two-magnon process by Kasuya and LeCraw28(1961). In this process, a uniformly-precessing magnon (k= 0) relaxes under absorption of a phonon to ak6= 0 magnon. If the thermal energy kBTis much larger than the energy of the involved magnons and phonons (T > 100 K) and low enough that no higher- order processes such as four-magnon scattering play a role (T < 350 K), the Kasuya-LeCraw process yields alinewidth that is linear in frequency and temperature:
fKL/T;f.28,30This microscopic process is therefore considered to be the physical process that explains the phenomenological Gilbert damping for low-damping bulk YIG. More recently, Cherepanov et al.34pointed out that the calculations by Kasuya and LeCraw28assume a quadratic magnon dispersion in k-space which is only correct for very small wave numbers k. Taking into ac- count a more realistic magnon dispersion (quadratic at lowk, linear to higher k), the Kasuya-LeCraw mechanism gives a value for the relaxation rate that is not in line with the experimental results. Cherepanov therefore de- veloped an alternative model that traces back the linear frequency and temperature dependence at high tempera- tures (150 K to 300 K) to the interaction of the uniform- precession mode with optical magnons of high frequency. Recently, atomistic calculations by Barker and Bauer35 conrmed the assumptions on the magnon spectrum that are necessary for the quantitative agreement of the latter theory with experiment. Both theories, the Kasuya-LeCraw theory and the Cherepanov theory, aim to describe the microscopic ori- gin of the intrinsic damping. They deviate in their prediction only in the low-temperature ( T < 100 K) behavior.30At these temperatures, however, impurities typically dominate the relaxation and mask the contri- bution of the intrinsic damping process. Therefore, the dominant microscopic origin of the YIG damping at tem- peratures above 150K has not been unambiguously de- termined to date. If rare-earth impurities with large orbital momentum exist in the crystal lattice, their exchange coupling with the iron ions introduces an additional relaxation chan- nel for the uniform precession mode of YIG. Depend- ing on the relaxation rate of the rare-earth impurities with respect to the magneto{dynamics of YIG, they are classied into slowly and fast relaxing rare-earth impu- rities. This is an important distinction as the eciency of the relaxation of the fundamental mode of YIG via the rare-earth ion to the lattice at a given frequency de- pends on the relaxation rate of the rare-earth ion and the strength of the exchange coupling. In both the slow and the fast relaxor case, a characteristic peak-like maximum is observed in the linewidth vs. temperature dependence at a characteristic, frequency-dependent temperature12. The frequency dependence of this peak temperature al- lows to distinguish fast and slowly relaxing rare-earth ions: The model of a fast relaxing impurity predicts that the peak temperature is constant, while in the case of slowly relaxing rare-earth ions the peak temperature is experted to increase with increasing magnetic eld (or frequency). The relaxation rate of rare-earths REis typ- ically modeled by a direct magnon to phonon relaxation, an Orbach processes36,37that involves two phonons, or a combination of both. The inverse relaxation rate of an Orbach process is described by1 Orbach =B e
=(kBT)1 with the crystal eld splitting
and a proportionality factorB. A direct process leads to an inverse relax-4 ation rate of1 direct =1 0coth 2kBTwith0, the relaxation time atT= 0 K. It has been found experimentally that most rare-earth impurities are to be classied as slow relaxors.30The sample investigated here is not intention- ally doped with a certain rare-earth element and the peak frequency and temperature dependence indicates a slow relaxor. We therefore focus on the slow relaxing rare- earth impurity model in the following. Deriving the theory of the slowly relaxing impurities has been performed comprehensively elsewhere.30The linewidth contribution caused by a slowly relaxing rare- earth impurity is given by31:
fSR=C 2fRE 1 + (fRE)2(5) withC/1 kBTsech a 2kBT . Therein, ais the splitting of the rare-earth Kramers doublet which is given by the temperature independent exchange interaction between the iron ions and the rare-earth ions. Also Fe2+impurities in YIG give rise to a process that leads to a linewidth peak at a certain temperature. The physical origin of this so-called valence exchange or charge-transfer linewidth broadening is electron hopping between the iron ions.30Simplied, it can be viewed as a two level system just like a rare-earth ion and thus re- sults in the same characteristic linewidth maximum as a slowly relaxing rare-earth ion. For valence exchange, the energy barrier
hopthat needs to be overcome for hopping determines the time scale of the process. The two processes, valence exchange and rare-earth impurity relaxation, can therefore typically not be told apart from FMR measurements only. In the following, we use the slow relaxor mechanism exclusively. This model consis- tently describes our measurement data and the resulting model parameters are in good agreement with literature. We would like to emphasize, however, that the valence exchange mechanism as the relevant microscopic process resulting for magnetization damping can not be ruled out from our measurements. IV. EXPERIMENTAL RESULTS AND DISCUSSION Two exemplary S21broadband spectra recorded at two distinct temperatures are shown in Fig. 2. The color- coded magnitude jS21jis a measure for the absorbed mi- crowave power. High absorption (bright color) indicates the resonant excitation of a MSM in the YIG sphere or the excitation of the electron paramagnetic resonance of the DPPH. In the color plot the color scale is truncated in order to improve visibility of small amplitude reso- nances. In addition, the frequency axis is shifted relative to the resonance frequency of a linear dispersion with g= 2:0054 (fg=2:0054 res =gB h0H) for each eld. In this way, modes with g= 2:0054 appear as vertical lines. A 0.20.40.60.81.01.21.4¹0H0 (T)290 K minmax Absorption −1.0 −0.5 0.0 0.5 1.0 f¡f(g=2:0054) res (GHz)0.20.40.60.81.01.21.4¹0H0 (T)20 KΔfAΔfM f (GHz)10.05 10.06 10.07 10.08 Im(S21) 2 046810 0 -4-224Re(S21) ȴffres(a) (b)FIG. 2. Eigenmode spectra of the YIG sphere at (a) 290 K and (b) 20 K. The (110) and (440) MSM are marked with red dashed lines. The change in their slope gives the change of the g-factor of YIG. Their splitting (
fM, red arrow) depends lin- early on the YIG magnetization. The increase in Msto lower temperatures is already apparent from the increased splitting
fM. Marked in orange is the oset of the resonance fre- quency
fAextrapolated to H0= 0 resulting from anisotropy eldsHanipresent in the sphere. The green marker denotes the position of the DPPH resonance line which increases in amplitude considerably to lower temperatures. Inset: S21pa- rameter (data points) and t (lines) at 0H= 321 mT and T= 20 K. deviatingg-factor is therefore easily visible as a dier- ent slope. Comparing the spectra at 290 K [Fig. 2 (a)] to the spectra at 20 K [Fig. 2 (b)], an increase of the g- factor is observed for all resonance modes upon reducing the temperature. The rich mode spectrum makes it nec- essary to carefully identify the modes and assign mode numbers. Note that the occurrence of a particular mode in the spectrum depends on the position of the sphere with respect to the CPW due to its inhomogeneous exci- tation eld. We employ the same method of identifying the modes as used in Ref. 16 and nd consistent mode spectra. As mentioned before, we do not use the DPPH resonance (green arrow in Fig. 2) but the (110) YIG mode as eld reference. For this eld reference, we take g(290 K) = 2 :0054 and 20Hani(290 K) = 68 :5 MHz de- termined for the same YIG sphere at room temperature in an electromagnet with more accurate knowledge of the applied external magnetic eld.16The discrepancy of the DPPHg-value from the literature values of g= 2:0036 is5 attributed to the non-optimal location of the DPPH spec- imen in the homogeneous region of the superconducting magnet coils. In Fig. 2, the tted dispersion of the (110) and (440) modes are shown as dashed red lines. As noted previ- ously, we only analyze these two modes in detail as all parameters can be extracted from just two modes. The (110) and (440) mode can be easily and unambiguously identied by simply comparing the spectra with the ones found in Ref 16. Furthermore, at high elds, both modes are clearly separated from other modes. This is necessary as modes can start hybridizing when their (unperturbed) resonance frequencies are very similar (cf. low-eld re- gion of Fig. 2 (b)) which makes a reliable determination of the linewidth and resonance frequency impossible. These attributes make the (110) and the (440) mode the ideal choice for the analysis. As described in Sec. II, we simultaneously t the (110) and the (440) dispersions with the same g-factor in or- der to extract Ms,Haniandg. In the t, we only take the high-eld dispersion of the modes into account where no other modes intersect the dispersion of the (110) and (440) modes. The results are shown in Fig. 3. Note that the statistical uncertainty from the t is not visible on the scale of any of the parameters Ms,Haniandg. Following the work of Solt38, we model the resulting temperature dependence of the magnetization (Fig. 3 (a)) with the Bloch-law taking only the rst order correction into ac- count: Ms=M0 1aT3 2bT5 2 : (6) The best t is obtained for 0M0 = 249:5(5) mT, a = (233)106K3=2and b= (1:080:11)107K5=2. The obtained t parameters depend strongly on the temperature window in which the data is tted. Hence, the underlying physics determining the constants aandbcannot be resolved.39Nevertheless, the temperature dependence ofMsis in good agreement with the results determined using a vibrating sample magnetometer.40 In particular, also the room temperature saturation magnetization of 0Ms(300 K) = (180 :00:8) mT is in perfect agreement with values reported in literature.41,42 Note that the splitting of the modes is purely in frequency space and thus errors in the eld do not add to the uncer- tainty. We detect a small non-linearity of the (110) and (440) mode dispersions that is most likely due to devia- tions from an ideal spherical shape or strain due to the YIG mounting. This results in a systematic, temperature independent residual of the linear ts to these disper- sions. This resulting systematic error of the magnetiza- tion is incorporated in the uncertainty given above. How- ever, a deviation from the ideal spherical shape, strain in the holder or a misalignment of the static magnetic eld can also modify the splitting of the modes and hence re- sult in a dierent Ms.43This fact may explain the small discrepancy of the value determined here and the valuedetermined for the same sphere in a dierent setup at room temperature.16 From the same t that we use to determine the mag- netization, we can deduce the temperature dependence of the anisotropy eld 0Hani[Fig. 3 (b)]. Most notably, Hanichanges sign at 200 K which has not been observed in literature before and can be an indication that the sample is slightly strained in the holder. The resonance frequency of DPPH extrapolated to 0H0= 0 (
fani, red squares) conrms that the error in the determined value Haniis indeed temperature independent and very close to zero. Thus, the extracted value for the anisotropy is not merely given by an oset in the static magnetic eld. The evolution of the g-factor with temperature is shown in Fig. 3 (c). It changes from 2 :005 at room tem- perature to 2 :010 at 10 K where it seems to approach a constant value. As mentioned before, the modes' disper- sion is slightly non-linear giving rise to a systematic, tem- perature independent uncertainty in the determination of gof0:0008. Theg-factor of YIG has been determined using the MSMs of a sphere for a few selected tempera- tures before.12Comparing our data to these results, one nds that the trend of the temperature dependence of g agrees. However, the absolute value of gand the mag- nitude of the variation dier. At the same time, we nd a change of the g-factor of DPPH that is on the scale of 0:0012. This may be attributed to a movement of the sample slightly away from the center of magnet with changing temperature due to thermal contraction of the dip stick. In this case, the YIG g-factor has to be cor- rected by this change. The magnitude of this eect on the YIGg-factor can not be estimated reliably from the change of the DPPH g-factor alone. Furthermore, the temperature dependence of the DPPH g-factor has not been investigated with the required accuracy in literature to date to allow excluding a temperature dependence of theg-factor of DPPH. We therefore do not present the corrected data but conclude that we observe a change in the YIGg-factor from room temperature to 10 K of at least 0.2 %. Next, we turn to the analysis of the damping properties of YIG. We will almost exclusively discuss the damping of the (110) mode in the following but the results also hold quantitatively and qualitatively for the other modes.16 Varying the applied microwave excitation power P(not shown) conrms that no nonlinear eects such as a power broadening of the modes are observed with P= 0:1 mW. Note that due to the microwave attenuation in the mi- crowave cabling, the microwave eld at the sample loca- tion decreases with increasing frequency for the constant excitation power. First, we evaluate the frequency dependent linewidth for several selected temperatures [Fig. 4 (a)]. At temper- atures above 100 K, a linear dependence of the linewidth with the resonance frequency is observed. This depen- dence is the usual so-called Gilbert-like damping and the slope is described by the Gilbert damping param- eter. A linear frequency dependence of the damping6 180200220240260¹0M (mT) (a)model −20246¹0Hani (mT) (b)YIG DPPH 0 50 100 150 200 250 300 Temperature (K)2.0042.0082.0122.0162.020g-factor (unitless) 0.08% 0.26% (c) FIG. 3. (a)YIG magnetization as function of tempera- ture extracted from the (110) and (440) mode dispersions us- ing Eq. 4. The purple line shows the t to a Bloch model (cf. parameters in the main text). (b)YIG anisotropy eld 0Hani(T) =2
fani. Red squares: Same procedure applied to the DPPH dispersion as reference. (c)YIGg-factor (blue circles). For reference, the extracted DPPH g-factor is also shown (red squares). The gray numbers indicate the rela- tive change of the g-factors from the lowest to the highest measured temperature (gray horizontal lines). As we use the YIG (110) mode as the magnetic eld reference, the extracted value ofgandHaniat 300 K are xed to the values determined in the room temperature setup.16 in bulk YIG has been described by the theory developed by Kasuya and LeCraw28and the theory developed by Cherepanov et al.34(cf. Sec. III). We extract from a global t of a linear model to the (110) and (440) linewidth with separate parameters for the inhomoge- neous linewidths
f110 0and
f440 0and a shared Gilbert damping parameter for all modes:16
f= 2f+
f110;440 0 (7) The t is shown exemplarily for the 290 K (red) data in Fig. 4 (a). The Gilbert damping parameter extracted using this tting routine for each temperature is shown in Fig. 5 (a). Consistently with both theories, increases with increas- ing temperature. The error bars in the gure correspond to the maximal deviation of extracted from separate ts for each mode. They therefore give a measure of howscatters in between modes. The statistical error of the t (typically 0:00001) is not visible on this scale. The Gilbert damping parameter linearly extrapolatedto zero temperature vanishes. Note that this is consis- tent with the magnon-phonon process described by Ka- suya and LeCraw28but not with the theory developed by Cherepanov et al.34. For room temperature, we ex- tract a Gilbert damping of 4 105which is in excel- lent agreement with the literature value.16,44From the t, we also extract the inhomogeneous linewidth
f0, which we primarily associate with surface pit scattering (Sec. III, Ref 16). In the data, a slight increase of
f0 towards lower temperatures is present [Fig. 5 (b)]. Such a change in the inhomogeneous linewidth can be caused by a change in the surface pit scattering contribution when the spin-wave manifold changes with Ms.16,25 Note that according to Fig. 5 (b)
f0is higher for the (440) mode than for the (110) mode. This is in agreement with the theoretical expectation that surface pit scatter- ing has a higher impact on
f0for modes that are more localized at the surface of the sphere like the (440) mode compared to the more bulk-like modes such as the (110) mode25.45 Turning back to Fig. 4 (a), for low temperatures (20 K, blue data points), a Gilbert-like damping model is obvi- ously not appropriate as the linewidth increases consider- ably towards lower frequencies instead of increasing lin- early with increasing frequency. Typically, one assumes that the damping at low frequencies is dominated by so- called low eld losses that may arise due to domain for- mation. The usual approach is then to t a linear trend to the high-frequency behavior only. Note, however, that even though the frequency range we use is already larger than usually reported13,14,46, this approach yields an un- physical, negative damping. We conclude that the model of a Gilbert-like damping is only valid for temperatures exceeding 100 K (Fig. 5) for the employed eld and fre- quency range. The linewidth data available in literature are typi- cally taken at a xed frequency and the linewidth is dis- played as a function of temperature12,29,30. We can ap- proximately reproduce these results by plotting the mea- sured linewidth at xed H0as a function of temperature [Fig. 4 (b)].47A peak-like maximum of the linewidth be- low 100 K is clearly visible. For increasing magnetic eld (frequency), the peak position shifts to higher tempera- tures. This is the signature of a slowly relaxing rare-earth impurity (Sec. III). A fast relaxing impurity is expected to result in a eld-independent linewidth vs. temperature peak and can thus be ruled out. At the peak position, the linewidth shows an increase by 2 :5 MHz which trans- lates with the gyromagnetic ratio to a eld linewidth in- crease of 0:08 mT. For 0.1 at. % Terbium doped YIG, a linewidth increase of 80 mT has been observed48. Con- sidering that the linewidth broadening is proportional to the impurity concentration and taking the specied pu- rity of the source material of 99.9999% used to grow the YIG sphere investigated here, we estimate an increase of the linewidth of 0 :08 mT, in excellent agreement with the observed value. Modeling the linewidth data is more challenging: The7 0 5 10 15 20 25 30 35 40 45 f110 res (GHz)0¢f023456¢f110 (MHz)(a) 20 K 290 K 0 50 100 150 200 Temperature (K)0.51.01.52.02.53.03.5¢f110 (MHz) (b) 341 mT, 9.6 GHz 1007 mT, 28.3 GHzTmax 0:3TTmax 1:0T FIG. 4. (a)Full width at half maximum (FWHM) linewidth
f110 resof the (110) mode as a function of frequency for dierent temperatures. A linear Gilbert-like interpretation is justied in the high- Tcase (T > 100 K) only. Below 100 K, the slope of
f110(f110 res) is not linear so that a Gilbert type interpretation is no longer applicable. (b)FWHM linewidth as a function of temperature for two dierent xed external magnetic elds. The linewidth peaks at a magnetic eld dependent temperature that can be modeled using the phenomenology of rare-earth impurities resulting in Tmax(vertical dotted lines). 01234® (unitless)£10−5 (a) 0 50 100 150 200 250 300 Temperature (K)0123¢f0 (MHz) (b)110 440 FIG. 5. (a)Gilbert damping parameter determined from the slope of a linear t to the
f(f;T) data for frequencies above 20 GHz. The red line shows the linear dependence of the linewidth with temperature expected from the Kasuya-LeCraw process. (b)Inhomogeneous linewidth
f0(intersect of the aforementioned t with f110 res= 0) as a function of temperature. The inhomogeneous linewidth shows a slight increase with decreasing temperature down to 100 K. In the region where the slow relaxor dominates the linewidth (gray shaded area, cf. Fig. 4), the linear t is not applicable and unphysical damping parameters and inhomogeneous linewidths are extracted. model of a slowly relaxing rare-earth ion contains the exchange coupling of the rare-earth ion and the iron sublattice, and its temperature dependent relaxation fre- quency as parameters. As noted before, typically a di- rect and an Orbach process model the relaxation rate, and both of these processes have two free parameters. Unless these parameters are known from other experi- ments for the specic impurity and its concentration in the sample, tting the model to the temperature behav- ior of the linewidth at just one xed frequency gives am- biguous parameters. In principle, frequency resolved ex- periments as presented in this work make the determina- tion of the parameters more robust as the mechanism re- sponsible for the rare-earth relaxation is expected not to vary as a function of frequency. The complete frequency and eld dependence of the linewidth is shown in Fig. 6.At temperatures above approx 100 K, the linewidth in- creases monotonically with eld, in agreement with a dominantly Gilbert-like damping mechanisms, which be- comes stronger for higher temperatures. On the same linear color scale, the linewidth peak below 100 K and its frequency evolution is apparent. Fig. 4 (b) corresponds to horizontal cuts of the data in Fig. 6 at 0H0= 341 and 1007 mT. For typical YIG spheres, that are not specically en- riched with only one rare-earth element, the composition of the impurities is unknown. Dierent rare-earth ions contribute almost additively to the linewidth and have their own characteristic temperature dependent relax- ation frequency respectively peak position. This is most probably the case for the YIG sphere of this study. The constant magnitude of the peak above 0 :3 T and the con-8 stant peak width indicates that fast relaxing rare-earth ions play a minor role. The evolution of the linewidth withH0andfcan therefore not be tted to one set of parameters. We thus take a dierent approach and model just the shift of the peak position in frequency and temperature as originating from a single slowly re- laxing rare-earth impurity. For this, we use a value for the exchange coupling energy between the rare-earth ions and the iron sublattice in a range compatible with literature30ofa= 2:50 meV. To model the rare-earth relaxation rate as a function of temperature, we use the values determined by Clarke et al.32for Neodymium doped YIG: 0= 2:51011s for the direct process and
= 10:54 meV and B= 91011s1for the Orbach pro- cess. The model result, i.e. the peak position, is shown as dashed line in Fig. 6 and shows good agreement with the data. This indicates that, even though valence exchange and other types of impurities cannot be rigourously ex- cluded, rare-earth ions are indeed the dominant source for the linewidth peak at low temperatures. V. CONCLUSIONS We determined the ferromagnetic dispersion and linewidth of the (110) magnetostatic mode of a polished YIG sphere as a function of temperature and frequency. From this data, we extract the Gilbert damping param- eter for temperatures above 100 K and nd that it varies linearly with temperature as expected according to the two competing theories of Kasuya and LeCraw28andCherepanov et al.34. At low temperatures, the temper- ature dependence of the linewidth measured at constant magnetic eld shows a peak that shifts to higher tempera- tures with increasing frequency. This indicates slowly re- laxing impurities as the dominant relaxation mechanism for the magnetostatic modes below 100 K. We model the shift of the peak position with temperature and fre- quency with values reported for Neodymium impurities32 in combination with a typical value for the impurity-ion to iron-ion exchange coupling. We nd that these param- eters can be used to describe the position of the linewidth peak. We thus directly show the implications of (rare earth) impurities as typically present in YIG samples on the dynamic magnetic properties of the ferrimagnetic garnet material. Furthermore, we extract the temper- ature dependence of the saturation magnetization, the anisotropy eld and the g-factor. ACKNOWLEDGEMENTS The authors thank M. S. Brandt for helping out with the microwave equipment. C.D. and S.O. would like to acknowledge R. Meyer, M. Reich, and B. Wenzel (IN- NOVENT e.V.) for technical assistance in the YIG crys- tal growth and sphere preparation. We gratefully ac- knowledge funding via the priority program Spin Caloric Transport (spinCAT), (Projects GO 944/4 and GR 1132/18), the priority program SPP 1601 (HU 1896/2- 1) and the collaborative research center SFB 631 of the Deutsche Forschungsgemeinschaft. hannes.maier- aig@wmi.badw.de 1X. Zhang, C.-l. Zou, L. Jiang, and H. X. Tang, Physical Review Letters 113, 156401 (2014). 2X. Zhang, C.-l. Zou, N. Zhu, F. Marquardt, L. Jiang, and H. X. Tang, Nature Communications 6, 8914 (2015). 3Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Physical Review Letters 113, 083603 (2014). 4L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and C.-M. Hu, Physical Review Letters 114, 227201 (2015). 5S. Klingler, H. Maier-Flaig, R. Gross, C.-M. Hu, H. Huebl, S. T. B. Goennenwein, and M. Weiler, Applied Physics Letters 109, 072402 (2016). 6S. Viola Kusminskiy, H. X. Tang, and F. Marquardt, Phys- ical Review A 94, 033821 (2016). 7R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Physical Review B 93, 174427 (2016). 8N. J. Lambert, J. A. Haigh, S. Langenfeld, A. C. Doherty, and A. J. Ferguson, Physical Review A 93, 021803 (2016). 9Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya- mazaki, K. Usami, and Y. Nakamura, Science 349, 405 (2015). 10E. G. Spencer, R. C. Lecraw, and A. M. Clogston, Physical Review Letters 3, 32 (1959).11M. Sparks and C. Kittel, Physical Review Letters 4, 232 (1960). 12K. P. Belov, L. A. Malevskaya, and V. I. Sokoldv, Soviet Physics JETP 12, 1074 (1961). 13M. Haidar, M. Ranjbar, M. Balinsky, R. K. Dumas, S. Khartsev, and J. Akerman, Journal of Applied Physics 117, 17D119 (2015). 14C. L. Jermain, S. V. Aradhya, J. T. Brangham, M. R. Page, N. D. Reynolds, P. C. Hammel, R. A. Buhrman, F. Y. Yang, and D. C. Ralph, arXiv preprint arXiv:1612.01954 (2016). 15H. Maier-Flaig, \Temperature dependent damp- ing of yttrium iron garnet spheres { Measure- ment data and analysis programs," (2017), https://dx.doi.org/10.17605/OSF.IO/7URPT. 16S. Klingler, H. Maier-Flaig, C. Dubs, O. Surzhenko, R. Gross, H. Huebl, S. T. B. Goennenwein, and M. Weiler, Applied Physics Letters 110, 092409 (2017). 17S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and J. P. Nibarger, Jour- nal of Applied Physics 99, 093909 (2006). 18M. L. Schneider, J. M. Shaw, A. B. Kos, T. Gerrits, T. J. Silva, and R. D. McMichael, Journal of Applied Physics 102, 103909 (2007).9 50 100 150 200 250 Temperature (K)0.20.40.60.81.01.21.4¹0H0 (T) 123456 ¢f110 (MHz) 5152535 f110 res (GHz) FIG. 6. Full map of the FWHM linewidth of the (110) mode as function of temperature and eld resp. resonance frequency f110 res. At low temperatures, only the slow relaxor peak is visible while at high temperatures the Gilbert-like damping becomes dominant. The position of the peak in the linewidth modeled by a slow relaxor is shown as dashed orange line. The model parameters are taken from Clarke31and taking a= 2:50 meV. The dotted lines indicate the deviation of the model for 0 :5a (lowerTmax) and 2a(higherTmax). 19H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider, M. J. Carey, S. Maat, and J. R. Childress, Physical Review B84, 054424 (2011). 20P. R oschmann and H. D otsch, physica status solidi (b) 82, 11 (1977). 21L. R. Walker, Physical Review 105, 390 (1957). 22Due to the comparable dimensions of center conductor width and sphere diameter, we expect that the sphere ex- periences an inhomogeneous microwave magnetic eld with its main component parallel to the surface of the CPW and perpendicular to its center conductor. As the microwave magnetic eld is suciently small to not cause any non- linear eects, a mode dependent excitation eciency is the only eect of the microwave magnetic eld inhomogeneity. 23C. Kittel, Physical Review 76, 743 (1949). 24P. Bruno, Physical Review B 39, 865 (1989). 25J. Nemarich, Physical Review 136, A1657 (1964). 26The resonance frequency of the DPPH sample that has been measured simultaneously was intended as a eld cal- ibration but can not be utilized due to the magnetic eld inhomogeneity. In particular, since the homogeneity of our superconducting magnet system is specied to 1 ‰for an o-axis deviation of 2 :5 mm, the spatial separation of 5 mm of the DPPH and the YIG sphere already falsies DPPH as an independent magnetic eld standard. Placing DPPH and YIG in closer proximity is problematic as the stray eld of the YIG sphere will aect the resonance frequency of the DPPH. Note further that we are not aware of any re- ports showing the temperature independence of the DPPH g-factor with the required accuracy. 27J. F. Dillon, Physical Review 111, 1476 (1958). 28T. Kasuya and R. C. LeCraw, Physical Review Letters 67, 223 (1961). 29E. G. Spencer, R. C. Lecraw, and J. Linares, Physical Review 123, 1937 (1961). 30M. Sparks, Ferromagnetic-Relaxation Theory , edited by W. A. Nierenberg (McGraw-Hill, 1964). 31B. H. Clarke, Physical Review 139, A1944 (1965). 32B. H. Clarke, K. Tweedale, and R. W. Teale, Physical Review 139, A1933 (1965).33M. Sparks, Journal of Applied Physics 38, 1031 (1967). 34V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re- ports 229, 81 (1993). 35J. Barker and G. E. W. Bauer, Physical Review Letters 117, 217201 (2016). 36R. Orbach, Proceedings of the Physical Society 77, 821 (1961). 37B. H. Clarke, Journal of Applied Physics 36, 1211 (1965). 38I. H. Solt, Journal of Applied Physics 33, 1189 (1962). 39Note that we failed to reproduce the t of Ref. 38 using the data provided in this paper and that the reasonable agreement with the there-reported t parameters might be coincidence. 40E. E. Anderson, Physical Review 134, A1581 (1964). 41P. Hansen, Journal of Applied Physics 45, 3638 (1974). 42G. Winkler, Magnetic Garnets , Tracts in pure and applied physics; Vol. 5 (Vieweg, 1981). 43R. L. White, Journal of Applied Physics 31, S86 (1960). 44P. R oschmann and W. Tolksdorf, Materials Research Bul- letin18, 449 (1983). 45In comparison to Klingler et al.16, here, we do not see an increased inhomogeneous linewidth of the (110) mode and no secondary mode that is almost degenerate with the (110) mode. The dierence can be explained by the ori- entation of the sphere which is very dicult to reproduce very accurately ( <1°) between the experimental setups: The change in orientation either separates the mode that is almost degenerate to the (110) mode or makes the degen- eracy more perfect in our setup. The dierent placement of the sphere on the CPW can also lead to a situation where the degenerate mode is not excited and therefore does not interfere with the t. 46Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz, W. Schneider, M. Wu, H. Schultheiss, and A. Homann, Applied Physics Letters 101, 152405 (2012). 47Naturally, the resonance frequency varies slightly (0:9 GHz) between the data points because the magneti- zation and the anisotropy changes with temperature. 48J. F. Dillon and J. W. Nielsen, Physical Review Letters 3, 30 (1959).

2017-03-28

We investigate the temperature dependent microwave absorption spectrum of an yttrium iron garnet sphere as a function of temperature (5 K to 300 K) and frequency (3 GHz to 43.5 GHz). At temperatures above 100 K, the magnetic resonance linewidth increases linearly with temperature and shows a Gilbert-like linear frequency dependence. At lower temperatures, the temperature dependence of the resonance linewidth at constant external magnetic fields exhibits a characteristic peak which coincides with a non-Gilbert-like frequency dependence. The complete temperature and frequency evolution of the linewidth can be modeled by the phenomenology of slowly relaxing rare-earth impurities and either the Kasuya-LeCraw mechanism or the scattering with optical magnons. Furthermore, we extract the temperature dependence of the saturation magnetization, the magnetic anisotropy and the g-factor.

Temperature dependent magnetic damping of yttrium iron garnet spheres

1703.09444v2

This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1 Nonlinear fluid damping of elastically mounted pitching wings in quiescent water Yuanhang Zhu1y, Varghese Mathai2and Kenneth Breuer1 1Center for Fluid Mechanics, School of Engineering, Brown University, Providence, RI 02912, USA 2Department of Physics, University of Massachusetts, Amherst, MA 01003, USA (Received xx; revised xx; accepted xx) We experimentally study the nonlinear fluid damping of a rigid but elastically mounted pitching wing in the absence of a freestream flow. The dynamics of the elastic mount are simulated using a cyber-physical system. We perturb the wing and measure the fluid damping coefficient from damped oscillations over a large range of pitching frequencies, pitching amplitudes, pivot locations and sweep angles. A universal fluid damping scaling is proposed to incorporate all these parameters. Flow fields obtained using particle image velocimetry are analyzed to explain the nonlinear behaviors of the fluid damping. 1. Introduction The interaction between elastically mounted pitching wings and unsteady flows is central to many applications. With a free-stream flow, this interaction can lead to self-sustained, flow-induced oscillations, which have been studied for understanding classic aeroelastic behaviour (Dowell et al.1989; Dugundji 2008), as well as in developing oscillating foil energy harvesting devices (Xiao & Zhu 2014; Young et al.2014). Without a free stream, but with prescribed heaving or flapping (i.e. hovering), the passive flow-induced pitching motionsareusedinmodellingthethrustgenerationandmaneuveringinanimalflight(Wang 2005; Bergou et al.2007; Shinde & Arakeri 2013; Kang & Shyy 2014; Beatus & Cohen 2015). One of the critical parameters that govern the flow-structure interactions of passively pitching wings is the fluid damping. According to the semi-empirical Morison equation (Morison etal.1950),thetotalfluidforceexertedonawingsubmergedinunsteadyviscous fluid can be divided into two parts – the force associated with fluid inertia (i.e. the added mass force), which is in phase with acceleration (Brennen 1982; Corkery et al.2019), and the force induced by vortices in the flow (i.e. the fluid damping force), which is in phase with velocity (Shih & Buchanan 1971; Kang & Shyy 2014; Su & Breuer 2019). While the structuraldampingforceistypicallyproportionaltovelocitybecauseoftheconstantstructural dampingcoefficient,thefluiddampingforceisexpectedtoscalequadraticallywithvelocity (Morison et al.1950; Keulegan & Carpenter 1958), and due to this nonlinearity, the fluid damping coefficient is usually obtained empirically as a function of the reduced frequency, the Reynolds number, the oscillation amplitude, etc (Shih & Buchanan 1971). For pitching flexible wings (Alben 2008) and heaving membrane wings (Tzezana & Breuer 2019), the fluid damping coefficient is found to scale inversely with the oscillation frequency. yEmail address for correspondence: yuanhang_zhu@brown.eduarXiv:2108.02090v1 [physics.flu-dyn] 4 Aug 20212 Y. Zhu, V. Mathai and K. Breuer For elastically mounted pitching wings with a free stream, the interplay between the fluid damping and the structural damping governs the flow-induced oscillation. By mapping out the cycle-averaged energy transfer between the elastic system and the ambient fluid using prescribedkinematics,Menon&Mittal(2019)andZhu etal.(2020)showedthattheenergy injectedbythenegativefluiddampingmustbeequaltotheenergydissipatedbythepositive structural damping in order for the flow-induced oscillations to sustain. In other words, the total damping of the system must be zero (Dugundji 2008). The negative fluid damping arisesprimarilyfromtheformationandsheddingofdynamicstallvortices(McCroskey1982; Corke&Thomas2015).Intheabsenceofafreestream,however,thefluiddampingbecomes positive and counteracts the pitching motion because of the drag effect. With both the fluid damping and the structural damping being positive, any perturbations to the system will be damped out. However, little is known about how the fluid damping shapes the damped oscillations, and understanding this is of critical importance for understanding the fluid- structure interactions of elastically mounted pitching wings under external perturbations such as gusts. In the present study, we use laboratory experiments to characterise the fluid damping of elastically mounted pitching wings in quiescent water, with the elastic mount simulated usingacyber-physicalsystem(§2).Weperform‘ringdown’experimentstoextractthefluid damping(§3.1).Theeffectsofmanyparametersareinvestigated,includingtheeffectsofthe pitching frequency, the pitching amplitude, the pivot location and the sweep angle (§3.2). We propose a universal fluid damping scaling to incorporate these parameters (§3.3), and correlate the nonlinear behaviour of the fluid damping with the dynamics of the vortical structures measured using particle image velocimetry (§3.4). Finally, the key findings are summarised in §4. 2. Experimental set-up Figure 1(a) shows a schematic of the experimental set-up. We conduct all the experiments in the Brown University free-surface water tunnel (test section widthdepthlength= 08 m06 m40 m), with the flow speed kept at zero ( 𝑈1=0m/s). A NACA 0012 wing, made of clear acrylic, is mounted vertically in the tunnel, with an endplate on the top to skim surface waves and eliminate wingtip vortices at the root. The wing is connected to a six-axis force/torque transducer (ATI 9105-TIF-Delta-IP65), which measures the fluid torque𝜏𝑓exerted on the wing. This 𝜏𝑓is then fed into the cyber-physical system (CPS). Depending on the input virtual structural parameters, specifically the torsional stiffness 𝑘𝑣, damping𝑏𝑣andinertia𝐼𝑣,theCPScalculatesthepitchingpositionofthewingandoutputs the signal to the servo motor (Parker SM233AE). An optical encoder (US Digital E3-2500) which is independent of the CPS is used to measure the pitching position 𝜃. The CPS is operated at 4000 Hz to minimise any phase delay between the input 𝜏𝑓and the output 𝜃. A detailed explanation of the CPS can be found in Zhu et al.(2020). Weusetwo-dimensionalparticleimagevelocimetry(PIV)tomeasuretheflowfieldaround thewing.Theflowisseededusing50 𝜇mdiameterhollowceramicspheresandilluminated byalasersheetatthemid-spanplane.Thelasersheetisgeneratedbyadouble-pulseNd:YAG laser(532nm,QuantelEverGreen)withLaVisionsheetoptics.Thetransparentwingenables flow field measurements on both sides of the wing. Due to the limitation of space beneath the tunnel, a 45mirror is used to reflect the images into two co-planar sCMOS cameras (LaVision).WeusetheDaVissoftware(LaVision)tocalculate(twopassesat 6464pixels, two passes at 3232pixels, both with 50% overlap) and stitch the velocity fields from the two cameras to form a field of view of 32𝑐32𝑐, where𝑐is the chord length of the wing. Figure 1(b) sketches the two types of wings we use in the present study. For the unsweptNonlinear fluid damping of pitching wings 3 Servo motor & gearbox Force transducer Laser sheet @ mid-spanEndplate 2×sCMOS with 35mm lensMirrorOptical encoder U = 0 m/sNACA 0012 (transparent)CPS Unswept wing Λx/c0 10.5 Swept wing(a) (b) cs csLE LE TE TE k b Iv v v ∞ F/i.pc/g.pc/u.pc/r.pc/e.pc 1. (a) A schematic of the experimental set-up. The structural dynamics of the wing is simulated by a cyber-physical system (CPS). ( b) Sketches of unswept and swept wings. The leading edge (LE) and the trailing edge (TE) are parallel. Dashed lines represent the pivot axis. wing, a wing holder mechanism (not shown) enables the pivot axis to be adjusted between 𝑥𝑐=0and 1 with a step size of 0.125. For the swept wings, the sweep angle Λis defined astheanglebetweentheleadingedgeandtheverticalaxis.Foursweptwingswith Λ=10, 15,20and25areused.Asshowninthefigure,thepivotaxisofsweptwingsisavertical line passing through the mid-chord point ( 𝑥𝑐=05) of the mid-span plane. All the wings haveaspanof 𝑠=03mandachordlengthof 𝑐=01m,whichresultsinanaspectratioof 𝐴𝑅=3. The governing equation of the system is 𝐼¥𝜃¸𝑏¤𝜃¸𝑘𝜃=𝜏𝑓 (2.1) where𝜃,¤𝜃,and¥𝜃aretheangularposition,velocityandacceleration,respectively. 𝐼,𝑏and𝑘 aretheeffectiveinertia,dampingandstiffnessofthesystem.Theeffectiveinertia 𝐼isthesum of the virtual inertia 𝐼𝑣, which we prescribe with the CPS, and the physical inertia 𝐼𝑝of the wing (i.e.𝐼=𝐼𝑣¸𝐼𝑝). The effective damping 𝑏equals the virtual damping 𝑏𝑣(i.e.𝑏=𝑏𝑣) because the friction in the system is negligible. The effective stiffness 𝑘equals the virtual stiffness(i.e. 𝑘=𝑘𝑣).𝜏𝑓isthenonlinearfluidtorqueexperiencedbythewing,whichcanbe divided into the added mass torque, 𝜏𝑎=𝐼𝑎¥𝜃, where𝐼𝑎is the added fluid inertia, and the fluid damping torque, for simplicity 𝜏𝑏=𝑏𝑓¤𝜃, where𝑏𝑓is the fluid damping coefficient (see §1). Note that 𝑏𝑓is expected to be a function of ¤𝜃(Mathaiet al.2019). Equation 2.1 can thus be rearranged as ¹𝐼¸𝐼𝑎º¥𝜃¸¹𝑏¸𝑏𝑓º¤𝜃¸𝑘𝜃=0 (2.2) After a perturbation of amplitude 𝐴0is applied at time 𝑡0, the damped oscillations of the system can be described as 𝜃=𝐴0𝑒𝛾¹𝑡𝑡0ºcos»2𝜋𝑓𝑝¹𝑡𝑡0º¼ (2.3) where 𝛾=𝑏¸𝑏𝑓 2¹𝐼¸𝐼𝑎ºand𝑓𝑝=1 2𝜋√︂ 𝑘 𝐼¸𝐼𝑎𝛾2¹24abº4 Y. Zhu, V. Mathai and K. Breuer 0 50 100 150-2-1012 0 0.5 1 1.5 2 2.5024610-3 51 5501 0 1 20.380.400.42(a) (b) n+1 n F/i.pc/g.pc/u.pc/r.pc/e.pc 2. (a) System response and amplitude decay in a typical ‘ring down’ test, where an elastically mounted unswept wing ( Λ=0) pivots around the mid-chord ( 𝑥𝑐=05) at a frequency of 𝑓𝑝=040Hz. The inset shows the measurements of the pitching amplitude 𝐴𝑛and the pitching frequency 𝑓𝑝of the𝑛-th peak. The fluid damping 𝑏𝑓at𝐴𝑛is extracted by fitting an exponential curve (i.e. the red solid line) to the adjacent three peaks. ( b) Extracted𝑏𝑓in air and in water. The zero value is indicated by the black dashed line. The inset compares the measured pitching frequency 𝑓𝑝in air and in water. 3. Results and Discussion 3.1.Extracting the fluid damping from ‘ring down’ experiments Weconduct‘ringdown’experimentstomeasurethefluiddampingexperiencedbyelastically mountedpitchingwings.Inthe‘ringdown’experiment,ashort-timeconstant-torqueimpulse isappliedtotheCPSastheperturbation,afterwhichthesystemresponseandtheamplitude decay of the wing is recorded and analysed. Figure 2( a) shows the results from a typical ‘ringdown’experiment.Inthisspecificcase,weuseanunsweptwing( Λ=0)whichpivots around the mid-chord ( 𝑥𝑐=05) at a frequency of 𝑓𝑝=040Hz. We conduct the ‘ring down’experimenttwice–onceinairandonceinwater.Thepitchingamplitudeofthewing decays faster in water than in air, indicating a higher total damping in water. Toquantifythisamplitudedecay,thepositivepeaksofthesystemresponseareidentified. Asshownintheinset,theamplitudeofthe 𝑛-thpeakisdenotedby 𝐴𝑛,andthecorresponding pitchingfrequencyismeasuredas 𝑓𝑝=2¹𝑡𝑛¸1𝑡𝑛1º.Tomeasurethetotaldamping 𝑏¸𝑏𝑓 at amplitude 𝐴𝑛, we fit an exponential, 𝑦=𝛼𝑒𝛾𝑡, to the three adjacent peaks, 𝑛1,𝑛and 𝑛¸1,andextractthecorresponding 𝛾(seeequation2.3).Nowtheonlyunknowninequation 2.4bistheaddedmass, 𝐼𝑎.Afterobtaining 𝐼𝑎,thefluiddamping, 𝑏𝑓,isthencalculatedusing equation 2.4 a(Rao 1995). Since 𝑓𝑝and𝛾are both measured, 𝐼𝑎and𝑏𝑓are alsomeasured quantities. Moreover, both 𝐼𝑎and𝑏𝑓arecycle-averaged , meaning they cannot reflect the instantaneousvariationofthefluidinertiaanddamping.Themeasuredfluiddamping, 𝑏𝑓,in bothairandwaterarecomparedinfigure2( b).Since𝜏𝑓inequation2.1isnegligibleinairas comparedtootherforcesintheequation, 𝑏𝑓staysnearzero,whichisindicatedbythegood agreementbetweentheredcirclesandtheblackdashedline.Asshownbythegreensquares, 𝑏𝑓in water is significant because of the existence of the fluid damping torque, 𝜏𝑏. It is also observed that 𝑏𝑓in water increases non-monotonically with 𝐴. This nonlinear behaviour willberevisitedlaterin§3.4.Theinsetoffigure2( b)showsthemeasuredpitchingfrequency, 𝑓𝑝, in both air and water. Due to the combined effect of the fluid inertia and damping, we see that𝑓𝑝is slightly lower in water than in air.Nonlinear fluid damping of pitching wings 5 0 0.5 1 1.5 2 2.5051015 0 0.5 1 1.5 2 2.504812(a) (b) 10-310-3 F/i.pc/g.pc/u.pc/r.pc/e.pc 3. (a) Extracted fluid damping 𝑏𝑓at different pitching frequencies for 𝑥𝑐=05. (b) A frequency scaling for the fluid damping which collapses 𝑏𝑓at different𝑓𝑝into one curve. Note that ( a) and (b) share the same legend. 3.2.Frequency scaling of the fluid damping We repeat the ‘ring down’ experiment for the unswept wing ( Λ = 0) pivoting at the mid- chord,𝑥𝑐=05, and change the pitching frequency by tuning the virtual inertia, 𝐼𝑣, and the virtual stiffness, 𝑘𝑣, while keeping the virtual damping 𝑏𝑣constant (Onoue & Breuer 2016, 2017). The extracted fluid damping, 𝑏𝑓, are shown in figure 3( a). Note that figure 3(a) and (b) share the same legend. We observe that 𝑏𝑓increases monotonically with the pitching frequency, 𝑓𝑝, and that the trend of 𝑏𝑓remains consistent for all frequencies. This observation agrees with those observed in heaving rigid plates (Keulegan & Carpenter 1958; Shih & Buchanan 1971), where the fluid damping coefficient scales inversely with the oscillation period. As we discussed earlier, 𝑏𝑓derives from the fluid damping torque 𝜏𝑏, which depends strongly on the vortex-induced forces on the wing (Kang & Shyy 2014). Onoue & Breuer (2016, 2017) have shown that the circulation of LEVs scales with the strength of the feeding shear-layer velocity. In our case without a free-stream flow, the feedingshear-layer velocityequals theleading-/trailing-edgevelocity, whichisproportional to𝑓𝑝. Based on this, we divide 𝑏𝑓by𝑓𝑝(figure 3b). It is seen that with this scaling, all of the fluid damping curves collapse nicely. We extend this frequency scaling to unswept wings with different pivot axes (figure 4 a) and to swept wings with different sweep angles (figure 4 b). For comparison, we include the previous results (figure 3 b) using purple circles in both figure 4( a) and (b). Note that each symbol shape in figure 4 contains fivedifferent pitching frequencies, 𝑓𝑝=020, 0.28, 0.40, 0.56 and 0.78 Hz. For the unswept wing ( Λ = 0), we change the pivot axis from 𝑥𝑐=0to 1 with a step sizeof0.125(seetheinsetoffigure4 a).Weobservethat 𝑏𝑓𝑓𝑝increasesasthepivotaxisis moved away from the mid-chord, 𝑥𝑐=05. For pivot axes that are symmetric with respect to the mid-chord (i.e. 𝑥𝑐=0375& 0.625, 0.25 & 0.75, 0.125 & 0.875 and 0 & 1), 𝑏𝑓𝑓𝑝 roughlyoverlap.Theslightinconsistencybetween 𝑏𝑓𝑓𝑝for𝑥𝑐¡05and𝑥𝑐05comes fromtheasymmetryoftheNACA0012winggeometrywithrespecttothemid-chord;wesee that the scaled damping, 𝑏𝑓𝑓𝑝, is always slightly higher for 𝑥𝑐 05. In these cases, the damping at the trailing edge dominates due to the higher velocity and longer moment arm, and is stronger than the cases when 𝑥𝑐 ¡05, where the leading-edge damping dominates. We will show in §3.4 that this is due to differences in the vortex structures generated by the sharp and rounded geometries. Thisfrequencyscaling, 𝑏𝑓𝑓𝑝,alsoholdsforthree-dimensional(3D)sweptwings(figure6 Y. Zhu, V. Mathai and K. Breuer 0 0.5 1 1.5 2 2.500.030.060.09 0 0.5 1 1.5 2 2.500.050.100.15 0 0.5 1(a) ( b) NACA 0012 ΛPivot axis F/i.pc/g.pc/u.pc/r.pc/e.pc 4. (a)𝑏𝑓𝑓𝑝for an unswept wing ( Λ=0) pivoting at 𝑥𝑐=0to 1 with a step size of 0.125. The pivotlocationforeachdatasetisshownbytheinset.( b)𝑏𝑓𝑓𝑝forsweptwingswith Λ=0,10,15,20 and25.Theinsetshowssideviewsofthefivesweptwingsandthedashedlineindicatesthepivotaxis.The colours of the wings correspond to the colours of 𝑏𝑓𝑓𝑝curves in the figure. The purple circles in ( a) and (b) are replotted from figure 3( b). Note that each dataset in ( a) and (b) includes fivedifferent𝑓𝑝. 4b). Again, each curve includes data from five pitching frequencies. Here, the pivot axes of swept wings are kept as a vertical line passing through the mid-chord of the mid-span plane (see the inset of figure 4 b). AsΛincreases, the average pivot axes of the top and the bottom portion of the swept wing move away from the mid-chord, leading to the increase of the scaled damping, 𝑏𝑓𝑓𝑝, in a manner similar to that observed for unswept wings with different pivot locations (figure 4 a). This argument will be revisited in the next section. 3.3.Universal fluid damping scaling for unswept and swept wings Figure 4( a) indicates that the pivot axis plays an important role in determining the fluid damping of unswept wings. We extend the frequency scaling of 𝑏𝑓to take into account this effect. First, we divide the wing into two parts, the fore part from LE to the pivot axis with a chord length of 𝑐𝐿𝐸, and the aft part from the pivot axis to TE with a chord length of𝑐𝑇𝐸(see the inset of figure 5 for an example when the wing pivots at 𝑥𝑐=05). The Morisonequation(Morison etal.1950)indicatesthatthefluiddampingforce 𝐹scaleswith 05𝜌𝑈2𝑠𝑐, where𝜌is the fluid density, 𝑈¤𝜃𝑐is the characteristic velocity and 𝑠𝑐is the wingarea.Wecanexpressthetotalfluiddampingtorqueasthesumofthetorqueexertedon the fore and aft portions of the wing, 𝜏𝑏𝐾𝐿𝐸𝐹𝐿𝐸𝑐𝐿𝐸¸𝐾𝑇𝐸𝐹𝑇𝐸𝑐𝑇𝐸 (3.1) where the subscripts 𝐿𝐸and𝑇𝐸refer to the leading- and trailing-edge contributions, and 𝐾𝐿𝐸and𝐾𝑇𝐸are empirical factors that account for the subtle differences in the damping associated with the specific geometries of the leading and trailing edges (figure 4 a). Since the differences are small, 𝐾𝐿𝐸and𝐾𝑇𝐸should be close to one, and for consistency, their average value must equal one ( ¹𝐾𝐿𝐸¸𝐾𝑇𝐸º2=1). Since the damped oscillations are observed to be near-sinusoidal (figure 2 a), the average angular velocity is given by 4𝑓𝑝𝐴. Simplifying, we arrive at an expression for the fluid damping: 𝑏𝑓2𝜌𝑓𝑝𝐴𝑠¹𝐾𝐿𝐸𝑐4 𝐿𝐸¸𝐾𝑇𝐸𝑐4 𝑇𝐸º (3.2)Nonlinear fluid damping of pitching wings 7 0 0.5 1 1.5 2 2.50123 cLE cTE s cLE,botcTE,bots/2 s/2 ΛcLE,topcTE,topFLE FTE F/i.pc/g.pc/u.pc/r.pc/e.pc 5. Non-dimensional fluid damping coefficient 𝐵 𝑓versus pitching amplitude 𝐴for unswept wings pivotingat𝑥𝑐=0to1andsweptwingswithsweepangles Λ=0to25.Theinsetshowsthedefinitionof the leading-edge chord 𝑐𝐿𝐸and the trailing-edge chord 𝑐𝑇𝐸, with black dashed lines indicating the pivot axes. The black dotted line indicates the small amplitude prediction for a drag coefficient of 𝐶𝐷=28. or, in non-dimensional form, 𝐵 𝑓𝑏𝑓 2𝜌𝑓𝑝𝑠¹𝐾𝐿𝐸𝑐4 𝐿𝐸¸𝐾𝑇𝐸𝑐4 𝑇𝐸º/𝐴 (3.3) Forsweptwings,becausethepivotaxispassesthrough 𝑥𝑐=05atthemid-span,thetop half of the wing has an average pivot axis 𝑥𝑐 ¡05, while the bottom half has an average pivot axis𝑥𝑐 05. Ignoring three-dimensional effects, we approximate the swept wing by two ‘equivalent’ unswept wing segments. We choose not to divide the wing into a large numberofnarrow‘bladeelements’(Glauert1983),becausethepivotaxisofsomeelements near the wing root/tip for large sweep angles may lie outside the range 𝑥𝑐=»01¼, where our scaling has not been tested. The inset of figure 5 shows how these two unswept wing segmentsareconfigured(rectangleswithreddottedlines).Basedonthewinggeometry,we see that 𝑐𝐿𝐸𝑡𝑜𝑝=𝑐𝑇𝐸𝑏𝑜𝑡=𝑐 2¸𝑠 4tanΛ 𝑐𝑇𝐸𝑡𝑜𝑝=𝑐𝐿𝐸𝑏𝑜𝑡=𝑐 2𝑠 4tanΛ(3.4) Following the same analysis as for the unswept wing, and adding the fluid damping of the topandthebottomwingsegmentstogether,wefindthatthefluiddampingforthefullswept wing is given by 𝑏𝑓𝜌𝑓𝑝𝐴𝑠¹𝐾𝐿𝐸𝑐4 𝐿𝐸𝑡𝑜𝑝¸𝐾𝑇𝐸𝑐4 𝑇𝐸𝑡𝑜𝑝¸𝐾𝐿𝐸𝑐4 𝐿𝐸𝑏𝑜𝑡¸𝐾𝑇𝐸𝑐4 𝑇𝐸𝑏𝑜𝑡º(3.5) If we define an effective leading-edge chord 𝑐𝐿𝐸=𝑐𝐿𝐸𝑡𝑜𝑝=𝑐𝑇𝐸𝑏𝑜𝑡and an effective trailing-edgechord 𝑐𝑇𝐸=𝑐𝑇𝐸𝑡𝑜𝑝=𝑐𝐿𝐸𝑏𝑜𝑡,thisscalingreducestoequation3.2with 𝐾𝐿𝐸 and𝐾𝑇𝐸cancelled out. This cancellation results because the effective pivot axes of the top and the bottom segments are symmetric about 𝑥𝑐=05at the mid-span, which averages out the slight differences in fluid damping experienced by the top and the bottom segments. For the same reason, 𝐾𝐿𝐸and𝐾𝑇𝐸also cancel out in equation 3.3 for swept wings.8 Y. Zhu, V. Mathai and K. Breuer (a) -1 0 1 (b) (c) (d) (e) -1 0 1 -1 0 1 (f) -1 0 1 (g) -1 0 1 (h) -1 0 1 -50-25025 50 LEVTEV LEVTEVLEVTEVLEVTEV LEVTEV LEV TEV LEVTEVLEV TEV= 0.52 = 1.05 = 1.57 = 2.09 F/i.pc/g.pc/u.pc/r.pc/e.pc 6. PIV flow field measurements for an unswept wing undergoing prescribed sinusoidal pitching motions in quiescent water. ( a–d) Pivot axis (shown by green dots) 𝑥𝑐=05, pitching frequency 𝑓𝑝=05 Hz, pitching amplitude 𝐴=052¹30º105¹60º157¹90ºand209¹120º. (e–h) Same as ( a–d), except that the pivot axis is at 𝑥𝑐=025. All the velocity fields are phase-averaged over 20 cycles. Only everyfifthvelocityvectorisshown.Spanwisevorticity 𝜔:positive(red),counterclockwise;negative(blue), clockwise. See supplementary materials for the full video. Figure 5 shows the non-dimensional fluid damping, 𝐵 𝑓, as a function of the pitching amplitude,𝐴,forunsweptandsweptwings.Here,wehaveused 𝐾𝐿𝐸=095and𝐾𝑇𝐸=105. We see that all of our measurements collapse remarkably well under the proposed scaling, especially for 𝐴 157¹90º, despite the wide range of pitching frequencies ( 𝑓𝑝=020 to 0.78 Hz), pivot axes ( 𝑥𝑐=0to 1) and sweep angles ( Λ = 0to25) tested in the experiments.Inthesmall-amplitudelimit( 𝐴05),𝐵 𝑓scaleslinearlywith 𝐴,withaslope that corresponds to the drag coefficient, 𝐶𝐷. We note that 𝐶𝐷28, which is comparable to that of an accelerated normal flat plate (Ringuette et al.2007). At higher pitching angles (𝐴¡05),however,thelinearapproximationnolongerholdsandweseeadecreasingslope of𝐵 𝑓as a function of 𝐴. This is presumably because the shed vortices no longer follow the rotating wing and the fluid force becomes non-perpendicular to the wing surface as 𝐴 increases.For 𝐴¡157¹90º,thescalingworksreasonablywellexceptforthecase Λ=0, 𝑥𝑐=05,whereadecreasing 𝐵 𝑓isobserved.Inthenextsection,wewilluseinsightsfrom the velocity fields to explain this non-monotonic behaviour. 3.4.Insights obtained from velocity fields To gain more insight regarding the nonlinear behaviour of 𝐵 𝑓, we conduct 2D PIV experiments to measure the surrounding flow fields of an unswept wing ( Λ = 0) with a prescribed pitching motion: 𝜃=𝐴sin¹2𝜋𝑓𝑝𝑡º. The results are shown in figure 6. The pitching frequency is kept at 𝑓𝑝=05Hz for all the cases and the pitching amplitude is variedfrom𝐴=052¹30ºto209¹120ºwithastepsizeof 052¹30º.Twopivotaxesare tested,𝑥𝑐=05(figure6a–d)and𝑥𝑐=025(figure6e–h).Notethattheflowfieldsshown in figure 6 are notsequential. Instead, all the snapshots are taken right before 𝑡𝑇=025 for different pitching amplitudes, where 𝑇is the pitching period. This specific time instant is chosen because it best reflects the difference in dynamics associated with the different pitching amplitudes and pivot axes. For both pivot locations (figure 6 a–d:𝑥𝑐=05ande–h:𝑥𝑐=025), the spanwise vorticity of the pitch-generated leading-edge vortex (LEV) and trailing-edge vortex (TEV)Nonlinear fluid damping of pitching wings 9 increases with the pitching amplitude, 𝐴. This can be explained by the increase in the feeding shear-layer velocities associated with the higher pitching amplitudes (Onoue & Breuer 2016). The boundary vortices near the wing surface, which are related to the added mass effect (Corkery et al.2019), also become more prominent due to the increase of the angular acceleration. When the wing pivots at 𝑥𝑐=05(figure 6a–d), the leading-edge velocityequalsthetrailing-edgevelocity.Asaresult,theLEVandTEVarefairlysymmetric about the pivot axis, with some subtle differences caused by the rounded and sharp edges, respectively. This confirms the arguments given earlier for the differences between 𝑏𝑓𝑓𝑝 for𝑥𝑐 ¡05and𝑥𝑐 05(figure 4a). For𝑥𝑐=025(figure 6e–h), however, the TEV is muchmoreprominentthantheLEVbecauseofthehighertrailing-edgevelocity.Duetothe low leading-edge velocity and the pitch-induced rotational flow, the sign of the LEV even reverses and becomes negative for 𝐴=105to209(figure 6f–h). Forbothpivotlocations,duetotheabsenceofaconvectivefreestream,andtheexistence of the pitch-induced rotational flow, the LEV and TEV (only the TEV for 𝑥𝑐=025) are entrainedclosertothewingsurfaceas 𝐴increases.For 𝑥𝑐=05,asshowninfigure6( c–d), the LEV moves towards the aft portion of the wing and the TEV moves towards the fore portionofthewingwhen 𝐴>157¹90º.Thetorquegeneratedbythesetwovortices,which counteracts the wing rotation for small 𝐴, now assists the rotation as the wing pitches up towardshigherangularpositions.Thisassistreducesthefluiddragexperiencedbythewing andthuslowersthefluiddamping.Thiseffectcanaccountforthenon-monotonicbehaviour of𝐵 𝑓for𝑥𝑐=05(figure 5). For 𝑥𝑐=025(figure 6g–h), a similar scenario is observed, in which the TEV moves towards the fore portion of the wing and gets closer to the wing surfaceas𝐴increases.However,becauseoftheexistenceofacounter-rotatingLEV,theTEV isnotabletoapproachthewingsurfaceascloselyasinthe 𝑥𝑐=05case.Thisexplainswhy a flattening behaviour, rather than a non-monotonic trend of 𝐵 𝑓, is observed for 𝑥𝑐=025 and presumably for other pivot locations at high pitching amplitudes. 4. Conclusions By utilising a cyber-physical control system to create an elastically mounted pitching wing, wehaveexperimentallymeasuredthenonlinearfluiddampingassociatedwithvorticesshed from a bluff body. A theoretical scaling has been proposed and validated, based on the Morison equation, which incorporates the frequency, amplitude, pivot location and sweep angle. The nonlinear behaviour of the scaled fluid damping has been correlated with the velocity fields measured using particle image velocimetry. One should note that our scaling may not be applicable for instantaneous fluid damping, becausethedampingcharacterisedinthepresentstudyiscycle-averagedovernear-sinusoidal oscillations.Inaddition,wehavenotconsideredthree-dimensionaleffects,whicharepresent due to the wing tip flows. Incorporating these may further improve the collapse of the fluid damping coefficient, 𝐵 𝑓(figure 5).Lastly, in §3.4, onlyqualitative analysis ofthe flow field hasbeenperformedthusfar.Inordertogetmoreaccuratecorrespondencebetweenthefluid dampingandtheflowdynamics,quantitativeanalysisofthevortextrajectoryandcirculation is needed, which will be the focus of future study. Despitetheselimitations,theproposedscalinghasbeenshowntocollapsethedataovera widerangeofoperatingconditions( 𝑓𝑝=020to0.78Hzand 𝐴=0to2.5)forbothunswept (𝑥𝑐=0to1)andsweptwings( Λ=0to25).Itcanbeusedtopredictdampingassociated withshedvortices,andthusbenefitthefuturemodellingofawidevarietyofflows,including unswept and swept wings in unsteady flows as well as other bluff body geometries. The universality of this scaling reinforces the underlying connection between swept wings and10 Y. Zhu, V. Mathai and K. Breuer unswept wings with different pivot locations. In addition, the results presented in this study will be of potential value as a source of experimental data for validation and comparison of future theoretical/computational models. Acknowledgments This work is funded by Air Force Office of Scientific Research, Grant FA9550-18-1-0322, managed by Dr. Gregg Abate. REFERENCES A/l.pc/b.pc/e.pc/n.pc, S. 2008 Optimal flexibility of a flapping appendage in an inviscid fluid. J. 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K. & W/i.pc/l.pc/d.pc/e.pc/m.pc/a.pc/n.pc, S. 2019Dynamicsofheavyandbuoyant underwater pendulums. J. Fluid Mech. 862, 348–363. M/c.pcC/r.pc/o.pc/s.pc/k.pc/e.pc/y.pc, W. J. 1982 Unsteady airfoils. Annu. Rev. Fluid Mech. 14(1), 285–311. M/e.pc/n.pc/o.pc/n.pc, K. & M/i.pc/t.pc/t.pc/a.pc/l.pc, R. 2019 Flow physics and dynamics of flow-induced pitch oscillations of an airfoil. J. Fluid Mech. 877, 582–613. M/o.pc/r.pc/i.pc/s.pc/o.pc/n.pc, J. R., J/o.pc/h.pc/n.pc/s.pc/o.pc/n.pc, J. W. & S/c.pc/h.pc/a.pc/a.pc/f.pc, S. A. 1950 The force exerted by surface waves on piles. J. Pet. Technol.2(05), 149–154. O/n.pc/o.pc/u.pc/e.pc, K. & B/r.pc/e.pc/u.pc/e.pc/r.pc, K. S. 2016 Vortex formation and shedding from a cyber-physical pitching plate. J. Fluid Mech. 793, 229–247. O/n.pc/o.pc/u.pc/e.pc, K. & B/r.pc/e.pc/u.pc/e.pc/r.pc, K. S. 2017 A scaling for vortex formation on swept and unswept pitching wings. J. Fluid Mech. 832, 697–720. R/a.pc/o.pc, S. S. 1995Mechanical Vibrations . Addison-Wesley. R/i.pc/n.pc/g.pc/u.pc/e.pc/t.pc/t.pc/e.pc, M. J., M/i.pc/l.pc/a.pc/n.pc/o.pc, M. & G/h.pc/a.pc/r.pc/i.pc/b.pc, M. 2007 Role of the tip vortex in the force generation of low-aspect-ratio normal flat plates. J. Fluid Mech. 581, 453–468. S/h.pc/i.pc/h.pc, C. C. & B/u.pc/c.pc/h.pc/a.pc/n.pc/a.pc/n.pc, H. J. 1971Thedragonoscillatingflatplatesinliquidsatlowreynoldsnumbers. J. Fluid Mech. 48(2), 229–239. S/h.pc/i.pc/n.pc/d.pc/e.pc, S. Y. & A/r.pc/a.pc/k.pc/e.pc/r.pc/i.pc, J. H. 2013 Jet meandering by a foil pitching in quiescent fluid. Phys. Fluids 25(4), 041701. S/u.pc, Y & B/r.pc/e.pc/u.pc/e.pc/r.pc, K. S. 2019 Resonant response and optimal energy harvesting of an elastically mounted pitching and heaving hydrofoil. Phys. Rev. Fluids 4(6), 064701. T/z.pc/e.pc/z.pc/a.pc/n.pc/a.pc, G. A. & B/r.pc/e.pc/u.pc/e.pc/r.pc, K. S. 2019 Thrust, drag and wake structure in flapping compliant membrane wings.J. Fluid Mech. 862, 871–888. W/a.pc/n.pc/g.pc, Z. J. 2005 Dissecting insect flight. Annu. Rev. Fluid Mech. 37, 183–210.Nonlinear fluid damping of pitching wings 11 X/i.pc/a.pc/o.pc, Q. & Z/h.pc/u.pc, Q. 2014 A review on flow energy harvesters based on flapping foils. J. Fluids Struct. 46, 174–191. Y/o.pc/u.pc/n.pc/g.pc, J., L/a.pc/i.pc, J. C. S. & P/l.pc/a.pc/t.pc/z.pc/e.pc/r.pc, M. F. 2014 A review of progress and challenges in flapping foil power generation. Prog. Aerosp. Sci. 67, 2–28. Z/h.pc/u.pc, Y., S/u.pc, Y. & B/r.pc/e.pc/u.pc/e.pc/r.pc, K. S. 2020Nonlinearflow-inducedinstabilityofanelasticallymountedpitching wing.J. Fluid Mech. 899, A35.

2021-08-04

We experimentally study the nonlinear fluid damping of a rigid but elastically mounted pitching wing in the absence of a freestream flow. The dynamics of the elastic mount are simulated using a cyber-physical system. We perturb the wing and measure the fluid damping coefficient from damped oscillations over a large range of pitching frequencies, pitching amplitudes, pivot locations and sweep angles. A universal fluid damping scaling is proposed to incorporate all these parameters. Flow fields obtained using particle image velocimetry are analyzed to explain the nonlinear behaviors of the fluid damping.

Nonlinear fluid damping of elastically mounted pitching wings in quiescent water

2108.02090v1

1 Study of spin dynamics and damping on the magnetic nanowire arrays with various nanowire widths Jaehun Cho a, Yuya Fujii b, Katsunori Konioshi b, Jungbum Yoon c, Nam -Hui Kim a, Jinyong Jung a, Shinji Miwa b, Myung -Hwa Jung d, Yoshishige Suzuki b, and Chun -Yeol You a,* a Department of Physics, Inha University , Inch eon, 402-751, South Korea b Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560 -8531, Japan c Department of Electrical and Computer Engineering , National University of Singapore , Singapore 117576 d Department of Physics, Sogang University, Seoul, 121 -742, South Korea Abstract We investigate the spin dynamics including Gilbert damping in the ferromagnetic nanowire array s. We have measured the ferromagnetic resonance of ferromagnetic nanowire arrays using vector -network analyzer ferromagnetic resonance (VNA -FMR) and analyzed the results with the micromagnetic si mulations . We find excellent agree ment between the experimental VNA -FMR spectra and micromagnetic simulations result for various applied magnetic fields . We find that the demagnetization factor for longitudinal conditions, Nz (Ny) increases (decreas es) as decreasing the nanowire width in the micromagnetic simulation s. For the transverse magnetic field , Nz (Ny) increas es (decreas es) as increasing the nanowire width . We also find that t he Gilbert damping constant increases from 0.018 to 0.051 as the incr easing nanowire width for the transverse case , while it is almost constant as 0.021 for the longitudinal case . 2 * Corresponding author. FAX: +82 32 872 7562. E-mail address: cyyou@inha.ac.kr Keywords : Nanowires , Ferromagnetic Resonance , Micromagnetic Simulations , Gilbert damping 3 Ferromagnetic nanostructures have recently attracted much interest for the wide potential applications in high density spintronic information storage , logic devices and various spin orbit torque phenomena .1,2,3,4,5 It is well known that the detail spin dynamics of nanostructure is far from the one of the bulk’s because of many reasons, different boundary conditions, changes of the magnetic properties including the saturat ion magnetization, anisotropy energy, and exchange stiffness constant, etc. Since the magnetic properties are usually sensitive functions of the sample fabrication conditions, it has been widely accepted that the detail sample fabrications are also importa nt in the study of spin dynamics. However, the relatively less caution has been made for the boundary conditions of the spin dynamics in the nanostructure. In the spin transfer torque magnetic random access memory (STT -MRAM), the magnetic damping constant is important because the switching current density is proportional to the damping constant .6 In the nanowire, damping constant also plays crucial role in the spin dynamics including domain wall motion with magnetic field7 and spin transfer torque .8 Furthermore, it is the most important material parameter in spin wave (SW) dynamics .9 Despite of the importance of the damping constant, many studies about spin dynamics in ferromagnetic nanowires have not taken into account the damping constant properly .10,11,12 Only a few studies paid attention to the magnetic damping in the nanowires spin 4 dynamics .13,14 In this study, arrays of CoFeB nanowire s are prepared by e -beam lithography , and they are covered coplanar wave -guide for the ferromagnetic resonance (FMR) measurement as shown in Fig. 1 . We measured FMR signal with longitudinal (wire direction) and transverse magnetic field s in order to investigate the spin dynamics with different boundary conditions. Also w e extract Gilbert damping constant using micromagnetic simulations with the different applied magnetic field directions in various nanowire arrays . We find the damping constant decreas es with increasing the nanowire width for the transverse magnetic field with constant input damping consta nt in micromagnetic simulations, while we obtain almost constant damping constant for the longitudinal field. The film s were prepared using DC magnetron sputtering. The stack s consist of Ta (5 nm)/Co 16Fe64B20 (30 nm)/Ta (5 nm) on single crystal MgO (001) substrate s. The film s are patterned as 100 -nm-width wire array s with 200 -nm-space each wires using e -beam lithography and an Ar ion milling technique as shown in Fig. 1. The width is determined with a scanning electron microscope (SEM). These nanowire arrays are covered by coplanar wave guide in order to characterized with the Vector Network Analyzer ( VNA )-FMR technique described elsewhere .15 We prepare nanowire arrays as shown in Fig. 1 , and external DC magnetic field direction for FMR measurement is also depicted. We use VNA -FMR spectra to measure imaginary parts of the susceptibility of the samples.16 5 The measured imaginary parts of the susceptibility raw data are calibrated with the careful calibration procedures .16 The calibrated imaginary parts of the susceptibility are shown in Fig 2(a) and (b) for an applied magnetic field at 0.194 T for the nanowire arrays . The un - patterned thin film is also examined for the reference. We find two resonance frequencies, 17.2 and 26 .4 GHz, as shown in Fig. 2(a) for the nanowire array, while there is only one peak at 16.8 GHz for the un-patterned thin film as shown in Fig 2(b). We believe that the smaller peak (17.2 GHz) in Fig. 2(a) is originated from the un-patterned part of the nanowire array, because the frequency is closed to the un -patterned thin film’s peak (16.8 GHz). Probably, the un-patterned part of the nanowire is formed due to poor e -beam lithography processes. On the other hand, t he resonance f requency near 26.0 GHz is calculated from micromagnetic simulation at an applied magnetic field at 0.200 T , as shown in Fig. 2 (c). We clarif y the source of the main peak (26.4 GHz) is nanowire arrays by using micromagnetic simulation. These two peaks name d as the uniform FMR mode (smaller peak position) and nanowire mode (higher peak position). In order to determine the saturation magnetization, the resonance frequencies are measured as a function of the applied magnetic field, and the results are fitted with the Kittel ’s equation .17 This equation employs the corresponding demagnetization factors of Nx = 0, Ny = 0 and Nz = 1 for un -patterned film, when applied magnetic field H is x- direction with following equations , 6 2y x s z x s f H N N M H N N M . (1) Here, is the gyromagnetic ratio, H is the applied magnetic field, Ms is saturated magnetization, Nx, Ny, and Nz are the demagnetization factor s applying the cyclic permutation for the applied magnetic field direction . The micromagnetic simulations are performed by using the Objective -Oriented - MicroMagnetic Framework (OOMMF)18 with 2-dimensional periodic boundary condition (PBC ).19 We select a square slat of 100 nm × 100 nm × 30 nm nanowire separated 200 nm in y- direction with a cell size of 5 nm × 5 nm × 30 nm. The material parameters of CoFeB used in our simulation are summarized as follows: Ms = 15.79 × 105 A/m, the exchange stiffness 1.5 × 1011 J/m, the gyromagnetic ratio 2.32 × 1011 m/(A ∙s) and we ignore the magneto - crystalline anisotropy. In this simulation, the Gilbert damping constant of 0.0 27 is fixed. The saturation magnetization and Gilbert damping constant are determined by using VNA -FMR measurement for un -patterned thin film . For t he exchange stiffness constant, experimentally determined values are range of 0.98 to 2.84 × 1011 J/m which value has dependence on the fabrication processes20 and composition of ferromagnetic materials ,21 while we have picked 1.5 × 1011 J/m as the exchange stiffness constant . The determination method of Gilbert damping constant will be described later. 7 In order to mimic FMR experiments in the micromagnetic simulations , a “sinc” function 0 0 0 ( ) sin 2 / 2y H HH t H f t t f t t , with H0 = 10 mT, and field frequency fH = 45 GHz, is applied the whole nanowire area.22 We obtain the FMR spectra in the corresponding frequency range from 0 to 45 GHz . The FMR spectra due to the RF -magnetic field are obtained by the fast Fourier transform (FFT) of stored My(x) (x, y, t ) in longitudinal (transverse) H0 field. More details can be described elsewhere .23 The closed blue circles in Fig. 3 is the calculated values with the fitting parameter using Eq. (1) which are fitted with the experimental data of un -patterned thin film. The obt ained Ms is 15.79 × 105 A/m while gyromagnetic ratio is fixed as 2.32 × 1011 m/(A∙s) . The obtained Ms value is similar with vibrating sample magnetometer method24 which CoFeB structure has Ta buffer layer. The resonance frequencies of uniform FMR mode in nanowire arrays are plotted as open red circles in Fig. 3. The resonance frequencies of uniform FMR mode is measured by VNA -FMR are agreed well with resonance freq uency of un-patterned thin film measured by VNA -FMR. In Fig. 3, the applied field dependences of the resonance frequencies Measured by VNA -FAM for the nanowire are plotted as open black rectangular , along with the result of micromagnetics calculated with E q. (1) as depicted closed black rectangular . It is also well agreed with the experimental result in nanowire mode and micromagnetic simulation result in the nanowire arrays. In order to reveal the effect s of spin dynamics properties with various nanowire width s, we 8 perform micromagnetic simulat ions. The nanowire width s are varied from 50 to 150 nm in 25-nm step for fixed 200 -nm-space with PBC , it causes changes of the demagnetization factor of the nanowire . In Fig. 4 (a) shows the nanowire width dependences of the resonance frequencies for the longitudinal magnetic field (open symbols) along with the resonance frequencies calculated with Eq. (1) (solid lines) . The demagnetization factors can be determined by fitting Eq. (1) while Nx is fixed as 0 to represent infinitely long wire . The agreements between the results of micromagnetic simulations (open circles) and Eq. (1) (solid lines) are excellent. For the transverse magnetic field, the direction of applied magnetic field is y - axis, Eq. (1) can be rewritten as follows: 2x y s z y s f H N N M H N N M . (2) In this equation, we use the relation of demagnetization factors , 1x y zN N N , in order to remove uncertainty in the fitting procedure . In the transverse field, the demagnetization factors are determined by Eq. (2). The resonance frequencies for transverse magnetic field which are obtained by micromagnetic simulation (open circles) and calculated by Eq. (2) (solid lines) as a function of the appl ied magnetic field with various nanowire width are displayed in Fig. 4(b). The longitudinal case, when the field direction is 9 easy axis, they are saturated with small field. However, the transverse case, when the field direction is hard axis, certain amoun t of field is necessary to saturate along the transverse directions. The narrower nanowire, the larger field is required as shown in Fig. 4 (b). Fig. 5(a) and (b) show the changes of demagnetization factors in longitudinal and transverse magnetic fields as a functi on of the nanowire width , respectively. The demagnetization factors play important role in the domain wall dynamics, for example the Walker breakdown is determined by the demagnetization factors .25 Furthermore, they are essential physical quantities to analyze the details of the spin dynamics. It is clear ly shown that the Nz (Ny) increase s (decrease s) with increasing the nanowire width in longitudinal magnetic field. For the transverse magnetic field, Nz (Ny) increase s (decrease s) with increasing the nanowire width , during Nx is almost zero value. The demagnetization factors both longitudinal and transverse have similar tendency with the effective demagnetization factors of dynamic origin26 and the static demagnetization factors for the prism geometry.27 Now, let us discuss about the Gilbert damping constant . The relation of the full width and half maxim a (f) of a resonance peak s as a function of applied field are shown in Fig. 6 for longitudinal (a) and transverse (b). The f is given by15: , ,2 22xy s z ex yxN f H M N N f . (3) 10 where, fex is the extrinsic line width contributions , when the applied magnetic field is x-(y- )axis for longitudinal (transverse) case . The symbol s are the result s of the micromagnetic simulations and the solid lines are the fitting result of Eq. (3) . We use pre -determined demagnetization factors (Fig. 5) during fitting procedures, and the agree ments are excellent. We have plotted the Gilbert damping constant as a function of the wavevector in nanowire width ( / qa , a is the nanowire width ) in Fig. 7. The black open rectangles are data extracted from the transverse field and the red open circles are longitudinal field data. We find that the Gilbert damping constant varied from 0.051 to 0.018 by changing the wavevector in nanowire width in transverse field. On the other hand, lon gitudinal field case the damping constant is almost constant as 0.021. Let us discuss about the un -expected behavior of the damping constant of transverse case. T he wire width acts as a kind of cut -off wavelength of the SW excitations in the confined geome try. SWs whose wavelength are larger than 2 a are not allowed in the nanowire. Therefore, only limited SW can be excited for the narrower wire, while more SW can be existed in the wider wire. For example, we show transverse standing SW as profiled in the inset of Fig. 6 for 150 -nm width nanowire in our micromagnetic simulations. More possible SW excitations imply more energy dissipation paths, it causes larger damping constant. For narrower nanowire (50 -nm), only limited SWs can be excited, so that the damping constant is smaller. However, for the limit case of infinite a case, it is the same with un -patterned thin films, there is no boundar y so that only uniform 11 mode can be excited, the obtained damping constant must be the input value. In summary, the VNA -FMR experiments is employed to investigate the magnetic properties of CoFeB nanowire arrays and the micromagnetic simulations is proposed to understand the magnetic properties including Gilbert damping constant of various CoFeB nanowire arrays width. We f ind that the demagnetization factors are similar with the dynamic origin and static for the prism geometry. The wire width or SW wavevector dependent damping constants can be explained with number of SW excitation modes. ACKNOWLEDGMENTS This work was supported by the National Research Foundation of Korea (NRF) Grants (Nos. 616-2011 -C00017 and 2013R1A12011936 ). References 1 S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008) . 2 D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, Science 309, 1688 (2005) . 3 I. M. Miron, G. Gaudin, S. Aufftet, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel and P. Gambardella, Nature Materials 9, 230 (2010) . 4 H-R Lee , K. Lee, J. Cho, Y . -H. Choi, C. -Y . You, M. -H. Jung, F. Bonell, Y. Shiota, S. Miwa and Y . Suzuki, Sci. Rep. 4, 6548 (2014). 5 J. Cho, et al. Nat. Commun. 6, 7635 (2015). 6 S. Ikeda , K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura and H. Ohno , Nat. Mater. 9, 721 (2010). 7 J.-S. Kim et al. , Nat. Commun . 5, 3429 (2014). 8 L. Thomas, R. Moriya, C. Rettner and S. S. P. Parkin , Science 330, 1810 (2010). 12 9 J.-S. Kim, M. Stark, M. Klaui, J. Yoon, C. -Y . You, L. Lopez -Diaz and E. Martinez , Phys. Rev. B 85, 174428 (2012). 10 B. Kuanr, R. E. Camleym, and Z. Celinski, Appl. Phys. Let t. 87, 012502 (2005) . 11 J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Le tt. 89, 277201 (2002). 12 L. Kraus, G. Infante, Z. Frait and M. Vazquez, Phys. Rev. B 83, 174438 (2011) . 13 C. T. Boone , J. A. Katine, J. R. Childress, V . Tiberkevich, A. Slavin, J. Zhu, X. Cheng and I. N. Krivorotov , Phys. Rev. Let t. 103, 167 601 (2009) . 14 M. Sharma, B. K. Kuanr, M. Sharma and Ananjan Nasu, IEEE Trans. Magn 50, 4300110 (2014). 15 D.-H. Kim, H .-H. Kim, and Chun -Yeol You, Appl. Phys. Lett. 99, 072502 (2011). 16 D.-H. Kim, H. -H. Kim, Chun -Yeol You and Hyungsuk Kim, J. Magn etics 16, 206 (2011). 17 C. Kittel, Introduction to Solid State Physics, 7th ed., pp. 504, (1996) . 18 M. J. Donahue and D. G. Porter: OOMMF User’ s Guide : Ver. 1.0, NISTIR 6376 (National Institute of Standards and Technology, Gaithersburg, Maryland, United States, 1999 ). 19 W. Wang, C. Mu, B. Zhang, Q. Liu, J. Wang, D. Xue, Comput. Mater. Sci. 49, 84 (2010) . See: http://oommf -2dpbc.sourceforge.net. 20 J. Cho, J . Jung, K .-E. Kim, S .-I. Kim, S .-Y. Park, M .-H. Jung, C .-Y. You, J. Magn. Magn. Mater. 339, 36 (2013). 21 C. Bilzer, T. Devolder, J -V . Kim, G. Counil, C. Chappert, S. Cardoso and P. P. Feitas , J. Appl. Phys. 100, 053903 (2006). 22 K.-S. Lee, D. -S. Han , S.-K. Kim, Phys. Rev. Let t. 102, 127202 (2009). 23 J. Yoon, C. -Y . You, Y . Jo, S. -Y. Park, M. H. Jung , J. Korean Phys. Soc. 57, 1594 (2010) . 24 Y . Shiota, F. Bonell, S. Miwa, N. Muzuochi, T. Shinjo and Y . Suzuki , Appl. Phys. Le tt. 103. 082410 (2013), 25 I. Purnama, I. S. Kerk, G. J. Lim and W. S. Lew , Sci. Rep. 5, 8754 (2015). 26 J. Ding, M. Kostylev, and A. O. Adeyeye, Phys. Rev. B. 84, 054425 (2011) . 27 A. Aharoni, J. Appl. Phys. 83, 3432 (2011) . 13 Figure Captions Fig. 1. Measurement geometry with SEM image s of the 100 -nm-width nanowires with a gap of 200 nm between nanowires . The longitudinal nanowire arrays are shown. After the nanowire patterns have been defined by e -beam lithography, they are covered by co -planar wave guides. Fig. 2. (a) The measured FMR spectrum of the CoFeB nanowire with H =0.194 T. The red (lower peak) and blue (higher peak) arrows indicate t he resonance frequencies of the uniform FMR mode and the nanowire mode, respectively. (b) The measured FMR spectrum of the CoFeB thin film with H =0.194 T. (c) Simulated FMR spectrum of the CoFeB nanowire with H= 0.200 T. Fig. 3. Measured and calculated FMR frequencies with the applied magnetic field for 100 - nm-width nanowire. The open black rectangles are nanowire mode and open red circles are the uniform FMR mode for CoFeB thin film. The closed black rectangles are calculated by OOMMF and the closed blue circles are theoretical ly calculated by Eq. (1) using fitted parameters form un -patterned film . Fig. 4. Variation of resonanc e frequencies with the applied magnetic field for the different PBC wire width for (a) longitudinal field and (b) transverse field. Inset: The geometry of 2 - dimensional PBC micromagnetic simulation with nanowire width a and a gap of 200 nm between nanowire s. The black open rectangles, red open circles, green open upper triangles, blue open down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, 100 nm, 125nm, and 150 nm, respectively. Fig. 5. Demagnetization factor with PBC wire widt h for (a) longitudinal and (b) transverse field. The black open circles, red open rectangles, blue open upper triangles represent as demagnetization factors, Ny, Nz, and Nx, respectively. Fig. 6. Full width and half maxim a with the applied magnetic field for (a) longitudinal and (b) transverse field. The black open rectangles, red open circles, green open upper triangles, blue open down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, 100 nm, 125nm, and 150 nm, respectively. Fig. 7. Damping constants with wavevector for transverse ( the black open rectangles ) and longitudinal ( the red open circles) field with errors. The black line is the input value which is determined from un -patterned film. Inset presents the profile of the trans verse spin density as SWs. Fig. 1 Fig. 2. Fig. 3. Fig. 4 ` Fig. 5 Fig. 6 Fig. 7

2015-09-06

We investigate the spin dynamics including Gilbert damping in the ferromagnetic nanowire arrays. We have measured the ferromagnetic resonance of ferromagnetic nanowire arrays using vector-network analyzer ferromagnetic resonance (VNA-FMR) and analyzed the results with the micromagnetic simulations. We find excellent agreement between the experimental VNA-FMR spectra and micromagnetic simulations result for various applied magnetic fields. We find that the demagnetization factor for longitudinal conditions, Nz (Ny) increases (decreases) as decreasing the nanowire width in the micromagnetic simulations. For the transverse magnetic field, Nz (Ny) increases (decreases) as increasing the nanowire width. We also find that the Gilbert damping constant increases from 0.018 to 0.051 as the increasing nanowire width for the transverse case, while it is almost constant as 0.021 for the longitudinal case.

Study of spin dynamics and damping on the magnetic nanowire arrays with various nanowire widths

1509.01807v1

Exact Intrinsic Localized Excitation of an Anisotropic Ferromagnetic Spin Chain in External Magnetic Field with Gilbert Damping, Spin Current and PT-Symmetry M. Lakshmanan1,a)and Avadh Saxena2,b) 1)Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli - 620 024, India 2)Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA We obtain the exact one-spin intrinsic localized excitation in an anisotropic Heisenberg ferromagnetic spin chain in a constant/variable external magnetic field with Gilbert damping included. We also point out how an appropriate magnitude spin current term in a spin transfer nano-oscillator (STNO) can stabilize the tendency towards damping. Further, we show how this excitation can be sustained in a recently suggested PT-symmetric magnetic nanostructure. We also briefly consider more general spin excitations. a)Electronic mail: lakshman.cnld@gmail.com b)Electronic mail: avadh@lanl.gov 1arXiv:1805.01230v1 [cond-mat.mes-hall] 3 May 2018I. INTRODUCTION The study of dynamics of classical Heisenberg ferromagnetic spin chain with anisotropic inter- action is of considerable importance in applied magnetism1,2and from application point of view3,4. While several continuum versions are known to be completely integrable soliton systems5–7, such as the isotropic case, no discrete integrable case is known in the literature, except for a modified version, namely the Ishimori lattice8. On the other hand, the present authors9have shown the exis- tence of several classes of exact solutions in terms of Jacobian elliptic functions which exist for the case of the discrete lattice including onsite anisotropy and external magnetic field. Identifying such interesting classes of solutions and their relevance in the context of appropriate physical situations constitute one of the important areas of investigation in spin dynamics9,10. From another point of view, occurrence of intrinsic localized breathers/oscillations in suitable anisotropic ferromagnetic spin chains is of practical relevance11,12and is being explored for the past several years. Apart from many numerical studies, in recent times the present authors and Subash13 have obtained explicit analytical solutions for the Heisenberg anisotropic spin chain with additional onsite anisotropy and constant external magnetic field corresponding to excitations of one, two and three spins and also investigated their stability. Additionally, relevant situations were pointed out where such excitations can be physically identified. Inrecenttimes,onehasalsoseenthatatnanoscalelevelspintransfernano-oscillator(STNO)14,15, which essentially consists of a trilayer structure of two nanoscale ferromagnetic films separated by a non-ferromagnetic but conducting layer, can lead to switching of spin angular momentum directions and allow for the generation of microwave oscillations16,17. The ferromagnetic film even when it is homogeneous is dominated by anisotropic interactions besides the presence of external magnetic fields (both dc and ac) and spin current terms. The equation of motion defining the evolution of the spins is the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation18where the spin current term is given by the Slonczewski form. One notices that the LLGS equation is a simple generalization of the Landau-Lifshitz-Gilbert (LLG) equation which describes the nonlinear magnetization dynamics 2in bulk materials as in the case of ferromagnetic lattices. Then it becomes important to ask what is the influence of spin current term on the spin excitations, particularly intrinsic localized oscillations (ILOs) and identify the conditions under which damping effect can be off-set by the spin current term. From yet another point of view, one may consider the possibility of designing a PT-symmetric ferromagnetic nanoscale device by considering two nano-film structures interspersed by a nonmag- netic but conducting thinner layer (i.e. a sandwich structure) as suggested by Lee, Kottos and Shapiro19very recently. These authors have proposed a class of synthetic magnetic nanostructures which utilize natural dissipation (loss) mechanisms along with suitable chosen gain mechanism so as to control the magnetization dynamics. We will also explore how the spin ILOs can be identified in these structures. In this paper we deduce an explicit one-spin excitation in an anisotropic ferromagnetic lattice (without onsite anisotropy, to start with) in the presence of external magnetic field and explore the effect of spin current term to maintain the oscillatory nature of the spin excitation. We then point out how this can be generalized to more general spin excitations and in PT-symmetric nanostruc- tures. The organization of the paper is as follows. In Sec. 2 we deduce the dynamical equation for an anisotropic ferromagnetic spin in the presence of external magnetic field and set up the appropriate equation for a one-spin excitation in the presence of Gilbert damping. In Sec. 3, we deduce the explicit one-spin excitation including the damping effect and analyze how the spin excitation gets affected by the damping. In Sec. 4, we incorporate the spin current term and point out how an appropriate strength of spin current can off-set the effect of damping so as to control the spin oscillations. In Sec. 5, we point out how the above analysis can be extended to a PT-symmetric nanostructure. We briefly indicate how this study can be extended to consider more general spin excitations in Sec. 6. Finally in Sec. 7, we present our conclusions. 3II. DYNAMICS OF THE ANISOTROPIC SPIN CHAIN AND ONE-SPIN EXCITATION Considering the evolution of spins of a one-dimensional anisotropic Heisenberg ferromagnetic spin chain modeled by the Hamiltonian12 H=−N/summationdisplay {n}(ASx nSx n+1+BSy nSy n+1+CSz nSz n+1)−D/summationdisplay n(Sz n)2−/vectorH·/summationdisplay n/vectorSn, (1) where the spin components /vectorSn= (Sx n,Sy n,Sz n)are classical unit vectors satisfying the constant length condition (Sx n)2+ (Sy n)2+ (Sz n)2= 1, n = 1,2,...,N. (2) HereA,B, andCare the exchange anisotropy parameters, Dis the onsite anisotropy parameter and the external magnetic field /vectorH= (H,0,0)is chosen along the x-axis for convenience. By introducing the appropriate spin-Poisson brackets and deducing the LLG spin evolution equation one can obtain the equation for the spin lattice (1) as d/vectorSn dt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff), (3) where /vectorHeff=A(Sx n+1+Sx n−1)ˆi+B(Sy n+1+Sy n−1)ˆj+C(Sz n+1+Sz n−1)ˆk+ 2DSz nˆk+/vectorH,(4) andαis the Gilbert damping parameter. In component form Eq. (3) with Eq. (4) reads as dSx n dt=CSy n(Sz n+1+Sz n−1)−BSz n(Sy n+1+Sy n−1)−2DSy nSz n+α/bracketleftbigg BSy nSz n(Sy n+1+Sy n−1) −A(Sx n+1+Sx n−1)((Sy n)2+ (Sz n)2) +CSz nSx n(Sz n+1+Sz n−1)−2DSx n(Sz n)2−H((Sx n)2+ (Sz n)2)/bracketrightbigg ,(5) dSy n dt=ASz n(Sx n+1+Sx n−1)−CSx n(Sz n+1+Sz n−1) + 2DSx nSz n+HSz n+α/bracketleftbigg CSz nSy n(Sz n+1+Sz n−1) −B((Sx n)2+ (Sz n)2)(Sy n+1+Sy n−1) +ASx nSy n(Sx n+1+Sx n−1)−2DSy n(Sz n)2+HSx nSy n/bracketrightbigg , (6) 4dSz n dt=BSx n(Sy n+1+Sy n−1)−ASy n(Sx n+1+Sx n−1)−HSy n+α/bracketleftbigg ASx nSy n(Sx n+1+Sx n−1) −C((Sx n)2+ (Sy n)2)(Sz n+1+Sz n−1) +BSy nSz n(Sy n+1+Sy n−1) + 2DSz n((Sx n)2+ (Sy n)2) +HSx nSz n/bracketrightbigg .(7) Now looking for the one spin excitation for (1) as /vectorSn=...,(1,0,0),(1,0,0),(Sx i(t),Sy i(t),Sz i(t)),(1,0,0),(1,0,0),..., (8) where we have used nto denote a general spin in the lattice and used ito specify the localized spin excitation, and redesignating (Sx i(t),Sy i(t),Sz i(t))as(Sx 0(t),Sy 0(t),Sz 0(t)), the equation of motion (LLG equation) for the excited spin can be given as dSx 0 dt=−2DSy 0Sz 0−α/bracketleftbigg (2A+H)((Sy 0)2+ (Sz 0)2) + 2DSx 0(Sz 0)2/bracketrightbigg , (9) dSy 0 dt= (2A+H)Sz 0+ 2DSx 0Sz 0+α/bracketleftbigg (2A+H)Sx 0Sy 0−2DSy 0(Sz 0)2/bracketrightbigg , (10) dSz 0 dt=−(2A+H)Sy 0+α/bracketleftbigg (2A+H)Sx 0Sz 0+ 2D((Sx 0)2+ (Sy 0)2)Sz 0/bracketrightbigg . (11) Note that from Eqs. (9) - (11), one can check that Sx 0dSx 0 dt+Sy 0dSy 0 dt+Sz 0dSz 0 dt= 0, (12) so that/vectorS2= (Sx o)2+ (Sy 0)2+ (Sz 0)2=Constant = 1is conserved. Next, further confining to the case where the onsite anisotropy vanishes, D= 0, we have the LLG equation for the one-spin excitation, dSx 0 dt=−α(2A+H)(1−(Sx 0)2), (13) dSy 0 dt= (2A+H)Sz 0+α(2A+H)Sx 0Sy 0, (14) dSz 0 dt=−(2A+H)Sy 0+α(2A+H)Sx 0Sz 0, (15) with the constraint /vectorS2= (Sx o)2+ (Sy 0)2+ (Sz 0)2= 1. The system (13) - (15) can be exactly solved as shown below. 5III. EXPLICIT ONE-SPIN EXCITATION Now the above system of nonlinear differential equations can be straightforwardly solved. Inte- grating (14) we obtain Sx 0(t) =c2e−2α(2A+H)t−1 c2e−2α(2A+H)t+ 1, (16) wherecis an arbitrary constant. We also note that when α= 0, that is no damping, Sx 0(t) = (c2−1)/(c2+ 1) =const =√ 1−a2as noted in ref. [13], Eq. (11). Also we note that Sx 0(0) = (c2−1)/(c2+ 1)andSx 0(∞) =−1, indicating a switching from a given initial value to the other ground state, Sx 0=−1. To findSy 0andSz 0, we proceed as follows. Considering Eq. (14) and differentiating once with respect toton both sides to obtain (d2Sy 0/dt2), after making use of the forms of (dSx 0/dt)and (dSz 0/dt)from (13) and (15), respectively, we have d2Sy 0 dt2=−(2A+H)2(1 +α2)Sy 0+ 2α(2A+H)2Sx 0Sz 0+ 2α2(2A+H)2(Sx 0)2Sy 0,(17) so that Sz 0(t) =1 2α(2A+H)2Sx 0/bracketleftBiggd2Sy 0 dt2+ (2A+H)2(1 +α2)Sy 0−2α2(2A+H)2(Sx 0)2Sy 0/bracketrightBigg .(18) Also from (14) we can write Sz 0(t) =1 2A+H/bracketleftBiggdSy 0 dt−αSx 0Sy 0/bracketrightBigg . (19) Equating the right hand sides of (18) and (19), we obtain d2Sy 0 dt2=−2α(2A+H)Sx 0dSy 0 dt+ (2A+H)2(1 +α2)Sy 0= 0. (20) Afterastandardtransformationandtwointegrations(asindicatedinAppendixA),wecanexplicitly write the solution for Sy 0as Sy 0=cexp(−α(2A+H)t) c2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (21) 6where ˆais an arbitrary constant. Also from (14) we have Sz 0(t) =1 (2A+H)/bracketleftBiggdSy 0 dt−αSx 0Sy 0/bracketrightBigg =−ˆasin(Ωt+δ)ce−α(2A+H)t c2e(−2α(2A+H)t+ 1. (22) Nowinordertofixtheconstant ˆawedemandthatthespinlengthconstraint (Sx 0)2+(Sy 0)2+(Sz 0)2= 1 be valid. This leads to ˆa2= 4orˆa= 2, (23) so that we have now the complete solution of the excited spin as Sx 0(t) =c2e−2α(2A+H)t−1 c2e−2α(2A+H)t+ 1, (24) Sy 0(t) =2ce−α(2A+H)t c2e−2α(2A+H)t+ 1cos[(2A+H)t+δ], (25) Sz 0(t) =−2ce−2α(2A+H)t c2e−2α(2A+H)t+ 1sin[(2A+H)t+δ]. (26) Note that the arbitrary constant corresponding to the undamped case ( α= 0) is ˆa=c2−1 c2+ 1. (27) It is obvious from the above that for α= 0,Sx 0=constant =c2−1 c2+1, whileSy 0andSz 0are periodic functions of t. In this case Eqs. (14)-(15) are linear in Sy 0andSz 0so that the perturbation around the origin in the ( Sy 0−Sz 0) plane admits pure imaginary eigenvalues. When α/negationslash= 0, they get damped as shown in Fig. 1 corresponding to the explicit forms (24)-(26). Note that in the above we have assumed the external magnetic field to be a constant in time. However, even in the case where the field is a variable function of time, say H(t) =h0+h1cosωt, (28) whereh0,h1andωare constants, we observe from the equations of motion of the spin components (13) - (15), that Hoccurs always as a linear combination 2A+H(t) = (2A+h0+h1cosωt). Therefore by redefining the time (2A+H)tas τ= (2A+h0)t−h1ωsinωt, (29) 7all the previous analysis goes through. The final spin excitations are of the same form as (24) - (26) but with the transformed time variable given by Eq. (29). tSx 0(t)(i) 2000 1500 1000 500 01 0.5 0 -0.5 -1 tSy 0(t)(ii) 2000 1500 1000 500 01 0.5 0 -0.5 -1 tSz 0(t)(iii) 2000 1500 1000 500 01 0.5 0 -0.5 -1 FIG. 1. Damped spin excitation: One-spin excitation (Eqs. (24)-(26)) showing the three spin components for the damped cases ( α= 0.005). IV. EFFECT OF SLONCZEWSKI SPIN CURRENT Next we consider the influence of spin current term in a trilayer structured STNO (see Fig. 2), where we consider the excitation of a single spin of magnetization in the outer uniformly magnetized layer under anisotropic interaction and external magnetic field in the presence of spin current. The corresponding spin excitation is given by the Landau-Lifshitz-Gilbert-Slonczewski equation for the spin as FIG. 2. A schematic representation of STNO. d/vectorSn dt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff) +j/vectorSn×(/vectorSn×/vectorSp), (30) 8where/vectorHeffis the effective field given by Eq. (4) and jis the magnitude of the spin current and the polarization vector /vectorSpis /vectorSp= (1,0,0), (31) corresponding to the flow of electrons in the x-direction. Consequently, /vectorSn×(/vectorSn×/vectorSp) =/vectorSn(/vectorSn·/vectorSp)−/vectorSp=−((Sy n)2+ (Sz n)2)ˆi+Sx nSy nˆj+Sx nSz nˆk, (32) where (ˆi,ˆj,ˆk)form the unit orthonormal trihedral. As a result, the equations for the one-spin excitations get modified from (13) - (15) as dSx 0 dt=−[α(2A+H)−j](1−(Sz 0)2), (33) dSy 0 dt= (2A+H)Sz 0+ [α(2A+H)−j]Sx 0Sy 0, (34) dSz 0 dt=−(2A+H)Sy 0+ [α(2A+H)−j]Sx 0Sz 0. (35) Now choosing the spin current as j=α(2A+H), (36) one can check that dSx 0 dt= 0, (37) dSy 0 dt= (2A+H)Sz 0, (38) dSz 0 dt=−(2A+H)Sz 0. (39) Consequently, the spin vector evolves as /vectorS0=/parenleftBig√ 1−ˆa2,ˆacos(Ωt+δ),−ˆasin(Ωt+δ/parenrightBig , (40) where ˆa=constant and Ω = (2A+H), and the effect of damping is exactly offset by the spin current term. Thus the spin current acts effectively as an “external magnetic field plus anisotropy" and the system can generate microwave oscillations. When j <α (2A+H), damping will overtake asymptotically and the spin will switch its direction. 9V.PT-SYMMETRIC MAGNETIC DEVICE Recentlyaclassofsyntheticmagneticnanostructuresthatmakesuseofthenatureofloss/dissipation mechanism together with appropriate amplification (gain) process has been suggested by Lee, Kot- tos and Shapiro19to control magnetization dynamics. The suggested arrangement consists of two coupled nano-ferromagnetic films, n= 1,2(when separated by a spacer) in the presence of an external magnetic field along the x-axis, for example out-of-plane geometry (so that the z-axis is perpendicular to the films) as shown in Fig. 3. FIG.3. APT-symmetrictrilayerstructurecomprisingtwomagneticthinfilmsandaspacerlayersuggested by Lee, Kottos and Shapiro19. Considering the effective instantaneous local fields as /vectorH1effand/vectorH2efffor the two layers 1 and 2, respectively, associated with the homogeneous magnetization vectors /vectorS1= (/vectorM1/|/vectorM1|)and/vectorS2= (/vectorM2/|/vectorM2|), we have the associated dynamical equations d/vectorS1 dt=/vectorS1×/vectorH1eff+k/vectorS1×/vectorS2+α/vectorS1×d/vectorS1 dt, (41) d/vectorS2 dt=/vectorS2×/vectorH2eff+k/vectorS2×/vectorS1−α/vectorS2×d/vectorS2 dt, (42) wherekistheferromagneticcouplingand αisthedamping/gaincoefficient. Notethatthecombined systems (41)-(42) are invariant under the simultaneous changes of the variables /vectorS1,2→ −/vectorS2,1, /vectorH1,2eff→−/vectorH2,1effandt→−t, which may be treated as equivalent to combined PT-symmetry 10operation19. Now we choose the two layers such that /vectorS2=/vectorS1×/vectorSp,/vectorS1=−/vectorS2×/vectorSp, (43) where/vectorSp= (1,0,0)is a fixed polarization vector. Equation (43) implies that /vectorSpis perpendicular to the plane of /vectorS1and/vectorS2. Then, similar to the analysis in Sec. 4, we can choose the ferromagnetic couplingksuch that for simple anisotropy as in Eqs. (13) - (15) and external magnetic field H, we can choose k=α(2A+H), (44) so that the gain/loss terms are exactly cancelled by the ferromagnetic coupling, leaving out d/vectorS1 dt=/vectorS1×/vectorH1eff, (45) d/vectorS2 dt=/vectorS2×/vectorH2eff, (46) leading to spin oscillations and thereby to an effective control of magnetization oscillations. VI. MORE GENERAL SPIN EXCITATIONS One can consider more general localized spin excitations like two, three, etc. spin excitations. For example, in the case of localized two-spin excitations, /vectorSn=...,(1,0,0),(1,0,0),(Sx i,Sy i,Sz i),(Sx i+1,Sy i+1,Sz i+1),(1,0,0),(1,0,0),... =...,(1,0,0),(1,0,0),(Sx 0,Sy 0,Sz 0),(Sx 1,Sy 1,Sz 1),(1,0,0),(1,0,0),..., (47) 11we obtain the dynamical equations from (5) - (7) as dSx 0 dt=CSy 0Sz 1−BSz 0Sy 1−2DSy 0Sz 0 +α[BSx 0Sy 0Sy 1−(A(Sx 1+ 1) +H)((Sy 0)2+ (Sz 0)2) +CSx 0Sz 0Sz 1−2DSx 0(Sz 0)2], (48) dSy 0 dt=ASz 0(Sx 1+ 1)−CSx 0Sz 1+ 2DSx 0Sz 0+HSz 0 +α[(A(Sx 1+ 1) +H)Sx 0Sy 0−BSy 1((Sx 0)2+ (Sz 0)2) +CSy 0Sz 0Sz 1−2DSy 0(Sz 0)2], (49) dSz 0 dt=BSx 0Sy 1−ASy 0(Sx 1+ 1)−HSy 0 +α[(A(Sx 1+ 1) +H)Sx 0Sz 0+BSy 0Sz 0Sy 1−CSz 1((Sx 0)2+ (Sy 0)2) + 2DSz 0((Sx 0)2+ (Sy 0)2)],(50) dSx 1 dt=CSz 0Sy 1−BSy 0Sz 1−2DSy 1Sz 1 +α[BSy 0Sx 1Sy 1−(A(Sx 0+ 1) +H)((Sy 1)2+ (Sz 1)2) +CSz 0Sx 1Sz 1−2DSx 1(Sz 1)2], (51) dSy 1 dt=ASz 1(Sx 0+ 1)−CSx 1Sz 0+ 2DSx 1Sz 1+HSz 1 +α[(A(Sx 0+ 1) +H)Sx 1Sy 1−BSy 0((Sx 1)2+ (Sz 1)2) +CSz 0Sy 1Sz 1−2DSy 1(Sz 1)2], (52) dSz 1 dt=BSx 1Sy 0−ASy 1(Sx 0+ 1)−HSy 1 +α[(A(Sx 0+ 1) +H)Sx 1Sz 1+BSy 0Sy 1Sz 1−CSz 0((Sx 1)2+ (Sy 1)2) + 2DSz 1((Sx 1)2+ (Sy 1)2)].(53) Note that the terms proportional to αare generalizations for the present two-spin excitation case compared to those given in Eqs. (9) - (11). As such the system (48) - (53) does not seem to be analytically solvable. In Fig. 4, we numerically integrate the system for both the undamped case (α= 0) and the damped case ( α/negationslash= 0) for nonzero Dand present the solutions in the undamped and damped cases to show the existence of more general internal localized excitations. The analysis can be extended to even more general situations, which will be presented elsewhere. VII. CONCLUSION By looking at the simplest internal localized excitations in an anisotropic Heisenberg ferromag- netic spin chain in external magnetic field with additional Gilbert damping, we deduced the explicit solutions which characteristically show the effect of damping. Then applying a spin current in an 12tSx 0(t)(i) 200 150 100 50 01 0.5 0 -0.5 -1 tSy 0(t)(ii) 200 150 100 50 01 0.5 0 -0.5 -1 tSz 0(t)(iii) 200 150 100 50 01 0.5 0 -0.5 -1 tSx 1(t)(iv) 200 150 100 50 01 0.5 0 -0.5 -1 tSy 1(t)(v) 200 150 100 50 01 0.5 0 -0.5 -1 tSz 1(t)(vi) 200 150 100 50 01 0.5 0 -0.5 -1Fig. 4 (a): Undamped two-spin excitations tSx 0(t)(i) 200 150 100 50 01 0.5 0 -0.5 -1 tSy 0(t)(ii) 200 150 100 50 01 0.5 0 -0.5 -1 tSz 0(t)(iii) 200 150 100 50 01 0.5 0 -0.5 -1 tSx 1(t)(iv) 200 150 100 50 01 0.5 0 -0.5 -1 tSy 1(t)(v) 200 150 100 50 01 0.5 0 -0.5 -1 tSz 1(t)(vi) 200 150 100 50 01 0.5 0 -0.5 -1 Fig. 4 (b): Damped two-spin excitations FIG. 4. Solution of Eqs. (48)-(53) for two-spin excitations S0andS1for the (a) undamped ( α= 0) and (b) damped cases ( α= 0.005), with the anisotropy parameters A= 0.1,B= 0.23,C= 1.0andD= 0.3 and the magnetic field H= 113Oe. STNO of appropriate magnitude, we pointed out how the tendency toward damping can be offset exactly and thereby sustaining the magnetic oscillations. Our prediction about a PT-symmetric 13STNO could be tested in magnetic multilayer structures with carefully balanced gain and loss. We have also pointed out how such controlled oscillations can be effected in a recently suggested nano- magnetic trilayer device. It will be insightful to observe these oscillations in appropriate magnetic systems experimentally. Finally, in a related context we note that nonreciprocal optical modes can exist at an interface between two PT-symmetric magnetic domains near a frequency corresponding to almost zero effective permeability20. VIII. ACKNOWLEDGMENTS The authors wish to thank Dr. D. Aravinthan for his help in the numerical analysis. The research work of ML was supported by a NASI Senior Scientist Platinum Jubilee Fellowship (NAS 69/5/2016-17) and a DST-SERB Distinguished Fellowship (No.: SERB/F/6717/2017-18). ML was also supported by a Council of Scientific and Industrial Research, India research project (No.: 03/1331/15/EMR-II) and a National Board for Higher Mathematics research project (No.: 2/48(5)/2015/NBHM(R.P.)/R&D II/14127). ML also wishes to thank the Center for Nonlinear Studies, Los Alamos National Laboratory, USA for its warm hospitality during his visit in the summer of 2017. This work was supported in part by the U.S. Department of Energy. APPENDIX A Here we briefly point out how to solve Eq. (20). Introducing the transformation Sy 0(t) =eα(2A+H)/integraltext Sx 0dt·ˆSy 0(t) (A. 1) into Eq. (20), we obtain d2ˆSy 0(t) dt2+ (2A+H)2ˆSy 0(t) = 0. (A. 2) Consequently, we have ˆSy 0(t) = ˆacos(Ωt+δ),Ω = 2A+H, (A. 3) 14where ˆaandδare arbitrary constants. Then, the prefactor on the right hand side of (21) can be deduced as follows. Since I=/integraldisplay Sx 0dt=/integraldisplayc2exp(−2α(2A+H)t)−1 c2exp(−2α(2A+H)t) + 1dt=−1 2α(2A+H)log(c2exp(−2α(2A+H)t) + 1)2 c2exp(−2α(2A+H)t, (A. 4) the prefactor becomes exp/bracketleftbigg α(2A+H)/integraldisplay Sx 0dt/bracketrightbigg =cexp(−α(2A+H)t) c2exp(−2α(2A+H)t) + 1. (A. 5) Correspondingly Sy 0=cexp(−α(2A+H)t) c2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (A. 6) which is Eq. (21). REFERENCES 1B. Hillerbrands and K. Ounadjela, Spin Dynamics in Confined Magnetic Structures , Vols. I & II (Springer, Berlin) 2002. 2M. Lakshmanan, Philos. Trans. R. Soc. A 369(2011) 1280. 3B. Georges, V. Cros and A. Fert, Phys. Rev. B 73(2006) 0604R. 4Z. Yang, S. Zhang and Y. C. Li, Phys. Rev. Lett. 99(2007) 134101. 5M. Lakshmanan, Phys. Lett. A 61(1977) 53. 6K. Nakamura and T. Sasada, J. Phys. C 15(1982) L915. 7E. K. Sklyanin, LOMI preprint E-3-79, Leningrad (1979). 8Y. Ishimori, Prog. Theor. Phys. 72(1984) 33. 9M. Lakshmanan and A. Saxena, Physica D 237(2008) 885. 10H. Zabel and M. Farle (Eds.), Magnetic Nanostructures: Spin Dynamics and Spin Transport (Springer, Berlin) 2013. 11A. Sievers and S. Takeno, Phys. Rev. Lett. 61(1988) 970. 1512Y. Zolotaryuk, S. Flach and V. Fleurov, Phys. Rev. B 63(2003) 214422. 13M. Lakshmanan, B. Subash and A. Saxena, Phys. Lett. A 378(2014) 1119. 14G. Bertotti, I. Mayergoyz and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems (Elsevier, Amsterdam) 2009. 15B. Georges, J. Grollier, V. Cros and A. Fert, Appl. Phys. Lett. 92(2008) 232504. 16B. Subash, V. K. Chandrasekar and M. Lakshmanan, Europhys. Lett. 102(2013) 17010; 109 (2015) 17009. 17J. Turtle, K. Beauvais, R. Shaffer, A. Palacios, V. In, T. Emery and P. Langhini, J. Appl. Phys. 113(2013) 114901. 18J. C. Slonczewski, J. Magn. & Magn. Mater. 159(1996) L261. 19J. M. Lee, T. Kottos and B. Shapiro, Phys. Rev. B 91(2015) 094416. 20J. Wang, H. Y. Dong, C. W. Ling, C. T. Chan and K. H. Fung, Phys. Rev. B 91(2015) 235410. 16

2018-05-03

We obtain the exact one-spin intrinsic localized excitation in an anisotropic Heisenberg ferromagnetic spin chain in a constant/variable external magnetic field with Gilbert damping included. We also point out how an appropriate magnitude spin current term in a spin transfer nano-oscillator (STNO) can stabilize the tendency towards damping. Further, we show how this excitation can be sustained in a recently suggested PT-symmetric magnetic nanostructure. We also briefly consider more general spin excitations.

Exact Intrinsic Localized Excitation of an Anisotropic Ferromagnetic Spin Chain in External Magnetic Field with Gilbert Damping, Spin Current and PT-Symmetry

1805.01230v1

Novel Bottomonium Results Ben Page,𝑎Chris Allton𝑎and Seyong Kim𝑏 𝑎Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom 𝑏Department of Physics, Sejong University, Seoul 143-747, Korea E-mail: {b.page.9/zero.alt3/zero.alt3727,c.r.allton}@swansea.ac.uk, skim@sejong.ac.kr We present the latest results from the use of the Backus-Gilbert method for reconstructing the spectra of NRQCD bottomonium mesons using anisotropic FASTSUM ensembles at non-zero temperature. We focus in particular on results from the 𝜂𝑏,Υ,𝜒𝑏1andℎ𝑏generated from Tikhonov-regularized Backus-Gilbert coefficient sets. We extend previous work on the Laplace shifting theorem as a means of resolution improvement and present new results from its use. We conclude with a discussion of the limitations of the improvement routine and elucidate a connection with Parisi-Lepage statistical scaling. The 39th International Symposium on Lattice Field Theory, 8th-13th August, 2022, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany Speaker ©Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2212.12016v1 [hep-lat] 22 Dec 2022Novel Bottomonium Results Ben Page 1. Introduction Bottomoniumhaspreviouslybeenstudiedasameansofestimatingthepropertiesofthequark- gluonplasmaasproducedinrelativisticheavy-ioncollisions. [1–3]. Theproblemofreconstructing thespectrumofbottomoniumstatesatnon-zerotemperatureisanill-posedoneduetothepresenceof finiteuncertaintiesindatacollectedfromlatticeQCDsimulations. Thisworkrepresentsoneofthe latestinacollectionofstudiesperformedbythe F/a.pc/s.pc/t.pc/s.pc/u.pc/m.pc Collaborationfocussingonbottomonium states simulated using anisotropic lattices at nonzero temperature. In the following, we build upon the results of previous work [4] and present a discussion on the use of the Backus-Gilbert method for reconstructing bottomonium spectra, showing how the Laplacian nature of the reconstruction formula my be exploited to give improved resolution. 2. Lattice details Non-relativisticQCD(NRQCD)isaneffectivefieldtheoryforheavyquarkonia,whichapprox- imates fully relativistic QCD by expanding the Lagrangian in powers of the heavy quark velocity [5]. One of the principal benefits of the NRQCD formulation is that the calculation of the time evolutionoftheheavyquarksreducestoaninitial-valueproblem,astheheavyquarksandantiquarks decouple in the non-relativistic regime. This decoupling effect turns the spectral representation of theEuclideancorrelator 𝐺¹𝜏ºintoaLaplacetransformationofthespectraldensityfunction 𝜌¹𝜔º: 𝐺¹𝜏;𝑇º=∫𝜔max 𝜔min𝑑𝜔 2𝜋𝐾¹𝜏𝜔º𝜌¹𝜔;𝑇º (1) where𝐾¹𝜏𝜔º=𝑒𝜔𝜏is the temperature independent kernel function and 𝑇=¹𝑎𝜏𝑁𝜏º1is the lattice temperature as a function of the temporal extent 𝑁𝜏. In order to relate NRQCD energies to physicalenergies,thereconstructionwindow 𝜔2»𝜔min𝜔max¼mustbeadditivelyrenormalisedby the NRQCD energy shift, Δ𝐸=746GeV. We make use of F/a.pc/s.pc/t.pc/s.pc/u.pc/m.pc’s Generation 2L ensembles, generated using anisotropic lattices (𝜉=𝑎𝑠𝑎𝜏35)with2+1flavour,clover-improvedWilsonfermionsusingaphysical 𝑠quarkand lighter, degenerate 𝑢and𝑑quarks[6]. The spatial extent of the lattice 𝑁𝑠=32and the temporal extent along with the corresponding lattice temperatures are detailed in Table 1. 𝑁𝜏 128645648403632282420 𝑇=1¹𝑎𝜏𝑁𝜏º[MeV] 4795109127152169190217253304 𝑁cfg 1024104110421123110211191090103110161030 Table 1: Temporal extent, corresponding lattice temperature in MeV and number of configurations for the F/a.pc/s.pc/t.pc/s.pc/u.pc/m.pc Generation 2L ensembles. The double vertical line mid-table represents our value of 𝑇pc[6]. 3. The Backus-Gilbert Method The Backus-Gilbert method [7] is a reconstruction technique which extracts regularised solu- tions from the ill-posed inverse problem by imposing constraints on the stability of its predictions 2Novel Bottomonium Results Ben Page under a change of input. Since 𝐺¹𝜏ºis only known to at most O¹100ºbut𝜌¹𝜔ºis continuous (O¹1000¸ºpoints), there are theoretically an infinite number of possible spectra 𝜌which produce the correct𝐺¹𝜏ºwithin numerical errors. Backus-Gilbert attempts to estimate a solution of Eq. 1, denoted ˆ𝜌, on a point-by-point basis by constructing averaging functions 𝐴¹𝜔𝜔 0ºcentred about some point𝜔0generated using the data kernel 𝐾¹𝜏𝜔º: ˆ𝜌¹𝜔0º=∫𝜔max 𝜔min𝐴¹𝜔𝜔 0º𝜌¹𝜔º𝑑𝜔 (2) with𝐴¹𝜔𝜔 0º=Í 𝜏𝑐𝜏¹𝜔0º𝐾¹𝜏𝜔º. In the limit 𝐴¹𝜔𝜔 0º!𝛿¹𝜔𝜔0º, we obtain a perfect reconstrucion of the target spectrum 𝜌. It is easily seen that plugging Eq. 1 into ˆ𝜌¹𝜔0º=∑︁ 𝜏𝑐𝜏¹𝜔0º𝐺¹𝜏º (3) gives Eq. 2. The coefficients 𝑐𝜏¹𝜔0ºcontrol the shape of 𝐴¹𝜔𝜔 0ºand are found by minimising thecostfunction 𝐽¹𝜔0ºrepresentingtheleast-squaresdistancebetweentheaveragingfunctionand the delta function at 𝜔0[8]: 𝐽¹𝜔0º=∫𝜔max 𝜔min»𝐴¹𝜔𝜔 0º𝛿¹𝜔𝜔0º¼2𝑑𝜔 (4) Setting𝜕𝑐𝜏𝐽¹𝜔0º=0reduces the problem to an inversion of a matrix-vector product: K𝜏𝜏0
𝑐𝜏0¹𝜔0º=𝐾¹𝜔0𝜏ºwhereK𝜏𝜏0=∫𝜔max 𝜔min𝐾¹𝜏𝜔º𝐾¹𝜏0𝜔º𝑑𝜔 (5) Thekernelwidthmatrix Kisnear-singular(worsenedbytheexponentialnatureofthekernel) andsomustbetreatedbyaconditioningroutinebeforeinversion. Onesuchconditioningroutineis the addition of a small constant term to the diagonal entries in a Tikhonov-like fashion[9]: K¹𝛼º=K¸𝛼𝐼 (6) where𝛼isaparameterwhichcontrolsthestrengthoftheregularisation. Thebenefitofsuchscalar conditioningoverothermethodsisthatthecoefficients 𝑐𝜏areconstructedwithoutpriorknowledge of𝐺¹𝜏º,enablingtheiruseinthereconstructionofanyspectraobeyingEq.1,regardlessofchoices of quantum numbers, see [4]. 4. Improvement via the Laplace Shift Transform It has been previously shown [10] that, for the case of the maximum entropy method (MEM), there exists a relationship between the choice of 𝜔minand the resolving power of the method. This effect also occurs in the Backus-Gilbert method which shares a similar basis-function mechanism of reconstruction as the MEM. The application of a Laplace shift transform: 𝐺0¹𝜏º=𝑒Δ
𝜏𝐺¹𝜏ºL=)𝜌0¹𝜔º=𝜌¹𝜔¸Δº (7) whereΔ¡0shifts the spectral features closer to 𝜔minwhere the resolving power of the method is improved [4] offering improved predictions for mass and width estimates. Building upon this feature, we have opted to combine 𝜔minand the Laplace shift transform parameter Δinto a single parameter eΔ=𝜔min¸Δ. 3Novel Bottomonium Results Ben Page 5. Systematic analysis: Removing eΔand𝛼dependence There is a systematic dependence of the ground state mass 𝑀and widthΓon the value of eΔ and𝛼which must be removed during analysis. This is illustrated in Fig. 1 where the mass (Left) and width (Right) are shown as a function of eΔfor various𝛼values. Ascanbeseen,forfixed 𝛼,themassincreasesandwidthdecreasesmonotonicallywith eΔdue to the resolution improvement. We also note that in the mass case, this slope approaches zero as 𝛼!0,indicatingthatthe eΔdependencefallsaway. Finally,wenotethatthemaximumtheoretical shift occurs when eΔequals the mass. This is indicated by the boundary of the hatched region in Fig.1(Left)). Wethereforeobtainourmassestimatebyfirstlyperformingalinearextrapolationof the mass, at fixed 𝛼, to the boundary of the hatched region, obtaining 𝑀¹𝛼º. The𝛼dependence is then removed by noting empirically that 𝑀¹𝛼ºbecomes independent of 𝛼, for small𝛼, and so can be fit with a constant. A bootstrap analysis produces the error estimate. Thegroundstatewidthisdeterminedbyasimilarprocedure. Firstalinearfitin eΔisperformed, atfixed𝛼,andextrapolatedtotheboundaryregion. The 𝛼dependenceisthenremovedinthesame manner as in the mass case. However we note that, due to the finite resolving power of the Backus Gilbert method, the widths obtained should be considered upper bounds on the physical width of the state. 6 7 8 9 10 (GeV) 8910Mass (GeV) (s-s) T=47 MeV PDG estimate BG estimate 109 107 105 103 101 log10() 6 7 8 9 (GeV) 050010001500 (GeV) (s-s) T=47 MeV 109 107 105 103 101 log10() Figure 1: Left:Plot of the ground state mass versus the shift parameter eΔfor the (smeared) Υmeson over a range of𝛼values. The hatched region represents the maximum possible shift, beyond which the ground state feature falls outside of the sampling window. The red dashed line is the PDG estimate [11] and the black dashed line represents our best estimate after extrapolating the eΔand𝛼hyper-parameters. Right: Plot ofourestimateoftheupperboundforthegroundstateFWHMwidthversus eΔfortheΥmesonoverarange of𝛼values. The black dashed line is our best estimate of the upper bound and the red band denotes its uncertainty. The temperature used in these plots is 𝑇=47MeV. The value of 𝜔minis held fixed in this analysis (at a value of 𝜔min=01𝑎1 𝜏68GeV). We have chosen to include all possible Euclidean times in our analysis, i.e. 0𝜏 𝑁 𝜏, because the resolving power of Backus Gilbert method is greatest for the largest time windows. 4Novel Bottomonium Results Ben Page 6. Results for the bottomonium sector: 𝜂𝑏,Υ,𝜒𝑏1andℎ𝑏 0 100 200 300 T (MeV)9.29.39.49.59.6Mass (GeV) (s-s) (2021) (s-s) 0 100 200 300 T (MeV)0250500750100012501500 (MeV) (s-s) (2021) (s-s) Figure 2: Comparisonbetweennewresultsfromthisworkandthosefrom[4],labelled(2021),forthemass (left)andupperboundonthewidth( right). ThemagentalineontheleftpaneisthePDGestimateforthe Υ mass[11]. Theredshadedbandontherightpaneindicatesourestimateofthemaximumresolvingpowerof the new method. We have improved our previous results [4]) by including the Laplace shifting (see §4) and an improved fitting analysis (see §5). In Fig. 2, we display our results for the Υmass and width as a function of temperature, including the results from our earlier analysis. We have also extended our previous work by including the 𝜂𝑏,𝜒𝑏1andℎ𝑏states, with results shown in Figs. 3 and 4. For this analysis, we restricted ourselves to data generated using smeared quark sources which have improved overlap with the ground state over local sources. We also conducted a more rigorous studyofthedependenceofthemassandwidthontheparameters 𝛼andeΔinanattempttomeasure their contribution to the systematic error. In particular, in the case of the Υmass, we wish to highlight in Fig.2 the comparison between the results of this work and the estimates presented in [4] which appear to be contaminated by systematics of the method, as the dependence of the the mass and width on 𝛼was not fully accounted for. We also note that below the pseudocritical temperature ( 𝑇pc=162MeV for our Gen-2L ensembles[6, 12]), the experimental widths for the Υand𝜂𝑏are5402125keV and 10¸4 5MeV respectively[11], over an order of magnitude smaller than even the minimum resolvable widthforourmethod(seeFig.4). Weonceagainreiteratethatourpresentedvaluesfortheground statewidthrepresentanupperboundonthetruevalue,andweleaveaninvestigationintotheeffect of changing the Euclidean time extent and the lower bound 𝜔minto a future study. 5Novel Bottomonium Results Ben Page 0.0 0.2 0.4 0.6 0.8 1.0 Temperature (MeV)0.00.20.40.60.81.0Mass (GeV)0 100 200 3009.09.29.49.6 b (s-s) 0 100 200 3009.09.29.49.6 (s-s) 0 100 200 3009.509.7510.0010.25 hb (s-s) 0 100 200 3009.509.7510.0010.25 b1 (s-s) Figure 3: Plots showing the estimate of the mass versus lattice temperature for select bottomonium states. The horizontal dashed line represents the PDG estimate for the given state [11]. 0.0 0.2 0.4 0.6 0.8 1.0 Temperature (MeV)0.00.20.40.60.81.0 (MeV) 0 100 200 300050010001500 b (s-s) 0 100 200 300050010001500 (s-s) 0 100 200 300050010001500 hb (s-s) 0 100 200 300050010001500 b1 (s-s) Figure 4: Plots showing the estimate of the upper bound on the width versus lattice temperature for select bottomonium states. The shaded region represents our best estimate of the resolving power of the method. 6Novel Bottomonium Results Ben Page 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 a/2 9.39.49.59.69.79.89.910.0Predicted Mass (GeV)T=47 MeV Predicted Masses b2 hb b1 b0 b PDG Masses b2 hb b1 b0 b Figure 5: Value of the energy shift Δsing(i.e. the predicted mass) which gives the most singular shifted covariance matrix (see Eq. 9) for a variety of bottomonium channels as a function of 1𝜏2. The covariance matricesaredefinedoverthetimeinterval 0𝜏 𝜏 2,andthereforethebestresultsareobtainedas 𝜏2!1. Thelatticetemperatureis47MeV.Thepredictedmasseseachmesontendstowardtheexperimentalestimate for the pseudoscalar mass. Experimental values for the meson masses are shown as horizontal lines [11]. 7. Connection with Parisi-Lepage Statistical Scaling Toestimatetheerrorintheresultingreconstruction,theuncertaintyintheEuclideancorrelator Δ𝐺¹𝜏ºmust be combined with the Backus Gilbert coefficients 𝑐𝜏. The uncertainty corresponding to Eq. 3 is simply Δˆ𝜌2=∑︁ 𝜏𝜏0𝑐𝜏Cov»𝐺¼𝜏𝜏0𝑐𝜏0 (8) where Cov»𝐺¼is the covariance in 𝐺¹𝜏º. Under the Laplace transformation outlined in Eq. 7, the covariance matrix transforms as Cov»𝐺;Δ¼𝜏𝜏0=𝑒Δ
𝜏Cov»𝐺¼𝜏𝜏0𝑒Δ
𝜏0(9) whichinturninfluencesthespectralerror Δˆ𝜌. Thiseffectcanbeprobedbymeasuringthecondition number of the resulting matrix, defined by 𝜅¹Cov»𝐺;Δ¼º=𝜎max 𝜎min(10) where𝜎are the singular values of the matrix. It is interesting to study 𝜅as a function of Δand determine the value, Δsingwhen Cov»𝐺;Δ¼ becomes singular. One may imagine that Δsingis the ground state mass of the 𝐺¹𝜏ºchannel. However,aspointedoutbyParisi[13]andelucidatedfurtherbyLepage[14],thecovariancematrix hasaspecialphysicalsignificance. Itcanbeexpressedasacorrelationfunctionofthe squareofthe 7Novel Bottomonium Results Ben Page interpolatingoperatorsoftheoriginalcorrelationfunction, 𝐺¹𝜏º. Analysingthisfurther,onefinds thatthelighteststatewhichcontributestothecovariancematrixisthepseudoscalar,nomatterwhat state was being probed by 𝐺¹𝜏º. This therefore implies that Cov »𝐺;Δ¼becomes singular when Δ=Δsingis the pseudoscalar mass (i.e. the 𝜂𝑏mass in our case) independent of the channel. We illustrate this in Fig. 5 where Δsingis plotted for a variety of channels. The covariance matrix was defined over the time interval 0𝜏 𝜏 2meaning that the large time limit (where the groundstatedominates)isobtainedas 𝜏2!1. Ascanbeseen,inthislimitwerecoverthe 𝜂𝑏(i.e. pseudoscalar) mass, thereby confirming the prediction of [13, 14]. 8. Summary Wehavepresentedresultsforthegroundstatemassandanestimatefortheupperboundonthe width for several bottomonium states using smeared quark sources, showing improved resolution compared to our previous results. We have demonstrated the ability of the Laplace shift to naively increase the resolving power of the method, but show that is still insufficient to resolve the ground state widths of the system. The effect of the Laplace shift transform on the covariance matrix of theEuclideancorrelationfunctionwasalsostudied,wheretheconditionnumberofthematrixwas found to confirm Parisi-Lepage statistical scaling in the long-time limit. Acknowledgments ThisworkissupportedbySTFCgrantST/T000813/1. SKissupportedbytheNationalResearch FoundationofKoreaundergrantNRF-2021R1A2C1092701andgrantNRF-2021K1A3A1A16096820, funded by the Korean government (MEST). BP has been supported by a Swansea University Re- search Excellence Scholarship (SURES). This work used the DiRAC Extreme Scaling service at theUniversityofEdinburgh,operatedbytheEdinburghParallelComputingCentreandtheDiRAC Data Intensive service operated by the University of Leicester IT Services on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS capital funding via STFC capital grants ST/R00238X/1, ST/K000373/1 and ST/R002363/1 and STFC DiRAC Opera- tionsgrantsST/R001006/1andST/R001014/1. DiRACispartoftheUKNationale-Infrastructure. This work was performed using PRACE resources at Cineca (Italy), CEA (France) and Stuttgart (Germany) via grants 2015133079, 2018194714, 2019214714 and 2020214714. We acknowledge thesupportoftheSwanseaAcademyforAdvancedComputing,theSupercomputingWalesproject, whichispart-fundedbytheEuropeanRegionalDevelopmentFund(ERDF)viaWelshGovernment, and the University of Southern Denmark and ICHEC, Ireland for use of computing facilities. We are grateful to the Hadron Spectrum Collaboration for the use of their zero temperature ensemble. References [1] G. Aarts, S. Kim, M. P. Lombardo, M. B. Oktay, S. M. Ryan, D. K. Sinclair et al., arXiv:1/zero.alt31/zero.alt3.3725 [hep-lat, physics:hep-ph, physics:nucl-th] . [2] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, M. B. Oktay, S. M. Ryan et al., arXiv:11/zero.alt39.4496 [hep-lat, physics:hep-ph, physics:nucl-th] . 8Novel Bottomonium Results Ben Page [3] G. Aarts, C. Allton, T. Harris, S. Kim, M. P. Lombardo, S. M. Ryan et al., arXiv:14/zero.alt32.621/zero.alt3 [hep-lat, physics:hep-ph] . [4] B. Page, G. Aarts, C. Allton, B. Jäger, S. Kim, M. P. Lombardo et al., arXiv:2112./zero.alt32/zero.alt375 [hep-lat] . [5] G. Lepage, in Nuclear Physics B - Proceedings Supplements , vol. 26, pp. 45–56. DOI. [6] G. Aarts, C. Allton, J. Glesaaen, S. Hands, B. Jäger, S. Kim et al., arXiv:2/zero.alt3/zero.alt37./zero.alt34188 [hep-lat, physics:hep-ph, physics:nucl-th] . [7] G. Backus and F. Gilbert, The Resolving Power of Gross Earth Data , vol. 16. 10.1111/j.1365-246X.1968.tb00216.x. [8] D. W. Oldenburg, An introduction to linear inverse theory , vol. GE-22. 10.1109/TGRS.1984.6499187. [9] A. N. Tikhonov, in On the Stability of Inverse Problems , vol. 39, pp. 195–198. [10] A. Rothkopf, arXiv:12/zero.alt38.5162 [hep-lat, physics:nucl-th, physics:physics] . [11] Particle Data Group, R. L. Workman, V. D. Burkert, V. Crede, E. Klempt, U. Thoma et al., . [12] G. Aarts, C. Allton, R. Bignell, T. J. Burns, S. C. García-Mascaraque, S. Hands et al., arXiv:22/zero.alt39.14681 [hep-lat, physics:nucl-th] . [13] G. Parisi, The strategy for computing the hadronic mass spectrum , vol. 103. 10.1016/0370-1573(84)90081-4. [14] G. P. Lepage, in The Analysis of Algorithms for Lattice Field Theory , pp. 97–120. 9

2022-12-22

We present the latest results from the use of the Backus-Gilbert method for reconstructing the spectra of NRQCD bottomonium mesons using anisotropic FASTSUM ensembles at non-zero temperature. We focus in particular on results from the $\eta_b$, $\Upsilon$, $\chi_{b1}$ and $h_b$ generated from Tikhonov-regularized Backus-Gilbert coefficient sets. We extend previous work on the Laplace shifting theorem as a means of resolution improvement and present new results from its use. We conclude with a discussion of the limitations of the improvement routine and elucidate a connection with Parisi-Lepage statistical scaling.

Novel Bottomonium Results

2212.12016v1

DECAY RATES FOR KELVIN-VOIGT DAMPED WAVE EQUATIONS II: THE GEOMETRIC CONTROL CONDITION NICOLAS BURQ AND CHENMIN SUN Abstract. We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. When the damping coecient is suciently smooth ( C1vanishing nicely, see (1.3)) we show that exponential decay follows from geometric control conditions (see [5, 12] for similar results under stronger assumptions on the damping function). 1.Introduction In this paper we investigate decay rates for Kelvin-Voigt damped wave equations under geometric control conditions. We work in a smooth bounded domain Rdand consider the following equation (1.1)8 >< >:(@2 t
)udiv(a(x)rx@tu) = 0 ujt=0=u02H1 0( ); @tujt=0=u12L2( ) uj@ = 0 with a non negative damping term a(x). The solution can be written as (1.2) U(t) =u @tu =eAtu0 u1 ; where the generator Aof the semi-group is given by A= 0 1
divar u0 u1 ; with domain D(A) =f(u0;u1)2H1 0L2;
u0+ divaru12L2;u12H1 0g: The energy of solutions E(u)(t) =Z (jrxuj2+j@tuj2)dx satises E((u0;u1))(t)E((u0;u1))(0) =Zt 0Z a(x)jrx@tuj2(s;x)ds: Our purpose here is to show that if the damping ais suciently smooth, the exponential decay rate holds, dropping some unnecessary assumptions on the behaviour of the damping term where it becomes positive in previous works [5]. Namely we shall assume a(x)>0 isC1( ) and satisfy the regularity hypothesis jraj6Ca1 2: (1.3) Our main result is Theorem 1. Assume that is a compact Riemannian manifold with smooth boundary. Let a2C1( )be a nonnegative function satisfying (1.3) , such that the following geometric control condition is satised: There exists  > 0such that all rays of geometric optics (straight lines) re ecting on the boundary according to the laws of geometric optics eventually reach the set !=fx2 :a(x)>gin nite time. 1arXiv:2010.05614v2 [math.AP] 19 Mar 20212 N. BURQ AND C-M. SUN Then there exists >0, such that for all t>0and every (u0;u1)2H1 0( )L2( ), the energy of solution u(t) of(1.1) with initial data (u0;u1)satises E[u](t)6etE[u](0): To prove this result, we rst reduce it very classicaly in Section 2 to resolvent estimates. Since the low frequency estimates are true, we are reduced to the high frequency regime. The proof relies on resolvent estimates which are proved through a contradiction argument that we establish in Section 2. In Section 3 we prove a priori estimates for our sequences. The main task then is to prove a propagation invariance for these measures. A main diculty to overcome is that it is not possible to put the damping term in the r.h.s. of the equation (1.1) and treat it as a perturbation . Instead we have to keep it on the left hand side and revisit the proof of the propagation property from [7]. In Section 4, we introduce the geometric tools necessary to tackle the boundary value problem and dene semi-classical measures associated to our sequences. In Section 5 we prove the interior propagation result for our measures. Finally, in Section 6, we nish the proof of the contradiction argument by establishing the invariance of the semi-classical measures we dened up to the boundary. Here the proof uses crucially the main result in [7, Th eor eme 1]. Remark 1.1. Throughout this note, we shall prove that some operators of the type PId,2R(resp. 2iR) are invertible with estimates on the inverse. All these operators share the feature that they have compact resolvent, i.e. 9z02C; (Pz0)1exists and is compact (or it will be possible to reduce the question to this situation). As a consequence, since (P) = (Pz0)1(Id + (z0))1); and (Id + (z0)1) is Fredholm with index 0, to show that ( P) is invertible with inverse bounded in norm byA, it is enough to bound the solutions of ( P)u=fand prove (P)u=f)kukL26AkfkL2: Remark 1.2. Assume that ais the restriction to of a nonnegative C2(Rd) function. Then the hypothesis (1.3) is satised. Proof. It is enough to prove (1.3) for = Rd,a2C2( ). Letx02Rdand denote by z0=ra(x0) From Taylor's formula, we have for any s2R, there exists 2(0;1), such that a(x0+sz0) =a(x0) +sjz0j2+s2 2a00(x0+sz0)(z0;z0)>0 Since this polynomial in sis non negative, we deduce tat its discriminant is non positive jz0j42ka00k1jz0j2a(z0)60)jrxa(x0)jj262ka00k1a(z0): Notice that in the above lemma, the condition cannot be relaxed to a2C2( );a>0. Indeed, consider the following example: = B(0;1) anda(x) = 1jxj2forjxj61. Then obviously a2C2( ),a>0 , but on the boundary,rxa6= 0, whilea= 0.  Acknowledgment. The rst author is supported by Institut Universitaire de France and ANR grant ISDEEC, ANR-16-CE40-0013. The second author is supported by the postdoc programe: \Initiative d'Excellence Paris Seine" of CY Cergy-Paris Universit e and ANR grant ODA (ANR-18-CE40- 0020-01). 2.Contradiction argument It is well known that decay estimates for the evolution semi-group follow from resolvent estimates [1, 2, 4]. Here we shall need the classical (see e.g. [6, Proposition A.1]) Theorem 2. The exponential decay of the Kelvin Voigt semi-group is equivalent to the following resolvent estimate: There exists Csuch that for all 2R, the operator (Ai)is invertible from D(A)toHand its inverse satises (2.1) k(Ai)1kL(H)6CKELVIN-VOIGT DAMPING 3 Let us rst recall that (2.2) ( Ai)u v =f g ,iu+v=f
u+ divarxviv=g From [8, Section 2], we have the following low frequencies estimates of the resolvent of the operator A: Proposition 2.1. Assume that a2L1is non negative a>0and non trivialR a(x)dx > 0. Then for any M > 0, there exists C > 0such that for all 2R;jj6M, the operatorAiis invertible from D(A)toH with estimate (2.3) k(Ai)1kL(H)6C: As a consequence, to prove Theorem 1 it is enough to study the high frequency regime !+1and prove Proposition 2.2. Assume that a2C1( )is a nonnegative function satisfying (1.3) . Then under the geometric control condition, there exists 0>0such that for any jj>0we have k(Ai)1kL(H)6C: By standard argument, we can reduce the proof of Proposition 2.2 to a semi-classical estimate. We denote by 0<h=jj11 and Ph=h2
1ihdiva(x)r: Proposition 2.3. There exists C > 0, such that for all 0<h1, kP1 hkL(L2)6Ch1: (2.4) For the proof of Proposition 2.3, we argue by contradiction. Assume that there exist sequences ( un) H2\H1 0;(fn)L2andhn!0, such that Phnun=fn,kunkL2= 1 andkfnkL2=o(hn). We will use a semi-classical notation and denote by ( uh;fh) the sequences with the properties kuhkL2= 1;kfhkL2=o(h); Phuh=fh: (2.5) Sometimes we even omit the subindex for uh;fh. In the following subsections, we will prove propagation estimates for such sequences. 3.A priori estimates In this section we establish a series of a priori estimates for the sequence dened in (2.5). Lemma 3.1. Assume that a2L1( )is a non-negative function, then (1)Z juhj2jhruhj2
= ReZ fhuh=o(h); (3.1) (2)Z a(x)jhruhj2=hImZ fhuh=o(h2):; (3.2) (3)h2kuhkH2=O(1): (3.3) Proof. We get (1) and (2) by multiplying the equation Phu=fbyu, integrating by part and taking the real and imaginary parts repectively. For (3), from the equation, we have h2
u+u+ihra
ru+iha
u=f;i.e.h2
u=u+f+ira
hru 1 +ih1a(x): From the global estimate of the Poisson equation kwkH26Ck
wkL2;8w2H2\H1 0; we obtain thatkh2ukH2=O(1).  Corollary 3.2. Assume that a2C1( )is a non-negative function satisfying (1.3) , then ka1 2uhkL2+ka1 2hruhk+ka1 2h2
uhkL2=o(h): (3.4)4 N. BURQ AND C-M. SUN Proof. We only need to estimateR a(x)juj2, sinceR a(x)jhruj2=o(h2) is just (3.2). Multiplying Phu=fby auand taking the real part, we haveZ a(x)juj2= ReZ hru
hr(au)ImhZ a(x)ru
r(a(x)u) + ImZ a(x)fu: Sincejraj6Ca1 2, the rst term on the r.h.s. can be bounded by ka1 2hruk2 L2+hkrahrukL2kukL2=o(h2): The third term of r.h.s is bounded by o(h)ka1 2ukL2, and the second term can be bounded by hZ ara
uru6hka1 2ukL2ka1 2rarukL26Cka1 2hrukka1 2ukL2=o(h)ka1 2ukL2: For the second derivative, we observe that a(x)1 2h2
u=a1 2u+a1 2f+iha1 2ra
hru 1 +ih1a(x); thuska1 2h2
ukL2=o(h). This completes the proof of Corollary 3.2.  Letbe the out-normal vector eld on @ . We denote by L2(@) =L2(@ ). The following hidden regularity holds: Lemma 3.3. Assume that a2C1( )is a nonnegative function satisfying (1.3) , then kh@ukL2(@)=O(1);ka1 2h@ukL2(@)=O(h1 2): Proof. We use the standard multiplier method. Let L=bj(x)@jbe anC2extension of the out-normal vector eld, wherebj's are supported in a neighborhood of @ . WritePh=Ph;0+iMh, where Ph;0=h2
1; Mh=hdiva(x)r are self-adjoint operators. Consider the commutator [ Ph;L] = [Ph;0;L] +i[Mh;L]. Note that [ Ph;0;L] = 1 h[Ph;0;hL] belongs to h2Op (S2), we deduce that [Ph;0;L]u;u
L2=O(1). By direct computation, we have [Mh;L]u;u
L2= h@k[a@k;bj@j]u;u
L2+h [@k;bj@j]a@ku;u
L2 = h@k(a(@kbj)@jubj(@ja)@ku);u
L2+ (@kbj)h@j(a@ku);u
L2 = (a@kbj)@jubj(@ja)@ku;h@ku
L2 a@ku;h@j((@kbj)u):
L2 From Corollary 3.2, the absolute value of the r.h.s. can be bounded by constant times karukL2+krarukL2=o(1): Therefore, [Ph;L]u;u
L2=O(1). On the other hand, by developing the commutator and exploiting the equation, we have [Ph;L]u;u
L2= PhLu;u
L2 Lf;u
L2 = Lu;P hu
L2 f;Lu
L2+h2k@uk2 L2(@)+ihka1 2@uk2 L2(@): Observe that Lu;P hu
L2= Lu;f2Mhu
L2=o(1)2 Lu;Mhu
L2=o(1)2 Lu;hra
ru+ha
u
L2: SinceLis a rst order dierential operator and from Corollary 3.2 that ka1 2h
ukL2=o(1), we have j Lu;hra
ru+ha
u
L2j6khrukL2krarukL2+ka1 2rukL2ka1 2h
ukL2=o(1): Therefore, kh@uk2 L2(@)+ihka1 2@uk2 L2(@)=O(1): Strictly speaking, the symbol of Lis not C1, but here we only need1 h[Ph;0; hL] =O(h2) onL2.KELVIN-VOIGT DAMPING 5 The proof of Lemma 3.3 is then completed by taking real and imaginary parts.  Let2C1 c(R) such that (z)1 forjzj61 and(z)0 forjzj>2. We decompose (3.5) uh=vh+wh; vh= ah1
uh; wh= 1 ah1

uh: In the rest of this note, we always assume that a2C1( ) is a nonnegative function satisfying (1.3). Lemma 3.4. We have kwhkL2+khrwhkL2=o(h1 2);ka1 2vhkL2+ka1 2hrvhkL2=o(h); and kuhkH1 h(a>ch)+kvhkH1 h(a>ch)=o(h1 2): Proof. By denition,Z jwj2+jhrwj2
6Z a>h juj2+jhruj2+jraj2juj2
: The conclusion then follows from Corollary 3.2 and the fact that jraj26Ca. Similarly, for any other cuto to the region a>ch, we deduce that kuhkH1 h(a>ch)=o(h1 2):For the estimate of v, note that a1 2hrv= a1 2hru+a1 2ra0v, from Corollary 3.2, we have ka1 2hrvkL26ka1 2hrukL2+k0a1 2(a1 2v)kL2=o(h): This completes the proof of Lemma 3.4.  4.Geometry, semi-classical measures Having the a priori estimates of the previous section at hand, we can now study vh. For some subsequence ofvh, we will associate it a semi-classical measure and then prove the invariance of the measure under the generalized geodesic ow. First recall some geometric preliminaries from [7]. 4.1.Geometry. Denote bybT the bundle of rank dwhose sections are the vector elds tangent to @ ,bT the dual bundle (Melrose's compressed cotangent bundle) and j:T !bT the canonical map. In any coordinate system where = fx= (xd>0;x0)g), the bundlebT is generated by the elds@ @x0,xd@ @xdandj is dened by (4.1) j(xd;x0;d;0) = (xd;x0;v=xdd;0): Denote by Car P0the semi-classical characteristic manifold of P0=h2
1 andZits projection (4.2) Car P0= (x;) = (x0;xd;0;d)2TRdj ;p(x;) = 0 ; Z =j(CarP0): The setZis a locally compact metric space. Consider, near a point x02@ a geodesic system of coordinates for which x0= (0;0), =f(xd;x0)2 R+Rd1gand the operator P0has the form (near x0) (4.3) Ph;0=h2
1 =h2D2 xdR(xd;x0;hDx0) +hQ(x;hDx); withRa second order tangential operator and Qa rst order operator. We recall now the usual decomposition of T@ (in this coordinate system). Denote by r(x0;xd;0) the semi-classical principal symbol of Randr0=rjxd=0. ThenT@ is the disjoint union of E[G[H with (4.4) E=fr0<0g;G=fr0= 0g;H=fr0>0g: Remark that jgives a natural identication between Zj@MandH[GT@M. InGwe distinguish between the diractive pointsG2;+=fr0= 0;r1=@xdrjxd=0>0gand the gliding pointsG=fr0= 0;r1=@xdrjxd=060g. We will make the assumption ( has no innite order contact with its tangents) that for any %02T@M, there existsN2Nsuch that HN r0(r1)6= 06 N. BURQ AND C-M. SUN The denition of the generalized bicharacteristic ow, 'sassociated to the operator P0is essentially the denition given in [11]: Denition 4.1. A generalized bicharacteristic curve (s) is a continuous curve from an interval IRtoZ such that (1) ifs02Iand (s0)2T then close to s0, is an integral curve of the Hamiltonian vector eld Hep (2) Ifs02Iand (s0)2H[G2;+then there exists ">0 such that for 0 <jss0j<",xd( (s))>0 (3) Ifs02Iand (s0)2Gthen for any function f2C1(TRdj ) satisfying the symmetry condition (4.5) 8%02Z;8b%0;e%02j1(%0)\Car(eP);f(b%0) =f(e%0) then d dsf(j( (s))js=s0=Hepjj1( (s0))f(j1( (s0))) It is proved in [11] that under the assumption of no innite order contact, through every point %o2bTMnf0g there exists a unique generalized bicharacteristic (which is furthermore a limit of bicharacteristics having only hyperbolic contacts with the boundary). This denes the ow . 4.2.Wigner measures. Consider functions a=ai+a@withai2C1 0(TM), anda@2C1 0(R2d1). Such symbols are quantized in the following way: take 'i2C1 0(M) (resp'@2C1 0(Rd)) equal to 1 near the x-projection of supp( ai) (resp the x-projection of supp( a@)) and dene (4.6) Op'i;'@ h(a)(x;hDx)f=1 (2h)dZ ei(xy)
=hai(x;)'i(y)f(y)dyd +1 (2h)d1Z ei(x0y0)
0=ha(xd;x0;)'(xd;y0)f(xd;y0)dy0d0: Remark that according to the symbolic semi-classical calculus, the operator Op'i;'@ h(a) does not depend on the choice of functions 'i;'@, modulo operators on L2of norms bounded by O(h1). For conciseness we shall in the sequel drop the index 'i;'@. Denote byAhthe space of the operators which are a nite sum of operators obtained as above in suitable coordinate systems near the boundary and for B2A, byb=(B) the semiclassical symbol of the operator A. For such functions bwe can dene (b)2C0(Z) by (4.7) (b)() =b(j1()) (the value is independent of the choice of j1() since the operator is tangential). The set (4.8) f(b);b=(B);B2Ahg is a locally dense subset of C0 c(Z). 4.2.1. Elliptic regularity. The sequence vhsatises (with h=(a=h)) Phvh=hPhuhh2div(rhuh)h2rhruhihdiv(arhuh)iharh
ruh: Sincerh=h10(a=h)raandjraj.a1 2, by Corollary 3.2, we have Phvh=oL2(h) +oH1(h2): Thus (h2
+ 1)vh=ihdiv(ar(hu)) +oL2(h) +oH1(h2): Using Corollary 3.2 again and the fact that a.hon the support of h, we deduce that har(hu) =oL2(h3 2), hence (h2
+ 1)vh=o(h3 2)H1( )+oL2(h): We deduce, by standard elliptic regularity resultsKELVIN-VOIGT DAMPING 7 Proposition 4.2. Ifaiis equal to 0near Car(P0)then (4.9) lim k!+1(Ophk(ai)vhk;vhk)L2= 0; while near the boundary (see e.g. [7, Appendice A.1] in a slightly dierent context) we get Proposition 4.3. Ifa@is equal to 0nearZ(i.e.aiis supported in the elliptic region) then (4.10) lim k!+1(a@(x0;xd;hkDx0)vhk;vhk)L2= 0: Remark 4.4. Note that if we regard the damping term hdiv(ar(hu)) =oH1(h3 2) as a source term, we are not able to use the classical propagation theorem for h2
+ 1 as a black box, as such a strategy would require smaller r.h.s., namely oH1(h2) +oL2(h). On the other hand, an integration by parts shows from Lemma 3.4  div(ar(hu));hu L2=ka1=2rxhuk2 L2=o(1); and this will ensure that in the propagation estimates such terms are invisible. The key of our analysis in the sequel will be to systematically uses this procedure: testing the damping term on expressions like Qhhu, doing the integration by part and then balancing a1 2to the other side. It is to perform this analysis that we need the conditionjraj6Ca1 2to ensure the gain O(h) from the commutator [ a1 2;Qh]. More precisely, we shall need the following lemma: Lemma 4.5. Assume that Qh;B0;h;B1;hare tangential h-pseudodierential operators of order 0andBh= B0;h+B1;hhDxd, then h1 QhMhu;Bhu
L2=o(1); whereMh=hdivar. Proof. SinceMhu=rahruah
u, from Corollary 3.2, we have h1 Qhrahru;Bhu
L26CkrarukL2kukH1 h=o(1): To estimate Qha
u;Bhu
L2, we writeQha
u=a1 2Qha1 2
u+ [Qh;a1 2]a1 2
u. From Corollary 3.2, a1 2
u= oL2(h1). By Corollary A.2, [ Qh;a1 2];[Bh;a1 2] =OL(L2)(h). Therefore, Qha
u;Bhu
L2= Qha1 2
u;Bha1 2u
L2+o(1): Again from Corollary 3.2, we have Bha1 2u=OL2(h), hence (Qha
u;Bhu)L2=o(1). The proof of Lemma 4.5 is complete.  4.2.2. Denition of the measure. The following results gives the existence of semi-classical measures. Proposition 4.6. Let(vhkp)be a sequence bounded in L2( ). There exists a subsequence (kp)and a Radon positive measure onZsuch that (4.11) 8Q2Ahkplim p!1(Qvhkp;vhkp)L2=h;((Q))i: The proof of this result relies on the G arding inequality for tangential operators (see G. Lebeau [10] for a proof in the classical context and [3, 9] for the semi-classical construction). As before, we drop the indexes kp and denote by ( vh) the extracted sequence. Proposition 4.7 (First properties of the measure ).We have (4.12) (H) = 0: Moreover, for any tangential symbol b, (4.13) lim sup k!+1j(Oph(b)hkDxdvhk;vhk)L2j6C sup %2supp(b)jrj1=2jbj:8 N. BURQ AND C-M. SUN Proof. The rst property (4.12) follows from the fact that the trajectories near a hyperbolic point is transversal to the boundary. It follows from [7], with additional attention to the damping term ihdivarvh. We factorize Ph;0=h2
1 as (hDxdL+ h)(hDxdL h)+OH1(h1) (see [7, Lemme 6.1]) near 02Hand choose L hwith principal symbols l(x0;xd;0) =p r(x0;xd;0). we denote by q0(x0;0)2C1 c(H) andq(y;x0;0) solutions of @xdqfl;qg= 0; qjxd=0=q0: Denote by u:= (xd)Q h(hDxdL h)u; where (xd)1 if 06xd60. We have (hDxdL h)u= (xd)[hDxdL h;Q h](hDxdL h)u+Q hfh+h i 0(xd)Q h(hDxdL h)u +iQ hMhu+OL2(h1) whereMh=hdivarandfh=oL2(h). By denition of q, the rst term of r.h.s. is O(h2), hence (hDxdL h)u=g hih 0(xd)Q h(hDxdL h)u+iQ hMhu; g h=oL2(h): (4.14) We have hd dxd(u;u)L2(@)=2 Im g hih 0(xd)Q h(hDxdL h)u+iQ hMhu;u
L2(@)+i (L hL; h)u;u
L2(@): Fory060, we have ku(y0)k2 L2(@)6ku(0)k2 L2(xd=0)+Ch1kg hkL2(xd6y0)kukL2(xd6y0)+Ckuk2 L2(xd6y0) +Ch1(Q hMhu;u)L2(xd6y0) The second line of r.h.s. is o(1), due to Lemma 4.5, and the rst line of r.h.s. can be bounded by ku(0)k2 L2(xd=0)+Ckuk2 L2(xd6y0)+o(1): Integrating both sides over y060, lettingh!0 and then 0!0, we deduce that h1y=0;q0i= 0. This proves (4.12). For (4.13), it suces to prove the inequality for uinstead ofv. By Cauchy-Schwarz,  Oph(b)h@xdu;u
L26 Oph(b)Oph(b)h@xdu;h@xdu
L21 2kukL2: Doing integration by part, we have Oph(b)Oph(b)h@xdu;h@xdu
L2= Oph(b)Oph(b)h2@2 xdu;u
L2+O(h): Replacingh2@2 xduby equation h2@2 xdu=RhuiMhu+OL2(h);we deduce that  Oph(b)Oph(b)h2@2 xdu;u
L26 Oph(jbj2)Rhu;u
L2+O(h) + Oph(b)Oph(b)Mhu;u
L2: From Lemma 4.5, the third term of r.h.s. is o(h). Passingh!0, we complete the proof of Proposition 4.7.  4.3.Invariance of the measure. The key to prove the invariance of the measure will be to apply the propa- gation theorem in [7, Th eor eme 1]. Theorem. The two following properties are equivalent (1)The measure is invariant along the generalised ow. (2)The measure satises _= 0and(G+ 2) = 0 in the sense thath;fp;qgi= 0holds for any even symbol q2C1 c(Car(P0)), i.e.q(x0;xd= 0;0;d) =q(x0;xd= 0;0;d).KELVIN-VOIGT DAMPING 9 Remark 4.8. Technically, Theorem 4.3 is proved in [7] for time dependent measures, i.e. measures depending in addition on two additional variables ( t;)2TR, andpis replaced by p2. However, it is easy to apply the results from [7] by considering the measure (4.15) e=x; dt =1; which is supported in Car(
+@2 t) and satises _e= 0 in the sense that h;fp2;qgi= 0 holds for any even symbolq2C1 c(Car(P02)), i.e.q(x0;xd= 0;t;0;d;) =q(x0;xd= 0;t;0;d;). Remark that though we shall not use it, the measure eis, in the sense of [7, Section 2], the microlocal defect measure on the sequence vn(t;x) =hneith1 nun(x) (the pre-factor hncomes from the H1normalisation of the sequence vnin [7]). Now, the generalised bicharacteristic ow for p02, sis given in terms of the generalised bicharacteristic ow for p0, sby s(t;x; = 1;) = (t2s;= 1; s(x;)); The set of diractive points eG2;+in the time dependent frame-work is given by eG2;+=G2;+Rf=1g and consequently, (G2;+) = 0,e(eG2;+) = 0; and in view of the particular form (4.15), the invariance of eby sis equivalent to the invariance of by s. Let us now brie y explain the procedure we are going to follow. First from Proposition 4.6 and the elllipticity (Proposition 4.9, Proposition 4.10), the measure is dened on Z=j(Car(P0)) by testing on symbols of the form q=qi+q@;qi2C1 c(T ) andq@ tangential (which is dense in C0(Z)). Using the fact (H) = 0 (Proposition 4.7), the measure can be extended to test on functions of Car(P0) which admits a representation (thanks to Malgrange's theorem) q(x0;xd;0;d) =q0(x0;xd;0) +dq1(x0;xd;0);on2 d=r(x0;xd;0): Then, we will show in Proposition 4.9 that for tangential h-pseudodierential operators B0;h;B1;h, the quadratic form ((B0;hk+B1;hk1 ihk@xd))vhk;vhk) converges toh;b0+b1d1=2Hi, by a suitable limit procedure for symbols in Ah. Consequently, for any q2C1 c(Car(P0)), we can make sense of the expression h;fp;qgi -a.e., by viewing fp;qg= 2d@xdq1=2Hfr;qg:We remark that to calculate fp;qg, it is enough to choose one representation q=q0+q1don Car(P0), sincefp;pg= 0 andp= 0 on supp( ). Finally, to prove that the measure is invariant along the Melrose-Sj ostrand ow, we apply Theorem 4.3, for which we need to verify the following conditions: (a)(G2;+) = 0 (b) _= 0, in the sense that h;fp;qgi= 0 holds for any even symbol q2C1 c(Car(P0)), i.e.q(x0;xd= 0;0;d) =q(x0;xd= 0;0;d). The verication of (a),(b) in our context is based on the propagation formula: Proposition 5.1 and Proposi- tion 6.1. Especially, starting from Proposition 6.1, by choosing suitable test symbols of the form q0+q1d, we are able to verify the conditions (a) and (b). Proposition 4.9. IfB0;h;B1;hare two tangential h-pseudodierential operators of with principal symbols b0;b1 of order 0, then we have lim k!1((B0;hk+B1;hk1 ihk@xd)vhk;vhk)L2=h;b0+b1d1=2Hi:10 N. BURQ AND C-M. SUN Proof. SinceB0;handB1;hare all tangential, by the denition of the measure, the rst term ( B0;hkvhk;vhk)L2 converges toh;b0i. It remains to prove the convergence of the second term ( B1;hk1 ihk@xdvhk;vhk)L2. For this, we pick>0;> 0 and dene B1;hk;= 1 xd 

B1;hk 1 xd 2

; B 1;hk=B1;hkB1;hk;; B; 1;hk= Ophk r 

B 1;hk; B 1;hk;=B 1;hkB; 1;hk; where is a cuto function which is 1 near 0. Now by the denition of and the dominating convergence, lim !0lim k!1(B1;hk;1 ihk@xdvhk;vhk)L2=h;b1d1xd>0i=h;b1d1=2Hi; since(E) =(H) = 0. Now from Proposition 4.7, the contribution of lim !0lim sup k!1j(B; 1;hkhk@xdvhk;vhk)L2j6C1 2; which converges to 0 if we let !0. Finally, by Cauchy-Schwarz, j(B 1;hk;hk@xdvhk;vhk)L2j6khk@xdvhkkL2k(B; 1;hk)vhkkL2: Notice that lim k!1k(B; 1;hk)vhkk2 L2= lim k!1(B; 1;hk(B; 1;hk)vhk;vhk)L2 =h; 1 r 

 xd 
1 xd 2

b1+ 1 xd 
xd 2

b1 i; taking the double limit lim sup!0lim sup!0, we obtain that lim sup !0lim sup !0lim k!1k(B; 1;hk)vhkk2 L26h;b2 11xd=01r6=0i= 0; since1E[H= 0. This completes the proof of Proposition 4.9.  5.Interior propagation estimate Proposition 5.1 (Interior propagation) .LetQh=eQhebe ah-pseudodierential operator of order 0, where e2C1 c( ), then we have 1 ih [h2
+ 1;Qh]vh;vh
L2=o(1): Proof. Denote byPh=Ph;0+iMhwithMh=hdivarandPh;0=h2
1, we have 1 ih [Ph;0;Qh]v;v
L2=1 ih Qhv;Ph;0v
L21 ih Ph;0v;Q hv
L2 =1 ih Qhv;Ph;0u
L21 ih Ph;0u;Q hv
L2+R1 withR1=1 ih Qhv;[Ph;0;]u
L21 ih [Ph;0;]u;Q hv
L2: By using the equation Ph;0u=fiMhu, we have 1 ih [Ph;0;Qh]v;v
L2=R1+R2+o(1); (5.1) where R2=1 h Qhv;Mhu
L2+1 h(Mhu;Q hv)L2: Note that [Ph;0;] =hr ra0(a=h)
+ 2ra0(a=h)hr; sincehr((a=h)) =ra0(a=h). Claim 1:R1=o(1)KELVIN-VOIGT DAMPING 11 It suces to show that ih1 Bhv;[Ph;0;]u
L2=o(1) for any compact supported h-pseudoBhof degree 0. By integration by part, 1 ih Bhv;[Ph;0;]u
L2=1 ih Bhv;hdiv (0(a h)rau) +0(a h)rahru
L2 and we simply apply Corollary 3.2, to get for each term o(1). Claim 2:R2=o(1) It suces to prove that ( Qhv;div (aru))L2=o(1). We write Qhv;divaru
L2= (r)Qhv;aru
L2 [r;Qh]v;aru
L2 Qhrv;aru
L2: Sinceja1 2rj=h1ja1 20raj6C, from Corollary 3.2, the rst term of r.h.s. can be bounded by kQhvkL2ka1 2rukL2=o(1): The second term of r.h.s. can be bounded by o(h). Observe that r(a1 2) =1 2a1 2rais bounded, thus from Corollary A.2, [a1 2;Qh] =OL(L2)(h): Therefore, Qhrv;aru
L26 Qha1 2rv;a1 2ru
L2+ [a1 2;Qh]rv;a1 2ru
L2: The second term is bounded by ChkrvkL2ka1 2rukL2=o(1), and the rst term can be bounded by o(1), due to Lemma 3.4. This completes the proof of Proposition 5.1.  6.Propagation near the boundary Recall that vh=(a=h)uh. Consider the operator Bh=B0;h+B1;hh i@xd whereBj;h=e1Oph(bj)e1,j= 0;1 are two tangential operators and e1has compact support near a point z02@ . Note that in the local coordinate system, Ph;0=h2
1 =1p jgjh@xdp jgjh@xdRh; whereRhis a self-adjoint tangential operator of order 2. The operator involving the damping can be written as Mh=hp jgj@xdp jgja@xdhp jgj@x0 kp jgjagjk@x0 j Proposition 6.1 (Boundary propagation) . 1 ih [Ph;0;Bh]v;v
L2= B1;hjxd=0(h@xdv)jxd=0;(h@xdv)jxd=0
L2(@)+o(1): Proof. We give the proof in the case B0;h= 0. TheB0;hterms are handled by slightly simpler versions of the same computations. By developing the commutator, we have 1 ih [Ph;0;Bh]v;v
L2=1 ih Bhv;Ph;0v
L21 ih BhPh;0v;v
L2+ B1;hjxd=0(h@xdv)jy=0;h@xdvjxd=0
L2(@); where the boundary term (the third) comes from the integration by part of the term 1 ih1p jgjh@xdp jgjh@xdv;v
L2; sinceRhis self-adjoint tangential operator. It suces to show that Ih:=1 ih B1;hh@xdv;Ph;0v
L21 ih B1;hh@xdPh;0v;v
L2=o(1): (6.1)12 N. BURQ AND C-M. SUN Sincev=uandPh;0u=PhuiMhu=fhiMhu, we have Ph;0v=Ph;0u+ [Ph;0;]u=fhiMhu+ [Ph;0;]u: Therefore, Ih=o(1) + Ih;1+ Ih;2; where Ih;1=1 ih B1;hh@xdv;[Ph;0;]u
L21 ih B1;hh@xd[Ph;0;]u;v
L2 and Ih;2=1 h B1;hh@xdv;Mhu
L21 h B1;hh@xdMhu;v
L2 Claim 1:Ih;1=o(1). Indeed, from integration by part, the second term ih1(B1;hh@xd[Ph;0;]u;v)L2=ih1([Ph;0;]u;h@xdAhv)L2 for some tangential operator Ah, hence it has the same structure as the rst term. It suces to show that h1 B1;hh@xdv;[Ph;0;]u
L2=o(1): Since [Ph;0;]u=hr
(ra0(a=h))u+ 2ra0(a=h)
hru; doing integration by part, we obtain that h1 B1;hh@xdv;[Ph;0;]u
L2= r(uB1;hh@xdv);ra0
L2+ 2h1 B1;hh@xdv;ra0hru
L2 = rB1;hh@xdv;ra0u
L2+h1 B1;hh@xdv;ra0hru
L2: Note thatv=u, if one of the derivatives h@xd,hrfall on(a=h) we can bound them from Corollary 3.2 by o(h). If all the derivatives fall on uin anyone of the two terms, by Lemma 3.1 and Corollary 3.2, these terms can be bounded by khr@xdukL2kraukL2+h1kh@xdukL2krahrukL2=o(1): Claim 2: Ih;2=o(1). It suces to prove that h1 B1;hh@xd(u);Mhu
L2=o(1): Note thatMhu=rahru+ah
u=oL2(1) andh@xd(u) =@xda0u+h@xdu. We have h1j B1;h@xda0u;Mhu
L2j6h1kraukL2kMhukL2=o(1); sincekraukL2=o(h) from Corollary 3.2. It remains to show that h1 B1;hh@xdu;(ra
hru+ah
u)
L2=o(1): SincekrahrukL2=o(h), we haveh1 B1;hh@xdu;(ra
hru)
L2=o(1). Finally, we show that h1 B1;hh@xdu;ah
u
L2=o(1): Recall that from jr(a1 2)j6Cand Corollary A.2, [B1;h;a1 2] =OL(L2)(h); we have h1j B1;hh@xdu;ah
u
L2j6h1j B1;ha1 2h@xdu;a1 2h
u
L2j+h1j [B1;h;a1 2]h@xdu;a1 2h
u
L2j 6Ch1ka1 2hrukL2ka1 2h
ukL2+CkhrukL2ka1 2h
ukL2=o(1): This completes the proof of Proposition 6.1. KELVIN-VOIGT DAMPING 13 To show that the semi-classical measure of (vhk) is invariant along the Melrose-Sj ostrand ow (to complete the proof of Proposition 2.3), we need to verify the condition (2) in Theorem 4.3. We will make use of the propagation formula, i.e. Proposition 6.1. Formally, for Bh=B0;h+B1;h1 ih@xd, the principal symbol of i h[Ph;0;Bh] is given by f2r;b0+b1dg=a0+a1d+a22 d; where a0=b1@xdrfr;b0g0; a 1= 2@xdb0fr;b1g0; a 2= 2@xdb1; (6.2) andf
;
g0is the Poisson bracket for ( x0;0) variables. On the other hand, by calculating the commutator, we nd i h[Ph;0;Bh] =A0+A1hDxd+A2h2D2 xd+hOph(S0 @+S0 @d); (6.3) whereA0;A1;A2are tangential operators with symbols a0;a1;a2, with respectively. We will prove the following propagation formula: Corollary 6.2. Assume that Bh=Bh;0+Bh;1hDxd, whereBh;0;Bh;1are tangential operators of order 0 with symbolsb0;b1, with respectively. Assume that b=b0+b1d. Dene the formal Poisson bracket fp;bg= (a0+a2r) +a1d1=2H; wherea0;a1;a2are given by (6.2) . Then the defect measure satises the equation h;fp;bgi=h@;b1i; where@is the semiclassical measure of (h@xdvhjxd=0):Moreover, if bis an even symbol (i.e. b(x0;xd= 0;0;d) =b(x0;xd= 0;0;d)), then we have h;fp;bgi= 0: In particular, by combining Proposition 5.1, we have _= 0. Proof. From Proposition 6.1 and the decomposition (6.3), we have (A0+A1hkDxd+A2h2 kD2 xd)vhk;vhk
L2=h@;b1i+o(1): (6.4) From Lemma 3.4, we can also replace the function vhkon the l.h.s. by uhk. Using the equation of uhk: (h2D2 xdRhk)uhk=iMhkuhkfhk+OL2(hk); we deduce that (A2h2 kD2 xduhk;uhk) = (A2Rhkuhk;uhk)L2+o(1); thanks to Lemma 4.5. Therefore, from Proposition 4.9, lim kh!1((A0+A1hkDxd+A2Rhk)uhk;uhk)L2=h;a0+a1d1=2H+a2ri=h;fp;bgi: Now ifb=b0+b1dis an even symbol, we must have b1jxd=0= 0, therefore,h;fp;bgi=h@;b1i= 0. The proof of Lemma 6.2 is complete.  Corollary 6.3. We have(G2;+) = 0 . Proof. We will make use of the formula h;fp;bgi=h@;b1iby choosing b=b1;with b1;(x0;xd;0) = xd 1 2
r(xd;x0;0) 
(xd;x0;0); where 2C1 c(R) equals to 1 near the origin and (y;x0;0)>0 near a point 02G2;+. Sincefp;bg= (a0+a2r) +a1d1=2H, anda0;a1;a2are given by the relation (6.2). In particular for our choice, by direct calculation we have a0=b1;@xdr; a 1=fr;g0 xd 1 2
r 
;14 N. BURQ AND C-M. SUN and a2= 2@xdb1;= 21 2 0xd 1 2
r 
+ 2@xdr  xd 1 2
0r 
+ 2 xd 1 2
r 
@xd: Note thata2is uniformly bounded in and for any xed ( y;x0;0),ra2!0 as!0. Thus by dominating convergence, we have lim !0h;fp;bgi=h;jxd=0@xdr1r=0i>0 since@xdr>0 onG2;+. However,h@;bi60, we must have 1G2;+= 0. This completes the proof of Lemma 6.3.  From Lemma 6.2 and Lemma 6.3, we have veried that _ = 0 and(G2;+) = 0, thus from Theorem 4.3, the semi-classical is invariant along the Melrose-Sj ostrand ow. Thanks to the geometric control condition and the fact that a1 2vhk=oL2(1), we deduce that = 0. This contradicts to the assumption that kvhkkL2= kuhkkL2= 1 +o(1), ask!1 . The proof of Proposition 2.3 is now complete. Appendix A.Some commutator estimates Lemma A.1. Assume that b(x;y; )2L1(R3d x;y;)such that j@ b(x;y; )j.hi(jj+1) for all multi-index 2Nd;jj6d+ 1. Then the operator Thassociated with the Schwartz kernel Kh(x;y) :=1 (2h)dZ Rdb(x;y; )ei(xy)
 hd is bounded on L2(Rd), uniformly in 0<h61. Proof. Using the Littlewood-Paley decomposition, we can decompose the operator Th=P j>0Th;jwhere each Th;jhas the Schwartz kernel Kh;j(x;y) =1 (2h)dZ Rdbj(x;y; )ei(xy)
 hd; withbj(x;y; ) =b(x;y; ) j() and j() = (2j) for some 2C1 c(1 26jj62), ifj>1 and 0() is supported onjj61. Note that (xy)Kh;j(x;y) =ijjhjj (2h)dZ RdD bj(x;y; )
ei(xy)
 hd; we have jKh;j(x;y)j.hjjd jxyjjj
2j(jj+1d): We have another trivial bound jKh;j(x;y)j.2jdhd:Therefore, for xed x2Rd, by choosingjj=d+ 1, we haveZ RdjKh;j(x;y)jdy.Z Rdmin 2j2jh jxyjd+1;2jdhd dy 6Z jzj62j d+1
2jh2jdhddz+Z jzj>2j d+1
2jh2j2jh jzjd+1dz.2jd d+1: Similarly, for xed y2Rd,Z RdjKh;j(x;y)jdx.2jd d+1: By Schur's test, we have kTh;jkL(L2).2jd d+1. Using the triangle inequality, we obtain that This bounded on L2(Rd), uniformly in 0 <h61. The proof of Lemma A.1 is now complete. KELVIN-VOIGT DAMPING 15 Corollary A.2. Assume that 2W1;1(Rd)andb2S0(R2d)is a symbol of order zero, then we have k[Oph(b);]kL(L2)=O(h): Proof. The kernel of [Oph(b);] is given by K(x;y) =1 (2h)dZ Rdb(x;)((y)(x))ei(xy)
 hd: Since2W1;1, there exists 2L1(Rd;Rd) such that (y)(x) = (yx)
(x;y): Thus K(x;y) =dX j=1h i(2h)dZ Rd@jb(x;) j(x;y))ei(xy) hd Applying Lemma A.1 to each @jb(x;) j(x;y), the proof of Corollary A.2 is complete.  References [1] Charles J. K. Batty and Thomas Duyckaerts. Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. , 8(4):765{780, 2008. [2] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann. , 347(2):455{478, 2010. [3] N. Burq. Mesures semi-classiques et mesures de d efaut. S eminaire Bourbaki , Mars 1997. [4] N. Burq. D ecroissance de l' energie locale de l' equation des ondes pour le probl eme ext erieur et absence de r esonance au voisinage du r eel. Acta Mathematica , 180:1{29, 1998. [5] N. Burq and H. Christianson. Imperfect geometric control and overdamping for the damped wave equation. Comm. Math. Phys. , 336(1):101{130, 2015. [6] N. Burq and P. G erard. Stabilization of wave equations on the torus with rough dampings. To appear Pure and Applied Analysis. preprint arxiv https://arxiv.org/abs/1801.00983 , 2020. [7] N. Burq and G. Lebeau. Mesures de d efaut de compacit e, application au syst eme de Lam e. Ann. Sci. Ecole Norm. Sup. (4) , 34(6):817{870, 2001. [8] Nicolas Burq. Decays for Kelvin-Voigt damped wave equations I: the black box perturbative method. SIAM J. Control Optim. , 58(4):1893{1905, 2020. [9] P. G erard and E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Mathematical Journal , 71:559{607, 1993. [10] G. Lebeau. Equation des ondes amorties. In A. Boutet de Monvel and V. Marchenko, editors, Algebraic and Geometric Methods in Mathematical Physics , pages 73{109. Kluwer Academic, The Netherlands, 1996. [11] R.B. Melrose and J. Sj ostrand. Singularities of boundary value problems II. Communications in Pure Applied Mathematics , 35:129{168, 1982. [12] Louis Tebou. A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping. C. R. Math. Acad. Sci. Paris , 350(11-12):603{608, 2012. Universit e Paris-Saclay, Laboratoire de math ematiques d'Orsay, UMR 8628 du CNRS, B ^atiment 307, 91405 Orsay Cedex, France and Institut Universitaire de France Email address :nicolas.burq@math.u-psud.fr Universit e de Cergy-Pontoise, Laboratoire de Math ematiques AGM, UMR 8088 du CNRS, 2 av. Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France Email address :chenmin.sun@u-cergy.fr

2020-10-12

We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. We prove that when the damping coefficient is sufficiently smooth ($C^1$ vanishing nicely) we show that exponential decay follows from geometric control conditions (see~\cite{BuCh, Te12} for similar results under stronger assumptions on the damping function).

Decays rates for Kelvin-Voigt damped wave equations II: the geometric control condition

2010.05614v2

arXiv:0811.0425v1 [cond-mat.mtrl-sci] 4 Nov 2008Amplitude-Phase Coupling in a Spin-Torque Nano-Oscillato r Kiwamu Kudo,∗Tazumi Nagasawa, Rie Sato, and Koichi Mizushima Corporate Research and Development Center, Toshiba Corpor ation, Kawasaki, 212-8582, Japan (Dated: October 31, 2018) The spin-torque nano-oscillator in the presence of thermal fluctuation is described by the normal form of the Hopf bifurcation with an additive white noise. By the application of the reduction method, the amplitude-phase coupling factor, which has a si gnificant effect on the power spectrum of the spin-torque nano-oscillator, is calculated from the La ndau-Lifshitz-Gilbert-Slonczewski equation with the nonlinear Gilbert damping. The amplitude-phase co upling factor exhibits a large variation depending on in-plane anisotropy under the practical exter nal fields. When a direct current Iflows into a magnetoresistive (MR) device, a stationary magnetic state becomes un- stable and a steady magnetic oscillation is excited by the spin-transfer torque. The oscillation is expected to be applicableto ananoscalemicrowavesource, i.e., the spin- torque nano-oscillator (STNO).1,2According to the the- ory based on the spin-wave Hamiltonian formalism,3,4,5,6 the frequency nonlinearity plays a key role in determin- ing the behavior of the oscillator. It has been shown that the strong frequency nonlinearity leads to significant ef- fects on the power spectrum of STNO in the presence of thermal fluctuation: a linewidth enhancement5and non-Lorentzian lineshapes6. In this paper, the impor- tantnonlinearityisexamined. FromtheLandau-Lifshitz- Gilbert-Slonczewski (LLGS) equation as the model of STNO,wecalculateexplicitlythemagnitudeofthequan- tity corresponding to the normalized frequency nonlin- earityN/Γeff(see, e.g., Eq. (4) in Ref. 6) of the spin- waveapproach. Inparticular,wetakeaccountofin-plane anisotropy of a magnetic film which has been neglected in the early studies3,4,5,6, finding the large effect of the anisotropy on the nonlinearity. We describe STNO by a generic oscillator model. It is known that small-amplitude oscillations near the Hopf bifurcation point are generally governed by the simple evolution equation for a complex variable W(t) known as the Stuart-Landau (SL) equation.7The SL equation is derived as a normal form of the supercritical Hopf bi- furcation from the general system of ordinary differen- tial equations. Accordingly, the LLGS equation similarly reduces to the SL equation in the case where the Hopf bifurcation,whichrepresentsagenerationofmagneticos- cillations in STNO, occurs. The reduction of the LLGS equation can be executed by the reduction method based on the center-manifold theorem. At finite temperature, there exists inevitable thermal magnetization fluctuation in STNO.8,9We include the thermal effect into the mag- netization dynamics by just adding white noise term to the SL equation, i.e., STNO in the presence of thermal fluctuation is described by the ‘noisy’ Hopf normal form: d˜W d˜t=i˜Ω˜W+(1+iδ)(p−|˜W|2)˜W+η(˜t),(1) where˜Wis the normalized complex variable represent- ing the amplitude and phase of a magnetization vec-torM(see Eq. (7) below). In Eq. (1), ˜Ω represents a fundamental frequency, ˜tis a normalized dimensionless time, andη(˜t) is the zero-mean, white Gaussian noise with the only non-vanishing second moment given by ∝an}bracketle{tη(˜t)¯η(˜t′)∝an}bracketri}ht= 4δ(˜t−˜t′).pis the bifurcation parame- ter. An oscillation is generated when pbecomes positive. In the context of STNO, p∝(I−Ic) whereIcis the threshold current. The parameter δquantifies the cou- pling between the amplitude and phase fluctuations and is called the amplitude-phase coupling factor . It isδthat we calculate numerically in this paper and that corre- sponds to the normalized frequency nonlinearity N/Γeff of the spin-wave approach. The amplitude-phase cou- pling factor δaffects the power spectrum of an oscillator and leads to linewidth enhancement and non-Lorentzian lineshapes.10,11Due toits effect, the factor δisalsocalled thelinewidth enhancement factor .12Eq. (1) is often used as the simplest model of a noisy auto-oscillator in many fields, for example, electrical engineering, chemical reac- tions, optics, biology, and so on.10,13Therefore, we can easily compare STNO with conventional oscillators and clarify its features. The amplitude-phase coupling factor δis obtained in the procedure of the reduction of the LLGS equation. In the following, we first explain the LLGS equation. Then, following Kuramoto’s monograph7, we consider an insta- bility of a steady solution and execute the reduction of the LLGS equation. The magnetic energy density of the free layer of STNO is assumed to have the form E=−M·Hext−Ku M2s(M·ˆx)2+1 24πM·N ·M,(2) whereMsisthesaturationmagnetization, Hext=Hxˆx+ Hyˆy+Hzˆzis an external field, Kuis uniaxial anisotropy along thexdirection, and Nis the demagnetizing ten- sor;N= diag(Nx,Ny,Nz). Using the spherical coordi- nate system (see Fig. 1), we describe the magnetization dynamics of STNO by the LLGS equation /braceleftBigg cosψ˙φ=−α(ξ)˙ψ−F1(φ,ψ,ω J) ˙ψ=α(ξ)cosψ˙φ+F2(φ,ψ,ω J),(3) whereF1(φ,ψ,ω J)≡(γ/Ms)∂E/∂ψ−a(φ,ωJ) and F2(φ,ψ,ω J)≡(γ/(Mscosψ))∂E/∂φ+b(φ,ψ,ω J).γis2 φHH M xHxyyz ψz uK pI FIG. 1: The spherical coordinate system ( φ,ψ) for the direc- tion of the free layer magnetization m=M/Msof STNO. pdenotes the direction of the pinned layer magnetization; p= (cosψpcosφp,cosψpsinφp,sinψp). the gyromagnetic ratio. The second terms of Fire- sult from the Slonczewski term TJ= (γaJ/Ms)M× (M×p) in which aJis proportional to the cur- rent density Jthrough the free layer14. Therefore, a(φ,ωJ)≡ωJcosψpsin(φ−φp) andb(φ,ψ,ω J)≡ ωJ[cosψpsinψcos(φ−φp)−sinψpcosψ], whereωJ= γaJ.α(ξ)-terms of Eqs. (3) are the generalized Gilbert damping terms proposed by Tiberkevich and Slavin.15 We take into account only the first non-trivial term of the Taylor series expansion for α(ξ) by the magneti- zation change rate ξ≡(∂m/∂t)2/(γ4πMs)2;α(ξ) = αG(1+q1ξ). According to Ref. 15, the nonlinear LLGS model with q1= 3 gives a good agreement with the ex- perimental results of Ref. 1 and Ref. 16. An instability of a steady solution of Eq. (3) is consid- ered. A steady solution ( φ0(ωJ),ψ0(ωJ)) is derived from Fi(φ0,ψ0,ωJ) = 0. Shifting the variables as u1≡φ−φ0 andu2≡ψ−ψ0, we have the Taylor series of Eq. (3) as follows, ˙u=Lu+N2uu+N3uuu+··· (4) whereu= (u1,u2)T. Here, the diadic and triadic nota- tions7have been used. The stability of a steady solution is determined by the eigenvalues of the linear coefficient matrixL:λ±= Γ±(Γ2−detL)1/2. Γ is defined as Γ = Γ(ωJ)≡(1/2)trLand plays the role as a control pa- rameter since it depends on ωJ. We confine ourselves to thecasewheretheHopfbifurcationoccurs. Then, λ±isa pair of complex-conjugate eigenvalues. The point, Γ = 0, is the Hopf bifurcation point; while a steady solution re- mains stable for Γ <0, it becomes unstable for Γ >0. The bifurcation point corresponds to the threshold ωc J which is determined by tr L= 0 andFi(φ0,ψ0,ωc J) = 0. Near the bifurcation point, we divide Linto the two parts;L=L0+ ΓL1, whereL0is the critical part and ΓL1istheremainingpart. Correspondingto L,λ+isalso divided into the two parts; λ+=λ0+Γλ1. Although L1 andλ1generally depend on Γ further, we neglect their dependence and evaluate them by the values at Γ = 0. Accordingly, λ0=iω0and λ1= 1−1 2iω0d dΓdetL/vextendsingle/vextendsingle/vextendsingle/vextendsingle Γ=0, (5) whereω0≡√detL0. The right and left eigenvector ofFIG.2: (Color online)(a)Power PdividedbyR0I2withR0= 13.6 Ω and (b) linewidth (FWHM) of the signal of STNO as a function of applied current I. Dots are experimental data atT= 150 K taken from Ref. 16. Red lines are theoretical fitting curves based on the model of Eq. (1). L0corresponding to the eigenvalue λ0are denoted as U andU∗, respectively. These are normalized as U∗U= ¯U∗¯U= 1 where ¯Umeans a complex conjugate of U. Let us apply the reduction method to Eq. (4). The SL equation for a complex amplitude W(t), ˙W= Γλ1W−g|W|2W (6) and the neutral solution for the magnetization dynamics, /parenleftbigg φ ψ/parenrightbigg =/parenleftbigg φ0 ψ0/parenrightbigg +W(t)eiω0tU+¯W(t)e−iω0t¯U(7) areobtainedwithinthelowestorderapproximation.7Un- der the approximation, only the Taylor expansion coef- ficients up to the third order are needed. The complex constantgin Eq. (6) is given by g≡ν1+iν2=−3(U∗,N3¯UUU) +4(U∗,N2UV0)+2(U∗,N2¯UV+),(8) whereV0=L−1 0N2U¯UandV+= (L0−2iω0)−1N2UU. The amplitude-phase coupling factor δis obtained from the complex constant gand is given by δ=ν2/ν1. (9) In this way, the factor δfor STNO can be calculated numerically from the parameters of the LLGS equation. The noisy Hopf normal form given by Eq. (1) is de- rived when we add the noise term f(t) with∝an}bracketle{tf(t)¯f(t′)∝an}bracketri}ht= 4Dγ2δ(t−t′) to the SL Eq. (6). f(t) has the di- mension of frequency. The components in Eq. (1) are defined as ˜W(t) = (Dγ2/ν1)−1/4W(t)ei(ω0+Γδ−ΓImλ1)t, ˜t≡/radicalbig Dγ2ν1t,p≡Γ//radicalbig Dγ2ν1, and˜Ω≡ω0//radicalbig Dγ2ν1. Therefore, we can make the most of many well-known properties of Eq. (1)10,11to examine the behavior of STNO. It is known, for example, that the spectrum linewidth ∆ ωFWHMfar abovethe threshold ( p≫0) is in- creased by a factor of (1+ δ2).10In the context of STNO, when Γ≫0, the linewidth can be expressed as ∆ωFWHM= ∆ωres×kBT Eosci×1 2(1+δ2),(10)3 FIG.3: (Color online)(a)Dependenceof δonthenonlinearity of the damping q1for various values of an external magnetic fieldH. An uniaxial anisotropy field is taken as Hk/4πMs= 0.04. (b) Dependence of δon an external magnetic field H for various values of an uniaxial anisotropy field Hk. which corresponds to Eq. (11) in Ref. 5. Here, kBTis the thermal energy. ∆ ωresis the linewidth at thermal equilibrium ( ωJ= 0) given by ∆ ωres= 2Γeq, where Γeq≡ −Γ(ωJ= 0). Moreover, Eosciis the magneti- zation oscillating energy and can be written as Eosci≃ 2U†[∂(∂u1E,∂u2E) ∂(u1,u2)]u=0UPWVfree=1 2ΓeqkBT Dγ2PWwhen it is assumed that Eosci≃kBTnear thermal equilibrium ( en- ergy equipartition ). Here,Vfreeis the volume of the free layer andPWis the total power of W(t) given byPW=/radicalbig Dγ2/ν1{p+2/F(p)}withF(p)≡√πep2/4[1+ erf(p/ 2)]. From the expression of Eq. (10), it is found that the MR device in STNO itself is nothing but a resonator on the analogy of electrical circuits. The other one of well-known properties of Eq. (1) is that the amplitude- phasecouplingfactordistortsthepowerspectrumtonon- Lorentzian lineshapes especially near the threshold (see, e.g., FIG. 5 of Ref. 11). The degree of the lineshape distortion is determined by the magnitude of δandp, corresponding to the calculation in Ref. 6. We com- ment on the validity of Eq. (1) for large-amplitude os- cillations. In Fig. 2, the theoretical fitting curves based on the model Eq. (1) are compared with the experimen- tal data of Ref. 16 and give a good agreement with them up toI≃5.6 mA (p≃8.2) beyond the threshold cur- rentIc= 4.8 mA (p= 0) estimated by the fitting.17 Therefore, although the derivation of Eq. (1) is based on a perturbation expansion around the bifurcation point, it is considered to be valid for rather large-amplitude os- cillations with p∼10. We briefly mention the oscillating frequency ωosci.From Eqs. (1) and (7), the oscillating frequency of a free layer magnetization far above threshold is written as ωosci=ω0−Γδ+ΓImλ1. Although the calculationresults for Imλ1of Eq. (5) are not shown here, we have found that this quantity has a small value with Im λ1∼αG for wide range of parameters of the LLGS equation. Ac- cordingly,ωosciisapproximatelygivenby ωosci≃ω0−Γδ. Since Γ∝(I−Ic), while the frequency ωoscidecreases as the current I(>Ic) increaseswhen δ>0 (red shift), ωosci increases when δ<0 (blue shift) in accordance with the spin-wave models3,4,5,6. Asillustratedabove,theamplitude-phasecouplingfac- torδplays a key role to determine the behavior of an oscillator. Therefore, the features of STNO can be found out by the calculation of δ. Some calculation examples of δare shown in Fig. 3. It is considered the case where a free layer is an in- plane magnetic film with an in-plane external field ap- plied along the xdirection, Hext=Hˆx. It is assumed thatN= diag(0,0,1),αG= 0.02, and (φp,ψp) = (0,0). In Fig. 3(a), the dependence of δon the nonlinearity of the damping q1is shown. It is found that δmono- tonically decreases for q1and the variation of δis very large. This result suggests that a nonlinear damping sig- nificantly changes the LLG dynamics.15In Fig. 3(b), the dependence of δon an external magnetic field Hfor var- ious values of an uniaxial anisotropy field Hk(= 2Ku/ Ms) is shown. The nonlinearity of the damping is taken asq1= 3.15In the practical external field region, δis very sensitive to an uniaxial anisotropy field and varies largely. Therefore, when the dynamics of STNO is con- sidered, it is necessary to take the effect of an uniaxial anisotropy field into account seriously. This is the main result of the present paper. In summary, we have considered the dynamics of STNO by reducing the LLGS equation to a generic oscil- latormodelandcalculatedexplicitlytheamplitude-phase coupling factor which is the key factor for the power spectrum. The amplitude-phase coupling factor δis very sensitive to magnetic fields, in-plane anisotropy, and the nonlinearity of damping. The large variation of δis the remarkable feature of STNO in comparison with conven- tional oscillators. The calculation way for δshown is ap- plicable for an arbitrarymagnetization configurationand is useful for finding a stable STNO with small ∆ ωFWHM (Eq. (10)), which is preferable for applications. ∗Electronic address: kiwamu.kudo@toshiba.co.jp 1S. I. Kiselev et al, Nature 425, 380 (2003). 2W. H. Rippard et al, Phys. Rev. Lett. 92, 027201 (2004). 3A. N. Slavin and P. Kabos, IEEE Trans. Magn. 41, 1264 (2005). 4V. Tiberkevich, A. N. Slavin, and J.-V. Kim, Appl. Phys. Lett.91, 192506 (2007). 5J.-V. Kim, V. Tiberkevich, and A. N. Slavin, Phys. Rev. Lett.100, 017207 (2008).6J.-V. Kim et al., Phys. Rev. Lett. 100, 167201 (2008). 7Y. Kuramoto, Chap. 2 of Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, Berlin, 1984). 8J.-V. Kim, Phys. Rev. B 73, 174412 (2006). 9K. Mizushima, K. Kudo, and R. Sato, J. Appl. Phys. 101, 113903 (2007). 10H. Risken, Chap. 12 of Fokker-Planck Equation (2nd Ed. Springer-Verlag, Berlin, 1989). 11J. P. Gleeson and F. O’Doherty, SIAM J. Appl. Math. 66,4 1669 (2006). 12C. H. Henry, IEEE Journal of Quantum Electronics, QE- 18, 259 (1982). 13H. Haken, Advanced Synergetics (Springer-Verlag, New York, 1993). 14J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 15V. Tiberkevich and A. Slavin, Phys. Rev. B 75, 014440 (2007). 16Q. Mistral et al., Appl. Phys. Lett. 88, 192507 (2006).17The dimensionless power in Fig. 2(a) is given by P/R0I2= a{p+2/F(p)}witha≃2.3063×10−9andp≃10.202(I− Ic). To obtain the linewidth in Fig. 2(b), we have used the parameters ofp Dγ2ν1/2π= 11.24 MHz and δ= 0.5, and have solved the eigenvalue problem of the Fokker-Planck equation corresponding to Eq. (1) as done in Ref. 10 or Ref. 6.

2008-11-04

The spin-torque nano-oscillator in the presence of thermal fluctuation is described by the normal form of the Hopf bifurcation with an additive white noise. By the application of the reduction method, the amplitude-phase coupling factor, which has a significant effect on the power spectrum of the spin-torque nano-oscillator, is calculated from the Landau-Lifshitz-Gilbert-Slonczewski equation with the nonlinear Gilbert damping. The amplitude-phase coupling factor exhibits a large variation depending on in-plane anisotropy under the practical external fields.

Amplitude-Phase Coupling in a Spin-Torque Nano-Oscillator

0811.0425v1

Parity-time symmetry breaking in magnetic systems Alexey Galda1and Valerii M. Vinokur1 1Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, USA (Dated: July 1, 2016) The understanding of out-of-equilibrium physics, especially dynamic instabilities and dynamic phase transi- tions, is one of the major challenges of contemporary science, spanning the broadest wealth of research areas that range from quantum optics to living organisms. Focusing on nonequilibrium dynamics of an open dissipative spin system, we introduce a non-Hermitian Hamiltonian approach, in which non-Hermiticity reflects dissipation and deviation from equilibrium. The imaginary part of the proposed spin Hamiltonian describes the e ects of Gilbert damping and applied Slonczewski spin-transfer torque. In the classical limit, our approach reproduces Landau-Lifshitz-Gilbert-Slonczewski dynamics of a large macrospin. We reveal the spin-transfer torque-driven parity-time symmetry-breaking phase transition corresponding to a transition from precessional to exponen- tially damped spin dynamics. Micromagnetic simulations for nanoscale ferromagnetic disks demonstrate the predicted e ect. Our findings can pave the way to a general quantitative description of out-of-equilibrium phase transitions driven by spontaneous parity-time symmetry breaking. INTRODUCTION A seminal idea of parity-time ( PT)-symmetric quantum mechanics [1,2], that has stated that the condition of Hermitic- ity in standard quantum mechanics required for physical ob- servables and energy spectrum to be real can be replaced by a less restrictive requirement of invariance under combined parity and time-reversal symmetry, triggered an explosive de- velopment of a new branch of science. The interpretation of PTsymmetry as “balanced loss and gain” [3] connected PT symmetry breaking to transitions between stationary and non- stationary dynamics and established its importance to under- standing of the applied field-driven instabilities. Experiments on a diverse variety of strongly correlated systems and phe- nomena including optics and photonics [4–10], superconduc- tivity [11–13], Bose-Einstein condensates [14], nuclear mag- netic resonance quantum systems [15], and coupled electronic and mechanical oscillators [16–18] revealed PT symmetry- breaking transitions driven by applied fields. These observa- tions stimulated theoretical focus on far-from-equilibrium in- stabilities of many-body systems [12,13,19] that are yet not thoroughly understood. Here we demonstrate that the non-Hermitian extension of classical Hamiltonian formalism provides quantitative de- scription of dissipative dynamics and dynamic phase transi- tions in out-of-equilibrium systems. Focusing on the case of spin systems, we consider the zero-temperature spin dynamics under the action of basic nonconservative forces: phenomeno- logical Gilbert damping [20] and Slonczewski spin-transfer torque [21] (STT). The latter serves as the most versatile way of directly manipulating magnetic textures by external cur- rents. We propose a general complex spin Hamiltonian, in which Slonczewski STT emerges from an imaginary magnetic field. ThePT-symmetric version of the Hamiltonian is shown to exhibit a phase transition associated with inability of the system to sustain the balance between loss and gain above a certain threshold of external nonconservative field. In the classical limit of a large spin, our formalism repro- duces the standard Landau-Lifshitz-Gilbert-Slonczewski [20–22] (LLGS) equation of spin dynamics and predicts the PT symmetry-breaking phase transition between stationary (con- servative) and dissipative (nonconservative) spin dynamics. In this Letter we focus on a single spin, yet our theory can be ex- tended to coupled spin systems in higher dimensions. More- over, as spin physics maps onto a wealth of strongly corre- lated systems and phenomena ranging from superconductiv- ity to cold-atom and two-level systems, our results provide quantitative perspectives on the nature of phase transitions as- sociated withPTsymmetry breaking in a broad class of far- from-equilibrium systems. We introduce the non-Hermitian Hamiltonian for a single spin operator ˆS: ˆH=EˆS
+ij
ˆS 1i; (1) where EˆS
denotes the standard Hermitian spin Hamil- tonian determined by the applied magnetic field Hand magnetic anisotropy constants kiin the x;y;zdirections: EˆS
=P ikiˆS2 i+ H
ˆS. A schematic system setup is shown in Fig. 1. The phenomenological constant >0 in Eq. (1) describes damping; the imaginary field ijis responsible for the applied Slonczewski STT, with jS=ep(~=2e)Jbeing the spin-angular momentum deposited per second in the di- rection epwith spin polarization =(J"J#)=(J"+J#) of the incident current J; and =gB=~is the absolute value of the gyromagnetic ratio; g'2,Bis the Bohr magneton, ~is the Planck’s constant, and eis the elementary charge. We conjec- ture that Eq. (1) serves as a fundamental generalization of the Hamiltonian description of both quantum and classical spin systems, which constitutes one of our core results. This form of the Hamiltonian proves extremely useful for the general un- derstanding of STT-driven dissipative spin dynamics. In this work we focus primarily on the classical limit of spin dynam- ics, while the semiclassical limit of finite spin will be consid- ered elsewhere. Spin dynamics in the classical limit is conveniently obtained by studying expectation value of the Hamilto- nian (1) with respect to SU(2) spin-coherent states [23,24]:arXiv:1512.05408v2 [cond-mat.other] 30 Jun 20162 FIG. 1. Schematic representation of the system setup. Ferromagnetic cylinder (blue) is placed in magnetic field Happlied along the xaxis, and STT-inducing electric current Jis polarized in the direction ep along theyaxis. Spin-polarized current passes through a nonmag- netic metallic spacer and induces torque (Slonczewski STT, shown by the small red arrow) on the total spin S. jzi=ezˆS+jS;Si, where ˆSˆSxiˆSy, and z2Cis the stan- dard stereographic projection of the spin direction on a unit sphere, z=(sx+isy)=(1sz), with siSi=S. Note that such parametrization of the phase space for a classical single-spin system (i.e., in the limit S!1 ) guarantees the invariance of the traditional equation of motion [24] under generalization to non-Hermitian Hamiltonians (see Appendix A): ˙z=i(1+¯zz)2 2S@H @¯z; (2) where zand ¯zform a complex conjugate pair of stereographic projection coordinates, and H(z;¯z)=hzjˆHjzi hzjzi(3) is the expectation value of the Hamiltonian (1) in spin- coherent states (for a detailed review see, e.g., Ref. [25]). In this formulation, the eigenstates of ˆHcorrespond to the fixed points ziof the equation of motion for H, while the eigen- values (i.e., energy values) are equal to Hevaluated at the corresponding fixed points, Ei=H(zi;¯zi). Assuming a constant magnitude of the total spin, ˙S=0, Eq. (2) reduces to the following equation of spin dynamics in the classical limit: ˙S=rS(ReH)S+1 SrS(ImH)SS: (4) Here we refer to the real and imaginary parts of the Hamil- tonian functionHwritten in the spin Srepresentation. For the non-Hermitian Hamiltonian (1), Eq. (4) reproduces theLLGS equation describing dissipative STT-driven dynamics of a macrospin:  1+2˙S= HeS+ S[ HeS]S+1 SS[Sj] +Sj; (5) He=rSE(S): (6) The first two terms in Eq. (5) describe the standard Landau- Lifshitz torque and dissipation, while the last two are respon- sible for the dissipative (antidamping) and conservative (e ec- tive field) Slonczewski STT contributions, correspondingly, both of which appear naturally from the imaginary magnetic field term in the Hamiltonian (1). PT-SYMMETRIC HAMILTONIAN Slonczewski STT turns the total spin-angular momentum, S, in the direction of spin-current polarization, ep, without changing its magnitude. On the S-sphere this can be repre- sented by a vector field converging in the direction of epand originating from the antipodal point. It is the imaginary mag- netic field ijthat produces exactly the same e ect on spin dy- namics, according to Eq. (2). The action of STT is invari- ant under the simultaneous operations of time reversal and reflection with respect to the direction ep, which is the un- derlying reason behind the inherent PTsymmetry of certain STT-driven magnetic systems, including the one considered below. Before turning to the PT-symmetric form of Hamilto- nian (1), we note that PT-symmetric systems play an im- portant role in the studies of nonequilibrium phenomena and provide a unique nonperturbative tool for examining the phase transition between stationary and nonstationary out- of-equilibrium dynamics. We show that despite being non- Hermitian, such systems can exhibit both of the above types of behavior, depending on the magnitude of the external non- conservative force. In the parametric regime of unbroken PTsymmetry, systems exhibit physical properties seemingly equivalent [26] to those of Hermitian systems: real energy spectrum, existence of integrals of motion (see Appendix C), and, notably, the validity of the quantum Jarzynski equal- ity [27]. However, in the regime of brokenPT symme- try, one observes complex energy spectrum and nonconser- vative dynamics. Therefore, the true transition between sta- tionary and nonstationary dynamics can be identified as the PTsymmetry-breaking phase transition. Spin systems are generally subject to various non- linear magnetic fields including ones originating from shape, exchange, and magnetocrystalline and magnetoelastic anisotropies. Restricting ourselves for simplicity to a second- order anisotropy term, we arrive at the following Hamiltonian for a nonlinear magnetic system with uniaxial anisotropy and applied Slonczewski STT: ˆHPT= H0 kzˆS2 z+hxˆSx+iˆSy ; (7)3 FIG. 2. Real (a) and imaginary (b) parts of energy spectrum of the Hamiltonian (7) as a function of the STT parameter forhx=1 and D=20. Blue and red lines correspond to the eigenvalues E1;2andE36, respectively. The first PTsymmetry-breaking transition occurs at jj=14:5. where the applied magnetic field hxis measured in units of some characteristic magnetic field H0, andis a dimension- less STT parameter determining the relative to Samount of angular momentum transfered in time ( H0)1(charac- teristic timescale of the dynamics, used as a unit of dimen- sionless time in what follows). The Hamiltonian (7) modeling the dynamics of the free magnetic layer in a typical nanopil- lar device with fixed polarizer layer (see Fig. 1) is PTsym- metric: It is invariant under simultaneous action of parity and time-reversal operators ( y!y,t!t,i!i). Because the Hamiltonian ˆHPTcommutes with an antilinear operator PT, its eigenvalues are guaranteed to appear in complex con- jugate pairs. Notice that PT-symmetric Hamiltonian (7) does not contain damping, which is assumed to be negligibly small, as is the case in many experimental systems. CLASSICAL SPIN SYSTEM In order to best illustrate the mechanism of PT symme- try breaking, we focus on the classical limit, S!1 and kzS!D=2, where Dis the dimensionless uniaxial anisotropy constant. Formula (2) then yields the following equation of motion for the Hamiltonian (7): ˙z(t)=i(hx+) 2 z2hx hx+! i D z1jzj2 1+jzj2; (8) with up to six fixed points zk,k=1;:::; 6. Shown in Fig. 2 are the real and imaginary parts of the en- ergy spectrum E16as functions of the STT amplitude . It reveals that in a system with strong anisotropy, D1,PT symmetry breaking occurs in three separate transitions, with the first one atjj=1=jhxjq1+p 1+(2D=jhxj)2=2, which corresponds to the smallest amplitude of STT at which Im(E),0. Therefore,PTsymmetry is not broken in the en- tire phase space of initial spin directions simultaneously, at variance to the linear spin system with D=0 (see AppendixB). Instead, the regions of broken and unbroken PTsymme- try may coexist in the phase diagram of a nonlinear spin sys- tem. In what follows we consider a system described by the Hamiltonian (7) with hx=1 and D=20. For all jj< 14:5,PT symmetry is unbroken and the character of spin (magnetization) dynamics is oscillatory in the entire phase diagram, i.e., for all possible initial conditions z. At jj=1the phase transition (first of the three, see Fig. 2) oc- curs sharply in a wide region around the easy plane, jzj=1, i.e. near the equator of the unit S-sphere, shown in gray in Figs. 3(a) and 3(b) in Cartesian and stereographic projection coordinates, respectively. It this region the nature of spin dy- namics becomes fundamentally di erent—all spin trajectories follow the lines connecting the fixed points z1andz2, where z1;2=Dhxip 42h2xD2h2x
=(hx+), and no closed trajectories are possible; see Fig. 3(b). Asjjis increased further, the region of broken PTsymme- try expands until it eventually closes around the fixed point z5 at29:3 (second bifurcation in Fig. 2) and, eventually, the last region of unbroken PT symmetry near z3disappears at 310:8. The second and third phase transitions are less rel- evant experimentally as they occur in the vicinity of the least favorable spin directions (parallel and antiparallel to the hard axisz) and at considerably higher applied currents. The predicted transition from precessional dynamics (un- brokenPTsymmetry) to exponentially fast saturation in the direction z1(hx;) for any initial spin position around the easy plane (brokenPT symmetry) occurs in the setup with mu- tually perpendicular applied magnetic field and Slonczewski STT. Such a transition in nanoscale magnetic structures can be used for STT- or magnetic-field-controlled magnetization switching in spin valves and a variety of other experimental systems. This e ect can further be used for direct measure- ments of the amplitude of the applied STT, which, unlike the applied current, can be hard to quantify experimentally.4 FIG. 3. (a, b) Spin dynamics described by Eq. (8) with hx=1,=4:7, and D=20.PTsymmetry is broken in the shaded region around the easy planejzj=1 (dashed line), encompassing two fixed points, z1;2(blue dots), appearing as source and sink nodes. The green line depicts a typical nonoscillatory spin trajectory in the region of broken PTsymmetry. Red dots represent the fixed points z36. (c) Results of micromagnetic simulations for as a function of stereographic projection of the initial spin direction z. In the blue region, 4 :6..4:8, the PTsymmetry is broken at all jj<, and the spin takes under 0 :5 ns to saturate in the direction of z1, which is in full agreement with the analytical result. NUMERICAL SIMULATIONS OF PTSYMMETRY BREAKING Here we present the results of numerical simulations con- firming thePT symmetry-breaking phase transition in the classical single-spin system (7) by modeling magnetization dynamics of a ferromagnetic disk 100 nm in diameter and d=5 nm thick, which is consistent with the anisotropy con- stant D=20 in Eq. (8). We used the following typical permalloy material parameters: damping constant =0:01, exchange constant Aex=131012J/m and saturation mag- netization Msat=800103A/m . The simulations were car- ried out using the open-source GPU-accelerated micromag- netic simulation program MuMax3 [28] based on the LLGS equation (5) discretized in space. We used a cubic discretiza- tion cell of 5 nm in size, which is smaller than the exchange length in permalloy, lex=(2Aex=0M2 sat)1=25:7 nm. The permalloy disk was simulated in an external magnetic field applied along the xaxis, H0=400 Oe, which corre- sponds to the characteristic time 0:14 ns. The STT was produced by applying electric current perpendicular to the disk in the zdirection with spin polarization =0:7 along ep=ˆy(see Fig. 1) and current density measured in dimen- sionless units of 2 eH0Msatd=~0:7108A/cm2. While such current density is comparable to typical switching cur- rent densities in STT-RAM devices [29,30], its magnitude can be optimized for various practical applications by changing H0and adjusting the size, shape, and material of the ferro- magnetic element. For all possible initial spin directions z, we calculated the critical amplitude of the applied STT, , for which the char- acter of spin dynamics changes from oscillatory (at jj< ) to exponential saturation. Shown in Fig. 3(c) is the color map ofas a function of zin complex stereographic coordinates. The region shown in the shades of blue corresponds to the initial conditions z, for which the minimum values of that would guarantee saturation of spin dynamics in the directionofz1in under 0:5 ns are between 4.6 and 4.8. This is in full agreement with the region of broken PTsymmetry at =4:7 calculated analytically, i.e., the shaded area in Fig. 3(b) [the outline is repeated in Fig. 3(c) for comparison]. Outside of this region, a considerably larger magnitude of the applied STT is required to break PTsymmetry. The agreement between theoretical results and micromag- netic simulations is remarkable considering the non-zero Gilbert damping parameter ( =0:01) and nonlinear e ects (demagnetizing field, finite size and boundary e ects, etc.) inherently present in the micromagnetic simulations but not included in the model Hamiltonian (7). CONCLUSION The presented non-Hermitian Hamiltonian formulation of dissipative nonequilibrium spin dynamics generalizes the pre- vious result [31], where the classical Landau-Lifshitz equa- tion was derived from a non-Hermitian Hamilton operator, to open STT-driven spin systems. The introduction of Slon- czewski STT in the imaginary part of the Hamiltonian re- vealed the possibility of STT-driven PTsymmetry-breaking phase transition. Micromagnetic simulations confirm the PTsymmetry-breaking phenomenon in realistic mesoscopic magnetic systems and its robustness against weak dissipation, indicating high potential for impacting spin-based informa- tion technology. The way STT enters the complex Hamilto- nian (1), i.e. as imaginary magnetic field, provides a unique tool for studying Lee-Yang zeros [32] in ferromagnetic Ising and Heisenberg models and, more generally, dynamics and thermodynamics in the complex plane of physical parame- ters. We envision further realizations of the PT symmetry- breaking phase transitions in diverse many-body condensed- matter systems and the expansion of practical implementa- tions of thePT symmetry beyond the present realm of op- tics [33] and acoustics [34].5 FIG. 4. Real (a) and imaginary (b) parts of energy spectrum of the linear Hamiltonian ˆH0PTas functions of forhx=1.PTsymmetry- breaking transition occurs at jj=1. ACKNOWLEDGEMENTS We thank Alex Kamenev for critical reading of the manuscript and valuable suggestions. This work was sup- ported by the U.S. Department of Energy, O ce of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. APPENDIX A. GENERALIZED EQUATION OF MOTION FOR NON-HERMITIAN SPIN HAMILTONIANS IN THE CLASSICAL LIMIT The remarkable simplicity of the equation of motion (2) for an arbitrary non-Hermitian spin Hamiltonian function Hstems from the choice of parametrization of the phase space, i.e., the complex stereographic projection coordinates fz;¯zg. The extension of classical equations of motion to non- Hermitian Hamiltonians in terms of canonical coordinates fq;pghas the following generalized form [35]: q p! = 1rq;p(ReH)G1rq;p(ImH); (A1) where andGare the symplectic structure and metric of the underlying classical phase space, respectively, which must satisfy the compatibility condition [36] written in the matrix form as 1= 1 1T: (A2) In the stereographic projection coordinates, one obtains the following symplectic structure and metric: =2 1+jzj2
2 0i i0! ; G=2 1+jzj2
2 0 1 1 0! :(A3) It is the form of these matrices that leads to Eq. (2), where the real and imaginary parts (as written in the Srepresentation) of the Hamiltonian combine naturally into a single complexfunctionH. Therefore, when written in stereographic projec- tion coordinates, the generalized classical equation of motion for non-Hermitian Hamiltonians coincides with that for tradi- tional Hermitian Hamiltonians. APPENDIX B.PTSYMMETRY BREAKING IN LINEAR SPIN SYSTEM In the absence of magnetic anisotropy fields, the Hamilto- nian (1) from the main text becomes linear: ˆH0= H+ij 1i!
ˆS; (B1) with e ects of applied magnetic field, damping and Slon- czewski STT contributions all incorporated in the complex magnetic field (in parentheses). The PT-symmetric version of this Hamiltonian has mutually perpendicular real and imag- inary parts of the complex magnetic field: ˆH0PT= H0 hxˆSx+iˆSy : (B2) The quantum spin-1 2version of this Hamiltonian describes a two-level quantum system with balanced loss and gain and is known [37,38] to exhibit PTsymmetry-breaking transition athx=. Whenjhxj>jj, the Hamiltonian ˆH0PThas real eigenvalues, 1;2=p h2x2, while in the parametric region jhxj<jjeigenvalues are imaginary, see Fig. 4. The generalized Hamilton’s equation of motion (2) for this Hamiltonian takes the form ˙z(t)=f(t)=i(hx+) 2 z2hx hx+! : (B3) It has two fixed points: z1;2=p (hx)=(hx+), which correspond to stereographic coordinates of the spin equi- libria directions on a unit Bloch sphere. The character of spin dynamics around the fixed points is fully deter- mined by the eigenvalues of the complex Jacobian matrix6 FIG. 5. Spin trajectories for classical linear Hamiltonian H0PTin the regime of unbroken, (a) and (c), (at =0:8) and broken, (b) and (d), (at =1:2)PTsymmetry for hx=1. JC=@(Ref;Imf)=@(Rez;Imz)in their vicinity. The solu- tion of Eq. (B3) takes the form of a M ¨obius transformation: z(t)=z(0)+ihx+p h2x2tan p h2x2 2t! ihxp h2x2tan p h2x2 2t! z(0)+1: (B4) The parametric region jhxj>jjdefines the regime of un- brokenPT symmetry with real Hamiltonian eigenvalues, E1;2=p h2x2. In the classical approximation, the spin performs persistent oscillations along circular orbits about the fixed points z1;2situated on the real axis, see Figs. 5(a) and 5(c). The eigenvalues of JCatz1;2are purely imaginary, iden- tifying the fixed points are centers , according to the stan- dard classification [39]. Closed trajectories represent PT- symmetric dynamics with balanced loss and gain: the spin system gains and loses equal amounts of energy from the non- conservative term iSyon they <0 andy >0 segments of trajectories, respectively.As the driving parameter jjis increased, z1;2move towards each other until they eventually collide at jj=jhxj, which marks the point of PT symmetry breaking. In the regime ofbrokenPT symmetry,jhxj<jj, the energy eigenvalues are imaginary, E1;2=ip 2h2x, and no closed trajectories are possible. The eigenvalues of the Jacobian JC(z1;2) in this regime are real and of the same sign, defining z1;2assink andsource nodes [39]. All trajectories follow circle arcs con- necting the fixed points with coordinates in three-dimensional space 0;p 1(hx=)2;hx= , which are now out of the xz plane, see Figs. 5(b) and 5(d). We emphasize that in the linear Hamiltonian (B1) the ef- fects of damping can always be fully compensated by the appropriate choice of the applied STT. For instance, the PT-symmetric Hamiltonian (B2) describes a single spin placed in external magnetic field H=H0(hx;; 0)and STT j= H0(hx;;0). Note that in the general case of nonlin- ear spin Hamiltonian, dissipation cannot be completely can- celed by STT. However, in many magnetic systems dissipative forces are extremely weak, which justifies the approximation of zero damping.7 APPENDIX C. INTEGRALS OF MOTION There exists a remarkable similarity between spin dynam- ics in the regime of unbroken PT symmetry and that of a fully Hermitian system. Indeed, it follows from Eq. (B3) that all spin trajectories are circular with the precession frequency equal top h2x2. The equivalent [26] to (B2) Hermitian Hamiltonian reads: H0= H0q h2x2S0 x; (B5)for which the equation of motion is obtained from Eq. (B3) by the circle-preserving M ¨obius transformation z0=s hx+ hxz: (B6) The relation between time evolution of complex linear spin Hamiltonians and M ¨obius transformations will be described elsewhere. The existence of this transformation leads to the non-Hermitian system (B2) having an integral of motion: I[z;¯z]=(hx+)z+¯z 1+hx+ hxjzj2; (B7) which is nothing but the magnetic energy conserved by the Hermitian Hamiltonian (B5): Iz0;¯z0=q h2x2s0 x=q h2x2z0+¯z0 1+jz0j2(B8) after the transformation (B6). 1C. M. Bender and S. Boettcher. Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998). 2C. M. Bender, S. Boettcher, and P. N. Meisinger. PT-symmetric quantum mechanics. J. Math. Phys. 40, 2201-2229 (1999). 3A. Ruschhaupt, F. Delgado, and J. G. Muga. Physical realization of PT-symmetric potential scattering in a planar slab waveguide. J. Phys. A 38, L171 (2005). 4S. Klaiman, U. G ¨unther, and N. Moiseev. Visualization of branch points in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008). 5S. Longhi. Bloch oscillations in complex crystals with PT symme- try.Phys. Rev. Lett. 103, 123601 (2009). 6S. Longhi. Optical realization of relativistic non-Hermitian quan- tum mechanics. Phys. Rev. Lett. 105, 013903 (2010). 7H. Schomerus. Quantum noise and self-sustained radiation of PT- symmetric systems. Phys. Rev. Lett. 104, 233601 (2010). 8H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides. Unidirectional nonlinear PT-symmetric optical structures. Phys. Rev. A 82, 043803 (2010). 9C. E. R ¨uteret al. Observation of paritytime symmetry in optics. Nature Phys. 6, 192-195 (2010). 10J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao. PT- symmetric optical potentials in a coherent atomic medium. Phys. Rev. A 88, 041803 (2013). 11J. Rubinstein, P. Sternberg, and Q. Ma. Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents. Phys. Rev. Lett. 99, 167003 (2007). 12J. Rubinstein, P. Sternberg, and K. Zumbrun. The resistive state in a superconducting wire: bifurcation from the normal state. Arch. Ration. Mech. Anal. 195, 117 (2010). 13N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and V . M. Vinokur. Stimulation of the fluctuation superconduc- tivity by PT symmetry. Phys. Rev. Lett. 109, 150405 (2012). 14E. M. Graefe, H. J. Korsch, and A. E. Niederle. Mean-field dynam- ics of a non-Hermitian Bose-Hubbard dimer. Phys. Rev. Lett. 101, 150408 (2008).15C. Zheng, L. Hao, and G. L. Long. Observation of fast evolu- tion in paritytime-symmetric system. Phil. Trans. R. Soc. A 371, 20120053 (2013). 16C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel. Observa- tion of PT phase transition in a simple mechanical system. Am. J. Phys. 81, 173 (2013). 17C. M. Bender et al. Twofold transition in PT-symmetric coupled oscillators. Phys. Rev. A. 88, 062111 (2013). 18J. Schindler et al. . Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A 84, 040101(R) (2011). 19N. Poccia et al. Critical behavior at a dynamic vortex insulator-to- metal transition. Science 349, 1202-1205 (2015). 20T. L. Gilbert. A phenomenological theory of damping in ferro- magnetic materials. IEEE Trans. Magn. 40, 3443 (2004). 21J. C. Slonczewski. Current-driven excitation of magnetic multilay- ers.J. Magn. Magn. Matter ,159, L1-L7 (1996). 22L. D. Landau and E. M. Lifshitz. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sow- jetunion 8, 101-114 (1935). 23E. H. Lieb. The classical limit of quantum spin systems. Commun. Math. Phys. 34, 327-340 (1973). 24M. Stone, K.-S. Park, and A. Garg. The semiclassical propagator for spin coherent states. J. Math. Phys. 41, 8025-8049 (2000). 25J. M. Radcli e. Some properties of coherent spin states. J. Phys. A 4, 313 (1971). 26A. Mostafazadeh. Exact PT-symmetry is equivalent to Hermitic- ity.J. Math. Phys. 36, 7081 (2003). 27S. De ner and A. Saxena. Jarzynski equality in PT-symmetric quantum mechanics. Phys. Rev. Lett. 114, 150601 (2015). 28A. Vansteenkiste et al. . The design and verification of MuMax3. AIP Adv. 4, 107133 (2014). 29K. L. Wang, J. G. Alzate, and P. K. Amiri. Low-power non-volatile spintronic memory: STT-RAM and beyond. J. Phys. D: Appl. Phys. 46, 074003 (2013). 30A. D. Kent, B. ¨Ozyilmaz, and E. Del Barco. Spin-transfer-induced precessional magnetization reversal. Appl. Phys. Lett. 84, 3897 (2004).8 31R. Wieser. Comparison of quantum and classical relaxation in spin dynamics. Phys. Rev. Lett. 110, 147201 (2013). 32C. N. Yang and T. D. Lee. Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87, 410 (1952). 33B. Peng et al. 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2015-12-16

The understanding of out-of-equilibrium physics, especially dynamic instabilities and dynamic phase transitions, is one of the major challenges of contemporary science, spanning the broadest wealth of research areas that range from quantum optics to living organisms. Focusing on nonequilibrium dynamics of an open dissipative spin system, we introduce a non-Hermitian Hamiltonian approach, in which non-Hermiticity reflects dissipation and deviation from equilibrium. The imaginary part of the proposed spin Hamiltonian describes the effects of Gilbert damping and applied Slonczewski spin-transfer torque. In the classical limit, our approach reproduces Landau-Lifshitz-Gilbert-Slonczewski dynamics of a large macrospin. We reveal the spin-transfer torque-driven parity-time symmetry-breaking phase transition corresponding to a transition from precessional to exponentially damped spin dynamics. Micromagnetic simulations for nanoscale ferromagnetic disks demonstrate the predicted effect. Our findings can pave the way to a general quantitative description of out-of-equilibrium phase transitions driven by spontaneous parity-time symmetry breaking.

Parity-time symmetry breaking in magnetic systems

1512.05408v2

Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque and damping on noncollinear classical magnetism due to electronic open quantum system Utkarsh Bajpai and Branislav K. Nikoli´ c Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA We analyze a quantum-classical hybrid system of steadily precessing slow classical localized mag- netic moments, forming a head-to-head domain wall, embedded into an open quantum system of fast nonequilibrium electrons. The electrons reside within a metallic wire connected to macroscopic reservoirs. The model captures the essence of dynamical noncollinear and noncoplanar magnetic textures in spintronics, while making it possible to obtain the exact time-dependent nonequilib- rium density matrix of electronic system and split it into four contributions. The Fermi surface contribution generates dissipative (or damping-like in spintronics terminology) spin torque on the moments, and one of the two Fermi sea contributions generates geometric torque dominating in the adiabatic regime. When the coupling to the reservoirs is reduced, the geometric torque is the only nonzero contribution. Locally it has both nondissipative (or field-like in spintronics terminology) and damping-like components, but with the sum of latter being zero, which act as the counter- parts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics. Such current-independent geometric torque is absent from widely used micromagnetics or atomistic spin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation, where previous analysis of Fermi surface-type torque has severely underestimated its magnitude. One of the most fruitful applications of geometric (or Berry) phase [1] concepts is encountered in quantum- classical hybrid systems where separation of time scales makes it possible to consider fast quantum degrees of free- dom interacting with the slow classical ones [2, 3]. The amply studied example of this kind are fast electrons in- teracting [4, 5] with slow nuclei in molecular dynamics (MD) [6–9] problems of physics, chemistry and biology. The parameters driving adiabatic evolution of quantum subsystem, with characteristic frequency smaller that its level spacing, are nuclear coordinates elevated to the status of dynamical variables. The electronic system then develops geometric phase in states evolving out of an instantaneous energy eigenstate, while also acquiring shifts in the energy levels. Conversely, nuclei experience forces due to back-action from electrons. The simplest force is the adiabatic Born-Oppenheimer (BO) force [4, 5] which depends only on the coordinates of the nuclei, and it is associated with electronic adiabatic potential sur- faces [6, 7]. Even small violation of BO approximation leads to additional forces—the first nonadiabatic correc- tion generates forces linear in the velocity of the nuclei, and being Lorentz-like they are dubbed [2, 10] “geomet- ric magnetism.” The “magnetism” is not a not a real magnetic field, but an emergent geometrical property of the Hilbert space [11], and akin to the true Lorentz force, the emergent geometric force is nondissipative . Additional forces appear upon making the quantum system open by coupling it to a thermal bath [10, 12] (usually modeled as an infinite set of harmonic oscilla- tors [13]) or to macroscopic reservoirs of particles [14]. In the latter case, one can also introduce chemical po- tential difference between the reservoirs to drive particle flux (i.e., current) through the quantum system which is, thereby, pushed out of equilibrium [14–16, 18, 19]. In FIG. 1. (a) Schematic view of a two-terminal system where a single classical LMM, precessing steadily with frequency ω and cone angle θ, interacts with an open quantum system of conduction electron spins. The electrons hop along 1D infinite tight-binding chain which terminates into the left and right macroscopic reservoirs kept at the same chemical potential µ. Panel (c) depicts 7 LMMs, M1–M7forming a head-to-head Bloch domain wall, which precess with the same frequency but are noncollinear andnoncoplanar . Both (a) and (c) can be mapped in the rotating frame to a time-independent four- terminal system in (b) with an effective bias voltage ~ω/e between the left or right pair of leads. both equilibrium and nonequilibrium cases, the energy spectrum of the quantum system is transformed into a continuous one, and frictional forces [8–10, 14–19] linear in the velocity of the nuclei become possible. Also, due to continuous spectrum, adiabaticity criterion has to be replaced by a different one [14]. Stochastic forces also ap- pear, both in equilibrium and in nonequilibrium, where in the former case [10, 12] they are due to fluctuations at finite temperature while in the latter case they includearXiv:2005.14153v2 [cond-mat.mes-hall] 14 Jun 20202 additional contribution from nonequilibrium noise [14– 16]. Finally, specific to nonequilibrium is the emergence of nonconservative forces [14–16, 18, 19]. The derivation of all of these forces is achieved by computing nonadia- batic corrections to the density matrix (DM) [10, 12, 14– 16, 18, 19]. This yields a non-Markovian stochastic Langevin equation, with nonlocal-in-time kernel describ- ing memory effects [20], as the most general [16, 19] equa- tion for nuclei in nonadiabatic MD. The analogous problem exists in spintronics, where the fast quantum system is comprised of conduction electron spins and slow classical system is comprised of localized- on-atoms spins and associated localized magnetic mo- ments (LMMs) described by unit vectors Mi(t). The dynamics of LMMs is accounted by the Landau-Lifshitz- Gilbert (LLG) type of equation [21] ∂Mi dt=−gM×Beff i+λMi×∂Mi ∂t +g µM/parenleftBig Ti/bracketleftBig ISα ext/bracketrightBig +Ti[∂Mi/∂t]/parenrightBig . (1) This includes phenomenological Gilbert damping, whose parameter λcan be measured or independently calcu- lated [22] by using electronic Hamiltonian with spin-orbit coupling and impurities. It can also include Slonczewski spin-transfer torque (STT) term Ti/bracketleftBig ISα ext/bracketrightBig due to exter- nally supplied spin current ISα ext. The STT is a phe- nomenon [28] in which spin angular momentum of con- duction electrons is transferred to local magnetization not aligned with electronic spin-polarization. Finally, some analyses [23–25] also consider current-independent torque Ti[∂Mi/∂t] as a back-action of electrons pushed out of equilibrium by time-dependent Mi(t). Neverthe- less, such effects have been deemed negligible [23, 26] or easily absorbed into Eq. (1) by renormalizing gandλ[23]. Heregis the gyromagnetic ratio; Beff i=−1 µM∂H/∂Mi is the effective magnetic field as the sum of external field, field due to interaction with other LMMs and magnetic anisotropy field in the classical Hamiltonian Hof LMMs; andµMis the magnitude of LMM [21]. The STT vector, T=TFL+TDL, can be decomposed [Fig. 1(a)] into: ( i) even under time-reversal or field-like (FL) torque, which affects precession of LMM around Beff i; and ( ii) odd under time-reversal or damping-like (DL) torque, which either enhances the Gilbert damp- ing by pushing LMM toward Beff ior competes with Gilbert term as “antidamping.” For example, negative values ofTDL=TDL·eDLin Figs. 2 and 3, where eDL= (Mi×∂Mi/∂t)|Mi×∂Mi/∂t|−1, means that TDL vector points away from the axis of precession which is antidamping action. Similarly, TFL=TFL·eFL, where eFL= (∂Mi/∂t)|∂Mi/∂t|−1, is plotted in Figs. 2 and 3. The current-driven STT Ti/bracketleftBig ISα ext/bracketrightBig acts as the coun- terpart of nonconservative force in nonadiabatic MD. The Gilbert damping plus current-independent torque FIG. 2. The FL and DL components [Fig. (1)] of three spin torques contributions in Eq. (4) exerted by nonequilibrium spin density of electrons onto a single localized precessing magnetic moment in the setup of Fig. 1(a) as a function of coupling to the leads. Black dotted line is the sum of the three torques. In panels (a) and (c) Jsd= 0.1γ, while in panels (b) and (d) Jsd= 20γensures perfectly adiabatic regime [32], Jsd/~ω/greatermuch1, for the chosen precession frequency ~ω= 0.001γ. Ti[∂Mi/∂t] appear as the counterpart of electronic friction [8, 9, 14–19], but Gilbert damping requires agents [22] other than electrons alone considered in nona- diabatic MD. Thus, the geometric torque and damping, as counterparts of geometric magnetism force [2] and fric- tion [10], are absent from standard modeling of classi- cal magnetization dynamics. Geometric torque has been added ad hoc into the LLG equation applied to spe- cific problems, such as spin waves within bulk magnetic materials [29–31]. A recent study [32] of a single clas- sical LMM embedded into a closed (i.e., finite length one-dimensional wire) electronic quantum system finds that nonequilibrium electronic spin density always gener- ates geometric torque, even in perfectly adiabatic regime where electron-spin/LMM interaction is orders of mag- nitude larger than the characteristic frequency of LMM dynamics. It acts as a purely FL torque causing anoma- lous frequency of precession that is higher than the Lar- mor frequency. By retracing the same steps [14, 15] in the derivation of the stochastic Langevin equation for electron-nuclei system connected to macroscopic reser- voirs, Ref. [33] derived the stochastic LLG equation [34– 37] for a single LMM embedded into an open electronic system out of equilibrium. The novelty in this derivation is damping, present even in the absence of traditional spin-flip relaxation mechanisms [23, 25], while the same3 FIG. 3. Spatial profile of FL and DL components of Tgeo i,Tsea iandTsurf ispin torques on precessing LMMs depicted in Fig. 1(c) for closed or open electronic quantum system and for two different values of Jsd. Insets on the top of each row mark positions and static configuration of LMMs within the Bloch DW, with their x-component depicted by the colorbar next to panel (a). conclusion about geometric torque changing only the pre- cession frequency of LMM has been reached (in some regimes, geometric phase can also affect the stochastic torque [38]). However, single LMM is a rather special case, which is illustrated in Fig. 1(a) and revisited in Fig. 2, and the most intriguing situations in spintronics involve dynamics of noncollinear textures of LMMs. This is exemplified by current- or magnetic-field driven dy- namics of domain walls (DWs) and skyrmions [25, 37, 39– 43] where a much richer panoply of back-action effects from fast electronic system can be expected. In this Letter, we analyze an exactly solvable model of seven steadily precessing LMMs, M1(t)–M7(t) [Fig. 1(c)], which are noncollinear and noncoplanar and embedded into a one-dimensional (1D) infinite wire host- ing conduction electrons. The model can be viewed as a segment of dynamical noncollinear magnetic texture, and it directly describes magnetic field-driven [43] head- to-head Bloch DW [44] but without allowing it to prop- agate [41, 43]. Its simplicity makes it exactly solvable— we fins the exact time-dependent DM via the nonequi- librium Green function (NEGF) formalism [45] and an- alyze its contributions in different regimes of the ratio Jsd/~ωofsdexchange interaction Jsd[23] between elec- tron spin and LMM and frequency of precession ω. In both Figs. 1(a) and 1(c), the electronic subsystem is an open quantum system and, although no bias voltage is applied between the macroscopic reservoirs, it is pushed into the nonequilibrium state by the dynamics of LMMs. For example, electronic quantum Hamiltonian becomes time-dependent due to M1(t) [Fig. 1(a)] or M1(t)– M7(t) [Fig. 1(c)], which leads to pumping [25, 27, 46] [Fig. 4(b),(c)] of spin current locally within the DW re- gion, as well as into the leads [Fig. 4(a)]. Pumping ofcharge current will also occur if the left-right symmetry of the device is broken statically [27] or dynamically [47]. The electrons are modeled on an infinite tight-binding (TB) clean chain with Hamiltonian in the lab frame ˆHlab(t) =−γ/summationdisplay /angbracketleftij/angbracketrightˆc† iˆcj−Jsd/summationdisplay iˆc† iˆˆci·Mi(t). (2) Here ˆc† i= (ˆc† i↑,ˆc† i↓) and ˆc† iσ(ˆciσ) creates (annihilates) an electron of spin σ=↑,↓at sitei. The nearest-neighbor hoppingγ= 1 eV sets the unit of energy. The active re- gion in Figs. 1(a) or 1(c) consists of one or seven sites, respectively, while the rest of infinite TB chain is taken into account as the left (L) and the right (R) semi-infinite leads described by the same Hamiltonian in Eq. (2), but withJsd= 0. The hopping between the leads and the active region is denoted by γc. The leads terminate at infinity into the macroscopic particle reservoirs with iden- tical chemical potentials µL=µR=EFdue to assumed absence of bias voltage, and EF= 0 is chosen as the Fermi energy. In contrast to traditional analysis in spin- tronics [23, 25], but akin to Refs. [32, 33], Hamiltonian in Eq. (2) does not contain any spin-orbit or impurity terms as generators of spin-flip relaxation. The spatial profile of Bloch DW is given by Mx i=−sech[(hDW−zi)/W] tanh[(ZDW−zi)],My i= sech2[(ZDW−zi)/W] andMz i= tanh[(ZDW−zi)/W], whereZDW= 4 andW= 0.9. Instead of solving LLG equations [Eq. (1)] for M1(t)–M7(t), we impose a so- lution where LMMs precess steadily around the z-axis: Mx i(t) = sinθicos(ωt+φi);My i(t) = sinθisin(ωt+ φi); andMz i(t) = cosθi. Using a unitary transfor- mation into the rotating frame (RF), the Hamiltonian in Eq. (2) becomes time-independent [25, 27], ˆHRF=4 ˆU†(t)ˆHlab(t)ˆU(t)−i~ˆU†∂ˆU/∂t =ˆHlab(t= 0)−~ωˆσα/2, with LMMs frozen at t= 0 configuration from the lab. The unitary operator is ˆU(t) = exp(−iωtˆσα/2) forα-axis of rotation. In the RF, the original two-terminal Lan- dauer setup for quantum transport in Figs. 1(a) and 1(c) is mapped, due to ~ωˆσα/2 term, onto an effective four-terminal setup [27] [illustrated for single LMM in Fig. 1(b)]. Each of its four leads is an effective half-metal ferromagnet which accepts only one spin species, ↑or↓ along theα-axis, and effective dc bias voltage ~ω/eacts between L or R pair of leads. In the RF, the presence of the leads and macro- scopic reservoirs can be taken into account exactly us- ing steady-state NEGFs [45] which depend on time differencet−t/primeand energy Eupon Fourier trans- form. Using the retarded, ˆG(E), and the lesser, ˆG<(E), Green functions (GFs), we find the exact nonequilib- rium DM of electrons in the RF, ˆ ρRF=1 2πi
dEˆG<(E). Here the two GFs are related by the Keldysh equa- tion, ˆG<(E) = ˆG(E)ˆΣ<(E)ˆG†(E), where ˆΣ<(E) is the lesser self-energy [45] due to semi-infinite leads and ˆG(E) = [E−ˆHRF−ˆΣ(E,~ω)]−1with ˆΣ(E,~ω) =/summationtext p=L,R,σ=↑,↓ˆΣσ p(E−Qσ α~ω) being the sum of retarded self-energies for each of the four leads p,σin RF. We use shorthand notation Q↑ p=−1/2 andQ↓ p= +1/2. Since typical frequency of magnetization dynamics is ~ω/lessmuchEF, we can expand [48] ˆ ρRFin small ~ω/EF and then transform it back to the lab frame, ˆ ρlab(t) = ˆU(t)ˆρRFˆU†(t) to obtain ˆ ρlab(t) = ˆρad t+ ˆρgeo(t)+ ˆρsea(t)+ ˆρsurf(t) where: ˆρad t=−1 πˆU+∞ −∞dEImˆG0f(E)ˆU†, (3a) ˆρgeo(t) =1 πˆU+∞ −∞dEIm/bracketleftbigg ˆG0/parenleftbigg i~ˆU†∂ˆU ∂t/parenrightbigg ˆG0/bracketrightbigg f(E)ˆU†,(3b) ˆρsea(t) =−~ω 2πˆU/summationdisplay p+∞ −∞dEIm/bracketleftbigg ˆG0/parenleftbigg∂ˆΣ↑ p ∂E−∂ˆΣ↓ p ∂E/parenrightbigg ˆG0/bracketrightbigg ×f(E)ˆU†, (3c) ˆρsurf(t) =~ω 4πˆU/summationdisplay p+∞ −∞dEˆG0(ˆΓ↑ p−ˆΓ↓ p)ˆG† 0∂f ∂EˆU†.(3d) We confirm by numerically exact calculations [39] that thus obtained ˆ ρlab(t) is identical to ~G<(t,t)/icomputed in the lab frame. Here ˆG0(E) = [E−ˆHRF−ˆΣ(E,0)]−1 isˆG(E) with ~ω= 0; ˆΓσ p(E) =i[ˆΣσ p(E)−ˆΣσ p(E)†] is the level broadening matrix due the leads; and fσ p(E) = f(E−[EF+Qσ α~ω]) is the the Fermi function of macro- scopic reservoir p,σin the RF. The total nonequilibrium spin density, /angbracketleftˆsi/angbracketright(t) = Tr[ˆρlab(t)|i/angbracketright/angbracketlefti|⊗ˆ] =/angbracketleftˆsi/angbracketrightad t+/angbracketleftˆsi/angbracketrightgeo(t) +/angbracketleftˆsi/angbracketrightsea(t) + /angbracketleftˆsi/angbracketrightsurf(t), has the corresponding four contributions fromDM contributions in Eq. (3). Here /angbracketleftˆsi/angbracketrightad tis the equilib- rium expectation value at an instantaneous time twhich defines ‘adiabatic spin density’ [23, 25, 30–32]. It is com- puted using ˆ ρad tas the grand canonical equilibrium DM expressed via the frozen (adiabatic) retarded GF [14, 15, 33], ˆGt(E) = [E−ˆHt−ˆΣ]−1, for instantaneous configu- ration of Mi(t) while assuming ∂Mi/∂t= 0 [subscript tsignifies parametric dependence on time through slow variation of Mi(t)]. The other three contributions—from ˆρgeo(t) and ˆρsea(t) governed by the Fermi sea and ˆ ρsurf(t) governed by the Fermi surface electronic states—contain first nonadiabatic correction [14, 15, 33] proportional to velocity∂Mi/∂t, as well as higher order terms due to ˆρlab(t) being exact. These three contributions define STT out of equilibrium [23, 39, 48] Ti=Jsd/angbracketleftˆsi/angbracketright(t)×Mi(t) =Tgeo i+Tsea i+Tsurf i.(4) Each term Tgeo i,Tsea i,Tsurf ican be additionally sepa- rated into its own DL and FL components [Fig. 1(a)], as plotted in Figs. 2 and 3. Note that Tsea iis insignificant in both Figs. 2 and 3, so we focus on Tgeo iandTsurf i. To gain transparent physical interpretation of Tgeo iand Tsurf i, we first consider the simplest case [32, 33]—a single M1(t) in setup of Fig. 1(a). The STT contributions as a function of the coupling γcto the leads (i.e., reservoirs) are shown in Fig. 2. We use two different values for Jsd, where large ratio of Jsd= 20 eV and ~ω= 0.001 eV is perfect adiabatic limit [30–32]. Nevertheless, even in this limit and for γc→0 we find Tgeo 1/negationslash= 0 in Fig. 2(b) as the only nonzero and purely FL torque. This is also found in closed system of Ref. [32] where Tgeo 1was expressed in terms of the spin Berry curvature. As the quantum system becomes open for γc>0,Tgeo 1is slightly reduced while Tsurf 1emerges with small FL [Fig. 2(b)] and large DL [Fig. 2(d)] components. The DL torque Tsurf,DL 1 points toward the z-axis and, therefore, enhances the Gilbert damping. In the wide-band approximation [49], the self-energy ˆΣ(E) =−iΓˆI2is energy-independent for Ewithin the bandwidth of the lead, which allows us to obtain analytical expression (at zero temperature) Tgeo 1(t) =~ω 2π/bracketleftbigg π−2 tan−1/parenleftbiggΓ Jsd/parenrightbigg/bracketrightbigg sinθeφ(t).(5) Here eφ(t) =−sinωtex+ cosωtey. Thus, in per- fect adiabatic limit, Jsd/~ω→∞ , or in closed system, Γ→0,Tgeo 1is independent of microscopic parameters as expected from its geometric nature [29]. The always present Tgeo i/negationslash= 0 means that electron spin is never along ‘adiabatic direction’ /angbracketleftˆsi/angbracketrightad t. Switching to DW [Fig. 1(c)] embedded into a closed quantum system ( γc= 0) shows in Fig. 3(a)–(d) that onlyTgeo i/negationslash= 0, which also acquires DL component lo- cally with damping or antidamping action depending on the position of LMM. Upon opening the quantum sys- tem (γc=γ), Fig. 3(e)–(h) shows emergence of ad- ditional Tsurf i/negationslash= 0 which, however, becomes negligible5 FIG. 4. (a) The z-component of total DL torques which act on DW in Fig. 1(c) as a function of Jsdforγc=γ. Cir- cles show that sum of spin currents pumped into the leads matches/parenleftbig/summationtext iTsurf,DL i/parenrightbig z≡ISz L+ISz R. Panel (b) and (c), which correspond to Fig. 3(g), show spatial profile of lo- cal spin currents ISz i→jpumped between sites iandjfor Jsd= 0.1γ, with their sum being identically zeroin panel (c). Dashed black line in panels (a) and (b) is pumped local spin current by SMF [24, 26], ISz SMF(x) =gµB~G0 4e2[∂M(x,t)/∂t× ∂M(x,t)/∂x]z, whereG0=G↑+G↓is the total conductivity. [Fig. 3(f),(h)] in the perfectly adiabatic limit Jsd/~ω/greatermuch1. At first sight, Tgeo,DL i/negationslash= 0 violates Berry and Robbins original analysis [2] according to which an isolated quan- tum system, with discrete energy spectrum, cannot exert friction onto the classical system. This apparent contra- diction is resolved in Fig. 4(a) where we show that total/summationtext iTgeo,DL i≡0 is always zero. Conversely, Fig. 4(a) con- firms that total/parenleftBig/summationtextTsurf,DL i/parenrightBig z≡ISz L+ISz Ris identical to net spin current pumped into the leads via which the conduction electrons carry away excess angular momen- tum of precessing LMMs [46]. Such identity underlies physical picture where spin current generated by time- dependent magnetization becomes DL torque [24, 46]. Note that pumped spin current ISz i→jdue to ˆρgeoor ˆρsea in Fig. 4(c) can be nonzero locally, but they sum to zero. The nonuniform pumped spin current due to spatially and time varying magnetization has prompted propos- als [24, 26] to amend the LLG equation by adding the corresponding DL torque M×D·∂M/∂twith 3×3 damp- ing tensorDwhose spatial dependence is given by the so- called spin-motive force (SMF) formula. However, SMF correction was estimated to be small [26] in the absence of spin-orbit coupling in the band structure. 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2020-05-28

We analyze a quantum-classical hybrid system of steadily precessing slow classical localized magnetic moments, forming a head-to-head domain wall, embedded into an open quantum system of fast nonequilibrium electrons. The electrons reside within a metallic wire connected to macroscopic reservoirs. The model captures the essence of dynamical noncollinear and noncoplanar magnetic textures in spintronics, while making it possible to obtain the exact time-dependent nonequilibrium density matrix of electronic system and split it into four contributions. The Fermi surface contribution generates dissipative (or damping-like in spintronics terminology) spin torque on the moments, and one of the two Fermi sea contributions generates geometric torque dominating in the adiabatic regime. When the coupling to the reservoirs is reduced, the geometric torque is the only nonzero contribution. Locally it has both nondissipative (or field-like in spintronics terminology) and damping-like components, but with the sum of latter being zero, which act as the counterparts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics. Such current-independent geometric torque is absent from widely used micromagnetics or atomistic spin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation, where previous analysis of Fermi surface-type torque has severely underestimated its magnitude.

Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque and damping on noncollinear classical magnetism due to electronic open quantum system

2005.14153v2

arXiv:1901.10470v1 [math.NA] 29 Jan 2019Bounding the spectral gap for an elliptic eigenvalue proble m with uniformly bounded stochastic coefficients A. D. Gilbert1I. G. Graham2R. Scheichl1,2I. H. Sloan3 January 31, 2019 Abstract A key quantity that occurs in the error analysis of several numeric al methods for eigenvalue problems is the distance between the eigenvalue of int erest and the next nearest eigenvalue. When we are interested in the smallest or f undamental eigenvalue, we call this the spectral orfundamental gap . In a recent manuscript [Gilbert et al., https://arxiv.org/abs/1808.02639 ], the current authors, to- gether with Frances Kuo, studied an elliptic eigenvalue problem with ho mogeneous Dirichlet boundary conditions, and with coefficients that depend on a n infinite num- ber of uniformly distributed stochastic parameters. In this settin g, the eigenvalues, and in turn the eigenvalue gap, also depend on the stochastic param eters. Hence, for a robust error analysis one needs to be able to bound the gap over a ll possible real- isations of the parameters, and because the gap depends on infinit ely-many random parameters, this is not trivial. This short note presents, in a simplifie d setting, an im- portant result that was shown in the paper above. Namely, that, u nder certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinit e-dimensional parameter space. 1 Introduction Eigenvalue problems are useful for modelling many phenomena from a pplications in engineering and physics, e.g, structural mechanics, acoustic sca ttering, elastic mem- branes, criticality of neutron transport/diffusion and band gap ca lculations for pho- tonic crystal fibres. In this work, we consider the following eigenvalue problem −∇·/parenleftbig a(x,y)∇u(x,y)/parenrightbig =λ(y)u(x,y), forx∈D,(1.1) u(x,y) = 0, forx∈∂D, where the derivatives are taken with respect to x, andy= (y1,y2,...) is a ran- dom, infinite-dimensional vector with independent uniformly distribu ted components yj∼U([−1 2,1 2]). We assume that the physical domainD⊂Rd, ford= 1,2,3, is bounded with Lipschitz boundary, and denote the stochastic/parameter domain by U:= [−1 2,1 2]N. 1Institute for Applied Mathematics and Interdisciplinary C enter for Scientific Computing, Universit¨ at Heidelberg, 69120 Heidelberg, Germany a.gilbert@uni-heidelberg.de ,r.scheichl@uni-heidelberg.de 2Department of Mathematical Sciences, University of Bath, B ath BA2 7AY UK i.g.graham@bath.ac.uk 3School of Mathematics and Statistics, University of New Sou th Wales, Sydney NSW 2052, Australia i.sloan@unsw.edu.au 1In many uncertainty quantification (UQ) applications, the coefficien ta(x,y) is given by a Karhunen–Lo` eve expansion of a random field. Taking this as motivation, we assume that the coefficient is an affine map of y, and satisfies 0< a(x,y) =a0(x)+∞/summationdisplay j=1yjaj(x)<∞,for allx∈D,y∈U. FurtherassumptionsonthecoefficientwillbegivenexplicitlyinAssump tionA1below. If we ignore the ydependence, then (1.1) is a self-adjoint eigenvalue problem, which has been studied extensively in the literature, see, e.g., [7, 9]. I n particular, it is well known that (1.1) has countably many eigenvalues 0< λ1< λ2≤λ3≤ ···, and that the smallest eigenvalue is simple. However, in our setting the eigenvalues depend on y:λk=λk(y), and since the parameter domain is infinite-dimensional, care must be taken when transferring classical results to our set ting. In particular, although it is well known that in the unparametrised setting the spec tral gap, λ2−λ1, is some fixed positive number, in our setting the spectral gap, λ2(y)−λ1(y), is a function defined on an infinite-dimensional domain, which could be arb itrarily close to 0. Our main result (see Theorem 3 for a full statement) is as follows. As suming that the terms ajin the coefficient decay sufficiently fast (in a suitable norm), then the re exists aδ >0, independent of y, such that the spectral gap of the eigenvalue problem (1.1) satisfies λ2(y)−λ1(y)≥δ,for ally∈U . As an example of the important role that the spectral gap plays in er ror analysis, consider the random elliptic eigenvalue problem from [8]. There, an algo rithm using dimension truncation, Quasi–Monte Carlo (QMC) quadrature and fin ite element (FE) methods was used to approximate the expectation with respect to the stochastic pa- rameters of the smallest eigenvalue. Throughout the error analys is the reciprocal of the spectral gap occurred in: 1) the bounds on the derivatives of the eigenvalues with respect to y(required for the QMC and dimension truncation error analysis); 2) the constants for the FE error; and 3) the convergence rate for th e eigensolver (by Arnoldi iteration). In short, the entire error analysis in [8] fails unless the g ap can be bounded from below uniformly in y. In the remainder of this section we frame (1.1) as a variational eigen value problem, introduce the function space setting and summarise some known pr operties of the eigenvalues. Then, in Section 2 we prove that the spectral gap is un iformly bounded. Finally, in Section 3we perform a numericalexperiment fora specific e xample of (1.1), and present results on the size of the gap over different realisation s of the parameter generated by a QMC pointset. 1.1 Variational eigenvalue problems It is often useful to study the eigenvalue problem (1.1) in its equivale nt variational form. In this section we introduce the variational eigenvalue proble m, then present some well known properties and tools that are required for our ana lysis. First, we clarify the assumptions on the coefficient aand our setting. Assumption A1. 1. The coefficient is of the form a(x,y) =a0(x)+∞/summationdisplay j=1yjaj(x), (1.2) withaj∈L∞(D), for allj≥0. 22. There exists 0< amin< amax<∞such that amin≤a(x,y)≤amax, for all x∈D,y∈U. 3. For some p∈(0,1), ∞/summationdisplay j=1/ba∇dblaj/ba∇dblp L∞<∞. The last condition (Assumption 1.3) is the same as is required for the Q MC and dimension truncation analysis for corresponding source problems ( see [11]). Note that in that paper they also allow p= 1, but with an extra condition on the size of the sum. Also, the assumption from the Introduction that each yjis uniformly distributed is not a restriction, the important point is that each yjbelongs to a bounded interval. LetV=H1 0(D) equipped with the norm /ba∇dblv/ba∇dblV:=/ba∇dbl∇v/ba∇dblL2, and let V∗denote the dual of V. We identify L2(D) with its dual, and denote the inner product on L2(D) by/an}b∇acketle{t·,·/an}b∇acket∇i}ht, which can be continuously extended to a duality pairing on V×V∗, also denoted /an}b∇acketle{t·,·/an}b∇acket∇i}ht. Note that we have the following chain of compact embeddings V⊂⊂L2(D)⊂⊂V∗. The parameter domain Uis equipped with the topology and metric of ℓ∞. Next, define the (parametric) symmetric bilinear form A:U×V×V→Rby A(y;w,v):=/integraldisplay Da(x,y)∇w(x)·∇v(x) dx, which is also an inner product on V. In this setting, for each y∈U, the variational eigenvalue problem equivalent to (1.1) is: Find 0 /ne}ationslash=u(y)∈Vandλ(y)∈Rsuch that A(y;u(y),v) =λ(y)/an}b∇acketle{tu(y),v/an}b∇acket∇i}ht,for allv∈V , (1.3) /ba∇dblu(y)/ba∇dblL2= 1. ItfollowsfromAssumptionA1thatthebilinearform A(y)iscoerciveandbounded, uniformly in y: A(y;v,v)≥amin/ba∇dblv/ba∇dbl2 V, for allv∈V ,and (1.4) A(y;w,v)≤amax/ba∇dblw/ba∇dblV/ba∇dblv/ba∇dblV, for allw,v∈V . (1.5) Asaconsequence,foreach ywehaveaself-adjointandcoerciveeigenvalueproblem. Therefore, it is well known (see, e.g., [3, 9]) that (1.3) has a countab le sequence of positive, real eigenvalues, which (counting multiplicities) we write as 0< λ1(y)≤λ2(y)≤ ···, and the corresponding eigenvectors are denoted by u1(y),u2(y),...∈V. Themin-max principle [3, (8.36)] λk(y) = min Vk⊂V dim(Vk)=kmax v∈VkA(y;v,v) /an}b∇acketle{tv,v/an}b∇acket∇i}ht, allows us to bound each eigenvalue above and below independently of y. Indeed, by (1.4) and (1.5) we have aminmin Vk⊂V dim(Vk)=kmax v∈Vk/an}b∇acketle{t∇v,∇v/an}b∇acket∇i}ht /an}b∇acketle{tv,v/an}b∇acket∇i}ht≤λk(y)≤amaxmin Vk⊂V dim(Vk)=kmax v∈Vk/an}b∇acketle{t∇v,∇v/an}b∇acket∇i}ht /an}b∇acketle{tv,v/an}b∇acket∇i}ht. Now, using the min-max properties of the kth eigenvalue of the Dirichlet Laplacian onD, which we denote by χk, the bounds above simplify to aminχk≤λk(y)≤amaxχk. (1.6) 3To consider our problem in the framework of Kato [10] for perturb ations of linear operators, we introduce, for each y∈U, the solution operator T(y) :V∗→V, which forf∈V∗is defined by A(y;T(y)f,v) =/an}b∇acketle{tf,v/an}b∇acket∇i}htfor allv∈V . (1.7) Clearly,µ= 1/λis an eigenvalue of T(y) if and only if λis an eigenvalue of (1.3), and their eigenspaces coincide. Alternatively, we can consider the o peratorT(y) : L2(D)→L2(D). In this case, T(y) is self-adjointwith respect to the L2inner product due to the symmetry of A(y); it is compact because V⊂⊂L2(D); and finally, it is bounded due to the Lax-Milgramtheorem, which states that for ea chf∈L2(D) there is a unique T(y)f∈Vsatisfying (1.7), with /ba∇dblT(y)f/ba∇dblV≤1 amin/ba∇dblf/ba∇dblV∗≤1 amin√χ1/ba∇dblf/ba∇dblL2. (1.8) In the last inequality, we have used the Poincar´ e inequality: /ba∇dblv/ba∇dblL2≤χ−1/2 1/ba∇dblv/ba∇dblV,forv∈V , (1.9) and a standard duality argument. Note that we have expressed th e Poincar´ e constant in terms of the smallest eigenvalue of the Dirichlet Laplacian on D, using again the min-max principle. 2 Bounding the spectral gap The Krein-Rutman theorem guarantees that for every ythe fundamental eigenvalue λ1(y) is simple, see, e.g., [9, Theorems 1.2.5 and 1.2.6]. However, it does not provide any quantitative statements about the size of the spectral gap, λ2(y)−λ1(y), for dif- ferent parameter values y. As discussed in the Introduction, when studying numerical methods for eigenvalue problems in a UQ setting (see, e.g., [8]) severa l areas of the error analysis require uniform positivity of this gap over all y∈U. Here, we prove the required uniform positivity under the conditions of Assumption A 1, in particular A1.3. An explicit bound on the spectral gap can be obtained in slightly differe nt settings or by assuming tighter restrictions on the coefficients. For example , for Schr¨ odinger operators ( −∆+V) onDwith a weakly convex potential Vand Dirichlet boundary conditions, [2] gives an explicit lower bound on the fundamental gap. Alternatively, using the upper and lower bounds on the eigenvalues (1.6), we can de termine restric- tions on aminandamaxsuch that the gap is bounded away from 0. Explicitly, if the coefficient ais such that aminandamaxsatisfy amin amax>χ1 χ2, (2.1) then, by (1.6), λ2(y)−λ1(y)≥aminχ2−amaxχ1>0. However, the condition (2.1) may prove to be too restrictive. The general idea of our proof is to use the continuity of the eigenva lues to show that a non-zero minimum of the gap exists. A complication that arises in this strategy is that the parameter domain Uis not compact, so we cannot immediately conclude the existence of such a minumum; we know that Ucannot be compact in the topology ofℓ∞because it is the unit ball of ℓ∞, and the unit ball of an infinite-dimensional Banachspace isnot compact. Our solutionis basedon the fact that althoughthere are infinitely-many parameters, because of the decay of the terms in t he coefficient (see Assumption A1.3), the contribution of a parameter yjdecreases as jincreases. Specif- ically, we reparametrise (1.3) as an equivalent eigenvalue problem who se parameters do belong to a compact set. The first step is the following elementary lemma, which shows that sub sets ofℓ∞ that are majorised by an ℓqsequence (for some 1 < q <∞) are compact. 4Lemma 1. Letα∈ℓqfor some 1< q <∞. The set U(α)⊂ℓ∞given by U(α):=/braceleftbigg w∈ℓ∞:|wj| ≤1 2|αj|/bracerightbigg is a compact subset of ℓ∞. Proof.Sinceℓ∞is a normed (and hence a metric) space, U(α) is compact if and only if it is sequentially compact. To show sequential compactness of U(α), take any sequence {y(n)}n≥1⊂U(α). Clearly, by definition of U(α), eachy(n)∈ℓqand moreover, /ba∇dbly(n)/ba∇dblℓq≤1 2/ba∇dblα/ba∇dblℓq<∞for alln∈N. Soy(n)is a bounded sequence in ℓq. Sinceq <∞,ℓqis a reflexive Banach space, and so by [4, Theorem 3.18] {y(n)}n≥1has a subsequence that converges weakly to a limit inℓq. We denote this limit by y∗, and, with a slight abuse of notation, we denote the convergent subsequence again by {y(n)}n≥1. We now prove that y∗∈U(α) and that the weak convergence is in fact strong, i.e. we show y(n)→y∗inℓ∞, asn→ ∞. For any j∈N, consider the linear functional fj:ℓq→Rgiven by fj(w) =wj, wherewjdenotes the jth element of the sequence w= (wj)j≥1∈ℓq. Clearly, fj∈(ℓq)∗(the dual space)and usingthe weakconvergence established above, it follows that y(n) j=fj(y(n))→fj(y∗) =y∗ jasn→ ∞,for each fixed j. That is, we have componentwise convergence. Furthermore, sinc e|y(n) j| ≤1 2|αj|it follows that |y∗ j| ≤1 2|αj|for each j, and hence y∗∈U(α). Now, for any J∈Nwe can write /ba∇dbly(n)−y∗/ba∇dblq ℓq=J/summationdisplay j=1|y(n) j−y∗ j|q+∞/summationdisplay j=J+1|y(n) j−y∗ j|q ≤Jmax j=1,2,...,J|y(n) j−y∗ j|q+∞/summationdisplay j=J+1|αj|q. (2.2) Letε >0. Since α∈ℓq, we can choose J∈Nsuch that ∞/summationdisplay j=J+1|αj|q≤εq 2, and since y(n)converges componentwise we can choose K∈Nsuch that |y(n) j−y∗ j| ≤(2J)−1/qεfor allj= 1,2,...,Jandn≥K. Thus, by (2.2) we have /ba∇dbly(n)−y∗/ba∇dblq ℓq≤εqfor alln≥K, and hence /ba∇dbly(n)−y∗/ba∇dblℓq→0 asn→ ∞. Because /ba∇dblw/ba∇dblℓ∞≤ /ba∇dblw/ba∇dblℓqwhenw∈ℓqand 1< q <∞, this also implies thaty(n)→y∗inℓ∞, completing the proof. A key property following from the perturbation theory of Kato [10] is that the eigenvalues λk(y) are continuous in y, which for completeness is shown below in Proposition 2. First, recall that T(y) is the solution operator as defined in (1.7), and let Σ(T(y)) denote the spectrum of T(y). Proposition 2. Let Assumption A1 hold. Then the eigenvalues λ1,λ2,...are Lips- chitz continuous in y. 5Proof.We prove the result by establishing the continuity of the eigenvalues µk(y) of T(y). Lety,y′∈Uand consider the operators T(y),T(y′) :L2(D)→L2(D) as defined in (1.7). Since T(y),T(y′) are bounded and self-adjoint with respect to /an}b∇acketle{t·,·/an}b∇acket∇i}ht, it follows from [10, V, §4.3 and Theorem 4.10] that we have the following notion of continuity of µ(·) in terms of T(·) sup µ∈Σ(T(y))dist(µ,Σ(T(y′)))≤ /ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.3) Foraneigenvalue µk(y)∈Σ(T(y)), (2.3)impliesthatthereexistsa µk′(y′)∈Σ(T(y′)) such that |µk(y)−µk′(y′)| ≤ /ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.4) Note that this means there exists an eigenvalue of T(y′) close to µk(y), but does not imply that the kth eigenvalue of T(y′) is close to µk(y), that is, in (2.4) k is not necessarily equal to k′. However, consider any µk(y) and let mdenote its multiplicity. Since m <∞, we can assume without loss of generality that the collection µk(y) =µk+1(y) =···=µk+m−1(y) is afinite system of eigenvalues in the sense of Kato. It then follows from the discussion in [10, IV, §3.5] that the eigenvalues in this system depend continuously on the operator with multiplic- ity preserved. This preservation of multiplicity is key to our argumen t, since it states that for T(y′) sufficiently close to T(y) there are mconsecutive eigenvalues µk′(y′),µk′+1(y′),...,µ k′+m−1(y′)∈Σ(T(y′)), no longer necessarily equal, that are close toµk(y). A simple argument then shows that each µkis continuous in the following sense |µk(y)−µk(y′)| ≤ /ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.5) To see this, consider, for k= 1,2,..., the graphs of µkonU. Note that the separate graphs can touch (and in principle can even coincide over some subse t ofU), but by definition cannot cross (since at every point in Uthe successive eigenvalues are nonincreasing); and by the preservation of multiplicity a graph cann ot terminate and a finite set of graphs cannot change multiplicity at an interior point. T hus by (2.23) the ordered eigenvalues µkmust be continuous for each k≥1 and satisfy (2.5). It then follows from the relationship µk(y) = 1/λk(y) along with the upper bound in (1.6) that we have a similar result for the eigenvalues λkof (1.3): |λk(y)−λk(y′)| ≤(amaxχk)2/ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.6) All that remains is to bound the right hand side of (2.6) by CLip/ba∇dbly−y′/ba∇dblℓ∞, with CLip>0 independent of yandy′. To this end, note that since the right hand side of (1.7) is independent of ywe have A(y;T(y)f,v) =A(y′;T(y′)f,v) for all f∈L2(D),v∈V . Rearranging and then expanding this gives A(y;(T(y)−T(y′))f,v)) =A(y′;T(y′)f,v)−A(y;T(y′)f,v) =/integraldisplay D[a(x,y′)−a(x,y)]∇[T(y′)f](x)·∇v(x) dx. Lettingv= (T(y)−T(y′))f∈V, the left hand side can be bounded from below using the coercivity (1.4) of A(y), and the right hand side can be bounded from above using the Cauchy-Schwarz inequality to give amin/ba∇dbl(T(y)−T(y′))f/ba∇dbl2 V≤ /ba∇dbla(y)−a(y′)/ba∇dblL∞/ba∇dblT(y′)f/ba∇dblV/ba∇dbl(T(y)−T(y′))f/ba∇dblV. Dividing by amin/ba∇dbl(T(y)−T(y′))f/ba∇dblVand using the upper bound in (1.8) we have /ba∇dbl(T(y)−T(y′))f/ba∇dblV≤1 a2 min√χ1/ba∇dblf/ba∇dblL2/ba∇dbla(y)−a(y′)/ba∇dblL∞. 6Then, applying the Poincar´ e inequality (1.9) to the left hand side and taking the supremum over f∈L2(D) with/ba∇dblf/ba∇dblL2≤1, in the operator norm we have /ba∇dblT(y)−T(y′)/ba∇dblL2→L2≤1 a2 minχ1/ba∇dbla(y)−a(y′)/ba∇dblL∞. Usingthis inequalityasanupperbound for(2.6)weseethatthe eigen valuesinherit the continuity of the coefficient, and so |λk(y)−λk(y′)| ≤a2 maxχ2 k a2 minχ1/ba∇dbla(y)−a(y′)/ba∇dblL∞. (2.7) where the constant is clearly independent of yandy′. Finally, to establish Lipschitz continuity with respect to y, we recall Assump- tions A1.1 and A1.3, expand the coefficients in (2.7) above and use the triangle in- equality to give |λk(y)−λk(y′)| ≤a2 maxχ2 k a2 minχ1∞/summationdisplay j=1|yj−y′ j|/ba∇dblaj/ba∇dblL∞≤a2 maxχ2 k a2 minχ1/parenleftBigg∞/summationdisplay j=1/ba∇dblaj/ba∇dblL∞/parenrightBigg /ba∇dbly−y′/ba∇dblℓ∞. By Assumption 1 the sum is finite, and hence the eigenvalue λk(y) is Lipschitz in y, with the constant clearly independent of y. Now that we have shown Lipschitz continuity of the eigenvalues and id entified suitable compact subsets, we can prove the main result of this pape r: namely, that the spectral gap is bounded away from 0 uniformly in y. The strategy of the proof is to rewrite the coefficient as a(x,y) =a0(x) +∞/summationdisplay j=1/tildewideyj/tildewideaj(x), with/tildewideyj=αjyjand/tildewideaj(x) =aj(x)/αj, choosing α∈ℓqto decay slowly enough such that/summationtext∞ j=1/ba∇dbl/tildewideaj/ba∇dblL∞<∞continues to hold. Then, using the intermediate re- sult (2.7) from the proof of Proposition 2 we can show that the eigen values of the “reparametrised” problem are continuous in the new parameter /tildewidey, which now ranges over the compact set U(α). The required bound on the spectral gap is obtained by using the equivalence of the eigenvalues of the original and reparam etrised problems. Theorem 3. Let Assumption A1 hold. Then there exists a δ >0, independent of y, such that λ2(y)−λ1(y)≥δ. (2.8) Proof.We can assume, without loss of generality, that p >1/2, because if Assump- tion A1.3 holds with exponent p′≤1/2 then it also holds for all p∈(p′,1). Conse- quently, set ε= 1−p∈(0,1/2) and consider the sequence αdefined by αj=/ba∇dblaj/ba∇dblε L∞+1/j,for each j∈N. (2.9) Settingq=p/ε=p/(1−p)∈(1,∞), using Assumption A1.3 and the triangle inequality, it is easy to see that α∈ℓq. Moreover, the inclusion of 1 /jin (2.9) ensures thatαj/ne}ationslash= 0, for all j≥1. Hence, from now on, for w= (wj)∞ j=1∈ℓ∞, we can define the sequences αw= (αjwj)∞ j=1andw/α= (wj/αj)∞ j=1. Then, recalling the definition of U(α) in Lemma 1, it is easy to see that /tildewidey∈U(α) if and only if /tildewidey/α∈U, and moreover, y∈Uif and only if αy∈U(α). Now for x∈Dand/tildewidey∈U(α), we define /tildewidea(x,/tildewidey) =a0(x)+∞/summationdisplay j=1/tildewideyjaj(x) αj, 7from which it is easily seen that /tildewidea(x,/tildewidey) =a(x,/tildewidey/α). (2.10) Then we set /tildewideA(/tildewidey;w,v):=/integraldisplay D/tildewidea(x,/tildewidey)∇w(x).∇v(x) dxforw,v∈V , and we consider the following reparametrised eigenvalue problem: Fin d/tildewideλ(/tildewidey)∈Rand 0/ne}ationslash=/tildewideu(/tildewidey)∈Vsuch that /tildewideA(/tildewidey;/tildewideu(/tildewidey),v) =/tildewideλk(/tildewidey)/an}b∇acketle{t/tildewideu(/tildewidey),v/an}b∇acket∇i}htfor allv∈V , /ba∇dbl/tildewideu(/tildewidey)/ba∇dblL2= 1. (2.11) Note that because we have equality between the original and repar ametrised coeffi- cients (2.10), for each y∈U, and corresponding /tildewidey=αy∈U(α), (2.10) implies that there is equality between eigenvalues λk(y) of (1.3) and /tildewideλk(/tildewidey) of the reparametrised eigenvalue problem (2.11) λk(y) =/tildewideλk(/tildewidey) fork∈N, (2.12) and their eigenspaces coincide. Moreover, for an eigenvalue /tildewideλk(/tildewidey) of (2.11), using (2.12) in the inequality (2.7) we have |/tildewideλk(/tildewidey)−/tildewideλk(/tildewidey′)| ≤a2 maxχ2 k a2 minχ1/ba∇dbla(/tildewidey/α)−a(/tildewidey/α)/ba∇dblL∞, which after expanding the coefficient and using the triangle inequality becomes |/tildewideλk(/tildewidey)−/tildewideλk(/tildewidey′)| ≤/parenleftBigg a2 maxχ2 k a2 minχ1∞/summationdisplay j=11 αj/ba∇dblaj/ba∇dblL∞ /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright /tildewideCLip/parenrightBigg /ba∇dbl/tildewidey−/tildewidey′/ba∇dblℓ∞, where/tildewideCLipis clearly independent of /tildewideyand/tildewidey′. Now by (2.9) together with Assump- tion A1, we have ∞/summationdisplay j=1/ba∇dblaj/ba∇dblL∞ αj≤∞/summationdisplay j=1/ba∇dblaj/ba∇dbl1−ε L∞=∞/summationdisplay j=1/ba∇dblaj/ba∇dblp L∞<∞. Thus,/tildewideCLip<∞and hence the reparametrised eigenvalues are continuous on U(α). It immediately follows that the spectral gap /tildewideλ2(/tildewidey)−/tildewideλ1(/tildewidey) is also continuous on U(α), whichbyLemma1isacompactsubset of ℓ∞. Therefore, thenon-zerominimum is attained giving that the spectral gap /tildewideλ2(/tildewidey)−/tildewideλ1(/tildewidey) is uniformly positive. Finally, because there is equality between the original and reparametrised eigenvalues (2.12) the result holds for the original problem over all y∈U. The paper [8] proves a similar result to Theorem 3 in a more general se tting, the same strategy is used there also. 3 Numerical results For our numerical results, we consider the 1-dimensional domain D= (0,1), and let the basis functions that define the coefficient in (1.2) be given by aj(x) = a0 forj= 0, c0 j2sin(jπx),forj= 1,2,...,s, 0 for j > s,(3.1) 8wherea0>0 andc0∈Rwill determine the values of aminandamax. Note that the stochastic dimension is here fixed at s= 100. Since multiplying the coefficient by any constant factor will simply rescale the eigenvalues by that same fac tor, without loss of generality we can henceforth set a0= 1 and vary c0. For a coefficient (1.2) given by the basis functions (3.1), we can obta in a formula for the bounds amin,amaxas follows amin= 1−c0 2s/summationdisplay j=11 j2= 1−0.81c0, amax= 1+c0 2s/summationdisplay j=11 j2= 1+0.81c0. We remark that these bounds are not sharp. We also consider the so-called log-normal coefficient: a(x,y) =a∗+exp/parenleftBiggs/summationdisplay j=1Φ−1(yj+1 2)aj(x)/parenrightBigg , (3.2) wherea∗≥0,ajare as in (3.1), and Φ−1is the inverse of the normal cumulative distribution function. In this way each Φ−1(yj+1 2)∼ N(0,1). Since Φ−1maps [0,1] toR, the coefficient (3.2) is unbounded, and although it is positive ( amin=a∗), for a∗= 0 it could be arbitrarily close to 0. So in this case Assumption 1.2 does n ot hold, our compactness argument fails and we have no theoretical predic tion. To approximate the eigenvalue problem in space we use piecewise linear finite elements on a uniform mesh, with a meshwidth of h= 1/64. Numerical tests for different meshwidths ( h= 1/8 toh= 1/128) produced qualitatively the same results. The purpose of this section is to provide supporting evidence that f or the problem above the gap remains bounded. We do this in a brute force manner b y studying the minimum of the gap over a large number of parameter realisations. Th e parameter realisations are generated by a QMC point set; specifically, a base-2 embedded lattice rule(see[5])with up to220points andasinglerandomshift. Togeneratethe pointswe use the generating vector exod2base2m20CKNfrom [12]. We choose QMC points as the test set because they can be shown to be well-distributed in h igh dimensions, see e.g. [5, 6]. The goal is not to find the minimum, but to provide eviden ce that as more and more of the parameter domain is searched (which corre sponds to more realisations),the minimum ofthe gap overallrealisationsapproache saconstant value. The results are given in Figures 1–5, where in each figure we plot the m inimum of the gap against the number of realisations N(blue circles) and an estimate (see the next paragraph) of the distance between the estimated minimu m and the true minimum (black triangles), along with a least-squares fit to αN−β(dashed red line) of this estimate. Each data point corresponds to a doubling of the n umber of QMC points:N= 1,2,4,...,220, and the axes are in loglog scale. LettingδNdenote the approximate minimum over the first Nrealisations, we estimate the distance to the true minimum by δN−min y∈U/parenleftbig λ2(y)−λ1(y)/parenrightbig ≈δN−δN∗, whereδN∗corresponds to the most accurate estimate of the minimum, with N∗= 220. The purpose of including such an estimate of the distance to the tru e minimum is to demonstrate that not only does the minimum of the gap appear to pla teau, but that the differences also decay like a power of N. First we considerthe affine coefficient in (1.2). The three different ch oices ofc0are: in Figure 1 c0= 1, which gives amin= 0.18 andamax= 1.82; in Figure 2 c0= 1.223, which gives amin= 2×10−4andamax= 2; and in Figure 3 c0= 0.5, which gives amin= 0.59 andamax= 1.41. Then, in Figure 4 we plot the log-normal coefficient (3.2) with amin=a∗= 0 and c0= 1, and in Figure 5 we plot the lognormal coefficient withamin=a∗= 0.18 andc0= 1. 9Foreachdifferentchoiceoftheaffinecoefficient(1.2)(Figures1,2, 3),theminimum value of the gap appears to plateau and approach a nonzero minimum . The minimum of the gap seems fairly insensitive to changes in c0. However, in Figures 4 and 5 for the log-normal coefficient (which, we recall is not covered by the th eory of the current work) the results are inconclusive. It appears that the gap tends to 0 in the case of a true lognormal coefficient ( a∗= 0, Figure 4), and that it is bounded away from 0 for the “regularised” lognormal coefficient with a∗= 0.18 in Figure 5. However, in Figure 4 the smallest computed value of the gap is close to 1, and it is po ssible that it will plateau for a denser set of QMC points. Also, in Figure 5 the plate au is not as clearly developed in as in Figures 1–3. It remains an open question if the gap can be bounded in the case of the lognormal coefficient (3.2), and wheth era∗needs to be strictly positive. 100102104106 Number of realisations (N)10-210-1100101102 Figure 1: Estimate of the minimum of the spectral gap and esti mate of the distance to the true minimum: affine coefficient (1.2) with a0=c0= 1,amin= 0.18,amax= 1.82. 100102104106 Number of realisations (N)10-1100101102 Figure 2: Estimate of the minimum of the spectral gap and esti mate of the distance to the true minimum: affine coefficient (1.2) with a0= 1,c0= 1.223,amin= 2×10−4,amax= 2. 10100102104106 Number of realisations (N)10-210-1100101102 Figure 3: Estimate of the minimum of the spectral gap and esti mate of the distance to the true minimum: affine coefficient (1.2) with a0= 1,c0= 0.5,amin= 0.59,amax= 1.41. 100102104106 Number of realisations (N)10-1100101102103 Figure 4: Estimate of the minimum of the spectral gap and esti mate of the distance to the true minimum: log-normal coefficient (3.2) with amin=a∗= 0 and c0= 1. 100102104106 Number of realisations (N)10-1100101102 Figure 5: Estimate of the minimum of the spectral gap and esti mate of the distance to the true minimum: log-normal coefficient (3.2) with amin=a∗= 0.18 andc0= 1. 114 Conclusion The spectral gap is an important quantity that occurs throughou t several areas of the numerical analysis of eigenvalue problems, and in this work we proved that, under certain conditions on the coefficient, the spectral gap of a random elliptic eigenvalue problem is uniformly bounded from below. In all of our numerical expe riments the results strongly suggest that the minimum of the gap approaches a nonzero constant value. The only exception is the lognormal coefficient, which is not cov ered by our theory. References [1] R. Andreev and Ch. Schwab. Sparse tensor approximation of pa rametric eigen- value problems. In I. G. Graham et al, editor, Numerical Analysis of Multiscale Problems, Lecture Notes in Computational Science and Engin eering, 203–241. Springer-Verlag, Berlin Heidelberg, Germany, 2012. [2] B. Andrews and J. Clutterbuck. Proof of the fundamental gap conjecture. J. Amer. Math. Soc. 24, 899–916, 2011. [3] I. Babuˇ ska and J. Osborn. Eigenvalue problems. In P. G. Ciarlet and J. L. Lions, editors,Handbook of Numerical Analysis, Volume 2: Finite Element Me thods (Part 1), 641–787. Elsevier Science, Amsterdam, The Netherlands, 1991. [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differenti al Equations . Universitext, Springer, New York, 2011. [5] R. Cools, F. Y. Kuo and D. Nuyens, Constructing embedded lattic e rules for multivariate integration. SIAM J. Sci. Comp. ,28, 2162–2188, 2006. [6] J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi- Monte Carlo way. Acta Numerica ,22, 133–288, 2013. [7] N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Self Adjoint Operators in Hilbert Space . Interscience Publishers, New York London, 1963. [8] A. D. Gilbert, I. G. Graham, F. Y. Kuo, R. Scheichl and I. H. Sloan , Analysis of quasi-Monte Carlo Methods for elliptic eigenvalue problems with stoch astic coef- ficients.Submitted, arXiv: https://arxiv.org/abs/1808.02639 , 2018. [9] A. Henrot. Extremum Problems for Eigenvalues of Elliptic Operators . Birkh¨ auser Verlag, Basel, Switzerland, 2006. [10] T. Kato. Perturbation Theory for Linear Operators . Springer-Verlag, Berlin Heidelberg, Germany, 1984. [11] F. Y. Kuo, Ch. Schwab, and I. H. Sloan. Quasi-Monte Carlo finite element meth- ods for a class of elliptic partial differential equations with random co efficients. SIAM J. Numer. Anal. ,50, 3351–3374, 2012. [12] D.Nuyens, https://people.cs.kuleuven.be/ ˜dirk.nuyens/qmc-generators/ , accessed November 2, 2018. 12

2019-01-29

A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental eigenvalue, we call this the spectral or fundamental gap. In a recent manuscript [Gilbert et al., arXiv:1808.02639], the current authors, together with Frances Kuo, studied an elliptic eigenvalue problem with homogeneous Dirichlet boundary conditions, and with coefficients that depend on an infinite number of uniformly distributed stochastic parameters. In this setting, the eigenvalues, and in turn the eigenvalue gap, also depend on the stochastic parameters. Hence, for a robust error analysis one needs to be able to bound the gap over all possible realisations of the parameters, and because the gap depends on infinitely-many random parameters, this is not trivial. This short note presents, in a simplified setting, an important result that was shown in the paper above. Namely, that, under certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinite-dimensional parameter space.

Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients

1901.10470v1

arXiv:1605.06797v1 [cond-mat.mtrl-sci] 22 May 2016Low Gilbert damping in Co 2FeSi and Fe 2CoSi films Christian Sterwerf,1,∗Soumalya Paul,2Behrouz Khodadadi,2Markus Meinert,1 Jan-Michael Schmalhorst,1Mathias Buchmeier,2Claudia K. A. Mewes,2Tim Mewes,2and G¨ unter Reiss1 1Center for Spinelectronic Materials and Devices, Physics Department, Bielefeld University, Germany 2Department of Physics and Astronomy/MINT Center, The University of Alabama, Tuscaloosa, AL 35487, USA (Dated: August 15, 2018) Thin highly textured Fe 1+xCo2−xSi (0≤x≤1) films were prepared on MgO (001) substrates by magnetron co-sputtering. The magneto-optic Kerr effect ( MOKE) and ferromagnetic resonance (FMR) measurements were used to investigate the compositio n dependence of the magnetization, the magnetic anisotropy, the gyromagnetic ratio and the rel axation of the films. The effective mag- netization for the thin Fe 1+xCo2−xSi films, determined by FMR measurements, are consistent wit h the Slater Pauling prediction. Both MOKE and FMR measuremen ts reveal a pronounced fourfold anisotropy distribution for all films. In addition we found a strong influence of the stoichiometry on the anisotropy as the cubic anisotropy strongly increase s with increasing Fe concentration. The gyromagnetic ratio is only weakly dependent on the composit ion. We find low Gilbert damping pa- rameters for all films with values down to 0 .0012±0.00012 for Fe 1.75Co1.25Si. The effective damping parameter for Co 2FeSi is found to be 0 .0018±0.0004. We also find a pronounced anisotropic relax- ation, which indicates significant contributions of two-ma gnon scattering processes that is strongest along the easy axes of the films. This makes thin Fe 1+xCo2−xSi films ideal materials for the appli- cation in STT-MRAM devices. I. INTRODUCTION Half-metallic ferromagnets have attracted great inter- est during the past few years because they promise to boost the performance of spintronic devices. High spin polarization at the Fermi level can generate high tun- nel magnetoresistance (TMR) ratios. A TMR effect can be measured in a magnetic tunnel junction (MTJ) that consists of two ferromagnetic films separated by a thin insulator. The same structures can also be utilized to spin transfer torque induced magnetization switching [1], howeverin this casea lowswitching currentdensity is de- sirable. Thus, low magnetic damping and a high spin po- larization are frequently required for spin transfer torque based devices [2]. A high spin polarization can be found in half-metals where one spin band structure is semicon- ducting while the other spin band structure is metallic. Co- and Fe-based Heusler compounds are good candi- dates for materials with high Curie temperatures and half-metallic behavior. Full Heusler compounds have the formula X 2YZ, where X and Y are transition metals and Z is a main group element. There are two different ordered structures: the L21structure and the X astructure with a different occu- pation sequence. Both structures consist of a four-atom basis and an fcc lattice. The prototype of the L2 1struc- ture is Cu 2MnAl (space group Fm ¯3m) with the occupa- tion sequence X-Y-X-Z [3]. The prototypes for the X a structure are Hg 2CuTi and Li 2AgSb with an occupation sequence Y-X-X-Z, with the two X-atoms at inequivalent positions in the lattice [4, 5]. In this work, we investigate ∗csterwerf@physik.uni-bielefeld.dethe magnetic properties of a stoichiometric series rang- ing from Co 2FeSi to Fe 2CoSi, where Co 2FeSi crystalizes in the L2 1structure and Fe 2CoSi in the X astructure, re- spectively. Both compounds should have a (pseudo-)gap in the minority states as predicted by first principle cal- culations. By substituting Co and Fe atoms the number of electrons varies and the Fermi level is expected to be shifted to lower energies when the Fe concentration is increased. As we reported previously, magnetic tunnel junctions based on the Fe 1+xCo2−xSi films exhibit very high TMR ratios for all stoichiometries [6]. At 15K a maximum TMR ratio of 262% was found for the inter- mediate stoichiometry Fe 1.75Co1.25Si, while the Co 2FeSi and Fe 2CoSi based MTJs showed a TMR ratio of 167% and 227%, respectively. One possible explanation for the high TMR ratio is that for Fe 1.75Co1.25Si the Fermi en- ergy is shifted inside the pseudo-gap. In this work we present results of the magnetic properties for the mag- netization dynamics in particular including anisotropy and the Gilbert damping parameter of the Fe 1+xCo2−xSi films, as the intrinsic relaxation is are expected to be low for half-metals [7]. II. PREPARATION AND CHARACTERIZATION TECHNIQUES Thin Fe 1+xCo2−xSi (x=0, 0 .25, 0.5, 0.75, 1) films were fabricated using co-sputtering in an UHV sputtering sys- temwithabasepressureof1 ·10−9mbar. TheArpressure duringsputteringwas2 ·10−3mbar. Thefilmsweregrown by dc- and rf-magnetron sputtering from elemental tar- gets ontoMgO (001) substrates. Additional MgOand Cr seed layers were used to accommodate small lattice mis-2 matches and to promote coherent and epitaxial growth, as the Cr seed layer grows in 45◦direction on the MgO layer, which has a lattice parameter of 4 .212˚A. The lat- tice mismatch between two unit cells of Cr (2 ×2.885˚A at 20◦C [8]) and one unit cell of Co 2FeSi (5.64˚A [9]) or Fe2CoSi (5.645˚A [10]) is about 2%. The 5nm thick MgO and Cr films were in-situ annealed at 700◦C to ob- tain smooth surfaces. Fe 1+xCo2−xSi films with a thick- ness of 20nm were deposited at room temperature and ex-situ vacuum annealed at 500◦C. A 2nm thick MgO capping layer was used to prevent oxidation of the films. To determine the stoichiometry and to adjust the sput- tering powers, x-rayfluorescencemeasurementswere car- ried out. To obtain information about the magnetization dynamics, in-plane ferromagneticresonance(FMR) mea- surements were performed using a broadband coplanar waveguide setup up to a maximum frequency of 40GHz. Least square fits of the raw data using a first derivative of a Lorenzian line shape were done to precisely deter- mine the resonance field and the peak-to-peak linewidth ∆H[11, 12]. For the FMR in-plane angle dependent measurements the samples were mounted on a rotating stage and the resonance spectra were measured at a fre- quencyof30GHz whilethe in-planeanglewaschangedin 5◦steps. In addition quasistatic magnetization reversal measurements were carried out using the magneto-optic Kerr effect (MOKE) in a vector MOKE setup with an s- polarized laser with a wavelength of 488nm. Anisotropy measurements were carried out using a rotating sample holder. The magnetic field was applied in the plane of the films. III. CRYSTALLOGRAPHIC PROPERTIES X-ray diffraction measurements were used to investi- gate the crystallographic properties of the Fe 1+xCo2−xSi films. Ordering parameters, determined from x-ray diffraction, were already discussed in our previous work [6] and found to be high for Co 2FeSi and decrease when going to Fe 2CoSi. In order to test the films for crystal- lographic symmetry ϕscans are performed on the (220) planes of the Fe 1+xCo2−xSi films. Figure 1 shows the results together with the (220) plane of the MgO (001) substrate. The result shows that the (100) Heusler plane is rotated by 45◦with respect to the MgO (100) plane. The fourfold symmetry of the ϕ-scans clearly verifies the highly textured growth of all Fe 1+xCo2−xSi films of this study. IV. MAGNETIZATION DYNAMICS In this section we present in-plane broadband FMR measurements for the Fe 1+xCo2−xSi samples to obtain information about the magnetic properties of the films. The Landau-Lifshitz-Gilbert equation describes the dy- namics of the magnetization vector /vectorMin the presencesqrt intensity (a.u.) 360 315 270 225 180 135 90 45 0 (°)x=0.75 x=0.5 x=0.25 x=0 MgO substrate x=1 FIG. 1. ϕ-scans of the (220) Fe 1+xCo2−xSi peak and (220) MgO substrate peak showing the fourfold symmetry of the films. 40 30 20 10 0f (GHz) 8 6 4 2 0 H (kOe) Fe2CoSi [100] Fe2CoSi [110] FIG. 2. Resonance frequency versus magnetic field (Kittel plot) along the in-plane magnetic hard [110] and the magneti c easy [100] axis for Fe 2CoSi. The experimental data are fitted using a combined fit (equations (3 and 4)) to determine Meff andγ′. of an effective field /vectorHeff, which contains both dc and ac fields. It is given by [13]: d/vectorM dt=−γ/vectorM×/vectorHeff+α M/parenleftBigg /vectorM×d/vectorM dt/parenrightBigg ,(1) whereγis the gyromagnetic ratio and the quantity pa- rameter αis the Gilbert damping parameter. Accord- ing to the Landau-Lifshitz-Gilbert equation (1), the res- onance condition can be expressed in terms of the sec- ond derivatives of the free-energy density Eby the Smit- Beljers formula [14]: /parenleftbiggf γ′/parenrightbigg2 =1 (Msinθ)2/bracketleftBigg ∂2E ∂θ2∂2E ∂ϕ2−/parenleftbigg∂2E ∂θ∂ϕ/parenrightbigg2/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle θ0,ϕ0,(2) whereγ′=γ/2π,θandϕare the polar and azimuthal angles of the magnetization /vectorMandθ0andϕ0the corre-3 Meff ( B / f.u.) 0 0.5 1 0.25 0.75 x7 6 5 4 3 2 1 07 6 5 4 3 2 1 0 MS ( B / f.u.) FIG. 3. Dependence of the fitted effective magnetic moment per formula unit for Fe 1+xCo2−xSi films with x=0, 0 .25, 0.5, 0.75, 1shownontheleftaxis. Thedashedlineshows theinter- polated expected magnetic moments according to the Slater- Pauling rule (right axis). sponding equilibrium values. Measurements of the mag- netic field dependent resonance frequency were carried out in two different orientations of the sample: in [100] and [110] direction of the Fe 1+xCo2−xSi Heusler alloy, as the [100] direction is the magnetic easy axis and the [110] direction the magnetic hard axis, respectively. Figure 2 showsthe exemplaryKittel plots along[100]and [110]di- rections for the Fe 2CoSi sample. The experimental data were fitted simultaneously using the Kittel equation for both easy and hard configurations [15]: f=γ′/radicalbigg (Hres−ha−H4)(Hres−ha+H4 2+4πMeff) (3) f=γ′/radicalbig (Hres−ea+H4)(Hres−ea+H4+4πMeff) (4) whereMeff,γ′andH4are shared fit parameters. H4de- scribes the magnitude of the in-plane fourfold anisotropy field.Hres−haandHres−eadenote the resonance field along the magnetic hard and the magnetic easy axis, respectively. The resulting fit parameters for the gyro- magnetic ratio γ′are presented in Fig. 6 a) for all xin Fe1+xCo2−xSi. Within the errorbarsitisnearlyconstant for x≥0.25 and slightly smaller for Co 2FeSi. The fitted effective magnetization, which includes any perpendicu- lar anisotropy present in the films, is shown in Fig. 3 for the Fe 1+xCo2−xSi samples. The error bars originate from fitting of the Kittel equations and the determina- tion of the volume of the unit cell. For bulk Co 2FeSi and Fe 2CoSi the experimentally determined magnetiza- tionsare5 .95µB/f.u.[9]and4 .99µB/f.u.[10], respectively, which match the expected magnetizations according to the Slater-Pauling rule (visualized by the dashed line in Fig. 3 on the right axis). The deviation from the expected values might be attributed to residual atomic disorder in the films or the presence of a perpendicular anisotropy caused by a small tetragonal distortion in the [001] direction.100 80 60 40 20 0H (Oe) 40 30 20 10 0 f (GHz) Co2FeSi Fe1.25Co 1.75Si Fe1.5Co 1.5Si Fe1.75 Co1.25Si Fe2CoSi FIG. 4. Frequency dependent FMR linewidth for all sam- ples measured along the magnetic hard axis [110] of the Fe1+xCo2−xSi films. The frequency dependence of the linewidth of the fer- romagnetic resonance absorption provides direct infor- mation about the magnetic relaxation. The frequency dependence of the linewidth [16, 17] can under certain conditions be characterized by an inhomogeneous resid- ual linewidth at zero field ∆ H0and an intrinsic contri- bution [18]: ∆H= ∆H0+2√ 3αeff γ′f. (5) For correct determination of the effective damping pa- rameter it is necessary to measure the linewidth over a wide frequency range to determine the slope. It is not sufficient to measure ∆ Hat a fixed frequency, because a non-zero extrinsic linewidth ∆ H0results in an over- estimated damping parameter αeff. Figure 4 shows the peak-to-peak linewidth ∆ Hfor all frequencies and all x. The measurements were performed in the direction of the magnetic hard axis of the Heusler films. The ex- perimental data were fitted by equation (5) to determine the effective damping parameters. The slope at higher frequencies was used to determine the damping parame- ters. The inhomogeneous residual linewidth at zero field ∆H0is presented in Fig. 6 b) for all stoichiometries. The error margins result from the different slopes in the ∆ H vs.fcurves. The residual linewidth decreases as the Fe concentration increases and reaches its lowest value of ∆H0= 12Oe for Fe 2CoSi. McMichael et al.[19] found that small grain size distributions can lead to low inho- mogeneous line broadening. The effective Gilbert damping parameter αeffis shown in Fig. 6 c). All damping parameters have the same order of magnitude and vary between 0 .0012±0.00012 to 0.0019±0.00013. The very upper limit of the er- ror margins was calculated assuming that the linewidth measured at 40GHz is caused solely by Gilbert type damping. Co 2FeSi exhibits a damping parameter of 0.0018±0.0004, while Fe 2CoSi shows a slightly larger value of 0 .0019±0.00013. Kasatani et al.found damping4 60 50 40 30 20 10 0) e O ( H 40 30 20 10 0 f (GHz)[100] [110]Fe 2CoSi magnetic hard axis magnetic easy axis FIG. 5. FMR linewidth for Fe 2CoSi measured along both the magnetic hard [110] and magnetic easy [100] axis. parameters from 0 .0023 to 0 .0061 for Co 2FeSi films and 0.002for Fe 2CoSi [20]. In general, the Gilbert damping is expected to be low in half-metallic materials, where spin- flip processes are suppressed [7, 21]. The small damping parameters of the metallic films show that a pseudo-gap aspresentinthe Fe 1+xCo2−xSisystemissufficienttogive rise to a low Gilbert damping. Figure 5 shows the frequency dependent linewidth along easy and hard axes for the Fe 2CoSi. The linewidth exhibits almost linearbehavior(the Gilbert model) along the hard axis. We observed non-linear behavior in the linewidth vs. frequency responsealongthe magneticeasy axis. ThisnonlineardependenceoftheFMRlinewidthon frequencyisatypicalobservationwhentwomagnonscat- tering contributes significantly to the relaxation [22, 23]. Two-magnon scattering is an extrinsic relaxation mecha- nism and can be induced by means of different scattering centers such as voids or pores [24], surface roughness [22] and grain size [25] or by network of misfit dislocations which causes scattering of the FMR mode (k=0) into propagating spin waves (k /negationslash=0). A. FMR in-plane rotation measurements To obtain further information about the magnetic anisotropies and magnetic relaxation additional FMR measurements were carried out as a function of the in-plane angle of the applied field with respect to the Fe1+xCo2−xSi[110]axis. Theoperatingfrequencyforthe rotation measurements was 30GHz. At this frequency the resonancefields are high enough to saturate the mag- netization along the easy and hard axes. All measure- ments were performed at room temperature. A fourfold symmetry is observed in the in-plane angle dependence of the ferromagnetic resonance field for all samples. Figure7a)exemplarilyshowstheferromagnetic resonancefield Hresversus the in-plane rotation angle for Fe2CoSi. The dependence of the resonance field on the in-plane angle was simulated numerically using equationH0) e O ( 2.94 2.92 2.90 2.88 2.86' (MHz/Oe) 60 40 20 0K4e k ( rmc/g3) 0 0.5 1 0.25 0.75 xa) b) c) d) eff 60 50 40 30 20 10 0 0.005 0.004 0.003 0.002 0.001 0.000 FIG. 6. a) Gyromagnetic ratio γ′, b)Extrinsic contribution to the linewidth ∆ H0of the FMR spectra, c) effective Gilbert damping parameter and d) cubic magnetic anisotropy con- stantK4for Fe 1+xCo2−xSi films with x= 0, 0.25, 0.5, 0.75 ,1. (2), assuming a cubic magnetic anisotropy contribution to the Gibbs free energy [26, 27]: Ecubic=−1 2K4/parenleftbig α4 1+α4 2+α4 3/parenrightbig , (6) whereK4is the cubic magnetic anisotropy constant and α1,α2,α3are the directional cosines with respect to the cubic principal axes. The experimentally determined in-plane angle dependent Hresdata were fitted with the numerical solution (red line in Fig. 7 a)) to determine the cubic anisotropy constant. Figure 7 b) shows the corresponding linewidth data, which also shows a clear fourfoldsymmetry. The linewidth exhibitsmaximaalong the easy axes and minima along the hard axes of the cubic magnetic anisotropy. Randomly distributed crys- talline defects oriented along the in-plane principal crys- tallographic axis [28] or a fourfold distribution in misfit dislocations [29] which induce the same symmetry on the strength of two magnon scattering can explain the ob- served anisotropic relaxation. The magnetic fourfold symmetry matches the crystal- lographic symmetry of the highly textured Fe 1+xCo2−xSi films mentioned before. A polar plot of the MOKE squareness versus the rotational angle for Fe 2CoSi is pre- sented in Fig. 8. This measurement confirms the cubic5 5100520053005400 5200 5300 5400045 90 135 180 225 270315 020 40 60 20 40 60 045 90 135 180 225 270315Hres (O e) [110] [110] [100][100] a) b) H (Oe) FIG. 7. Polar plots of a) the resonance fields H resand b) the linewidth ∆ Has a function of the in-plane angle of the applied field with respect to the [110] axis of a 20nm thick Fe2CoSi film measured at a microwave frequency of 30GHz.0.60.81 0.8 1045 90 135 180 225 270315[110] [100] MR/M S FIG.8. Polar plotsofthesquarenessMR MSforFe 2CoSiobtained by MOKE measurements. anisotropy present in the films as seen in the FMR mea- surement. The magnetic easy axis is located along the [100] crystallographic axis and the magnetic hard axis is located along the [110] crystallographic axis. A cubic anisotropy with the easy magnetic axis in the Heusler [100] direction is found for all samples. The cubic mag- netic anisotropy constant K4obtained from the FMR measurements changes significantly in this series from 55.8kerg cm3for Fe 2CoSi to 16 .6kerg cm3for Co 2FeSi, respec- tively. The cubic anisotropy constants for all stoichiome- tries are presented in Fig. 6 d). Hashimoto et al.found a similarcubicanisotropyconstantof18kerg cm3forcrystalline Co2FeSi with a film thickness of 18 .5nm [30]. Some films showanadditionaluniaxialanisotropycomponent,which can originate from miscut substrates. V. CONCLUSION In summary we found very small damping parame- tersforthehalf-metallicFe 1+xCo2−xSifilmsvaryingfrom 0.0012±0.00012 to 0 .0019±0.00013. Co 2FeSi exhibits a damping parameter of 0 .0018±0.0004. Thus, the films are suitable for the use in STT-MRAMs. FMR andMOKEmeasurementsrevealafourfoldmagnetocrys- talline anisotropy for all films in accordance with the fourfold crystalline symmetry in the highly textured films. The need for frequency dependent FMR measure- ments was exemplified by the finding that the residual linewidth changes both with composition and with the measurement direction. ACKNOWLEDGMENTS The authors gratefully acknowledge financial support from Bundesministerium f¨ ur Bildung und Forschung (BMBF) and Deutsche Forschungsgemeinschaft (DFG, contract no. RE 1052/32-1) as well as support through the MINT Center summer program. S. Paul, B. Kho- dadadi and T. Mewes would like to acknowledge sup- port by the NSF-CAREER Award No. 0952929, C.K.A. Mewes would like to acknowledge support by the NSF- CAREER Award No. 1452670.6 [1] L. Berger, Physical Review B 54, 9353 (1996). [2] J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 159, L1 (1996). [3] A. J. Bradley and J. W. Rodgers, Proceedings of the Royal Society of London Series A 144, 340 (1934). [4] M. Puselj and Z. Ban, Croat. Chem. Acta 41, 79 (1969). [5] H. Pauly, A. Weiss, and H. Witte, Z. Metallkunde 59, 47 (1968). [6] C. Sterwerf, M. Meinert, J.-M. Schmalhorst, and G. Reiss, IEEE Transactions on Magnetics 49, 4386 (2013). [7] C. Liu, C. Mewes, M. Chshiev, and T. 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Mewes, Relaxation in Magnetic Ma- terials for Spintronics, in: Handbook of Nanomagnetism (Pan Stanford, 2015) p. 74. [19] R. D. McMichael, D. J. Twisselmann, and A. Kunz, Phys. Rev. Lett. 90, 227601 (2003). [20] Y. Kasatani, S. Yamada, H. Itoh, M. Miyao, K. Hamaya, and Y. Nozaki, Applied Physics Express 7, 123001 (2014). [21] G. M. M¨ uller, J. Walowski, M. Djordjevic, and G. X. Miao, Nature Materials 8, 56 (2009). [22] H. Lee, Y. Wang, C. Mewes, and W. H. Butler, Applied Physics Letters 95, 082502 (2009). [23] P. Landeros, R. E. Arias, and D. L. Mills, Physical Re- view B77, 214405 (2008). [24] M. J. Hurben and C. E. Patton, Journal of Applied Physics83, 4344 (1998). [25] R. D. McMichael, M. D. Stiles, and P. J. Chen, Journal of Applied Physics 83, 7037 (1998). [26] M. Farle, Reports on Progress in Physics 61, 755 (1998). [27] B. Heinrich and J. Bland, eds., Radio Frequency Tech- niques, in: Ultrathin Magnetic Structures II (Springer, 1994) p. 195. [28] I. Barsukov, P. Landeros, R. Meckenstock, J. Lindner, D. Spoddig, Z.-A. Li, B. Krumme, H. Wende, D. L. Mills, and M. Farle, Phys. Rev. B 85, 014420 (2012). [29] G. Woltersdorf and B. Heinrich, Physical Review B 69, 184417 (2004). [30] M. Hashimoto, J. Herfort, H. P. Schonherr, and K. H. Ploog, Applied Physics Letters 87, 102506 (2005).

2016-05-22

Thin highly textured Fe$_{\mathrm{1+x}}$Co$_{\mathrm{2-x}}$Si ($0 \leq$ x $\leq 1$) films were prepared on MgO (001) substrates by magnetron co-sputtering. The magneto-optic Kerr effect (MOKE) and ferromagnetic resonance (FMR) measurements were used to investigate the composition dependence of the magnetization, the magnetic anisotropy, the gyromagnetic ratio and the relaxation of the films. The effective magnetization for the thin Fe$_{\mathrm{1+x}}$Co$_{\mathrm{2-x}}$Si films, determined by FMR measurements, are consistent with the Slater Pauling prediction. Both MOKE and FMR measurements reveal a pronounced fourfold anisotropy distribution for all films. In addition we found a strong influence of the stoichiometry on the anisotropy as the cubic anisotropy strongly increases with increasing Fe concentration. The gyromagnetic ratio is only weakly dependent on the composition. We find low Gilbert damping parameters for all films with values down to $0.0012\pm0.00012$ for Fe$_{1.75}$Co$_{1.25}$Si. The effective damping parameter for Co$_2$FeSi is found to be $0.0018\pm 0.0004$. We also find a pronounced anisotropic relaxation, which indicates significant contributions of two-magnon scattering processes that is strongest along the easy axes of the films. This makes thin Fe$_{\mathrm{1+x}}$Co$_{\mathrm{2-x}}$Si films ideal materials for the application in STT-MRAM devices.

Low Gilbert damping in Co2FeSi and Fe2CoSi films

1605.06797v1

Theory of Spin Magnetohydrodynamics Yaroslav Tserkovnyak and Clement H. Wong Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA (Dated: June 10, 2021) We develop a phenomenological hydrodynamic theory of coherent magnetic precession coupled to electric currents. Exchange interaction between electron spin and collective magnetic texture produces two reciprocal eects: spin-transfer torque on the magnetic order parameter and the Berry- phase gauge eld experienced by the itinerant electrons. The dissipative processes are governed by three coecients: the ohmic resistance, Gilbert damping of the magnetization, and the \  coecient" describing viscous coupling between magnetic dynamics and electric current, which stems from spin mistracking of the magnetic order. We develop general magnetohydrodynamic equations and discuss the net dissipation produced by the coupled dynamics. The latter in particular allows us to determine a lower bound on the magnetic-texture resistivity. PACS numbers: 72.15.Gd,72.25.-b,75.75.+a I. INTRODUCTION Conduction electrons moving in a ferromagnet interact with the magnetization through the exchange interaction. If the exchange eld is strong and slowly varying in space and time, the electron spin will adiabatically follow the direction of the magnetization. We may then consider electrons with spins up and down along the magnetiza- tion direction as two distinct species of particles, and for convenience call them spin up/down electrons. As is well known, a spin up/down electron wave packet acquires a Berry phase1that in uences their orbital motion. In ef- fect, the electrons experience a Lorentz force due to \c- titious" electromagnetic elds which are local functions of the magnetization.2 In this ctitious electrodynamics, spin up/down elec- trons have opposite charges and dierent conductivities. Their motion and associated currents interact with the magnetization through what is commonly called current- driven spin-transfer torques. We call this interplay be- tween spin currents and magnetization spin magneto- hydrodynamics , in analogy to the classical theory of magnetohydrodynamics,3where the magnetic elds cou- ple to electric currents in conducting uids, and the cur- rents in turn generate magnetic elds. In our spin magne- tohydrodynamics, the Maxwell's equations for the mag- netic eld are replaced by the Landau-Lifshitz-Gilbert (LLG) equation for the magnetization. In this paper, we neglect full dynamics of the real electromagnetic elds, focusing on the spin-related phenomena. The electron spin follows the magnetization direction perfectly only in the limit of an innitely large exchange eld. In reality, there will be some misalignment and associated spin relaxation. This is usually described phenomenologically as a dissipative spin torque with a coecientin the Landau-Lifshitz equation.4,5,6In a one-dimensional ring geometry, we will derive the com- plete set of coupled spin-magnetohydrodynamical equa- tions, starting from the semi-phenomenological dynami- cal equations for nonequilibrium currents and magneti- zation. We recast the reactive spin torque mediated bythe Berry phase in this thermodynamic context. In our theory, we take an alternative view that the term arises from a correction to the Berry-phase electromotive force (EMF) in the equation of motion for the charge current, with the appropriate dissipative spin torque established by the Onsager reciprocity. This physics is presently vigorously studied (exper- imentally as well as theoretically) in the contexts of current-driven magnetic excitations and domain-wall motion4,5,6,7,8,9,10and the reciprocal spin accumula- tions and voltages generated by the ctitious gauge elds.11,12,13,14,15,16Since the mesoscopic regime (mainly dealing with variants of magnetic spin valves, tunnel junctions, and magnetic multilayers) is at present well explored,17we will limit our attention here to the case of continuous magnetic systems. II. NONDISSIPATIVE SPIN TORQUE Since the underlying physics is rich and complex in the most general setting, we will limit our discussion to a simple setting, which we believe captures all the es- sential ingredients of the spin magnetohydrodynamics. Consider a uniform current in a ferromagnetic ring, as- suming for simplicity incompressible electric ows (the continuity equation prohibits current inhomogeneities for an incompressible electron uid). The electric current is then the only dynamical variable describing the electron uid. The magnetic texture here could be a domain wall or magnetic spiral, for example (in higher dimensions we could have topological twists and kinks such as vortices, hedgehogs, or skyrmions). See Fig. 1 for a schematic of the setup. In the Landau-Lifshitz phenomenology of ferromagnetic dynamics well below the Curie tempera- ture, only the instantaneous direction of the magnetiza- tionm(x;t) (or, equivalently, spin density) is assumed to be a dynamic variable. The magnitude of the spin densitySalong mis assumed to be uniform and con- stant in time. We will separately drive the current with a time-dependent external magnetic ux ( t) inside thearXiv:0806.4656v2 [cond-mat.mes-hall] 6 Jan 20092 !!J(t)m!H(x,t)!(t)e" FIG. 1: (color online). Schematics of our principal \study case:" Uniform electric current J(t) carried by itinerant elec- trons can be driven by the external magnetic ux ( t) gen- erating the EMF E=@t=c. The magnetic texture m(x;t) responds to the eective eld H(x;t), which may have an ex- ternal contribution applied to the wire independently of . The reactive magnetohydrodynamic coupling stems from the Berry phase 0, which is acquired by the electron spin (shown in blue) following the instantaneous magnetic prole (shown in red) around the loop. 0corresponds geometrically to the solid angle enclosed by the electron spin. Coupled dissipative processes arise once we relax the projection approximation, allowing for some orientational spin mistracking and dephas- ing as electrons propagate through the magnetic texture. ring, and the magnetic dynamics with a magnetic eld h(x;t) applied directly to the wire. The rst step in our phenomenology is to identify the free energyFas a function of the thermodynamic variablesJandm(x;t) (or their thermodynamic con- jugates), which completely determine the macroscopic state of our system, assuming local thermal equilibrium. Neglecting spin, the gauge-invariant free energy associ- ated with an electric current in the ring is given by F(J;) = (J =c)2=2L, where we dene LJto be the current corresponding to the canonical momentum of the electrons. Lis the self-inductance of the ring and cis the speed of light. However, spin up/down electrons propagating through a quasistatic magnetic texture18ac- cumulate also a Berry phase,1which gives a ctitious contribution to the vector potential associated with a ctitious EMF.11This vector potential is given (in some convenient gauge) by14A0 x= (~c=e) sin2(=2)@x, pro- ducing gauge-invariant ctitious ux, 0=I dxA0 x=~c 2eI dx(1cos)@x: (1) (;) are the spherical angles parametrizing m(x).e>0 is minus the electron charge. Eq. (1) is the ux associated with spin-up electrons adiabatically following magnetic texture, with the opposite result for spin-down electrons. The free energy accounting for the Berry phase be- comes F0(J;;0[m(x;t)]) = [J ( +p0)=c]2=2L; (2)wherepis the polarization of the spin s-dependent con- ductivitys:p= ("#)=("+#) (assuming fast spin relaxation or halfmetallic ferromagnets). The elec- tric current is given by Jc@F0= [J ( +p0)=c]=L=@JF0;(3) which is thus the thermodynamic conjugate of J. The equation of motion for current in our simple electric cir- cuit is given by Ohm's law, @tJL@tJ+@t( +p0)=c=RJ: (4) whereRis the resistance of the wire. Naturally, the dynamic Berry phase is seen to give a contribution to the EMF:11 E0p@t0=c=PI dxm
(@xm@tm);(5) which is a well-known result.2(We denedP=p~=2e.) Now that the free energy of the current is coupled to the magnetization of the ring through the Berry-phase ux, there will be a corresponding reactive coupling of the magnetization to the current. We describe magnetic dynamics by the Landau-Lifshitz-Gilbert equation19 @tm=Hm=Sm@tm; (6) where the eective eld His dened by the functional derivative, H@mF(so that locally H?m), and is the dimensionless Gilbert damping20parameter. The total free energy of our magnetoelectric system is F(m;J;) =F(m)+F0(J;;0[m(x;t)]), whereF(m) is a standard free energy of the ferromagnet. Variation of theF0with respect to mgives current-driven spin torque applied to the magnetic dynamics:210@mF0m, where@mF0@m0@0F0=pJ@ m0=c. Dierentiat- ing Berry phase (1) with respect to m, we nd 0=PJ@xm: (7) Since ~=2eis the electron spin-charge conversion factor, we can give another interpretation of this term. It is sim- ply the rate of change of the angular momentum of the conducting electrons with spins locked to the magnetic prole. The spins of the up/down electrons rotate in the opposite directions so that, if the spin up/down conduc- tivities are the same (and thus P= 0), the net change in their angular momentum vanishes. Putting this term on the left-hand side, we get @tmPJ@xm=S=@mF(m)m=Sm@tm:(8) The left-hand side of this equation is the rate of change of the total angular-momentum density of the magneto- electric system,2while the right-hand side gives the usual LLG torque on the system.3 III. DISSIPATIVE SPIN TORQUE LLG equation (6) with torque (7) and Ohm's law (4) with the ctitious EMF (5) now constitute coupled equa- tions of our spin magnetohydrodynamic theory, with the reactive coupling mediated by Berry phase (1). We re- produce them here for clarity (after putting the magne- tization equation in the Landau-Lifshitz form): @tJ=RJ; @ tm=HmH (1 +2)S: (9) These are the equations of motion for a quasistationary, thermodynamic system near equilibrium.22In equilib- rium, the current Jis zero and magnetization is static. Out of equilibrium, the rst-order time derivatives of (J;m) are completely specied by the instantaneous val- ues of their thermodynamic conjugates ( J;H). The right- hand side is a linear expansion in these conjugates with dissipative coecients Randthat cause the system to relax back to equilibrium. So far, the dissipation in the current and magnetization is separate and physically un- related. We now add the dissipative couplings which will be key results of this paper. We proceed phenomenologically by adding to the cur- rent equation (4) correction
E0to the Berry-phase EMF and correction R0to resistance, due to coupling with the magnetic texture m(x;t). The modied Ohm's law then becomes: @tJ=(R+R0)J+
E0(10) To avoid a slew of uninteresting coecients and anisotropies, we will constrain the phenomenology by as- suming spin-rotational symmetry of the magnetic texture and the inversion symmetry of the wire. Under the lat- ter,m!m,J!J,@x!@x, andE0!E0. In the spirit of the standard quasistationary description,22we expand only up to the linear order in the nonequilibrium quantitiesJand@tm, so that terms of the form, e.g., J2@tm
@xmare excluded. To the second order in @xm, the only possible terms satisfying these requirements are:
E0R0J= PI dx@xm
@tm2P2 SJI dx(@xm)2:(11) The rst term stems physically from a spin mistracking of electrons propagating through the magnetic texture.14 Since the mistracking should scale as 1 =
xc(vanishing in the limit of innite exchange
xc), we may anticipate the dissipative coupling to be governed by a small parame- ter~=s
xc, wheresis a characteristic (transverse) spin-dephasing time. The term in Eq. (10) describes the resistance associated with magnetic texture, which is of- ten discussed in the context of magnetic domain walls.23 Both terms in Eq. (11) are odd under time reversal, like ohmic resistance and Gilbert damping. Finally, we notethat including in Eq. (11) a reactive term of the form (5) would not add anything new to the following consid- erations, as long as we treat Pas a phenomenological coecient. Our modication of Ohm's law must respect the Onsager reciprocity principle.22Substituting @tmfrom Eqs. (9) into Eq. (11), we see how the eective eld H (which is conjugate to m) aects the dynamics of J. The Onsager theorem is now readily applied to determine how the electric current J(which is conjugate to J) should modify the dynamics of m. We write the nal result as a correction to the spin torque (7):
0=PJm@xm: (12) The complete equation of motion of the magnetic texture in the LLG form thus becomes @tm=Hm=Sm@tm+
0=S; (13) with 0implicitly included in H. Eqs. (10) and (13) are our nal coupled deterministic equations. We can rewrite them in a more explicit form as L@tJ+ (R+R0)J+@t=c= PI dx@xm
(m)@tm; S(1 +m)@tm+mH=PJ(1 +m)@xm:(14) Here, the deterministic spin-torque contribution (7) is for clarity separated out of the eective eld H, which here consists of the usual purely magnetic contributions. The left-hand sides in these equations contain the ordinary Ohm's law (corrected for the magnetic-texture resistance R0) and the LLG terms, respectively, while the right-hand sides describe the reactive Berry-phase coupling and its dissipative correction. Eq. (12) was derived microscopically in Refs. 4,6,24,25, relatingto electron spin dephasing: ~=s
xc(con- sistent with our anticipation above). Its Onsager coun- terpart in Eq. (11) was rst obtained phenomenologi- cally in Ref. 14 and microscopically in Ref. 13. These \terms" are now accepted to be crucial in understand- ing current-driven magnetic dynamics and the reciprocal gauge elds. IV. DISSIPATION POWER Suppose we perturb our system with some nonequi- librium current and magnetic texture, after which the system evolves back toward equilibrium according to the equations of motion, producing entropy. If the system is steadily driven, the heat will be dissipated to the envi- ronment at some nite rate. From standard thermody- namics, the dissipation power is4 P[m(x;t);J(t)]J@tJI dxH
@tm=RJ2+I dx S(@tm)22PJ@xm
@tm+2P2 SJ2(@xm)2 :(15) According to the second law of thermodynamics, the dis- sipation (15) must always be positive, which means that 1. This gives us the lower bound on the resistivity of the magnetic texture: =2P2 S(@xm)22P2 S(@xm)2: (16) In models where comes solely from the coupling of the magnetization to the conducting electrons (which is in fact believed to be the dominant cause for Gilbert damping in metallic ferromagnets), we may expect the lower bound (16) to give an estimate for the texture re- sistivity. For a mean-eld Stoner-model treatment of Gilbert damping, we found =, while for an sd model we had = (s=S), wheresis the portion of spin density carried by the selectrons,Sis the total spin density, and =~=s
xcin both cases (with the spin- dephasing time sgoverned by the magnetic and spin- orbit impurities).6In both models, therefore, S=s, giving for the resistivity estimate (up to the second order in spatial derivative) &(P2=s)(@xm)2; (17) which involves only quantities related to conducting elec- trons. Taking parameters relevant to Permalloy wires:7 p1,102, domain-wall width of 20 nm, and the magnetization of 103emu=cm3, we nd the resistiv- ity (17) to be 104
cm. This is smaller than the domain-wall resistivity calculated to the (1 =
xc)2 order in spin mistracking of the magnetic prole (but still quadratic order in texture), in the absence of spinrelaxation,23whose overall prefactor appears to be larger than in our Eq. (17) for transition metals. We thus con- clude that our may in practice be much larger than unity (which is the lower bound necessary for the consis- tency of our phenomenology). Let us also note in the passing that in the special case of=and= 1, the magnetic dissipation (15) ac- quires a very simple form: P[m(x;t)]!SI dx @tmPJ S@xm2 ; (18) which is nothing but the Gilbert dissipation with the ad- vective time derivative Dt=@t+v@x(v=PJ=S). It is clear that this limit describes dissipative magnetic dynamics that are simply carried by the electric ow at speedv. In this case, the spin torques disappear if we write the LLG equation (6) with Dtin the place of @t.8 V. THERMAL NOISE At nite temperatures, thermal agitation causes uctu- ations of the current and magnetization, which are cor- related due to their coupling. A complete description requires that we supplement the stochastic equations of motion with the correlators of these uctuations. It is convenient to regard these uctuations as being due to a stochastic external magnetic eld hand a stochastic current source J: their noise correlators are then related to the dissipative coecients of the theory according to the uctuation-dissipation theorem (FDT). Constructing the noise sources by following the standard procedure,22 our nal coupled stochastic equations become: L@tJ+~R(J+J) +@t=c=PI dx@xm
(m)@tm; (19) S(1 +m)@tm+m(H+h) =PJ@xm+P(J+J)m@xm; (20) where we have explicitly separated the deterministic spin-torque contribution PJ@xmout of the eective eld H, which here consists of the usual purely magnetic contributions. The left-hand sides in these equations contain the ordinary Ohm's law (corrected for the magnetic-texture resistance: ~R=R+R0) and the LLG terms, respectively, while the right-hand sides describe the reactive Berry-phase coupling and its dissipative correction. WritingfJ;Hg=^ f@tJ;@tmg, we read out the \matrix" ^ from Eqs. (19) and (20): ^ J;J=1 R0;^ J;h(x)=P R0@xm;^ h(x);J=P R0@xm; ^ hi(x);hi0(x0)=Sii0jmj(x)(xx0) +Sii0(xx0)2P2 R0@xmi(x)@xmi0(x0) (21) whereijkis the antisymmetric Levi-Civita tensor. Symmetrizing matrix ^ immediately produces Langevin sources5 satisfying the FDT,22in the limit that ~!kBT: hJ(t)J(t0)i= 2kBT(tt0)=~R;hJ(t)h(t0)i= 0; hhi(x)hi0(x0)i= 2kBTh Sii0(xx0)(2P2=~R)@xmi@x0mi0i (tt0): (22) Apart from the obvious contributions, we have a magnetic eld noise proportional to 2, in the form of a nonlocal tensor Gilbert damping. The nonlocal Gilbert damping is apparent, if the electrons are not externally driven, @t = 0, in the limit L!0 of a large ring, in which case the magnetic equation decouples to give S(1 +m)@tm+m(H+h+h0) =P2 ~R(1 +m)@xmI dx0@x0m
(m)@tm: (23) Here, we moved the spin torque driven by the Nyquist noise to the left as h0=PJm@xm: (24) h0thus enters the equation as a statistically independent current-driven noise source. Writing the right-hand side of Eq. (23) as mI dx0$K(x;x0)@tm(x0); (25) where Kii0(x;x0) =P2 R0(m@xm@xm)i(m@x0m+@x0m)i0; (26) and extracting the symmetric part of the tensor Kii0(x;x0), we arrive at the total Gilbert damping tensor Gii0(x;x0) =ii0(xx0) +P2 S~R (m@xm)i(m@x0m)i02@xmi@x0mi0 : (27) This is exactly the form required by the FDT, consistent with the correlator for h+h0. The eective Gilbert damping can thus appear both negative and positive in dierent regions. The minimal texture resistivity (16), however, insures that we have a nonnegative damping globally. This Gilbert damping originates physically in the spin torques that are generated by the magnetically- driven ctitious EMF. Nonlocal @x@x0magnetic noise was recently constructed in Ref. 26 (neglecting spin relax- ation and) by heuristically converting Nyquist current noise into magnetic uctuations via adiabatic spin trans- fer. Although the DFT-required nonlocal @x@x0Gilbert tensor (27) was established in that paper (apart from the 2piece), only here we are able to derive it directly from the fundamental Langevin sources of the coupled mag- netohydrodynamic theory, dictated by the FDT. As es- timated in Ref. 26, this nonlocal contribution to Gilbert damping is in practice important (in comparison to ) in nanoscale magnetic structures. VI. SUMMARY We developed a general phenomenological theory of magnetohydrodynamic coupling in isotropic metallic fer-romagnets. The reactive coupling between magnetic tex- ture dynamics on the one hand and electric ows on the other stems from the Berry phase accumulated by elec- tron spin following the quasistationary magnetic texture. Dissipative terms of the coupled dynamic equations orig- inate in the electron spin mistracking of the magnetic or- der parameter and the associated spin dephasing. Apart from the usual Gilbert damping, the latter leads to a viscous coupling between electric currents and magnetic texture dynamics, parametrized by a single parameter . We also obtain a small correction to the texture resistiv- ity at order 2. Finally, our thermodynamic description of the magnetohydrodynamic coupling allows us to de- rive the stochastic Langevin contributions to the eective eld and electric current, according to the uctuation- dissipation theorem. Acknowledgments We acknowledge stimulating discussions with Gerrit E. W. Bauer, Arne Brataas, and Mark D. Stiles. This work was supported in part by the Alfred P. Sloan Foundation. 1M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).2G. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83 (1987).6 3L. D. Landau, E. M. Lifshitz, and L. P. 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2008-06-28

We develop a phenomenological hydrodynamic theory of coherent magnetic precession coupled to electric currents. Exchange interaction between electron spin and collective magnetic texture produces two reciprocal effects: spin-transfer torque on the magnetic order parameter and the Berry-phase gauge field experienced by the itinerant electrons. The dissipative processes are governed by three coefficients: the ohmic resistance, Gilbert damping of the magnetization, and the "beta coefficient" describing viscous coupling between magnetic dynamics and electric current, which stems from spin mistracking of the magnetic order. We develop general magnetohydrodynamic equations and discuss the net dissipation produced by the coupled dynamics. The latter in particular allows us to determine a lower bound on the magnetic-texture resistivity.

Theory of spin magnetohydrodynamics

0806.4656v2

arXiv:1502.01420v2 [cond-mat.mes-hall] 12 Mar 2015Nonlinear analysis of magnetization dynamics excited by sp in Hall effect Tomohiro Taniguchi National Institute of Advanced Industrial Science and Tech nology (AIST), Spintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan. (Dated: October 6, 2018) We investigate the possibility of exciting self-oscillati on in a perpendicular ferromagnet by the spin Hall effect on the basis of a nonlinear analysis of the Lan dau-Lifshitz-Gilbert (LLG) equation. In the self-oscillation state, the energy supplied by the sp in torque during a precession on a constant energy curve should equal the dissipation due to damping. Al so, the current to balance the spin torque and the damping torque in the self-oscillation state should be larger than the critical current to destabilize the initial state. We find that these conditio ns in the spin Hall system are not satisfied by deriving analytical solutions of the energy supplied by t he spin transfer effect and the dissipation due to the damping from the nonlinear LLG equation. This indi cates that the self-oscillation of a perpendicular ferromagnet cannot be excited solely by the s pin Hall torque. PACS numbers: 75.78.-n, 05.45.-a, 75.78.Jp, 75.76.+j I. INTRODUCTION Nonlinear dynamics such as fast switching and self- oscillation (limit cycle) has been a fascinating topic in physics1,2. Magnetization dynamics excited by the spin transfer effect3,4in a nanostructured ferromagnet5–12 provide fundamentally important examples of such non- linear dynamics. The magnetization switching was first observed in Co/Cu metallic multilayer in 20005. Three years later, self-oscillation was reported in a similar system6. In these early experiments on the spin transfer effect, linear analysis was used to estimate, for exam- ple, the critical current destabilizing the magnetization in equilibrium13,14. However, recently it became clear that nonlinear analysis is necessary to quantitatively an- alyze the magnetization dynamics2,15–26. For example, current density to excite self-oscillation can be evaluated by solvinga nonlinearvectorequation calledthe Landau- Lifshitz-Gilbert (LLG) equation23,24. Originally, the spin transfer effect was studied by ap- plyinganelectriccurrentdirectlytoaferromagneticmul- tilayer. Recently, however, an alternative method em- ploying the spin Hall effect has been used to observe the spin transfer effect27–40. The spin-orbit interaction in a nonmagnetic heavy metal scatters the spin-up and spin- down electrons to the opposite directions, producing a pure spin current flowing in the direction perpendicular to an applied current. The pure spin current excites the spin torque, called spin Hall torque, on a magnetization in a ferromagnet attached to a nonmagnet. The direc- tionofthespinHalltorqueisgeometricallydetermined27, and its magnitude shows a different angular dependence than the spin torque in the ferromagnetic multilayer3. Therefore, it is fundamentally unclear whether the phys- ical phenomena observed in the multilayer5–12can be re- produced in the spin Hall system, and thus, new phys- ical analysis is necessary. The magnetization switching of both in-plane magnetized and perpendicularly mag- netized ferromagnets by spin Hall torque was recently reported28–31,36,37. Accordingly, it might be reasonableto expect reports on self-oscillation by spin Hall torque. However, whereas self-oscillation has been observed in the in-plane magnetized system32, it has not been re- ported yet in the perpendicularly magnetized system. The purpose of this paper is to investigate the possibil- ityofexcitingself-oscillationbyspinHalltorquebasedon a nonlinear analysis of the LLG equation. We argue that two physical conditions should be satisfied to excite self- oscillation. The first condition isthat the energythat the spintorquesuppliesduringaprecessiononaconstanten- ergy curve should equal the dissipation due to damping. The second condition is that the current to balance the spin torque and the damping torquein the self-oscillation state should be larger than the critical current to desta- bilize the initial state. This is because the magnetization initially stays at the minimum energy state, whereas the self-oscillation corresponds to a higher energy state. We derive exact solutions of the energy supplied by the spin transfer effect and the dissipation due to damping in the spin Hall system by solving the nonlinear LLG equation, andfindthat theseconditionsarenotsatisfied. Thus, the self-oscillation of a perpendicular ferromagnet cannot be excited solely by the spin Hall torque. The paper is organized as follows. The physical condi- tions to excite a self-oscillation is summarized in Sec. II. These conditions are applied to the spin Hall system in Sec. III. Section IV is devoted to the conclusions. II. PHYSICAL CONDITIONS TO EXCITE SELF-OSCILLATION Let us first summarize the physical conditions neces- saryto excite self-oscillation. The magnetization dynam- ics are described by the LLG equation dm dt=−γm×H−γHsm×(p×m)+αm×dm dt,(1) wheremandpare the unit vectors pointing in the directions of the magnetization and the spin polariza-2 tion of the spin current, respectively. The gyromag- netic ratio and the Gilbert damping constant are de- noted as γandα, respectively. The magnetic field H relates to the energy density of the ferromagnet Evia H=−∂E/∂(Mm), where Mis the saturation magneti- zation. The strength of the spin torque, Hs, is propor- tional to the current density j. Since the LLG equation conserves the norm of the magnetization, the magnetiza- tion dynamics can be described as a trajectory on a unit sphere. The energy density Eshows constant energy curves on this sphere. For example, when the system has uniaxial anisotropy, the constant energy curves are lati- tudelines. Theself-oscillationisasteadyprecessionstate on a constant energy curve excited by the field torque, the first term on the right-hand side of Eq. (1). This means that the second and third terms of Eq. (1), aver- aged over the constant energy curve, cancel each other. In other words, the energy supplied by the spin trans- fer effect during the precession on the constant energy curve equals the dissipation due to the damping. This condition can be expressed as2,24 /contintegraldisplay dtdE dt=Ws+Wα= 0, (2) where the energy supplied by the spin transfer effect and the dissipation due to the damping during the precession on the constant energy curve of Eare given by2,15–26 Ws(E) =γM/contintegraldisplay dtHs[p·H−(m·p)(m·H)],(3) Wα(E) =−αγM/contintegraldisplay dt/bracketleftBig H2−(m·H)2/bracketrightBig .(4) The time integral is over a precession period on a con- stant energy curve. We emphasize that Eqs. (3) and (4) are functions of the energy density E. We denote the minimum and maximum values of EasEminandEmax, respectively. When the energy density also has saddle pointsEsaddle,Emaxin the following discussion can be replaced by Esaddle. To excite the self-oscillation, there should be a certain value of the electric current density that satisfies Eq. (2) for Emin< E < E maxin a set of real numbers. Therefore, Eq. (2) can be rewritten as ∃j∈R,Ws+Wα= 0. (5) We denote the current satisfying the first condition, Eq. (2), or equivalently Eq. (5), as j(E). Another condition necessary to excite self-oscillation relatestothefactthatthemagnetizationinitiallystaysat the minimum energy state. To excite any kind of magne- tization dynamics, the spin torque should destabilize the initial state, which means that a current density larger than the critical current density, jc=j(Emin), should be injected. Then, the condition j(E)> j(Emin), (6)z xy jmspin Hall torque Ht(a) (b) spin Hall torquespin Hall torquedamping dampingz xyHt // xHt // y FIG. 1: (a) Schematic view of system. The current density jflows in the nonmagnet along the x-axis, exciting the spin Hall torque pointing in the y-direction on the magnetization min the ferromagnet. The applied magnetic field is denoted asHt. (b) Schematic view of the precession trajectory of the magnetization on the constant energy curve. The solid circle is the trajectory in the absence of the magnetic field or in the presence of the field along the z-axis, whereas the dashed elliptical lines are those in the presence of the field in thexandy-axes. The solid and dotted arrows represent the directions of the spin Hall torque and the damping torque, respectively. should be satisfied to excite the self-oscillation. If this condition is not satisfied, the magnetization directly moves to a constant energy curve including the saddle point without showing a stable steady precession, and stops dynamics because the spin torque does not balance the damping torquefor Emin< E < E saddle. An example of such dynamics is shown below; see Fig. 3. We empha- size that Eqs. (5) and (6) are applicable to any kind of physical system showing a self-oscillation. III. SPIN HALL SYSTEM Let us apply the abovediscussions to the spin Hall sys- tem schematically shown in Fig. 1 (a), where the electric current flows in the nonmagnet along the xdirection, whereas the ferromagnet is attached along the zdirec- tion. The spin polarization of the spin current is geomet- rically determined as p=ey. In the spin Hall system, the spin torque strength Hsis given by Hs=/planckover2pi1ϑj 2eMd, (7)3 whereϑanddare the spin Hall angle and the thickness of the ferromagnet, respectively. The magnetic field H consists of the applied field Htand the perpendicular anisotropy field HKmzez. We can assume that Ht>0 without losing generality because the sign of Htonly af- fects the sign of j(E) derived below. Since we are in- terested in a perpendicular ferromagnet, we assume that HK> Ht>0. Figure 1 (b) schematically shows the precession trajectory of the magnetization on a constant energycurve, where the directions ofthe spin Hall torque and the damping torque are represented by the solid and dotted arrows, respectively. The spin Hall torque is par- allel to the damping torque for my>0, whereas it is anti-parallel to the damping torque for my<0. This means that the spin Hall torque dissipates energy from the ferromagnetwhen my>0, andsuppliesthe energyto the ferromagnet when my<0. Then, due to the symme- try of the trajectory, the net energy supplied by the spin Hall torque, Ws, is zero when the applied magnetic field points to the x- orz-direction. This means that Eq. (2) cannot be satisfied, and thus, self-oscillation cannot be excited in the spin Hall system in the absence of the ap- plied magnetic field, or in the presence of the field point- ing in the x- orz-direction. Therefore, in the following we focus on the applied magnetic field pointing in the y-direction. The magnetic field and the energy density are given by H=Htey+HKmzez, (8) E=−MHtmy−MHK 2m2 z. (9) The minimum energy of Eq. (9) is Emin=−MHK 2/bracketleftBigg 1+/parenleftbiggHt HK/parenrightbigg2/bracketrightBigg ,(10) which corresponds to a point mstable = (0,Ht/HK,/radicalbig 1−(Ht/HK)2). On the other hand, Eq. (9) has a saddle point at msaddle= (0,1,0), corresponding to the energy density Esaddle=−MHt. (11) Since the magnetization initially stays at the minimum energy state, and the magnetization dynamics stops whenmreaches the saddle point msaddle, we consider the energy region of Emin< E < E saddle. To calculate Eqs. (3) and (4), it isnecessarytosolveanonlinearequa- tiondm/dt=−γm×H, whichdetermines the precession trajectory of mon the constant energy curve. Since the constant energy curve of Eq. (9) is symmetric with re- spect to the yz-plane, it is sufficient for the calculation of Eqs. (3) and (4) to derive the solutions of mfor half of the trajectory in the region of mx>0, which are exactly given by mx(E) = (r2−r3)sn(u,k)cn(u,k),(12)my(E) =r3+(r2−r3)sn2(u,k),(13) mz(E) =/radicalBig 1−r2 3−(r2 2−r2 3)sn2(u,k),(14) whereu=γ/radicalbig HtHK/2√r1−r3t, andrℓare given by r1(E) =−E MHt, (15) r2(E) =Ht HK+/radicalBigg 1+/parenleftbiggHt HK/parenrightbigg2 +2E MHK,(16) r3(E) =Ht HK−/radicalBigg 1+/parenleftbiggHt HK/parenrightbigg2 +2E MHK.(17) The modulus of Jacobi elliptic functions, sn( u,k) and cn(u,k), is k=/radicalbiggr2−r3 r1−r3. (18) The derivations of Eqs. (12), (13), and (14) are shown in Appendix A. The precession period is τ(E) =2K(k) γ/radicalbig HtHK/2√r1−r3, (19) whereK(k) is the first kind of complete elliptic integral. The work done by spin torque and the dissipation due to damping, WsandWα, are obtained by substituting Eqs. (12), (13), and (14) into Eqs. (3) and (4), integrating over [0,τ/2], and multiplying a numerical factor 2 be- cause Eqs. (12), (13), and (14) are the solution of the precession trajectory for a half period. Then, WsandWα forEmin< E < E saddleare exactly given by Ws=8MHs√r1−r3 3Ht/radicalbig HK/(2Ht)Hs, (20) Wα=−4αM√r1−r3 3/radicalbig HK/(2Ht)Hα, (21) whereHsandHαare given by Hs=Ht/parenleftbigg1−r2 1 r1−r3/parenrightbigg K(k)−/parenleftbiggE M+H2 t HK/parenrightbigg E(k),(22) Hα=Ht/parenleftbigg1−r2 1 r1−r3/parenrightbigg K(k)+/parenleftbigg5E M+3HK+2H2 t HK/parenrightbigg E(k). (23) Here,E(k) is the second kind of complete elliptic inte- gral. The derivations of Eqs. (20) and (21) are shown in4 Ht/H K=0.1, 0.3, 0.5, 0.7, 0.9current, j(E) energy, E0 0.2 0.4 0.6 0.8 1.000.20.40.60.81.0 FIG. 2: The dependence of the current j(E), Eq. (24), for several values of Ht/HKon the energy density E. For simplicity, the horizontal and vertical axes are normalize d as j(E)/jcandE/(Esaddle−Emin)−[Emin/(Esaddle−Emin)] to makej(Emin) = 1,Emin= 0, and Esaddle= 1. time (μs)0 0.2 0.4 0.6 0.8 1.001.0 -1.0magnetization mz mymx FIG. 3: Typical magnetization dynamics excited by the spin Hall effect. The parameter values are taken from experiments36–38,42asM= 1500 emu/c.c., HK= 540 Oe, α= 0.005,γ= 1.764×107rad/(Oe·s),d= 1 nm, ϑ= 0.1, andHt= 50 Oe. The current magnitude is 14 ×106A/cm2, while the critical current, Eq. (25), is 13 ×106A/cm2. Appendix B. The current j(E) forEmin< E < E saddle is given by j(E) =2αeMd /planckover2pi1ϑHtHα 2Hs. (24) The currents for E→EminandE→Esaddleare41 j(Emin) =2αeMd /planckover2pi1ϑHK Ht/HK/bracketleftBigg 1−1 2/parenleftbiggHt HK/parenrightbigg2/bracketrightBigg ,(25) j(Esaddle) =2αeMd /planckover2pi1ϑ/parenleftbigg3HK−2Ht 2/parenrightbigg .(26) Equation (24) is the current density satisfying Eq. (2), or equivalently Eq. (5). Then, let us investigate whetherEq. (24) satisfies Eq. (6). It is mathematically difficult to calculate the derivative of Eq. (24) with respect to Efor an arbitrary value of E, although we can confirm thatj(Emin)> j(Esaddle) forHt< HK. We note that a parameter determining whether Eq. (6) is satisfied is onlyHt/HKbecause the otherparameters, such as αand M, are just common prefactors for any j(E). As shown in Fig. 2, j(E) is a monotonically decreasing function of Efor a wide range of Ht/HK, i.e., Eq. (6) is not satis- fied. This result indicates that the magnetization stays in the equilibrium state when j < jc=j(Emin), whereas it moves to the constant energy curve of Esaddlewithout showing stable self-oscillation when j > jcbecause the spin Hall torque does not balance the damping torque on any constant energy curve between EminandEsaddle. The magnetizationfinally stopsits dynamics at ±msaddle because all torques become zero at these points. Figure 3 shows a typical example of such dynamics, in which the time evolution of each component is shown. Therefore, self-oscillation solely by the spin Hall torque cannot be excitedin the perpendicularferromagnet. Thisis apossi- ble reason why the self-oscillation has not been reported yet. Recently, many kinds of other torques pointing in different directions or having different angular depen- dencies, such as field-like and Rashba torques, have been proposed28,29,36,37,40,43–45. These effects might change the above conclusions. Adding an in-plane anisotropy21,22, tilting the perpendicular anisotropy40, or using higher order anisotropy might be another candidate. Spin pumping is also an interesting phe- nomenon because it modifies the Gilbert damping constant46–49. It was shown in Refs.48,50that the en- hancement of the Gilbert damping constant in a fer- romagnetic/nonmagnetic/ferromagnetic trilayer system depends on the relative angle of the magnetization. This means that the Gilbert damping constant has an angular dependence. In a such case, it might be possible to sat- isfy Eqs. (5) and (6) by attaching another ferromagnet to the spin Hall system and by choosing an appropriate alignment of the magnetizations. The above formulas also apply to these studies. In Appendix C, we briefly discuss a technical difficulty to include the effect of the field-like torque or Rashba torque. IV. CONCLUSION In conclusion, wedevelopedamethod forthe nonlinear analysisofthe LLGequationinthe spinHall systemwith a perpendicular ferromagnet. We summarized physical conditions to excite self-oscillation by the spin transfer effect. The first condition, Eq. (2), or equivalently Eq. (5), implies that the energy supplied by the spin torque during a precession on a constant energy curve should equal the dissipation due to damping. The second con- dition, Eq. (6), implies that the current to balance the spin torque and the damping torquein the self-oscillation5 state should be larger than the critical current to desta- bilize the initial state. By solving the nonlinear LLG equation, we derived exact solutions of the energy sup- plied bythe spintransfereffect andthe dissipationdue to damping, and showed that these conditions are not sat- isfied. These results indicate that self-oscillation cannot be excited solely by the spin Hall torque. The author would like to acknowledge T. Yorozu for his great constructive help on this work. The author also thanks M. Hayashi, H. Kubota, and A. Emura for their kind supports. This work was supported by JSPS KAK- ENHI Grant-in-Aid for Young Scientists (B) 25790044. Appendix A: Precession trajectory on a constant energy curve Here, we show the derivation of Eqs. (12), (13), and (14). The precession trajectory on a constant energy curve is determined by dm/dt=−γm×H. They- component of this equation is dmy/dt=γHKmxmz. Thus, we find /integraldisplay dt=1 γHK/integraldisplaydmy mxmz. (A1) As mentioned in Sec. III, since the constant energycurve of Eq. (9) is symmetric with respect to the yz-plane, it is sufficient to derive the solutions of mfor half of the trajectory in the region of mx>0. Using Eandmy,mx andmzare expressed as mx=/radicalbigg 1−m2y+2E MHK+2Ht HKmy,(A2) mz=/radicalbigg −2E MHK−2Ht HKmy. (A3) The initial state of myis chosen as my(0) =r3, where r3is given by Eq. (17). Then, myat a certain time tis determined from Eq. (A1) as γ/radicalbig 2HtHK/integraldisplayt 0dt =/integraldisplaymy r3dm′ y/radicalBig (m′y−r1)(m′y−r2)(m′y−r3).(A4) We introduce a new parameter sasmy=r3+(r2−r3)s2. Then, we find γ/radicalbigg HtHK 2√r1−r3t=/integraldisplays 0ds′ /radicalbig (1−s′2)(1−k2s′2),(A5) where the modulus kis given by Eq. (18). The solution ofsiss= sn(u,k). Therefore, myis given by Eq. (13). Equations (12) and (14) are obtained by substituting Eq. (13) into Eqs. (A2) and (A3).We note that Eqs. (12), (13), and (14) are periodic functions with the period given by Eq. (19). On the other hand, when E=Esaddle, the magnetization stops itsdynamicsfinallyatthesaddlepoint m= (0,1,0). The solution of the constant energy curve of Esaddlewith the initial condition my(0) =r3can be obtained by similar calculations, and are given by mx= 2/parenleftbigg 1−Ht HK/parenrightbiggtanh(νt) cosh(νt), (A6) my=−1+2Ht HK+2/parenleftbigg 1−Ht HK/parenrightbigg tanh2(νt),(A7) mz= 2/radicalBigg Ht HK/parenleftbigg 1−Ht HK/parenrightbigg1 cosh(νt),(A8) whereν=γ/radicalbig Ht(HK−Ht). Appendix B: Derivation of Eqs. (20) and (21) Using Eqs. (12), (13), and (14), the explicit form of Eq. (3) for the spin Hall system is given by Ws= γMHs/integraltext dtws, wherewsis given by ws= (Ht−HKr3)(1−r2 3) +/braceleftbig −2Htr3+HK/bracketleftbig r3(r2+r3)−(1−r2 3)/bracketrightbig/bracerightbig (r2−r3)sn2(u,k) +{−Ht+HK(r2+r3)}(r2−r3)2sn4(u,k). (B1) Similarly, Eq. (21) for the spin Hall system is given by Wα=−αγM/integraltext dtwα, wherewαis given by wα= (1−r2 3)(Ht−HKr3)2 −/bracketleftbig 2H2 tr3−H2 K(r2+r3)(1−2r2 3)+2HtHK(1−r2r3−2r2 3)/bracketrightbig ×(r2−r3)sn2(u,k) −[Ht−HK(r2+r3)]2(r2−r3)2sn4(u,k). (B2) Then, WsandWαare obtained by integrating over [0,τ/2], and multiplying a numerical factor 2. The fol- lowing integral formulas are useful, /integraldisplayu du′sn2(u′,k) =u−E[am(u,k),k] k2,(B3) /integraldisplayu du′sn4(u′,k) =sn(u,k)cn(u,k)dn(u,k) 3k2 +2+k2 3k4u −2(1+k2) 3k4E[am(u,k),k],(B4) whereE(u,k), am(u,k), and dn( u,k) are the second kind ofincomplete elliptic integral,Jacobiamplitude function, and Jacobi elliptic function, respectively.6 Appendix C: The effect of the field-like torque or Rashba torque The direction of the field-like torque or the Rashba torque is given by m×p, wherepis the direction of the spin polarization. This means that the effects of these torques can be regarded as a normalization of the field torquem×H. Then, the energy density Eand the magnetic field Hin the calculations of WsandWαshould be replaced with an effective energy density Eand an effective field Bgiven by E=E−βMHsm·p, (C1)B=H+βHsp, (C2) where a dimensionless parameter βcharacterizes the ratio of the field-like torque or Rashba torque to the spin Hall torque. We neglect higher order terms of the torque44for simplicity, because these do not change the main discussion here. In principle, j(E) satisfying Eq. (5) can be obtained by a similar calculation shown in Sec. III. However, for example, the right-hand-side of Eq. (24) now depends on the current through EandH. Thus, Eq. (24) should be solved self-consistently with respect to the current j, which is technically difficult. 1S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, 2003), 2nd ed. 2G. Bertotti, I. Mayergoyz, and C. Serpico, Nonlinear mag- netization Dynamics in Nanosystems (Elsevier, Oxford, 2009). 3J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 4L. Berger, Phys. Rev. B 54, 9353 (1996). 5J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). 6S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature 425, 380 (2003). 7W. H. 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On the other hand, when Ht= 0, the magnetization can- not move at any point on the constant energy curve of Esaddle, and thus, Wsis zero. 42Private communication with M. Hayashi. 43C. O. Pauyac, X. Wang, M. Chshiev, and A. Manchon, Appl. Phys. Lett. 102, 252403 (2013). 44K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin,and P. Gambardella, Nat. Nanotech. 8, 587 (2013). 45X. Qiu, P. Deorani, K. Narayanapillai, K.-S. Lee, K.-J. Lee, H.-W. Lee, and H. Yang, Sci. Rep. 4, 4491 (2014). 46Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). 47Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B66, 224403 (2002). 48Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B67, 140404 (2003). 49C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas, and A. Fer- guson, Nat. Nanotech. 10, 50 (2015). 50T. Taniguchi and H. Imamura, Phys. Rev. B 76, 092402 (2007).

2015-02-05

We investigate the possibility of exciting self-oscillation in a perpendicular ferromagnet by the spin Hall effect on the basis of a nonlinear analysis of the Landau-Lifshitz-Gilbert (LLG) equation. In the self-oscillation state, the energy supplied by the spin torque during a precession on a constant energy curve should equal the dissipation due to damping. Also, the current to balance the spin torque and the damping torque in the self-oscillation state should be larger than the critical current to destabilize the initial state. We find that the second condition in the spin Hall system is not satisfied by deriving analytical solutions of the energy supplied by the spin transfer effect and the dissipation due to the damping from the nonlinear LLG equation. This indicates that the self-oscillation of a perpendicular ferromagnet cannot be excited solely by the spin Hall torque.

Nonlinear analysis of magnetization dynamics excited by spin Hall effect

1502.01420v2

Buckling Metamaterials for Extreme Vibration Damping David M.J. Dykstra,Coen Lenting, Alexandre Masurier, and Corentin Coulaisy Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, the Netherlands Damping mechanical resonances is a formidable challenge in an increasing number of applications. Many of the passive damping methods rely on using low stiness dissipative elements, complex mechanical structures or electrical systems, while active vibration damping systems typically add an additional layer of complexity. However, in many cases, the reduced stiness or additional complexity and mass render these vibration damping methods unfeasible. This article introduces a new method for passive vibration damping by allowing buckling of the primary load path, which sets an upper limit for vibration transmission: the transmitted acceleration saturates at a maximum value, no matter what the input acceleration is. This nonlinear mechanism leads to an extreme damping coecient tan 0:23 in a metal metamaterial|orders of magnitude larger than the linear damping of traditional lightweight structural materials. This article demonstrates this principle experimentally and numerically in free-standing rubber and metal mechanical metamaterials over a range of accelerations, and shows that bi-directional buckling can further improve its performance. Buckling metamaterials pave the way towards extreme vibration damping without mass or stiness penalty, and as such could be applicable in a multitude of high-tech applications, including aerospace structures, vehicles and sensitive instruments. Any mechanical system will exhibit a resonance. At this resonance, the transmission of force and accelera- tion is maximal. Limiting the amplication of accel- eration is paramount in a wide range of applications where vibrations can cause unwanted noise and failure. A paradigmatic example is that of a mass and spring damper shaken from the bottom (Fig. 1A): at reso- nance, the mass will vibrate with a much higher acceler- ation than the input acceleration provided by the shaker. While myriad strategies using highly viscoelastic materi- als [1, 2], negative stiness components [3{8], band-gap metamaterials [9{15] and active control [2, 16, 17] have been proposed, they typically suer from added mass or loss in stiness. Here, we propose to use Euler buckling as a functional mechanism to create vibrations absorbers (Fig. 1B): buckling structures are simultaneously sti thanks to their high stiness prior to buckling, yet limit the trans- mission of acceleration under post-buckling since they exhibit a force plateau under compression. Although the idea is best illustrated with a single column under axial vibrations (Fig. 1B), such a structure would collapse in post-buckling and would not be suitable for applications that require freestanding unsupported structures. This is precisely where metamaterials could come to the res- cue, as they can combine buckling in one direction and structural stiness in the other directions [18{24]. While their shock response has been well studied, their nonlin- ear vibration response remains poorly understood. In this article, we demonstrate that free-standing load- carrying metamaterials that undergo a buckling insta- dmj.dykstra@gmail.com ycoulais@uva.nlbility provide an upper limit for transmission of vibra- tions. We show that this ecient vibration absorption stems from elastic and damping nonlinearities that are in- duced by buckling. We further generalize the concept to metallic metamaterials and to metamaterials that buckle both under compression and tension. We obtain extreme damping with respect to ordinary lightweight structural materials. Our work demonstrates that buckling meta- materials are a competitive solution for lightweight struc- tures combining high damping and high specic stiness in high-tech applications. To create a structure that can maintain its own lat- eral stability in post-buckling, we rst turn to one of the most common designs in exible mechanical metamate- rials [25{27]: a polymeric slab that is patterned with a square array of circular holes (Fig. 1CD). This metama- terial exhibits a global buckling mode, where the pattern of pores becomes an array of ellipses with alternating orientations (Fig. 1D, centre). To determine its nonlin- ear vibration characteristics, we mount a mass on top and subject the sample to a base excitation around the eigenfrequency at 2 dierent levels: a low level of 0.26 G (Fig. 1CE) and a high level of 0.89 G (Fig. 1df), where G=9.81 m/s2is the acceleration of gravity. At the lower excitation level, we observe that the holes remain close to circular (Fig. 1C, see also Supplemen- tary Video 1) and we observe both a sinusoidal output response, which has a =2 phase lag with respect to the input excitation (Fig. 1E). This was to be expected based on linear vibrations [28]. At the higher excitation level, the sample buckles as seen in Fig. 1D, see also Supple- mentary Video 1. As a result, the output response in Fig. 1F is no longer perfectly sinusoidal. Importantly, despite the input level increasing by more than a factor three (from 0.26 G to 0.89 G), the output level in com- pressive direction merely changes by a third (from 4.3 GarXiv:2302.11968v1 [cond-mat.soft] 23 Feb 20232 B A MM MMM D C F EMMMMM 1 cm FIG. 1: Damping vibrations with buckling. (A) A mass ( M) spring damper system, with base excitation (blue) can show a large amplied response (orange) around resonance. (B) When the spring is a slender beam, which can buckle when subjected to a sucient compressive load from the base excitation, the amplied response may be lower. (C,D) show the deformation of a holar sample with mass mounted on top when subjected to a base excitation from the bottom around the eigenfrequency. (C) corresponds to a base excitation acceleration of 0.26 G at 33.8 Hz, while (D) corresponds to a base excitation acceleration of 0.89 G at 33.0 Hz. The ellipticity of the holes, , is tracked with red and blue ellipses (Section IV D, color bar). (E,F) Base excitations (blue) of 0.26 G (E) and 0.89 G (F) induce output accelerations (orange) of 4.3 G (E) and 5.7 G (F) respectively. to 5.7 G). More surprisingly, the maximum acceleration in tensile direction also only increases by a factor of two instead of three (from 4.3 G to 8.9 G). This suggests that compressive buckling also dampens vibrations in tensile direction. This ecient vibration damping stems from nonlineari- ties induced by buckling. To quantify such nonlinearities, we perform compression and tension mechanical tests at various strain rates (Fig. 2A, see Methods for details). As expected [26, 29], while the response is nearly linear in tension and for compression less than 2 mm, a buck- ling instability occurs at a compressive displacement of 2 mm. This instability induces a force plateau, a key nonlinearity that explains the saturation of acceleration under compression seen in Fig. 1F. However, this nonlin- earity alone does not suce to explain the reduction of acceleration in the tensile direction. The missing ingredient is an additional damping non- linearity that is also rooted in buckling. Indeed, when we vary the loading rate, we observe that the amount of hys- teresis increases signicantly around the point of buck- ling when we increase the loading rate from 1 mm/min to 1000 mm/min (Fig. 2A-inset). To better quantify thiseect, we compress and extend the sample at a loading rate of 1000 mm/min up to dierent compression levels and measure the average hysteretic force (dierence be- tween loading and unloading) across the loading regime as function of the compression. We see that the hysteresis is non-monotonic with a maximum at a compression of 4 mm, which corresponds to the buckling point. This non- monotonic damping diers from that of linear viscoelastic materials (e.g. the Voigt damper in Fig. 1A), where the average hysteretic force is constant. One concludes that buckling amplies viscoelastic eects. This can be inter- preted by the fact that the material that makes up the slender parts of the metamaterial undergoes much larger strain rates than the full structure does. Moreover, an- other key component is present specically in vibrations with a base excitation. Eectively speaking, nonlinear- ities break resonance. As the peak force levels do not increase linearly with acceleration exciting level, the en- ergy injected in the system does not increase linearly with acceleration exciting level either. These combined elastic and damping nonlinearities both team up to eciently dissipate vibrations. To quan- tify such dissipation, we measure the amplication fac-3 C D A B H IE F G FIG. 2: Vibration damping performance of buckling metamaterials. (A) Force ( F) displacement ( d) curves during compression-tension tests of the sample of Fig. 1DE. The four curves in D and E correspond to loading rates of 1, 10, 100 and 1000 mm/min from yellow to red respectively. (E) Equivalent force-displacement curve of simplied model (see Section IV E). (B,F) Average hysteretic force over compression range when performing tension-compression experiments at 1000 mm/min: (B) experimental, (F) numerical, normalized with the average hysteretic force for 0.1mm compression-tension. (G) Numerical Voigt damper strength as function of the compressive displacement, normalized by the linear Voigt damper strength (Methods). (C,H) Maximum acceleration amplication factor Aas function of frequency fduring a frequency sweep with rising and (inset) dropping frequency: (C) experimental and (H) numerical. Solid lines correspond to tension while dashed lines correspond to compression. The color of the curve indicates the input acceleration ain. (D,I) Maximum output acceleration aout across the frequency range: (E) experimental and (I) numerical. Blue (orange) curves correspond to rising (dropping) frequencies. Circles and solid lines (squares and dashed lines) correspond to compression (tension) in E and I respectively. Grey lines show the linearized trend. tor (ratio between output and input acceleration) as a function of frequency: we perform frequency sweeps at various input levels with both rising (Fig. 2C) and drop- ping (Fig. 2C-inset) frequency levels, passing the reso- nance. For rising frequencies, buckling is very ecient at limiting the vibration transmission. For small excita- tions (blue in Fig. 2C), amplication factors up to 19 are found. During post-buckling the peak amplication factors across the frequency domain become as low as 6 in compression (bordeaux solid lines) and 9 in tension (bordeaux dashed lines). This corresponds to more than doubling and tripling the amount of damping for tension and compression respectively. The peak amplication factors also shift to lower frequencies. For dropping fre- quencies, we experience a dierent trend. We observe larger amplication factors (Fig. 2C-inset) at smaller frequencies compared to a frequency sweep with a rising frequency. This dierence between rising and dropping frequency sweeps demonstrates bistability post-buckling at frequency ranges below that of the linear resonance.Equivalently, this reduction of the amplication factor corresponds to a saturation of the maximum output ac- celeration, aout, across the frequency range as function of the input acceleration, ain(Fig. 2D). In the com- pressive direction (circles), we observe a hard limit on the maximum vibration transmission at 5.9 G. In tension (squares), we observe that the trend becomes markedly lower than the linear trendline (grey) post-buckling. This behavior is consistent with analytical examples of vibra- tions of mass-spring dampers with softening springs such as quadratic or cubic examples [30{32]. However, while analytical examples of vibrations of mass-spring dampers with weakly nonlinear softening springs typically demon- strate increased amplication factors at larger excita- tions [30, 31], Fig. 2C demonstrates the opposite: the drastic nonlinearities spawned by buckling shave o the resonance peak and eciently limit vibration transmis- sion. However, the fact that buckling dampens eectively simple frequency sweeps does not yet guarantee that it4 dampens more complex vibrations. After all, nonlinear vibration responses can not be linearly combined like lin- ear vibration responses can be combined. Therefore, we also subject the sample of Fig. 1 to random vibrations (See Section IV G) and we nd again that the maxi- mum transmission of acceleration saturates at 6.0 G. This demonstrates that buckling based vibration damp- ing works eectively regardless of the type of vibration: controlled or random. In order to design buckling-based vibration damping, it is also necessary to be able to predict it. In theory, nite element methods could be used to model the response of buckling based vibration damping. However, nonlin- ear dynamic nite element methods are notoriously ex- pensive computationally, especially for a very large num- ber of cycles [33]. For this purpose, we develop a sim- ple numerical model, based on a nonlinear mass spring damper, similar to the simplied representation of Fig. 1A. However, instead of a linear model, we use nonlin- ear force-displacement (Fig. 2E) and dashpot strength- displacement (Fig. 2FG) curves. We tune these param- eters to t the experimental results of Fig. 2AB and the low excitation eigenfrequency of Fig. 2C (Methods). We then subject the model numerically to the same fre- quency sweeps as Fig. 2CD (Methods) and obtain the equivalent numerical results in Fig. 2HI. We observe that the results in Fig. 2HI match the re- sults of Fig. 2CD well: not only qualitatively but also quantitatively. The only signicant dierence found is that the numerical model predicts slightly higher output accelerations post-buckling than the experiments. This could suggest that the bilinear approximations of Fig. 2EG oversimplify the nonlinear dissipation of the exper- iments. However, the fact that the numerical predic- tions are slightly higher than the experimental results also show that the numerical results slightly underesti- mate the damping performance. Moreover, while the t- ting parameters of Fig. 2EG have been obtained based on the experiments (See Section IV E), these could also be generated using nonlinear nite element methods, with- out modeling the entire frequency sweep tests with these same nite element methods. Together, this implies that buckling metamaterials with target stiness and target damping could conservatively be designed and predicted using a very simple single element numerical model. I. HIGH STIFFNESS MATERIALS So far, we have demonstrated that vibration damping using buckling metamaterials made out of an elastomer is ecient. However, our results suer from three main shortcomings. First of all, elastomers inherently suer from a low specic stiness, which makes them inher- ently unsuited for load bearing structures. Second, it is far from obvious that buckling metamaterials can be gen- eralized to sti materials, such as metals, bre reinforced composites or ceramics, which have a low yield or failurestrain [34]. This implies that sti metamaterials cannot undergo repeated thick-walled buckling. Finally, while viscoelastic buckling will increase the amount of dissi- pation (Fig. 1D (inset)), the resulting extra amount of dissipated energy can still be relatively small compared to the total strain energy in the system for base materials with low damping coecients. To overcome the rst two problems, we can use thin- walled metal metamaterials. We can prevent yielding while buckling by letting the entire mechanism bend, as is demonstrated in Fig. 3EF, as opposed to localized mechanism deformation in Fig. 1D. Furthermore, while straight thin-walled beams inherently exhibit a low buck- ling load, the buckling load can be increased and tailored using curvature about the longitudinal axis of each beam. In turn, by using the thin-walled design of Fig. 3EF, the design becomes inherently sensitive to shear. We can re- move this compliance to shear by creating a 3D structure made out of two copies of the 2D mechanism, wherein the two 2D mechanism stabilize one another. We construct the metamaterial sample of Fig. 3AEF. This sample consists of 0.15 mm thin laser cut steel sheets (AISI 301 Full Hard) of 90 50 mm, bolted to 3D printed aluminium connectors (AlSi10Mg, selective laser sinter- ing). The steel sheets are pre-curved by 1.5 mm. The top and bottom crosses are made from 5 mm thick CNC milled aluminium, while the cross sheets in the middle are made from 0.25 mm thin laser cut steel sheets (AISI 301 Full Hard). The geometry has been selected with the help of static nonlinear nite element methods in Abaqus. The sample measures 333 333328 mm and has a mass of 1.9 kg. When we compress the sample slowly (1 mm/min) and plot the force-displacement curve in Fig. 3B, we ob- serve a buckling force plateau at 78 N. While buckling, we observe snap-through instabilities (Fig. 3B inset). When the pre-curved members buckle, they snap to an uncurved state about their longitudinal axis. In particu- lar, we observe that snap-through releases strain energy and excites local vibration modes of the metal sheets. This immediately helps to solve the third problem: insuf- cient dissipation for low damping base materials, which was identied in the beginning of this section. The snap- through instabilities show negative stiness and dissipate energy [3, 5, 35]. In turn, this dissipated energy as function of the dis- placement is plotted in Fig. 3C. It is also noteworthy that we observe that the force-displacement curve before buckling is not entirely linear due to imperfections in the sample. Similar to what we did for the elastomeric sample in Fig. 1 and Fig. 2, we add a mass of 1.5 kg on the top of the sample and subject the metal sample to a vibrational base excitation, see also Supplementary Video 2. We sub- ject the sample to frequency sweeps from low to high fre- quencies and back at several acceleration excitation levels (See Section IV F). We track the acceleration response at the base and top of the sample, similar to what we did5 E F A B G C D H I 333 328 FIG. 3: Metallic buckling metamaterial for vibration damping. (A) Metallic metamaterial consisting of two crossing 33 unit cells, allowing buckling, with side view: (E) unbuckled and uncompressed, (F) buckled and compressed by 1.2mm. (B) Force displacement curve of metal metamaterial with zoomed insert. (C) Cumulative dissipated energy, H, corresponding to (B). (D) When subjected to a base vibration, with a 1.4kg mounted on top: as function of the average input acceleration ain, the peak output acceleration aoutacross the frequency domain. Orange (blue) curves correspond to rising (dropping) frequencies. Circles (squares) correspond to compression (tension). Grey lines show the average linearized trend. (G-I) Numerical results for simulations, equivalent to the experimental results of B-D (See Methods). for the elastomeric sample in Fig. 2. We also compute the maximum output acceleration across the frequency domain as function of the input acceleration level in Fig. 3D. Again, we nd that buckling based vibration damp- ing works. More specically, the metal sample features a loss coecient tan 1=Amax0:23 atain= 1 G, a tripling of the damping coecient with respect to the low excitation response. . This loss coecient greatly surpasses those of traditional light-weight structural ma- terials such as aluminium alloys (1
1042
103), steels (2
1043
103) or carbon bre reinforced polymers (1
1033
103) [34]. This shows that buckling based vibration damping can be used to surpass the Ashby lim- its of loss coecient versus specic modulus [34]. Moreover, it is noteworthy that the metal sample did not suer from fatigue damage during the performed vi- bration tests. Over the course of the test campaign, the metal sample has been subjected to around 105vibra- tion cycles, across a variety of frequency and accelera- tion levels. At no point during this testing campaign has any visual damage been identied. This is all the more impressive considering that snap-through induces local modes, which may even vibrate at a higher frequency. This shows that it is also possible to produce buckling metamaterials for vibration damping with a high specicstiness without being fatigue sensitive. Similar to what we did for the elastomeric sample, we can also model the performance of the metal sample using a numerical model in Fig. 3GHI (See Section IV E). We can approximate the force-displacement curve of Fig. 3B using the bilinear force-displacement curve of Fig. 3G (blue). Here, we omit the imperfections in the force- displacement curve of Fig. 3B to see how a sample with little imperfections would respond. However, unlike what we considered for the numerical model of the elastomeric sample in Fig. 2EFG, we will omit nonlinear viscos- ity and instead only consider the dissipation due to the snap-through instabilities. To do this, we smear the hys- teresis of Fig. 3B linearly over the range between the start of buckling in Fig. 3G and the distance of the last snap-through instability in Fig. 3B. As such, we obtain the unbuckling force-displacement curve of Fig. 3G (or- ange), with corresponding hysteresis-displacement curve in Fig. 3H. We then subject the model numerically to the same back-and-forth frequency sweeps as the exper- imental experimental of Fig. 3A and obtain the equiva- lent numerical results of Fig. 3D in Fig. 3I. Again, we also nd that buckling based vibration damping works numerically, particularly in compression. We do however obtain some signicant dierences be-6 tween the experiments and numerics. Namely, the nu- merics predict lower peak accelerations in compression than the experiments and higher peak accelerations in tension, especially for dropping frequencies. The lower peak acceleration in compression could be because the metal sample also has additional local modes, which the numerical model does not take into account. Further- more, viscoelasticity can delay buckling, which can in- crease the peak load [23]. A similar eect in turn explains why the numerical mode predicts higher accelerations in tensile direction: delayed buckling and unbuckling induced by viscoelasticity can increase dissipation [23], which can lead to lower amplication factors, as discussed above in the case of the elastomeric sample. Despite these discrepancies, even without considering nonlinear damp- ing, but when considering snap-through induced dissi- pation, the numerical model shows valid trends. This demonstrates that such a simple numerical model can still be used to predict and design the performance of buckling based vibration damping metamaterials, even when they are made from thin-walled structures with high specic moduli and snap-through instabilities. II. BI-DIRECTIONAL BUCKLING So far, we have demonstrated that buckling metama- terials dampen vibrations for both soft and sti materi- als. However, all of the cases analyzed so far consider buckling exclusively in the compressive direction. Yet, it is also possible to realize geometries that can buckle in multiple directions, such as the geometries considered in Fig. 4. In Fig. 4A, we have created a sample, which can buckle in both the compressive (Fig. 4B) and the tensile (Fig. 4C) direction(See section IV A 2 for manufacturing details). This sample has in fact been optimized to buckle in tension and compression at the same strain level using Bayesian optimisation and nonlinear nite element meth- ods with Python and Abaqus [36]. The sample can even buckle simultaneously in tension and compression when subjected to more complex loading cases, such as bending in Fig. 4D. While this sample demonstrates that buckling can be achieved simultaneously in both tension and compres- sion, it does not oer sucient stiness by itself to be used as a load bearing vibration damping structure. However, we can generalize our numerical model to sim- ulate the vibration response of structures that oer snap- through buckling in both tension and compression. To do so, we adjust the model we used to simulate Fig. 3G-I, and apply the force-displacement curve of Fig. 4E in- stead. This force-displacement curve is in fact the sym- metrically buckling version of Fig. 3G, with the numeri- cally considered mass subtracted (See Section IV E). We then use our model to subject the sample numerically to the same up and down frequency sweeps at dierent base accelerations as we did for Fig. 3I. When we track themaximum output acceleration across the frequency do- main,aoutas function of the input acceleration, ain, we obtain the results of Fig. 4F. Here, we nd that buckling based vibration damping is very eective to set an up- per limit for vibration transmission. In compression, it is already more eective than what was the case when buck- ling only occurred in compressive direction, as in Fig. 3I. This is because the snap-through induced hysteresis per cycle is twice as large, as snap-through occurs in both compression and tension. Furthermore, Fig. 4F demon- strates that the same upper level for vibration transmis- sion can be set in tensile direction, making it much more eective in tensile direction than what was the case in Fig. 3I. This shows that buckling metamaterials oering buckling in both tension and compression can oer in- creased vibration damping over those that only buckle in compression. III. OUTLOOK Of course, the geometry of Fig. 4A-D is not the only metamaterial design, which can oer buckling in both tensile and compressive direction. For instance, kirigami forms a common alternative [37{39]. However, if we look carefully at the structure of Fig. 4A, we can see that it is very similar to lightweight lattice structures [40, 41]. Such bre-reinforced lattice structures are specically de- signed for their high strength and stiness to weight ratio and can be made in both 2D or 3D geometries. Typically in such structures, the members have been sized such that they do not buckle under load [40, 41]. However, their members could also be sized instead to allow for elastic buckling, possibly accompanied by snap-through, such as the sample shown in Fig. 3. A specic advantage of such lattice structures, but also of the metal sample of Fig. 3, is that they are stretching dominated pre- buckling. This implies that a much higher specic sti- ness can be achieved than in many bending dominated structures [19{21, 24, 42]. As such, it may be possible to produce buckling based vibration damping metamateri- als with a very high stiness to weight ratio. Furthermore, while the current research demonstrates a threefold improvement in damping for both samples, we expect that much larger improvements in damping coef- cient could be obtained in systems with large amplica- tion factors at small excitations. More specically, if the sample of Fig. 3, or another similar structure produced out of high stiness materials, had less imperfections, we would expect a much larger amplication at small excitation. Meanwhile, we would not expect the ampli- cation factors of excitations that induce post-buckling to increase signicantly. As such, the increase in damp- ing performance of buckling metamaterials for vibration damping could even be much larger than reported in this manuscript. Finally, while this manuscript has focused on limiting the transmission of a specic resonance, the method itself does not depend on a frequency. In fact,7 CB E D FA 24 cm FIG. 4: Buckling based vibration damping in tension and compression . (A) 3D printed TPU sample allowing buckling in two directions, buckling respectively in (B) compression, (C) tension and (D) bending. (E) Force-displacement for featuring tensile and compressive snap-through buckling, based on the force-displacement curve of Fig. 3G. (F) For numerical up and down frequency sweeps at various acceleration levels, the maximum output acceleration aoutacross the frequency range. Blue (orange) curves correspond to rising (dropping) frequencies. Solid (dashed) lines correspond to compression (tension). Grey lines show the linearized trend. this method should work to restrict transmissions of any vibration. This could be a big advantage in cases with multiple modes. Open challenges remain however. Specically, it still has to be demonstrated that buckling based vibration damping metamaterials can be produced with an even higher specic stiness, without signicant imperfec- tions. Similarly, the possible ceiling limits of performance still need to be identied. When these points are addressed, we anticipate ap- plications in any situation, where a high specic sti- ness is required along with high damping, including aerospace, sensitive instruments and high-tech machin- ery [1, 2, 8, 43{45].8 IV. MATERIALS AND METHODS A. Sample design and fabrication 1. Elastomeric holar sample fabrication To create the sample of Fig. 1CD, we pour a two component silicone rubber (Zhermack Elite Double 32) in a 3D printed mold, manufactured with a Stratasys Objet Connex 500 3D printer. The mold contains a pattern of 55 circular holes in which we place steel rods. The holes have a diameter of 10mm and wall thickness of 1.5 mm, implying a sample width and height of 59 mm. The depth of the sample is 50 mm. The sample is connected to two perspex plates using silicone rubber glue. The perspex plates can be used to mount the sample to a test rig at the bottom and to mount additional weight and an accelerometer at the top. 2. Multi-directional buckling sample fabrication The sample of Fig. 4A-D has been manufactured from a 2-component slow curing silicone rubber using an En- visionTEC 3D Bioplotter. The sample measures 240  11522 mm. B. Experimental methods 1. Uniaxial testing To perform the tests of Fig. 2AB and Fig. 3B, we compress and extend the samples using a uniaxial testing device (Instron 5943 with a 500 N load cell). We do so at constant rates of 1 mm/min (Fig. 3B) and 1, 10, 100 and 1000 mm/min (Fig. 2AB). 2. Vibration testing To perform the vibration tests of Fig. 1, Fig. 2 and Fig. 6, we vibrate the sample using an Instron Elec- tropuls 3000. We suspend a rectangular steel frame from the machine and mount the sample on the bottom of this frame, such that we eectively obtain a base exciting from the bottom. A mass is mounted on top of the frame to eectively obtain a mass-spring system with base ex- citation (Fig. 1CD. For the vibration tests with the metal sample of Fig. 3, we vibrate the sample using a Tira Vibration Test Sys- tem TV 5220-120 instead, where we control the frequency with an Aim-TTi TG5011 function generator. We add a large mass of 4.5 kg on the shaker next to the sample to keep the vibration input acceleration level more constant during a frequency sweep. The set-up is shown in Fig. 5. ShakerStabilising massAccelerometer 2 Accelerometer 1Top mass 328333FIG. 5: Metal buckling based vibration damping sample mounted on shaker table. In both cases, we measure the accelerations at 5000 Hz with two PCB Piezotronics 352C33 accelerometers: one at the bottom, capturing the base excitation and one on top, capturing the output accelerations. The tests of Fig. 1, Fig. 2 are also recorded using a Phantom VEO 640 monochrome high-speed CMOS camera with Nikon 200mm f/4 Macro lens, at 250 frames per second at a resolution of 1024 1024, inducing a spatial resolution of 0:10 mm. C. Processing accelerometer data To process the accelerometer data, we rst apply a fourth order Butterworth bandpass lter from 4 to 60 Hz for the elastomeric sample and from 5 to 100 Hz for the metal sample. This way, we lter out low frequency sensor drift and high frequency noise generated by the test set-up. We then use the Python scipy.signal.nd peaks algo- rithm to nd the trends of the peak acceleration and as function of time and frequency. We also apply a Savitzky- Golay lter to smooth the resulting peak acceleration trends. D. Image analysis To get a quantitative understanding of the degree of buckling of the sample used in Fig. 1 and Fig. 2, we use particle tracking (OPENCV and Python) and custom- made tracking algorithms to quantify the attening f and orientation w.r.t. the horizontal of each pore and calculate the polarisation [18, 23, 46]: nxny:= (1)nx+nyfcos 2; (1) wherenx(ny) is the hole's column (row). We then plot9 these tracked ellipses along with their polarisation in red and blue in Fig. 1CD. E. Numerical model The numerical model consists of a nonlinear mass spring damper system (Fig. 1A). The system contains a massM, output displacement y(t) and acceleration __y (orange in Fig. 1A),coupled through a spring to a base excitationu(t),__u(blue in Fig. 1A). The spring extension s=yu. The spring stiness Kis multilinear, while the damper Cis multilinear only in the spring extension s.KandCcan be normalized with the mass to obtain thekandc. These can then be coupled to the linear system response as follows, where !n=fn 2is the angu- lar eigenfrequency, is the damping coecient, Gis the amplication factor and fnis the eigenfrequency: k=K m=!2 n (2) c=C m= 2!n (3) =1 2G(4) We derive the linearized spring stiness K, eigenfre- quencyfnand amplication factor Gfrom the experi- mental data of Fig. 2A,C for the elastomeric sample and Fig. 3B as well as the underlying data at small excita- tions of Fig. 3D for the metal sample. We derive for small excitations, K= 11:38 N/mm, fn= 32 andG= 19 for the elastomeric sample and K= 368:8 N/mm,fn= 48 Hz andG= 13:2 for the metal sample. We use these values to derive M= 0:28 kg andC= 2:83 Ns/m for the elastomeric sample and M= 4:0 kg andC= 92:6 Ns/m for the metal sample. These masses are higher than the actual masses mounted on top as the sample mass also aects the eigenfrequency. We also derive the buckling force plateaus to be 25 N and 78 N, for the elastomeric (Fig. 2E) and metal (Fig. 3) samples respectively. To account for gravity, we subtract the gravitational accel- eration,gM, whereg= 9:81 m/s2in the model to end up with the base state at s= 0 without excitation in the model. The force-displacement curve for the model with buckling in two directions, seen in Fig. 4E presents the same force plateau as the one for the metal sample in Fig. 3G, with gMsubtracted. The system is governed by the following dierential equation. __y+c( _y_u) +k(yu) = 0 (5)When the system is subjected to a base excitation u(t), __u, the system can be solved numerically in the time do- main using a forward Taylor series, which we do at a frequency of 2000 Hz: __y=c( _y_u)k(yu) (6) ___y=c__y__u
k( _y_u) (7) Compared to some alternative ways to numerically solve the system, a forward Taylor series has a number of distinct advantages. Compared to closed form solu- tions, such as piecewise solutions [47, 48], a forward Tay- lor series has the distinct advantage that it can easily be modied to allow for more complicated stiness and damping characteristics. Compared to Euler's method, which is a rst order Taylor series, a higher order forward Taylor series oers more numerical stability and oers a much higher speed of convergence. Finally, compared to explicit nonlinear nite element methods, a simple single degree of freedom system oers large advantages in sim- plicity, computational needs and numerical stability, at the cost of a lower accuracy. F. Dening the vibration signal To obtain the frequency response, we subject the nu- merical and experimental sample to a sinusoidal vibra- tion with constant acceleration amplitude, where the sine slightly changes frequency after every cycle. At each start of a full cycle, implying the acceleration and displacement are 0, the frequency of the sine cycle is updated. This implies that the input acceleration, velocity and displace- ment are dened as follow, with Bthe input acceleration amplitude, fthe frequency and tcthe time in the cycle betweentc= 0s totc=1 f: __u=Bsin (2ftc) (8) _u=B 2fcos (2ftc) (9) u=B (2f)2sin (2ftc) (10) G. Random vibration We rst dene a random vibration spectrum. We start from the acceleration Power Spectral Density Function of Fig. 6A, calculated at 1 standard deviation (1 ).PSDn10 EC D F A B FIG. 6: Random vibration. (A) Power Spectral Density (PSD) prole of the input acceleration, normalized with the peak PSD level. (B) Corresponding time-displacement history applied in the tests. (C,D) Experimental time-acceleration history for input accelerations of 0.06 gRMS(c) and 0.30 gRMS(d). Orange and blue curves correspond to the input and out accelerations respectively. (E) Output acceleration (gRMSout) as function of input acceleration (gRMS in). (F) Maximum acceleration, amax, as function of the input acceleration. Blue and orange correspond to compression and tension respectively. All RMS values are shown at 1 . is the PSD value, normalized with the maximum PSD value. This PSD shows a constant PSD value between 12.5 Hz and 50 Hz, which also includes the area surround- ing the eigenfrequency of the sample of Fig. 1. This im- plies that vibration with this PSD will only excite the frequencies around the eigenfrequency. The 1root-mean-square of any PSD spectrum can be calculated as [49]:RMS =sZ1 0PSDdf: (11) We can convert the acceleration PSD to a time sig- nal using an inverse Fourier transform [49], where we assume random values for the starting phase of the equivalent wave at each frequency. We can integrate the time-acceleration signal to a time-velocity and time- displacement signal and use a high-pass Butterworth l- ter to remove low-frequency drift. As such, we can ob- tain the time-displacement signal of Fig. 1B. We can also convert time signals to PSD signals using a Fourier transform. We can then shake the sample, as described in the Main Text and in Section IV B 2, using this time signal as an input. If we subject the sample to a low random base excitation of 0.06 gRMS in Fig. 6D, we observe that the output acceleration in blue is much larger than the input acceleration in orange and that acceleration levels are similar in tension and compression. However, when we do the same at a higher base excitation of 0.30 gRMS in Fig. 6D, we observe a clear upper limit in compression, due to buckling. We can also track how the output accelerations change with the applied input acceleration. In Fig. 6E, we ob- serve forgRMSin>0:15 thatgRMSoutstarts to go lower than the linear trend, which suggests that buckling based vibration damping also works in random vibra- tions. However, to convert a time signal to a frequency based signal, including a gRMS value, the implicit as- sumption is made that the signal is a sum of sinusoidal signals, which is approximately symmetric about 0. That is not the case here, as seen in Fig. 6D. As such, it is more representative to look at the peak accelerations instead. These are plotted in Fig. 6F. Here we observe a clear up- per limit in compression (blue) for gRMSin>0:1. How- ever, forgRMSin>0:2, we observe a similar plateau in tension (orange). We presume that the reason why it requires a much larger base excitation to set an upper limit in tension, is because only a few cycles induce buck- ling and therefore induce additional dissipation. This is dierent from vibrations at constant frequency and ac- celeration, where buckling occurs every cycle, along with the accompanying increased dissipation. However, both Fig. 6E and Fig. 6F demonstrate that buckling based vibration damping also works in tension. H. Design & Finite Element Analysis For the sample of Fig. 4, we designed the sample us- ing a form of Bayesian optimisation combined with nite element simulations in Abaqus [50]. To predict the response of the metamaterial samples of Fig. 3 and Fig. 4, we use nonlinear nite element simulations in Abaqus 2021 (Dassault Syst emes).11 B A C D E F G a/2tbeam/2 thinge/2thinge/2 thinge α FIG. 7: Simulating and optimising bi-directional buckling metamaterials . (A) Finite element mesh. (B) Tensile response. (C) Compressive response. (D) Quarter unit cell. (E) Stiness ratios for training points before and after buckling: minimum of tension of compression. (F) Stiness ratios interpolated with Gaussian Processes Regression before and after buckling: minimum of tension of compression. (G) Normalized stress ( )-strain () curve in tension (blue) and compression (orange) of sample in A. 1. Metal sample We designed the sample of Fig. 3 using a combination of nite element analyses and experiments. We started designing the sample using 3D analyses with 2D plane stress plate elements (CPS8R). We used a combination of single strip analyses, 3 3 and 55 metamaterial anal- yses. We modelled the entire sample using aluminium with a Young's modulus of 70 GPa, Poisson's ratio of 0.3 and density of 2800 kg/m3. We performed various sets of analyses, varying the aspect ratio, thickness and out-of- plane curvature. We compressed and uncompressed the sample numerically using a nonlinear quasi-static analy- sis. In some cases, snap-through instabilities made the analyses unstable, in which cases we opted for dynamic explicit analyses instead. We sized the design using two main criteria: 1. At ves times the post-buckling strain, the force- drop from peak is less than 50% and a 5 5 alu- minium sample still has at least a 40 N force. This prevents unstable collapse post-buckling. 2. At ves times the post-buckling strain, the max- imum Von Mises Stress is less than 100 MPa in aluminium. This prevents fatigue damage in high strength metals. After analysis, we opted to make the sample out of steel sheets instead due to the higher load carrying capac- ity and better availability of high-strength variants with small thicknesses (AISI 301 Full Hard). After producingthe rst sample, we opted to produce the nal sample of Fig. 3 out of a higher thickness than originally analysed as the produced load carrying capacity was lower than predicted due to imperfections. 2. Bi-directional buckling sample The nal design and mesh of the sample of Fig. 4 is given in Fig. 7A. We modelled the sample in 2D using quadratic quad-dominated CPE8R plane strain elements. For the material, we assumed a Neo-Hookean material model with a Poisson's of 0.48. We fully constrained the left and right side of Fig. 7A and compressed and extended the sample numerically using a nonlinear quasi- static analysis. To maximise the eciency of bi-directional buckling based vibration damping, the sample had to buckle in both directions at an equal strain. To obtain a suitable design, we used Bayesian optimisation [36, 50]. First, we specied the design using the parameters as shown in the quarter unit cell of Fig. 7D. We optimized the design for three normalized design parameters: t beam =tbeam a, t hinge =thinge aandangle =. We ran 500 analyses with a variety of combinations of these three variables. We populated the design space using a Sobol sequence, an ecient method to populate a multivariable design space with an arbitrary number of data points [51]. We tracked two outputs: (i) the ratio in stiness before and after buckling,K1=K2: minimum of tension and compression, and (ii) the ratio of the buckling force in tension and12 compressive direction, Fb;t=Fb;c. ForK1=K2, we plotted the results in Fig. 7E. We then used Bayesian machine learning (Gaussian Processes Regression) to more densely interpolate the design space as seen in Fig. 7F for K1=K2. In doing so, we also obtained the standard deviation, , of the uncertainty of the interpolation. We then dened the following criteria to dene designs of sucient quality: 1.K1=K2>5 + 1:96K1=K2 2.jFb;t=Fb;cj1<0:151:96Fb;t=Fb;c This presents us with several designs which adhere to our requirements. Around these locations, we then re- ned our data by running 100 additional analyses using a Sobol sequence, which then accurately provided us with a variety of designs adhering to our requirements. One of those is the design of Fig. 7A-C, which we produced in Fig. 4. The nonlinear normalized stress-strain curve as calculated by Abaqus, is presented in Fig. 7G, wherethe stress has been normalized by the cross-section and Young's modulus. V. DATA AND CODE AVAILABILITY The data and codes that support the gures within this paper are publicly available on a Zenodo repository [52]. Two videos that support this article can also be found in the supplementary information. 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Dykstra, J. van der Laan, and C. Coulais, Oligomodal metamaterials with multifunc- tional mechanics, Proceedings of the National Academy of Sciences 118(2021). [47] S. W. Shaw and P. Holmes, A periodically forced piece- wise linear oscillator, Journal of sound and vibration 90, 129 (1983). [48] C. Jung, B. I. Epureanu, et al. , Nonlinear amplitude ap- proximation for bilinear systems, Journal of Sound and Vibration 333, 2909 (2014). [49] T. Irvine, An introduction to the vibration response spec- trum, Revision D, Vibrationdata (2009). [50] M. A. Bessa, P. Glowacki, and M. Houlder, Bayesian ma- chine learning in metamaterial design: Fragile becomes supercompressible, Advanced Materials 31, 1904845 (2019). [51] I. M. Sobol, On the distribution of points in a cube and the approximate evaluation of integrals, Zhurnal Vy- chislitel'noi Matematiki i Matematicheskoi Fiziki 7, 784 (1967). [52] D. M. J. Dykstra, C. Lenting, A. Masurier, and C. Coulais, Buckling metamaterials for extreme vibra- tion damping, Zenodo, 10.5281/zenodo.7661958 (2023).

2023-02-23

Damping mechanical resonances is a formidable challenge in an increasing number of applications. Many of the passive damping methods rely on using low stiffness dissipative elements, complex mechanical structures or electrical systems, while active vibration damping systems typically add an additional layer of complexity. However, in many cases, the reduced stiffness or additional complexity and mass render these vibration damping methods unfeasible. Here, we introduce a method for passive vibration damping by allowing buckling of the primary load path, which sets an upper limit for vibration transmission: the transmitted acceleration saturates at a maximum value, no matter what the input acceleration is. This nonlinear mechanism leads to an extreme damping coefficient tan delta ~0.23 in our metal metamaterial|orders of magnitude larger than the linear damping of traditional lightweight structural materials. We demonstrate this principle experimentally and numerically in free-standing rubber and metal mechanical metamaterials over a range of accelerations, and show that bi-directional buckling can further improve its performance. Buckling metamaterials pave the way towards extreme vibration damping without mass or stiffness penalty, and as such could be applicable in a multitude of high-tech applications, including aerospace structures, vehicles and sensitive instruments.

Buckling Metamaterials for Extreme Vibration Damping

2302.11968v1

arXiv:1809.09429v1 [cond-mat.mtrl-sci] 25 Sep 2018Theory of damping in magnetization dynamics, dispelling a m yth and pointing a way forward D M Edwards Department of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom There is a widely-held belief amongst theoreticians that th e Gilbert damping parameter αin magnetization dynamics is infinite for a pure metal at T=0. Th e basic error leading to this belief is pointed out explicitly and the various methods of calcula tion used are viewed in a unified way based on the Lorentzian lineshape of ferromagnetic resonan ce spectra. A general torque formula forαis proposed as a good starting-point for treating inhomogen eous materials such as alloys, compounds and layered structures. Local spin density funct ional theory provides a simple physical picture, in terms of a non-uniform precessional cone angle i n ferromagnetic resonance, of how such inhomogeneity contributes to the damping. In acomplementa ry many-bodytheory this contribution is given by a vertex correction to the torque-torque respons e function. The damping of magnetization dynamics in ferromagnetic metals and a lloys is of critical importance in spintronic devices. Damping largely controls the speed at which a device can ope rate and its energy requirement. In device physics damping is usually treated phenomenologically by means of a Gilb ert term in the Landau-Lifshitz-Gilbert equation [1, 2] and many quantum-mechanical calculations of the Gilb ert parameter have been made for specific materials [3–10]. A reliable treatment of damping in transition metals an d alloys would be an invaluable guide in the search for materials with very low damping [11, 12], as required fo r the future development of devices such as magnetic random access memory(MRAM). Most recent work in th is direction is concerned with the important intrinsic contribution arising from spin-orbit coupling(SOC) and it is th is which concerns us here. A satisfactory theory should work in the limit of a pure metal but almost all existing ca lculations predict that the Gilbert damping parameter αdiverges to infinity for a pure metal at T=0. This would mean that in th e pure metals Fe, Co and Ni at low temperature the linewidth in a ferromagnetic resonance (FMR ) experiment would be much too large for the resonance to be observed. The prediction or acceptance of infinit e damping has been made by so many authors [3– 10, 13, 14] over the last forty years that it has acquired the stat us of a myth. A very recent paper [15] repeats it once again. It is noteworthy that no experimentalist seems to have troubled to investigate the problem by work on high purity metals and dilute alloys at low temperature. The aim of th is article is not only to dispel the myth but to formulate a firm starting-point for future calculations of αin technically important materials such as alloys, compounds and layered structures. The most direct method to investigate damping,both experimentally a nd theoretically, is to study the ferromagnetic resonance (FMR) linewidth. In FMR a uniform static magnetic field His applied and the absorption of a transverse microwave field of angular frequency ωpeaks around the frequency bex//planckover2pi1wherebex= 2µBHis the Zeeman energy. ForHin the z direction the absorption is determined by the imaginary part o f the dynamical transverse susceptibility χ−+(ω). This susceptibility, which must include the effect of SOC, can be calc ulated by standard many-body theory using the Kubo formula or by time-dependent spin-density function al theory (SDFT). In practice the many-body method is usually based on a tight-binding approximation and employs t he random phase approximation (RPA) with a short-range screened Coulomb interaction. This is then equiv alent to a time-dependent Hartee-Fock mean- field theory. The long-range interaction can also be included if care is taken that it does not enter the exchange terms [16, 17]. SDFT is approximated similarly as a time-dependent mea n-field theory in the local spin density approximation (LSDA) and the long-range Coulomb interaction pres ents no problem since it is effectively screened in the exchange-correlation functional. It is useful to consider bo th these methods in parallel. In a system with varying direction of magnetization SDFT is based on a density matrix o f order 2 [18] rather than just spin and particle densities. χ−+(ω) is then coupled to fifteen other response functions which determ ine the longitudinal spin susceptibility as well as the charge response and mixed charge-spin responses [19, 20]. These last relate to phenomena like the spin-Hall effect. Some of these response functions, includin g the longitudinal spin susceptibility, involve the long-range Coulomb interaction importantly even in the absence of S OC [16, 17, 20]. Costa and Muniz [22], following an earlier paper [23], show how SOC produces mode coupling in the RPA m any-body approach. However the long- range Coulomb interaction is left untreated. Their paper is particula rly important for being the first to challenge the myth of infinite damping. We firstdiscussthe caseofaBravaislattice whichisappropriatefor puremetalswith acubic structurelikeFe andNi at T=0. In both the approaches described above the dynamical su sceptibility is related to mean-field susceptibilities of the general form χ0(ω) =N−1/summationdisplay kmnMmn(k)fkn−fkm Ekm−Ekn−/planckover2pi1ω+iη. (1)2 HereEkmis the energy of the one-electron state with wave-vector kin bandm, calculated in the presence of SOC, fkmis the corresponding occupation number, Mmn(k) is a product of matrix elements and ηis a small positive constant which ultimately tends to zero. As in usual time-dependen t perturbation theory equation (1) represents the response to a perturbing field of angular frequency ωin which transitions occur between occupied and unoccupied states. ”Intraband transitions” with m=nclearly do not occur for ω/negationslash= 0 owing to the cancellation of the Fermi functions. These transitions between identical states, which are not really transitions at all, can play no role in a dynamical process. Hankiewicz et al have made a similar point [24]. How ever, in nearly all calculations of the Gilbert damping parameter α, intraband transitions appear and lead to the infinite damping discus sed above. To dispel a myth effectively it is necessary to see how it has arisen. It is instructive to review, in a unified way, some methods which have been used to calculate α. We start from the Lorentzian form of the FMR lineshape which is well-established experimentally [21] and theoretically [22]. Near the re sonance the dynamical transverse susceptibility is dominated by a pole at /planckover2pi1ω=bex+/planckover2pi1∆ωwhere ∆ω∼ξ2,ξbeing the SOC parameter, so that χ−+(ω) =−2/angbracketleftSz/angbracketright/N /planckover2pi1(ω−∆ω)−bex. (2) HereSzis thezcomponent of total spin and Nis the number of atoms in the crystal. Near the resonance the FMR absorption is determined by ℑ(χ−+(ω)) =−2(/angbracketleftSz/angbracketright/N)ℑ(/planckover2pi1∆ω) (/planckover2pi1ω−ℜ(/planckover2pi1∆ω)−bex)2+(ℑ(/planckover2pi1∆ω))2. (3) ℜ(/planckover2pi1∆ω) corresponds to a shift in the resonance frequency and ℑ(/planckover2pi1∆ω) determines the linewidth, both due to SOC. The Gilbert damping factor αis given by ℑ(/planckover2pi1∆ω)/bex(e.g. [25]). The most direct way to calculate αis a brute-force numerical RPA calculation of ℑ(χ−+(ω)), with SOC included, as a function of ωaround the resonance. Costa and Muniz [22] performed such calculations using the tight-binding appro ximation and found perfect Lorentzians from which they deduced α. Taking a monnolayer of Co as an example they found no tendency fo rαto diverge in the pure limit of sharp electronic states. This method of calculating αis very computer intensive and more economic methods exist if one assumes a Lorentzian curve from the outset. It follows immediately from (2) that α=ℑ(/planckover2pi1∆ω)/bex=2/angbracketleftSz/angbracketright Nbexℑ(1 χ−+(bex//planckover2pi1)). (4) This new formula for αmay be regarded as exact. A full treatment of the transverse su sceptibility includes coupling to other modes and leads to a rather complex expression in terms of sixteen mean-field susceptibilities of the form (1) with different sets of matrix elements [20]. There is an enormous simplifi cation in the case of a Bravais lattice if we calculate αonly to second order in the SOC parameter ξ. Following the arguments of [20] it is readily found that coupling of the transverse susceptibility to other modes is then elimin ated and that χ−+in (4) may be replaced by the mean-field susceptibility χ0 −+. This elimination depends on inversion symmetry, which is a property o f a Bravais lattice. Without this symmetry, coupling of the transverse suscep tibility to other modes occurs in general even to orderξ2, as discussed later. It follows further that to order ξ2 α= (N∆2/2/angbracketleftSz/angbracketrightbex)ℑ(χ0 −+(bex//planckover2pi1)) (5) where ∆ is the exchange splitting in the band structure. It is usually s ufficient to calculate the last factor to first order in bexso we may take the unphysical limit bex→0, but with due care as discussed below. Then α= (N∆2/2/angbracketleftSz/angbracketright)[∂ωℑ(χ0 −+(ω)]ω=0 (6) where the electronic state energiesand matrixelements in ℑ(χ0 −+(ω) are calculatedwith bex= 0. Beforeproceedingto the static ω→0 limit it is essential not to include contributions from ”intraband tran sitions”, as pointed out after (1). This precaution was not taken in [14], where a similar formula was obtain ed, so the spurious infinite damping for a pure metal appeared. Sometimes it is preferable to keep the physic al non-zero Zeeman field to remove all danger of including intraband transitions. This also gives the option of calculatin g the frequency-swept FMR linewidth as a function of Zeeman field. This has been measured [21] and can be con verted to a frequency dependence of α. Such a dependence has been discussed by Costa and Muniz [22]. However the low-field limit is usually sufficient and here we take the limit bex→0, with the precaution mentioned above, to compare with other the oretical work. Following [14], but excluding intraband terms, we find the following two express ions forαat T=0: α= (π∆2/2/angbracketleftSz/angbracketright)/summationdisplay k/summationdisplay′ mn|/angbracketleftkm|S−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF) = (πξ2/2/angbracketleftSz/angbracketright)/summationdisplay k/summationdisplay′ mn|/angbracketleftkm|T−|kn/angbracketright|2δ(Ekm−EF)δ(Ekn−EF).(7)3 HereSSS= (Sx,Sy,Sz) is the total spin operator, S−=Sx−iSy,ξhsois the total spin-orbit interaction, T−= [S−,hso] is a torque operator and EFis the Fermi energy. The prime on the sum over bands means m/negationslash=nand the sum over k is to be carried out as an integral over the Brillouin zone as usual. As p ointed out these expressions are only correct to orderξ2so that in the second expression we must evaluate the electronic st ates and energies with ξ= 0. The prime on the summation sign may then be omitted since the m=nterms are zero owing to inversion symmetry [10]. The resulting expression, which can now be written in terms of one-part icle Green functions if desired, is just the version of Kambersky’s torque formula [13] for a Bravais lattice derived in tw o ways by Edwards [20]. It is the mean-field approximation to a much more general formula [20], valid for an orde red or disordered system, α=−(ξ2/2bex/angbracketleftSz/angbracketright)ℑ[χξ=0 T(bex//planckover2pi1)]. (8) We shall refer to this as the general torque formula. It is exact to orderξ2and we have left open the option of taking the limit bex→0. Here the torque-torque response function is given by the Four ier transform of a retarded Green function using the Kubo formula χT(ω) =/integraldisplay /angbracketleft/angbracketleftT−(t),T+/angbracketright/angbracketrighte−iωtdt. (9) The wide application of (8) is discussed later and we recall that the se cond expression in (7), corresponding to the mean-field approximation to χT, is only valid for a Bravais lattice. To evaluate the integral over kin the formula (7) numerically it is usual to replace the delta-functions by Lorentzians of width proportional to an inverse relaxation time parameter τ−1. This broadening of the electron states may be regarded as a crud e representation of the effect of impurity and/or phonon scattering. The limit τ−1→0 of a perfect crystal at T=0 leads to a finite value of αbut is quite tricky to perform numerically [26]. If we wrongly retain SOC in ca lculating the electron states in the second expression of (7) the diagonal matrix elements are non-zero and le ad to the notorious infinite damping parameter α. The only work which deals correctly with αin pure metals is reported in fig.1 of [10] and in [22, 26].(In [26] the caption of fig.1 should read ”with and without SOC included in calculating electronic states”). Wenowturntothetaskofestablishingafirmbasisforcalculatingthe dampingparameter αintechnicallyimportant materials, which are typically random alloys or layered structures. T his task is greatly simplified if we are satisfied with calculating αto second order in the SOC parameter ξ. This should be sufficient in nearly all systems of interest. At room temperature the ξ2dependence of αis well-established experimentally in several alloy systems, including some containing Pt with its large SOC [27, 28]. The general torque for mula (8) is a very convenient starting-point. Its derivation in Appendix A of [20] is for a completely general ferromag netic material, either ordered or disordered, and again relies only on the universal FMR Lorentzian lineshape. The deriv ation proceeds by comparing an exact relation between χ−+(ω) andχT(ω) with an expansion of (2) in the limit /planckover2pi1∆ω/(/planckover2pi1ω−bex)→0 followed by /planckover2pi1ω→bex. This order of limits is essential and results in the form (8) where χTis evaluated in the absence of SOC. A similar formula was derived by Kambersky [13] in another way where crucially the pre scription ξ= 0 did not become apparent. The formula is remarkable for describing the essence of a phenomenon a rising solely from SOC without the need to include SOC in the calculation. The calculation of χTin a disordered system is still a very demanding problem. It may be app roached using the RPA of standard many-body theory or, less obviously, using time-d ependent LSDA. A diagrammatic RPA treatment ofχTinvolves a sum of ladder diagrams and the first term, without an inter action line, corresponds to the mean-field approximation χ0 T. The remaining terms constitute a vertex correction and we have s hown above that in a monatomic Bravais lattice this vanishes. In a disordered system like an alloy, or a metal at finite temperature in a frozen phonon picture, this is not the case. However this great simplification persis ts if, in a very crude approximation, the system is replaced, at the outset, by an effective medium with the full transla tional symmetry of the lattice but finite electron lifetime. We are then led to the Kambersky-like formula (7) for αwith a Lorentzian broadening of the delta-functions determined by relaxation times which may be dependent on spin and te mperature. This is the background to a recent calculation of αin bulk Ni at room temperature [26] which is in reasonable agreement w ith experiment. A proper treatment of χTin a disordered material must deal simultaneously with the RPA verte x correction and any vertex corrections which arise in connection with methods of taking a configurational average, such as the coherent potential approximation (CPA). There is a small literature on this pr oblem as applied to χ−+, notχT, in a one-band model [29, 30]. Santos and Costa [30] find that for dilute non-magne tic impurities the RPA vertex correction is particularly important. However as yet the many-body approach is far from being able to provide reliable results for αin real disordered materials. The time-dependent LSDA method see ms more promising. As shown below, it gives a clear physical picture of the RPA vertex correction and separat es it from the configurational averaging problem. In a FMR experiment the local magnetization vector sweeps out a co ne as it precesses around the Zeeman field direction and in the presence of SOC the cone angle θ(rrr) is a function of position. In the time-dependent LSDA θ(rrr) satisfies an integral equation whose solution is avoided in [14] by tak ing a spatially-independent averaged cone angle4 θ(rrr). This approximation enforces a uniform precession, as occurs in t he absence of SOC, and removes the possibility of coupling between transverse and longitudinal susceptibilities. It is very reasonable for a monatomic Bravais lattice where the variation of θ(rrr) within a unit cell is largely an artificial consequence of the local appr oximation. In the tight-binding framework of [20] it would not be an approximation at all for a monatomic Bravais lattice. However, in compounds, alloys and layered structures, variation of the cone a ngle between different types of atom and different layers may be very important. In the many-body approach the ver tex correction in χTis the difference between the fullχTand the mean-field approximation χ0 T. Since we have seen that the mean-field approximation works well for a homogeneous system, like a monatomic Bravais lattice, we conc lude that the vertex correction corresponds to the effect of the spatial variation of the cone angle, which can be st udied with the LSDA approach. This will be demonstrated explicitly in a forthcoming publication. This productive interplay between standard many-body theory and density-functional theory is quite unusual. I would like to acknowledge a useful exchange of e-mails with Filipe Guima res on the subject-matter of this paper. [1] L.D. Landau, E.M. Lifshitz and L.P. Pitaevski, Statistical Physics, Part 2 (Oxford: Pergamon 1980). [2] T.L. Gilbert, Phys. Rev. 1001243 (1955) [3] K. Gilmore, Y.U. Idzerda and M.D. Stiles, Phys. Rev. Lett .99027204 (2007). [4] C. Liu, C.K.A. Mewes, M. Chshiev, T. Mewes and W.H. Butler , Appl. Phys. Lett. 95022509 (2009). [5] Y.Liu, A.A. Starikov, Z. Yuan and P.J. Kelly, Phys. Rev. B 84014412 (2011). [6] H. Ebert, S. Mankovsky, and D. K¨ odderitzsch, Phys. Rev. Lett.107066603 (2011). [7] A. Sakuma, J. Phys. Soc. Japan 81084701 (2012). [8] A. Starikov, P.J.Kelly, A. Brataas, Y.Tserkovnyak and G .E.W. Bauer, Phys. Rev. Lett. 105236601 (2010). [9] S. Mankovsky, D. K¨ odderitzsch, G Woltersdorf and H. Ebe rt, Phys. Rev. B 87014430 (2013). [10] E. Barati, M. Cinal, D.M. Edwards and A. Umerski, Phys. R ev. B90014420 (2014). [11] M.A.W. Schoen, D. Thonig, M.L. Schneider, T.J. Silva, H .T. Nembach, O. Eriksson, O. Karis and J.M. Shaw, Nat. Phys. 12839 (2016). [12] A.J. Lee, J.T. Brangham, Y. Cheng, S.P. White, W.T. Ruan e, B.D. Esser, D.W. McComb, P.C. Hammel and F. Yang, Nat. Commun. 8234 (2017). [13] V. Kambersky, Czech. J. Phys. B 261366 (1976). [14] I. Garate and A. MacDonald, Phys. Rev. B 79, 064403 (2009). [15] F.S.M. Guimares, J.R. Suckert, J. Chico, J. Bouaziz, M. dos Santos Dias and S. Lounis, arXiv:1807.11808v1. [16] D.J. Kim, H.C. Praddaude and B.B. Schwartz, Phys. Rev. L ett.23419 (1969). [17] D.J. Kim, B.B. Schwartz and H.C. Praddaude, Phys. Rev. B 7205 (1973). [18] U. von Barth and L. Hedin, J. Phys. C 51629 (1972). [19] A.R .Williams and U. von Barth, in Theory of the Inhomogeneous Electron Gas ed. S. Lundqvist and N.H. March (Plenum 1983) [20] D.M. Edwards, J. Phys. Condens. Mater. 28086004 (2016). [21] S.S. Kalarickai, P. Krivosik, M. Wu, C.E. Patton, M.L. S chneider, P. Kabos, T.J. Silva and J.P. Nibarger, J. Appl. Ph ys. 99093909 (2006). [22] A.T. Costa and R.B. Muniz, Phys. Rev. B 92014419 (2015). [23] A.T. Costa, R.B. Muniz, S. Lounis, A.B. Klautau and D.L. Mills, Phys. Rev. B 82014428 (2010). [24] E.M. Hankievicz, G. Vignale and Y. Tserkovnyak, Phys. R ev. B75174434 (2007). [25] D.M. Edwards and O. Wessely, J. Phys. Condens. Matter 21146002 (2009). [26] A. Umerski and D.M. Edwards, IOP Conf. Series: Journal o f Physics: Conf. Series 903012056 (2017). [27] C. Scheck, L. Cheng, I. Barsukov, Z. Frait and W.E. Baile y, Phys. Rev. Lett. 98117601 (2007). [28] P. He, X. Ma, J.W. Zhang, H.B. Zhao, G. L¨ upke, Z. Shi and S .M. Zhou, Phys. Rev. Lett. 110077203 (2013). [29] H. Yamada and M. Shimizu, J. Phys. Soc. Japan 311344 (1971). [30] E.B. Santos and A.T. Costa, J. Mag. Magn. Mat. 46969 (2019).

2018-09-25

There is a widely-held belief amongst theoreticians that the Gilbert damping parameter {\alpha} in magnetization dynamics is infinite for a pure metal at T=0. The basic error leading to this belief is pointed out explicitly and the various methods of calculation used are viewed in a unified way based on the Lorentzian lineshape of ferromagnetic resonance spectra. A general torque formula for {\alpha} is proposed as a good starting-point for treating inhomogeneous materials such as alloys, compounds and layered structures. Local spin density functional theory provides a simple physical picture, in terms of a non-uniform precessional cone angle in ferromagnetic resonance, of how such inhomogeneity contributes to the damping. In a complementary many-body theory this contribution is given by a vertex correction to the torque-torque response function.

Theory of damping in magnetization dynamics, dispelling a myth and pointing a way forward

1809.09429v1

1 Enhancement of u ltrafast demagnetization rate and Gilbert damping driven by femtosecond laser -induced spin currents in Fe81Ga 19/Ir20Mn 80 bilayers Wei Zhang1,2, Qian Li u3, Zhe Yuan3, Ke Xia3, Wei He1, Qing -feng Zhan4, Xiang -qun Zhang1, and Zhao -hua Cheng1,2,5* 1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School o f Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 China 4State Key Laboratory of Precision Spectroscopy, School of Phy sics and Materials Science, East China Normal University, Shanghai 200241, China 5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China 2 Abstract In spintronics application s, ultrafast spin dynamics have to be controlled at femtosecond (fs) timescale s via f s-laser radiation. At such ultrafast timescale s, the effect of the Gilbert damping factor α on ultrafast demagnetization time M should be considered. In previous explorations for the relationship between these two parameters, it was found that the theoretical calculations based on the local spin -flip scattering model do not agree with the experimental results. Here, we find that in Fe81Ga19(FeGa) /Ir20Mn 80(IrMn) bilayers, the unconventional IrMn thickness dependence of α results from the competition between spin currents pumped from the ferromagnetic (FM) FeGa layer to the antiferromagnetic (AFM) IrMn layer and those pumped from the AFM layer to the FM layer. More importantly , we establish a proportional relationship between the change of the ultrafast demagnetization rate and the enhancement of Gilbert damping induced by the spin currents via interfacial spin chemical potential s . Our work build s a bridge to connect the ul trafast demagnetization time and Gilbert damping in ultrafast photo -induced spin currents dominated systems , which not only explains the disagreement between experimental and theoretical results in the relation of 𝜏𝑀 with α, but pr ovides further insight into ultrafast spin dynamics as well . PACS numbers: 75.78.Jp, 75.40.Gb, 76.50.+g, 78.47.+p *To whom all correspondence should be addressed. zhcheng@iphy.ac.cn 3 I. INTRODUCTION The understanding of spin dynamics from nanosecond (ns) down to femtosecond (fs) timescales is an essential task toward s the realization of ultrafast spintronic devices in the frequency range from GHz to THz [1,2] . The study of ultrafast demagnetization time, M, is one of the most challenging problem s in laser -induced ultrafast spin dynamics . The Gilbert damping factor , α , is of the utmost importance for high frequency switching of spintronic devices. Since both M and α require a transfer of angular momentum from the electronic system to the lattice, the unification of these two seemingly unrelated parameters can facilitate the exploration of the microscopic mechanism of laser -induced ultrafast spin dynamics. An inverse ly proportional relation ship between M and α was predicted by theoretical calculations based on the local phonon -mediated Elliott -Yafet scattering mechanism [3-5] as well as the stochastic Landau -Lifshitz -Bloch (LLB) model [6]. However, the relationship between M and α has been debated for over one decade [7]. Until now, all experimental results have show n that M increases with α [8-12]. Apart from the local spin -flip scattering mechanism [13], we proposed that the non-local spin current s should be taken into account to coordinate the contradiction in the relationship between 𝜏𝑀 and α. Previous work suggest ed that the superdiffusive spin current contribut ed to ultrafast demagnetization [14], whilst the Gilbert damping could also be enhanced via non -local spin currents in ferromagnetic (FM)/nonmagnetic (NM) [15 ] and FM/antiferromagnetic (AFM) heterostructures [16]. Femtosecond laser 4 irradiation of ferromagnetic thin films is a fascinating novel approach to create large spin currents [17,18 ]. Figure 1(a) shows that i n the case of time -resolved magneto - optical Kerr effect (TRMOKE) experiments, hot electrons excited by fs -laser pulses can travel at high velocities and over tens of nanometers through the films. The difference of mean free path between spin majority and spin minority hot electrons in ferromagnetic thin film s generates superdiffusive spin currents on fs timescales. Such spin current s dissipated at the interface of the heterostructure result in the out -of- equilibrium spin accumulation represented by spin chemical potential 𝜇𝑠. Moreover, figure 1(b) shows the damped magnetization precession around the effective field could be influenced via spin current. Tveten et al. [19] predicted that the ultrafast demagnetization time M could be described in the language of spin current -induced damping 𝛼𝑠𝑝 in magnetic heterostructures based on the electron -magnon scattering theory. However, the experimental evidence on the connection of ultrafast demagnetization time with damping driven by f s laser-induced spin currents is not yet understood. II. RESULTS A. Sample properties Ir20Mn 80 (tIrMn)/Fe 81Ga19 (10 nm) bilayers [20 ] were deposited on optically transparent single -crystalline MgO (001) substrates in a magnetron sputtering system with a base pressure below 3×10−7 Torr. The substrates were annealed at 700 °C for 1 h in a vacuum chamber and then held at 250 °C during deposition. FeGa layers were 5 obliquely deposited at an incidence angle of 45°. The IrMn layers were deposited while continuously rotating the substrates. In order to induce an exchange bias (EB) along the FeGa [010] direction, a magnetic field of 500 Oe provided by a permanent magnet was applied along the MgO [110] axis during growth. After deposition, a 3 nm protective Ta layer was deposited on the samples to avoid oxidation. The static longitudinal Kerr loops of Fe 81Ga19 (10 nm)/Ir 20Mn 80 (𝑡𝐼𝑟𝑀𝑛) along FeGa [010] direction with various AFM IrMn thicknesses (𝑡𝐼𝑟𝑀𝑛) at room temperature were acquired using a laser diode with a wavelength of 650 nm. Figure 2(a) shows the longitudinal Kerr loops of Fe 81Ga19 (10 nm)/Ir 20Mn 80 ( IrMnt nm) along FeGa [010] direction with various AFM IrMn thicknesses ( IrMnt ) at room temperature, whereas the thickness of FM FeGa layer was fixed at 10 nm. For nm2IrMnt , the width of the hysteresis loops is enlarged with no obvious shift along the x-axis, implying that the thickness of IrMn layer is too thin to form an antiferromagnetic order for pinning the magnetization reversal of FeGa [21] (Insert in Fig. 2(b) (left)). For nm2IrMnt , the antiferromagnetic orders are well established, and consequently the antiferromagnetic moments pin FM ones reversal to induce a unidirectio nal anisotropy (Insert in Fig. 2 (b) (Right)). The loops therefore exhibit evidently exchange bias behavior. The exchange bias field achieves a value of about 60 Oe when nm2IrMnt , whilst the largest value of coercivity (~72 Oe) occurs at IrMnt 2 nm. B. TRMOKE measurements for ultrafast demagnetization and Gilbert damping 6 We performed the polar TRMOKE experiment to measure ultrafast demagnetization time under a saturated applied field of 20 kOe in the normal direction of the samples [22]. The details of the TRMOKE experiment are described in APPENDIX A. Figure 3(a) shows th e demagnetization curves for various IrMn thicknesses with a maximum magnetization quenching of ~10% [23,24 ]. The temporal changes of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just before the pump laser excitation. The time evolution of magnetization on sub - picosecond timescales can be fitted according to Eq. (1) in terms of the three - temperature model (3TM) [17]. −∆𝑀(𝑡) 𝑀={{[𝐴1 (𝑡𝜏0+1⁄)0.5−𝐴2𝜏𝐸−𝐴1𝜏𝑀 𝜏𝐸−𝜏𝑀𝑒−𝑡 𝜏𝑀−𝜏𝐸(𝐴1−𝐴2) 𝜏𝐸−𝜏𝑀𝑒−𝑡 𝜏𝐸]Θ(𝑡)}∗𝐺(𝑡,𝜏𝐺)}∗𝐺(𝑡,𝜏𝐺) (1) where ),(*GtG represents the convolution product with the Gaussian laser pulse profile, G is the full width at half maximum (FWHM) of the laser pulses , )(t is a step function , )(t is the Dirac delta function . 𝐴1 represents the value of ∆𝑀(𝑡) 𝑀 after equilibrium between electrons, spins , and lattices . 𝐴2 is proportional to the initial electrons temperature rise. Here, we used the 780 nm laser as the pump pulse to excite the magnetic system out of equilibrium, while the 390 nm laser pulse was used as a probe beam. Therefore, i n Eq. (1), t he state filling e ffects during pump probe experiment are neglected due to the different wavelength of pump and probe beams used in this study. The cooling time by heat diffusion is described by 𝜏0, which should be about one order of magnitude larger than 𝜏𝐸 representing the timescale of electron - phonon interactions. The best -fitted value of 𝜏𝐸=500 𝑓𝑠 for all samples is in good 7 agreement with that of previous reports [18]. The fitting parameters in Eq. (1) are shown in Table I, from which one notes the pulse width is 350 fs for all the samples. In our experimental setup, the time -resolution is about 80 fs. In order to obtain a high time resolution, we measured the ultrafast demagnetization with very fine step of time delay (15 fs). The values of ultrafast dem agnetization time (120 -220 fs) obtained from Eq. (1) are defined as the time needed for the magnetization to reach a level of 𝑒−1 of its maximum demagnetization. The time needed for magnetization to reach its maximum demagnetization (>500fs) should be longer than the time extracted from Eq. (1). A similar result was reported by B. V odungbo et al .[25]. The very large temporal stretchi ng of the laser pulse up to 430 fs was attributed to the conversion of the incident laser pulse into a cascade of hot electrons. This could be one of the possible reasons resulting in the spread of laser pulse up on the samples in this study. Via changing the single parameter ， , we can accurately reproduce the experimental results for various samples. The ultrafast demagnetization time M was observed to decrease from 220± 10 fs for IrMnt = 0 nm to 120±10 fs for IrMnt = 2 nm, then increase with further increasing 𝑡𝐼𝑟𝑀𝑛 [Fig. 3(b)]. The precessional frequency and damping factor can be derived by means of the TRMOKE signal s as well [26, 27]. Figure 4(a) shows the typical time evolution of the polar component of magnetization after pump laser excitation at different fields applied along with the [110] direction of F eGa for 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. It is observed clearly that the spin precession process can be influenced obviously by applied fields. The exact M8 values for 𝑓 with various applied fields can be obtained using the damped harmonic function added to an ex ponential -decaying background : 𝛥𝑀(𝑡)=𝐴+𝐵𝑒𝑥𝑝(−𝑣𝑡)+𝐶𝑒𝑥𝑝(−𝑡 𝜏)𝑠𝑖𝑛 (2𝜋𝑓𝑡+𝜑) (2) where 𝐴 and 𝐵 are the background magnitudes, and 𝑣 is the background recovery rate. 𝐶,𝜏,𝑓 and 𝜑are the magnetization precession amplitude, relaxation time, frequency and phase, respectively. The field dependence of frequency 𝑓 extracted from the fitting procedure is shown in Fig. 4(b). We note that the experimental f-H relation can be reproduced very well by Kittel equation ( 3) [27 ]. . (3) with and . And γ=𝛾𝑒𝑔2⁄ is the gyromagnetic ratio. 𝜑𝑀 and 𝜑𝐻 are the angles of in -plane equilibrium M and H respect to the FeGa [010] easy axis. 𝐾1,𝐾𝑢,𝐾𝑒𝑏 and 𝐾𝑂𝑢𝑡 are the in -plane magneto crystalline, uniaxial , unidirectional and out -of-plane magnetic anisotrop y constants of FeGa films, respectively . The value of magnetocrystalline anisotropy constant is 𝐾1=4.5×105 𝑒𝑟𝑔/𝑐𝑚3 for the samples with various AFM layer thickness during the fitting procedure and the uniaxial magnetic anisotropy constant𝐾𝑢=(1.5±0.3)×105𝑒𝑟𝑔/𝑐𝑚3. For 𝑡𝐼𝑟𝑀𝑛=3 𝑛𝑚 and 5 nm, the unidirectional magnetic anisotropy constant of 𝐾𝑒𝑏=3×104𝑒𝑟𝑔/𝑐𝑚3 has to be included for more accurate fitting, although it is one order magnitude smaller than those 2 1 22 1)2( HHMf s M eb H M s M M u s O K HM KK K M K H cos ) cos( 2sin 2 cos2 4 2-2 1 12 2 ut 1 M eb H M s M u M K HM K K H cos ) cos( 2cos2 4cos21 2 9 of magneto crystalline and uniaxial anisotropy . The effective G ilbert damping factor 𝛼𝑒𝑓𝑓 shown in Fig. 4 (c) is determined from the relaxation time 𝜏 by Eq. ( 4) [28]: ) (/22 1H Heff (4) Since the overall effective damping factor 𝛼𝑒𝑓𝑓 consists of intrinsic damping and extrinsic damping whereby the second one arises from both the two-magnon -scattering and the dephasing effect in the sample s, the overall effective Gilbert damping factor decreases monotonously to a constant value with increasing the applied field (Fig . 4(c)). As one of the main ly extrinsic contributions , the two -magnon -scattering induced damping has been extensively studied in exchange biased heterostructures [29-34]. The mature theory was developed to explain the two-magnon scattering process due to spatial fluctuations of anisotropy and exchange bias field [30,35 ]. The two -magnon scattering process comes from the scatterings of the uniform ( 𝑘=0) precession mode into nonuniform modes ( k≠0 magnons) that are degenerate in frequency. This process is described by the Hamiltonian, in which the spatial fluctuation in the exchange coupling caused by interface roughness determines the scattering strength. The roughness gives rise to a large fluctuating field because the FM magnetization interacts alternatively with one or the other AF sublattice via the atomic exchange coupling. It is a well -known relaxation mechanism effective in exchange biased heterostructur es due to the interface roughness occurring on the short length scales. When a low external field comparable with the exchange bias field was applied, the two-magnon scattering 10 effect will result in the increase of Gilbert damping with the exchange bias fi eld according to previous reports [33, 34 ]. However, as shown in Ref. 36, a strong enough applied field can be used to exclude the contributions from the two-magnon -scattering, where the value of Gilbert dam ping factor keeps as a constant with various two - magnon -scattering strength. Based on this result , a similar method using strong enough external fields was applied in this study to exclude the two -magnon -scattering effect. Moreover, previous works show that the two -magnon -scattering induced damping increases with precession frequency because of the increased degeneracy of spin waves [37, 38]. Our work demonstrated that the damping factor keeps almost a constant value at high enough applied fields, i ndicating the minor contributions from the two-magnon - scattering to Gilbert damping. Besides, it has been demonstrated previously that the two-magnon -scattering contri butions decrease monotonously with increasing the film thickness [33, 34 ]. This again disagrees with the tendency of thickness dependence of damping at high applied field shown in Fig. 5(c). Therefore, in this study, the two - magnon -scattering strength was suppressed effectively by applying a high enough external field. On the other hand, inhomogeneities in FeGa thin film may cause variations in the local magnetic anisotropy field. It leads to the variations of spin orientations when the external field is not large enough, and gives rise to t he enhanced damping arising from spin dephasing e ffect [28]. However, an applied field (~ kOe) much la rger than the anisotropy field makes the spin orientation uniform, as a result, the dephasing effect was suppressed largely. Based on the above analysis, t he intrinsic part of damping is independent of t he external field or precession frequency, while the 11 extrinsic part including both the depha sing effect and the two-magnon -scattering effect are field-dependent. In order to avoid the effect of the extrinsic damping factor , the intrinsic damping factors were obtained by fitting the overall damping factor as the function of applied fields with the Eq. (5) [39, 40 ] shown as the red line in Fig. 4(c): 0/ 1HH eff e (5) where α and 𝛼1𝑒−𝐻𝐻0⁄ are the intrinsic and extrinsic parts of the damping factor, respectively. For the derivation of spin precessional frequency as well as the Gilbert damping, the similar producers as shown above were adapted to various samples. Fig ure 5(a) shows the precessional frequency from oscillation curves with various IrMn thicknesses. Since the exchange bias field and coercivity are much weaker tha n applied fields, the f- H curves of FeGa films are therefore slightly different with various AFM layer thicknesses , which is in contra st to the observation that the enhanced uniaxial anisotropy of Fe/CoO bilayers [28] increases the precessional frequency largely. More importantly, w e find th e effective damping factor eff decreases with applied fields [Fig. 5(b)]. The solid lines represent the fittin g expression shown as the Eq. (5 ). Interestingly, the effective Gilbert damping factors drop to a nearly constant value as the intrinsic damping factor when the applied fields increase strong enough to suppress the extrinsic contributions as stated above . The value s of the intrinsic damping factor as a function of the thickness of the IrMn layer are illustrated in Fig. 5(c ). It increases firstly and reaches the maximum 12 value when the thickness of the IrMn layer at IrMnt = 2 nm, and finally decreases with further increasing the thickness of the IrMn AFM layer. A drastic change of 2.5 times for damping occurs at IrMnt = 2 nm . Similarly, S. Azzawi et al. showed around 2 times enhancement of damping in NiFe/Pt bilayers when a continuous Pt capping layer is just forming at 0.6 nm by TRMOKE measurements [41]. Moreover, once a continuous IrMn layer is forming at 2 nm, the accompanied strong intrinsic anisotropy of AFM would contribute partly t o the damping enhancement superimposed to spin pumping effect. This has been demonstrated previously by W. Zhang et al where the damping of Py/IrMn bilayers is 3 or 4 times larger than that in the Py/Cu/IrMn samples [42]. Based on the discussions in Fig. 4 , we can exclude the extrinsic mechanisms such as the two- magnon -scattering and the dephasing effect as the dominant contributions to the damping process when the external fields are high enough [43]. Besides, FeGa alloys are particularly interesting because of their magneto -elastic properties [44]. The acoust ic waves are possible to be tri ggered by ultrashort laser and as a result, spin precession would be excited non-thermally via a magnetoelastic effect [45]. However, this effect can be excluded based on the following reasons: firstly, the external field has to be applied along with the hard axis of FeGa, otherwise , the magnetization precession cannot be induced. It agrees with the fact that the canted magnetization from the easy axis is necessary when the spin precession arising from instantaneous anisotropy change accompanied by ultrafast demagnetization occurs [26]. In contrast, the occurrence of spin precession from the magnetoelastic effect is independent of initial magnetization orientation. Secondly, in order to check the contribution of resonance 13 mode from the magnetoelastic effect, we per formed a fast Fourier transform in APPENDIX B . Only the uniform field-dependent precession mode was excited at present study. This is not the expected behavior for the acoustically induced modulation of the magneto -optical effects. Therefore, the magnetoelastic effect of FeGa was suppressed largely in this study. It probably because the laser fluence of around 1 mJ/cm2 is not high enough to induce a large amplitude of strain pulse. According to Ref. 45 , the oscillations amplitude of acoustic mode increases linearly with the laser energy density within the probed range . Moreover, the FeGa material with a thickness as thic k as 60 nm is preferred to induce an obvious magnetoelastic behavior [46], while 10 nm at the present experiment is probably too thin. As a result, t he intrinsic damping can be influenced by the following paramenters : (1) magnetocrystalline anisotropy of FM [47]; (2) exchange bias field [30, 31, 36 ], and (3) spin pumping effect at the interface between FM and the AFM [ 15, 16, 42, 48 ]. In the case of FeGa/IrMn bilayers , the magnetocrystalline anisotropy constant of FeGa 1K = 3 5/ 105.4 cm erg , which is obtained fro m Fig. 4 and Fig. 5 , is invariant with AFM layer thickness. Moreover , referring to Fig. 2 (b), it seems that there is no direct relationship between the intrinsic damping factor and the exchange bias field Heb. When the applied field is far higher than the exchange bias field, both the precessional frequency and the damping factor show independence of excha nge bias field [36]. Therefore, the IrMn thickness dependence of the intrinsic damping is not attributed to the magnetocrystalline anisotropy and the exchange bias field . Due to the strong spin-orbit coupling of the heavy metal (HM) Ir in the IrMn alloy, the contribution of spin pumping to the damping 14 factor must be taken into account. It is noteworthy that the IrMn thickness dependence of damping in FeGa/IrMn is different from that in other normal FM/HM bilayers, where the damping factor increases monotonically with the thickness of HM layer and approa ches a saturation valu e [49]. However, the damping of FeGa ferromagnetic layer decreases again after reaching a peak value at 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. The change of the damping factor is always accompanied by the spin currents transfer between FM and AFM layers. More spin currents absorbed by the neighbor ing layer result in larger damping in the FM layer. A n unconventional decrease of the damping factor implies that not only the effect of heavy metal Ir in IrMn alloy has to be taken into account, but also the ant iferromagnetic magnetization. The heavy metal Ir serves as a perfect spin sink to absorb the spin currents , and consequently increases the damping in FeGa, while the antiferromagnetic magnetization in IrMn serves as a new source to compensate the dissipati on of magnetization precession and decrease the damping of FeGa. C. First -principle calculations for IrMn layer thickness dependence of Gilbert damping To understand the behavior of the IrMn thickness -dependent damping factor, we calculated the damping factor using the scattering theory of magnetization dissipation combined with the first -principles electronic structure [50]. The calculated FM/AFM bilayer structure shown in Fig. 6(a) are the same as that in the experiment. Here, the magnetic moments of AFM sublattices serve as not only a spin sink to absorb the spin current pumped from the adjacent FM layer, but also a spin current emitter to partly 15 cance l the spin pumping effect of the FM. The interfacial exchange coupling force s the magnetic moments of the IrMn sublattices in a few layers near the interface to preces s following the adjacent FM , generat ing spin current s back into the FM layer [Fig. 6 (b)]. Based on this model, t he enhancement of damping due to the spin current 𝛼𝑠𝑝=∆𝛼= 𝛼𝑡𝐼𝑟𝑀𝑛−𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚 as a function of IrMn thickness was calculated and shown as the solid circle in Fig. 6(c). It increases firstly to a peak value at IrMnt = 2 nm, and then drops with further increasing the IrMn layer thickness . When IrMnt 2 nm , the thickness of the IrMn layer is too thin to establish the antiferromagnetic order, which can be supported by the negligible exchang e bias as shown in Fig. 2 (b). In this case, the pumped spin current from the AFM back into the FM to partially cancel the spin pumping effect by the FM is largely reduced because of the disorder of the antiferromagnetic moments as illust rated on the left side in Fig. 6 (b). In this re gion, therefore, the magnetic moments in the AFM serve as a perfect spin sink to absorb the spin current pumped from the adjacent FM resulting in a significant enhancement in the damping factor . For the samples with the thickness of IrMn 𝑡𝐼𝑟𝑀𝑛>2𝑛𝑚, however , the antiferromag netic order is well established and the accompanied exchange bias is remarkably large ( See Fig . 2(b) and its insert ). Because of the exchange coupling between FM and AFM at the interface, the magneti c moments of the AFM sublattices in a few layers near the interface is forced to precess following the magnetic moment of the FM, while those far away from the interface would keep static. Such an exchange spring effect at the interface caused spin precession in the AFM layer, and consequently , spin currents would be transfe rred from AFM to the FM layer. Moreover, these spin 16 current s from the AFM would be enhanced due to the coherent precession of magnetization in different sublattices as illustr ated in the right side of Fig. 6 (b). The exchange spring effect induced precession of the AFM has two effects: (1) the AFM has intrinsic damping that increases the overall dam ping of the FM/AFM bilayer. (2) the precessional motion of magnetic moments in AFM sublattices pumps spin current s into the FM, which cancels partly the spin pumping by the FM. As a result, the overall damping of the bilayers is reduced. From the solid circles in Fig. 6(c), one can find that the damping decreases with increasing IrMnt when IrMnt 2 nm, indicating that the latter effect of the pumped spin currents is dominant over the intrinsic damping. Besides, by comparing the calculated and experimental values [Fig. 6(c) and (d)], one can find that the calculated Gilbert damping is larger than the experiment al one for IrMnt 1 nm . The reason for t he deviation is the assumption of a perfect ly flat FeGa/ IrMn interface in the calculation, which leads to a larger spin current pumped from the FM . Unfortunately, it is almost impossible to fabricate the perfect ly flat film when the thickness is less than 1 nm. In order to separate the contribution of the precession of the magnetic moment of the AFM sublattice to damping, we also calculated the damping by assuming perfectly static AFM ordered IrMn without precession (solid diamonds in Fig. 6 (c)) and a paramagnetic IrMn layer with vanishing Néel order (solid triangles in Fig. 6(c)). The calculated results demonstrate that if the magnetic moments of the AFM sublattice either do not precess or align randomly, the IrMn layers serve only as a perfect spin 17 sink to absorb the spin current s pumped from the adjacent FM resulting in a significant enhancement of damping . The damping increases monotonically to a saturation value with IrMn thickness, which is similar to that of heavy metals [49]. D. Relationship between ultrafast demagnetization rate and Gilbert damping induced by non -local spin currents The central strategy of our study is to establish a direct correlation between M and α. According to Fig. 3(b ) and Fig. 5(c), we find that the femtosecond laser -induced ultrafast demagnetization time M and the Gilbert damping α show an opposite IrMn thickness dependence in FeGa/IrMn bilayers. By plotting M versus α as shown in Fig. 7 (a), one can clearly observe that the value of M decreases with α, suggesting that spin transport plays an additional dissipation channel for accelerating the ultrafast demagnetization and enhanc ing the d amping. The damping factor 𝛼𝑡𝐼𝑟𝑀𝑛 for IrMnt 0 nm is ascribed to the spin pumping effect induced by various AFM thicknesses 𝛼𝑠𝑝 and the contribution from the FM itself 𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚.To give further insight into th e relationship, we replo tted Fig. 7(a) by using t he change of the ultrafast demagnetization rate ∆1 𝜏𝑀=1 𝜏𝑀|𝑡𝐼𝑟𝑀𝑛−1 𝜏𝑀|𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚 versus the enhancement of Gilbert damping 𝛼𝑠𝑝=∆𝛼=𝛼𝑡𝐼𝑟𝑀𝑛−𝛼𝑡𝐼𝑟𝑀𝑛=0𝑛𝑚 induced by the spin current. An approximately linear relationship is confirmed and shown in Fig. 7 (b), which can be fitted using Eq. (6): (For the de rivation of Eq. (6 ), please see APPENDIX D for details) ∆1 𝜏𝑀=𝜇𝑠 ℏ∆𝛼, (6) 18 Where ∆1 𝜏𝑀 , ∆α represents the enhancement of ultrafast demagnetization rate and Gilbert damping induced by the spin current, respectively, 𝜇𝑠 is the spin chemical potential, and ℏ is the Planck constant. A reasonable value of 𝜇𝑠≈1 𝑒𝑉 which is similar to that of spin splitting in 3d transition metals was obtained by the linear fitting using Eq. (6). The spin chemical potential 𝜇𝑠 is proportional to spin accumulation s at the interface between different layers . It contributes largely to ultrafast demagnetization according to the model of laser -induced ultrafast superdiffusive spin transpor t in layered heterostructures [14, 5 1]. There is a large difference in velocities or lifetimes for spin -dependent hot electrons [52]. As a result, the transport properties of ho t electrons are spin -dependent. For instance , the minority -spin electrons excited by ultrashort laser survive for a quite short time an d they decay to non -mobile bands approximately at the position they were excited. Instead, majority -spin electrons have longer lifetimes and higher velocities. So they leave fast from the excitation reg ion after being created, resulting in part of the demagnetization process. Because the directions of motion for all the electrons are random , they can obtain a velocity directed back towards the ferromagnetic film . A second part of the demagnetization is ascribed to the backflow of spin -minority elec trons from the substrate or the neighbor layer. Spin - majority electrons entering the ferromagnetic layer will find good transport properties and continu e diffusing without severely decay ing. However , spin-minority electrons experience a considerable worsen ing of the transport properties as soon as they enter 19 the ferromagnetic layer. The consequence is that they are trapped at the entrance of the ferromagnetic layer , giving rise to the spin accumulations at the interface . Nevertheless, the quantitative description for spin accumulations during ultrashort laser -induced demagnetization in heterostructures is still lacking. This work aims at filling this gap by relating ultrafast demagnetization time and Gilbert damping. A de tailed calculation for the value of 1 eV for spin chemical potential obtained in this experiment is highly desirable. The non -local spin current s dissipated at the interface of FeGa/IrMn open an additional channel to accelerate the ultrafast demagnetization and enhance the Gilbert damping. However , in the case of the sample with IrMnt = 0 nm without the assistant AFM layer , both the local spin -flip and non -local spin transport mechanism s probably contribute to the ultrafast demagnetization in the ferromagnetic layer. For instance, based on the breathing Fermi -surface model of the Gilbert damping and the Elliott - Yafet relation for the spin -relaxation time, a relation shown as Eq. (7 ) is established between the conductivity -like Gilbert damping and ultrafast demagnetization time 𝜏𝑀 [10]. (7) Taking the values of nm tMIrMn 0 and nm tIrMn 0 are 220 fs and 0.004, respectively , a value of 𝛼𝜏𝑀⁄=1.8×1010𝑠−1 is derived. This value is reasonable and agrees well with that of 3d transition metal Ni calculated by the breathing Fermi -surface model [ 53], 2pbFM elM20 indicating that the ultrafast demagnetization of ferromagnetic FeGa film itself is mainly governed by the local spin -flip scattering events . Nonetheless, we have to address that, ultrafast demagnetization in the ferromagnetic layer was accelerated and the Gilbert damping was enhanced via the interfacial spin accumulations o nce the IrMn layer was attached . III. CONCLUSIONS The unconventional IrMn thickness dependence of α is attributed to the cancellation of the spin currents pumped from the AFM IrMn layer to the FM FeGa layer. We establish a proportional relationship between the change of ultrafast demagnetization rate and the enhancement of Gilbert damping induced by the spin currents via interfacial spin chemical potential . This result can facili tate the utilization of ultrafast spintronic devices in the THz region. 21 Acknowledgments This work is supported by the National Key Research Program of China (Grant Nos. 2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural Sciences Foundation of China (Grant Nos. 91622126, 51427801, and 51671212) and the Key Research Program of Fro ntier Sciences, CAS (Grant Nos. QYZDJ -SSW - JSC023, KJZD -SW-M01 and ZDYZ2012 -2). The work at Beijing Normal University is partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the Recruitment Program of Global Youth Experts and the Fundamental Research Funds for the Central Universities (Grant No. 2018EYT03). The work at East China Normal University is partly supported by the National Natural Sciences F oundation of China (Grant No. 11874150). 22 APPENDIX A: TRMOKE MEASUREMENTS In this study, the dynamical process of fast and ultrafast spin dynamics was measured by TRMOKE. The experiments were carried out using an all -optical pump - probe technique. A train of optical pulses with a wavelength of 780 nm, 55 fs duration , and 100 nJ/pulse is generated at 5.2 MHz repetition rate by a Ti: sapphire oscillator (FEMTOLASER, XL -100). A 200 μm thickness BBO crystal was used to double the frequency of femtosecond laser. The laser beam from the source is split into both 780 nm and 390 nm beams. We use the 780 nm laser as the pump pulse to excite the magnetic system out of equilibrium, while the 3 90 nm laser pulse was used as a probe beam to measure the subsequent magnetization dynamics with the timescale from sub - picosecond to nanosecond. The pump laser beam is much stronger than the probe with an intensity ratio of about 100 for all the measureme nts. Both the pump and probe beam s are incident along the normal axis (z -axis) of the sample s. The detection geometry is only sensitive to the out -of-plane component of the magnetization Mz. For fast spin dynamics, we applied various external fields along the Fe 81Ga19 [110] direction to trigger the spin precession, while a large enough field about 20 kOe was applied along the Fe 81Ga19 [001] direction to obtain the ultrafast demagnetization curves. W e adjusted the pump laser fluence from 1 mJ/cm2 to 1.25 mJ/cm2 to obtain the same maximum quenching for various samples. The pump and probe beams are focused onto the sample s with spot diameters of ~10 μm and ~5μm via an objective lens, respectively. For the spin precession measurements, t he scheme of the TRMOKE 23 experiment is illustrated in Fi g. 8. The signal s are sensitive with the polar component of magnetization after pump laser excitation at different fields applied along the [110] direction of FeGa . APPENDIX B: FAST FOURIER TRANSFORM ANALYSIS The ferromagnetic FeGa is a famous material for its magneto -elastic properties. After femtosecond laser irradiation, an external field -independent resonance mode would be triggered due to the excitation of coherent acoustic phonons. However, only one f ield-dependent resonance mode was excited in this study according to fast Fourier transform analysis in Fig. 9. APPENDIX C: FIRST -PRINCIPLE CALCULATIONS The electronic structure of FeGa /IrMn bilayer is calculated self -consistently using the local density approximation of the density functional theory. The spin -dependent potentials, charge and spin densities are obtained with the minimal basis of tight - binding linear muffin -tin orbitals. In the calculation of the total damping, the scattering region consisting of the repeated FeGa /IrMn bilayers are connected to two semi -infinite Cu leads. We have introduced the thermal lattice disorder into a 4x4 supercell a nd displaced the atoms in the scattering region randomly away from their equilibrium positions with a Gaussian distribution. The root -mean -square atomic displacements of the Gaussian distribution are determined using a simple Debye model with the Debye 24 temperature of 470 K. The two -dimensional Brillouin zone of the supercell is sampled by a 24x24 k -mesh corresponding to the 96x96 mesh for the Brillouin zone for the 1x1 unit cell. The effect of magnons in the FM FeGa is neglected in our calculation. This is because the magnetic damping is dominated by electrons at the Fermi level in metals, which can efficiently transfer spin angular momentum into the orbital motion via spin - orbit interaction. In metals and alloys, the influence of magnon -phonon coupling is negligible except for near the Curie temperature [54]. If magnetization precession occurs only in the FM FeGa layer, the calculated damping enhancement does not sensitively depend on the specific order of the AFM IrMn. Here we take two limits: the perfectly antiferromagnetic ordered IrMn and the paramagnetic IrMn (the magnetic moments of Mn are rand omly distributed such that both the Néel order and total magnetization vanish). The damping enhancements calculated for the two cases are nearly identical, where the damping factor is enhanced and saturates at the thickness of 2 nm. It indicates that the p umped spin current by the precessional FeGa is immediately absorbed by the IrMn layer. The large moment on the Mn atom can absorb the pumped transverse spin current efficiently. On the other hand, the AFM IrMn is forced to precess due to the interfac ial exchange coupling, however, the efficient of the spin current generation by AFM depends on its specific order . It is suppressed largely in the case of paramagnetic IrMn because of the cancellation via magnetic moments with various orientations shown on the left side of Fig. 6(b) in the main text . In contrast, the efficient of the spin current generation by the AFM is enhanced remarkably by the coherent precession of the ordered magnetic moments 25 shown in the right side of Fig. 6(b) in the main text . And the cone angle of precessional IrMn is modeled to exponentially decay from the interface with a typical decay length of 2 nm. The precessional AFM has mainly two contributions to the damping enhancement of the bilayer. First, the AFM has intrinsic damping that increases the total energy loss during the magnetization dynamics. The second effect is that the precessional AFM pumps spin current into the FM that cancels partly the spin pumping by the FM and decreases the damping enhancement. APPENDIX D: DERIV ATION OF EQ. (6 ) IN THE MAIN TEXT It is well known that the magnetic moment sM is proportional to the spin angular momentum S via gyromagnetic ratio Bg ： S Ms (8) where g is Lande factor, B is Bohr magneton. Normally, we take sMVm as the total magnetic moments , where V is the volume of the atom. M is the ultrafast demagnetization time. Therefore, t he value of M1 is taken as the demagnetization rate. The demagnetization is alw ays accompanied by the dissipation of spin angular momentum, and hence the rate of spin angular momentum dissipation is : Mm 1 . (9) On the other hand, the spin current 𝑗𝑠⃗⃗⃗ of per unit area generated by spin pumping effect 26 reads: j𝑠⃗⃗ =1 4𝜋𝑔𝑒𝑓𝑓𝜇𝑠⃗⃗⃗ , (10) where 𝑔𝑒𝑓𝑓is the effective interfacial spin -mixing conductance including the influence of the backflow spin current from the AFM IrMn to FeGa , s is the spin accumulation - driven chemical potential. The pumped spin current across the interface is 𝐼𝑠⃗⃗⃗ =𝑗𝑠⃗⃗⃗ 𝐴, where A is the area of the interface. 𝑔𝑒𝑓𝑓=4𝜋𝑀𝑠𝑑∆𝛼 𝑔𝜇𝐵, (11) where 𝑑is the thickness of the ferromagnetic layer, nm t tIrMn IrMn 0 is the enhancement of Gilbert damping induced by the absorption and generation of spin current via various IrMn thicknesses. Therefore, if we correlate the spin angular momentum dissipated by the ultrafast demagnetization and that induced by spin pumping, the relationship reads: s MIm1 (12) And then we take Eq. 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Values of the main fit parameters of ultrafast demagnetizations curves for various thicknesses of the samples. FIG.1. (color online ) Basic concept of both ultrafast demagnetization and spin precession induced by spin currents . (a) The excitation of fs laser pulse transforms slow majority -spin d electrons (red) into fast sp electrons, thereby launching a sp in current toward s the AFM layer. The spin current crossing the interface results in the spin accumulation at the interface represented by spin chemical potential 𝜇𝑠. (b) The typical time evolu tion of magnetization after femtosecond laser irradiation measured by TRMOKE experiment. FIG. 2. ( Color online) Static magnetic properties of of MgO/Fe 81Ga19 (10 nm)/Ir 20Mn 80 (t nm) bilayers. (a) Longitudinal -MOKE loops with various thicknesses of IrMn layer IrMnt . (b) Coercivity Hc and exchange bias field ebH as a function of IrMn layer thickness IrMnt . FIG.3. (Color online ) Ultrafast demagnetization. (a) Ultrafast demagnetization curves with various IrMn layer thickness es. The solid lines represent the fitting results by Eq. (1) in the text. The insert shows the configuration of the measurement for ultrafast 33 demagnetization. (b) Ultrafast demagnetization time as a function of IrMn layer thickness. FIG.4. (Color Online) Spin precession. (a) TRMOKE signals of FeGa/IrMn bilayers with MnIrt = 2 nm in various applied fields. (b) Precessional frequency as a function of applied fields . (c) Effective Gilbert damping constant as a function of applied fields. FIG. 5 . (Color Online) Frequency and damping of spin precession. (a) Fr equency of spin precession as a function of applied fields with various IrMn thickness. The solid lines represent the fitting results by Kittle equations. (b) Effective Gilbert damping constants as a function of applied fields with various IrMn thicknesses. (c) Intrinsic Gilbert damping as a function of IrMn thickness. FIG.6. (Color online) Results of First -principle calculation s. (a) Illustration of the ferromagnet (FM)/antiferromagnet (AFM) structure employe d to investigate the spin transport. (b) The configuration of the IrMn magnetic moments located at the first layer near the interface. (c) The calculated damping enhancement as a function of the thickness of the antiferromagnetic IrMn. The solid circles sh ow the calculated damping enhancement with the precession of AFM magnetic moments. The solid diamonds show the calculated damping enhancement with perfectly static AFM ordered IrMn without precession, while the solid triangles correspond to the calculated values using a static paramagnetic IrMn layer with vanishing Néel order. (d) The experimental damping enhancement as a function of the thickness of antiferromagnetic IrMn. 34 FIG.7. (Color online ) (a) Ultrafast demagnetization time as a function of Gilbert damping . (b) The variation of ultrafast demagnetization rate as a function of Gilbert damping enhancement . The red line indicates the fitting via Eq. (6 ) in the text. FIG. 8. (Color online ) Scheme of TRMOKE experiment for spin precession dynamics. FIG. 9. (Color online ) Fourier transform spectra measured between 0.85 kOe and 3.0 kOe for 𝑡𝐼𝑟𝑀𝑛=2 𝑛𝑚. 35 Table I . tIrMn (nm) A1 A2 0 220±10 500 5 350 0.8 2 1 160±10 500 6 350 0.8 2 2 120±10 500 7 350 0.8 2 3 145±10 500 4 350 0.8 2 5 200±10 500 5 350 0.8 2 )(fsM )fsE（ )(0ps )(fsG36 FIG.1. 37 FIG. 2. 38 FIG.3. 39 FIG.4. 40 FIG. 5 . 41 FIG.6. 42 FIG.7. 43 FIG. 8. 44 FIG.9.

2019-08-29

In spintronics applications, ultrafast spin dynamics have to be controlled at femtosecond (fs) timescales via fs-laser radiation. At such ultrafast timescales, the effect of the Gilbert damping factor {\alpha} on ultrafast demagnetization time should be considered. In previous explorations for the relationship between these two parameters, it was found that the theoretical calculations based on the local spin-flip scattering model do not agree with the experimental results. Here, we find that in Fe81Ga19(FeGa)/Ir20Mn80(IrMn) bilayers, the unconventional IrMn thickness dependence of {\alpha} results from the competition between spin currents pumped from the ferromagnetic (FM) FeGa layer to the antiferromagnetic (AFM) IrMn layer and those pumped from the AFM layer to the FM layer. More importantly, we establish a proportional relationship between the change of the ultrafast demagnetization rate and the enhancement of Gilbert damping induced by the spin currents via interfacial spin chemical potential . Our work builds a bridge to connect the ultrafast demagnetization time and Gilbert damping in ultrafast photo-induced spin currents dominated systems, which not only explains the disagreement between experimental and theoretical results in the relation of {\tau}_M with {\alpha}, but provides further insight into ultrafast spin dynamics as well.

Enhancement of ultrafast demagnetization rate and Gilbert damping driven by femtosecond laser-induced spin currents in Fe81Ga19/Ir20Mn80 bilayers

1908.11084v1

Nonlocal damping of spin waves in a magnetic insulator induced by normal, heavy, or altermagnetic metallic overlayer: A Schwinger-Keldysh field theory approach Felipe Reyes-Osorio and Branislav K. Nikoli´ c∗ Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA (Dated: December 15, 2023) Understanding spin wave (SW) damping, and how to control it to the point of being able to amplify SW-mediated signals, is one of the key requirements to bring the envisaged magnonic tech- nologies to fruition. Even widely used magnetic insulators with low magnetization damping in their bulk, such as yttrium iron garnet, exhibit 100-fold increase in SW damping due to inevitable con- tact with metallic layers in magnonic circuits, as observed in very recent experiments [I. Bertelli et al. , Adv. Quantum Technol. 4, 2100094 (2021)] mapping SW damping in spatially-resolved fashion. Here, we provide microscopic and rigorous understanding of wavevector-dependent SW damping using extended Landau-Lifshitz-Gilbert equation with nonlocal damping tensor , instead of conventional local scalar Gilbert damping, as derived from Schwinger-Keldysh nonequilibrium quantum field theory. In this picture, the origin of nonlocal magnetization damping and thereby in- duced wavevector-dependent SW damping is interaction of localized magnetic moments of magnetic insulator with conduction electrons from the examined three different types of metallic overlayers— normal, heavy, and altermagnetic. Due to spin-split energy-momentum dispersion of conduction electrons in the latter two cases, the nonlocal damping is anisotropic in spin and space, and it can be dramatically reduced by changing the relative orientation of the two layers when compared to the usage of normal metal overlayer. Introduction. —Spin wave (SW) or magnon damping is a problem of great interest to both basic and ap- plied research. For basic research, its measurements [1–4] can reveal microscopic details of boson-boson or boson- fermion quasiparticle interactions in solids, such as: magnon-magnon interactions (as described by second- quantized Hamiltonians containing products of three or more bosonic operators [5, 6]), which are frequently en- countered in antiferromagnets [4, 5] and quantum spin liquids [7], wherein they play a much more important role [8] than boson-boson interactions in other condensed phases, like anharmonic crystalline lattices or superflu- ids [5]; magnon-phonon interactions [3], especially rel- evant for recently discovered two-dimensional magnetic materials [2]; and magnon-electron interactions in mag- netic metals [1, 9–12]. For the envisaged magnon- based digital and analog computing technologies [13– 17], understanding magnon damping makes it possible to develop schemes to suppress [18] it, and, further- more, achieve amplification of nonequilibrium fluxes of magnons [19–22]. In fact, overcoming damping and achieving amplification is the keyto enable complex magnon circuits where, e.g., a logic gate output must be able to drive the input of multiple follow-up gates. Let us recall that the concept of SW was introduced by Bloch [23] as a wave-like disturbance in the local mag- netic ordering of a magnetic material. The quanta [6] of energy of SWs of frequency ωbehave as quasiparticles termed magnons, each of which carries energy ℏωand spin ℏ. As regards terminology, we note that in magnon- ics [13] SW is often used for excitations driven by an- tennas [24–27] and/or described by the classical Landau- Lifshitz-Gilbert (LLG) equation [9, 10, 28, 29], whereas magnon is used for the quantized version of the same ex- e ee ee eFIG. 1. (a) Schematic view of bilayers where a metallic over- layer covers the top surface of magnetic insulator, as often encountered in spintronics and magnonics [13, 30]. Three different energy-momentum dispersion of conduction elec- trons at the interface are considered, with their Fermi sur- faces shown in panel (b)—normal metal (NM); heavy metal (HM) with the Rashba SOC [31, 32], and altermagnetic metal (AM) [33, 34]—with the latter two being spin-split. The rel- ative alignment of the layers is labeled by an angle θ[33, 34], meaning that the wavevector qof SWs within FI is at an an- gleθaway from the kx-axis. citation [5], or these two terms are used interchangingly. In particular, experiments focused on SW damp- ing in metallic ferromagnets have observed [1] its de- pendence on the wavevector qwhich cannot be ex- plained by using the standard LLG equation [28, 29],∂tMn=−Mn×Beff n+αGMn×∂tMn(where ∂t≡ ∂/∂t), describing dynamics of localized magnetic mo- ments (LMMs) Mnat site nof crystalline lattice (also used in atomistic spin dynamics [28]) viewed as classi- cal vectors of unit length. This is because αG, as the Gilbert damping parameter [35, 36], is a local scalar (i.e., position-independent constant). Instead, various forms of spatially nonuniform (i.e., coordinate-dependent) and nonlocal (i.e., magnetization-texture-dependent) damp- ing due to conduction electrons have been proposed [9,arXiv:2312.09140v1 [cond-mat.mes-hall] 14 Dec 20232 10, 37–39], or extracted from first-principles calcula- tions [40], to account for observed wavevector-dependent damping of SWs, such as ∝q2(q=|q|) measured in Ref. [1]. The nonlocal damping terms require neither spin-orbit coupling (SOC) nor magnetic disorder scatter- ing, in contrast to αGwhich is considered to vanish [41] in their absence. Thus, in magnonics, it has been considered [30] that usage of magnetic insulators, such as yttrium iron gar- net (YIG) exhibiting ultralow αG≃10−4(achieved on a proper substrate [42]), is critical to evade much larger and/or nonlocal damping of SWs found in ferromagnetic metals. However, very recent experiments [24–27] have observed 100-fold increase of SW damping in the segment of YIG thin film that was covered by a metallic overlayer. Such spatially-resolved measurement [24] of SW damp- ing was made possible by the advent of quantum sensing based on nitrogen vacancy (NV) centers in diamond [43], and it was also subsequently confirmed by other meth- ods [25–27]. Since excitation, control, and detection of SWs requires to couple YIG to metallic electrodes [13], understanding the origin and means to control/suppress large increase in SW damping underneath metallic over- layer is crucial for realizing magnonic technologies. To explain their experiments, Refs. [24–27] have employed the LLG equation with ad hoc intuitively-justified terms (such as, effective magnetic field due to SW induced eddy currents within metallic overlayer [24]) that can fit the experimental data, which is nonuniversal and unsatisfac- tory (many other examples of similar phenomenological strategy exist [1, 44]). In contrast, in this Letter we employ recently derived ∂tMn=−Mn×Beff n+Mn×X n′(αGδnn′+λR)·∂tMn′, (1) extended LLG equation with all terms obtained [45] microscopically from Schwinger-Keldysh nonequilibrium quantum field theory [46] and confirmed [45] via exact quantum-classical numerics [47–50]. It includes nonlo- cal damping as the third term on the right-hand side (RHS), where its nonlocality is signified by dependence onR=rn−rn′, where rnis the position vector of lat- tice site n. Equation (1) is applied to a setup depicted in Fig. 1 where conduction electron spins from three differ- ent choices for metallic overlayer are assumed to interact with LMMs of ferromagnetic insulator (FI) at the inter- face via sdexchange interaction of strength Jsd, as well as possibly underneath the top surface of FI because of electronic evanescent wavefunction penetrating into it. Note that FI/normal metal (NM) bilayer directly mod- els recent experiments [24] where FI was a thin film of YIG and NM was Au, and SW damping within FI was quantified using quantum magnetometry via NV centers in diamond. Next, the FI/heavy metal (HM) bilayer, such as YIG/Pt [18, 27], is frequently encountered in 0 2 4 K/J0.51.0q (1/a) kF= 0 .92kF= 0 .99kF= 1 .08kF= 1 .15kF= 1 .22kF= 1 .30kF= 1 .38(a) 1.0 1.2 1.4 kF0.81.01.21.4qmax(1/a) ∝kF(b)FIG. 2. (a) Wavevector qof SW generated by injecting spin- polarized current in TDNEGF+LLG simulations of NM over- layer on the top of 1D FI [Fig. 1(a)] as a function of anisotropy K[Eq. (3)] for different electronic Fermi wavevectors kF. (b) Maximum wavevector qmaxof SWs that can be generated by current injection [21, 57] before wavevector-dependent SW damping becomes operative, as signified by the drop around kFin curves plotted in panel (a). various spintronics and magnonics phenomena [13, 30]. Finally, due to recent explosion of interest in altermag- nets [33, 34], the FI/altermagnetic metal (AM) bilay- ers, such as YIG/RuO 2, have been explored experimen- tally to characterize RuO 2as a spin-to-charge conver- sion medium [51]. The Schwinger-Keldysh field theory (SKFT), commonly used in high energy physics and cos- mology [52–54], allows one to “integrate out” unobserved degrees of freedom, such as the conduction electrons in the setup of Fig. 1, leaving behind a time-retarded dis- sipation kernel [48, 55, 56] that encompasses electronic effects on the remaining degrees of freedom. This ap- proach then rigorously yields the effective equation for LMMs only , such as Eq. (1) [45, 56] which bypasses the need for adding [1, 24, 44] phenomenological wavevector- dependent terms into the standard LLG equation. In our approach, the nonlocal damping is extracted from the time-retarded dissipation kernel [45]. SKFT-based theory of SW damping in FI/metal bilay- ers.—The nonlocal damping [45] λRin the third term on the RHS of extended LLG Eq. (1) stems from back- action of conduction electrons responding nonadiabati- cally [48, 59]—i.e., with electronic spin expectation value ⟨ˆsn⟩being always somewhat behind LMM which gener- ates spin torque [60] ∝ ⟨ˆsn⟩×Mn—to dynamics of LMMs. It is, in general, a nearly symmetric 3 ×3 tensor whose components are given by [45] λαβ R=−J2 sd 2πZ dε∂f ∂εTr σαAnn′σβAn′n . (2) Here, f(ε) is the Fermi function; α, β =x, y, z ;σα is the Pauli matrix; and A(ε) = i GR(ε)−GA(ε) is the spectral function in the position representation ob- tained from the retarded/advanced Green’s functions (GFs) GR/A(ε) = ε−H±iη
−1. Thus, the calcula- tion of λRrequires only an electronic Hamiltonian H as input, which makes theory fully microscopic (i.e.,3 −5 0 5 X−505Y λNM R(a)NM −1 0 1 λα R −5 0 5 X−505Y λxx R(b)HM t SOC= 0.3t0 −5 0 5 X−505Y λzz R(c) −5 0 5 X−505Y λ⊥ R(d)AM t AM= 0.5t0 0.0 0.5 1.0 1.5 q (1/a)0123Γq≡Im(ωq) (J/¯ h)×10−1 Eq.(6)(e) η= 0.1 η= 0 −5 0 5 X−505Y λyy R(f) −5 0 5 X−505Y λxy R(g) −5 0 5 X−505Y λ/bardbl R(h) FIG. 3. (a)–(d) and (f)–(h) Elements of SKFT-derived nonlocal damping tensor in 2D FI, λRwhere R= (X, Y, Z ) is the relative vector between two sites within FI, covered by NM [Eq. (5)], HM [Eqs. (8)] or AM [Eqs. (9)] metallic overlayer. (e) Wavevector-dependent damping Γ qof SWs due to NM overlayer, where the gray line is based on Eq. (6) in the continuous limit [58] and the other two lines are numerical solutions of extended LLG Eq. (1) for discrete lattices of LMMs within FI. The dotted line in (e) is obtained in the absence of nonlocal damping ( η= 0), which is flat at small q. Hamiltonian-based). Although the SKFT-based deriva- tion [45] yields an additional antisymmetric term, not displayed in Eq. (2), such term vanishes if the system has inversion symmetry. Even when this symmetry is broken, like in the presence of SOC, the antisymmet- ric component is often orders of magnitude smaller [56], therefore, we neglect it. The first term on the RHS of ex- tended LLG Eq. (1) is the usual one [28, 29], describing precession of LMMs in the effective magnetic field, Beff n, which is the sum of both internal and external ( Bextez) fields. It is obtained as Beff n=−∂H/∂Mnwhere His the classical Hamiltonian of LMMs H=−JX ⟨nn′⟩Mn·Mn′+K 2X n(Mz n)2−BextX nMz n.(3) Here we use g= 1 for gyromagnetic ratio, which sim- plifies Eq. (1); Jis the Heisenberg exchange coupling between the nearest-neighbors (NN) sites; and Kis the magnetic anisotropy. When nonlocal damping tensor, λRis proportional to 3×3 identity matrix, I3, a closed formula for the SW dispersion can be obtained via hydrodynamic the- ory [58]. In this theory, the localized spins in Eq. (1), Mn= (Re ϕn,Imϕn,1−m)T, are expressed using com- plex field ϕnand uniform spin density m≪1. Then, using the SW ansatz ϕn(t) =P qUqei(q·rn−ωqt), we ob- tain the dispersion relation for the SWs ωq= (Jq2+K−B) 1 +i(αG+˜λq) , (4) where qis the wavevector and ωis their frequency. Thedamping of the SW is then given by the imaginary part of the dispersion in Eq. (4), Γ q≡Imωq. It is comprised by contributions from the local scalar Gilbert damping αGand the Fourier transform of the nonlocal damping tensor, ˜λq=R drnλrneiq·rn. Results for FI/NM bilayer. —We warm up by extract- ing Γ qfor the simplest of the three cases in Fig. 1, a one-dimensional (1D) FI chain under a 1D NM over- layer with spin-degenerate quadratic electronic energy- momentum dispersion, ϵkσ=t0k2 x, where t0=ℏ2/2m. The GFs and spectral functions in Eq. (2), can be calculated in the momentum representation, yielding λ1D R=2J2 sd πv2 Fcos2(kFR)I3, where vFis the Fermi velocity, R≡ |R|, and kFis the Fermi wavevector. Moreover, its Fourier transform, ˜λq=2J2 sd v2 F[δ(q) +δ(q−2kF)/2], dic- tates additional damping to SWs of wavevector q= 0,±2kF. Although the Dirac delta function in this ex- pression is unbounded, this unphysical feature is an ar- tifact of the small amplitude, m≪1, approximation within the hydrodynamic approach [58]. The features of such wavevector-dependent damping in 1D can be cor- roborated via TDNEGF+LLG numerically exact simu- lations [47–50] of a finite-size nanowire, similar to the setup depicted in Fig. 1(a) but sandwiched between two NM semi-infinite leads. For example, by exciting SWs via injection of spin-polarized current into the metallic overlayer of such a system, as pursued experimentally in spintronics and magnonics [21, 57], we find in Fig. 2(a) that wavevector qof thereby excited coherent SW in-4 creases with increasing anisotropy K. However, the max- imum wavevector qmaxis limited by kF[Fig. 2(b)]. This means that SWs with q≳kFare subjected to additional damping, inhibiting their generation. Although our an- alytical results predict extra damping at q= 2kF, finite size effects and the inclusion of semi-infinite leads in TD- NEGF+LLG simulations lower this cutoff to kF. Since SW experiments are conducted on higher- dimensional systems, we also investigate damping on SWs in a two-dimensional (2D) FI/NM bilayer. The electronic energy-momentum dispersion is then ϵkσ=t0(k2 x+k2 y), and the nonlocal damping and its Fourier transform are given by λNM R=k2 FJ2 sd 2πv2 FJ2 0(kFR)I3, (5) ˜λNM q=kFJ2 sdΘ(2kF−q) 2πv2 Fqp 1−(q/2kF)2, (6) where Jn(x) is the n-th Bessel function of the first kind, and Θ( x) is the Heaviside step function. The nonlo- cal damping in Eqs. (5) and (6) is plotted in Fig. 3(a), showing realistic decay with increasing R, in contrast to unphysical infinite range found in 1D case. Addition-ally, SW damping in Eq. (6) is operative for wavectors 0≤q≤2kF, again diverging for q= 0,2kFdue to arti- facts of hydrodynamic theory [58]. Therefore, unphysical divergence can be removed by going back to discrete lat- tice, such as solid curves in Fig. 3(e) obtained for n=1– 100 LMMs by solving numerically a system of coupled LLG Eq. (1) where λRin 2D is used [45]. In this numer- ical treatment, we use kF= 0.5a−1where ais the lattice spacing; k2 FJ2 sd/2πv2 F=η= 0.1;K= 0; Bext= 0.1J; andαG= 0.1. Results for FI/HM bilayer. —Heavy metals (such as of- ten employed Pt, W, Ta) exhibit strong SOC effects due to their large atomic number. We mimic their presence at the FI/HM interface [31] by using 2D energy-momentum dispersion ϵk=t0(k2 x+k2 y) +tSOC(σxky−σykx), which includes spin-splitting due to the Rashba SOC [31, 32]. Using this dispersion, Eq. (2) yields λHM R= λxx Rλxy R0 λxy Rλyy R0 0 0 λzz R , (7) for the nonlocal damping tensor. Its components are, in general, different from each other λxx R=J2 sd 4πkF↑ vF↑J0(kF↑R) +kF↓ vF↓J0(kF↓R)2 + cos(2 θ)kF↑ vF↑J1(kF↑R)−kF↓ vF↓J1(kF↓R)2 , (8a) λyy R=J2 sd 4πkF↑ vF↑J0(kF↑R) +kF↓ vF↓J0(kF↓R)2 −cos(2 θ)kF↑ vF↑J1(kF↑R)−kF↓ vF↓J1(kF↓R)2 , (8b) λzz R=J2 sd 4πkF↑ vF↑J0(kF↑R) +kF↓ vF↓J0(kF↓R)2 −kF↑ vF↑J1(kF↑R)−kF↓ vF↓J1(kF↓R)2 , (8c) λxy R=−J2 sdsin(2θ) 4πkF↑ vF↑J1(kF↑R)−kF↓ vF↓J1(kF↓R)2 , (8d) where kF↑andkF↓are the spin-split Fermi wavevec- tors [Fig. 1(b)], and θis the relative orientation angle [Fig. 1(b)] between the SW wavevector qand the kxdi- rection. Thus, the nonlocal damping tensor in Eq. (7) generated by HM overlayer is anisotropic in spin due to its different diagonal elements, as well as nonzero off- diagonal elements. It is also anisotropic in space due to its dependence on the angle θ. Its elements [Eqs. (8)] are plotted in Figs. 3(b), 3(c), 3(f), and 3(g) using tSOC= 0.3t0. They may become negative, signifying the possibility of antidamping torque [21] exerted by conduc- tion electrons. However, the dominant effect of nearby LMMs and the presence of local scalar αGensures that LMM dynamics is damped overall. Although there is no closed expression for the SW dispersion in the presence of anisotropic λHM R, we can still extract SW damping Γ qin-duced by an HM overlayer from the exponential decay of the SW amplitude in numerical integration of extended LLG Eq. (1) using SW initial conditions with varying q. For an HM overlayer with realistic [31, 32] tSOC= 0.1t0 the results in Fig. (4)(a) are very similar to those ob- tained for NM overlayer with the same Fermi energy. Also, the spatial anisotropy of λHM Rdid not translate into θ-dependence of the SW damping. Results for FI/AM bilayer. —Altermagnets [33, 34] are a novel class of antiferromagnets with spin-split elec- tronic energy-momentum dispersion despite zero net magnetization or lack of SOC. They are currently in- tensely explored as a new resource for spintronics [51, 61, 62] and magnonics [63, 64]. A simple model for an AM overlayer employs energy-momentum dispersion ϵkσ=t0(k2 x+k2 y)−tAMσ(k2 x−k2 y) [33, 34], where tAMis5 0.0 0.5 1.0 1.5 q (1/a)1.01.52.02.5Γq≡Im(ωq) (J/¯ h)×10−1 (a)NM HM 0.0 0.5 1.0 1.5 q (1/a)1234Γq≡Im(ωq) (J/¯ h)×10−1 (b)NM AM :θ= 45◦ AM :θ= 0◦ FIG. 4. (a) Wavevector-dependent damping Γ qof SWs under NM or HM overlayer with the Rashba SOC of strength tSOC= 0.1t0. (b) Γ qof SWs under AM overlayer with tAM= 0.8t0 and for different relative orientations of FI and AM layers measured by angle θ[Fig. 1]. All calculations employ η= 0.1 and Fermi energy εF= 0.25t0. the parameter characterizing anisotropy in the AM. The corresponding λAM R= diag( λ⊥ R, λ⊥ R, λ∥ R) tensor has three components, which we derive from Eq. (2) as λ⊥ R=J2 sd 4πA+A− J2 0rϵF t0R+ +J2 0rϵF t0R− , (9a) λ∥ R=J2 sd 2πA+A−J0rϵF t0R+ J0rϵF t0R− , (9b) where A±=t0±tAMandR2 ±=X2/A±+Y2/A∓is the anisotropically rescaled norm of R. They are plot- ted in Figs. 3(d) and 3(h), demonstrating that λAM Ris highly anisotropic in space and spin due to the impor- tance of angle θ[61, 65, 66]. Its components can also take negative values, akin to the case of λHM R. It is inter- esting to note that along the direction of θ= 45◦[gray dashed line in Figs. 3(d) and 3(h)], λ⊥ R=λ∥ Rso that nonlocal damping tensor is isotropic in spin. The SW damping Γ qinduced by an AM overlayer is extracted from numerical integration of extended LLG Eq. (1) and plotted in Fig. (4)(b). Using a relatively large, but real- istic [33], AM parameter tAM= 0.8t0, the SW damping for experimentally relevant small wavevectors is reduced when compared to the one due to NM overlayer by up to 65% for θ= 0◦[Fig. 4(b)]. Additional nontrivial features are observed at higher |q|, such as being operative for a greater range of wavevectors and with maxima around |q|= 2p ϵF/t0and|q|= 3p ϵF/t0. Remarkably, these peaks vanish for wavevectors along the isotropic direction θ= 45◦[Fig. 4(b)]. Conclusions. —In conclusion, using SKFT-derived non- local damping tensor [45], we demonstrated a rigorous path to obtain wavevector damping of SWs in magnetic insulator due to interaction with conduction electrons of metallic overlayer, as a setup often encountered in magnonics [13–17, 30] where such SW damping was di- rectly measured in very recent experiments [24–27]. Ouranalytical expressions [Eqs. (5), (7), and (9)] for nonlo- cal damping tensor—using simple models of NM, HM, and AM overlayers as an input—can be directly plugged into atomistic spin dynamics simulations [28]. For more complicated band structures of metallic overlayers, one can compute λRnumerically via Eq. 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2023-12-14

Understanding spin wave (SW) damping, and how to control it to the point of being able to amplify SW-mediated signals, is one of the key requirements to bring the envisaged magnonic technologies to fruition. Even widely used magnetic insulators with low magnetization damping in their bulk, such as yttrium iron garnet, exhibit 100-fold increase in SW damping due to inevitable contact with metallic layers in magnonic circuits, as observed in very recent experiments [I. Bertelli et al., Adv. Quantum Technol. 4, 2100094 (2021)] mapping SW damping in spatially-resolved fashion. Here, we provide microscopic and rigorous understanding of wavevector-dependent SW damping using extended Landau-Lifshitz-Gilbert equation with nonlocal damping tensor, instead of conventional local scalar Gilbert damping, as derived from Schwinger-Keldysh nonequilibrium quantum field theory. In this picture, the origin of nonlocal magnetization damping and thereby induced wavevector-dependent SW damping is interaction of localized magnetic moments of magnetic insulator with conduction electrons from the examined three different types of metallic overlayers -- normal, heavy, and altermagnetic. Due to spin-split energy-momentum dispersion of conduction electrons in the latter two cases, the nonlocal damping is anisotropic in spin and space, and it can be dramatically reduced by changing the relative orientation of the two layers when compared to the usage of normal metal overlayer.

Nonlocal damping of spin waves in a magnetic insulator induced by normal, heavy, or altermagnetic metallic overlayer: a Schwinger-Keldysh field theory approach

2312.09140v1

arXiv:1701.03201v2 [cond-mat.mes-hall] 20 Mar 2017Dynamic coupling of ferromagnets via spin Hall magnetoresi stance Tomohiro Taniguchi National Institute of Advanced Industrial Science and Tech nology (AIST), Spintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan (Dated: September 17, 2018) The synchronized magnetization dynamics in ferromagnets o n a nonmagnetic heavy metal caused by the spin Hall effect is investigated theoretically. The di rect and inverse spin Hall effects near the ferromagnetic/nonmagnetic interface generate longit udinal and transverse electric currents. The phenomenon is known as the spin Hall magnetoresistance effec t, whose magnitude depends on the magnetization direction in the ferromagnet due to the spin t ransfer effect. When another ferro- magnet is placed onto the same nonmagnet, these currents are again converted to the spin current by the spin Hall effect and excite the spin torque to this addit ional ferromagnet, resulting in the excitation of the coupled motions of the magnetizations. Th e in-phase or antiphase synchronization of the magnetization oscillations, depending on the value o f the Gilbert damping constant and the field-like torque strength, is found in the transverse geome try by solving the Landau-Lifshitz-Gilbert equation numerically. On the other hand, in addition to thes e synchronizations, the synchroniza- tion having a phase difference of a quarter of a period is also f ound in the longitudinal geometry. The analytical theory clarifying the relation among the cur rent, frequency, and phase difference is also developed, where it is shown that the phase differences o bserved in the numerical simulations correspond to that giving the fixed points of the energy suppl ied by the coupling torque. PACS numbers: 85.75.-d, 75.78.-n, 05.45.Xt, 72.25.-b I. INTRODUCTION Dynamic coupling of ferromagnets has been of interest inthefieldofmagnetism. Thedipoleinteractionhasbeen the basic interaction to excite the coupled motion of the magnetizations, and is applied to magnetic recording1,2. Another method to realizethe couplingis to usethe spin- transfer effect3,4, where the applicationof an electric cur- rent to ferromagnetic/nonmagnetic multilayers results in the magnetization switching and self-oscillation5–14. The coupled dynamics through pure spin current generated in spin pumping15,16and nonlocal17geometries have also been observed. It should be emphasized that these cou- plings are strongly restricted by the characteristic length scales. Forexample, thedipolecouplingdecaysaccording to the inversecube detection law, whereas the spin trans- fer effect by a spin-polarized electric or pure spin current occurs in a system smaller than the spin diffusion length. Recently, physical phenomena, such as the spin torque18–25and magnetoresistance effects26–34, due to the spin Hall effect35–37in bilayers consisting of an insu- lating or metallic ferromagnet and a nonmagnetic heavy metal have attracted much attention. The latter, known as the spin Hall magnetoresistance, originates from the charge-spin conversion of an external electric current by the direct and inverse spin Hall effects, and has been observed by measuring the longitudinal and transverse electric currents, which are given by Jcx J0= 1+χ′′+χm2 y, (1) Jcy J0=−χmxmy−χ′mz, (2)respectively, where J0is the electric current density gen- erated by the external electric field. The definitions of the dimensionless coefficients, χ,χ′, andχ′′, are given below. It should be emphasized that the currents given by Eqs. (1) and (2) depend on the magnetization direc- tionm= (mx,my,mz). When another ferromagnet is placed onto the same nonmagnet, these currents will be converted to spin current by the spin Hall effect again, and excite spin torque on this additional ferromagnet. Then, the magnetization dynamics of two ferromagnets will be coupled through the angular dependencies of the electric current given by Eqs. (1) and (2). This coupling is unavoidable whenever several ferromagnets are placed onto the same nonmagnet, and is not restricted by the distance between the ferromagnets because it is carried by the electric current. Since the structure consisting of several ferromagnets on the nonmagnetic heavy metal will be important from the viewpoints of both fundamen- tal physics and practical applications based on the spin Halleffect, suchasmagneticrandomaccessmemory, spin torque oscillators, and bio-inspired computing38,39, it is of interest to clarify the role of this coupling. In this paper, we investigate the coupled dynamics of magnetizations in ferromagnets in the presence of the spin Hall effect by solving the Landau-Lifshitz-Gilbert (LLG) equation numerically for both longitudinal and transversegeometries. Inadditiontothe externalelectric current, the current contributing to the spin Hall mag- netoresistance also excites the spin torque. The strength of this additional torque is estimated from the theory of the spin Hall magnetoresistance extended to the system consistingofseveralferromagnets. The conventionalspin torque is proportional to the spin Hall angle ϑ, whereas the new torque is on the order of ϑ3, and therefore, its value is two orders of magnitude smaller than the con-2 x yzlongitudinal coupling transverse coupling ExF2 F1 F3 Ex (b)(a) F1 F2 σNEx+j1+j2SHE SHE ISHE ISHEN N FIG. 1: (a) Schematic view of the system in this study. Three ferromagnets F ℓ(ℓ= 1,2,3) are placed onto the same nonmagnet N. The external electric field is applied to the x direction. (b) Schematic view of the generations of electri c currents by the direct and inverse spin Hall effect (SHE and ISHE)in the longitudinal geometry. The total electric curr ent is the sum of the conventional electric current J0=σNExand the current generated near the F 1/N and F 2/N interfaces, j1 andj2. ventional spin torque. Nevertheless, it is found that this additional new torque affects the phase difference of the magnetizationin the self-oscillationstate. The numerical simulationrevealsthatthein-phaseorantiphasesynchro- nization is observed in the transverse geometry. On the other hand, in addition to them, the phase difference be- comes a quarter of a period in the longitudinal geometry. It is found that these phase differences depend on the values of the Gilbert damping constant and the dimen- sionless field-like torque strength. An analytical theory clarifying the relation among the current, frequency, and phase difference is also developed. The paper is organized as follows. In Sec. II, we de- scribe the system under consideration, and discuss the theoretical formula of the spin torque excited by the spin Halleffect inthepresenceoftheseveralferromagnetsand the spin Hall magnetoresistance effect. In Sec. III, we study the phase differences in the synchronized state of the magnetizations for both the longitudinal and trans- verse geometries by solving the LLG equation numeri- cally. InSec. IV,thetheoryclarifyingtherelationamong the current, frequency, and phase difference is developed based on the LLG equation averaged over constant en- ergy curves. The summary of the paper is given in Sec. V. II. SYSTEM DESCRIPTION AND LLG EQUATION In this section, we describe the system adopted in this study, and show the spin torque formulas including the coupling torques between the ferromagnets.A. System description The system we consider is schematically shown in Fig. 1(a), where three ferromagnets F ℓ(ℓ= 1,2,3) are placed onto the same nonmagnet N. We assume that the ma- terial parameters in the ferromagnets are identical, for simplicity. The external electric field, Ex, is applied to thexdirection, inducing the electric current density J0=σNEx, where σNis the conductivity of the non- magnet. The direct and inverse spin Hall effects produce electriccurrentsinthelongitudinal( x)andtransverse( y) directions. These electric currents are converted to the spin current and injected into the F 2and F 3layers due to the spin Hall effect, resulting in the excitation of the spin torque. Then, the magnetization dynamics in the F ℓ (ℓ= 2,3)layerisaffectedbythatintheF 1layer,andvice versa. We call the coupling between the F 1and F2layers the longitudinal coupling, whereas that between the F 1 and F 3layers the transverse coupling. We assume that both the ferromagnet and nonmagnet are metallic be- cause metallic bilayers are generally used to measure the magnetization switching and oscillation by the spin Hall effect18–21,24. Although the spin Hall magnetoresistance was originally studied for insulating ferromagnets26–29, a large spin Hall magnetoresistance in metallic system has also been reported recently32–34. The dimensionless coefficients, χ,χ′, andχ′′, in Eqs. (1) and (2) for single ferromagnets have been derived for an insulating30or metallic34,40ferromagnet, which are given by χ=ϑ2λN dN/bracketleftbigg Reg↑↓ gN+g↑↓coth(dN/λN)−g∗ gN/bracketrightbigg tanh2/parenleftbiggdN 2λN/parenrightbigg , (3) χ′=−ϑ2λN dNImg↑↓ gN+g↑↓coth(dN/λN)tanh2/parenleftbiggdN 2λN/parenrightbigg , (4) χ′′=2ϑ2λN dNtanh/parenleftbiggdN 2λN/parenrightbigg −ϑ2λN dNReg↑↓ gN+g↑↓coth(dN/λN)tanh2/parenleftbiggdN 2λN/parenrightbigg , (5) wheredNandλNare thickness and spin diffusion length of the nonmagnet, respectively, whereas gN/S= hσN//parenleftbig 2e2λN/parenrightbig with the cross section area of the ferro- magnetic/nonmagnetic interface S. The dimensionless mixing conductance g↑↓=g↑↓ r+ig↑↓ iconsists of its real and imaginary parts41–43, andg∗is defined as 1 g∗=2 (1−p2g)g+1 gFtanh(dF/λF)+1 gNtanh(dN/λN). (6) Here,g=g↑↑+g↓↓is the sum of the conductances of the spin-up and spin-down electrons, whereas pg=3 (g↑↑−g↓↓)/gis its spin polarization. The ferromag- netic/nonmagnetic interface resistance ris related to gviag/S= (h/e2)/r. We also introduce gF/S= h/parenleftbig 1−p2 σ/parenrightbig σF//parenleftbig 2e2λF/parenrightbig , whereσFis the conductivity of the ferromagnet and pσis its spin polarization. The thickness and spin diffusion length of the ferromagnet are denoted as dFandλF, respectively. The term related tog∗is neglected when the ferromagnetis an insulator30, i.e.,r→ ∞. The following quantities correspond to the effective spin polarizations of the damping-like (or Slonczewski3) torque and the field-like torque, respec- tively, ϑR(I)=ϑtanh/parenleftbiggdN 2λN/parenrightbigg Re(Im)g↑↓ gN+g↑↓coth(dN/λN). (7) For the later discussion, we introduce β=−ϑI ϑR, (8) which corresponds to the ratio of the field-like torque to the damping-like torque. The values of the parameters used in the following calculations are derived from recent experiments on the W/CoFeB heterostructure34, where ρF= 1/σF= 1.6 kΩnm,pσ= 0.72,λF= 1.0 nm,ρN= 1/σN= 1.25 kΩnm,λN= 1.2 nm, and ϑ= 0.27, whereas the thick- nesses areassumed to be dF= 2 nm and dN= 3 nm. The interfaceconductanceswerenotevaluatedinRef.34byas- suming a transparent interface. Instead, we use typical values of the interface conductances obtained from the first-principles calculations43,r= 0.25 kΩnm2,pg= 0.5, andg↑↓ r/S= 25 nm−2. We note that the imaginary part of the mixing conductance, g↑↓ i, is either positive or neg- ative, depending on the material and thickness43. The sign ofg↑↓ idetermines those of χ′andϑI, or equivalently, β. For example, when g↑↓ i/S= 1 nm−2,χ≃0.010, χ′≃ −0.0002,χ′′≃0.035,θR≃0.167, and θI≃0.002 (β≃ −0.010); see Appendix A. In the following calcula- tions, we study the magnetization dynamics for several values of β. B. Spin torques in longitudinal and transverse geometries Equation (1) was derived for a system having single ferromagnet. In this case, J0is the external electric cur- rent density, whereas ( χ′′+χmy)J0is the current density generated as a result of the charge-spinconversionby the direct and inverse spin Hall effects. In the longitudinal geometryinthepresentstudy, ontheotherhand, twofer- romagnetic/nonmagnetic interfaces, i.e., F 1/N and F 2/N interfaces, contribute to the generation of the longitudi- nal current through the direct and inverse spin Hall ef- fects, as schematically shown in Fig. 1(b). Let us denote the electric current density generated by these effects near the F ℓ/N interface as jℓx(ℓ= 1,2). This currentdensity is determined by the conservation law of the elec- tric current, as follows. Considering in a similar manner to the case of the single ferromagnet, the electric current J0+j1xis converted to the spin current by the spin Hall effect near the F 2/N interface, and this spin current pro- duces an additionalelectric current ( χ′′+χm2 2y)(J0+j1x) by the inverse spin Hall effect. Therefore, the total lon- gitudinal electric current density near the F 2/N interface is (1+χ′′+χm2 2y)(J0+j1x). Similarly, the electric cur- rent density near the F 1/N interface can be expressed as (1+χ′′+χm2 1y)(J0+j2x). These currents should be equal to the total electric current density, J0+j1x+j2x, according to the conservation law of the electric current. Then, we find that jℓxis given by jℓx=/parenleftBig χ′′+χm2 ℓy/parenrightBig/parenleftBig 1+χ′′+χm2 ℓ′y/parenrightBig 1−/parenleftBig χ′′+χ′m2 ℓy/parenrightBig/parenleftBig χ′′+χm2 ℓ′y/parenrightBigJ0 ≃/parenleftbig χ′′+χm2 ℓy/parenrightbig J0,(9) where we neglect the higher order terms of ϑ. Then, the spin torque excited on the F ℓlayer by the spin Hall effect in the longitudinal geometry is obtained by replacing the external electric current density J0=σNExin the previ- ous work30withJ0+jℓ′x≃/parenleftBig 1+χ′′+χm2 ℓ′y/parenrightBig J0, and is given by TL ℓ=−γ/planckover2pi1ϑRJ0 2eMdF/parenleftbig 1+χ′′+χm2 ℓ′y/parenrightbig mℓ×(ey×mℓ) −γ/planckover2pi1ϑIJ0 2eMdF/parenleftbig 1+χ′′+χm2 ℓ′y/parenrightbig ey×mℓ, (10) where (ℓ,ℓ′) = (1,2) or (2,1). The unit vector pointing in the magnetization direction of the F ℓlayer ismℓ, and γ,M, anddare the gyromagnetic ratio, the saturation magnetization, and the thickness of the ferromagnet, re- spectively. Note that the terms, T(0) ℓ=−γ/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ)−γ/planckover2pi1ϑIJ0 2eMdFey×mℓ, (11) in Eq. (10) are the conventional spin torques generated by the external electric current J0and the spin Hall ef- fect, and are often referred to as the damping-like torque andthefield-liketorque, respectively. Ontheotherhand, the terms −γ/planckover2pi1ϑRJ0 2eMdF/parenleftbig χ′′+χm2 ℓ′y/parenrightbig mℓ×(ey×mℓ) −γ/planckover2pi1ϑIJ0 2eMdF/parenleftbig χ′′+χm2 ℓ′y/parenrightbig ey×mℓ,(12) in Eq. (10) originate from the current jℓ′x, and are newly introduced in this study. It should be emphasized that Eq. (12) depends on the magnetization direction of the4 other ferromagnet F ℓ′(ℓ′= 1 or 2), mℓ′, resulting in the coupling between the F 1and F2layers. Let us next show the spin torque formulasin the trans- verse geometry. We denote the electric current den- sity flowing in the ydirection generated near the F ℓ/N (ℓ= 1,3) interface by the inverse spin Hall effect as jℓy. The conservation law of the total electric current density, j1y+j3y, gives (see also Appendix B) jℓy=−/bracketleftbigg(χmℓxmℓy+χ′mℓz) 1−(χ′′+χm2 ℓx)(χ′′+χm2 ℓ′x) +/parenleftbig χ′′+χm2 ℓx/parenrightbig (χmℓ′xmℓ′y+χ′mℓ′z) 1−(χ′′+χm2 ℓx)(χ′′+χm2 ℓ′x)/bracketrightBigg J0 ≃ −(χmℓxmℓy+χ′mℓz)J0.(13) Inadditiontotheconventionalspintorque,Eq. (11), this transverse electric current also excites the spin torque on the other ferromagnet. In total, the spin torque acting on the F ℓlayer is given by [( ℓ,ℓ′) = (1,3) or (3,1)] TT ℓ=−γ/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ)−γ/planckover2pi1ϑIJ0 2eMdFey×mℓ −γ/planckover2pi1ϑR(χmℓ′xmℓ′y+χ′mℓ′z)J0 2eMdFmℓ×(ex×mℓ) −γ/planckover2pi1ϑI(χmℓ′xmℓ′y+χ′mℓ′z)J0 2eMdFex×mℓ, (14) Thelasttwotermsrepresentthe couplingtorquebetween the F1and F 3layers. Note that the direction of this coupling torque is different from that of the conventional torque because the currents J0andjℓyflow in different directions. In the following, the torques related to χ,χ′, andχ′′ in Eqs. (10) and (14) are referred to as coupling torque. The ratio of these new torques to the conventional spin torque is on the order of χ∝ϑ2∼10−2. Since the conventional spin torque due is proportional to the spin Hall angle ϑR∝ϑ, the coupling torque is proportional toϑ3. Although the spin Hall angle is usually a small quantity, it will be shown that the coupling torques play a non-negligible role on the magnetization dynamics, as shown below. C. LLG equation In the following sections, we study the magnetization dynamics excited by the spin torque given by Eq. (10) or (14) both numerically and analytically. We neglect the transverse coupling when the role of the longitudinal coupling is studied, and vice versa, for simplicity, which corresponds to considering a system consisting of the F 1 and F 2layers, or F 1and F 3layers. The magnetization dynamics in the F ℓ(ℓ= 1,2,3) is described by the LLG equation, dmℓ dt=−γmℓ×Hℓ+αmℓ×dmℓ dt+TL,T ℓ,(15)where the Gilbert damping constant is denoted as α. In the following calculations, we use the values of the material parameters, γ= 1.764×107rad/(Oe s) and α= 0.005, derived from the experiments44. For the later discussion, it is useful to show the explicit forms of the LLG equation in the longitudinal and transverse geome- tries. Equation (15) for the longitudinal geometry is /parenleftbig 1+α2/parenrightbigdmℓ dt=−γmℓ×Hℓ −γ(1+χ′′)/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ)−αγmℓ×(mℓ×Hℓ) −γχm2 ℓ′y/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ) −γ(1+χ′′)(α+β)/planckover2pi1ϑRJ0 2eMdFmℓ×ey +γ(1+χ′′)αβ/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ) −γ(α+β)χm2 ℓ′y/planckover2pi1ϑRJ0 2eMdFmℓ×ey +γαβχm2 ℓ′y/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ), (16) whereas that for the transverse geometry is /parenleftbig 1+α2/parenrightbigdmℓ dt=−γmℓ×Hℓ −γ/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ)−αγmℓ×(mℓ×Hℓ) −γ(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0 2eMdFmℓ×(ex×mℓ) −γ(α+β)/planckover2pi1ϑRJ0 2eMdFmℓ×ey +γαβ/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ) −γ(α+β)(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0 2eMdFmℓ×ex +γαβ(χmℓ′xmℓ′y+χ′mℓ′z)/planckover2pi1ϑRJ0 2eMdFmℓ×(ex×mℓ). (17) III. NUMERICAL ANALYSIS OF SYNCHRONIZATION In this section, we study the magnetization dynam- ics in the ferromagnets in the presence of the coupling torques by solving Eq. (15) numerically. The self- oscillation of the magnetization provides an interesting example to understand the role of the coupling torque. Note that the coupling torques are proportional to the parameters χ,χ′, andχ′′, and their products to other pa- rameters such as αβχ, the values of which are relatively small, as can be seen in Eqs. (16) and (17). Nevertheless, the coupling torques lead to the phase synchronization, as shown below.5 The self-oscillation of the magnetization in single fer- romagnets by the spin Hall effect has been observed for in-plane magnetized ferromagnets19, and is induced by the conventional spin torque given by Eq. (11). There- fore, in the following, we assume that the magnetic field, Hℓ=HKmℓyey−4πMmℓzez, (18) consists of an in-plane anisotropy field HKalong the y direction and a demagnetization field 4 πMin thezdi- rection, which we assumeas HK= 200Oe and M= 1500 emu/c.c.34in the following calculations. It is known for the case of the single ferromagnet45that the in-plane self-oscillation can be excited when the electric current densityJ0is in the range of Jc< J0< J∗, where Jc=2αeMd F /planckover2pi1ϑR(HK+2πM), (19) J∗=4αeMd F π/planckover2pi1ϑR/radicalbig 4πM(HK+4πM),(20) which in this study are Jc≃26 andJ∗≃33 MA/cm2. A. Transverse geometry Let us first investigate the magnetization dynamics in the transverse geometry because this geometry pro- vides a simple example of the coupled motion. We start with solving the LLG equation given by Eq. (17) for the F1and F 3layers. Figure 2(a) shows an exam- ple of the trajectory of the magnetization dynamics ob- tained by solving Eq. (17) numerically, where J0= 30 MA/cm2. As shown, the in-plane oscillation is observed in the steady state. The initial conditions set for m1 andm3are different as m1(0) = (cos80◦,sin80◦,0) and m3(0) = (cos95◦,sin95◦,0). Therefore, the dynamics of m1andm3near the initial time are different, as shown in Fig. 2(b), where the time evolutions of m1x(t) and m3x(t) in 0≤t≤1 ns are shown. Nevertheless, the dy- namics of m1andm3synchronize gradually, and finally, synchronizationof m1x(t) =m3x(t) andm1y(t) =m3y(t) is realized, as shown in Figs. 2(c) and 2(d). We empha- size here that the dynamics shown in these figures are obtained for β=−0.01. The mutual, as well as self, synchronization of the spin torque induced magnetization oscillation by using spin waves, electric current, microwave field, or dipole coupling has been an exciting topic from the viewpoints of both nonlinear science and practical applications46–62. The key quantity of the synchronization is the phase dif- ference ∆ ϕbetween each magnetization to enhance the emission power from the spin torque oscillators and to investigate new practical applications such as neuromor- phic architectures38,39. The synchronization found here, i.e.,mℓx(t) =mℓ′x(t), is called the in-phase synchro- nization. We should emphasize here that, although the(a) mxmymz 1.00-1.0 -1.01.0 0-1.001.0 time (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1 :F3:F1 :F3 01.0 -1.0 01.0 0.5mx my(c) (d) time (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1 :F301.0 -1.0 01.0 0.5mx my(e) (f)time (ns)0 0.2 0.4 0.6 0.8 1.001.0 -1.0mx(b) FIG. 2: (a) A typical in-plane self-oscillation trajectory of the magnetization obtained by solving Eq. (15) numerically for the transverse geometry. The dotted lines indicate the oscillation direction. (b) The time evolutions of m1x(t) and m1y(t) near the initial state. The time variations of the x andycomponents of the magnetizations in the steady state are shown in (c) and (d) for β=−0.01 and (e) and (f) for β= +0.01. The solid red lines correspond to time evolution in F1, and dotted blue lines to those in F 3for (b) through (f). results shown in Figs. 2(b) and 2(c) are shown for one certain initial condition, the in-phase synchronizations are confirmed for the present set of the parameters even when the initial conditions are changed. On the other hand, it was shown for the case of the current-injectionlockingofthespin torqueoscillatorthat the phase difference between the magnetization oscilla- tion and the alternative current depends on the strength of the field-like torque53. The field-like torque in the present system can be either positive or negative, as mentioned above. These facts motivate us to study the magnetization dynamics for different values of β. When β= +0.01, synchronized dynamics is observed in a sim- ilar manner, but in this case, the phase difference is an- tiphase, i.e., m1x(t) =−m3x(t), as shown in Figs. 2(e) and 2(f). Figure 3 summarizes the dependences of the phase dif- ference, ∆ ϕ, between the magnetizations on the field-like torque strength βfor the several values of the damping constant α. The vertical axis in this figure represents the phase difference in the unit of the oscillation period; i.e., ∆ϕ= 0correspondsto the in-phase synchronization. On the other hand, ∆ ϕ= 0.50 means that the phase dif-6 field-like torque, βphase difference , Δφ -0.030 -0.020 -0.010 0.010 0.020 0.030 000.250.50 α=0.005, 0.010, 0.015, 0.020, 0.025, 0.030 FIG. 3: Dependencesofthephase differences, ∆ ϕ, for several values of αon the field-like torquestrength βin the transverse geometry. ∆ ϕ= 0 and 0 .5 correspond to the in-phase and antiphase, respectively. The values of the current density ,J0, is increased linearly, where J0= 30 MA/cm2forα= 0.005. ference is half of a period, and thus, is antiphase. The algorithm evaluating ∆ ϕin the numerical simulation is summarized in Appendix A. The results indicate that the phase difference is antiphase for positive β, whereas it becomes in-phase when βbecomes smaller than a cer- tainvalue, exceptforthenarrowintermediateregionnear β∼ −α/2, where the phase difference is in between in- phase and antiphase. B. Longitudinal geometry Next, we study the magnetization dynamics in the lon- gitudinal geometry between F 1and F 2layers by solv- ing Eq. (16) numerically. Figure 4(a) and 4(b) show mℓx(t) andmℓy(t) in a steady state, where β=−0.01. The initial conditions in these figures are m1(0) = (cos80◦,sin80◦,0) and m2(0) = (cos95◦,sin95◦,0). An antiphase synchronization is observed in this case. We notice, however, that the in-phase synchronization can also be realized when the initial conditions are changed. Figure 4(c) and 4(d) show such an exam- ple, where m1(0) = (cos80◦,sin80◦,0) andm2(0) = (cos85◦,sin85◦,0) are assumed. These numerical results indicate that both the in-phase and antiphase synchro- nizations are stable in this case, and whether the phase difference finally becomes in-phase or antiphase depends on the initial conditions, material parameters, and cur- rent magnitude. We also notice that the phase difference is changed when the value of βis changed. Figures 4(e) and 4(f) show mℓxandmℓyforβ= +0.01. In this case, the phase difference of the magnetizations is a quarter of a precession period. The dependences of the phase difference on the field- like torque strength βfor several values of the Gilbert damping constant αare summarized in Fig. 5, where ∆ϕ= 0.25 in this figure means that the phase differencetime (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1 :F201.0 -1.0 01.0 0.5mx my(a) (b) time (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1 :F201.0 -1.0 01.0 0.5mx my(c) (d) time (ns) time (ns)999.0 999.2 999.4 999.6 999.8 1000 999.0 999.2 999.4 999.6 999.8 1000:F1 :F201.0 -1.0 01.0 0.5mx my(e) (f) FIG. 4: The variations of the xandycomponents of the magnetizations in the F 1(solid, red) and F 2(dotted, blue) in the longitudinal geometry are shown in (a), (c) and (b), (d) respectively, where β=−0.01. The initial conditions of m2 are different between (a), (b) and (c), (d). The magnetizatio n dynamics for β= +0.01 are shown in (e) and (f) is a quarter of a period. The phase difference is found to become a quarter of the period for positive β, whereas it becomes in-phase or antiphase for negative β, depending on the initial states of the magnetizations. C. Summary of numerical simulations Let us summarize the results of the numerical simu- lations here. In the transverse geometry, the coupling torque induces the synchronized oscillation of the mag- netizations and finally stabilizes the configuration in the in-phase or antiphase state, depending on the values of the field-liketorque strength βand the damping constant α. The phase difference in the stable synchronized state in the longitudinal geometry also depends on the values ofβandα, as well as the initial conditions. In this case, however, in addition to the in-phase or antiphase state, a phase difference with a quarter of a period is generated. The phase difference can be measured from the spin Hallmagnetoresistanceeffect. AccordingtoEq. (13), the total electric current density in the transverse direction7 field-like torque, β-0.030 -0.020 -0.010 0.010 0.020 0.030 000.250.50 α=0.005 α=0.010 α=0.015 α=0.020α=0.030α=0.025phase difference , Δφ FIG. 5: Dependencesofthephase differences, ∆ ϕ, for several values of αon the field-like torque strength βin the longitu- dinal geometry, where ∆ ϕ= 0 and 0 .5 correspond to the in-phase and antiphase, respectively. ∆ ϕ= 0.25 means that the phase difference is a quarter of a period. The values of the current density, J0, is increased linearly, where J0= 30 MA/cm2forα= 0.005. time (ns)999.0 999.2 999.4 999.6 999.8 100000.3 0.2 0.1 -0.3-0.2-0.1 current density (106 A/cm2)(a) time (ns)999.0 999.2 999.4 999.6 999.8 10003233current density (106 A/cm2)(b) FIG. 6: (a) The transverse electric current densities given by Eq. (21) for in-phase synchronization (solid line) and an - tiphase synchronization (dotted line). (b) The longitudin al electric current densities given by Eq. (22) for in-phase or antiphase synchronization (solid line) and phase differenc e of a quarter of a period (dotted line). is JT c=j1y+j3y =−(χm1xm1y+χ′m1z)J0 −(χm3xm3y+χ′m3z)J0.(21) Then, an oscillating current appears in the transverse di- rection for in-phase synchronization, whereas the trans- versecurrentbecomeszeroforantiphasesynchronization, as shown in Fig. 6(a). The Fourier transformation of Eq. (21) for in-phase synchronization has peaks at the frequencies of fn= (2n−1)f0(n= 1,2,3,···), where f0 is the lowest frequency; see Appendix C. Similarly, the total electric current density in the longitudinal direction is JL c=J0+j1x+j2x =J0+/parenleftbig χ′′+χm2 1y/parenrightbig J0+/parenleftbig χ′′+χm2 2y/parenrightbig J0.(22) The oscillation frequency of this current is different for the synchronizations having the phase difference of in- phase or antiphase and that of a quarter of a period,as shown in Fig. 6(b). The Fourier transformation of Eq.(22) has the peaks at fn= 2nf0for in-phase or an- tiphase and fn= 4nf0when the phase difference is a quarter of a period. An interesting question regarding these numerical re- sults is to clarify the reason why the phase difference finally becomes a certain value for a given set of the parameters, i.e., which phase difference is an attrac- tor of the limit cycle. It is, however, difficult to an- swer this question directly due to the following rea- son. We note that the present model includes sev- eral small-valued parameters, α,β,χ,χ′, andχ′′, as shown in Eqs. (16) and (17), and is complicated. The torque related to the lowest order of βin these equations is the conventional field-like torque given by [γβ/planckover2pi1ϑRJ0/(2eMdF)]mℓ×ey. It should be noted that the phase difference is not determined solely by this term because this lowest order term of the field-like torque does not include the coupling between the magnetiza- tions. Similarly, the attractor is not determined solely by the lowest order term of the coupling torque, which is [γ/planckover2pi1ϑRJ0/(2eMdF)]χm2 ℓ′ymℓ×(ey×mℓ) in Eq. (16) and [γ/planckover2pi1ϑRJ0/(2eMdF)]χmℓ′xmℓ′ymℓ×(ex×mℓ) in Eq. (17), because this torque does not include the field-like torque. Their combinations or the higher order terms including both βand the coupling torques related to χ, χ′, andχ′′should be taken into account to answer the question, which is difficult due to the nonlinearity and complexity of the LLG equation. Nevertheless, it is possible to reveal the relation be- tween the current and frequency in the synchronized os- cillation state by assuming a certain value of the phase difference between the magnetizations. The current- frequency relation has been often measured in the ex- periments, and therefore, it will be useful to develop a theory clarifying the role of the coupling on the current- frequency relation. In the next section, we will investi- gate this subject. IV. THEORETICAL ANALYSIS OF CURRENT-FREQUENCY RELATION The purpose of this section is to develop an analyti- cal theory of the synchronization revealing the relation amongthe current, frequency, and the phase difference of the magnetizations in the synchronized oscillation state. A. Basis of analysis Here, let us mention the basis of our theoretical anal- ysis. It is difficult to solve the LLG equation exactly because of its nonlinearity. Instead, we employ the aver- aging technique of the LLG equation on constant energy curves63. This approach has been used to study the spin torque switching in thermally activated regions64–68and spin torque oscillators69–77, as well as the microwave as-8 sisted magnetization reversal78,79, but has not been ap- plied to the coupled system. This approach is valid when the magnetic energy is changed slowly compared with the oscillation period. We note that only the lowest or- der terms in the LLG equation is necessary to derive the current-frequency relation, as far as several parameters such asβare small. Thus, we use the simplified LLG equationmaintainingonlythedominantterms. TheLLG equation used in this section for the longitudinal geome- try is dmℓ dt≃−γmℓ×Hℓ−αγmℓ×(mℓ×Hℓ) −γ/planckover2pi1ϑRJ0 2eMdF/parenleftbig 1+χ′′+χm2 ℓ′y/parenrightbig mℓ×(ey×mℓ), (23) whereas that for the transverse geometry is dmℓ dt≃−γmℓ×Hℓ−αγmℓ×(mℓ×Hℓ) −γ/planckover2pi1ϑRJ0 2eMdFmℓ×(ey×mℓ) −γ/planckover2pi1ϑRJ0 2eMdFχmℓ′xmℓ′ymℓ×(ex×mℓ).(24) In the self-oscillation state, the damping torque dur- ing the precession is balanced with the spin torque, and thetorqueduetothemagneticfield, correspondingtothe first term on the right hand side of Eq. (15), becomes the dominantterm determining the magnetizationdynamics. The dynamic trajectory given by this field torque corre- sponds to constant energy curves of the energy density, E=−M/integraltext dmℓ·Hℓ, where its explicit form is E=−MHK 2m2 ℓy+2πM2m2 ℓz. (25) The minimum and saddle points of Eq. (25) are mmin= ±eyandmsaddle=±ex, where the corresponding energy densities are Emin=−MHK/2 andEsaddle= 0. The solution of mℓprecessing on a constant energy curve is described by the Jacobi elliptic function80as (see also Appendix C summarizing the derivations) mx=/radicalbigg 1+2E MHKsn/bracketleftbigg4K(k) τ(E)t+ϕ0,k/bracketrightbigg ,(26) my=/radicalBigg 4πM−2E/M HK+4πMdn/bracketleftbigg4K(k) τ(E)t+ϕ0,k/bracketrightbigg ,(27) mz=/radicalBigg HK+2E/M HK+4πMcn/bracketleftbigg4K(k) τ(E)t+ϕ0,k/bracketrightbigg ,(28) whereϕ0is the initial phase. The period of mxandmzis τ(E), whereasthat of myisτ(E)/2, wherethe precession frequency f(E) = 1/τ(E) is given by f(E) =γ/radicalbig HK(4πM−2E/M) 4K(k),(29)whereK(k) is the first kind of complete elliptic integral with the modulus k=/radicalBigg 4πM(HK+2E/M) HK(4πM−2E/M). (30) Note that Eq. (29) reproduces the ferromagnetic res- onance (FMR) frequency, γ/radicalbig HK(HK+4πM)/(2π), in the limit of E→Emin. Identifying Eandϕ0corresponds to the determination of the initial condition. The averagedtechnique investigates the energy change during a precession on a constant energy curve, which is obtained from the LLG equation as /contintegraldisplay dtdE dt=Ws+WL,T s+Wα, (31) where Wsis the energy change by the conventional spin torque due to the spin Hall effect, whereas Wαis the dis- sipation due to the damping torque. The integral range is over the precession period. The explicit forms of Ws andWαare given by67 Ws=/contintegraldisplay dtγ/planckover2pi1ϑRJ0 2edF[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)] =π/planckover2pi1θRJ0(HK+2E/M) edF/radicalbig HK(HK+4πM),(32) Wα=−/contintegraldisplay dtαγM/bracketleftBig H2 ℓ−(mℓ·Hℓ)2/bracketrightBig =−4αM/radicalBigg 4πM−2E/M HK/bracketleftbigg2E MK(k)+HKE(k)/bracketrightbigg , (33) whereE(k)isthesecondkindofcompleteellipticintegral. On the other hand, WL,T srepresents the work done by the coupling torque in the longitudinal or transverse geometry, corresponding to the last term in Eq. (23) or (24). The explicit forms of WL sandWT sare shown in the following sections. For both cases, the relation between the current and frequency is clarified as follows. Inthe self-oscillationstate, sincethespin torquebalances the damping torque, the following condition should be satisfied, /contintegraldisplay dtdE dt= 0. (34) The current density J0satisfying Eq. (34) is the current necessaryto excite the self-oscillationon the constanten- ergy curve of E, and is denoted as J0(E). The relation between the current and frequency in the self-oscillation state is given by this J0(E) andf(E) given by Eq. (29). It should be emphasized that the current density J0(E) depends on the phase difference between the magnetiza- tions through WL,T s. We will therefore study the relation between the phase difference and the current-frequency relation in line with this deduction.9 B. Transverse geometry In this section, we investigate the current-frequency relation in the transverse geometry. The work done by the coupling torque is defined as WT s=/contintegraldisplay dtγ/planckover2pi1ϑRχJ0 2edFmℓ′xmℓ′y[ex·Hℓ−(mℓ·ex)(mℓ·Hℓ)]. (35) Before advancing the calculation, let us briefly men- tion the definition of the phase difference in the present approach. If the oscillation trajectory is a circle, the phase difference is easily defined, i.e., the antiphase cor- responds to ∆ ϕ=π, whereas ∆ ϕ= 0 is the in-phase. In the present case, on the other hand, the oscillation trajectory is described by the elliptic function, as shown in Eqs. (26), (27), and (28). In this case, the phase dif- ference is defined using the elliptic integral K(k), where ∆ϕ= 0 for the in-phase synchronization, and the an- tiphase synchronization corresponds to ∆ ϕ= 2K(k). It is useful to note that ∆ ϕ= 2K(k) becomes πin the limit ofk→0, corresponding to the case that the oscillation trajectory becomes a circle. Similarly, ∆ ϕ=K(k) means that the phase difference is a quarter of a period. Equation (35) for an arbitrary phase difference is eval- uated by numerically calculating the integral with the solutions of mℓandmℓ′shown in Appendix D. It is, however, useful to derive the analytical solutions of Eq. (35) for specific values of the phase difference. Equa- tion (35) for both the in-phase (∆ ϕ= 0) and antiphase [∆ϕ= 2K(k)] becomes WT s=∓π/planckover2pi1ϑRχJ0 2edF/radicalbig HK(HK+4πM)/parenleftbigg −2E M/parenrightbigg/parenleftbigg 1+2E MHK/parenrightbigg , (36) where the double sign means the upper for the in-phase (∆ϕ= 0)synchronizationandthelowerfortheantiphase [∆ϕ= 2K(k)] synchronization. Equation (36) is zero at E=EminandEsaddle, and is negative (positive) for the energy density Ein the rage of Emin< E < E saddle when ∆ϕ= 0 [2K(k)]. This means that the coupling torque acts as a damping (an antidamping) torque when the phase differenceis in-phase(antiphase). We alsonote thatWT s= 0 when ∆ ϕ=K(k); i.e., the phase difference is a quarter of a period. The calculations necessary to obtain these specific values of WT sare also summarized in Appendix D. We note that the sign changeof WT swith respect to the phase difference is related to the fact that the coupling torque in the transverse geometry, Eq. (14), has the angular dependence of mℓ′xmℓ′ymℓ×(ex×mℓ). Because of this angular dependence, the coupling torque acts as an anti-damping (a damping) torque when mℓx andmℓ′xhave the opposite (same) signs, resulting in the increase (decrease) of the energy supplied to the ferro- magnets by the coupling torque. In summary, the work done by the coupling torque, WT s, isnegativeandminimizedatthe in-phase(∆ ϕ= 0), zero for ∆ ϕ=K(k), and positive and maximized at the antiphase [∆ ϕ= 2K(k)].32 30 28 26 6.05.04.03.02.0 01.02.03.04.0current density, J 0(E) (106 A/cm 2)(a) frequnecy (GHz) phase difference, Δφ/K(k) 31 29 27 25current density, J 0(E) (106 A/cm 2)(c) (d) phase difference, Δ φ/K(k)0 1.0 2.0 3.0 4.027.0127.0227.03current density, J 0(E) (106 A/cm 2) 6.05.04.03.02.0 01.02.03.04.0 frequnecy (GHz) phase difference, Δφ/K(k)(b) phase difference, Δ φ/K(k)0 1.0 2.0 3.0 4.028.0028.1028.2028.30current density, J 0(E) (106 A/cm 2) FIG. 7: (a) The current density, J0(E), necessary to excite the self-oscillation in the transverse geometry as a functi on of the oscillation frequency f(E) and the phase difference ∆ ϕ of the magnetizations. The phase difference is in the unit ofK(k). (b) Dependence of J0(E) for the transverse geome- try, on ∆ ϕatf(E) = 4.6 GHz. The dotted line represents J0(E) in the absence of the coupling. (c) The relation among J0(E),f(E), and ∆ ϕin the longitudinal geometry. (d) The current density J0(E) forf(E) = 4.6 GHz in the longitudinal geometry. The current J0in the transverse geometry is defined as J0(E) =2αeMd F /planckover2pi1ϑRN DT, (37) whereNandDTare defined as N=γ/contintegraldisplay dt/bracketleftBig H2 ℓ−(mℓ·Hℓ)2/bracketrightBig , (38) DT=γ/contintegraldisplay dt[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)] +γχ/contintegraldisplay dtmℓ′xmℓ′y[ex·Hℓ−(mℓ·ex)(mℓ·Hℓ)]. (39) The explicit form of Nis obtained from Eq. (33) as N= 4/radicalBigg 4πM−2E/M HK/bracketleftbigg2E MK(k)+HKE(k)/bracketrightbigg .(40) On the other hand, DTfor the in-phase or antiphase is obtained from Eqs. (32) and (36) as DT=2π(HK+2E/M)/radicalbig HK(HK+4πM) ∓πχ(−2E/M)[1+2E/(MHK)]/radicalbig HK(HK+4πM),(41)10 where the double sign means the upper for the in-phase synchronization and the lower for the antiphase synchro- nization. Figure 7(a) shows J0(E) as functions of the oscilla- tion frequency f(E) and the phase difference ∆ ϕ. The current-frequencyrelationinthetransversegeometrycan be obtained from this figure. To reveal the role of the phase difference more clearly, we show J0(E) for a cer- tain value of f(E)(= 4.6GHz) in Fig. 7(b). Note that J0(E) is smaller than that in the absence of the coupling, which is shown by the dotted line, and minimized when ∆ϕ= 2K(k), i.e., theantiphase. Thisisbecausethework done by the coupling torque is positive and maximized at the antiphase. On the other hand, J0(E) is maximized at the in-phase, and is larger than that in the absence of the coupling because the work done by the coupling torqueis negativeand minimized at the in-phase. We no- tice that the phase differences observed in the numerical simulation in Sec. III, i.e., the in-phase and antiphase, correspond to ∆ ϕsatisfying ∂J0 ∂∆ϕ= 0, (42) or equivalently, ∂WT s ∂∆ϕ= 0. (43) In other words, the phase differences observed in the nu- merical simulations correspond to those giving the ex- trema of J0(WT s). C. Longitudinal geometry Let us investigate the theoretical relation between the current and frequency in the longitudinal geometry. In this case, the averaged LLG equation is given by /contintegraldisplay dtdE dt= (1+χ′′)Ws+WL s+Wα,(44) where WsandWαare given by Eqs. (32) and (33). On the other hand, WL srepresenting the energy change due to the longitudinal coupling is defined as WL s=/contintegraldisplay dtγ/planckover2pi1ϑRχJ0 2edFm2 ℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]. (45) For both the in-phase and antiphase, Eq. (45) becomes (see Appendix D) WL s=π/planckover2pi1ϑRχJ0 2edF/radicalbig HK(HK+4πM) ×(HK+2E/M)[4πM(HK−2E/M)−2HK(2E/M)] HK(HK+4πM). (46)Onthe other hand, when the phasedifference isaquarter of a period [∆ ϕ=K(k)], Eq. (45) becomes WL s=π/planckover2pi1ϑRχJ0 edFHK+2E/M HK(HK+4πM)/radicalBigg −2E M/parenleftbigg 4πM−2E M/parenrightbigg . (47) Itshouldbeemphasizedthat WL sisalwayspositiveforan arbitrary phase difference. This is because the coupling torqueinEq. (10)alwaysactsasananti-dampingtorque. We can determine the current density J0(E) satisfying Eq. (34) in the longitudinal geometry, as in the case of the transverse geometry, by replacing DTin Eq. (37) with DL=γ(1+χ′′)/contintegraldisplay dt[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)] +γχ/contintegraldisplay dtm2 ℓ′y[ey·Hℓ−(mℓ·ey)(mℓ·Hℓ)]. (48) The explicit form of DLfor both the in-phase and an- tiphase is obtained from Eqs. (32) and (46) as DL=2π(1+χ′′)(HK+2E/M)/radicalbig HK(HK+4πM) +πχ(HK+2E/M)[4πM(HK−2E/M)−2HK(2E/M)] [HK(HK+4πM)]3/2, (49) whereas that when the phase difference is a quarter of a period is obtained from Eqs. (32) and (47) as DL=2π(1+χ′′)(HK+2E/M)/radicalbig HK(HK+4πM) +2πχ(HK+2E/M) HK(HK+4πM)/radicalBigg −2E M/parenleftbigg 4πM−2E M/parenrightbigg . (50) Figure 7(c) shows J0(E) as functions of f(E) and ∆ϕ. The current-frequency relation in the longitudinal geom- etry can be obtained from this figure. It is noted that J0(E) is always smaller than that in the absence of the coupling because the coupling torque in the longitudinal geometry always points to the anti-damping direction, and therefore, the work done by the coupling torque is always positive. Figure 7(b) shows J0(E) as a function of ∆ϕat a certain value of f(E). As shown, J0(E) has minima at both the in-phase (∆ ϕ= 0) and the antiphase [∆ϕ= 2K(k)], whereas it is maximized when the phase difference is a quarter of a period [∆ ϕ=K(k)]. We again notice that these phase differences found in the numer- ical simulations in Sec. III correspond to ∆ ϕsatisfying ∂J0(E)/∂∆ϕ= 0, or equivalently, ∂WL s ∂∆ϕ= 0. (51)11 D. Phase differences in stable synchronization and fixed points of effective potential Equation (31) describes the slow change of the mag- netic energy in the oscillation state. The magnetization dynamics is regarded as a motion of a point particle in an effective potential given by its right-hand side. Equa- tions (43) and (51) correspond to the stability conditions ofthe point particlein this effective potential. Therefore, the phase difference found in the numerical simulation fi- nally converges to one of these values satisfying Eq. (43) or Eq. (51). Whether the in-phase, antiphase, or the phase difference with a quarter of a period becomes the attractor depends on the higher order terms of the small parameters, as well as the initial states of the magnetiza- tions, asmentioned at the end ofSec. III. This discussion is beyond the scope of this paper. E. Instability threshold At mentioned at the beginning of Sec. III, the in-plane self-oscillation for a single ferromagnet is stabilized when the currentdensity is in the rangeof Jc< J0< J∗, where JcandJ∗are given by Eqs. (19) and (20), respectively. At the end of this section, let us briefly discuss the effect of the coupling on these scaling currents. Let us remindthe readersthat Jcis the currentdensity necessarytodestabilizethemagnetizationinequilibrium, whereas J∗is the current necessary to overcome the en- ergy barrier, Esaddle−Emin. These current densities are theoretically defined as67 Jc= lim E→EminJ0(E), (52) J∗= lim E→EsaddleJ0(E). (53) It is confirmed that Eqs. (19) and (20) are reproduced by substituting Eqs. (32) and (33) in the definition of J0(E) in the absence of the coupling. On the other hand, in the presence of the transverse coupling, it is confirmed from Eq. (36) that a factor [1−(χ/2)] should be multiplied to the denominator of Eq. (19) when the phase difference between the mag- netizations is in-phase, whereas this factor is replaced by [1+( χ/2)] when the phase difference is antiphase. The other scaling current, J∗, is unchanged for these phase differences. In the longitudinal geometry, we see from Eqs. (46) and (47) that the factor (1+ χ+χ′′) should be multiplied to the denominator of Eq. (19) when the phase difference is in-phase, antiphase, or a quarter of a period, whereas, for J∗, the factor becomes 1+χ′′+(χ/2)[4πM/(HK+4πM)] for in-phase and an- tiphase, and 1+ χ′′when the phase difference is a quarter of a period.V. CONCLUSION In conclusion, the coupled magnetization dynamics in the ferromagnets through the spin Hall magnetoresis- tance effect was investigated. The coupling appears in both the longitudinal and transverse directions of the alignment of the ferromagnets. The in-phase or an- tiphase synchronization of the magnetization oscillation was found in the transversegeometry by solving the LLG equation numerically. On the other hand, in addition to them, the synchronization having the phase difference of a quarter of a period is also found in the longitudinal ge- ometry. It wasshownthatthesephasedifferencesdepend on the values of the damping constant and the field-like torque strength. The analytical theory revealing the re- lationamongthe current, frequency, and phasedifference was also developed. It was shown that the phase differ- ences observed in the numerical simulations correspond to that giving the fixed points of the energy supplied by the coupling torque. Acknowledgement The author is grateful to Takehiko Yorozu for his con- tributions to the analytical calculations, and to Hitoshi Kubota, Sumito Tsunegi, Yoichi Shiota, Shingo Tamaru, Tazumi Nagasawa, Kiwamu Kudo, and Yoji Kawamura for valuable discussions. The author is also thankful to SatoshiIba, AurelieSpiesser,AtsushiSugihara, Takahide Kubota, Hiroki Maehara, and Ai Emura for their sup- port and encouragement. This work was supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B) 16K17486. Appendix A: Values of parameters in numerical simulations The exact values of the parameters used in the sim- ulations, evaluated from the parameters found in the experiment34, areϑR= 0.16680863, β=−0.00973617, χ= 0.009525272, χ′=−0.000152995, and χ′′= 0.03516089for g↑↓ i/S= 1.0 nm−2. In the main text, β= −0.01 andβ= +0.01 correspond to β=−0.00973617 andβ= +0.00973617, respectively. Strictly speaking, the change of the value of g↑↓ iaffects not only βbut also other quantities such as ϑR,χ, andχ′. We, how- ever, change the value of βonly in the numerical simu- lation, for simplicity, because the results do not change significantly unless |g↑↓ i/g↑↓ r| ≪1. The LLG equation with these parameters is solved by using the fourth-order Runge-Kutta method from t= 0 tot= 1µs with the time step of ∆ t= 20 fs; i.e., the number of the time mesh isNt= 5×107. The presentsystem has twostable states at mℓ=±ey. For convention, we assume that the magnetizations ini- tially stay near one equilibrium, mℓ= +ey. For in-phase12 synchronizations, such as those shown in Figs. 2(c) and 2(d), the zcomponents are also synchronized with in- phase, i.e., mℓz(t) =mℓ′z(t). On the other hand, for antiphase synchronization shown in, for example in Figs. 2(e) and 2(f), the zcomponents are also synchronized with antiphase, mℓz(t) =−mℓ′z(t). The algorithm evaluating the phase differences shown in Figs. 3 and 5 from the discrete numerical data is as follows. We gathered Ni= 216= 65536 data ofmℓ(t) (ℓ= 1,2,3) from t= (Nt−Ni+1)∆tto t=Nt∆t= 1µs. Then, the averaged periods Tℓof the oscillation of each magnetization were eval- uated from the peaks of mℓ(t) in this time range asTℓ= [(tℓ,a−tℓ,a−1)+···+(tℓ,2−tℓ,1)]/(Nℓ−1) = (tℓ,a−tℓ,1)/(Nℓ−1), where Nℓis the number of the peaks in mℓ(t), whereas tℓ,ais the time corresponding to thea-th peak. Then, the phase difference is evalu- ated as ∆ ϕ=/summationtextN′ a=1|tℓ,a−tℓ′,a|//parenleftbig N′¯T/parenrightbig , where N′= min[Nℓ,Nℓ′] and¯T= (Tℓ+Tℓ′)/2 with (ℓ,ℓ′) = (1,3) for the transverse geometry, whereas that is (1 ,2) for the longitudinal geometry. For the in-phase synchro- nization, this ∆ ϕis zero because tℓ,a=tℓ′,a. When the phase difference is antiphase, ∆ ϕ= 0.50 because |tℓ,a−tℓ′,a|=¯T/2 in this case. Similarly, ∆ ϕis 0.25 when the phase difference is a quarter of a period. Note that the critical current density to excite the self- oscillation, given by Eq. (19), is proportional to the damping constant α. Therefore, the value of the current density should be increased to observe the self-oscillation whenαis varied, as in the case of Figs. 3 and 5. In these figures,J0isassumedas n×30MA/cm2forα=n×0.005 (n= 1−6). The numerical simulation in Fig. 2(e) indicates that the antiphase synchronization is an attractor for β= +0.01. An exception is that if the initial conditions are set to be identical, the final state becomes in-phase syn- chronization due to the symmetry of the LLG equation with respect to the change of ( ℓ,ℓ′)→(ℓ′,ℓ). Since Eq. (43) is satisfied, the phase difference is fixed to in-phase even if it is unstable. Similar situations occur in other cases for such specific initial conditions. Appendix B: Derivation of coupling torque in transverse geometry In a ferromagnetic/nonmagnetic bilayer, the spin cur- rent density flowing in the idirection ( i=x,y,z) with the spin polarization in the νdirection is related to the electrochemical potential ¯ µNand the spin accumulation δµNvia Jsiν,N=−/planckover2pi1σN 2e2∂iδµN,ν−/planckover2pi1ϑσN 2e2ǫiνj∂j¯µN,(B1) where∂j¯µN/eis the electric field in the jdirection, and therefore, σN∂j¯µN/eis the electric current density. Weassume that this equation is extended to J(ℓ) szν,N=−/planckover2pi1σN 2e2∂zδµ(ℓ) N,ν+/planckover2pi1ϑ 2e(J0δνy−jℓ′yδνx),(B2) in the transverse geometry, where J(ℓ) szν,Nis the spin cur- rent density flowing near the F ℓ/N interface in the zdi- rection with the spin polarization in the νdirection. The spin accumulation obeys the diffusion equation, and the boundary conditions of the diffusion equation are given by the spin current density at the boundaries. Using Eq. (B2), the solution of the spin accumulation is given by40 δµ(ℓ) N,ν=2π (gN/S)sinh(dN/λN)/braceleftbigg −JFℓ/N szνcosh/parenleftbiggz+dN λN/parenrightbigg −/planckover2pi1ϑ 2e(J0δνy−jℓ′yδνx)/bracketleftbigg cosh/parenleftbiggz λN/parenrightbigg −cosh/parenleftbiggz+dN λN/parenrightbigg/bracketrightbigg/bracerightbigg , (B3) where we assume that the nonmagnet is in the region of −dN≤z≤0. The spin current density, JFℓ/N szν, at the Fℓ/N interface is given by30,40 JFℓ/N s=/planckover2pi1ϑg∗ 2egNtanh/parenleftbiggdN 2λN/parenrightbigg (J0mℓy−jℓ′ymℓx)mℓ +/planckover2pi1 2eJ0[ϑRmℓ×(ey×mℓ)+ϑIey×mℓ] −/planckover2pi1 2ejℓ′y[ϑRmℓ×(ex×mℓ)+ϑIex×mℓ], (B4) where the vector notation in boldface represents the di- rection of the spin polarization, whereas the spatial di- rection of Eq. (B4) is defined as the positive direction, i.e., from the nonmagnet to the ferromagnet. On the other hand, the electric current density in the nonmagnet flowing in the idirection is given by Jci,N=σN e∂i¯µN−ϑσN eǫijν∂jδµN,ν.(B5) In the present case, the electric current density near the Fℓ/N interface flowing in the ydirection becomes J(ℓ) cy,N=jℓ′y−ϑσN e∂zδµ(ℓ) N,x. (B6) Substituting Eqs. (B3) and (B4) into Eq. (B6), and averaging along the zdirection as J(ℓ) cy,N= (1/dN)/integraltext0 −dNJ(ℓ) cy,Ndz, we find that J(ℓ) cy,N=/parenleftbig 1+χ′′+χm2 ℓx/parenrightbig jℓ′y −(χmℓxmℓy+χ′mℓz)J0.(B7) The conservation law of the electric current along the y direction requires that J(ℓ) cy,N=jℓy+jℓ′y. Solving this equation for ℓ= 1 and 3, we obtain Eq. (13). The spin torque is defined from Eq. (B4) as Tℓ=−γ MdFmℓ×/parenleftBig JFℓ/N s×mℓ/parenrightBig .(B8)13 Appendix C: Analytical solution of magnetization on a constant energy curve In this appendix, we shows the derivation of the ana- lyticalsolutionofthe magnetizationonaconstantenergy curve. Forsimplicity, weremovethesubscript ℓ(= 1,2,3) distinguishing the ferromagnets. The magnetization dy- namics on a constant energy curve is described by the Landau-Lifshitz (LL) equation dm dt=−γm×H. (C1) The magnetic field, Eq. (18), is related to the magnetic energy density EviaE=−M/integraltext dm·H, as mentioned in the main text. Using the relation m2 x+m2 y+m2 z= 1, Eq. (25) is rewritten as m2 z+HK HK+4πMm2 x=2E/M+HK HK+4πM.(C2) Thisequationindicatesthat mzandmxcanbe expressed asmz=v′cosuandmx= (v′/v)sinu, respectively, wherevandv′are defined as v2=HK/(HK+4πM) andv′= (2E/M+HK)/(HK+4πM). Then, du/dt= (du/dsinu)(dsinu/dt) = (1/cosu)[d(v/v′)mx/dt] = (v/mz)(dmx/dt), which becomes, from Eq. (C1), du dt=γ(HK+4πM)vmy. (C3) Introducing new variable w= sinu, this equation gives dw/radicalbig (1−w2)(1−k2w2)=γ(HK+4πM)v/radicalbig 1−v′2dt, (C4) where the modulus kis given by Eq. (30). The mod- ulus monotonically varies from 0 to 1 by changing the energy density Efrom its minimum Eminto saddle Esaddle. We also notice that ( HK+4πM)v√ 1−v′2=/radicalbig HK(4πM−2E/M). Equation (C4) indicates that w is given by w= sn/bracketleftBig γ/radicalbig HK(4πM−2E/M)t+ϕ0,k/bracketrightBig ,(C5) wheresn( u,k)istheJacobiellipticfunction, and ϕ0isthe initial phase determined by the initial condition. Using the relations sn2(u,k) + cn2(u,k) = 1 and dn2(u,k) =/radicalbig 1−k2sn2(u,k), we find that the solution of mon the constant energy curve is given by Eqs. (26), (27), and (28). The peak frequencies of the Fourier transformation of Eq. (21) in the transverse geometry are discussed as fol- lows. We note that Eq. (21) for in-phase synchronization is proportional to mℓx(t)mℓy(t) andmℓz(t). Substituting the following formulas80, sn(u,k) =2π kK(k)∞/summationdisplay m=0qm+1/2 1−q2m+1sin/bracketleftbigg(2m+1)πu 2K(k)/bracketrightbigg , (C6)cn(u,k) =2π kK(k)∞/summationdisplay m=0qm+1/2 1+q2m+1cos/bracketleftbigg(2m+1)πu 2K(k)/bracketrightbigg , (C7) dn(u,k) =π 2K(k) +2π K(k)∞/summationdisplay m=0qm+1 1+q2(m+1)cos/bracketleftbigg(m+1)πu K(k)/bracketrightbigg , (C8) to Eqs. (26), (27), (28), where q= exp/bracketleftbig −πK/parenleftbig√ 1−k2/parenrightbig /K(k)/bracketrightbig , it is found that the peak frequencies of Eq. (21) appear at fn= (2n−1)f0 (n= 1,2,3,···), where the lowest frequency f0is given by Eq. (29). In the longitudinal geometry, Eq. (22) is proportional tom2 1y+m2 2y. When the phase difference of the magne- tizations is in-phase or antiphase, it becomes 2 m2 1y. In this case, using the formula81 dn2(u,k) =E(k) K(k)+2π2 K2(k)∞/summationdisplay m=1mqm 1−q2mcos/bracketleftbiggmπu K(k)/bracketrightbigg , (C9) it is found that the Fourier transformation of Eq. (22) has the peaks at fn= 2nf0. On the other hand, when the phase difference is a quarter of a pe- riod, Eq. (22) is proportional to g(u)≡dn2(u,k) + dn[u+K(k),k] = dn2(u,k) +/bracketleftbig/parenleftbig 1−k2/parenrightbig /dn2(u,k)/bracketrightbig . We notice that g[u+K(k)] =g(u), indicating that the Fourier transformation of Eq. (22) in this case has the peaks at fn= 4nf0. Appendix D: Details of calculations of Eqs. (35) and (45) Equations (35) and (45) can be calculated by substi- tuting the solution of mℓon a constant energy curve to the integrals. As emphasized in the main text, the phase difference ∆ ϕbetween the magnetizations is an impor- tant quantity. According to Eqs. (26), (27), and (28), we setmℓandmℓ′as mℓx=/radicalbigg 1+2E MHKsn/bracketleftbigg4K(k) τ(E)t,k/bracketrightbigg ,(D1) mℓy=/radicalBigg 4πM−2E/M HK+4πMdn/bracketleftbigg4K(k) τ(E)t,k/bracketrightbigg ,(D2) mℓz=/radicalBigg HK+2E/M HK+4πMcn/bracketleftbigg4K(k) τ(E)t,k/bracketrightbigg ,(D3) and mℓ′x=/radicalbigg 1+2E MHKsn/bracketleftbigg4K(k) τ(E)t+∆ϕ,k/bracketrightbigg ,(D4)14 mℓ′y=/radicalBigg 4πM−2E/M HK+4πMdn/bracketleftbigg4K(k) τ(E)t+∆ϕ,k/bracketrightbigg ,(D5) mℓ′z=/radicalBigg HK+2E/M HK+4πMcn/bracketleftbigg4K(k) τ(E)t+∆ϕ,k/bracketrightbigg .(D6) The value of ∆ ϕvaries in the rage of 0 ≤∆ϕ <4K(k). ∆ϕ= 0 corresponds to the in-phase synchronization, whereas ∆ ϕis 2K(k) for the antiphase synchronization. The analytical formulas of Eqs. (35) and (45) for the in-phase and antiphase synchronizations can be ob- tained as follows. First, since the elliptic functions sat- isfy sn[u+2K(k),k] =−sn(u,k), cn[u+2K(k),k] = −cn(u,k), and dn[ u+2K(k),k] = dn( u,k),WT shas the same magnitude but different sign for ∆ ϕ= 0 and ∆ϕ= 2K(k), whereas WL sis the same for the in-phase and antiphase. Therefore, it is sufficient to calculate WT s andWL sfor the in-phase case. In this case, it is unneces- sary to distinguish mℓandmℓ′. Next, it should be noted that Eq. (35) includes the following two integrals, /contintegraldisplay dtm2 ℓxm3 ℓy∝/integraldisplay dusn2(u,k)dn3(u,k),(D7) /contintegraldisplay dtm2 ℓxmℓym2 ℓz∝/integraldisplay dusn2(u,k)cn2(u,k)dn(u,k). (D8) By replacing the integral variable from uwithx= sn(u,k), and noting that du=dx//radicalbig (1−x2)(1−k2x2), these integrals are calculated as /integraldisplay dusn2(u,k)dn3(u,k) =/integraldisplay dxx2(1−k2x2)√ 1−x2 =x√ 1−x2/bracketleftbig −4+k2/parenleftbig 3+2x2/parenrightbig/bracketrightbig +/parenleftbig 4−3k2/parenrightbig sin−1x 8, (D9) /integraldisplay dusn2(u,k)cn2(u,k)dn(u,k) =/integraldisplay dxx2/radicalbig 1−x2 =x√ 1−x2/parenleftbig −1+2x2/parenrightbig +sin−1x 8. (D10) Using these integrals, Eq. (36) is obtained. On the other hand, Eq. (45) includes the following three integrals, /contintegraldisplay dtm3 ℓy∝/integraldisplay dudn3(u,k) =/integraldisplay dx1−k2x2 √ 1−x2 =k2x√ 1−x2+/parenleftbig 2−k2/parenrightbig sin−1x 2,(D11) /contintegraldisplay dtm5 ℓy∝/integraldisplay dudn5(u,k) =/integraldisplay dx/parenleftbig 1−k2x2/parenrightbig2 √ 1−x2 =k2x√ 1−x2/bracketleftbig 8−k2/parenleftbig 3+2x2/parenrightbig/bracketrightbig +/parenleftbig 8−8k2+3k4/parenrightbig sin−1x 8, (D12)/contintegraldisplay dtm3 ℓym2 ℓz∝/integraldisplay dudn3(u,k)cn2(u,k) =/integraldisplay dx/radicalbig 1−x2/parenleftbig 1−k2x2/parenrightbig =x√ 1−x2/bracketleftbig 4+k2/parenleftbig 1−2x2/parenrightbig/bracketrightbig +/parenleftbig 4−k2/parenrightbig sin−1x 8. (D13) Using these integrals, Eq. (46) is obtained. When the phase difference is a quarter of a period (∆ ϕ=K(k)), the relations, sn[u+K(k),k] = cn(u,k)/dn(u,k), cn[u+K(k),k] = −√ 1−k2cn(u,k)/dn(u,k), dn[ u+K(k),k] =√ 1−k2/dn(u,k), are used to evaluate Eqs. (35) and (45). 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2017-01-12

The synchronized magnetization dynamics in ferromagnets on a nonmagnetic heavy metal caused by the spin Hall effect is investigated theoretically. The direct and inverse spin Hall effects near the ferromagnetic/nonmagnetic interface generate longitudinal and transverse electric currents. The phenomenon is known as the spin Hall magnetoresistance effect, whose magnitude depends on the magnetization direction in the ferromagnet due to the spin transfer effect. When another ferromagnet is placed onto the same nonmagnet, these currents are again converted to the spin current by the spin Hall effect and excite the spin torque to this additional ferromagnet, resulting in the excitation of the coupled motions of the magnetizations. The in-phase or antiphase synchronization of the magnetization oscillations, depending on the value of the Gilbert damping constant and the field-like torque strength, is found in the transverse geometry by solving the Landau-Lifshitz-Gilbert equation numerically. On the other hand, in addition to these synchronizations, the synchronization having a phase difference of a quarter of a period is also found in the longitudinal geometry. The analytical theory clarifying the relation among the current, frequency, and phase difference is also developed, where it is shown that the phase differences observed in the numerical simulations correspond to that giving the fixed points of the energy supplied by the coupling torque.

Dynamic coupling of ferromagnets via spin Hall magnetoresistance

1701.03201v2

arXiv:1003.3769v1 [cond-mat.mes-hall] 19 Mar 2010Dynamics of magnetization on the topological surface Takehito Yokoyama1, Jiadong Zang2,3, and Naoto Nagaosa2,4 1Department of Physics, Tokyo Institute of Technology, Toky o 152-8551, Japan 2Department of Applied Physics, University of Tokyo, Tokyo 1 13-8656, Japan 3Department of Physics, Fudan University, Shanghai 200433, China 4Cross Correlated Materials Research Group (CMRG), ASI, RIK EN, WAKO 351-0198, Japan (Dated: October 17, 2018) We investigate theoretically the dynamics of magnetizatio n coupled to the surface Dirac fermions of athree dimensional topological insulator, byderiving t heLandau-Lifshitz-Gilbert (LLG) equation in the presence of charge current. Both the inverse spin-Gal vanic effect and the Gilbert damping coefficient αare related to the two-dimensional diagonal conductivity σxxof the Dirac fermion, while the Berry phase of the ferromagnetic moment to the Hall conductivity σxy. The spin transfer torque and the so-called β-terms are shown to be negligibly small. Anomalous behavior s in various phenomena including the ferromagnetic resonance are predi cted in terms of this LLG equation. PACS numbers: 73.43.Nq, 72.25.Dc, 85.75.-d Topologicalinsulator(TI) providesa new state of mat- ter topologically distinct from the conventional band in- sulator[1]. In particular,the edge channelsorthe surface states are described by Dirac fermions and protected by the band gap in the bulk states, and backward scatter- ing is forbidden by the time-reversal symmetry. From the viewpoint of the spintronics, it offers a unique op- portunity to pursue novel functions since the relativistic spin-orbit interaction plays an essential role there. Actu- ally, several proposals have been made such as the quan- tized magneto-electric effect [2], giant spin rotation [3], magneto-transport phenomena [4], and superconducting proximity effect including Majorana fermions [5–7]. Also, a recent study focuses on the inverse spin- Galvanic effect in a TI/ferromagnet interface, predicting the current-induced magnetization reversal due to the Hall current on the TI [8]. In Ref. [8], the Fermi energy is assumed to be in the gap of the Dirac dispersion opened by the exchange coupling. In this case, the quantized Hall liquid is realized, and there occurs no dissipation coming from the surface Dirac fermions. However, in realistic systems, it is rather difficult to tune the Fermi energy in the gap since the proximity- inducedexchangefieldisexpectedtobearound5-50meV. Therefore, it is important to consider the generic case where the Fermi energy is at the finite density of states of Dirac fermions, where the diagonal conductivity is much larger than the transverse one, and the damping of the magnetizationbecomes appreciable. Related systems are semiconductors and metals with Rashba spin-orbit interaction, where the spin-Galvanic effect and current induced magnetization reversal have been predicted [9] and experimentally observed [10, 11]. Compared with these systems where the Rashba coupling constant is a key parameter, the spin and momentum in TI is tightly related to each other corresponding to the strong cou- pling limit of spin-orbit interaction, and hence the gigan- tic spin-Galvanic effect is expected.
                  FIG. 1: (Color online) (a) Illustration of the Dirac dispers ion on top of TI. The Fermi level εFis far above the surface gap opened by magnetization in the ferromagnetic layer. (b) Sketch of FMR experiment in the soft magnetic layer. The substrate in the figure is TI, which is capped by a layer of soft ferromagnet. The magnetization precesses around the external magnetic field Heff. In this letter, we study the dynamics of the magnetiza- tion coupled to the surface Dirac fermion of TI. Landau- Lifshitz-Gilbert (LLG) equationin the presenceofcharge current is derived microscopically, and (i) inverse spin- Galvanic effect, (ii) Gilbert damping coefficient α, (iii) theso-called β-terms, and(iv)thecorrectiontotheBerry phase, are derived in a unified fashion. It is found that these are expressed by relatively small number of param- eters, i.e., the velocity vF, Fermi wave number kF, ex- change coupling M, and the transport lifetime τof the Dirac fermions. It is also clarified that the terms re- lated to the spatial gradient are negligibly small when the surface state is a good metal. With this LLG equa- tion, we propose a ferromagnetic resonance (FMR) ex- periment, wheremodificationsoftheresonancefrequency and Gilbert damping are predicted. Combined with the transport measurement of the Hall conductivity, FMR provide several tests of our theory.2 Derivation of LLG equation. — By attaching a ferro- magnet on the TI as shown in Fig. 1, we can consider a topological surface state where conducting electrons in- teract with localized spins, S, through the exchange field Hex=−M/integraldisplay drn(r)·ˆσ(r). (1) Here, we set S=Snwith a unit vector npointing in the direction of spin, ˆσ(r) =c†(r)σc(r) represents (twice) the electronspindensity, with c†= (c† ↑,c† ↓) beingelectron creation operators, σthe Pauli spin-matrix vector, and Mbeing the exchange coupling energy. The total Hamil- tonian of the system is given by Htot=HS+Hel+Hex, whereHSandHelare those for localized spins and con- ducting electrons, respectively. The dynamics of magnetization can be described by the LLG equation ˙n=γ0Heff×n+α0˙n×n+t′ el, (2) whereγ0Heffandα0are an effective field and a Gilbert damping constant, respectively, both coming from HS. Effects of conducting electrons are contained in the spin torque tel(r)≡s0t′ el(r) =Mn(r)×∝angbracketleftˆσ(r)∝angbracketrightne,(3) which arises from Hex. Here, s0≡S/a2is the local- ized spin per area a2. In the following, we thus calculate spin polarization of conducting electrons perpendicular ton,∝angbracketleftˆσ⊥(r)∝angbracketrightne, in such nonequilibrium states with cur- rent flow and spatially varying magnetization to derive theβ-term, or with time-dependent magnetization for Gilbert damping. Here and hereafter, ∝angbracketleft···∝angbracketrightnerepresents statistical average in such nonequilibrium states. Following Refs. [12–14] we consider a small transverse fluctuation, u= (ux,uy,0),|u| ≪1, around a uniformly magnetized state, n= ˆz, such that n= ˆz+u. In the ‘unperturbed’ state, n= ˆz, the electrons are described by the Hamiltonian H0=/summationdisplay kvF(kyσx−kxσy)−Mσz−εF+Vimp(4) whereVimpis the impurity potential given by Vimp= u/summationtext iδ(r−Ri) in the first-quantization form. We take a quenched average for the impurity positions Ri. The electron damping rate is then given by γ= 1/(2τ) = πniu2νFin the first Born approximation. Here, niis the concentration of impurities, and νF=εF/(2πv2 F) is the density of states at εF. We assume that γ≪vFkF=/radicalbig ε2 F−M2,M, and calculate spin transfer torque in the lowest non-trivial order. In the presence of u(r,t) =u(q,ω)ei(q·r−ωt), the con- ducting electrons feel a perturbation (note that Hel+ Hex=H0+H1) H1=−M/summationdisplay kσc† k+qσck·u(q,ω)e−iωt,(5)and acquires a transverse component ∝angbracketleftˆσ′α ⊥(q,ω)∝angbracketrightne=Mχαβ ⊥(q,ω+i0)uβ(q,ω) (6) in the first order in uin the momentum and frequency representation. Here, χαβ ⊥is the transverse spin suscep- tibility in a uniformly magnetized state with α,β=x,y, and summing over βis implied. Now, we study the ω-linear terms in the uniform ( q= 0) part of the transverse spin susceptibility, χαβ ⊥(q= 0,ω+i0). We make the following transformation of the operator: c=U˜c=1/radicalbig 2ε(ε+M)/parenleftbigg vF(ky+ikx) ε+M/parenrightbigg ˜c(7) withε=/radicalbig (vFk)2+M2. Note U†U= 1,U†σxU= vFky/ε,andU†σyU=−vFkx/ε. This transformation maps two component operator cinto one component op- erator on the upper Dirac cone ˜ c. With this new op- erator, we calculate the transverse spin susceptibility in Matsubara form χαβ ⊥(0,iωλ) =/integraldisplayβ 0dτeiωλτ/angbracketleftbig Tτσα(0,τ)σβ(0,0)/angbracketrightbig =−T/summationdisplay k,nU†σαU˜G(k,iεn+iωλ)U†σβU˜G(k,iεn) (8) with˜G(k,iεn) = (iεn−ε+εF+iγsgn(εn))−1. By sym- metry consideration of the integrand in k-integral, we findχαβ ⊥(0,iωλ)∝δαβ. After some calculations, we ob- tain the torque stemming from the time evolution: tα el=M2iω 2π1 2v2 F/parenleftbiggvFkF εF/parenrightbigg2 εFτn×u (9) =1 2/parenleftbiggMvFkF εF/parenrightbigg2 νFτ˙ n×n.(10) This result fits the conventional Gilbert damping with α=1 2/parenleftbiggMvFkF εF/parenrightbigg2 νFτa2 ¯hS. (11) We next examine the case of finite current by applying a d.c. electric field E, and calculate a linear response of σα ⊥toE, i.e.,< σα ⊥(q)>ne=Kα i(q)Ei. First, it is clear thatKα i(q=0) =−εiασxx/(evF) where εiαandσxx are the anti-symmetric tensor and diagonal conductivity, respectively, because electron’s spin is ”attached” to its momentum. This representsthe inversespin-Galvanicef- fect, i.e., chargecurrentinduces magneticmoment. Since we assume that Fermi level is far away from the surface gap,σxx≫σxywhereσxyis the Hall conductivity. The dominant term in χis thusχxy∝σxx. This is quite different from the case studied in Ref. [8], where Fermi level lies inside the surface gap and therefore σxxis van- ishing. Hence, the only contribution to the inverse spin- Galvanic effect is χxx∝σxy, which is much smaller than3 the effect proposed in this letter. Compared with the in- verse spin-Galvanic effect in Rashba system [9–11], this effect is much stronger since the small Rashba coupling constant, i.e., the small factor αRkF/EFin Eq. (16) of Ref. [9], does not appear in the present case. Taking into account the realistic numbers with α= 10−11eVmandvF= 3×105m/s, onefindsthat theinversespin-Galvanic effect in the present system is ∼50times largerthan that in Rashba systems. The next leading order terms of the expansion in uβ andqjcan be obtained by considering the four-point ver- tices [12] as ∝angbracketleftˆσα ⊥(q)∝angbracketrightne=−eMπ 45i 8πε2 Fεikεjl[δαβδkl+δαkδβl+δαlδβk]qjuβEi (12) =−eM5i 32ε2 F[q·Euα−q·(u׈z)(E׈z)α+u·(E׈z)(q׈z)α]. (13) Therefore, the spin torque steming from the spatial gradient has the form: tβ el=−β1 2e[n×(j·∇)n−(j−(j·n)ˆz)∇·(n׈z)+(∇−(n·∇)ˆz)n·(j׈z)] (14) wherej=σCEwith charge current jand conductivity σC=e2 4π/parenleftBig vFkF εF/parenrightBig2 εFτ. and β=5π 4εFτ/parenleftbiggM vFkF/parenrightbigg2 . (15) From Eq.(14), one can find the followings: (i) The spin transfer torque of the form ( j·∇)nis missing since we consider the upper Dirac cone only. (ii) The β- term has a form essentially different from that in the conventioal one.[12, 15, 16] In contrast to the conven- tionalferromagnet,[12] thisconstantcomesfromthe non- magnetic impurity. Considering vFkF∼=εF, we get α/β∼(εFτ)2from Eqs. (11) and (15). Therefore, theβ-terms are negligible for a good surface metal, i.e., εFτ≫1. Up to now, we consider only one branch of the band where the Fermi energy is sitting. When we consider the 2-band structure, i.e., the 2 ×2 matrix Hamiltonian H= vF[(ky+Mnx vF)σx−(kx−Mny vF)σy], we have the correctionto the Berry phase term. In analogy with the minimal coupling of electromagnetic field, A=−M evF(−ny,nx) plays the same role as the U(1) gauge. By integrating the fermions out, one can get a Chern-Simons term in termsofthemagnetization LCS=σxyǫµνρAµ∂νAρwhere µ,ν,ρ=t,x,y. When the gradient of magnetization van- ishes, it can be rewritten as LCS=σxy(M evF)2(nx˙ny−ny˙nx).(16) This additional term can be interpreted as an additional Berryphase for the magnetization. In fact, as nzremains constant in the present case, we have [ nx,ny] =inz. Therefore, nxandnybecome conjugate variables up to a factor, which naturally leads to a Berry phase: nx˙ny−ny˙nx. This term is exactly equivalent to the Chern-Simons term. Including all the terms derived above, we finally arrive at a modified LLG equation: ˙n−2σxy(M evF)2˙n/(s0N) =γ0Heff×n+/parenleftbiggM evFs0N/parenrightbigg (−j+(n·j)ˆz)+(α0+α/N)˙n×n+tβ el/(s0N) (17) whereNisthenumberofferromagneticlayers. Notethat α-,β- andBerryphaseterms originatefromthe interplay between Dirac fermions and local magnetization which persists over a few layers of the ferromagnet. Therefore, the overall coefficients are divided by the number of fer- romagnetic layers N.Ferromagnetic resonance. —Observingthe smallvalue ofβ, the spatial gradient of magnetization can be ne- glected for the time being. Only one uniform domain in the absence of current is taken into account for simplic- ity. Without loss of generality, assume that an external magnetic field is applied along zdirection, and consider4 the ferromagnet precession around that field. ˙ nz= 0 is kept in the first order approximation, namely nzis a constant in the time evolution. By inserting the ansatz nx(y)(t) =nx(y)e−iωtinto the modified LLG equation, one obtains ℜω=ξ ξ2+η2ω0,ℑω=−η ξ2+η2ω0(18) whereη= (α0+α/N),ω0=γ0Heffandξ= 1− 2σxy(M evF)2/(s0N). Expanding up to the first order in σxyandη, one gets ℜω=ω0+ 2σxy(M evF)2ω0/(s0N) andℑω=ηω0. Therefore, the precession frequency ac- quires a shift proportional to σxyin the presence of in- terplay between Dirac fermions and the ferromagnetic layer. The relative shift of ℜωis 2σxy(M evF)2ω0/(s0N) = 1 πSNM εF(Ma vF)2∼1 N(M εF)3[17]. By tuning the Fermi level, this shift can be accessible experimentally. Meanwhile, the Gilbert damping constant αcan be measured directly without referring to the theoretical expression in Eq. (11). One can investigate the fer- romagnetic layer thickness dependence of FMR line- width. While increasing the thickness Nof ferromagnet, the Gilbert damping constant stemming from the Dirac fermions decreases inversely proportional to the thick- ness. Taking into account the realistic estimation with εFτ∼100 and M/εF∼0.3, one has α/s0∼1, while α0∼0.001 usually. Therefore, even for a hundred of lay- ers of ferromagnet, the contribution from the proximity effect is still significant compared to the one coming from theferromagnetitself. Observingthattheimaginarypart of resonance frequency in Eq. (18) is proportional to η, one may plot the relation between the FMR peak broad- ening, namely ℑω, and 1/N. The broadening is a linear function of 1 /N, and approaches the value of the ferro- magnet at large thickness limit. We can find the value of αfrom the slope of the plot. On the other hand, the real part of FMR frequency provides rich physics as well. Since in the presence of additional Berry phase, the frequency shift is propor- tional to the Hall conductivity on the surface of TI, it leads to a new method to measure the Hall conductiv- ity without four-terminal probe. In an ideal case when the Fermi level lies inside the surface gap, this quantity is quantized as σ0 xy=e2 2h. However, in realistic case, Fermi level is away from the surface gap, and therefore the Hall conductivity is reduced to σxy=e2 2hMnz εF[17]. As a result, the shift of resonance frequency is proportional ton2 z∝cos2θ, and the FMR isotropy is broken. Here, θis the angle between effective magnetic field and the normal to the surface of TI. One can perform an angle resolved FMR measurement. The signal proportional to cos2θcomes from additional Berry phase. Since parameters αandβdepend on Mandτ, it is quite important to measure these quantities directly.Molecular-beam epitaxy method can be applied to grow TI coated by a thin layer of soft ferromagnet. As is re- quired in the above calculation, Fermi level of TI should lie inside the bulk band gap. Also, the soft ferromag- net should be an insulator or a metal with proper work function. One may employ angular resolved photoemis- sion spectroscopy(ARPES) or scanning tunneling micro- scope techniques to measure the surface gap ∆ opened by the ferromagnet, which is given by ∆ = Mnz. As the easy axis nzcan be found experimentally, Mcan be fixed as well. On the other hand, the lifetime τis indirectly determined by measuring the diagonal conductivity σxx viaσxx=e2 4π/parenleftBig vFkF εF/parenrightBig2 εFτ. Finally, Fermi surface can be determined by ARPES, and all parameters in LLG equation Eq.(17) can be obtained. In summary, we have investigated theoretically the dynamics of magnetization on the surface of a three dimensional topological insulator. We have derived the Landau-Lifshitz-Gilbert equation in the presence of charge current, and analyzed the inverse spin-Galvanic effect and ferromegnetic resonance predicting anomalous features of these phenomena. This work is supported by Grant-in-Aid for Scientific Research (Grants No. 17071007, 17071005, 19048008 19048015, and 21244053) from the Ministry of Educa- tion, Culture, Sports, Science and Technology of Japan. [1] M. Z. Hasan and C. L. Kane, arXiv:1002.3895; X. L. Qi and S. C. Zhang, Physics Today, 63, 33 (2010) and references therein. [2] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008). [3] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. Lett.102, 166801 (2009). [4] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. B 81, 121401(R) (2010). [5] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008). [6] Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev. Lett.103, 107002 (2009). [7] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbø, and N. Nagaosa, Phys. Rev. Lett. 104, 067001 (2010). [8] I. Garate and M. Franz, arXiv:0911.0106. [9] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). [10] A. Chernyshov et al., Nature Phys. 5, 656 (2009). [11] I.M. Miron et al., Nature Materials 9, 230 (2010). [12] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006). [13] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W. Bauer, Phys. Rev. B 74, 144405 (2006); Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn. Magn. Mater. 320, 1282 (2008). [14] Y. Tserkovnyak, G.A. Fiete and B.I. Halperin, Appl. Phys. Lett. 84, 5234 (2004). [15] Clement H. Wong and Y. Tserkovnyak, Phys. Rev. B 80,5 184411 (2009). [16] S. Zhang and Steven S.-L. Zhang, Phys. Rev. Lett. 102, 086601 (2009).[17] J. Zang, and N. Nagaosa, arXiv:1001.1578

2010-03-19

We investigate theoretically the dynamics of magnetization coupled to the surface Dirac fermions of a three dimensional topological insulator, by deriving the Landau-Lifshitz-Gilbert (LLG) equation in the presence of charge current. Both the inverse spin-Galvanic effect and the Gilbert damping coefficient $\alpha$ are related to the two-dimensional diagonal conductivity $\sigma_{xx}$ of the Dirac fermion, while the Berry phase of the ferromagnetic moment to the Hall conductivity $\sigma_{xy}$. The spin transfer torque and the so-called $\beta$-terms are shown to be negligibly small. Anomalous behaviors in various phenomena including the ferromagnetic resonance are predicted in terms of this LLG equation.

Dynamics of magnetization on the topological surface

1003.3769v1

arXiv:0805.0147v1 [nlin.CD] 1 May 2008Chaotic Spin Dynamics of a Long Nanomagnet Driven by a Current Yueheng Lan and Y. Charles Li Abstract. We study the spin dynamics of a long nanomagnet driven by an electrical current. In the case of only DC current, the spin d ynamics has a sophisticated bifurcation diagram of attractors. One type of attractors is a weak chaos. On the other hand, in the case of only AC current, t he spin dynamics has a rather simple bifurcation diagram of attract ors. That is, for small Gilbert damping, when the AC current is below a critica l value, the attractor is a limit cycle; above the critical value, the att ractor is chaotic (turbulent). For normal Gilbert damping, the attractor is a lways a limit cycle in the physically interesting range of the AC current. We als o developed a Melnikov integral theory for a theoretical prediction on t he occurrence of chaos. Our Melnikov prediction seems performing quite well in the DC case. In the AC case, our Melnikov prediction seems predicting tra nsient chaos. The sustained chaotic attractor seems to have extra support from parametric resonance leading to a turbulent state. Contents 1. Introduction 2 2. Mathematical Formulation 3 3. Isospectral Integrable Theory for the Heisenberg Equation 4 4. A Melnikov Function 14 5. Numerical Simulation 17 6. Appendix: The Connection Between the Heisenberg Equation and the NLS Equation 23 References 25 1991Mathematics Subject Classification. Primary 35, 65, 37; Secondary 78. Key words and phrases. Magnetization reversal, spin-polarized current, chaos, D arboux transformation, Melnikov function. c/circlecopyrt2008 (copyright holder) 12 YUEHENG LAN AND Y. CHARLES LI 1. Introduction The greatest potential of the theory of chaos in partial different ial equations lies in its abundant applications in science and engineering. The variety of the spe- cific problems demands continuing innovation of the theory [ 17] [16] [18] [23] [24] [15] [20] [21]. In these representative publications, two theories were develop ed. The theory developed in [ 17] [16] [18] involves transversal homoclinic orbits, and shadowing technique is used to prove the existence of chaos. This t heory is very complete. The theory in [ 23] [24] [15] [20] [21] deals with Silnikov homoclinic orbits, and geometric construction of Smale horseshoes is employe d. This theory is not very complete. The main machineries for locating homoclinic orbit s are (1). Darbouxtransformations,(2). Isospectraltheory,(3). Per sistenceofinvariantman- ifolds and Fenichel fibers, (4). Melnikov analysis and shooting techn ique. Overall, the two theories on chaos in partial differential equations are resu lts of combining Integrable Theory, Dynamical System Theory, and Partial Differe ntial Equations [19]. In this article, we are interested in the chaotic spin dynamics in a long n ano- magnet diven by an electrical current. We hope that the abundant spin dynamics revealed by this study can generate experimental studies on long n anomagets. To illustrate the general significance of the spin dynamics, in particular the magneti- zation reversal issue, we use a daily example: The memory of the har d drive of a computer. The magnetization is polarized along the direction of the e xternal mag- netic field. By reversing the external magnetic field, magnetization reversal can be accomplished; thereby, generating 0 and 1 binary sequence and accomplishing memory purpose. Memory capacity and speed via such a technique h ave reached their limits. The “bit” writing scheme based on such Oersted-Maxwell magnetic field(generatedbyanelectricalcurrent)encountersfundamen talproblemfromclas- sical electromagnetism: the long range magnetic field leads to unwan ted writing or erasing of closely packed neighboring magnetic elements in the extre mely high den- sity memory device and the induction laws place an upper limit on the mem ory speed due to slow rise-and-decay-time imposed by the law of inductio n. Discovered by Slonczewski [ 35] and Berger [ 1], electrical current can directly apply a large torque to a ferromagnet. If electrical current can be directly ap plied to achieve magnetization reversal, such a technique will dramatically increase t he memory capacity and speed of a hard drive. The magnetization can then be s witched on the scale of nanoseconds and nanometers [ 39]. The industrial value will be tremendous. Nanomagnets driven by currents has been intensive ly studied recently [13, 10, 34, 7, 14, 12, 33, 8, 39, 11, 32, 37, 38, 4, 3, 26, 27, 29, 4 2, 31]. The researches have gone beyond the original spin valve system [ 35] [1]. For instance, current driven torques have been applied to magnetic tunnel junc tions [36] [6], di- lute magnetic semiconductors [ 40], multi-magnet couplings [ 10] [14]. AC currents were also applied to generate spin torque [ 34] [7]. Such AC current can be used to generate the external magnetic field [ 34] or applied directly to generate spin torque [7]. Mathematically, the electrical current introduces a spin torque fo rcing term in the conventional Landau-Lifshitz-Gilbert (LLG) equation. The A C current can induce novel dynamics of the LLG equation, like synchronization [ 34] [7] and chaos [25] [41]. Both synchronization and chaos are important phenomena to und erstand before implementing the memory technology. In [ 25] [41], we studied the dynamicsCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 3 of synchronization and chaos for the LLG equation by ignoring the e xchange field (i.e. LLG ordinary differential equations). When the nanomagnetic d evice has the same order of length along every direction, exchange field is not impo rtant, and we have a so-called single domain situation where the spin dynamics is gove rned by the LLG ordinary differential equations. In this article, we will study what we call “long nanomagnet” which is much longer along one direction than othe r directions. In such a situation, the exchange field will be important. This leads to a LLG partial differential equation. In fact, we will study the case where the exchange field plays a dominant role. The article is organized as follows: Section 2 presents the mathemat ical formu- lation of the problem. Section 3 is an integrable study on the Heisenbe rg equation. Based upon Section 3, Section 4 builds the Melnikov integral theory f or predicting chaos. Section 5 presents the numerical simulations. Section 6 is an appendix to Section 3. 2. Mathematical Formulation To simplify the study, we will investigate the case that the magnetiza tion de- pends on only one spatial variable, and has periodic boundary condit ion in this spatial variable. The application of this situation will be a large ring sha pe nano- magnet. Thus, we shall study the following forced Landau-Lifshitz -Gilbert (LLG) equation in the dimensionless form, (2.1)∂tm=−m×H−ǫαm×(m×H)+ǫ(β1+β2cosω0t)m×(m×ex), subject to the periodic boundary condition (2.2) m(t,x+2π) =m(t,x), wheremis a unit magnetization vector m= (m1,m2,m3) in which the three components are along( x,y,z) directions with unit vectors( ex,ey,ez),|m|(t,x) = 1, the effective magnetic field Hhas several terms H=Hexch+Hext+Hdem+Hani =∂2 xm+ǫaex−ǫm3ez+ǫbm1ex, (2.3) whereHexch=∂2 xmis the exchange field, Hext=ǫaexis the external field, Hdem=−ǫm3ezisthe demagnetizationfield, and Hani=ǫbm1existhe anisotropy field. For the materials of the experimental interest, the dimension less parameters are in the ranges a≈0.05, b≈0.025, α≈0.02, β1∈[0.01,0.3], β2∈[0.01,0.3] ; (2.4) andǫis a small parameter measuring the length scale of the exchange field . One can also add an AC current effect in the external field Hext, but the results on the dynamics are similar. Our goal is to build a Melnikov function for the LLG equation around do main walls. The roots of such a Melnikov function provide a good indication o f chaos. For the rest of this section, we will introduce a few interesting nota tions. The Pauli matrices are: (2.5)σ1=/parenleftbigg0 1 1 0/parenrightbigg , σ2=/parenleftbigg0−i i0/parenrightbigg , σ3=/parenleftbigg1 0 0−1/parenrightbigg .4 YUEHENG LAN AND Y. CHARLES LI Let (2.6) m+=m1+im2, m−=m1−im2, i.e.m+=m−. Let (2.7) Γ = mjσj=/parenleftbiggm3m− m+−m3/parenrightbigg . Thus, Γ2=I(the identity matrix). Let ˆH=−H−αm×H+βm×ex,Π =/parenleftbiggˆH3ˆH1−iˆH2 ˆH1+iˆH2−ˆH3/parenrightbigg . Then the LLG can be written in the form (2.8) i∂tΓ =1 2[Γ,Π], where [Γ,Π] = ΓΠ −ΠΓ. 3. Isospectral Integrable Theory for the Heisenberg Equati on Settingǫtozero,theLLG(2.1)reducestotheHeisenbergferromagneteq uation, (3.1) ∂tm=−m×mxx. Using the matrix Γ introduced in (2.7), the Heisenberg equation (3.1) has the form (3.2) i∂tΓ =−1 2[Γ,Γxx], where the bracket [ ,] is defined in (2.8). Obvious constants of motion of the Heisenberg equation (3.1) are the Hamiltonian, 1 2/integraldisplay2π 0|mx|2dx , the momentum,/integraldisplay2π 0m1m2x−m2m1x 1+m3dx , and the total spin,/integraldisplay2π 0mdx . The Heisenberg equation (3.1) is an integrable system with the followin g Lax pair, ∂xψ=iλΓψ , (3.3) ∂tψ=−λ 2(4iλΓ+[Γ,Γx])ψ , (3.4) whereψ= (ψ1,ψ2)Tis complex-valued, λis a complex parameter, Γ is the matrix defined in (2.7), and [Γ ,Γx] = ΓΓ x−ΓxΓ. In fact, there is a connection be- tween the Heisenberg equation (3.1) and the 1D integrable focusing cubic nonlinear Schr¨ odinger (NLS) equation via a nontrivial gauge transformatio n. The details of this connection are given in the Appendix.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 5 3.1. A Simple Linear Stability Calculation. As shown in the Appendix, the temporally periodic solutions of the NLS equation correspond to the domain walls of the Heisenberg equation. Consider the domain wall Γ0=/parenleftbigg0e−iξx eiξx0/parenrightbigg , ξ∈Z; i.e.m1= cosξx, m 2= sinξx, m 3= 0, which is a fixed point of the Heisenberg equation. Linearizing the Heise nberg equa- tion at this fixed point, one gets i∂tΓ =−1 2[Γ0,Γxx]−1 2[Γ,Γ0xx]. Let Γ =/parenleftbigg m3 e−iξx(m1−im2) eiξx(m1+im2) −m3/parenrightbigg , we get ∂tm1= 0, ∂tm2=m3xx+ξ2m3, ∂tm3=−m2xx−2ξm1x. Let mj=∞/summationdisplay k=0(m+ jk(t)coskx+m− jk(t)sinkx), wherem± jk(t) =c± jkeΩt,c± jkand Ω are constants. We obtain that (3.5) Ω =/radicalbig k2(ξ2−k2) which shows that only the modes 0 <|k|<|ξ|are unstable. Such instability is called a modulational instability, also called a side-band instability. Comp aring the Heisenberg ferromagnet equation (3.1) and the Landau-Lifsh itz-Gilbert equa- tion (2.1), we see that if we drop the exchange field Hexch=∂2 xmin the effective magnetic field H(2.3), such a modulational instability will disappear, and the Landau-Lifshitz-Gilbert equation (2.1) reduces to a system of thr ee ordinary differ- ential equations, which has no chaos as verified numerically. Thus th e modulational instability is the source of the chaotic magnetization dynamics. In terms of m± jk, we have d dtm± 1k= 0,d dtm± 2k= (ξ2−k2)m± 3k,d dtm± 3k=k2m± 2k∓2ξkm∓ 1k. Choosingξ= 2, we have for k= 0, m∓ 10 m± 20 m± 30 =c1 1 0 0 +c2 0 1 0 +c3 0 4t 1 ; fork= 1, m∓ 11 m± 21 m± 31 =c1 1 ±4 0 +c2 0√ 3 1 e√ 3t+c3 0 −√ 3 1 e−√ 3t; fork= 2, m∓ 12 m± 22 m± 32 =c1 0 0 1 +c2 1 0 ∓8t +c3 0 1 4t ;6 YUEHENG LAN AND Y. CHARLES LI fork>2, m∓ 1k m± 2k m± 3k =c1 1 ±4/k 0 +c2 0√ k2−4cos(k√ k2−4t) ksin(k√ k2−4t) +c3 0 −√ k2−4sin(k√ k2−4t) kcos(k√ k2−4t) ; wherec1,c2andc3are arbitrary constants. The nonlinear foliation of the above linear modulational instability can b e es- tablished via a Darboux transformation. 3.2. A Darboux Transformation. A Darboux transformation for (3.3)- (3.4) can be obtained. Theorem 3.1. Letφ= (φ1,φ2)Tbe a solution to the Lax pair (3.3)-(3.4) at ( Γ,ν). Define the matrix G=N/parenleftbigg(ν−λ)/ν 0 0 (¯ν−λ)/¯ν/parenrightbigg N−1, where N=/parenleftbigg φ1−φ2 φ2φ1/parenrightbigg . Then ifψsolves the Lax pair (3.3)-(3.4) at ( Γ,λ), (3.6) ˆψ=Gψ solves the Lax pair (3.3)-(3.4) at ( ˆΓ,λ), where ˆΓis given by (3.7) ˆΓ =N/parenleftbigg e−iθ0 0eiθ/parenrightbigg N−1ΓN/parenleftbigg eiθ0 0e−iθ/parenrightbigg N−1, whereeiθ=ν/|ν|. The transformation (3.6)-(3.7) is called a Darboux transformation . This theo- rem can be proved either through the connection between the Heis enberg equation and the NLS equation (with a well-known Darboux transformation) [ 2], or through a direct calculation. Notice also that ˆΓ2=I. Let/parenleftbigg Φ1−Φ2 Φ2Φ1/parenrightbigg =N/parenleftbigg e−iθ0 0eiθ/parenrightbigg N−1 =1 |φ1|2+|φ2|2/parenleftbigge−iθ|φ1|2+eiθ|φ2|2−2isinθ φ1φ2 −2isinθφ1φ2eiθ|φ1|2+e−iθ|φ2|2/parenrightbigg . (3.8) Then (3.9) ˆΓ =/parenleftbigg ˆm3 ˆm1−iˆm2 ˆm1+iˆm2−ˆm3/parenrightbigg , where ˆm+= ˆm1+iˆm2=Φ12(m1+im2)−Φ2 2(m1−im2)+2Φ1Φ2m3, ˆm3=/parenleftbig |Φ1|2−|Φ2|2/parenrightbig m3−Φ1Φ2(m1+im2)−Φ1Φ2(m1−im2).CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 7 One can generate the figure eight connecting to the domain wall, as t he nonlinear foliation of the modulational instability, via the above Darboux trans fomation. 3.3. Figure Eight Connecting to the Domain Wall . LetΓbethedomain wall Γ =/parenleftbigg 0e−i2x ei2x0/parenrightbigg , i.e.m1= cos2x,m2= sin2x, andm3= 0. Solving the Lax pair (3.3)-(3.4), one gets two Bloch eigenfunctions (3.10)ψ=eΩt/parenleftbigg 2λexp{i 2(k−2)x} (k−2)exp{i 2(k+2)x}/parenrightbigg ,Ω =−iλk , k =±2/radicalbig 1+λ2. To apply the Darboux transformation (3.7), we start with the two B loch functions withk=±1, φ+=/parenleftbigg√ 3e−ix ieix/parenrightbigg exp/braceleftBigg√ 3 2t+i1 2x/bracerightBigg , (3.11) φ−=/parenleftbigg−ie−ix√ 3eix/parenrightbigg exp/braceleftBigg −√ 3 2t−i1 2x/bracerightBigg . The wise choice for φused in (3.7) is: (3.12) φ=/radicalbigg c+ c−φ++/radicalbigg c− c+φ−=/parenleftbigg/parenleftbig√ 3eτ+iχ−ie−τ−iχ/parenrightbig e−ix /parenleftbig ieτ+iχ+√ 3e−τ−iχ/parenrightbig eix/parenrightbigg , wherec+/c−= exp{σ+iγ},τ=1 2(√ 3t+σ), andχ=1 2(x+γ). Then from the Darboux transformation (3.7), one gets ˆm1+iˆm2=−ei2x/braceleftbigg 1−2 sech2τcos2χ (2−√ 3 sech2τsin2χ)2/bracketleftbigg sech2τcos2χ +i/parenleftBig√ 3−2 sech2τsin2χ/parenrightBig/bracketrightbigg/bracerightbigg , (3.13) ˆm3=2 sech2τtanh2τcos2χ (2−√ 3 sech2τsin2χ)2. (3.14) Ast→ ±∞, ˆm1→ −cos2x ,ˆm2→ −sin2x ,ˆm3→0. The expressions (3.13)-(3.14) represent the two dimensional figu re eight separatrix connecting to the domain wall ( m+=−ei2x,m3= 0), parametrized by σand γ. See Figure 1 for an illustration. Choosing γ= 0,π, one gets the figure eight curve section of Figure 1. The spatial-temporal profiles correspo nding to the two lobes of the figure eight curve are shown in Figure 2. In fact, the tw o profiles corresponding the two lobes are spatial translates of each other byπ. Inside one of the lobe, the spatial-temporal profile is shown in Figure 3(a). Out side the figure eight curve, the spatial-temporal profile is shown in Figure 3(b). He re the inside and outside spatial-temporal profiles are calculated by using the int egrable finite difference discretization [ 9] of the Heisenberg equation (3.1), (3.15)d dtm(j) =−2 h2m(j)×/parenleftbiggm(j+1) 1+m(j)·m(j+1)+m(j−1) 1+m(j−1)·m(j)/parenrightbigg ,8 YUEHENG LAN AND Y. CHARLES LI wherem(j) =m(t,jh),j= 1,···,N,Nh= 2π, andhis the spatial mesh size. For the computation of Figure 3, we choose N= 128. γ σ Figure 1. The separatrix connecting to the domain wall m+= −ei2x,m3= 0. 0246 0102030−101 xtm1 (a)γ= 00246 0102030−101 xtm1 (b)γ=π Figure 2. The spatial-temporal profiles corresponding to the two lobes of the figure eight curve. By a translation x→x+θ, one can generate a circle of domain walls: m+=−ei2(x+θ), m3= 0, whereθis the phase parameter. The three dimensional figure eight separa trix connecting to the circle of domain walls, parametrized by σ,γandθ; is illustrated in Figure 4. In general, the unimodal equilibrium manifold can be sought as follows: Let mj=cjcos2x+sjsin2x , j= 1,2,3, then the uni-length condition |m|(x) = 1 leads to |c|= 1,|s|= 1, c·s= 0, wherecandsare the two vectors with components cjandsj. Thus the unimodal equilibrium manifold is three dimensional and can be represented as in F igure 5.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 9 0246 10203040−101 xtm1 (a) inside0123456 10152025303540−101 xtm1 (b) outside Figure 3. The spatial-temporal profiles corresponding to the in- side and outside of the figure eight curve. γ σ θ Figure 4. The separatrix connecting to the circle of domain walls m+=ei2(x+θ),m3= 0. Using the formulae (3.13)-(3.14), we want to build a Melnikov integral. The zeros of the Melnikov integral will give a prediction on the existence o f chaos. To build such a Melnikov integral, we need to first develop a Melnikov vecto r. This requires Floquet theory of (3.3). 3.4. Floquet Theory. Focusing on the spatial part (3.3) of the Lax pair (3.3)-(3.4), let Y(x) be the fundamental matrix solution of (3.3), Y(0) =I(2×2 identity matrix), then the Floquet discriminant is defined by ∆ = trace Y(2π). The Floquet spectrum is given by σ={λ∈C| −2≤∆(λ)≤2}.10 YUEHENG LAN AND Y. CHARLES LI s c Figure 5. A representation of the 3 dimensional unimodal equi- librium manifold. Periodicandanti-periodicpoints λ±(whichcorrespondtoperiodicandanti-periodic eigenfunctions respectively) are defined by ∆(λ±) =±2. A critical point λ(c)is defined by d∆ dλ(λ(c)) = 0. A multiple point λ(n)is a periodic or anti-periodic point which is also a critical point. The algebraic multiplicity of λ(n)is defined as the order of the zero of ∆(λ)±2 atλ(n). When the order is 2, we call the multiple point a double point, and denote it by λ(d). The order can exceed 2. The geometric multiplicity of λ(n) is defined as the dimension of the periodic or anti-periodic eigenspace atλ(n), and is either 1 or 2. Counting lemmas for λ±andλ(c)can be established as in [ 30] [23], which lead to the existence of the sequences {λ± j}and{λ(c) j}and their approximate locations. Nevertheless, counting lemmas are not necessary here. For any λ∈C, ∆(λ) is a constantofmotionofthe Heisenbergequation(3.1). Thisistheso- calledisospectral theory. Example 3.2. For the domain wall m1= cos2x,m2= sin2x, andm3= 0; the two Bloch eigenfunctions are given in (3.10). The Floquet discriminant is given by ∆ = 2cos/bracketleftBig 2π/radicalbig 1+λ2/bracketrightBig . The periodic points are given by λ=±/radicalbigg n2 4−1, n∈Z, nis even. The anti-periodic points are given by λ=±/radicalbigg n2 4−1, n∈Z, nis odd. The choice of φ+andφ−correspond to n=±1 andλ=ν=i√ 3/2 withk=±1. ∆′=−4πλ√ 1+λ2sin/bracketleftBig 2π/radicalbig 1+λ2/bracketrightBig . ∆′′=−4π(1+λ2)−3/2sin/bracketleftBig 2π/radicalbig 1+λ2/bracketrightBig −8π2λ2 1+λ2cos/bracketleftBig 2π/radicalbig 1+λ2/bracketrightBig .CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 11 λ Figure 6. The periodic and anti-periodic points corresponding to the potential of domain wall m+=ei2x,m3= 0. The open circles are double points, the solid circle at the origin is a multiple point of order 4, and the two bars intersect the imaginary axis at two periodic points which are not critical points. Whenn= 0, i.e.√ 1+λ2= 0, by L’Hospital’s rule ∆′→ −8π2λ , λ=±i . That is,λ=±iare periodic points, not critical points. When n=±1, we have two imaginary double points λ=±i√ 3/2. Whenn=±2,λ= 0 is a multiple point of order 4. The rest periodic and anti- periodicpoints areallrealdouble points. Figure 6is anillustrationoft hese spectral points. 3.5. Melnikov Vectors. Starting from the Floquet theory, one can build Melnikov vectors. Definition 3.3. An importantsequenceofinvariants Fjofthe Heisenbergequation is defined by Fj(m) = ∆(λ(c) j(m),m). Lemma 3.4. If{λ(c) j}is a simple critical point of ∆, then ∂Fj ∂m=∂∆ ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j. Proof. We know that ∂Fj ∂m=∂∆ ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j+∂∆ ∂λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j∂λ(c) j ∂m. Since ∂∆ ∂λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j= 0, we have ∂2∆ ∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j∂λ(c) j ∂m+∂2∆ ∂λ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j= 0.12 YUEHENG LAN AND Y. CHARLES LI Sinceλ(c) jis a simple critical point of ∆, ∂2∆ ∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j/ne}ationslash= 0. Thus ∂λ(c) j ∂m=−/bracketleftBigg ∂2∆ ∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j/bracketrightBigg−1 ∂2∆ ∂λ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j. Notice that ∆ is an entire function of λandm[23], then we know that∂λ(c) j ∂mis bounded, and ∂Fj ∂m=∂∆ ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle λ=λ(c) j. /square Theorem 3.5. As a function of two variables, ∆ = ∆(λ,m)has the partial deriva- tives given by Bloch functions ψ±(i.e.ψ±(x) =e±Λx˜ψ±(x), where˜ψ±are periodic inxof period 2π, andΛis a complex constant): ∂∆ ∂m+=−iλ√ ∆2−4 W(ψ+,ψ−)ψ+ 1ψ− 1, ∂∆ ∂m−=iλ√ ∆2−4 W(ψ+,ψ−)ψ+ 2ψ− 2, ∂∆ ∂m3=iλ√ ∆2−4 W(ψ+,ψ−)/parenleftbig ψ+ 1ψ− 2+ψ+ 2ψ− 1/parenrightbig , ∂∆ ∂λ=i√ ∆2−4 W(ψ+,ψ−)/integraldisplay2π 0/bracketleftbig m3/parenleftbig ψ+ 1ψ− 2+ψ+ 2ψ− 1/parenrightbig −m+ψ+ 1ψ− 1+m−ψ+ 2ψ− 2/bracketrightbig dx , whereW(ψ+,ψ−) =ψ+ 1ψ− 2−ψ+ 2ψ− 1is the Wronskian. Proof. Recall that Yis the fundamental matrix solution of (3.3), we have the equation for the differential of Y ∂xdY=iλΓdY+i(dλΓ+λdΓ)Y , dY (0) = 0. Using the method of variation of parameters, we let dY=YQ , Q (0) = 0. Thus Q(x) =i/integraldisplayx 0Y−1(dλΓ+λdΓ)Ydx , and dY(x) =iY/integraldisplayx 0Y−1(dλΓ+λdΓ)Ydx . Finally d∆ = trace dY(2π) =itrace/braceleftbigg Y(2π)/integraldisplay2π 0Y−1(dλΓ+λdΓ)Ydx/bracerightbigg . (3.16) Let Z= (ψ+ψ−)CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 13 whereψ±are two linearly independent Bloch functions (For the case that the re is only one linearly independent Bloch function, L’Hospital’s rule has to be used, for details, see [ 23]), such that ψ±=e±Λx˜ψ±, where˜ψ±are periodic in xof period 2πand Λ is a complex constant (The existence of such functions is the result of the well known Floquet theorem). Then Z(x) =Y(x)Z(0), Y(x) =Z(x)[Z(0)]−1. Notice that Z(2π) =Z(0)E ,whereE=/parenleftbigg eΛ2π0 0e−Λ2π/parenrightbigg . Then Y(2π) =Z(0)E[Z(0)]−1. Thus ∆ = trace Y(2π) = traceE=eΛ2π+e−Λ2π, and e±Λ2π=1 2[∆±/radicalbig ∆2−4]. In terms of Z,d∆ as given in (3.16) takes the form d∆ =itrace/braceleftbigg Z(0)E[Z(0)]−1/integraldisplay2π 0Z(0)[Z(x)]−1(dλΓ+λdΓ)Z(x)[Z(0)]−1dx/bracerightbigg =itrace/braceleftbigg E/integraldisplay2π 0[Z(x)]−1(dλΓ+λdΓ)Z(x)dx/bracerightbigg , from which one obtains the partial derivatives of ∆ as stated in the t heorem. /square It turns out that the partial derivatives of Fjprovide the perfect Melnikov vectors rather than those of the Hamiltonian or other invariants [ 23], in the sense thatFjis the invariant whose level sets are the separatrices. 3.6. An Explicit Expression of the Melnikov Vector Along the Figure Eight Connecting to the Domain Wall. We continue the calculation in sub- section 3.3 to obtain an explicit expression of the Melnikov vector alon g the figure eight connecting to the domain wall. Apply the Darboux transformat ion (3.6) to φ±(3.11) atλ=ν, we obtain ˆφ±=±¯ν−ν ¯νexp{∓1 2σ∓i1 2γ}W(φ+,φ−) |φ1|2+|φ2|2 φ2 −φ1 . In the formula (3.6), for general λ, detG=(ν−λ)(¯ν−λ) |ν|2, W(ˆψ+,ˆψ−) = detG W(ψ+,ψ−). In a neighborhood of λ=ν, ∆2−4 = ∆(ν)∆′′(ν)(λ−ν)2+ higher order terms in ( λ−ν). Asλ→ν, by L’Hospital’s rule √ ∆2−4 W(ˆψ+,ˆψ−)→/radicalbig ∆(ν)∆′′(ν) ν−¯ν |ν|2W(φ+,φ−).14 YUEHENG LAN AND Y. CHARLES LI Notice, by the calculation in Example 3.2, that ν=i√ 3 2,∆(ν) =−2,∆′′(ν) =−24π2, then by Theorem 3.5, ∂∆ ∂m+/vextendsingle/vextendsingle/vextendsingle/vextendsingle m=ˆm= 12√ 3πi (|φ1|2+|φ2|2)2φ22, ∂∆ ∂m−/vextendsingle/vextendsingle/vextendsingle/vextendsingle m=ˆm= 12√ 3π−i (|φ1|2+|φ2|2)2φ12, ∂∆ ∂m3/vextendsingle/vextendsingle/vextendsingle/vextendsingle m=ˆm= 12√ 3π2i (|φ1|2+|φ2|2)2φ1φ2, where ˆmis given in (3.13)-(3.14). With the explicit expression (3.12) of φ, we obtain the explicit expressions of the Melnikov vector, ∂∆ ∂m+/vextendsingle/vextendsingle/vextendsingle/vextendsingle m=ˆm=3√ 3π 2isech2τ (2−√ 3 sech2τsin2χ)2/bracketleftbigg (1−2tanh2τ)cos2χ +i(2−tanh2τ)sin2χ−i√ 3 sech2τ/bracketrightbigg e−i2x, (3.17) ∂∆ ∂m−/vextendsingle/vextendsingle/vextendsingle/vextendsingle m=ˆm=3√ 3π 2−isech2τ (2−√ 3 sech2τsin2χ)2/bracketleftbigg (1+2tanh2 τ)cos2χ −i(2+tanh2τ)sin2χ+i√ 3 sech2τ/bracketrightbigg ei2x, (3.18) ∂∆ ∂m3/vextendsingle/vextendsingle/vextendsingle/vextendsingle m=ˆm=3√ 3π 22isech2τ (2−√ 3 sech2τsin2χ)2/bracketleftbigg 2 sech2τ−√ 3sin2χ −i√ 3tanh2τcos2χ/bracketrightbigg , (3.19) where again m±=m1±im2, τ=√ 3 2t+σ 2, χ=1 2(x+γ), andσandγare two real parameters. 4. A Melnikov Function The forced Landau-Lifshitz-Gilbert (LLG) equation (2.1) can be re written in the form, (4.1) ∂tm=−m×mxx+ǫf+ǫ2g wherefis the perturbation f=−am×ex+m3(m×ez)−bm1(m×ex) −αm×(m×mxx)+(β1+β2cosω0t)m×(m×ex), g=−αm×[m×(aex−m3ez+bm1ex)]. The Melnikov function for the forced LLG (2.1) is given as M=/integraldisplay∞ −∞/integraldisplay2π 0/bracketleftbigg∂∆ ∂m+(f1+if2)+∂∆ ∂m−(f1−if2)+∂∆ ∂m3f3/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle m=ˆmdxdt ,CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 15 where ˆmis given in (3.13)-(3.14), and∂∆ ∂w(w=m+,m−,m3) are given in (3.17)- (3.19). The Melnikov function depends on several external and int ernal parameters M=M(a,b,α,β 1,β2,ω0,σ,γ) whereσandγare internal parameters. We can split fas follows: f=af(a)+f(0)+bf(b)+αf(α)+β1f(β1) +β2/bracketleftbigg cos/parenleftbiggσω0√ 3/parenrightbigg f(c)+sin/parenleftbiggσω0√ 3/parenrightbigg f(s)/bracketrightbigg , where f(a)=−m×ex, f(0)=m3(m×ez), f(b)=−m1(m×ex), f(α)=−m×(m×mxx), f(β1)=m×(m×ex), f(c)= cos/parenleftbigg2√ 3ω0τ/parenrightbigg m×(m×ex), f(s)= sin/parenleftbigg2√ 3ω0τ/parenrightbigg m×(m×ex). ThusMcan be splitted as M=aM(a)+M(0)+bM(b)+αM(α)+β1M(β1) +β2/bracketleftbigg cos/parenleftbiggσω0√ 3/parenrightbigg M(c)+sin/parenleftbiggσω0√ 3/parenrightbigg M(s)/bracketrightbigg , (4.2) whereM(ζ)=M(ζ)(γ),ζ=a,0,b,α,β 1, andM(ζ)=M(ζ)(γ,ω0),ζ=c,s. In general [ 19], the zeros of the Melnikov function indicate the intersection of certain center-unstable and center-stable manifolds. In fact, t he Melnikov function is the leading order term of the distance between the center-unst able and center- stable manifolds. In some cases, such an intersection can lead to ho moclinic orbits and homoclinic chaos. Here in the current problem, we do not have an invariant manifold result. Therefore, our calculation on the Melnikov function is purely from a physics, rather than rigorous mathematics, point of view. In terms ofthe variables m+andm3, the forced Landau-Lifshitz-Gilbert (LLG) equation (2.1) can be rewritten in the form that will be more convenie nt for the calculation of the Melnikov function, ∂tm+=i(m+m3xx−m3m+xx)+ǫf++ǫ2g+, (4.3) ∂tm3=1 2i(m+m+xx−m+m+xx)+ǫf3+ǫ2g3, (4.4)16 YUEHENG LAN AND Y. CHARLES LI where f+=f1+if2=af(a) ++f(0) ++bf(b) ++αf(α) ++β1f(β1) + +β2/bracketleftbigg cos/parenleftbiggσω0√ 3/parenrightbigg f(c) ++sin/parenleftbiggσω0√ 3/parenrightbigg f(s) +/bracketrightbigg , f3=af(a) 3+f(0) 3+bf(b) 3+αf(α) 3+β1f(β1) 3 +β2/bracketleftbigg cos/parenleftbiggσω0√ 3/parenrightbigg f(c) 3+sin/parenleftbiggσω0√ 3/parenrightbigg f(s) 3/bracketrightbigg , g+=g1+ig2=αag(a) ++αg(0) ++αbg(b) +, g3=αag(a) 3+αg(0) 3+αbg(b) 3, f(a) +=−im3, f(0) +=−im3m+, f(b) +=−i1 2m3(m++m+), f(α) +=1 2m+(m+m+xx−m+m+xx)+m3(m3m+xx−m+m3xx), f(β1) +=1 2m+(m+−m+)−m2 3, f(c) += cos/parenleftbigg2√ 3ω0τ/parenrightbigg/bracketleftbigg1 2m+(m+−m+)−m2 3/bracketrightbigg , f(s) += sin/parenleftbigg2√ 3ω0τ/parenrightbigg/bracketleftbigg1 2m+(m+−m+)−m2 3/bracketrightbigg , f(a) 3=1 2i(m+−m+), f(0) 3= 0, f(b) 3=1 4i(m2 +−m+2), f(α) 3=m3xx|m+|2−1 2m3(m+m+xx+m+m+xx), f(β1) 3=1 2m3(m++m+), f(c) 3=1 2cos/parenleftbigg2√ 3ω0τ/parenrightbigg m3(m++m+), f(s) 3=1 2sin/parenleftbigg2√ 3ω0τ/parenrightbigg m3(m++m+),CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 17 g(a) +=m2 3−1 2m+(m+−m+), g(0) +=m2 3m+, g(b) +=1 2m2 3(m++m+)−1 4m+(m2 +−m+2), g(a) 3=−1 2m3(m++m+), g(0) 3=−m3|m+|2, g(b) 3=−1 4m3(m++m+)2. Direct calculation gives that M(a)(γ) =M(0)(γ) =M(b)(γ) = 0, M(α)(γ) = 91.3343, andM(β1)andM(c)are real, while M(s)is imaginary. The graph of M(β1)is shown in Figure 7(a) (Notice that M(β1)is independent of ω0). The graph of M(c) is shown in Figure 7(b). The imaginary part of M(s)is shown in Figure 7(c). In the case of only DC current ( β2= 0),M= 0 (4.2) leads to (4.5) α=−β1M(β1)/91.3343, whereM(β1)(γ) is a function of the internal parameter γas shown in Figure 7(a). In the general case ( β2/ne}ationslash= 0),M(s)(γ,ω0) = 0 determines curves (4.6) γ=γ(ω0) = 0, π/2, π,3π/2, andM= 0 (4.2) leads to (4.7) |β2|>/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig 91.3343α+β1M(β1)/parenrightBig/slashbigg M(c)/vextendsingle/vextendsingle/vextendsingle/vextendsingle, whereM(β1)andM(c)are evaluated along the curve (4.6), M(β1)=±43.858 (‘+’ forγ=π/2,3π/2; ‘−’ forγ= 0, π),M(c)is plotted in Figure 8 (upper curve corresponds to γ=π/2,3π/2; lower curve corresponds to γ= 0, π), and cos/parenleftbiggσω0√ 3/parenrightbigg =−91.3343α+β1M(β1) β2M(c). 5. Numerical Simulation In the entire article, we use the finite difference method to numerica lly simulate the LLG (2.1). Due to an integrable discretization [ 9] of the Heisenberg equation (3.1), the finite difference performs much better than Galerkin Fou rier mode trun- cations. As in (3.15), let m(j) =m(t,jh),j= 1,···,N,Nh= 2π, andhis the spatial mesh size. Without further notice, we always choose N= 128 (which pro- vides enough precision). The only tricky part in the finite difference d iscretization of (2.1) is the second derivative term in H, for the rest terms, just evaluate mat m(j): ∂2 xm(j) =2 h2/parenleftbiggm(j+1) 1+m(j)·m(j+1)+m(j−1) 1+m(j−1)·m(j)/parenrightbigg .18 YUEHENG LAN AND Y. CHARLES LI 0246 024−50050 γω0Mβ1 (a)0246 024−50050 γω0M(c) (b) 0246 024−50050 γω0M(s) (c) Figure 7. (a). The graph of M(β1)as a function of γ, andM(β1) is independent of ω0. (b). The graph of M(c)as a function of γ andω0. (c). The graph of the imaginarypart of M(s)as a function ofγandω0. 5.1. Only DC Current Case. In this case, β2= 0 in (2.1), and we choose β1as the bifurcation parameter, and the rest parameters as: (5.1) a= 0.05,b= 0.025,α= 0.02,ǫ= 0.01. The computation is first run for the time interval [0 ,8120π], then the figures are plotted starting from t= 8120π. The bifurcation diagram for the attractors, and the typical spatial profiles on the attractors are shown in Figure 9 . This figure indiactes that interesting bifurcations happen over the interval β1∈[0,0.15] which is the physically important regime where β1is comparable with values of other parameters. There are six bifurcation thresholds c1···c6(Fig. 9). When β1<c1, the attractor is the spatially uniform fixed point m1= 1 (m2=m3= 0). WhenCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 19 00.511.522.533.54−50−40−30−20−1001020304050 ω0M(c) Figure 8. The graphs of M(c)along the curves (4.6). β1c2c3c1c c c4 5 6I II III IV V VI VII Figure 9. The bifurcation diagram for the attractors and typ- ical spatial profiles on the attractors in the case of only DC current where β1is the bifurcation parameter, c1···c6are the bifurcation thresholds, and c1∈[0,0.01],c2∈[0.0205,0.021], c3∈[0.0231,0.0232],c4∈[0.025,0.026],c5∈[0.08,0.1],c6∈ [0.13,0.15]. c1≤β1≤c2, the attractoris a spatially non-uniform fixed point as shown in Figur e 10(a). When c2< β1< c3, the attractor is spatially non-uniform and temporally periodic (a limit cycle) or quasiperiodic (a limit torus) as shown in Figure 1 0(b). Here as the value of β1is increased, first there is one basic temporal frequence, then more frequencies enter and the temporal oscillation amplitude becomes bigger and bigger. When c3≤β1< c4, the attractor is chaotic, i.e. a strange attractor as shown in Figure 10(c). Even though the chaotic nature is not ver y apparent in Figure 10(c), due to the smallness of the perturbation paramete r together with smallness of all other parameters, the temporal evolution is chaot ic, and we have used Liapunov exponent and power spectrum devices to verify this . Whenc4≤ β1< c5, the attractor is spatially non-uniform and temporally periodic (a limit cycle) as shown in Figure 11(a). The spatial modulation is small, and it is even not apparentin Figure11(a). But itisapparentonthe individualtypical spatialprofiles as seen in region V in Figure 9. With the increase of β1, the spatial modulation becomes smaller and smaller. When c5≤β1<c6, the attractor is spatially uniform and temporally periodic (a limit cycle, the so-called procession) as sho wn in Figure20 YUEHENG LAN AND Y. CHARLES LI 0123456 0510152025300.60.81 xtm1 (a)β1= 0.0205 spatially non-uniform fixed point0123456 05101520253000.51 xtm1 (b)β1= 0.023 spatially non-uniform and temporally periodic or quasiperiodic attrac- tor 0123456 051015202530−101 xtm1 (c)β1= 0.0235 weak chaotic attractor Figure 10. The spatio-temporal profiles of solutions in the at- tractors in the case of only DC current. 11(b). When β1≥c6, the attractor is the spatially uniform fixed point m1=−1 (m2=m3= 0). Whenβ2= 0, the Melnikov function predicts that around β1= 0.041 (4.5), there is probably chaos; while the numerical calculation shows that t here is an interval [0.0231,0.026] forβ1where chaos is the attractor. Since the perturbation parameterǫ= 0.01 in the numerical calculation, the Melnikov function prediction seems in agreement with the numerical calculation. Some of the attractors in Figure 9 are attractors of the corresp onding ordinary differential equations by setting ∂x= 0 in (2.1), i.e. the single domain case. These attractors are the ones in regions I, VI and VII in Figure 9 [ 25] [41]. Because weCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 21 0123456 0100200300400500600−101 xtm1 (a)β1= 0.026 spatially non-uniform and temporally periodic attractor0123456 0100200300400500600−0.500.5 xtm1 (b)β1= 0.1 spatially uniform and tempo- rally periodic attractor Figure 11. The spatio-temporal profiles of solutions in the at- tractors in the case of only DC current (continued). are studying the only DC current case, the ordinary differential eq uations do not have any chaotic attractor [ 25] [41]. On the other hand, the partial differential equations (2.1) does ha ve a chaotic attractor (region IV in Figure 9). In general, when β1<0, the Gilbert damping dominates the spin torque driven by DC current and m1= 1 is the attractor. When β1>0.15, the spin torque driven by DC current dominates the Gilbert damping, m1=−1 is the attractor, and we have a magnetization reversal. In some technological applications, β1>0.15 may correspond too high DC current that can burn the device. On the o ther hand, in the technologically advantageousinterval β1∈[0,0.15], magnetization reversalmay be hard to achieve due to the sophisticated bifurcations in Figure 9. 5.2. Only AC Current Case. In this case, β1= 0 in (2.1), and we choose β2as the bifurcation parameter, and the rest parameters as: (5.2) a= 0.05,b= 0.025,α= 0.0015,ǫ= 0.01,ω0= 0.2. Unlike the DC case, here the figures are plotted starting from t= 0. It turns out that the types of attractors in the AC case are simpler than th ose of the DC case. When β2= 0, the attractor is a spatially non-uniform fixed point as shown in Figure 12. In this case, the only perturbation is the Gilbert d amping which damps the evolution to such a fixed point. When 0 < β2< β∗ 2where β∗ 2∈[0.18,0.19], the attractor is a spatially non-uniform and temporally periodic solution. When β2≥β∗ 2, the attractor is chaotic as shown in Figure 12. Our Melnikov prediction (4.7) predicts that when |β2|>0.003, certain center-unstable andcenter-stablemanifolds intersect. Ournumericsshowsthat s uch anintersection seems leading to transient chaos. Only when |β2|>β∗ 2, the chaos can be sustained as an attractor. It seems that such sustained chaotic attracto r gains extra support from parametric resonance due to the AC current driving [ 22], as can be seen from22 YUEHENG LAN AND Y. CHARLES LI 0123456 050001000015000−101 xtm1 (a)β2= 0 spatially non-uniform fixed point020004000600080001000012000−1−0.95−0.9−0.85−0.8−0.75−0.7−0.65−0.6−0.55−0.5 tm1(x1) (b)β2= 0 temporal evolution at one spatial location 0123456 050001000015000−101 xtm1 (c)β2= 0.21 chaotic attractor020004000600080001000012000−1−0.8−0.6−0.4−0.200.20.40.60.81 tm1(x1) (d)β2= 0.21 temporal evolution at one spa- tial location Figure 12. The attractors in the case of only AC current. the turbulent spatial structure of the chaotic attractor (Fig. 1 2), which diverges quite far away from the initial condition. Another factor that may b e relevant is the fact that higher-frequency spatially oscillating domain walls hav e more and stronger linearly unstable modes (3.5). By properly choosing initial c onditions, one can find the homotopy deformation from the ODE limit cycle (process ion) [25] [41] to the current PDE chaos as shown in Figure 13 at the same paramet er values. We also simulated the case of normal Gilbert damping α= 0.02. For all values ofβ2∈[0.01,0.3], the attractor is always non-chaotic. That is, the only attracto r we can find is a spatially uniform limit cycle with small temporal oscillation a s shown in Figure 14. Of course, when neither β1norβ2is zero, the bifurcation diagram is a combi- nation of the DC only and AC only diagrams.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 23 Figure 13. Homotopy deformation of the attractors under differ- ent initial conditions. See http://www.math.missouri.edu/˜cli Figure 14. The attractor when α= 0.02,β2= 0.21 and all other parameters’ values are the same with Figure 12. See http://www.math.missouri.edu/˜cli 6. Appendix: The Connection Between the Heisenberg Equatio n and the NLS Equation In this appendix, we will show the details on the connection between t he 1D cubic focusing nonlinear Schr¨ odinger (NLS) equation and the Heise nberg equation (3.1). The nonlinear Schr¨ odinger (NLS) equation (6.1) iqt+qxx+2|q|2q= 0, is a well-known integrable system with the Lax pair ∂xφ= (iλσ3+U)φ , (6.2) ∂tφ=−(2iλ2σ3+2λU+V)φ , (6.3)24 YUEHENG LAN AND Y. CHARLES LI whereλis the complex spectral parameter, σ3is defined in (2.5), and U=/parenleftbigg0iq i¯q0/parenrightbigg , V=−i|q|2σ3+/parenleftbigg0qx −¯qx0/parenrightbigg . Lemma 6.1. Ifφ= (φ1,φ2)Tsolves the Lax pair (6.2)-(6.3) at λ, then(−φ2,φ1)T solves the Lax pair (6.2)-(6.3) at ¯λ. Whenqis even, i.e. q(−x) =q(x), then (φ2(−x),φ1(−x))Tsolves the Lax pair (6.2)-(6.3) at −¯λ. Whenλis real, and φis a nonzero solution, then (−φ2,φ1)Tis another linearly independent solution. For any two solutions of the Lax pair, their Wronskian is indepen dent ofxandt. For any real λ0, by the well-known Floquet theorem [ 28] and Lemma 6.1, there are always two linearly independent Floquet (or Bloch) eigenfunction sφ±to the Lax pair (6.2)-(6.3) at λ=λ0, such that φ+=/parenleftbiggϕ1 ϕ2/parenrightbigg , φ−=/parenleftbigg−ϕ2 ϕ1/parenrightbigg , φ+(x+2π) =ρφ+(x), φ−(x+2π) = ¯ρφ−(x),|ρ|2= 1. SincetheWronskian W(φ+,φ−)isindependentof xandt, withoutlossofgenerality, we chooseW(φ+,φ−) = 1. Then S=/parenleftbigg ϕ1−ϕ2 ϕ2ϕ1/parenrightbigg is a unitary solution to the Lax pair at λ=λ0: S−1=SH=/parenleftbiggϕ1ϕ2 −ϕ2ϕ1/parenrightbigg ,|ϕ1|2+|ϕ2|2= 1. Recall the definition of Γ (2.7), let Γ =S−1σ3S=/parenleftbigg |ϕ1|2−|ϕ2|2−2ϕ1ϕ2 −2ϕ1ϕ2|ϕ2|2−|ϕ1|2/parenrightbigg , i.e. m1+im2=−2ϕ1ϕ2, m3=|ϕ1|2−|ϕ2|2. Now for any φsolving the Lax pair (6.2)-(6.3) at λ, defineψas ψ=S−1φ . Thenψsolves the pair ψx=i(λ−λ0)Γψ , (6.4) ψt=−/braceleftbigg 2i(λ2−λ2 0)Γ+1 2(λ−λ0)[Γ,Γx]/bracerightbigg ψ . (6.5) The compatibility condition of this pair leads to the equation (6.6) Γ t=−/braceleftbigg 4λ0Γx+1 2i[Γ,Γxx]/bracerightbigg . Settingλ0= 0 or performing the translation t=t,ˆx=x−4λ0t, equation (6.6) reduces to the Heisenberg equation (3.2). Therefore, the Gauge transformStrans- forms NLS equation into the Heisenberg equation. Periodicity in xmay not persist.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 25 Example 6.2. Consider the temporally periodic solution of the NLS equation (6.1), q=aeiθ(t), θ(t) = 2a2t+γ . The corresponding Bloch eigenfunction of the Lax pair (6.2)-(6.3) a tλ= 0 is ϕ=1√ 2/parenleftbigg eiθ/2 e−iθ/2/parenrightbigg eiax. Then Γ =S−1σ3S=/parenleftbigg 0 −e−i2ax −ei2ax0/parenrightbigg , which is called a domain wall. Acknowledgment : The second author Y. Charles Li is grateful to Professor Shufeng Zhang, Drs. Zhanjie Li and ZhaoyangYang, and Mr. Jiexu an He for many helpful discussions. References [1] L. Berger. Emission of Spin Waves by a Magnetic Multilaye r.Phys. Rev. B , 54:9353–9358, 1996. [2] A. Calini. A Note on a B¨ acklund Transformation for the Co ntinuous Heisenberg Model. Phys. Lett. A, 203: 333–344, 1995. [3] M. Covington et al. Current-Induced Magnetization Dyna mics in Current Perpendicular to the Plane Spin Valves. Phy. Rev. B , 69:184406, 2004. [4] A. Fabian et al. Current-Induced Two-Level Fluctuation s in Pseudo-Spin-Valve (Co/Cu/Co) Nanostructures. Phys. Rev. Lett. , 91:257209, 2004. [5] L. D. Faddeev, L. A. Takhtajan. Hamiltonian Methods in the Theory of Solitons . Springer- Verlag, Berlin, 1987. [6] G. D. Fuchs et al. Spin-Transfer Effects in Nanoscale Magn etic Tunnel Junctions. Appl. Phys. Lett., 85:1205, 2004. [7] J. Grollier, et al. Synchronization of Spin-Transfer Os cillators Driven by Stimulated Mi- crowave Currents. arXiv:cond-mat/0509326 , 2005. [8] Y. Ji et al. Current-Induced Spin-Wave Excitations in a S ingle Ferromagnetic Layer. Phys. Rev. Lett. , 90:106601, 2003. [9] M. Kaashoek, A. Sakhnovich. Discrete Skew Selfadjoint C anonical Systems and the Isotropic Heisenberg Magnet Model. J. Funct. Anal. , 228: 207-233, 2005. [10] S. Kaka, et al. Mutual Phase-Locking of Microwave Spin T orque Nano-Oscillators. Nature, 437: 389–392, 2005. [11] S. I. Kiselev et al. Microwave Oscillations of a Nanomag net Driven by a Spin-Polarized Current. Nature, 425:380–383, 2003. [12] R. H. Koch et al. Time-Resolved Reversal of Spin-Transf er Switching in a Nanomagnet. Phys. Rev. Lett. , 92:088302, 2004. [13] I. Krivorotov, et al. Time-Domain Measurements of Nano magnet Dynamics Driven by Spin- Transfer Torques. Science, 307: 228–231, 2005. [14] K. Kudo, et al., Synchronized Magnetization Oscillati ons in F/N/F Nanopillars, arXiv:cond-mat/0511095 , 2005. [15] Y. Li. Smale Horseshoes and Symbolic Dynamics in Pertur bed Nonlinear Schr¨ odinger Equa- tions.Journal of Nonlinear Sciences , 9, no.4:363–415, 1999. [16] Y. Li. Chaos and Shadowing Around a Homoclinic Tube. Abstract and Applied Analysis , 2003, no.16:923–931, 2003. [17] Y. Li. Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions. J. Dynamics and Differential Equations , 15, no.4:699–730, 2003. [18] Y. Li. Homoclinic Tubes and Chaos in Perturbed Sine-Gor don Equation. Chaos, Solitons and Fractals , 20, no.4:791–798, 2004. [19] Y. Li. Chaos in Partial Differential Equations . International Press, 2004.26 YUEHENG LAN AND Y. CHARLES LI [20] Y. Li. Persistent Homoclinic Orbits for Nonlinear Schr ¨ odinger Equation Under Singular Per- turbation. Dynamics of PDE , vol.1, no.1, 87-123, 2004. [21] Y. Li. 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Current-Driven Magnetic Switching in Mangan ite TrilayerJunctions. J. Mag. Mag. Mater., 202:157, 1999. [37] S. Urazhdin et al. Current-Driven Magnetic Excitation s in Permalloy-Based Multilayer Nanopillars. Phys. Rev. Lett. , 91:146803, 2003. [38] J. E. Wegrowe. Magnetization Reversal and Two-Level Fl uctuations by Spin Injection in a Ferromagnetic Metallic Layer. Phy. Rev. B , 68:2144xx, 2003. [39] S. A. Wolf et al. Spintronics: a Spin-Based Electronics Vision for the Future. Science, 294:1488, 2001. [40] M. Yamanouchi et al. Current-Induced Domain-Wall Swit ching in a Ferromagnetic Semicon- ductor Structure. Nature, 428:539, 2004. [41] Z. Yang, S. Zhang, Y. Li. Chaotic Dynamics of Spin Valve O scillators. Phys. Rev. Lett. , 99: 134101, 2007. [42] J. G. Zhu et al. Spin Transfer Induced Noise in CPP Read He ads.IEEE Trans. on Magn. , 40:182, 2004. Department of Mechanical Engineering, University of Califo rnia, Santa Barbara, CA 93106 E-mail address :yueheng lan@yahoo.com Department of Mathematics, University of Missouri, Columbi a, MO 65211 E-mail address :cli@math.missouri.edu

2008-05-01

We study the spin dynamics of a long nanomagnet driven by an electrical current. In the case of only DC current, the spin dynamics has a sophisticated bifurcation diagram of attractors. One type of attractors is a weak chaos. On the other hand, in the case of only AC current, the spin dynamics has a rather simple bifurcation diagram of attractors. That is, for small Gilbert damping, when the AC current is below a critical value, the attractor is a limit cycle; above the critical value, the attractor is chaotic (turbulent). For normal Gilbert damping, the attractor is always a limit cycle in the physically interesting range of the AC current. We also developed a Melnikov integral theory for a theoretical prediction on the occurrence of chaos. Our Melnikov prediction seems performing quite well in the DC case. In the AC case, our Melnikov prediction seems predicting transient chaos. The sustained chaotic attractor seems to have extra support from parametric resonance leading to a turbulent state.

Chaotic Spin Dynamics of a Long Nanomagnet Driven by a Current

0805.0147v1

arXiv:1905.13487v2 [math.NT] 30 Mar 2020CHARACTERIZING THE MOD- ℓLOCAL LANGLANDS CORRESPONDENCE BY NILPOTENT GAMMA FACTORS GILBERT MOSS Abstract. LetFbe ap-adic field and choose kan algebraic closure of Fℓ, withℓdifferent from p. We define “nilpotent lifts” of irreducible generic k- representations of GLn(F), which take coefficients in Artin local k-algebras. We show that an irreducible generic ℓ-modular representation πofGLn(F) is uniquely determined by its collection of Rankin–Selberg ga mma factors γ(π× /tildewideτ,X,ψ) as/tildewideτvaries over nilpotent lifts of irreducible generic k-representations τofGLt(F) fort= 1,...,⌊n 2⌋. This gives a characterization of the mod- ℓ local Langlands correspondence in terms of gamma factors, a ssuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts. 1.Introduction 1.1.Notation. LetFbe a finite extension of Qpwith finite residue field oforder q, letG:=Gn:=GLn(F), letRbe a commutative ring with unit, and let RepR(Gn) be the category of smooth R[Gn]-modules (the stabilizer of any element is open). Letℓbe a prime different from p, letk:=Fℓ, and letW(k) be the ring of Witt vectors(it is the ℓ-adic completion of the ring of integers in the maximal unramified extension of Qℓ). LetK= Frac(W(k)) =W(k)[1 ℓ], letKbe an algebraic closure, and letObe the ring of integers in K. After fixing a field isomorphism K∼=Cwe may translate the properties of RepC(Gn) to RepK(Gn). Fix a nontrival additive character ψ:F→W(k)×. For any W(k)-algebra W(k)→Rwe denote by ψRthe extension F→W(k)×→R×. ExtendψRto the subgroupUofunipotent upper triangularmatricesby ψ(u) =ψ(u1,2+···+un−1,n). GivenaW(k)-algebraR, wesayπ∈RepR(Gn) isgenericifthere existsa nontrivial homomorphism π→IndG UψR:={W:G→Rsmooth :W(ug) =ψR(u)W(g), u∈U, g∈G}. IfRis an algebraically closed field of characteristic different from p,Agen R(n) will denote the set of isomorphism classes of irreducible generic object s in RepR(G). Givenπ∈Agen R(n), themapπ→IndG UψRisunique uptoscaling([Vig96, III.1.11]); its image is denoted W(π,ψ) and called the Whittaker model ofπ, andπ∼=W(π,ψ) by irreducibility. 1.2.On the mod- ℓconverse theorem. Givenintegers n,t≥1,andπ∈Agen K(n), τ∈Agen K(t), Jacquet, Piatetski-Shapiro, and Shalika in [JPSS79] construct g amma factors of pairs γ(π×τ,X,ψ)∈K(X) satisfying a certain functional equation. If π1,π2are isomorphic, then γ(π1×τ,X,ψ) =γ(π2×τ,X,ψ) for allτ∈Agen K(t), for allt≥1. A “local converse theorem” identifies a collection of representa tions Date: April 1, 2020. 12 GILBERT MOSS τsuch that the converse statement holds, i.e. such that the collect ion ofγ(π× τ,X,ψ) uniquely determines π∈Agen K(n). The first converse theorems appear in [JL70, JPSS83] for n= 2 and 3, respectively; in both cases τruns over characters ofG1. In [Hen93] Henniart proves a converse theorem for arbitrary n, whereτruns over irreducible generic objects in RepK(Gt) fort= 1,2,...,n−1. In [Hen93], Henniart applies the converse theorem to prove that th e local Lang- lands correspondence over Kis uniquely characterized by the property that it equatestheDeligne–Langlandsgammafactors([Del73])withtheJac quet–Piatetski- Shapiro–Shalikagammafactors([JPSS83]). Vign´ erasprovesin[Vig0 1]theexistence of a local Langlands correspondence over k. It is uniquely characterized not by gamma factors but by its compatibility with the K-correspondence under reduction mod-ℓ(see§1.3 below for more details). The goal of this article is to prove a mod- ℓconverse theorem and use it to uniquely characterize the mod- ℓlocal Langlands correspondence intrinsically by mod-ℓgamma factors, without referring to the K-setting. We will now describe our mod-ℓconverse theorem, and postpone until §1.3 the question of characterizing the mod-ℓlocal Langlands correspondence. The construction of gamma factors associated to objects in Agen R(n) is already established in the literature when R=k=Fℓ. Gamma factors of pairs are devel- oped over Fℓby Kurinczuk–Matringe in [KM17], and shown to be compatible with the reduction mod- ℓof the gamma factors of [JPSS83]. At first, one might hope for a mod-ℓconverse theorem that parallels the situation over K: that the gamma factorsγ(π×τ,X,ψ) uniquely characterize π∈Agen k(n) asτranges overAgen k(t) fort= 1,...,n−1. However, we give a counterexample in Section 2 when n= 2 showing this hope is false. Thus a new framework is needed to achieve a mod-ℓ converse theorem. When examining the counterexample of Section 2, one finds that the lack of nontrivialℓ-power roots of unity in kleads to a lack of tamely ramified k-valued characters. This leads to an excess of congruences mod ℓbetween gamma factors γ(πθ×χ,X,ψ) asχvaries, where πθis a certain integral object in Agen K(2) con- structed in§2. The same problem arises when trying to adapt the traditional method of proving converse theorems over C, where a crucial ingredient is the com- pleteness of Whittaker models with respect to the L2-inner product. We give an example in Section 4 showing that completeness of Whittaker models f ails overk, again due to the lack of nontrivial ℓ-power roots of unity in k. Their appearence in the mod-ℓBernstein center (c.f. Example 3.4) is anotherexample ofthe import ance ofℓ-power roots of unity to the mod- ℓrepresentation theory. To recover the converse theorem, we pass from kto a larger class of k-algebras that has an ample supply of ℓ-power roots of unity: Artin local k-algebras, or equivalently, finite-dimensional local k-algebras. Such rings Rhave only a single prime ideal (namely, their nilradical) they have residue field k, and the composition k→R→kis the identity map. Definition 1.1. Given an object τ∈Agen k(t), anilpotent lift ofτto an Artin local k-algebraRis an admissible R[Gt]-submodule/tildewideτofIndG UψRfor which /tildewideτ⊗Rk=W(τ,ψ). For an Artin local k-algebraR, letAgen R(n)denote the set of isomorphism classes of nilpotent lifts to Rof objects inAgen k(t).CHARACTERIZING MOD- ℓLOCAL LANGLANDS 3 LetAgen nil(t)denote the set of isomorphism classes of all nilpotent lifts of objects inAgen k(t), to anyR. In other words, a nilpotent lift is an infinitesimal deformation of the W hittaker model ofτ. Givenτ∈Agen k(t), the isomorphism τ∼=W(τ,ψ) allows us to identify τwith a nilpotent lift of itself, so we have an “inclusion” Agen k(t)⊂Agen nil(t). An extension of the theory of gamma factors that encompasses nilpotent lifts is carried out in the author’s thesis ([Mos16b, Mos16a]). We refer to §3 and§6 for a summary of these gamma factors. Our main result is the following converse theo rem. Theorem 1.2. Letπ1,π2be inAgen k(n),n≥2. Suppose (1) γ(π1×/tildewideτ,X,ψ) =γ(π2×/tildewideτ,X,ψ) for all nilpotent lifts /tildewideτ∈Agen nil(t), for allt= 1,2,...,⌊n 2⌋. Thenπ1∼=π2. In fact, the theorem is still true if π1andπ2are inAgen nil(n) (see Theorem 7.1). Theorem 1.2 addresses a conjecture of Vign´ eras from 2000. In [V ig00], Vign´ eras defines modified gamma factors for cuspidal objects in Agen k(2) and proves they satisfy a mod- ℓconverse theorem. In the same article, she conjectures that an ℓ- modular converse theorem holds for all n>2, using some suitably modified version of the gamma factor. This is discussed in more detail in Section 1.4. Whenn>2 there is another aspect of the converse theorem, which is obtain ing theminimal range oftsuch that the collection γ(π×τ,X,ψ) determines πasτ varies. For example, over K, Henniart’s proof[Hen93] requires AK(t) for alltwithin the range 1≤t≤n−1. Whenn= 3, it is shown in [JPSS83] that t= 1 suffices, and Jacquet conjectured in 1999 that 1 ≤t≤⌊n 2⌋should suffice for arbitrary n. Jacquet’s conjecture was proven in [Cha19] and [JL16], independen tly, for all n. We thus present Theorem 1.2 as a mod- ℓanalogue of Jacquet’s conjecture. It is shown in [ALST18] that the bound t≤⌊n 2⌋cannot be improved in the K-setting. 1.3.Characterizing the mod- ℓLanglands correspondence. Suppose for the time being that Ris any algebraically closed field of characteristic ℓdifferent from p. An irreducible representation in RepR(Gn) issupercuspidal (respectively, cusp- idal) if it is not isomorphic to a subquotient (respectively, quotient) of a ny para- bolic induction from a proper Levi subgroup (the two notions are eq uivalent when the characteristic of Ris zero). LetA0 R(n) denote the set of isomorphism classes of irreducible supercuspidal representations in RepR(Gn). Givenπ∈RepR(Gn) irreducible, the supercuspidal support ofπ, denoted scs( π), is the unique multi- set{π1,...,π k}such thatπis a subquotient of the normalized parabolic induc- tionπ1×···×πk. IfAgen R(n) denotes the set of irreducible generic representa- tions in RepR(Gn), andAscs R(n) denotes the set of supercuspidal supports of ir- reducible representations in RepR(Gn), the map τ/mapsto→scs(τ) defines a bijection Agen R(n)→Ascs R(n) ([Vig96, III.1.11]). Choose an algebraic closure of F, letWFbe the absolute Weil group, let G0 R(n) denotethesetofisomorphismclassesof n-dimensionalirreduciblesmooth R-representations ofWF. A local Langlands correspondence over Rstarts with a sequence of bijec- tionsL0 R,n:G0 R(n)→A0 R(n),n= 1,2,.... IfGss R(n) denotes the set of isomor- phism classes of semisimple n-dimensional R-representations of WF, asemisim- plelocal Langlands correspondence over Ris a sequence ( Lss R,n)n≥1of bijections4 GILBERT MOSS Lss R,n:Gss R(n)→Ascs R(n) extending a sequence L0 R,n:G0 R(n)→A0 R(n) according to the relation Lss R,n(ρ1⊕···⊕ρk) ={L0 R,n(ρ1),...,L0 R,n(ρk)}. Composingwith the inverseofthe bijection Agen R(n)∼→Ascs R(n) givesa genericlocal Langlands correspondence Lgen R,n:Gss R(n)→Agen R(n). In [HT01] (and [LRS93] for positive characteristic F), it is shown for R=Kthat there exists a sequence L0 K,n:G0 K(n)→A0 K(n) satisfying: (i)L0 K,1is given by local class field theory, (ii)L0 K,n(ρ∨) =L0 K,n(ρ)∨, and (iii) for all pairs of integers n,t≥1,ρ∈G0 K(n),ρ′∈G0 K(t), γ(L0 K,n(ρ)×L0 K,t(ρ),s,ψ) =γ(ρ⊗ρ′,s,ψ). whereγ(ρ⊗ρ′,s,ψ) denotesthe Deligne–Langlandslocalfactorof[Del73]. Sincethe Deligne–Langlands gamma factor is multiplicative in direct sums, and th e Jacquet– Piatetski-Shapiro–Shalika gamma factor is multiplicative in normalized p arabolic inductions, the genericLanglandscorrespondence Lgen K,n:Gss K(n)→Agen K(n) induced by (L0 K,n)n≥1satisfies: (i)Lgen K,1is given by local class field theory on G0 K(1) (=Gss K(1)), (ii) for all pairs n>t≥1,ρ∈Gss F(n),ρ′∈Gss K(t), γ(Lgen K,n(ρ)×Lgen K,t(ρ′),s,ψ) =γ(ρ⊗ρ′,s,ψ), It then follows immediately from (ii) and the converse theorem of [Hen 93] that there can be at most one sequence ( Lgen K,n)n≥1with these properties. Forℓ>0 a prime,ℓ/ne}ationslash=p, Vign´ eras proves in [Vig01] the existence of a sequence ofbijections L0 k,n:G0 k(n)→A0 k(n), for alln≥1, which is uniquely characterizedby the property that its induced semisimple correspondence ( Lss k,n)n≥1is compatible with (L0 K,n)n≥1under reduction mod- ℓ. More precisely, a finite length object πin RepK(Gn) (or RepK(WF)) isintegralif it admits a model Loverthe ring of integers OEinafinite extension E/QℓthatisfreeoverOEandfiniteoverOE[G]; writerℓ(π) to denote the semisimplified mod- ℓreduction (L⊗OEk)ss. The reduction rℓ(π) of π∈A0 K(Gn) remains irreducible and cuspidal but may no longer be supercuspidal, and the reduction rℓ(ρ) ofρ∈G0 K(n) may no longer be irreducible. Compatibility with (L0 K,n)n≥1under reduction mod- ℓmeans that, for ρ,ρ′∈G0 Kintegral, •L0 K,n(ρ) is integral •rℓ(L0 K,n(ρ)) =rℓ(L0 K,n(ρ′))⇐⇒rℓ(ρ) =rℓ(ρ′), and •Lss k,n(rℓ(ρ)) = scs(rℓ(L0 K,n(ρ))). The uniqueness of the mod- ℓcorrespondence ( Lss k,n)n≥1(and hence ( Lgen k,n)n≥1) then follows from the uniqueness of ( Lss K,n)n≥1. Our goal is to use Theorem 1.2 to characterize ( Lgen k,n)n≥1directly, in analogy with the characterization of ( Lgen K,n)n≥1. To accomplish this, we define nilpotent lifts of objects in Gss k:CHARACTERIZING MOD- ℓLOCAL LANGLANDS 5 Definition 1.3. Givenρ∈Gss k(n), anilpotent lift ofρto an Artin local k-algebra Ris a smooth R[WF]-module/tildewideρ, free of rank noverR, such that there exists an isomorphism /tildewideρ⊗Rk∼=ρ. For an Artin local k-algebraR, letGR(n)denote the set of isomorphism classes of nilpotent lifts to Rof objects inGss k(n). LetGnil(n)denote the set of all isomorphism classes of nilpotent lifts of objects inGss k(n), to anyR. In other words, a nilpotent lift is an infinitesimal deformation. We may identify Gss k(n) with a subset of Gnil(n), and ofGR(n) for anyR. The theory of gamma factors of objects in Gss k(n) has long been established by work of Deligne [Del73], but the classical theory does not accommoda te nilpotent lifts. Fortunately, by previous work of the author and Helm ([HM15]) , the Deligne– Langlands gamma factor generalizes to arbitrary Noetherian W(k)-algebras (see Theorem 8.1 for a summary). Our second main result characterizes the sequence ( Lgen k,n)n≥1by gamma factors, so far as it can be extended to a correspondence on nilpotent lifts. Theorem 1.4. There exists at most one sequence of maps Lgen k,n:Gss k(n)→Agen k(n), n≥1 that admits, for every Artin local k-algebraR, an extension to a sequence of sur- jections LR,n:GR(n)→Agen R(n), n≥1 satisfying (1)LR,1is given by local class field theory, (2) For all n > t,ρ∈GR(n),ρ′∈GR′(t), we have the following equality in (R⊗kR′)[[X]][X−1]: γ(ρ⊗ρ′,X,ψ) =γ(LR,n(ρ)×LR′,t(ρ′),X,ψ) In fact, if all of the extensions ( LR,n)n≥1exist, each one is unique (Corollary 9.1). There remains the question of whether the sequence of bijections (Lgen k,n)n≥1of Vign´ eras actually admits the surjective extensions LR,nsatisfying the conditions of Theorem 1.4. The existence of the extensions satisfying conditions (1) and (2) of Theorem 1.4 follows from the machinery of the local Langlands corre spondence “in families,” as it is stated in [HM18], however its surjectivity is not immediat e. This will be addressed in future work. 1.4.Relation to other work and further questions. Recently, the proof of Jacquet’s conjecture was extended to the setting of reduced ℓ-torsion free W(k)- algebras in the preprint [LM19] of Liu and the author. As the count erexample in Section 2 of the present article shows, the converse theorem in th eℓ-torsion setting requires a different framework than [LM19]. In Section 4 we show that the point of failure in the traditional conve rse theorem proof method is the completeness of Whittaker models over k. We recover com- pleteness of Whittaker models by including nilpotent lifts and using the geometry of the integral Bernstein variety, as developed in [Hel16a, Hel16b].6 GILBERT MOSS The gamma factors used in Theorem 1.2 are different from the “new” gamma factors proposed by Vign´ eras in [Vig00]. Vign´ eras’ new gamma fa ctors take coef- ficients ink, and to each cuspidal τinAgen k(2),χinAgen k(1), there is attached a collection of factors ε(τ⊗χ,y) which determine τasχandyare allowed to vary (yis an element ofO× Fofℓ-power order). Givenπ∈Agen k(n),τ∈Agen k(t), we speculate that there might be a way to con- struct a collection of gamma factors γ1(π×τ,X,ψ),γ2(π×τ,X,ψ),...∈k(X) for which the mod- ℓconverse theorem holds, for which there is an analogous collection forGss k(n), and for which there is an equality of sets {γ1(Lgen k,n(ρ)×Lgen k,t(ρ′),X,ψ),γ2(Lgen k,n(ρ)×Lgen k,t(ρ′),X,ψ),...} ={γ1(ρ⊗ρ′,X,ψ),γ2(ρ⊗ρ′,X,ψ),...}. This would more closely resemble Vign´ eras’ construction of “new” g amma factors forG2in [Vig00], and would eliminate the need for nilpotent lifts in Theorem 1.4. It seems plausible that the set γ1(π×τ,X,ψ),γ2(π×τ,X,ψ),···∈k(X) could be formed by taking appropriate linear combinations of the collection of γ(π×/tildewideτ,X,ψ) as/tildewideτvaries over nilpotent lifts of τ, and mapping them to k(X) in a clever way. In the recent preprint [KM18], Kurinczuk and Matringe demonstra te the failure of preservation of L-factors of pairs in Lgen k,n. We remark that the putative corre- spondenceLR,nappearing in Theorem 1.4 would preserve gamma factors of pairs (and hence so would Lgen k,n), where the gamma factors are those of [HM15]. The reason for assuming Fhas characteristic zero is this paper’s dependence on the results of [Hel16a, Hel16b, HM18], where the same assumption is made but almost certainly not required. 1.5.Acknowledgements. The ideas appearing here were influenced by several discussions with David Helm during and after the author’s work on his P hD the- sis. The interest and encouragement of Rob Kurinczuk and Nadir Ma tringe were invaluable. The author would also like to thank Jean-Fran¸ cois Dat, G uy Henniart, Marie-France Vign´ eras, and an anonymous referee for their help ful comments at various stages. 2.A counterexample to the naive mod- ℓconverse theorem In this section we give an example of two irreducible generic k-representations ofG2:=GL2(F) with distinct mod- ℓsupercuspidal supports (in fact, in different ℓ-blocks), having the same gamma factors for all twists by charact ers. When writing [Vig00], Vign´ eraswas no doubt awareof an example similar to the one presented below, but it does not appear in the literature. We als o note that our counterexample is different from that in [M ´12], where M´ ınguez gives two distinct irreducible k-represntations with the same Godement–Jacquet gamma factor s, but one is nongeneric and they have the same supercuspidal support. In this section, we use the gamma factors as constructed in [KM17]. Givenπ,τ inAgen k(n), choose/tildewideπ,/tildewideτsubrepresentations of IndG UψOlifting the Whittaker models ofπ,τ, respectively. The classicalgammafactor of[JPSS83], γ(/tildewideπ⊗K×/tildewideτ⊗K,X,ψ), defines a formal Laurent series with coefficients in O. Thenγ(π×τ,X,ψ) is defined as the reduction of γ(/tildewideπ×/tildewideτ,X,ψ) modulo the maximal ideal of O. It is uniquely determined by a functional equation (c.f. [KM17, Cor 3.11]). Later, we will require gamma factors in a broader context, see §6.CHARACTERIZING MOD- ℓLOCAL LANGLANDS 7 Letq=p= 5,ℓ= 2, andn= 2. Letθ:F× q2→Zℓ×be the character that sends a primitive 24’th root of unity to ζ3, a primitive 3rd root of unity in Zℓ. Since θq/ne}ationslash=θ, this is a regular character of F× q2, and therefore gives rise to an irreducible cuspidal representation λθofGL2(F) ([BH06,§6.4]). LetK0=GL2(OF). Inflate λθto a representation of K0, thenλθhas trivial central character, since θ|F× qis trivial ([BH06,§6.4(1)]). Thus we may extend λθto a representation Λ of F×K0 by declaring that Λ |F×is trivial. The triple ( M2(OF),F×K0,Λ) is acuspidal typeof level zero, in the language of [BH06, 15.5], and the representatio nπθ:= c-IndG2 F×K0Λ is an irreducible cuspidal representation of G2. In fact,πθis integral and, sinceθq/ne}ationslash≡θmodℓ, its mod-ℓreductionπθissupercuspidal, see[Vig96, III.3.3], or [KM17, Thm§2.7]. Lemma 2.1. Ifχ:F×→K×is an unramified or a tamely ramified character, thenγ(χπθ,X,ψ)≡1modℓ. Iflevel(χ)≥1, thenγ(χπθ,X,ψ) =γ(χ◦det,X,ψ). Proof.Ifχis tamely ramified, χ|UFis trivial mod- ℓ, so there exists an unramified characterχ′such thatχ≡χ′modℓ. Henceχπθ≡χ′πθmodℓ, andγ(χπθ,X,ψ)≡ γ(χ′πθ,X,ψ) modℓ([KM17, Thm 3.13(2)]). Thus we may assume without loss of generality that χis unramified. Sinceχπθis cuspidal, γ(χπθ,X,ψ) =ε(χπθ,X,ψ). Sinceχis unramified, the cuspidal type of χπθis the triple ( M2(OF),F×K0,χΛ), and the restrictions χΛ|K0 andΛ|K0arebothequivalentto λθ. Itfollowsfromthedescriptionin[BH06,Section 25.4]thatε(χπθ,X,ψ) andε(πθ,X,ψ)areequivalentandgivenby −qτ(θ,˜ψ), where ˜ψ(x) :=ψ(x+xq),x∈Fq2, andτ(θ,˜ψ) is the Gauss sum/summationtext x∈F× 25θ(x)ψ(x+xq). Sincex/mapsto→xqis a field automorphism, we have τ(θ,˜ψ) =/summationdisplay x∈F× 25θ(xq)ψ(xq+(xq)q) =/summationdisplay x∈F× 25θ(x)qψ(x+xq) =/summationdisplay x∈F× 25θ(x)2ψ(x+xq) =τ(θ−1,˜ψ). Therefore,τ(θ,˜ψ)2=τ(θ,˜ψ)τ(θ−1,˜ψ) which we can compute as /summationdisplay x,y∈F× 25θ(xy−1)˜ψ(x+y) =/summationdisplay u∈F× 25θ(u)/summationdisplay y∈F× 25˜ψ(y(u+1)). Separating terms into u=−1 andu/ne}ationslash=−1, θ(−1)(25−1)+/summationdisplay u/\e}atio\slash=−1θ(u)/summationdisplay y∈F× 25˜ψ(y(u+1)) =θ(−1)(25−1)+/summationdisplay u/\e}atio\slash=−1−θ(u) =θ(−1)(25)−/summationdisplay u∈F× 25θ(u) =θ(−1)(25). It follows that ε(πθ,X,ψ) is congruent to 1 mod ℓ. Since the central character of πθis trivial, the result for ramified characters of level≥1 follows immediately from the stability of gamma factors. For example ,8 GILBERT MOSS we can use the explicit formulation in [BH06, 25.7], after observing tha tε(χ◦ det,X,ψ) =γ(χ◦det,X,ψ) for charactersof level ≥1 (c.f. [BH06, 26.6 Prop]). /square LetBdenote the subgroup of upper triangular matrices of G2, and1:B→k× thetrivialcharacter. The“special”representationSp2isthecuspidal k-representation ofG2thatoccursasasubquotientof iG B(1)(notethatnormalizedandnon-normalized parabolic induction coincide since q≡1 modℓ). As Sp 2is a subquotient of the in- duction of an unramified characterofthe torus, its supercuspida l support is distinct (in fact, inertially inequivalent) from πθ. Lemma 2.2. If¯χ:F×→k×is an unramified or tamely ramified character, γ(¯χSp2,X,ψ)≡1modℓ. Iflevel(¯χ)≥1, thenγ(¯χSp2,X,ψ) =γ(¯χ◦det,X,ψ). Proof.By the multiplicativity of gamma factors, γ(¯χSp2,X,ψ) =γ(¯χ◦det,X,ψ) =γ(iG B¯χ,X,ψ) =γ(¯χ,X,ψ)2. If ¯χis unramified (which is equivalent to tamely ramified since q−1 is a power of ℓ), we may choose an unramified lift χ:F×→K×and compute γ(χ,X,ψ) =ε(χ,X,ψ)L(χ−1,1/(q1/2X)) L(χ,q−1/2X) =χ(̟)−1X−11−χ(̟)q−1/2X 1−χ(̟)−1q−1/2X−1 =χ(̟)−1X−1−q−1/2 1−χ(̟)−1q−1/2X−1, which is equivalent to −1 modℓ, sinceq≡1 modℓ. It remains to show that γ(¯χ◦det,X,ψ) =ε(¯χ◦det,X,ψ) for allχof level≥1. This is shown in [M ´12,§6]. /square Corollary 2.3. For every character ¯χ:F×→k×,γ(¯χπθ,X,ψ) =γ(¯χSp2,X,ψ), butπθandSp2have distinct supercuspidal supports (in fact, inertially distinct). Remark 2.4. There are analogues of the local converse theorem for repres entations ofGLn(Fq)overC([Nie14]). Granted the extension of the theory of GLn(Fq) gamma factors to k-representations, the Gauss sum calculations in this examp le could be adapted to illustrate the failure of the naive mod- ℓconverse theorem over finite fields. Remark 2.5. Finding similar examples when ℓ/ne}ationslash= 2appears to be nontrivial. It could be interesting to describe the “gamma-factor ℓ-blocks,” i.e. the sets of in- ertial supercuspidal supports whose twisted gamma factors become equivalent upon reduction modulo ℓ. Thelackoftamelyramified k-valuedcharacterswasimportantinestablishingthe congruences in Lemmas 2.1, 2.2. Let O0be the sub- W(k)-algebra of Zℓgenerated by the values of a tamely ramified χ, which sends a primitive 24’th root of unity inF× 25to a primitive 4’th root of unity. In particular, O0is a obtained from W(k) by adjoining a 4’th root of unity, or equivalently a 4’th root of ℓ, and therefore R=O0/ℓO0is isomorphic to the four-dimensional local k-algebrak[Y]/Y4, andCHARACTERIZING MOD- ℓLOCAL LANGLANDS 9 ζ:=Y+1 is a fourth root of unity such that ζ2/ne}ationslash= 1. Now let χbe the reduction modℓO0ofχ,insteadof the reduction modulo the maximal ideal of O0. Nowχ is a nilpotent lift of the reduction of χmodulo the maximal ideal of O0. With the notation of§2,γ(χπθ,X,ψ) is given by the reduction mod ℓof−qτ(χEθ,˜ψ), whereχE(x) :=χ(xq+1) onF× 25. The same calculation as in the proofof Lemma 2.1 givesγ(χπθ,X,ψ)2=χ(−1) =ζ2/ne}ationslash= 1, where we consider −1 as an element of F× 25. On the other hand, the same calculation as in the proof of Lemma 2.2 s hows that γ(χSp2,X,ψ) is the reduction of τ(χEθ,˜ψ)2=χ(−1) ([BH06, 23.6.2]). Thus, γ(χπθ,X,ψ)2=γ(χSp2,X,ψ) =ζ2. It follows that γ(χπθ,X,ψ)/ne}ationslash=γ(χSp2,X,ψ). This illustrates, for our particular example, how finite-dimensional k-algebras are large enough to distinguish twisted gamma factors in characteristic ℓ. 3.Co-Whittaker modules and the integral Bernstein center To recoverthe conversetheoremin characteristic ℓ, we will passto R-coefficients, whereRis an Artin local k-algebra. For this we need the theory of co-Whittaker R[Gn]-modules, where Ris aW(k)-algebra. IfVis asmooth R[Gn]-module, define V(n)tobe theψ-coinvariants V/V(Un,ψ), whereV(Un,ψ) is theR-submodule generated by {ψ(u)v−uv:u∈Un,v∈V}. This functor is exact and, for any R-moduleMthere is a natural isomorphism (V⊗RM)(n)∼=V(n)⊗RM. IfV(n)is nonzero, Frobenius reciprocity produces a canonical nonzero m ap (2) V→IndG UψV(n):v/mapsto→Wv, whereψV(n):=ψ⊗W(k)V(n). The quotient map V→V(n)is equivalent to v/mapsto→Wv(1). Definition 3.1. LetRbe a Noetherian W(k)-algebra. A smooth R[Gn]-moduleV is co-Whittaker if the following conditions hold (1)Vis admissible as an R[Gn]-module, (2)V(n)is a freeR-module of rank one, (3) ifQis a quotient of Vsuch thatQ(n)= 0, thenQ= 0. For example, when R=C,n= 2,Bis the Borel subgroup, and χ=χ1⊗χ2is a character of the torus T, the normalized parabolic induction iG B(χ) is co-Whittaker so long asχ1χ−1 2/ne}ationslash=|·|. IfVandV′are co-Whittaker R[G]-modules, any nonzero G-equivariant map V→V′is surjective, as otherwise the cokernel would be a nongeneric quo tient. In this caseVis said to dominate V′. We sayVandV′areequivalent if there exists a co-Whittaker R[Gn]-moduleV′′dominating both VandV′. This is an equivalence relation on isomorphism classes of co-Whittaker modules. By definition a co-Whittaker module admits an isomorphism V(n)∼=R, which induces an isomorphism IndG UψV(n)∼=IndG UψR. The image of Vin the composition V→IndG UψV(n)∼=IndG UψR will be denoted by W(V,ψ) and called the ( R-valued) Whittaker model of Vwith respect toψ. It is independent of the choice of isomorphism V(n)∼=R. Lemma 3.2. IfVis a co-Whittaker R[Gn]-module, its Whittaker model W(V,ψ) is an equivalent co-Whittaker R[Gn]-module.10 GILBERT MOSS Proof.SinceW(V,ψ) is a quotient of V, we need only show W(V,ψ)(n)is free of rank one. LetWbe the image of Vin the map (2). Choosing an isomorphism V(n)∼=Rinduces an isomorphism W∼=W(V,ψ), so it suffices to prove W(n)∼= V(n). But the map V→V(n)factors asV→W→V(n), the second map being evalution at 1. This induces a map W(n)→V(n), which is surjective. On the other hand, the natural surjection V→Winduces a surjection V(n)→W(n)which is its inverse. /square In [Hel16b], Helm constructs a co-Whittaker module which is “universa l” up to this notion of equivalence. The key tool is the integral Bernstein ce nter ofGn, i.e. the center of the category RepW(k)(Gn). The center of an abelian category is the endomorphism ring of the ide ntity functor, in otherwordsthe ringofnaturaltransformationsfro mthe identity functor to itself. It acts on every object in the category in a way compatible with all morphisms. We denote by Znthe center of RepW(k)(Gn). Schur’slemma holdsforanyco-Whittaker R[Gn]-moduleV, meaningthe natural mapR→EndA[Gn](V) is an isomorphism (c.f. [Hel16b, Prop 6.2]), and thus there exists a map fV:Zn→R, which we call the supercuspidal support ofV. Note thatValso admits a central character ωV:F×→R×. IfRis a field and V,V′are objects inAgen R(n), thenfV=fV′if and only scs( V) = scs(V′) in the traditional sense ([Hel16a, 12.12]). Aprimitiveidempotent eofZngivesrisetoadirectfactorcategory eRepW(k)(Gn), which is the full subcategory of RepW(k)(Gn) on which eacts as the identity. As described in [Hel16a], the primitive idempotents in Znare in bijection with inertial equivalence classes of pairs ( L,π), whereLis a Levi subgroup of Gnandπis an irreducible supercuspidal k-representation of L. IfVis a simple object in RepW(k)(Gn) that is killed by ℓ, the mod-ℓinertial supercuspidal support is defined to be the usual inertial supercu spidal support of Vin Repk(Gn). IfVis an integral simple object in RepK(Gn), its mod-ℓreduction may no longer be simple. The mod- ℓinertial supercuspidal support of Vis defined by combining the inertial supercuspidal supports of the constitue nts of its mod- ℓ reduction (c.f. [Hel16a, 4.12,4.13]). Ifeis the idempotent corresponding to the pair ( L,π), then a representation V in RepW(k)(Gn) lies ineRepW(k)(Gn) if and only if every simple subquotient of V has mod-ℓinertial supercuspidal support [ L,π]. If [M′,π′] is an inertial cuspidal support in RepK(Gn), letZM′,π′denote the center of the subcategory of RepK(Gn) corresponding to [ M′,π′]. Theorem 3.3 ([Hel16a], Thm 10.8, 12.1) .Letebe any primitive idempotent of Zn, corresponding to a mod- ℓinertial supercuspidal support [L,π]. (1) The ring eZnis a finitely generated, reduced, ℓ-torsion free W(k)-algebra. (2) There is an isomorphism eZ⊗K∼=/productdisplay [M′,π′]ZM′,π′, where[M′,π′]varies over pairs consisting of a Levi M′⊃Mandπ′a cus- pidal integral object in RepK(M′)with mod-ℓinertial supercuspidal support equivalent to [L,π].CHARACTERIZING MOD- ℓLOCAL LANGLANDS 11 The following example indicates that, in characteristic ℓ, valuable information is lost by restricting one’s attention only to the set of k-pointseZn→krather than, say, the set of points eZn→k[ε]/(ε)2. Example 3.4. Letn= 2,ℓ= 3,q= 5, and letebe the primitive idempotent corresponding to the mod- ℓinertial supercuspidal support [T,1], whereTis the maximal torus and 1is its trivial character. If ζis a nontrivial cube root of unity inK, thenζ+ζ−1=−1, and it follows from [Hel16a, 13.11] , combined with [Pai14, Thm 4.11] that eZ2∼=W(k)[Θ1,Θ±1 2,Y]/((Y−2)Θ1,(Y−2)(Y+1)). In particular, its spectrum possesses two irreducible comp onents:{Y= 2}and {Θ1= 0}, which become the principal block and a cuspidal block, resp ectively, over Kin the sense of Theorem 3.3. On the other hand, since −1≡2modℓ, the special fibereZ2/(ℓ)contains the nontrivial cube-root of unity ζ=Y−1. Now letRbe aW(k)-algebra, and let Vbe a co-Whittaker R[Gn]-module. Suppose further that Vlies ineRepW(k)(Gn) for some primitive idempotent e(so the supercuspidal support map fVfactors through the projection Zn→eZn). Let Wnbe the smooth W(k)[Gn]-module c-IndGn Unψ. For any primitive idempotent e ofZn, we have an action of eZnoneWn. Theorem 3.5 ([Hel16b], Theorem 6.3) .Letebe any primitive idempotent of Zn. The smooth eZn[Gn]-moduleeWnis a co-Whittaker eZn[Gn]-module. If Ris Noe- therian and has an eZn-algebra structure, the module eWn⊗eZnAis a co-Whittaker R[Gn]-module. Conversely, Vis dominated by eWn⊗eZn,fVR. We thus say that, up to the equivalence relation induced by dominanc e,eWnis the universal co-Whittaker module in eRepW(k)(Gn). Lemma 3.6 ([LM19] Lemma 2.4) .LetRbe a Noetherian ring and suppose V1and V2are two co-Whittaker R[Gn]-modules. The following are equivalent: (1) There exists WinW(V1,ψ)∩W(V2,ψ)such thatW(g)∈R×for some g∈G. (2)W(V1,ψ) =W(V2,ψ). (3)fV1=fV2. (4)V1andV2are equivalent. Corollary 3.7. LetRbe a complete local Noetherian W(k)-algebra with residue fieldk(e.g. an Artin local k-algebra). There is a bijection from the set of equivalence classes of co-Whittaker R[Gn]-modules to the set Agen R(n), sending an equivalence class to the Whittaker model of any representative. Proof.If/tildewideτis inAgen R(n), we must show it is co-Whittaker. But this follows imme- diately from [EH14, 6.3.2 Lemma, 6.3.4 Proposition]. Conversely, let Vbe a representative of an equivalence class of co-Whittaker R[Gn]-modules. By Lemma 3.6, each equivalence class has a unique Whittake r modelW ⊂IndG UψR. The cosocle (i.e. largest semisimple quotient) of V⊗R kis irreducible and generic (e.g. [EH14, 6.3.5 Lemma]). Hence cosoc( V⊗Rk) isomorphic to its Whittaker model. Thus the reduction W⊗Rkis isomorphic to the Whittaker model ofcosoc( V⊗Rk), soWis a nilpotent lift of cosoc( V⊗Rk)./square12 GILBERT MOSS There is a duality operation on co-Whittaker modules which interpolat es the contragredient across a co-Whittaker family ([HM18, Prop 2.6]). If Vis a smooth W(k)[Gn]-module, let Vιdenote the W(k)[Gn]-module with the same underlying W(k)-module structure, and for which the Gnaction, which we will denote by g·v, is given by g·v=gιv, wheregι=tg−1. This duality has a very concrete interpretation in terms of Whittaker functions. Let ωn,m=/parenleftbiggIn−m0 0ωm/parenrightbigg , whereωmis the longest Weyl element of Gm. For any function WonGn, let /tildewiderW(g) =W(ωngι). IfWis inW(V,ψ), then/tildewiderWis inW(Vι,ψ−1). 4.Failure of completeness of Whittaker models over Fℓ OverC∼=K, ifHis in c-IndGt UtψK, and /an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht:=/integraldisplay U\GH(x)W(x)dx= 0 for all Whittaker functions WinW(τ,ψ−1), for allτ∈ Agen K(t), thenH= 0. In other words, the Whittaker models of Agen K(t) arecomplete with respect to this integral pairing. In all known proofs of converse theorems, c ompleteness of Whittaker models is used in an essential way. Question4.1. LetHbe an element of c-IndGt Utψk. Suppose that/integraltext Ut\GtH(x)W(x)dx= 0for allW∈W(τ,ψ−1 k)for allτ∈Agen t(t). MustHbe zero? The answerto Question4.1is “no,”asthe followingexampleshowsalrea dywhen t= 1. Suppose that q≡1 modℓandq−1 =ℓamwheremis relatively prime to ℓ. DecomposeO× F=X×YwhereXis subgroup ofO× Fwhose pro-order is prime-to- ℓ andYis a cyclic group of order ℓa. Sinceqis invertible in k, there exists a k-valued Haar measure µonF×, which we may normalize so that µ(X) = 1 (or any unit). We remark that µ(O× F) = 0. Letn= 1, so that U={1}andψis trivial, and every smooth character χ:F×→k×is irreducible generic with Whittaker space c·χfor constantsc∈k. TakeHin c-IndG1 U1ψ=C∞ c(F×,W(k)) to be the characteristic function ofO× F. Given a character χ, we have /an}⌊ra⌋ketle{tH,χ/an}⌊ra⌋ketri}ht=/integraldisplay F×H(x)χ(x)dx=/integraldisplay O× Fχ(x)dx. Since each y∈Yhasℓ-power order, so does χ(y)∈k, and thusχ(y) = 1. Let K be the largest finite index subgroup of Xsuch thatK⊂ker(χ). Then for any χ we have /an}⌊ra⌋ketle{tH,χ/an}⌊ra⌋ketri}ht=µ(K)/summationdisplay x∈O× F/Kχ(x) =ℓa[X:K]−1/summationdisplay x∈X/Kχ(x) = 0. However, if we expand the coefficient ring of τenough, the answer becomes “yes”, as we will now describe. The property of being a co-Whittaker model does not depend on th e choice ofψ in the definition of ( −)(n). The Whittaker model of eWnwith respect to ψ−1has the following W(k)-module structure, as observed in [HM18, p. 1010].CHARACTERIZING MOD- ℓLOCAL LANGLANDS 13 Proposition 4.2 ([HM18]) .There is a map of W(k)-modulesθe:eZn→W(k) which induces an isomorphism of W(k)-modules eWn∼→W(eWn,ψ−1) f/mapsto→Wf θe◦W←/mapsfromcharW The following Corollary is essentially a duality statement: Corollary 4.3. LetRbe aW(k)-algebra and suppose His a nonzero element of c-IndψRThen there is a primitive idempotent eofZand a Whittaker function W inW(eWn,ψ−1)such that /an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht:=/integraldisplay U\GH(x)⊗W(x)dx is nonzero in R⊗W(k)eZ. Proof.As a function on Gwe haveH(g)/ne}ationslash= 0 inRfor someg∈G. Now choose some compact open subgroup Ksuch thatHis invariant under K(note: we must takeKto be small enough that its Haar measure is nonzero) and such that ψis trivial onU∩gKg−1. We build the function φin c-IndG Uψ−1as follows. φwill be supported on the double coset UgKand will satisfy φ(ngk) =1 meas(K)ψ(n)−1. In particular,/an}⌊ra⌋ketle{tH,φ/an}⌊ra⌋ketri}ht=H(g) is nonzero in R. Now as c-IndG Uψ−1is the direct sum of e(c-Indψ−1) for primitive idempotents e, there must be some primitive idempotent esuch that/an}⌊ra⌋ketle{tH,eφ/an}⌊ra⌋ketri}ht/ne}ationslash= 0. In the isomorphism of Proposition 4.2 there is a Whittaker function Weφ∈ W(eWn,ψ−1) corresponding to the function eφ, and we have θe(Weφ(x)) =eφ(x). Now/an}⌊ra⌋ketle{tH,Weφ/an}⌊ra⌋ketri}htcannot be zero, since otherwise its image /an}⌊ra⌋ketle{tH,eφ/an}⌊ra⌋ketri}htunder the map id⊗θewould be zero. /square 5.Completeness of Whittaker models over Artin local k-algebras In the previous section we showed that completeness of Whittaker models holds after allowing representations with coefficients in Z, which is quite large. The main result of this paper is that it suffices to allow coefficients only in finite-d imensional localk-algebras. Theorem5.1. LetRbe an Artinlocal k-algebra. Let Hbe an element of c-IndGt UtψR. Then the following implication holds: if /an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht= 0for allW∈W(/tildewideτ,ψ), for all /tildewideτ∈Agen nil(t), thenH= 0. We state the following special case of Theorem 5.1, where Htakes values in k. Corollary 5.2. LetHbe an element of c-IndGt Utψk. Then the following implication holds: if/an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht= 0, for allW∈W(/tildewideτ,ψ), for all/tildewideτ∈Agen nil(t), thenH= 0. The proof of Theorem 5.1 uses a simple geometric argument on the mo d-ℓBern- stein center, which has a subtlety arising from the presence of nilpo tents. Consider, for the sake of illustration, the special case where Htakes values in k. To prove Theorem 5.1, we will prove the contrapositive. If H/ne}ationslash= 0, consider the Whittaker functionWguaranteed by Lemma 4.3, so that /an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}htis nonzero in eZ⊗k. If we knew that/an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}htwere non-nilpotent, the principal open set D(/an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht) would be nonempty, and would necessarily intersect the dense set of clos ed points, in14 GILBERT MOSS other words there is a map eZ⊗k→kthat does not kill /an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht. However, eZ⊗khas many nilpotents in general, and /an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}htmay be among them. Thus, instead of closed points, we instead look for points eZ⊗k→Rto non-reduced finite-dimensional k-algebrasRin which the image of /an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}htis nonzero. The author is grateful to David Helm for communicating the following le mma. Lemma 5.3. LetSbe a Noetherian ring, and let xbe a nonzero element of S. Then there exists a maximal ideal mofSand an integer isuch thatxis not in mi. Proof.We would like to show that M=/intersectiontext mmax’l/parenleftBig/intersectiontext i≥1mi/parenrightBig is zero. For a fixed m, there is an inclusion M ֒→/intersectiontext imi,i≥1. TheS-module/intersectiontext imiis finitely generated overSand satisfies m·/parenleftBigg/intersectiondisplay imi/parenrightBigg =/intersectiondisplay imi. The same is true ofits localization/parenleftbig/intersectiontext imi/parenrightbig mat the ideal m. It follows by Nakayama that the localization/parenleftbig/intersectiontext imi/parenrightbig mis zero. But the map on localizations Mm→/parenleftbig/intersectiontext imi/parenrightbig mremains injective, and we conclude that Mm= 0. /square Proof of Thm 5.1. TakeSin Lemma 5.3 to be eZ⊗W(k)R, and takex∈Sto be the element/an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht, whereWis the Whittaker function guaranteed by Lemma 4.3. In particular, xis nonzero. By Lemma 5.3, there is a maximal ideal mofSsuch thatxis not killed in the map S→S/mi. The ringS/mihas a single prime ideal, m/mi, which is the nilradical of S/mi. SinceSis a finitely generated k-algebra,S/m=k. The successive quotients of the filtrationS/mi⊃m/mi⊃···⊃ mi/mi= 0 are one-dimensional k-vector spaces, thereforeS/mihaslengthiasamoduleoveritself. Inparticular S/miisanArtinian localk-algebra. Letφ:S→S/mibethequotientmap, let fbethecomposition eZ→S→S/mi and letgbe the composition R→S→S/mi. Then/an}⌊ra⌋ketle{tH,g◦W/an}⌊ra⌋ketri}htmust be nonzero in (S/mi)⊗W(k)R, since otherwise its image /an}⌊ra⌋ketle{tf◦H,g◦W/an}⌊ra⌋ketri}ht=φ(/an}⌊ra⌋ketle{tH,W/an}⌊ra⌋ketri}ht) inS/miwould have to be zero, contrary to our construction of φ. The Theorem now follows since g◦Wis a Whittaker function in the S/mi-valued Whittaker space of/tildewideτ:=eW⊗eZ,f(S/mi), which is a co-Whittaker ( S/mi)[G]-module (c.f. Corollary 3.7). /square 6.Rankin–Selberg theory and gamma factors This section closely follows [Mos16a], and the reader should refer th ere for more details. Let AandBbe Noetherian W(k)-algebras and let R=A⊗W(k)B. Let VandV′be co-Whittaker A[Gn]- andB[Gt]-modules respectively, where t < n, with central characters ωV,ωV′. ForW∈W(V,ψ) andW′∈W(V′,ψ−1), and for 0≤j≤n−t−1, we define the formal series in the variable Xwith coefficients in R: Ψ(W,W′,X;j) :=/summationdisplay r∈Z/integraldisplay Mj,t(F)/integraldisplay Ut\{g∈Gt:v(detg)=r}/parenleftbigg W/parenleftbiggg x Ij In−t−j/parenrightbigg ⊗W′(g)/parenrightbigg Xrdgdx.CHARACTERIZING MOD- ℓLOCAL LANGLANDS 15 It is straightforward to show that there are finitely many negative -power terms, so Ψ(W,W′,X) := Ψ(W,W′,X;0) defines an element of R[[X]][X−1]. LetSbe the multiplicative system in R[X,X−1] consisting of all polynomials whose leading and trailing coefficients are units. Theorem 6.1 ([Mos16a]) .(1) Both Ψ(W,W′,X)andΨ(W,W′,X;j)are ele- ments inS−1(R[X,X−1]). (2) There exists a unique element γ(V×V′,X,ψ)∈S−1(R[X,X−1])such that Ψ(W,W′,X;j)γ(V×V′,X,ψ)ωV′(−1)n−1 = Ψ(ωn,t/tildewiderW,/tildewiderW′,qn−t−1 X;n−t−1−j), for anyW∈W(V,ψ),W′∈W(V′,ψ)and for any 0≤j≤n−t−1. Proposition 6.2 ([LM19] Prop 3.3) .LetAbe a Noetherian W(k)-algebra, let π1,π2be co-Whittaker A[Gn]-modules with n≥2. Assume that γ(π1×χ,X,ψ) =γ(π2×χ,X,ψ), for any character χ:F×→W(k)×. Thenωπ1=ωπ2. 7.The mod-ℓconverse theorem In this section we deduce the following mod- ℓconverse theorem. Theorem 7.1. LetRbe an Artin local k-algebra, let n≥2, and let/tildewiderπ1,/tildewiderπ2be in Agen R(n). If γ(/tildewiderπ1×/tildewideτ,X,ψ) =γ(/tildewiderπ2×/tildewideτ,X,ψ) for all nilpotent lifts /tildewideτ∈Agen nil(t), for all1≤t≤⌊n 2⌋, then/tildewiderπ1=/tildewiderπ2. Givenπ1,π2inAgen k(n), we can identify them with their trivial nilpotent lifts (i.e. their Whittaker models), which gives Theorem 1.2 from the introd uction. Note that Proposition 6.2 removes the need (when n≥2) for the hypothesis that/tildewiderπ1and/tildewiderπ2have the same central character. Proof of Theorem 7.1. The only part of the proof of the main theorem in [LM19] that requires reducedness and ℓ-torsion free-ness is the completeness of Whittaker models statement [LM19, Thm 4.1]. After replacing Theorem 4.1 in [LM1 9] with Theorem 5.1 of the present paper, and applying the argument of [L M19,§4-6] mutatis mutandis, we obtain Theorem 7.1. /square Remark 7.2. From the proof of Theorem 5.1, we see that in Theorems 5.1 and 7 .1 we can restrict the range of /tildewideτtoAgen R′(t), for ringsR′specifically of the form (eZ⊗W(k)R)/mi, whereeis a primitive idempotent of Z,mis a maximal ideal of eZ⊗W(k)R, and iis a positive integer. Maybe it is possible to classify such r ings.16 GILBERT MOSS 8.Deligne–Langlands gamma factors of nilpotent lifts Ifκis a field of characteristic zero containing W(k), andρ:WF→GLn(κ) is a Weil group representation, then there is a rational function γ(ρ,X,ψ) inκ(X), called the Deligne–Langlands γ-factor ofρ. The main result of [HM15] extends this construction to families of Weil group representations. In particu lar, one has: Theorem 8.1 ([HM15], Theorem 1.1) .LetRbe a Noetherian W(k)-algebra and letρ:WF→GLn(R)be a representation that is ℓ-adically continuous in the sense of[HM15,§2]. Then there exists an element γR(ρ,X,ψ)ofS−1R[X,X−1]with the following properties: (1) Iff:R→R′is a homomorphism of Noetherian W(k)-algebras, then one has: f(γR(ρ,X,ψ)) =γR′(ρ⊗RR′,X,ψ), where we have extended fto a mapS−1R[X,X−1]→(S′)−1R′[X,X−1]in the obvious way. (2) IfRis a field of characteristic zero, then γR(ρ,X,ψ)coincides with the Deligne-Langlands gamma factor γ(ρ,X,ψ). Notethatif Risreducedand ℓ-torsionfreethenthesecondpropertycharacterizes γR(ρ,X,ψ) uniquely. But any ρarises by base change from some finite collection of “universal” representations ρνover reduced ℓ-torsion free rings Rν– see [HM15] for more details. Thus the two properties of Theorem 8.1 uniquely ch aracterize the association ρ/mapsto→γ(ρ,X,ψ). AnArtinlocal k-algebraRisanℓ-adicallycompleteandseparated W(k)-algebra. Since a nilpotent lift ρ∈GR(n) is smooth, it is ℓ-adically continuous over Rin the sense of [HM15,§2]. Therefore Theorem 8.1 applies to nilpotent lifts. 9.Characterizing the mod- ℓlocal Langlands correspondence with nilpotent gamma factors Corollary 9.1. Suppose there exists, for every Artin local k-algebraR, a sequence of surjections LR,n:GR(n)→Agen R(n), n≥1, satisfying (1)LR,1is given by local class field theory, (2) For all ρinGR(n), for all Artin local k-algebrasR′, for allρ′∈GR′(t), for allt<n, we have the following equality in (R⊗R′)[[X]][X−1]: γ(ρ⊗ρ′,X,ψ) =γ(LR,n(ρ)×LR′,t(ρ′),X,ψ). Then the sequence (LR,n)n≥1is uniquely determined for every R. Proof.Fix an Artin local k-algebraR. Suppose ( ˜LR,n)n≥1is another sequence of maps satisfying (1) and (2). We will show that ˜LR,n=LR,nfor alln. The map ˜LR,1is uniquely determined by condition (1), so that ˜LR,1=LR,1. Now suppose n≥2. Givenρ∈GR(n), we have by condition (2) that γ(˜LR,n(ρ)×LR′,t(ρ′),X,ψ) =γ(ρ⊗ρ′,X,ψ) =γ(LR,n(ρ)×LR′,t(ρ′),X,ψ) for all Artin local k-algebrasR′, for allρ′∈GR′,t, for allt < n. SinceLR′,tis surjective by assumption, LR′,t(ρ′) runs over all of Agen nil(t) asρ′andR′vary. By Theorem 7.1, we conclude that ˜LR,n(ρ) =LR,n(ρ). /squareCHARACTERIZING MOD- ℓLOCAL LANGLANDS 17 Proof of Theorem 1.4. This follows immediately from Corollary 9.1 taking R=k, sinceGk(n) =Gss k(n). Note that Lgen k,nis thus the restriction to Gss k(n)⊂GR(n) of any of the maps LR,nappearing in the system of maps of Corollary 9.1, for any R. /square References [ALST18] Moshe Adrian, Baiying Liu, Shaun Stevens, and Geo K am-Fai Tam. On the sharpness of the bound for the local converse theorem of p-adicGLprime.Proceedings of the AMS, Series B , 5:6–17, 2018. [BH06] Colin Bushnell and Guy Henniart. The Local Langlands Conjecture for GL(2). Berlin Heidelberg: Springer-Verlag, 2006. [Cha19] Jingsong Chai. Bessel functions and local converse conjecture of jacquet. Journal of the EMS, 21:17031728, 2019. [Del73] Pierre Deligne. Les constantes des ´ equations fonctionelles des fonctions L. Spring- Verlag, 1973. [EH14] Matthew Emerton and David Helm. The local Langlands c orrespondence for GL(n) in families. Ann. Sci. E.N.S. , 2014. [Hel16a] David Helm. The Bernstein center of the category of smoothW(k)[GLn(F)]-modules. Forum of math, sigma , 4, 2016. [Hel16b] David Helm. Whittaker models and the integral Bern stein center for GL(n).Duke Math. J. , 165(9):1597–1628, 2016. [Hen93] Guy Henniart. Caracterisation de la correspondanc e de langlands locale par les facteurs ǫde paires. Inventiones Mathematicae , 1993. [HM15] David Helm and Gilbert Moss. Deligne–Langlands gamm a factors in families. arXiv:1510.08743 , 2015. [HM18] David Helm and Gilbert Moss. Converse theorems and th e local Langlands correspon- dence in families. Invent. Math. , 214:999–1022, 2018. [HT01] Michael Harris and Richard Taylor. On the geometry an d cohomology of some simple Shimura varieties. Annals of Math. Studies , 2001. [JL70] Herve Jacquet and Robert Langlands. Automorphic Forms on GL(2). Springer Lecture notes in Mathematics, 1970. [JL16] Herv´ e Jacquet and Baiying Liu. On the local converse theorem for p-adicGLn.Amer. J. Math. , 2016. [JPSS79] Herve Jacquet, Ilja Iosifovitch Piatetski-Shapi ro, and Joseph Shalika. Automorphic forms onGL(3) I.The Annals of Mathematics , 1979. [JPSS83] Herve Jacquet, Ilja Iosifovitch Piatetski-Shapi ro, and Joseph Shalika. Rankin-Selberg convolutions. American Journal of Mathematics , 1983. [KM17] Robert Kurinczuk and Nadir Matringe. Rankin-selber g local factors modulo ℓ.Selecta Math., 23:767–811, 2017. [KM18] Robert Kurinczuk and Nadir Matringe. The ℓ-modular local Langlands correspondence and local factors. arXiv:1805.05888 , 2018. [LM19] Baiying Liu and Gilbert Moss. On the local converse th eorem and descent theorem in families. Math. Z. , 2019. to appear (arXiv:1711.11159). [LRS93] G. Laumon, M. Rapoport, and U. Stuhler. D-elliptic sheaves and the Langlands corre- spondence. Inventiones Mathematicae , 113:217–338, 1993. [M´12] Alberto M´ ınguez. Fonctions Zˆ eta ℓ-modulaires. Nagoya Math. J. , 208, 2012. [Mos16a] Gilbert Moss. Gamma factors of pairs and a local con verse theorem in families. Int Math Res Notices , 2016(16):4903–4936, 2016. [Mos16b] Gilbert Moss. Interpolating local constants in fa milies.Math Res. Lett. , 23, 2016. [Nie14] Chufeng Nien. A proof of the finite field analogue of Ja cquets conjecture. American Journal of Mathematics , 136(3):653–674, 2014. [Pai14] David Paige. The projective envelope of a cuspidal r epresentation of a finite linear group.Journal of number theory , 136:354–374, 2014. [Vig96] Marie-France Vigneras. Representations ℓ-modulaires d’un groupe reductif p-adique avecℓdifferent de p. Boston: Birkhauser, 1996. [Vig00] Marie-France Vigneras. Congruences modulo ℓbetweenǫfactors for cuspidal represen- tations ofGL(2).Journal de theorie des nombres de Bordeaux , 2000.18 GILBERT MOSS [Vig01] Marie-France Vign´ eras. Correspondance de Langla nds semi-simple pour GLn(F) mod- uloℓ/negationslash=p.Inventiones mathematicae , 144:177–223, 2001.

2019-05-31

Let $F$ be a $p$-adic field and choose $k$ an algebraic closure of $\mathbb{F}_{\ell}$, with $\ell$ different from $p$. We define ``nilpotent lifts'' of irreducible generic $k$-representations of $GL_n(F)$, which take coefficients in Artin local $k$-algebras. We show that an irreducible generic $\ell$-modular representation $\pi$ of $GL_n(F)$ is uniquely determined by its collection of Rankin--Selberg gamma factors $\gamma(\pi\times \widetilde{\tau},X,\psi)$ as $\widetilde{\tau}$ varies over nilpotent lifts of irreducible generic $k$-representations $\tau$ of $GL_t(F)$ for $t=1,\dots, \lfloor \frac{n}{2}\rfloor$. This gives a characterization of the mod-$\ell$ local Langlands correspondence in terms of gamma factors, assuming it can be extended to a surjective local Langlands correspondence on nilpotent lifts.

Characterizing the mod-$\ell$ local Langlands correspondence by nilpotent gamma factors

1905.13487v2

Technology to Counter Online Flaming Based on the Frequency-Dependent Damping Coefficient in the Oscillation Model Shinichi Kikuchi Tokyo Metropolitan University Tokyo 191–0065, Japan kikuchi-shinichi@ed.tmu.ac.jpChisa Takano Hiroshima City University Hiroshima, 731–3194 Japan takano@hiroshinma-cu.ac.jpMasaki Aida Tokyo Metropolitan University Tokyo 191–0065, Japan aida@tmu.ac.jp Abstract —Online social networks, which are remarkably ac- tive, often experience explosive user dynamics such as online flaming, which can significantly impact the real world. However, countermeasures based on social analyses of the individuals causing flaming are too slow to be effective because of the rapidity with which the influence of online user dynamics propagates. A countermeasure technology for the flaming phenomena based on the oscillation model, which describes online user dynamics, has been proposed; it is an immediate solution as it does not depend on social analyses of individuals. Conventional coun- termeasures based on the oscillation model assume that the damping coefficient is a constant regardless of the eigenfrequency. This assumption is, however, problematic as the damping co- efficients are, in general, inherently frequency-dependent; the theory underlying the dependence is being elucidated. This paper discusses a design method that uses the damping coefficient to prevent flaming under general conditions considering the frequency-dependence of the damping coefficient and proposes a countermeasure technology for the flaming phenomena. Index Terms —online flaming, user dynamics, oscillation model, damping coefficient I. I NTRODUCTION In recent years, with the spread of social networking sites such as Twitter and Facebook, users’ activities in online social networks have come to be closely connected to social activities in the real world, not only in online communities. As a result, the effects of explosive online user dynamics, including the flaming phenomena, are becoming more serious, and countermeasures are needed [1], [2]. Although it is desirable to respond immediately with direct countermeasures to eliminate the factors that cause the flaming phenomena, analyzing each event in detail, one by one, will be too slow to prevent the damage from spreading [3], [4]. Thus we need an engineering framework for flaming countermea- sures that does not depend on the details of each individual event. One such framework has been proposed [5]. This is based on the oscillation model on networks [6], [7] which is used to describe online user dynamics. Conventional countermeasures for the flaming phenomena have been discussed under the assumption that the damp- ing coefficient [5] is a constant, and independent of the eigenfrequency. However, it is known that regardless of thephenomenon, the damping coefficient generally depends on the eigenfrequency. In fact, the theoretical characteristics of the frequency dependence of the damping coefficient have recently been clarified [8]. Based on these insights, we can consider countermeasures for the flaming phenomena based on the oscillation model, even in general cases where the damping coefficient does depend on the eigenfrequency. In this paper, we introduce a design methodology that allows the damping coefficient to be used to counter the flaming phenomena even when the damping coefficient depends on the eigenfrequency; we use it to propose a countermeasure technology for the flaming phenomena. II. O SCILLATION MODEL FOR DESCRIBING ONLINE USER DYNAMICS Let the Laplacian matrix of the online social network (OSN) withnnodes be L, which is an nnsquare matrix, and the weight of the link from node ito nodej(i!j)bewij. In addition, let the eigenvalues of Lbe(= 0;1; :::; n1) and the eigenvectors associated with bev. We assume the eigenvalues are not duplicated. The eigenvalues of Lare generally complex numbers, whose range of existence is given by the largest Gershgorin disk [9] of Las Dmax G(L) =fz2C:jzdmaxjdmaxg; (1) wheredmax is the maximum weighted out-degree of the network. It is known that all the eigenvalues of Llie within its largest Gershgorin disk [5]. The oscillation model [5], [6] is a minimal model for de- scribing user dynamics in OSNs. Let xi(t)be the state of node (user)iat timet. Since the influence of interaction between users must propagate through any OSN at a finite speed, its description by the wave equation should be possible, which is the equation for describing the propagation of something at finite speed. For the state vector x(t) :=t(x1(t); :::; xn(t)), the wave equation in the OSN is written as d2 dt2x(t) +d dtx(t) =Lx(t); (2)arXiv:2011.08117v1 [cs.SI] 16 Nov 2020where is the matrix expressing the strength of the damping. Substituting the expansion of x(t)byvas x(t) =n1X =0a(t)v; (3) into the wave equation (4), yields the equation of motion for each oscillation mode a(t) (= 0;1; :::; n1)as d2 dt2a(t) + (!)d dta(t) =a(t); (4) where (!)is the damping coefficient; it depends on !=p and is expressed as (!) := 0+ 1 with the constant 0and 1[8]. Note that Re[ (!)] = 0+ 1Re[]0. The solution of (4) is written as a(t) =c+ exp (!) 2t+ iprexp i 2 t +c exp (!) 2tiprexp i 2 t ;(5) wherec+ andc are constants that depend on , andrand ( < )are, respectively, the absolute value and the argument of the following complex number: rexp(i) := (!) 22 = + (!) 22 :(6) In the oscillation model, the oscillation energy can be considered as the strength of the activity of user dynamics [10], [11]. Also, the situation in which oscillation energy diverges over time is considered to describe explosive user dynamics, which include the flaming phenomena. Therefore, in order to prevent explosive user dynamics, it is necessary to consider the conditions under which the oscillation energy does not diverge. By deriving the strength of the damping that satisfies this condition, we can obtain a framework in which the strength of damping can be adjusted to prevents the flaming phenomena. The conventional solutions to the flaming phenomena as- sume that the damping coefficient is a constant and indepen- dent of eigenfrequency. This corresponds to the special case of 1= 0in (4). Following [5], the condition under which the oscillation energy does not diverge is given as 8; 0 2prsin 2; (7) and the value of the damping coefficient required to satisfy this condition is given by 0p 2dmax: (8) In the next section, in order to consider countermeasures for the flaming phenomena, we discuss the conditions under which the oscillation energy does not diverge in the case of 16= 0.III. M ODEL OF EXPLOSIVE USERDYNAMICS CONSIDERING FREQUENCY -DEPENDENT DAMPING COEFFICIENT Since the oscillation energy is proportional to the square of the absolute value of a(t), we derive the condition under whicha(t)does not diverge and then the condition under which flaming does not occur. By decomposing the damping coefficient (!)into real and imaginary parts as in (!) = ( 0+ 1Re[]) + i 1Im[]: (9) a(t)is written as a(t) =c+ exp 0+ 1Re[] 2tprsin 2 t c+ exp i 1Im[] 2t+ iprcos 2 t +c exp 0+ 1Re[] 2t+prsin 2 t c exp i 1Im[] 2tiprcos 2 t : (10) To determine if a(t)diverges or not, we need to check whether the real components of the exponent of the expo- nential function in (10) are positive or negative, and if a(t) diverges, the following condition is satisfied: Re[ (!)] 2pr+ sin 2Re[ (!)] 2prsin 2 <0: This means the one of real components of the exponent is positive and the other is negative. Consequently, the condition under which the oscillation energy diverges is given by 9; 0+ 1Re[] 2pr<sin 2; (11) and the condition that the oscillation energy does not diverge is obtained as 8; 0+ 1Re[] 2prsin 2: (12) Since the conventional condition (7) that the oscillation energy does not diverge corresponds to the case of 1= 0 as per condition (12), the condition (12) for the frequency-dependent damping coefficient is a generalization of the conventional result. IV. C OUNTERMEASURE FOR FLAMING PHENOMENA GIVEN THE FREQUENCY -DEPENDENT DAMPING COEFFICIENT Based on condition (12), i.e., the oscillation energy does not diverge, we consider a design method for the damping coefficient to satisfy (12), and consider a countermeasure technology for the flaming phenomena by adjusting the value of the damping coefficient.A. Adjusting the Damping Coefficient Among parameters 0and 1, which determine the strength of damping, 1is the eigenfrequency dependent term. This means that the value of 1is a parameter predetermined by the structure of the network. Therefore, 0is the only parameter that can be manipulated independently of the network struc- ture. In this framework, even if various values are given as 1, we can counter the flaming phenomena by adjusting the value of 0. Here, the actual action to adjust the value of 0includes disseminating other information to attract users’ attention. In the following, we consider the value of 0necessary to prevent the triggering of explosive user dynamics, and we use its value in a flaming countermeasure. The range of eigenvalues of Lis the interior of the largest Gershgorin disk (including its boundaries) of radius dmax with center (dmax;0). From condition (12), the oscillation energy does not diverge, we can consider the range satisfying condition (12) on the complex plane. Then, if the largest Gershgorin disk of Llies completely within the range on the complex plane, the oscillation energy never diverges regardless of the network structure. In order to clarify the region on the complex plane in which condition (12) ensures that the oscillation energy does not diverge, the inequality of condition (12) is transformed as follows: Im[]2Z 44 0 14 2 1Re[]; (13) where Z= 4 0+ 4 3 0 1Re[] + 6 2 0 2 1Re[]2 + 4 2 0X+ 4 0 3 1Re[]3+ 8 0 1Re[]X + 4 1Re[]4+ 4 2 1Re[]2X; andXis the real part of (6). The range of eigenvalues of L, determined by the Gersh- gorin theorem, are written as Im[]2d2 max(Re[]dmax)2: (14) We compare (14) and the condition (13) that the oscillation energy does not diverge. Because both of them are axially sym- metric on by the real axis, we consider the upper-half plane. The condition that the largest Gershgorin disk is completely included the range of (13) is expressed as d2 max(Re[]dmax)2Z 4 (1 0 1 2 1Re[]); which can be transformed to 0 1+ 1 + 2 2 1dmax
Re[] ( 2 0+ 2 0 1dmax2dmax); (15) by considering Re[]0. We consider the conditions for satisfying inequality (15) in the following three cases. if 0 1+ 1 + 2 2 1dmax= 0, 2 0+ 2 0 1dmax2dmax0: (16)The range of 0that satisfies the above is as follows from 00 0q 2 1d2max+ 2dmax 1dmax: (17) if 0 1+ 1 + 2 2 1dmax>0, 2 0+ 2 0 1dmax2dmax 0 1+ 1 + 2 2 1dmaxRe[]: (18) In order for this inequality to hold, the numerator needs to be non-negative, so we obtain 2 0+ 2 0 1dmax2dmax0: (19) Considering 00, the condition of 0to ensure the non-divergence of oscillation energy for all eigenvalues is written as 0q 2 1d2max+ 2dmax 1dmax: (20) if 0 1+ 1 + 2 2 1dmax<0, 2 0+ 2 0 1dmax2dmax 0 1+ 1 + 2 2 1dmaxRe[]: (21) Inequality (21) is transformed to ( 0+ 2 1dmax)20 (22) Therefore, inequality (21) always holds in this case. To summarize the above results, the 0condition that en- sures the oscillation energy does not diverge for all eigenvalues is obtained as 0q 2 1d2max+ 2dmax 1dmax: (23) Therefore, given the maximum weighted out-degree of the network,dmax, and parameter 1of the damping coefficient, adjusting the value of 0to satisfy (23) will counter the flaming phenomena. B. Case Studies Using an example network with dmax= 100 , this section considers three cases of different values of 1, the frequency dependence of the damping coefficient: 1= 0, 1>0or 1<0. In all cases, we confirm that the region of the condition that the oscillation energy does not diverge includes the largest Gershgorin disk of Lby satisfying the condition (23) of 0. First, we confirm the case of complete flaming prevention with 0=p 2 1d2max+ 2dmax 1dmax, where 0is the minimum value that satisfies condition (23). Figure 1 shows the regions in which the oscillation energy does not diverge as given by condition (12), for the cases of 1= 0:1, 1= 0 and 1=0:1. These regions are depicted in blue. In addition, the largest Gershgorin disk of Lis depicted in red. In all figures, it can be seen that the regions wherein the oscillation energy does not diverge completely include the largest Gershgorin disk, so that no divergence of oscillation energy occurs regardless of the details of the network structure. Next, we show the case of incomplete flaming prevention by using 0=p 2 1d2max+ 2dmax 1dmax5, in which 0Fig. 1. Examples of complete flaming prevention Fig. 2. Examples of incomplete flaming prevention is less than the minimum value that satisfies condition (23). Figure 2 shows the regions wherein the oscillation energy does not diverge as indicated by condition (12), for the cases of 1= 0:1, 1= 0, and 1=0:1. In these cases, the regions cannot completely enclose the Gershgorin disk. If even just one eigenvalue appears outside of the region, the oscil- lation energy diverges and the flaming phenomenon occurs. Therefore, depending on the position of the eigenvalues of L, flaming prevention is not assured. V. C ONCLUSION In this paper, we proposed a countermeasure technology for the flaming phenomena based on the oscillation model with the frequency-dependent damping coefficient. The design method that yields the damping coefficients using condition (23) is a generalized version of the conventional countermeasure technology for the flaming phenomena. Regardless of the value of parameter 1, which is the strength of the frequency dependence of damping coefficient, we can prevent explosive user dynamics by setting parameter 0to be an appropriate value. Furthermore, the required value of 0can be determined from justdmax, which is the maximum weighted out-degree of the network, and 1, which is the strength of the frequency- dependence of the damping coefficient. One of the methods for increasing the value of 0in the actual OSNs is that to disseminate other information to attract users’ attention is mentioned.ACKNOWLEDGEMENT This research was supported by Grant-in-Aid for Scientific Research 19H04096 and 20H04179 from the Japan Society for the Promotion of Science (JSPS). REFERENCES [1] M. Alonzo and M. Aiken, “Flaming in electronic communication,” Decision Support Systems , vol. 36, no. 3, pp. 205–213, 2004. [2] Y . Adachi and F. Takeda, “The impact of online flaming on firm value: The evidence from Japan,” IPRC Discussion Paper Series , no. 14, 2014. [3] P. Moor, A. Heuvelman and R. Verleur, “Flaming on YouTube,” Com- puters in human behavior , vol. 26, no. 6, pp. 1536–1546, 2010. [4] S. Tsugawa and H. Ohsaki, “Negative messages spread rapidly and widely on social media,” The 2015 ACM on Conference on Online Social Networks , pp. 151–160, 2015. [5] M. Aida, Introduction to Network Dynamics , Morikita Publishing, 2020 (in Japanese). [6] M. Aida, C. Takano and M. Murata, “Oscillation model for describing network dynamics caused by asymmetric node interaction,” IEICE Transactions on Communications , vol. E101–B, no. 1, pp. 123–136, 2018. [7] M. Aida, C. Takano and M. Murata, “Oscillation model for network dynamics caused by asymmetric node interaction based on the sym- metric scaled Laplacian matrix,” The 12th International Conference on Foundations of Computer Science (FCS 2016) , pp. 38–44, 2016. [8] C. Takano and M. Aida, “Decay characteristics of user dynamics in online social networks,” IEEE Access , vol. 8, pp. 73986–73991, 2020. [9] R. Varga, Gershgorin and His Circles , Springer-Verlag, 2004. [10] C. Takano and M. Aida, “Revealing of the underlying mechanism of different node centralities based on oscillation dynamics on networks,” IEICE Transactions on Communications , vol. E101-B, no. 8, pp. 1820– 1832, 2018. [11] C. Takano and M. Aida, “Proposal of new index for describing node centralities based on oscillation dynamics on networks,” 2016 IEEE Global Communications Conference (GLOBECOM) , Washington, DC, pp. 1–7, 2016.

2020-11-16

Online social networks, which are remarkably active, often experience explosive user dynamics such as online flaming, which can significantly impact the real world. However, countermeasures based on social analyses of the individuals causing flaming are too slow to be effective because of the rapidity with which the influence of online user dynamics propagates. A countermeasure technology for the flaming phenomena based on the oscillation model, which describes online user dynamics, has been proposed; it is an immediate solution as it does not depend on social analyses of individuals. Conventional countermeasures based on the oscillation model assume that the damping coefficient is a constant regardless of the eigenfrequency. This assumption is, however, problematic as the damping coefficients are, in general, inherently frequency-dependent; the theory underlying the dependence is being elucidated. This paper discusses a design method that uses the damping coefficient to prevent flaming under general conditions considering the frequency-dependence of the damping coefficient and proposes a countermeasure technology for the flaming phenomena.

Technology to Counter Online Flaming Based on the Frequency-Dependent Damping Coefficient in the Oscillation Model

2011.08117v1

arXiv:2008.04523v1 [math.NA] 11 Aug 2020AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR GANG BAO∗, XIANG XU†,AND JIAN ZHAI‡ Abstract. This paper proposes a new and efficient numerical algorithm f or recovering the damping coefficient from the spectrum of a damped wave operator, which is a classical Borg-Levinson inverse spectral problem . The algorithm is based on inverting a sequence oftrace formulas , which are deduced by a recursive formula, bridging geometr ical and spectrum information explicitly in terms of Fredholm integral equations. Numerical examples are prese nted to illustrate the efficiency of the proposed algorithm. Key words. trace formulas, damped wave operator, inverse spectral pro blem AMS subject classifications. 11F72, 35R30, 65F18 1. Introduction. A damped wave equation describes a wave whose amplitude of os cillation decreases with time. It has far-ranging applications in many directio ns such as electromagnetic waves, acoustic waves and elastic waves. For instance, it was the first practical mo del to describe the radio transmission by spark gap transmitters during the wireless telegraphy era, which is now generally referred to as “Class B” emis- sion. In [ 1], the authors studied the harmonics on stringed instrument s and the damping coefficient was considered as the frictional resistance of the string, whic h may be caused by external forces. Moreover, similar mathematical models with damping term are proposed for linear elastic systems in [ 3,7], where the damping coefficient was considered as viscosity. More appli cations can be found in the survey [ 10] and the references cited therein. Consider the one-dimensional da mped wave equation with unit wave speed and viscous damping α(x): utt(x,t)−uxx(x,t)+α(x)ut(x,t)=0,(x,t)∈(0,1)×[0,∞), u(·,0)=f0,ut(·,0)=f1,t>0, u(·,t)satisfies certain boundary conditions at x=0 and x=1 for t∈R+.(1.1) Assume that α(x)∈L∞(0,1)is real-valued and 0 ≤2a≤α(x)≤2b<+∞. We can rewrite ( 1.1) in a vector form: (1.2) Vt=A(α)V, where V=(u,ut)and (1.3) A(α)=/parenleftBigg0 I/parenleftBig d2 dx2/parenrightBig bc−α(x)/parenrightBigg . Here the subscript “bc” represents appropriate boundary co nditions at x=0,1, eg. Dirichlet, etc., to be described in details in Section 2. The initial condition (f0,f1)needs to be consistent with the boundary condition. The well-posedness of the initial boundary valu e problem for ( 1.2) with initial value V(0) = (u(·,0),ut(·,0))=( f0,f1)∈L2(0,1)2can be obtained by the standard semigroup approach [ 6,8]. Moreover, ∗School of Mathematical Sciences, Zhejiang University, Han gzhou, Zhejiang, China. G. Bao’s research was supported in p art by NSFC 11621101. ( baog@zju.edu.cn ). †School of Mathematical Sciences, Zhejiang University, Han gzhou, Zhejiang, China. X. Xu’s research was supported in pa rt by the Fundamental Research Funds for the Central Universities. ( xxu@zju.edu.cn ). ‡Institute for Advanced Study, The Hong Kong University of Sc ience and Technology, Hong Kong, China. ( iasjzhai@ust.hk ). 12 G. BAO, X. XU AND J. ZHAI it is known that if α(x)∈L∞(0,1)then A(α)has a compact inverse and hence a discrete spectrum, consist ing of countably many eigenvalues, denoted by σp(A(α))={λj(A(α))}j∈J. The present work is devoted to the inverse problem of recover ing the damped coefficient α(x)from the spectrum σp(A(α)). This is a classical inverse spectral problem in mathematic al physics and relates to a variety of vibration absorption problems in the engineerin g literature, see [ 13]. Mathematically, it can be viewed as a classical Borg-Levinson inverse spectral problem . The uniqueness on determination of α(x) from the Dirichlet eigenvalues was established for αeven, with respect to x=1/2, see [ 4]. In [ 14], for weakly damped strings, i.e., with no purely imaginary eigen values, the determination of the potential and the boundary conditions were considered by the given spectr um and length of the string. In [ 2], Borisov et al showed the criterion for the damping term to be constant an d expect this inverse problem to be more rigid than Sturm-Liouville problem since there is no other smooth damping term yielding the same spectrum as constant damping. For numerical reconstruction of the damp ing coefficient, to the authors’ best knowledge, the only available approach was introduced by Cox and Embree [4], which was based on a refined asymptotic formula for the large eigenvalues. However, it is known that for inverse Sturm-Liouville operators, there are many works on numerical algorithms, see [ 12,15,16] and the references therein for an overview on numerical progress. Moreover, Xu and Zhai [ 18] have developed a numerical scheme for recovering a density in the Sturm-Liouville operator based on a sequence oftrace formulas which give an explicit relation between the eigenvalues and the unknown coefficient recentl y. In this paper, we propose a novel numerical scheme for recove ring the damping coefficient α(x)from the spectrum σp(A(α))in a similar framework as [ 18]. The scheme is based on the explicit formulas which will be derived in the next section for the following maps (1.4) α→∑ j∈Jλj(A(α))−s,fors=1,···,∞. where ∑j∈Jλj(A(α))−sare traces of (A(α))−s. It has been shown in [ 18] that inverting the above maps are severely ill-posed when Ais a Laplacian operator with Dirichelt bounary conditions. According to the property of trace class operators of (A(α))−s(s=2,···,∞), it makes sense to reduce the numerical instability by inverting the following maps α→/braceleftBig ∑ j∈JTn(λj(A(α))−1)/bracerightBig∞ n=1, with a collection of carefully chosen polynomials {Tn(z)}∞ n=1,z∈C. It should be noted that due to the inherent difficulties for damped wave operator, the two ingr edients of the numerical algorithm in [ 18], i.e., trace formulas and stabilizing polynomials are completely different. Due to the model difference, the trace formulas are derived based on the resolvent of A(α), instead of the Green function for the Sturm-Liouville operator. Moreover, since the eigenvalue distribution is n o longer in the real axis as in the previous case, the choice of stabilizing polynomial which depends on the spect rum distribution becomes more complicated. The rest of the paper is organized as follows. Section 2is devoted to establishing the desired trace formulas. By analyzing the resolvent of A(α), we arrive at some explicit recursive formulas. In Section 3, we show the injectivity of the Fr´ echet derivative of the map (1.4) at a constant damping. In Section 4, we present the algorithm with implementation details. In Sect ion5, we conduct several numerical experiments to illustrate the efficiency of our algorithm. Impacts of dif ferent parameters are also discussed in this section. 2. Trace formulas. In this section, we derive a sequence of trace formulas usefu l for inverting α(x). LetTbe an unbounded operator on L2(0,1)such that T f=id dxfwith appropriate boundary conditions at x=0,1. The operator Tneeds to be densely defined and closed. Here, we list some exam ples of T, namely,AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 3 Tmin,T0,T1,Tω, which are carefully defined and characterized in [ 10]. For the convenience of readers, we summarize some results here. The domains of these operators are: dom(Tmin)={f∈L2(0,1)|f∈AC([0,1]);f(0)=f(1)=0}, dom(T0)={f∈L2(0,1)|f∈AC([0,1]);f(0)=0;f′∈L2(0,1)}, dom(T1)={f∈L2(0,1)|f∈AC([0,1]);f(1)=0;f′∈L2(0,1)}, dom(Tω)={f∈L2(0,1)|f∈AC([0,1]);f(1)=ωf(0);f′∈L2(0,1)}. Here AC([0,1])denotes the space of absolutely continuous functions on [0,1]. For ω∈R\{0,1}, we have ker(Tmin)=ker(T0)=ker(T1)=ker(Tω)={0}. Then T∗T f=−f′′for any T=Tmin,T0,T1,Tωwith dom(Tmin∗Tmin)={f∈L2(0,1)|f,f′∈AC([0,1]),f(0)=f(1)=0,f′′∈L2}, dom(T0∗T0)={f∈L2(0,1)|f,f′∈AC([0,1]),f(0)=f′(1)=0,f′′∈L2}, dom(T1∗T1)={f∈L2(0,1)|f,f′∈AC([0,1]),f′(0)=f(1)=0,f′′∈L2}, dom(Tω∗Tω)={f∈L2(0,1)|f,f′∈AC([0,1]),f(1)=ωf(0),ωf′(1)=f′(0);f′′∈L2}. By the fact ker(T∗T)=ker(T), we have the invertibility of T∗TforT=Tmin,T0,T1,Tωwith ω∈R\{0,1}. Remark 2.1. Notice that Tmin∗Tmin=−∆D=−/parenleftBig d2 dx2/parenrightBig Dis the Dirichlet Laplacian. We take Tto be any of the above defined operators. Define (2.1) A(α)=/parenleftbigg0 I −T∗T−α(x)/parenrightbigg on the space L2([0,1])2. Since T∗Tis coercive, then 0 ∈ρ(A(α))[10, Theorem 2.3]. It is easy to see that if λis an eigenvalue of A(α)with eigenvector u=[y,z], then z=λyand (2.2) y′′−λ αy−λ2y=0, with ysatisfying suitable boundary conditions. It is clear that λis also an eigenvalue of A(α)with eigen- vector u= [y,z] = [ y,λy]. Moreover, by [ 10, Lemma 2.5], the two eigenvalues λandλhave the same geometric and algebraic multiplicities. Actually, the spe ctrum σp(A(α))consists of two infinite sequences {λ±j(A(α))}∞ j=1, where Im λ−j=−Imλj. We denote J=Z\{0}, and σp(A(α))={λj}j∈J={λj(A(α))}j∈J. The eigenvalues are ordered as follows ···≤ Imλ−2≤Imλ−1≤Imλ1≤Imλ2≤···4 G. BAO, X. XU AND J. ZHAI counting algebraic multiplicities. If the spectrum does no t contain real eigenvalues, this labeling of eigen- values is clear and λ−j=λjfor any j. If real eigenvalues exist, one can invoke [ 2, Lemma 4.1] and [ 5, Theorem 5.3]. Next we give a sufficient condition for the none xistence of real eigenvalues. LEMMA 2.2. If b</radicalbig µ1(T∗T), where µ1(T∗T)is the smallest eigenvalue of T∗T , then σp(A(α))∩ R=/ 0. Proof. Integrating ( 2.2) against y, we obtain /integraldisplay1 0|y′|2dx+λ/integraldisplay1 0α|y|2dx+λ2/integraldisplay1 0|y|2dx=0 fory∈dom(T∗T),T=Tmin,T0,T1,Tωwith ω∈R\{0,1}. Then we find λ=−/integraltext1 0α|y|2dx±/parenleftBig (/integraltext1 0α|y|2dx)2−4/integraltext1 0|y′|2dx/integraltext1 0|y|2dx/parenrightBig1/2 2/integraltext1 0|y|2dx. Using α≤2b, we have /parenleftbigg/integraldisplay1 0α|y|2dx/parenrightbigg2 −4/integraldisplay1 0|y′|2dx/integraldisplay1 0|y|2dx ≤4b2/parenleftbigg/integraldisplay1 0|y|2dx/parenrightbigg2 −4/integraldisplay1 0|y′|2dx/integraldisplay1 0|y|2dx ≤4/parenleftbigg/integraldisplay1 0|y|2dx/parenrightbigg2/parenleftBigg b2−/integraltext1 0|y′|2dx /integraltext1 0|y|2dx/parenrightBigg . Notice that the smallest eigenvalue of T∗Tisµ1(T∗T)=infy∈dom(T∗T)/integraltext1 0|y′|2dx/integraltext1 0|y|2dx. Therefore if b</radicalbig µ1(T∗T), λis not real-valued. Note that µ1(Tmin∗Tmin)=π2,µ1(T0∗T0)=µ1(T1∗T1)=1 4π2. Now we proceed to deriving the trace formulas for (A(α))−n−1,n=0,1,2···. All the trace formulas can be generated by a recursive relation, which is used for th e inversion algorithm. The trace formulas for n=2kare obtained in [ 10], but in a less explicit form. Denote R(ζ)=−(2ζ+α)Q(ζ), Q(ζ)=( T∗T+ζ2+ζα)−1. Note that Q(ζ),R(ζ)are of trace class in the separable Hilbert space L2(0,1). Some useful properties of trace-class operators are summarized in [ 18]. We denote Atr=Bif the operators AandBhave the same trace. By simple calculations, we have the following explicit expr ession for the resolvent of A(α)(cf. [ 5]) (A(α)−ζ)−1=/parenleftbigg −Q(ζ)(ζ+α)−Q(ζ) I−ζQ(ζ)(ζ+α)−ζQ(ζ)/parenrightbigg , forζ∈R\{0}with|ζ|sufficiently small. Notice that the operator (A(α)−ζ)−1is not of trace class (The only “bad” term is the identity operator in the lower left ent ry). However it is clear that ∂ ∂ζ(A(α)−ζ)−1=∂ ∂ζ/parenleftbigg−Q(ζ)(ζ+α)−Q(ζ) −ζQ(ζ)(ζ+α)−ζQ(ζ)/parenrightbiggAN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 5 is of trace class. Moreover, we have ∂ ∂ζ(A(α)−ζ)−1tr=∂ ∂ζ[−Q(ζ)(ζ+α)−ζQ(ζ)]tr=∂ ∂ζR(ζ). We note here that although (A(α)−ζ)−1is not of trace class, the operator R(ζ)is. Next we derive a sequence of trace formulas associated with R(ζ). First notice LEMMA 2.3. Q′(ζ)=−Q(ζ)(2ζ+α)Q(ζ). Proof. To prove this, we only need to directly calculate Q′(ζ)=lim h→0Q(ζ+h)−Q(ζ) h =lim h→0/parenleftbig T∗T+(ζ+h)2+(ζ+h)α/parenrightbig−1−/parenleftbig T∗T−ζ2−ζα/parenrightbig−1 h =lim h→01 h/parenleftbig T∗T+(ζ+h)2+(ζ+h)α/parenrightbig−1/parenleftbig T∗T+ζ2+ζα−T∗T−(ζ+h)2−(ζ+h)α/parenrightbig /parenleftbig T∗T+ζ2+ζα/parenrightbig−1 =lim h→01 h/parenleftbig T∗T+(ζ+h)2+(ζ+h)α/parenrightbig−1/parenleftbig −2hζ+h2−hα/parenrightbig/parenleftbig T∗T+ζ2+ζα/parenrightbig−1 =−/parenleftbig T∗T+ζ2+ζα/parenrightbig−1(2ζ+α)/parenleftbig T∗T+ζ2+ζα/parenrightbig−1 =−Q(ζ)(2ζ+α)Q(ζ). To derive trace formulas, we start with R(ζ)=−(2ζ+α)Q(ζ). By the chain rule, we can calculate the derivatives of R(ζ)with respect to ζas follows R′(ζ)=−2Q(ζ)+(2ζ+α)Q(ζ)(2ζ+α)Q(ζ), and R′′(ζ)=2Q(ζ)(2ζ+α)Q(ζ)+2Q(ζ)(2ζ+α)Q(ζ)+2(2ζ+α)Q(ζ)Q(ζ) −2(2ζ+α)Q(ζ)(2ζ+α)Q(ζ)(2ζ+α)Q(ζ) =2(2ζ+α)Q(ζ)Q(ζ)+2/parenleftbig 2Q(ζ)−(2ζ+α)Q(ζ)(2ζ+α)Q(ζ)/parenrightbig (2ζ+α)Q(ζ) =−2R(ζ)Q(ζ)−2R′(ζ)(2ζ+α)Q(ζ). We can continue and obtain R′′′(ζ)=−2R′(ζ)Q(ζ)+2R(ζ)Q(ζ)(2ζ+α)Q(ζ)−2R′′(ζ)(2ζ+α)Q(ζ) −4R′(ζ)Q(ζ)+2R′(ζ)(2ζ+α)Q(ζ)(2ζ+α)Q(ζ) =−6R′(ζ)Q(ζ)−R′′(ζ)(2ζ+α)Q(ζ)−2R′′(ζ)(2ζ+α)Q(ζ) =−6R′(ζ)Q(ζ)−3R′′(ζ)(2ζ+α)Q(ζ).6 G. BAO, X. XU AND J. ZHAI We observe that 1 2!R′′(ζ)=−R(ζ)Q(ζ)−R′(ζ)(2ζ+α)Q(ζ), 1 3!R′′′(ζ)=−R′(ζ)Q(ζ)−1 2!R′′(ζ)(2ζ+α)Q(ζ). Generally, we have the following recursive relation: LEMMA 2.4. Assume ζ∈R\{0}with|ζ|sufficiently small, such that ζ∈ρ(A(α)), then (2.3)1 n!R(n)(ζ)=−1 (n−2)!R(n−2)(ζ)Q(ζ)−1 (n−1)!R(n−1)(ζ)(2ζ+α)Q(ζ). Proof. We prove it by induction. We have already seen that ( 2.3) holds true for the case n=2. Assume that for n, the above ( 2.3) holds true. Then we proceed 1 (n+1)!R(n+1)(ζ) =1 n+11 n!d dζR(n)(ζ) =1 n+11 (n−2)!R(n−2)(ζ)Q(ζ)(2ζ+α)Q(ζ)−1 n+11 (n−2)!R(n−1)(ζ)Q(ζ) −1 n+11 (n−1)!R(n)(ζ)(2ζ+α)Q(ζ)−2 n+11 (n−1)!R(n−1)(ζ)Q(ζ) +1 n+11 (n−1)!R(n−1)(ζ)(2ζ+α)Q(ζ)(2ζ+α)Q(ζ) =1 n+1/parenleftbigg1 (n−2)!R(n−2)(ζ)Q(ζ)+1 (n−1)!R(n−1)(ζ)(2ζ+α)Q(ζ)/parenrightbigg (2ζ+α)Q(ζ) −1 n+11 (n−1)!R(n)(ζ)(2ζ+α)Q(ζ)−1 n+11 (n−2)!R(n−1)(ζ)Q(ζ) −2 n+11 (n−1)!R(n−1)(ζ)Q(ζ) =−1 (n+1)!R(n)(ζ)(2ζ+α)Q(ζ)−1 n+11 (n−1)!R(n)(ζ)(2ζ+α)Q(ζ) −/parenleftbigg1 n+1+2 (n+1)(n−1)/parenrightbigg1 (n−2)!R(n−1)(ζ)Q(ζ) =−1 n!R(n)(ζ)(2ζ+α)Q(ζ)−1 (n−1)!R(n−1)(ζ)Q(ζ). The lemma is proved. We use the notation Rn(α) =1 n!R(n)(0). Evaluating the recursive relation ( 2.3) atζ=0 gives the following proposition. PROPOSITION 2.5. The follow recursive formula holds: Rn(α)=−Rn−2(α)(T∗T)−1−Rn−1(α)α(T∗T)−1(2.4)AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 7 for n=2,3,···, with R0(α)=R(0)=−αQ(0)=−α(T∗T)−1, R1(α)=R′(0)=−2Q(0)+αQ(0)αQ(0)=−2(T∗T)−1+α(T∗T)−1α(T∗T)−1. The following lemma is similar to [ 10, Theorem 5.11]. LEMMA 2.6. Denote λj=λj(A(α)). We have that for any n =1,2,···, Im∑ j∈Jλj(A(α))−n−1=0 (2.5) and trace(Rn(α))=∑ j∈Jλj(A(α))−n−1=Re∑ j∈Jλj(A(α))−n−1. (2.6) Proof. Assume ζ∈(−ε0,ε0)\{0},ε0>0 sufficiently small such that (−ε0,ε0)⊂ρ(A(α)). Then, we have trace/parenleftbigg∂ ∂ζ[(A(α)−ζ)−1]/parenrightbigg =∑ j∈J∂ ∂ζ[(λj(A(α))−ζ)−1] =∑ j∈J∂ ∂ζ[λj(A(α))−1/parenleftbig 1−ζλj(A(α))−1/parenrightbig−1] =∑ j∈J∂ ∂ζ[∞ ∑ n=0λj(A(α))−1/parenleftbig ζλj(A(α))−1/parenrightbign] =∞ ∑ n=1/parenleftBigg ∑ j∈Jλj(A(α))−n−1/parenrightBigg ζn−1. Notices that trace/parenleftbigg∂ ∂ζ[(A(α)−ζ)−1]/parenrightbigg =trace(R′(ζ))=∞ ∑ n=1trace(Rn(α))ζn−1. thus the lemma is proved. Remark 2.7. Note that ∑j∈Jλj(A(α))−1is not summable. However, it is proved in [ 10, Theorem 5.11] that ∑ j∈JReλj(A(α))−1=∑ j∈JReλj |λj|2=trace(R0(α)), where the sum is convergent. Also, it is clear that (∑−1 j=−N+∑N j=1)Imλ−1 j=0 for any N, since Im λ−j= −ImλjTherefore, the identities ( 2.5) and ( 2.6) are also valid for n=0 when using the regularized sum limN→+∞(∑−1 j=−N+∑N j=1).8 G. BAO, X. XU AND J. ZHAI One can use Proposition 2.5and Lemma 2.6to derive an infinite sequence of trace formulas. Let us write down a few ones. ∑ j∈Jλj(A(α))−1=trace(R0(α))= trace(−α(T∗T)−1), ∑ j∈Jλj(A(α))−2=trace(R1(α))= trace(−2(T∗T)−1+α(T∗T)−1α(T∗T)−1), ∑ j∈Jλj(A(α))−3=trace(R2(α)) =trace(α(T∗T)−2+2(T∗T)−1α(T∗T)−1−α(T∗T)−1α(T∗T)−1α(T∗T)−1) =trace(3(T∗T)−1α(T∗T)−1−α(T∗T)−1α(T∗T)−1α(T∗T)−1), ∑ j∈Jλj(A(α))−4=trace(R3(α)) =trace/parenleftBig 2(T∗T)−2−α(T∗T)−1α(T∗T)−2−α(T∗T)−2α(T∗T)−1 −2(T∗T)−1α(T∗T)−1α(T∗T)−1 +α(T∗T)−1α(T∗T)−1α(T∗T)−1α(T∗T)−1/parenrightBig =trace/parenleftbig 2(T∗T)−2−4(T∗T)−1α(T∗T)−1α(T∗T)−1 +α(T∗T)−1α(T∗T)−1α(T∗T)−1α(T∗T)−1/parenrightbig , etc. We see that the above trace formulas establish a very clear re lation between the damping coefficient αand the spectrum of A(α). We propose an inversion scheme for the map (2.7) F:α→{tn(α)}∞ n=0:={trace(Rn(α))}∞ n=0={∑ j∈Jλj(A(α))−n−1}∞ n=0 for the recovery of α(x). 3. Injectivity of a linearized map. The unique determination of an even damping α(x) =α(1−x) from the Dirichlet eigenvalues {λj(A(α))}j∈Jis known (cf. [ 4]). However, it is not clear whether there is a one-to-one correspondence between {λj(A(α))}j∈Jand{∑j∈Jλj(A(α))−n−1}∞ n=0. It is also not clear whether the map ( 2.7) is injective. In the next section, we consider the lineariz ation of the map Fat constant damping and show the injectivity of the linearized map. THEOREM 3.1. Assume T =Tmin. The Fr ´echet derivative of the map Fatα=α0, where α0is a constant, F′[α0]:δα→{t′ n[α0](δα)}∞ n=0={trace(R′ n[α0](δα))}∞ n=0 is injective for δα(x)=δα(1−x). Proof. We calculate R′ 0[α0](δα)=−δα(T∗T)−1,AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 9 R0(α0)=−α0(T∗T)−1, and R′ 1[α0](δα)=α0δα(T∗T)−2+α0(T∗T)−1δα(T∗T)−1, R1(α0)=α2 0(T∗T)−2−2(T∗T)−1. We claim that R′ n−1[α0](δα)=(−1)nαn−1 0/parenleftBig δα(T∗T)−n+(T∗T)−1δα(T∗T)−n+1+···+ (T∗T)−n+1δα(T∗T)−1/parenrightBig +∑ m+ℓ≤n−1cn−1,m,ℓ(α0)(T∗T)−mδα(T∗T)−ℓ, Rn−1(α0)=(−1)nαn 0(T∗T)−n+∑ k≤n−1dn−1,k(α0)(T∗T)−k, and prove by induction. Here cn−1,m,ℓ(α0)anddn−1,k(α0)are some constants depending on α0. Using the recursive formula ( 2.4), we have R′ n[α0](δα)=−R′ n−2[α0](δα)(T∗T)−1−R′ n−1[α0](δα)α0(T∗T)−1−Rn−1(α0)δα(T∗T)−1 =(−1)n+1αn−1 0/parenleftBig δα(T∗T)−n+(T∗T)−1δα(T∗T)−n+1+··· +(T∗T)−n+1δα(T∗T)−1/parenrightBig α0(T∗T)−1+(−1)n+1αn 0(T∗T)−nδα(T∗T)−1 +∑ m+ℓ≤ncn,m,ℓ(α0)(T∗T)−mδα(T∗T)−ℓ =(−1)n+1αn 0/parenleftBig δα(T∗T)−n−1+(T∗T)−1δα(T∗T)−n+··· +(T∗T)−nδα(T∗T)−1/parenrightBig +∑ m+ℓ≤ncn,m,ℓ(α0)(T∗T)−mδα(T∗T)−ℓ. Similarly, we can prove Rn(α0)=(−1)n+1αn+1 0(T∗T)−n−1+∑ k≤ndn,k(α0)(T∗T)−k. The claim is proved. This implies that trace(R′ n−1[α0](δα))=(−1)n+1nαn−1 0trace(δα(T∗T)−n)+n−1 ∑ k=1cn,k(α0)trace(δα(T∗T)−k) with some constants cn,k(α0)depending on α0. Therefore, if F′[α0](δα)=0, we have trace (R′ n−1[α0](δα))= 0 for n=1,2,···, and thus trace(δα(T∗T)−n)=trace(δα(−∆D)−n)=0.10 G. BAO, X. XU AND J. ZHAI Equivalently, we have /integraldisplay1 0gn(x,x)δα(x)dx=0, where gn(x,y)is the Green’s function for (−∆D)n, and by Mercer’s Theorem (see, for example, [ 11]) we have gn(x,y)=∞ ∑ m=12 m2nπ2nsinmπxsinmπy, and thus gn(x,x)=∞ ∑ m=12 m2nπ2n(sinmπx)2=∞ ∑ m=12 m2nπ2n1−cos2 mπx 2. Notice that 0=lim n→+∞π2n/integraldisplay1 0gn(x,x)δα(x)dx=/integraldisplay1 0δα(x)(1−cos2 πx)dx. Then /integraldisplay1 0/parenleftBigg ∞ ∑ m=22 m2nπ2n1−cos2 mπx 2/parenrightBigg δα(x)dx=0, for every n. Then 0=lim n→+∞22nπ2n/integraldisplay1 0/parenleftBigg ∞ ∑ m=22 m2nπ2n1−cos2 mπx 2/parenrightBigg δα(x)dx=/integraldisplay1 0δα(x)(1−cos4 πx)dx. Continuing this process, we have /integraldisplay1 0δα(x)(1−cos2 mπx)dx=0 for each m. Taking the limit m→+∞, and invoking the Riemann-Lebesgue Lemma lim m→+∞/integraldisplay1 0δα(x)cos2 mπxdx=0, we have /integraldisplay1 0δα(x)dx=0. Thus we end up with /integraldisplay1 0δα(x)cos2 mπxdx=0 form=0,1,2,···. Then δα=0, and the injectivity is proved.AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 11 4. Inversion Algorithm. We derive an algorithm for recovering α(x)from the spectral of A(α)based on the trace formulas derived in Section 2. We only describe the algorithm for the operator with Dirich let boundary condition, that is T=Tmin. Other boundary conditions can be dealt with in the same way. Assume{µℓ,φℓ(x)}∞ ℓ=1are the eigenvalues and eigenfunctions of −∆D=Tmin∗Tmin, where −∆Dφℓ=−φ′′ ℓ=µℓφℓ, µℓ=ℓ2π2,φℓ(x)=√ 2sinℓπx. Define the unitary operator W:L2(0,1)→l2such that Wf={f1,f2,···}, where fadmits the decomposition under the basis {φℓ}∞ ℓ=1ofL2(0,1): f(x)=∞ ∑ ℓ=1fℓφℓ(x). Then we have the spectral decomposition of (−∆D)−1as (−∆D)−1=W−1diag/parenleftbig µ−1 1,µ−1 2,···,µ−1 ℓ,···/parenrightbig W. Similarly, the multiplication operator Mα:L2(0,1)→L2(0,1):(Mαf)(x) =α(x)f(x)also can be decom- posed as follows Mα=W−1M(α)W, where (M(α))i j=/integraldisplay1 0α(x)φi(x)φj(x)dx. To see this, one only needs to notice (Mαf)(x)=∞ ∑ i=1/parenleftbigg/integraldisplay1 0αf(x)φi(x)dx/parenrightbigg φi(x) =∞ ∑ i=1/parenleftBigg/integraldisplay1 0α(x)∞ ∑ j=1fjφj(x)φi(x)dx/parenrightBigg φi(x) =∞ ∑ i=1/parenleftBigg ∞ ∑ j=1/parenleftbigg/integraldisplay1 0α(x)φj(x)φi(x)dx/parenrightbigg fj/parenrightBigg φi(x). Denote (M1(α))i j=−µ−1 j/integraldisplay1 0α(x)φi(x)φj(x)dx, M2(α)=2M1(1)+M1(α)2.12 G. BAO, X. XU AND J. ZHAI By direct calculation, it is easy to see that R0(α)=−α(T∗T)−1=−Mα(−∆D)−1 =−W−1M(α)WW−1diag/parenleftbig µ−1 1,µ−1 2,···,µ−1 n,···/parenrightbig W =W−1M1(α)W, and therefore R1(α)=−2(−∆D)−1+Mα(−∆D)−1Mα(−∆D)−1 =2W−1M1(1)W+W−1M1(α)WW−1M1(α)W =W−1M2(α)W. Generally, we define Mn(α)=Mn−1(α)M1(α)+Mn−2(α)M1(1). (4.1) Then, one can verify that Rn−1(α)=W−1Mn(α)W. (4.2) Since Wis a unitary operator which can be viewed as a rotation transf ormation and keeps eigenvalues invariant when both WandW−1are applied, thus we have PROPOSITION 4.1. The following relations hold: trace(Mn(α))= trace(Rn−1(α))=∑ j∈Jλ−n j,for n=1,2,···. (4.3) When n=1 in the above formula, we need to use the regularized summati on as in Remark 2.7. Remark 4.2. The proposition gives an explicit expression between t he damping coefficient α(x)and the spectral data {λj}j∈Jin terms of a series of Fredholm equations. For example if n=1, then we have ∞ ∑ ℓ=1−µ−1 ℓ/integraldisplay1 0α(x)φ2 ℓ(x)dx=∑ j∈Jλ−1 j. Solving an infinite series of Fredholm integral equations ( 4.3) is severely ill-posed. The main reason is that N ∑ n=1/parenleftBigg ∑ j∈Jλ−n j(α)−∑ j∈Jλ−n j(αtrue)/parenrightBigg2 (4.4) is not a good choice to measure the misfit. As in [ 18], we need to use a sequence of “proper” polynomials {Tn}N n=1and measure the misfit as N ∑ n=1/parenleftBigg ∑ j∈JTn(λ−1 j(α))−∑ j∈JTn(λ−1 j(αtrue))/parenrightBigg2 . (4.5)AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 13 0 0.2 0.4 0.6 0.8 10.811.21.41.61.822.22.42.6 (a) the damping coefficient−3 −2.5 −2 −1.5 −1 −0.5 0−100−80−60−40−20020406080100 (b) distribution of the eigenvalues−0.8 −0.6 −0.4 −0.2 00.2 0.4 0.6−0.6−0.4−0.200.20.40.6 (c) the reciprocal of the eigenvalues Fig. 1: Distribution of eigenvalues Before proceeding to seeking proper polynomials, which is c ritical to the success of the inversion, let us first summarize some properties of the spectrum of A(α). We refer to [ 2,5] for more details. Assume α0=/integraltext1 0α(x)dx. Then 1. The spectrum of A(α)is symmetric about the real axis, i.e., σp(A(α))= σp(A(α)); 2. The spectrum of A(α)is contained in {λ∈C:|λ|≥π,−b≤Reλ≤−a}∪[−b−(b2−π2)1/2 +,−a+(b2−π2)1/2 +]; 3. The eigenvalue λj(A(α))has the asymptotic behavior (4.6) λj(A(α))=−α0 2+jπi+O/parenleftbigg1 j/parenrightbigg . The distribution of a sample damping coefficient is depicted in Figure 1. Recall that the conformal mapping1 zon the complex plane Cmaps the line {ℜz=−α0 2}to the circle B(−1 α0,0)(1 α0), then{λj(A(α))−1}j∈Jscatter near that circle if the damping αis not large, see Figure 1(c). We need the polynomials to be well-behaved on the circle, and create enough oscillations near z=0 to discriminate the measured eigenvalues. We use the polynomi als Tn(z)=z(α0z+1)n−1, where α0is approximated using the asymptotics ( 4.6). Moreover, denote /tildewideTn(z)=z2(α0z+1)n−2=zTn−1(z)=1 α0Tn(z)−1 α0Tn−1(z).14 G. BAO, X. XU AND J. ZHAI 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1012345678910 (a) the damping coefficient-4 -3.5 -3 -2.5 -2 -1.5 -1-60-40-200204060 (b) distribution of the eigenvalues-0.4 -0.3 -0.2 -0.1 0 0.1 0.2-0.25-0.2-0.15-0.1-0.0500.050.10.150.20.25 (c) the reciprocal of the eigenvalues Fig. 2: Distribution of eigenvalues for large damping We use the following recursive relation for the polynomials ofTn. Tn+1(z)=(α0z+1)Tn(z) =(α2 0z2+2α0z+1)Tn−1(z) =α2 0z2Tn−1(z)+2α0z2(α0z+1)n−2+Tn−1(z) =α2 0z2Tn−1(z)+2α0/tildewideTn(z)+Tn−1(z) =α2 0z2Tn−1(z)+2(Tn(z)−Tn−1(z))+Tn−1(z) =2Tn(z)−Tn−1(z)+α2 0z2Tn−1(z).(4.7) We note z2Tn−1(z)=z/tildewideTn(z)for later use. Remark 4.3. This choice of polynomials does not work well for large d ampings, for which the eigen- values λ−1 jforjsmall might be far away from the circle B(−1 α0,0)(1 α0). See Figure 2for the distribution of the eigenvalues for an example of Freitas [ 9], (4.8) α(x)=3.1133 π 2+1.4896 πcos2 πx. Notice that we actually have 2 b=supx∈[0,1]α(x)>2π=2/radicalbig µ1(T∗T). However, it still can be used for low frequency approximation, which will be demonstrated by Exa mple 5.5in the next section. Also, a more complicated strategy for choosing polynomials might enabl e one to go to higher frequencies. We use truncated Fourier cosine series to approximate an eve n damping coefficient, (4.9) αM(x)=M ∑ m=1amcos2(m−1)πx, and denote a={a1,a2,···,aM}. With a little abuse of notations, we use Mn(a)in place of Mn(α)in the following. Then M1(a)=M ∑ m=1amM1(em),AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 15 where (M1(em))i j=−2 π2j2/integraldisplay1 0siniπxsinjπxcos2(m−1)πxdx = 1 2π2j2, i+j+2m−2=0, 1 2π2j2, i+j−2m+2=0, −1 2π2j2,i−j+2m−2=0,m/\e}atio\slash=1, −1 2π2j2,i−j−2m+2=0,m/\e}atio\slash=1, −1 π2j2,i=j,m=1, 0, otherwise , andem={a1=0,···,am−1=0,am=1,am+1=0,···,aM=0}. Remark 4.4. The matrix M1(em)here is not a symmetric matrix, in contrast to the one defined i n [18]. Next we define T1(a)=M1(a), T2(a)=α0M2(a)+M1(a)=α0(2M1(e1)+M1(a)2)+M1(a), and /tildewideTn(a)=1 α0(Tn(a)−Tn−1(a)), Tn+1(a)=2Tn(a)−Tn−1(a)+α2 0/parenleftBig Tn−1(a)M1(e1)+/tildewideTn(a)M1(a)/parenrightBig =2Tn(a)−Tn−1(a)+α2 0Tn−1(a)M1(e1)+α0(Tn(a)−Tn−1(a))M1(a), forn=2,3,···, in parallel with ( 4.7). One can check that if T(z) =∑n i=1bizi, then Tn(a) =∑n i=1biMi(a). Therefore, we obtain PROPOSITION 4.5. For n=1,2,···, trace(Tn(a))=∑ j∈JTn(λ−1 j). In light of the above proposition, we invert the map a→{trace(Tn(a))}N n=1. Applying the chain rule and the recursive formula for Tn(a), we have the following recursive formula for the Fr´ echet derivatives ∂T1(a) ∂am=M1(em), ∂T2(a) ∂am=M1(em)+α0M1(a)M1(em)+α0M1(em)M1(a),16 G. BAO, X. XU AND J. ZHAI and ∂Tn+1(a) ∂am=2∂Tn(a) ∂am−∂Tn−1(a) ∂am+α2 0∂Tn−1(a) ∂amM1(e1) +α0/parenleftbigg∂Tn(a) ∂am−∂Tn−1(a) ∂am/parenrightbigg M1(a)+α0(Tn(a)−Tn−1(a))∂M1(a) ∂am. Now we can summarize the algorithm in Algorithm 4.1. Algorithm 4.1 Inversion of trace formulas for the damped wave operator 1:precompute M(em), traces rtrue 1=∑∞ k=1T1(λ−1 k),..., rtrue N=∑∞ k=1TN(λ−1 k) 2:get an approximate value of α0from measured eigenvalues 3:given initial guess a0 4:for1≤n≤max number of iterations do 5: form T1(an−1),T2(an−1),∂T1(an−1) ∂am,∂T2(an−1) ∂am 6: r1=trace(T1(an−1)) 7: r2=trace(T2(an−1)) 8: for1≤m≤Mdo 9: J1,m=trace/parenleftBig ∂T1(an−1) ∂am/parenrightBig 10: J2,m=trace/parenleftBig ∂T2(an−1) ∂am/parenrightBig 11: end for 12: for2≤j≤N−1do 13: Tj+1(an−1) = 2Tj(an−1)−Tj−1(a) + α2 0Tj−1(an−1)M1(e1) + α0(Tj(an−1)− Tj−1(an−1))M1(an−1) 14: for1≤m≤Mdo 15:∂Tj+1(an−1) ∂am=2∂Tj(an−1) ∂am−∂Tj−1(an−1) ∂am+α0(Tj(an−1)−Tj−1(an−1))∂M1(an−1) ∂am +α0/parenleftBig∂Tj(an−1) ∂am−∂Tj−1(an−1) ∂am/parenrightBig M1(an−1)+α2 0∂Tj−1(an−1) ∂amM1(e1) 16: end for 17: rj+1=trace(Tj+1(an−1)) 18: for1≤m≤Mdo 19: Jj+1,m=trace/parenleftBig∂Tj+1(an−1) ∂am/parenrightBig 20: end for 21: end for 22: compute δausing Jacobi J=(Jj,m)N×Mand residual rtrue−r=(rtrue j−rj)N×1 23: an=an−1+δa 24:end for Remark 4.6. Note that the trace formulas involve infinite sums. But r ealistically we can only have a finite number of measured eigenvalues. Assume we have 2 Kmeasured eigenvalues, say {λj}K j=−K, we can approximate the infinite sum ∑ j∈JTn(λ−1 j)=K ∑ j=−KTn(λ−1 j)+∑ |j|≥K+1Tn(λ−1 j),AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 17 by ∑ j∈JTn(λ−1 j)≈K ∑ j=−KTn(λ−1 j)+ ∑ K+1≤|j|≤K1Tn((−α0 2+jπi)−1), noticing λj≈−α0 2+jπi (cf. ( 4.6)). 5. Numerical experiments. In this section we conduct some numerical experiments to ill ustrate the efficiency of Algorithm 4.1. We design five examples to show reconstructions for smooth o r non-smooth damping coefficients with accurate or inaccurate data. To ge nerate synthetic data, we use Chebyshev pseudo- spectral collocation method to discretize the Laplacian op erator ∆=d2 dx2, using Trefethen’s cheb.mroutine [17]. We use 400 Chebyshev points to discretize the Laplacian. F or all computations, Gauss-Newton is used as the optimization algorithm with tolerance set to 10−5×N. The parameters in the algorithm are listed in Table 1. We discuss the impacts of different choices of these parameters on the performance of the algorithm. notation parameter K 2K: number of “true” eigenvalues measured M number of basis functions J J×J: size of the truncated matrix M N highest degree of the polynomials K1 2K1: total number of eigenvalues utilized in traces i.e., 2(K1−K)“approximated” eigenvalues Table 1: Parameters for the algorithm It is learned from [ 4] that the m-th eigenvalue may encode the m-th Fourier modes information of α. Hence, we usually take M=Kfor numerical reconstructions. Example 5.1. Set the damping coefficient as follows: α(x)=−exp(−(x−1 2)2)+8(x−1 2)4+6(x−1 2)2+1.25. In Table 2, we list the first K=4 eigenvalues with positive imaginary parts for the true dam ping α, the reconstructed one αMand the Fourier approximation αF. We see that when the number K=Mincreases from 4 to 8 simultaneously, the accuracy of reconstruction w ill be improved. However, there exists a balance between different paramete rs. When we fix KandK1and then increase M, it does not always give a better result, see Table 3where the error is defined in L2-norm, i.e.,/integraltext1 0|α(x)− αM(x)|2dx. For instance, from Table 3, we can find that when K1andKare fixed and Mis increasing, the error decreases at the beginning and then increases. It indi cates that Mdoes play the role as a regularization parameter and depend on the accuracy of trace formulas, whic h is in fact determined by the number of known eigenvalues K1andJ. In the following numerical simulations, we take a reasonab le choice of K1=J to avoid rounding error which may affect the accuracy of appr oximation of trace formulas.18 G. BAO, X. XU AND J. ZHAI Table 2: True eigenvalues vseigenvalues for reconstructed aM(x)and the Fourier approximation λ1 λ2 λ3 λ4 trueλj -0.2493 + 3.1335i -0.3996 + 6.2742i -0.4343 + 9.4142i -0.4469 +12.5566i ˜λj,K=M=4 -0.2493 + 3.1335i -0.3997 + 6.2744i -0.4380 + 9.4141i -0.4483 +12.5560i |λj−˜λj| 0.0000 0.0002 0.0036 0.0015 ˜λj,K=M=8 -0.2493 + 3.1335i -0.3996 + 6.2742i -0.4342 + 9.4143i -0.4487 +12.5563i |λj−˜λj| 0.0000 0.0000 0.0001 0.0018 ˜λj(A(αF)) -0.2493 + 3.1335i -0.3996 + 6.2742i -0.4343 + 9.4142i -0.4469 +12.5566i |λj−˜λj| 0.0000 0.0000 0.0000 0.0000 20 40 60 80 100 120 140 1600.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022 M=K=3 M=K=4 M=K=5 M=K=6 M=K=7 (a)N=50, Error with respect to J=K120 40 60 80 100 120 140 160123456789x 10−3 M=K=3 M=K=4 M=K=5 M=K=6 M=K=7 (b)N=200, Error with respect to J Fig. 3: Impact of K1,Jfor fixed M,KandN Table 3: Inversion errors of damping coefficients in L2norm. K=8 M=3 M=4 M=5 M=6 M=7 M =8 K1=J=25,N=25 0.0051 0.0144 0.0216 0 .0242 0.0247 0.0248 K1=J=50,N=50 0.0071 0.0052 0.0194 0 .0206 0.0209 0.0209 K1=J=100, N=100 0.0080 0.0032 0.0061 0 .0090 0.0209 0.0294 K1=J=150, N=150 0.0081 0.0052 0.0025 0 .0023 0.0021 0.0114 When M=KandNare fixed, it is shown from Figure 3(a-b) that K1andJactually do not affect the final reconstruction too much. The curves in Figure 3are almost flat for different cases. However, the gaps between different cases are large, which indicates that the number of measured spectral data Kis of more importance than other parameters in reconstruction. Moreo ver, when Nis small, the error may increase with larger M=K, see Figure 3. The reason lies in the fact that small Ndoes not discriminate enough eigenvalues in reconstruction. When Nis large in Figure 3(b), it is clear that the error decreases with M. Figure 4actually shows part of numerical inversion results for K=8,K1=J=N=150, where the dashed lines represents the initial guess of αM, the orange solid line represents the exact α(x)and the blueAN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 19 0 0.2 0.4 0.6 0.8 100.511.522.5 true reconstruction initial (a) M=50 0.2 0.4 0.6 0.8 100.511.522.5 true reconstruction initial (b) M=70 0.2 0.4 0.6 0.8 100.511.522.5 true reconstruction initial (c) M=9 Fig. 4: Reconstruction of αMin Example 5.1with K=8,K1=J=N=150. solid line represents the reconstruction. Example 5.2. In this example, we set α(x)=1.4062−0.6951cos(2πx)+0.2967cos(4πx)+0.1368cos(6πx)−0.2103cos(8πx) +0.031cos(10πx)+0.153cos(12πx)−0.0718cos(14πx)−0.0512cos(16πx)+0.1258cos(18πx) +0.04cos(20πx)+0.02cos(22πx)−0.0132cos(24πx)+0.02cos(26πx)+0.02cos(28πx). Notice that this function is highly oscillatory, and thus th e reconstruction needs more Fourier basis functions to see the fine structure. Therefore, in contrast to Example 5.1, we need to have more eigenvalues to get an accurate reconstruction. In the numerical experiments, we fix K1=J=100 and N=300. To illustrate the impact of the number of “accurate” eigenvalues 2 Kon the performance, we test three cases: K=4,K=10 and K=50. See Figure 5(a-c), (d-f) and (g-i) respectively. One can see that for the first case K=4, we can only recover lower frequency information of α. Though we can set M>K, i.e., Figure 5(a-c), the fine structure can not be recovered as not sufficient information is given. For the s imilar reason of K=10, the reconstruction forM=12 and M=8 are both worse than for M=10, see Figure 5(d-f). However, for K=50, the reconstruction for M=12 is better than for M=10 and M=8, which indicates more “accurate” measured eigenvalues give a better reconstruction. Example 5.3. In this example, we show a non-smooth damping coefficien t reconstruction. Here we set α(x)= 2,x∈[0,0.3] 3,x∈(0.3,0.7) 2,x∈[0.7,1] The non-smoothness inevitably results in more difficulties for reconstruction. In order to capture the discon- tinuity, we actually need quite a lot modes in Fourier expans ion. However, on the other hand, the number M needs to be chosen as a regularization parameter. The result s are shown in Figure 6. Example 5.4. In this example, we test the stability of the algorithm w ith noisy data. Suppose the spectral data is polluted by random noise λδ j=λj+δ×rand(0,1)×(1+i),j=1,2,···20 G. BAO, X. XU AND J. ZHAI 0 0.2 0.4 0.6 0.8 10.811.21.41.61.822.22.42.6 true reconstruction initial (a)K=4,M=80 0.2 0.4 0.6 0.8 10.511.522.53 true reconstruction initial (b)K=4,M=100 0.2 0.4 0.6 0.8 10.40.60.811.21.41.61.822.22.4 true reconstruction initial (c)K=4,M=12 0 0.2 0.4 0.6 0.8 10.811.21.41.61.822.22.42.6 true reconstruction initial (d)K=10,M=80 0.2 0.4 0.6 0.8 10.811.21.41.61.822.22.42.6 true reconstruction initial (e)K=10,M=100 0.2 0.4 0.6 0.8 10.40.60.811.21.41.61.822.22.4 true reconstruction initial (f)K=10,M=12 0 0.2 0.4 0.6 0.8 10.811.21.41.61.822.22.42.6 true reconstruction initial (g)K=50,M=80 0.2 0.4 0.6 0.8 10.811.21.41.61.822.22.42.6 true reconstruction initial (h)K=50,M=100 0.2 0.4 0.6 0.8 10.811.21.41.61.822.22.42.6 true reconstruction initial (i)K=50,M=12 Fig. 5: Reconstruction of αMin Example 5.2with K1=J=100 and N=300. where δis noise level and rand(0,1) represents the standard unifor m distribution on the open interval (0,1). Moreover, we set the damping coefficient α(x)as follows α(x)=1.5+0.2cos(2πx)+0.1cos(4πx)−0.04cos(6πx)+0.03cos(8πx) As we know that both the noisy spectral data and the finite trun cated series of eigenvalues Mresult in approx- imation error in trace formulas. Hence the reconstruction o f the damping coefficient is definitely influenced by these two parameters. Figure 7(a-c) shows numerical inversion results when δ=0.1%,0.5%,1%, re- spectively. It is clear that for δ=0.1%, when Mincreases from 3 to 6, the reconstruction becomes better and better. However, when δ=1% and Mincreases, the reconstruction becomes better first and then worse,AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 21 0 0.2 0.4 0.6 0.8 11.61.822.22.42.62.833.2 true reconstruction initial (a) M=40 0.2 0.4 0.6 0.8 11.61.822.22.42.62.833.2 true reconstruction initial (b) M=50 0.2 0.4 0.6 0.8 11.41.61.822.22.42.62.833.2 true reconstruction initial (c) M=6 0 0.2 0.4 0.6 0.8 11.822.22.42.62.833.2 true reconstruction initial (d) M=70 0.2 0.4 0.6 0.8 11.822.22.42.62.833.2 true reconstruction initial (e) M=80 0.2 0.4 0.6 0.8 11.822.22.42.62.833.2 true reconstruction initial (f) M=9 Fig. 6: Reconstruction of αMin Example 5.3with K=10,K1=J=100 and N=100. 0.10.20.30.40.50.60.70.80.91.31.41.51.61.71.81.9 M=3 M=4 M=5 M=6 Exact α Initial α (a)δ=0.1%0.10.20.30.40.50.60.70.80.91.31.41.51.61.71.81.9 M=3 M=4 M=5 M=6 Exact α Initial α (b)δ= 0.5%0.10.20.30.40.50.60.70.80.91.31.41.51.61.71.81.9 M=3 M=4 M=5 M=6 Exact α Initial α (c)δ= 1% Fig. 7: Reconstruction of αMin Example 5.4with K=M,K1=J=75 and N=75. hence the optimal choice of MisM=4 in Figure 7(c). We believe that if we utilize clean spectral data, i.e., δ=0, the optimal Mshould be larger.22 G. BAO, X. XU AND J. ZHAI Example 5.5. In this example, we reconstruct a large damping coeffici ent. We set αtrue(x)=π/parenleftBig 1.5567+1.4896cos2 πx+0.3cos4 πx+0.1cos6 πx +0.2cos8 πx+0.2cos10 πx+0.2cos12 πx/parenrightBig . which can be viewed as a perturbation of ( 4.8). According to the discussion for previous examples, we choose K=M,K1=J=75. We remark here that the parameter N, the highest degree of the polynomials used in the algorithm, can not be large. The underlying reaso n lies in the behaviors of the chosen polynomi- als. From Figure 2, we see that the reciprocal of some eigenvalues, i.e., z=λ−1are not close to the circle B(−1 α0,0)(1 α0), and Tn(z) =z(α0z+1)n−1changes rapidly away from the circle when nis large. Since the limited number of polynomials can not discriminate enough e igenvalues, the number of basis functions M can not be large either. Also, for large damping term, the con vergence of the algorithm is very sensitive to the initial guess. However, one can adopt a multi-step optim ization scheme as mentioned in [ 18]: starting with small Mand use the reconstructed profile as the initial guess for the reconstruction with a slightly larger M, and so forth. The results of numerical experiments are shown in Figure 8. We test for different MandN. Since the true damping has 7 modes, it is clearly that the reconstructi on for M=K=7 is better than M=K=6 and M=K=8 for the same N. 6. Conclusion. We have developed a novel inversion algorithm to recover the damping coefficient in a wave operator. A sequence of trace formulas are derived in a recursive form by investigating the resolvent properties of the damped wave operator, for which the invers ion scheme is devised. Moreover, a class of polynomials needs to be chosen for the success of the inversi on. Based on the distribution of eigenvalues and the properties of trace class operators, a sequence of pr oper polynomials is used. Numerical examples in Section 5illustrate the efficiency of the Algorithm. Acknowledgements. JZ thanks the many stimulating discussions with Steven Cox a nd Julio Moro. REFERENCES [1] A. B AMBERGER , J. R AUCH ,AND M. T AYLOR ,A model for harmonics on stringed instruments , Arch. Ration. Mech. Anal., 79 (1982), pp. 267–290. [2] D. B ORISOV AND P. F REITAS ,Eigenvalue asymptotics, inverse problems and a trace formu la for the linear damped wave equation , J. Differential Equations, 247 (2009), pp. 3028–3039. [3] G. C HEN AND D. R USSELL ,A mathematical model for linear elastic systems with struct ural damping , Quart. Appl. Math., 39 (1982), pp. 433–454. [4] S. C OX AND M. E MBREE ,Reconstructing an even damping from a single spectrum , Inverse Problems, 27 (2011), p. 035012. [5] S. C OX AND E. Z UAZUA ,The rate at which energy decays in a damped string , Commu. Part. Diff. Eq., 19 (1994), pp. 213–243. [6] K.-J. E NGEL AND R. N AGEL ,One-Parameter Semigroups for Linear Evolution Equations , Springer, New York, NY , 2000. [7] H. F ALUN ,Some problems for linear elastic systems with damping , Acta Math. Sci., 10 (1990), pp. 319–326. [8] H. F ATTORINI ,Second Order Linear Differential Equations in Banach Space s, vol. 108, Elsevier, 1985. [9] P. F REITAS ,Optimizing the rate of decay of solutions of the wave equatio n using genetic algorithms: a counterexample to the constant damping conjecture , SIAM J. Control Optim., 37 (1999), pp. 376–387. [10] F. G ESZTESY AND H. H OLDEN ,The damped string problem revisited , J. Differential Equations, 251 (2011), pp. 1086–1127. [11] P. L AX,Functional Analysis , Pure and Applied Mathematics, Wiley, 2002. [12] B. D. L OWE , M. P ILANT ,AND W. R UNDELL ,The recovery of potentials from finite spectral data , SIAM J. Math. Anal., 23 (1992), pp. 482–504. [13] J. E. M OTTERSHEAD AND Y. M. R AM,Inverse eigenvalue problems in vibration absorption: pass ive modification and active control , Mech. Syst. Signal Process., 20 (2006), pp. 5–44. [14] V. P IVOVARCHIK ,Direct and inverse problems for a damped string , J. Operator Theory, 42 (1999), pp. 189–220.AN INVERSE SPECTRAL PROBLEM FOR A DAMPED WA VE OPERATOR 23 0 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (a) K=6,N=250 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (b) K=6, N=300 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (c) K=6, N=35 0 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (d) K=7,N=250 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (e) K=7, N=300 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (f) K=7, N=35 0 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (g) K=8,N=250 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (h) K=8, N=300 0.2 0.4 0.6 0.8 102468101214 true reconstruction initial (i) K=8, N=35 Fig. 8: Reconstruction of large damping αMin Example 5.5with K=M,K1=J=75. [15] W. R UNDELL AND P. E. S ACKS ,Reconstruction techniques for classical inverse sturm-li ouville problems , Math. Comp., 58 (1992), pp. 161–183. [16] P. E. S ACKS ,Inverse Spectral Problems: 1-D, Algorithms , Springer Berlin Heidelberg, Berlin, Heidelberg, 2015, pp . 735–740. [17] L. N. T REFETHEN ,Spectral Methods in MATLAB , vol. 10, SIAM, 2000. [18] X. X U AND J. Z HAI,Inversion of trace formulas for a Sturm-Liouville operator , arXiv preprint arXiv:1906.12108, (2019).

2020-08-11

This paper proposes a new and efficient numerical algorithm for recovering the damping coefficient from the spectrum of a damped wave operator, which is a classical Borg-Levinson inverse spectral problem. The algorithm is based on inverting a sequence of trace formulas, which are deduced by a recursive formula, bridging geometrical and spectrum information explicitly in terms of Fredholm integral equations. Numerical examples are presented to illustrate the efficiency of the proposed algorithm.

An inverse spectral problem for a damped wave operator

2008.04523v1

Anisotropic Gilbert damping in perovskite La 0:7Sr0:3MnO 3thin film Qing Qin,1Shikun He*,2Haijun Wu,1Ping Yang,1, 3Liang Liu,1 Wendong Song,2Stephen John Pennycook,1and Jingsheng Chen*1 1Department of Materials Science and Engineering, National University of Singapore, Singapore 117575 2Data Storage Institute, Agency for Science, Technology and Research (A*STAR), 2 Fusionopolis Way 08-01 Innovis, Singapore 138634 3Singapore Synchrotron Light Source (SSLS), National University of Singapore, 5 Research Link, Singapore 117603 1arXiv:1804.00554v1 [cond-mat.mtrl-sci] 2 Apr 2018Abstract The viscous Gilbert damping parameter governing magnetization dynamics is of primary importance for various spintronics applications. Although, the damping constant is believed to be anisotropic by theories. It is commonly treated as a scalar due to lack of experimental evidence. Here, we present an elaborate angle dependent broadband ferromagnetic resonance study of high quality epitaxial La 0:7Sr0:3MnO 3films. Extrinsic effects are suppressed and we show convincing evidence of anisotropic damping with twofold symmetry at room temperature. The observed anisotropic relaxation is attributed to the magnetization orientation dependence of the band structure. In addition, we demonstrated that such anisotropy can be tailored by manipulating the stain. This work provides new insights to understand the mechanism of magnetization relaxation. A. INTRODUCTION The magnetization relaxation process determines the speed of magnetization relaxation and the energy required for current-induced magnetization reversal [1–6]. Understanding the mechanism and controlling of magnetization relaxation [7–12], including intrinsic Gilbert damping and extrinsic effects, pave the way for ultra-low power and high performance spintronic devices based on spin transferandspinorbittorques[13–15]. IthasbeendemonstratedthatGilbertdampingconstant( ) canbetunedeffectivelybyengineeringthedensityofstatesandspinorbitcoupling(SOC)[9,16–18]. In addition, magnetization relaxations subjected to finite size and interfacial effects have also been extensively investigated [8, 19, 20]. However, it is still an open question that if magnetic damping is anisotropic. In principle, is magnetization orientation dependent and should be a 3 3 tensor in the phenomenological Gilbert equation [21, 22], yet it is often treated as a scalar (isotropic). In the case of polycrystalline thin films prepared by sputtering, such treatment is reasonable due to the smearing of long range structural order. Whereas for single crystal thin films, it is still difficult to draw a conclusion due to the lack of convincing experimental evidence. From the view of theo- ries, the Gilbert damping is determined by two scattering processes, the interband resistivity-like scattering and the intraband conductivity-like scattering [12]. Both terms vary with temperature through their dependence on electron relaxation time. The interband scattering which dominates damping in most ferromagnets becomes isotropic at room temperature [23]. Therefore, anisotropic linewidth in 3d magnetic metals was only observed at low temperature[24]. From the aspect of experimental technique, Seib et al. have predicted that the precession trajectory of magnetization in a ferromagnetic resonance (FMR) measurement (standard technique for measuring damping) may partially average out the anisotropy [25]. Hence, detecting the anisotropy in Gilbert damping is extremely difficult. Furthermore, the existence of several angle dependent extrinsic contributions to damping in most materials further hinders the determination of a possible weak anisotropic damping [11, 26–28]. We note that in a ferromagnet with nearly half-metallic band structure, the isotropic interband term is suppressed [29] and the damping can be dominated by the anisotropic intraband contribution[23]. Recent reports have claimed the observation of anisotropic damping in half-metallic Heusler alloy[30, 31]. However, unavoidable chemical disorder [32, 33]of Heusler alloy introduces extrinsic effects such as spin wave scattering hence complicates the verification procedure of such anisotropy. heshikun@gmail.com msecj@nus.edu.sg 2La0:7Sr0:3MnO 3(LSMO) is an oxide perovskite material exhibited half-metallic band structure and ultra-low damping at room temperature [34, 35]. In this work, we studied the magnetiza- tion relaxation of LSMO films deposited on NdGaO 3(NGO) (110) substrates using angle-resolved broadband ferromagnetic resonance. The purpose of choosing NGO (110) substrates is to utilize its non-equalaandbaxis value. Such asymmetry will potentially lead to non-spherical Fermi surface. Two types of high quality samples with different static magnetic anisotropies were investigated. The normal LSMO film (hereafter denoted as S-LSMO) exhibited weak uniaxial magnetic anisotropy whereas the other with modulated strain relaxation mode (hereafter denoted as W-LSMO) have both uniaxial and cubic anisotropy fields. The angle dependence of the in-plane intrinsic Gilbert dampingshowedtwo-foldsymmetryinbothtypeofsamples. Strikingly, theorientationofminimum damping differs 90 degree. This work provided strong evidence of anisotropic nature of magneti- zation relaxation and demonstrated the tuning of anisotropy in damping through stress relaxation engineering. B. RESULTS Epitaxial growth of LSMO Pulsed laser deposition (PLD) was used to deposit LSMO thin films with a thickness of 25nm on (110) NGO substrates. The energy and repetition frequency of KrF laser (248nm) were 225mJ and 2Hz, respectively. During deposition, the substrate temperature was fixed at 950C. The oxygen pressure was 225mTorr for S-LSMO and 200mTorr for W-LSMOAfter deposition, S-LSMO was cooled down to room temperature at 10K/min under the oxygen pressure of 1 Torr, whereas W-LSMO at 5K/min under the oxygen pressure of 100 Torr in order to promote the modification of strain hence micro-structurestructure. Crystalline quality analysis The crystallographic structures of the films were characterized by synchrotron high resolution X-ray diffraction. Reciprocal space maps (RSMs) taken at room temperature around {013} pc(here the subscript pc stands for pseudocubic) reflections confirm the epitaxial growth of LSMO layers on the NGO substrate as shown in Fig. 1 (a). The vertical alignment of LSMO and NGO reciprocal latticepointclearlyshowsthattheLSMOfilmiscompletelystrainedontheNGOsubstrate. Lattice mismatch along [100] pcand [010] pcare 1.03% and 0.8%, respectively. Considering the position of the LSMO reciprocal lattice point in the {013} pcmappings, equal Lvalues of (103) pcand (-103) pc indicates the perpendicular relation between vector aandcin the lattice, whereas different L values for (013) pcand (0-13) pcshows that the angle between bandcis not equal to 90Âř. Thus, the LSMO is monoclinic phase which is consistent with previous reports [36]. The good crystalline quality was further verified by aberration-corrected scanning transmission electron microscopy (AC-STEM). Fig. 1 (b, c) are the simultaneously acquired high angle annular dark field (HAADF) and annular bright field (ABF) images of S-LSMO along [100]pc direction, while Fig. 1 (d, e) are for [010] pc direction. The measurement directions can be differentiated from the diffraction of NGO substrate: 31/2[010] superlattices for [100] pcdirection (inset of Fig. 1(c)) and 1/2[101] superlattices for [100] pc direction (inset of Fig. 1(e)). High quality single crystalline films are essential for the present purposes because high density of defects will result in spin wave scattering [26]. Magnetic anisotropy fields The magnetic dynamic properties were investigated by a home-built angle-resolved broadband FMR with magnetic field up to 1.5T. All measurements were performed at room temperature. Shown in Fig. 2(a) is the color-coded plot of the transmission coefficient S21 of the S-LSMO sample measured at 10GHz. 'His the in-plane azimuth angle of the external magnetic field counted from [010] pcdirection (Fig. 2(b)). This relative orientation was controlled by a sample mounting manipulator with a precision of less than 0.1. The olive shape of the color region indicates the existence of anisotropy field, whereas the very narrow field region of resonances is an evidence of low damping. Three line cuts at 'H=0, 45 and 90 degrees are plotted in Fig. 2(c), showing the variation of both FMR resonance field ( Hres) and line shape with 'H. All curves are well fitted hence both theHresand resonance linewidth
H are determined. The 'Hdependence of H resat two selected frequencies (20 and 40 GHz) are shown in Fig. 2(d) for S-LSMO. The angle dependencies of the resonance field Hres('H)is calculated starting from the total energy [37]: E=MH [cosHcosM+ sinHsinMcos('M'H)] + 2M2cos2M1 2MH 2?cos2M 1 4MH 4?cos4M1 2MH 2ksin2Mcos2('M2IP)1 4MH 4k3+cos 4('M4IP) 4sin4M(1) whereMand'Mare the polar angle and the azimuth angle of the magnetization ( M),H2?, H4?,H2k,H4kare the uniaxial and cubic out-fo-plane and in-plane anisotropy fields. The easy axes of in-plane anisotropies are along 2IPand4IP, respectively. According to Smit-Beljers equation the resonance condition for M=/2 is [38]: 2f= Msinp EE'' (2) Here,E=Hrescos('M'H) + 4MeH2kcos2('M2IP) +H4k(3 + cos 4('M4IP)=4)and E''=Hrescos('M'H)+H2kcos 2('M2IP)+H4kcos 4('M4IP)aresecondpartialderivatives of the total energy with respect to the polar and azimuth angles. =1.76107s1G1denotes the gyromagnetic ratio, 4Me= 4MH2?is the effective magnetization. The resonance field of S-LSMO shows pronounced minimum at 'H=n
, indicating the existence of uniaxial magnetic anisotropy with easy axis along 2IP= 0or [010] pcdirection. Cubic anisotropy is negligible hence H4k=0. Such uniaxial anisotropy observed in S-LSMO is consistent with previous reports [39], which is attributed to anisotropic strain produced by the NGO(110) substrate [40–42]. Compared to the resonance fields in our measurement, the magnetic anisotropy fields are orders of magnitude smaller. Therefore, the calculated difference between 'Hand'Mare always smaller than 1and '='H='Mis assumed in the following discussion. 4Magnetization orientation dependence of Gilbert damping In order to study the symmetry of magnetization relaxation of the sample. The FMR linewidth
Hfor a matrix of parameter list (72 field orientations and 36 frequency values) are extracted. The results are shown by 3-D plots in Fig. 3(a) . Here, zaxis is
Handx,yaxes aref
cos' andf
sin', respectively. The figure clearly shows that the linewidth depends on magnetization orientation. At a given frequency, the linewidth is maximum (minimum) at '= 0('==2) for S-LSMO. Fig. 3(c) shows the
Hversus frequency for three field orientations. The FMR linewidth due to intrinsic magnetic damping scales linearly with frequency
HGL= 4f= cos ('M'H) according to Laudau-Lifshitz-Gilbert phenomenological theory [43, 44]. However, a weak non- linearity in the low frequency range can be identified. In general, extrinsic linewidth contributions such as inhomogeneity and magnon scattering will broaden the FMR spectrum hence result in additionallinewidthcontributionsscalesnon-linearlywithfrequency[9,11]. Theinterfacialmagnon scattering is suppressed due to relative large film thickness (25 nm) and the bulk magnon scattering contribution to the linewidth is negligible in our samples with very good atomic order. However, the static magnetic properties of the thin film may vary slightly in the millimeter scale. Since the FMR signal is an averaged response detected by the coplanar waveguide (5mm long), a superposition of location resonance modes broadens the FMR spectrum. Such well-known contribution to linewidth, defined as
Hinhom, are generally treated as a constant [9, 44, 45]. However, it is frequency dependent for in-plane configuration and need to be treated carefully for samples with ultra-low damping. Here, we fit the data with
H=
HGL+
Hinhom, taking into account the frequency and orientation dependence of
Hinhom. As can be seen from Fig. 3(c), the data are well reproduced for every field orientations. Hence, the magnetization orientation dependence of intrinsic damping constant is determined and plotted in Fig. 3(e). Remarkably, the damping constant shows two-fold symmetry. The lowest damping of S-LSMO with in-plane magnetization, observed at '= 0and '=, is(8:40:3)104and comparable to the value measured under a perpendicular field (Tbl. I). The maximum damping at '==2and'= 3=2is about 25% higher. Since the magnetization damping and resonance field of the S-LSMO sample exhibited identical symmetry (Fig. 2 (d) and Fig. 2(e)), it seems that the observed anisotropic damping is directly relatedtocrystallineanisotropy. Therefore, wepreparedtheW-LSMOsamplewithslightlydifferent structureandhencemodifiedstaticmagneticanisotropyproperties. TheW-LSMOsampleexhibited 1D long range atomic wave-like modulation [36] (twining domain motif) along [100] pcaxis near the interface between substrate and film. Due to different strain relaxation mechanism as compared to S-LSMO, the 'Hdependence of Hresfor the W-LSMO have additional features and can only be reproduced by including both H2k(13:90:9Oe) andH4k(11:81:2Oe) terms. The easy axis of the uniaxial anisotropy ( 2IP=0 ) is the same as S-LSMO whereas the additional cubic anisotropy is minimum at 4IP=45Âř. The magnetization orientation dependence of the FMR linewidth for W-LSMO is significantly different (Fig. 3(b)) as compared to S-LSMO. Such change in trend can be clearlyidentifiedfromthefrequencydependenceoflinewidthforselectedmagnetizationorientations shown in Fig. 3(d). Magnetization damping values are extracted using the same procedure as S- LSMObecausethespinwavecontributionisexcluded. Thedampingconstantagainshowedtwo-fold in-plane symmetry. However, in contrast to S-LSMO, the maximum damping value of W-LSMO is observed at '= 0and'=. 54Meff(T)H2k(Oe)H4k(Oe) ? ('= 0)('==2) S-LSMO 0.3280 0.0011 3740 (8:60:5)104(8:40:3)104(110:6)104 W-LSMO 0.3620 0.002513.90:911.81:2(4:70:7)104(6:50:3)104(5:30:3)104 Table I. Summary of the parameters for S-LSMO and W-LSMO samples. C. DISCUSSION Anisotropy in linewidth at low temperatures have been reported decades ago, however, data in most early publications were taken at a fixed frequency in a cavity-based FMR [24, 46]. Due to lack of frequency dependence information, it is not clear if the anisotropy in linewidth is due to intrinsic damping or extrinsic effects [47–49]. In this study, besides wide range of frequencies, we also adopted samples with effective anisotropy orders of magnitude smaller than the external field. Therefore, the field dragging effect and mosaicity broadening, both of which are anisotropic in natur e[50], are negligibly small and the Gilbert damping constant is determined reliably. Furthermore, themechanisminthissimplesystemisdifferentfrompreviousreportsrelatedtointerfacialexchange coupling and spin pumping[51, 52]. Since both S-LSMO and W-LSMO exhibited in-plane uniaxial magnetic anisotropy, the opposite trends observed in these two samples exclude the existence of a direct link between anisotropic damping and effective field. Both magnetic anisotropy and damping are related to the band structure but in quite different ways. According to perturbation theory, the magnetic anisotropy energy is determined by the matrix elements of the spin-orbit interaction between occupied states. Hence, the contributions from all the filled bands must be considered to calculate the absolute value of magnetic anisotropy. On the other hand, the magnetic damping is related to the density of states at the Fermi level. The damping term in the Landau-Lifshitz-Gilbert equation of motion is jMj MdM dt
, there- fore, anisotropy in damping can have two origins, one related to the equilibrium orientation of magnetization M(orientation anisotropy) and the other depends on the instantaneous change in magnetization dM=dt(rotational anisotropy). In FMR experiments the magnetization vector ro- tates around its equilibrium position, therefore, the rotational anisotropy may be smeared out [25]. The orientation anisotropy is described by both interband and intraband scattering process. Ac- cording to Gilmore et al.[23], the latter is isotropic at sufficiently high scattering rates at room temperature. We suspect that the anisotropic damping in LSMO is due to its half-metallic band structure. As a result of high spin polarization, interband scattering is suppressed and the room temperature damping is dominated by intraband scattering. The intraband contribution to damp- ing exhibit anisotropy for all scattering rates [23] which agree well with our experiments. The suppression of interband scattering is evidenced by the ultra-low damping in the order of 104. Notably, the absolute value of the observed anisotropy, 2.6 104for S-LSMO and 1.2 104for W-LSMO, is so small that could not be identified reliably for a material with typical damping values between 5 103to 2102. In a microscopic picture, the Gilbert damping is proportional to the square of SOC constant ( ) and density of states at the Fermi level, 2D(EF). The shape of the Fermi surface depends on the orientation of the magnetization due to SOC. Hence, the anisotropy can be attributed to the angle dependence of D(EF)which is in turn induced by the substrate. The trend reversal in the damping anisotropy of the two LSMO samples can be explained by the modification of the Fermi 6surface and thus D(EF)by strain relaxation. During the preparation of this paper, we noticed a similar work in ultra-thin Fe layers deposited on GaAs substrate[53]. There, the anisotropy is attributed to interfacial SOC. This work suggests that anisotropic damping can exist in bulk samples. ACKNOWLEDGMENTS The research is supported by the Singapore National Research Foundation under CRP Award No. NRF-CRP10-2012-02. P. Yang is supported from SSLS via NUS Core Support C-380-003-003- 001. S.J.P is grateful to the National University of Singapore for funding. [1] J. C. Slonczewski, Journal of Magnetism and Magnetic Materials 159, L1 (1996). [2] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Physical Review Letters 84, 3149 (2000). [3] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, Nat Mater 9, 721 (2010). [4] D. C. Ralph and M. D. Stiles, Journal of Magnetism and Magnetic Materials 320, 1190 (2008). [5] A. Brataas, A. D. Kent, and H. Ohno, Nat Mater 11, 372 (2012). [6] K. Watanabe, B. Jinnai, S. Fukami, H. Sato, and H. Ohno, Nat Commun 9, 663 (2018). [7] A. 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Baberschke, and Z. Frait, Physical Review B 76, 104416 (2007). [51] C. Le Graët, D. Spenato, S. P. Pogossian, D. T. Dekadjevi, and J. Ben Youssef, Physical Review B 82, 1 (2010). [52] A. A. Baker, A. I. Figueroa, C. J. Love, S. A. Cavill, T. Hesjedal, and G. van der Laan, Phys Rev Lett116, 047201 (2016). [53] L. Chen, S. Mankovsky, S. Wimmer, M. A. W. Schoen, H. S. KÃűrner, M. Kronseder, D. Schuh, 8D. Bougeard, H. Ebert, D. Weiss, and C. H. Back, Nature Physics (2018). 9Figure 1. Structure characterization of S-LSMO sample. (a) and (b) XRD profiles around S-LSMO (00L) reflections (L=1,2,3,4) with the incident beam aligned along the [100] pcand [010] pc, respectively. (b) and (c) STEM-HAADF/ABF lattice images of S-LSMO along [100] pcdirection. (d) and (e) STEM- HAADF/ABF images of S-LSMO along [010] pcdirection. the insets are the intensity profile and FFT image; The red dashed line indicates the interface. 10(b) (c) 10GHz(a) 30 21060 24090 270120 300150 330180 010GHz 242022002420Hres(Oe) 5.625.70 12.6512.712.75 Hres(kOe)(d) 40GHz 20GHzS-LSMO 0 90 180 270 360 0 0 fit 45 45 fit 90 90 fitS21 (a.u.) 2200 2300 2400 Hres(Oe)Figure 2. Magnetic anisotropy characterization. (a) The 2D polar color plot of the FMR spectra of S-LSMO. The frequency is 10GHz. (b) Schematics of the FMR setup and the definition field orientation. (c) FMR spectra for 'H=0, 45 and 90 degrees for S-LSMO. (d) Field orientation ( 'H) dependence of the resonance fields ( Hres) of the S-LSMO sample at f=20 and 40GHz. The solid lines in (c) and (d) are calculated values. 11404040 0 060 -40 40 4020 0 030 -40 f (GHz) f (GHz)0 10 20 30 40204060ΔH (Oe) 102030ΔH (Oe) 0 10 20 30 40 810120 fit 45 90fit fit0 fit 45 90fit fit(a) (b) (c) (d) (e) (f)ΔH (Oe) ΔH (Oe) f (GHz) f (GHz)α (10-4) 0 360 270 180 90 φ567 0 90 180 270 360α (10-4) φf (GHz) f (GHz)Figure 3. Anisotropic linewidth and damping: (a)-(b) 3-D plot of frequency and in-plane field ori- entation dependence of FMR linewidth. (c)-(d) frequency dependence of FMR linewidth for seleted field orientations. Solid symbols are experimental data and the lines are calculated value. (e)-(f) Damping constant as a function of '. (a),(c), (e) are for S-LSMO and (b),(d), (f) are for W-LSMO. 12

2018-04-02

The viscous Gilbert damping parameter governing magnetization dynamics is of primary importance for various spintronics applications. Although, the damping constant is believed to be anisotropic by theories. It is commonly treated as a scalar due to lack of experimental evidence. Here, we present an elaborate angle dependent broadband ferromagnetic resonance study of high quality epitaxial La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ films. Extrinsic effects are suppressed and we show convincing evidence of anisotropic damping with twofold symmetry at room temperature. The observed anisotropic relaxation is attributed to the magnetization orientation dependence of the band structure. In addition, we demonstrated that such anisotropy can be tailored by manipulating the stain. This work provides new insights to understand the mechanism of magnetization relaxation.

Anisotropic Gilbert damping in perovskite La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ thin film

1804.00554v1

Isotropic non -local Gilbert damping d riven by spin current s in epitaxial Pd/Fe/MgO(001) film s Yan Li1,2,Yang Li1,2,Qian Liu3, Zhe Yuan3, Wei He1,Hao -Liang Liu1, Ke Xia3,Wei Yu1, Xiang- Qun Zhang1, and Zhao- Hua Cheng1,2, * 1State Key Laboratory of Magnetism and Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, 100875 China ABSTRACT Although both theoretical predications and experimental observations demonstrate d that the damping factor is anisotropic at ferromagne t/semiconductor interface with robust interfacial spin- orbit coupling , it is not well understood whether non-local Gilbert damping driven by spin current s in heavy metal /ferromagnetic metal (HM/FM) bilayers is anisotropic or not. H ere, we investigated the in -plane angular - and frequenc y- dependen ce of magnetic relaxation of epitaxial Fe /MgO(001) films with different capping layers of Pd and Cu. After disentangl ing the parasitic contributions, such as two -magnon scattering (TMS) , mosaicity, and field-dragging effect, we unambiguously observed that both local and non- local Gilbert damping are isotropic in Fe(001) plane , suggest ing that the pure spin current s absorption is independent of Fe magnetization orientation in the epitaxial Pd/Fe heterostructure. First principles calculation reveal s that the effective spin mixing conductance of Pd/Fe interface is nearly invariant for different magnetization directions in good agreement with the experimental observation s. These results offer a valuable insight into the transmission and absorption of pure spin currents, and facilitate us to utilize next-generation spintronic devices. PACS number s: 72.25.Mk, 75.78.- n, 76.50.+g *Corresponding author E-mail: zhcheng@iphy.ac.cn I. INTRODUCTION The rapid development of spintronic devices inquires deeper understanding of the magnetization relaxation mechanism1-3. The Gilbert damping factor, one of key parameter s in spin dynamics , characterizes the energy transfer from the spin subsystem to the lattice and governs the magnetization switching time and the critical current density in spin transfer torque devices4-6. Since the shape of Fermi surface depends on the orientation of the mag netization direction due to the spin- orbit interaction, an anisotropic Gilbert damping is expected in single crystal ultrathin films7-10. Chen et al. discove red an anisotropic damping in the Fe/GaAs(001) ultrathin films where an robust interfacial spin -orbit field exists , due to GaAs substrate . The magnitude of damping anisotropy , however, decreases with increasing Fe thickness , and disappears when the Fe thickness is larger than 1.9 nm11-13. Besides intrinsic Gilbert damping in ferromagnetic materials (FM) , spin current s sink into heavy metal s (HM) or other magnetic layer s importing non -local Gilbert damping in HM/FM bilayer s or spin valve structure according to spin pumping model14-16. Although anisotropic magnetization relaxation in ferromagnetic multilayers w as observed, it is debated whether the absorption of pure spin currents is anisotropic or isotropic in ferromagnetic multilay ers17-21. This is because the frequency - and angular -dependent ferromagnetic resonance (FMR) linewidth results are often contaminated by parasitic contributions , such as two -magnon scattering (TMS), mosaicity, and field -dragging effect . Li et al. found that nearly isotropic absorption of pure spin current in Co in Py1-xCux/Cu(5 nm)/Co(5 nm) trilayers using spin pumping technique22. Meanwhile, Baker et al. found an anisotropic absorption of pure spin currents in Co 50Fe50/Cr/Ni 81Fe19 spin valves with variable Cr thickness, while the anisotropy is suppressed above the spin diffusion length23. Here, we investigated spin pumping and clarified the dependence of diverse magnetic relaxations on Fe magnetic orientation using Vector Network Analyzer ferromagnetic resonance (VNA- FMR) of epitaxial Fe /MgO(001) films capped by Pd and Cu layers. Simple FM/HM bilayers would be a more convincing candidate to explore the non-local relaxation mechanism. Exclu ding the misleading dragging effect and the deceitful extrinsic terms, we unambiguously observed that both local and non- local Gilbert damping are isotropic in Fe(001) plane . The i sotropic non- local Gilbert damping suggest s that the pure spin current s abso rption is independent of Fe magnetization orientation , which is supported by the first principle s calculation. II. EXPERIMENTS Sample s were prepared in molecular beam epitaxy chambers with a basic pressure-102 10× mbar24. Prior to deposition, MgO(001) substrate was annealed at 700 ℃ for 2 hours, and then 6 nm Fe film was deposited on a MgO(001) substrate using electron -beam gun , and finally 5 nm Pd w as covered on Fe films . The crystalline quality and epitaxial relationship was confirmed by high- resolution transmissio n electron micro scopy (HRTEM), as shown in Fig . 1(a) and (b). It has been revealed that the films were grown with the epitaxial relationship Pd(001)<110>||Fe(001)<100>||MgO(001)<110> (see the inset of Fig . 1(b)). For comparison, Cu(3.5 nm)/Fe (6 nm)/MgO(001) sample was also prepared. In-plane VNA- FMR measurements were performed by facing the sample down on employing a co-planar waveguide (CPW) and recording the transmission coefficient S 2125-27. All depositions and measurements were performed at room temperature. III. RESULTS AND DISCUSSI ON Fig. 2(a) s hows schematically the stacked sample and the measured configurat ion. The representative FMR spectra at fixed frequency 13.4 GHz and various magnetic field angle s Hϕ are illustrated in Fig. 2(b). T he FMR signal (the transmission parameter S 21) is a superposition of symmetric and antisymmetric Lorentzian functions . The following equation could be used to extract the resonance field H r and the resonance line width H∆: 2 21 0 22 22( / 2)( ) ( / 2)Re ( ) +( ) ( / 2) ( ) ( / 2)r rrH HH HSH SL DHH H HH H∆ − ∆= −− +∆ − +∆. (1) Here, Re S21, S0, H, L and D are the real part of transmission parameter, the offset, the external magnetic field, the symmetric and antisymmetric magnitude , respectively25-27 . The resonance frequency f is given by Kittel formula28 0=2RR ab f HHγµ π (2) with2 42 cos( ) (3 cos 4 ) / 4 sin ( 45 )R ar MH M M d H H HH H ϕϕ ϕ ϕ = −++ + − −a, 42 cos( ) cos 4 sin 2R b r MH M M HH H H ϕϕ ϕ ϕ= −+ − and 02=out ds sKHMMµ− . Here, γ and 0µ are the gyromagnetic ratio and the vacuum permeability. H , H2, H4 and Ms are the applied magnetic field, the uniaxial and four-fold magnetic anisotropy field s and saturation magnetization , respectively. outK is the out -of-plane uniaxial magnetic anisotropy constant. The equilibrium azimuthal angle of magnetization Mϕis determined by the following equation: 42 sin( ) ( / 4)sin 4 ( / 2)cos 2 0r MH M M H HHϕϕ ϕ ϕ −+ + = . (3) The angular dependent FMR measurements were performed by rotating the samples in plane while sweeping the applied magnetic field. At a fixed frequen cy of 13.4 GHz, the angular dependence of H r can be derived from Eq. (2) and plotted in Fig. 2(c) and 2(d) for Fe/MgO(001) sample s capped by Pd and Cu, respectively . It can be seen clearly that the angular dependence of H r demonstrates a four -fold symmetry and the values of 2=0H Oe, 4=625H Oe and 0 2.0dHµ= T for Pd/ Fe/MgO(001) and2=0H Oe, 4=625H Oe and 0 1.9dHµ= T for Cu /Fe/MgO(001) , respectivel y. Compar ing to the sample with Cu c apping l ayer, Pd/Fe interface modifies the out-of-plane uniaxial magnetic anisotropy, and has a negligible contribution to the in-plane uniaxial magnetic anisotropy. In cont rast to the four -fold symmetry of H r, the angul ar dependence of H∆for the samples with Pd and Cu capping layers indicates to be superposition of four-fold and quasi -eight -fold contributions , as shown in Fig. 3(a) and 3(b) , respectively . In fact, the quasi -eight -fold broadening also represent s a four-fold symmetry with multiple extreme value point s. In the case of the sample with Pd capping layer, H∆exhibits two peaks around the hard magnetization direction s Fe<1 10>, and the values of H∆ for Fe<100> and Fe<110 > direction s are almost the same (58 Oe). On the other hand, a larger difference in the magnitude of H∆ was observed along these two directions of Cu/Fe/MgO(001) sample , i.e. 71 Oe and 4 9 Oe for Fe<100> and Fe<110> axes, respectively . In order to understand the mechanism of anisotropic magnetic relaxation, we must take both intrinsic and extrinsic contributions into account29-34. H∆ is follow ed by the expression32 : _ =mosaicity TMS Gilbert dragging HH H H∆ ∆ +∆ +∆ . ( 4) The first term denotes TMS, represent ing that a uniform prerecession magnon ( 0k=) is scattered into a degenerate magno n ( 0k≠) due to imperfect crystal structure. Therefore, the contribution of TMS to the linewidth reli es heavily on the symmetrical distribution of defects and manifest s anisotropic feature accordingly . The second term describes the mo saicity contribution in a film plane, which is caused by a slightly spread of magnetic parameters on a very large scale. The last term _ Gilbert draggingH∆ is the Gilbert damping contribution with field -dragging . In the case of Fe/ MgO( 001) epitaxial film, the contribution of TMS to FMR linewidth composes of numerous two-fold and four -fold TMS channel s31-34, j,max 4 j,max j,max 2 j,maxcos ( ) cos 2( )TMS twofold M twofold fourfold M fourfold jjH ϕϕ ϕϕ ∆ =Γ − +Γ − ∑∑ . ( 5) Here, j,max twofoldϕ and j,max fourfoldϕ represent angle of the maximum scattering rate in two-fold and four -fold scatterings along the direction j. However, the same values of H∆between Fe<100> and Fe<110> directions suggest that the TMS can be neglected in Pd/Fe/MgO(001 ) epitaxial film. On the other hand, the larger difference in the magnitude of H∆ was observed along these two directions , suggesting that either significant TMS contribution or anisotropic Gilbert damping exists in Cu/Fe/MgO(001) s ample13, 32, 33. The angular dependence of mosaicity contribution can be described as32, 34 =r mosaicity H HHH ϕϕ∂∆∆∂, ( 6) where Hϕ∆ represents an in plane variation of mosaicity. 0mosaicityH∆= Oe should be hold along easy magnetization direction s and hard directions where =0r HH ϕ∂ ∂. Due to magnetocrystalline anisotropy, magn etization would not always align at the direction of the applied field when the field is weaker than the saturation field. We evaluate the field -dragging effect during rotation of the sample or frequency -swept based on the numerical calculation using Eq. ( 3). Fig. 4(a) shows Hϕdependence on Hϕ at 13.4 GHz. The relation reveals a conspicuous dragging effect with a four-fold symmetry. At 25Hϕ=a, HMϕϕ− is as high as 12a. Fig. 4(b) sh ows Mϕ dependence on f at various Hϕ. When the magnetic field is applied along Fe<100> or Fe<110> directions , the magnetization is always aligned along the applied magnetic field. However, there is a conspicuous angle between the magnetization and the magnetic field with the field along intermediate axis. Owing to the angle between magnetization and applied field , H∆ corresponding to Gilbert contributio n with the field-dragging could be disclose d according to the following equations12, 13 _ = [Im( )]Gilbert draggingH χ ∆∆ ( 7) and 22 2[]Im( )( ) ()RR RR e f f ab a a ab s RR RR a b ab e f f ab a bHH H H HH M H H HH HH H+ Haχa+=−+ , (8) where aH and bH are R aH and R bH in non- resonance condition. The effective parameter effa consist s of the intrinsic Gilbert damping an d the non- local one driven by spin currents . Generally , effa was obtained by the slope of the linear dependence of H∆ on frequency f along the directions without field -dragging 28: 0 04efffHHπa µγ∆ = +∆ , (9) where 0H∆ is inhomogeneous non- Gilbert linewidth at zero -frequency25-27. Fig. 5 shows H∆ dependence on frequency at various Hϕ. Obviously , H∆ versus f can be fitted linear ly with 3 / 6.0 10Pd Fea−= × and 3 /=4.2 10Cu Fea−× for magnetic field along easy axes Fe<1 00> or hard axes Fe<1 10> of the samples with Pd and Cu capping layers, respectively , indicating isotropic damping (Fig. 5(f) and 5( j)). By using the aforementioned isotropic damping factor s, the contributions of TMS, mosaicity, and field -dragging effect are separated from the angular dependence of H∆ (Fig. 3 a-b). Table I summar izes the fitted parameters in the two samples. Compared with Cu/Fe sample , one observes a significant reduction of mosaicity broadening and a negligible TMS term in Pd/Fe bilayers. In fact, due to high mobilit y, the capping layer Cu forms nanocrystallites on Fe film, which causes interfacial defects dependence on the crystallographic ax es35-38. The interfacial defects will impact a four-fold linewidth broadening due to TM S. In contrast , the excellent epitaxial quality at Pd/Fe interface not only ensures a sharp interfacial structure , but also reduces defect density to decrease TMS contribution. Moreover, the mosaicity contribution, indicat ing the fluctuation of the magne tic anisotropy field, could be strengthen by the interfacial stacking faults. C onsequently , a fully epitaxial structure could significantly decrease the extrinsic contributions, especially TMS and mosaicity terms . Taking these contributions to magnetizatio n relaxation into account, the frequency dependence of H∆ at various directions can be well reproduced, as shown in Fig. 5(f)- (j). For other directions rather than Fe<1 00> and Fe<1 10>, nonlinear relationship between H∆ and f are evident and illustrated in Fig . 5(g-i). At =20Hϕa, the H∆ vs f curve brings out a slight bump comparing to the linear ones along hard or easy ax es. At =27Hϕa, H∆ has a rapid decrease after H∆ experiencing an abrupt enhancement . At =33Hϕa, H∆ decreases more sharply after 11 GHz. The nonlinearity can be ascrib ed to the parasitic contrib utions, such as TMS, mosaicity, and field -dragging effect . It is virtually impossible to stem from TMS for the d istorted curves because a nonlinear linewidth broadening due to TMS increases as frequency increases, and approach es to saturation at high frequency31. According to the calculation in Fig. 4(b), there is a huge field -dragging effect except the applied magnetic field H along hard and easy ax es. The field -dragging will make H∆ vs f deviate from the linear relationship . As expected , we could effectively fit the experimental data H∆ vs f using the following equation in association with the original formula s (7), 0 [Im( )] HH χ ∆ =∆ +∆ . ( 10) Eq. (10) converges to the Eq. (9) with the applied magnetic field along the directions without field -dragging, i.e. easy axes Fe<1 00> or hard axes Fe<1 10>13. After distinguish ing the contributions of extrinsic terms and field -dragging effect, the Gilbert damping factors effa along various direc tions are show n in Fig. 6(a). According to the classical spin pumping model14, precess ional magnetization in FM layer will pump spins into adjacent nonmagnetic metals across interface. Cu with only s conduction band has a smaller spin- flip probability and a large r spin diffusion length than 500 nm39, therefore, the reference sample Cu/Fe cannot increase the Gilbert damping due to a capp ing layer Cu. In contrast , Pd-layer with stron g spin- orbit coupling has a larger spin- flip probability , the injected spin currents are dissipated in Pd-layer , and enhance the intrinsic Gilbert damping of Fe film. The enhancement of the Gilbert damping allows us to comprehend the non- local relaxation m echanism. Obviously, it can be seen from Fig. 6(a) that there is no strong relation between the non-local Gilbert damping and the magnetization orientation in epitaxial film Pd/Fe. The parameters-3 /=4.2 10Cu Fea × and -3 /=6.0 10Pd Fea × are the Gilbert damping of Pd/Fe and Cu/Fe , respectively. The non-local Gilbert damping could be evaluated using the effective spin mixing conductance effg↑↓14 // =4B Pd Fe Cu Fe eff s FeggMtµaa aπ↑↓∆ −= . ( 11) The obtained isotropic value 19 2=1.23 10effgm↑↓ −× is comparable to the literature s40-42. In order to theoretically investigate the dependence of the non- local Gilbert damping on the magnetization orientation, the first principles calculation was performed to calculate the total Gilbert damping of the Pd /Fe/Pd multilayer on the basis of the scattering theory43-45. The electronic structure of the Pd/Fe interface was calculated self -consistently using the surface Green’s function technique implemented with the tight- binding linearized muffin -tin orbitals method. Within the atomic sphere approximation, the charge and spin densities and the effective Kohn- Sham potentials were evaluat ed inside atomic spheres. The total Gilbert damping was then calculated using the scattering theory of magnetization dissipation45. We simulate d the room tempe rature via introducing frozen thermal lattice disorder into a 5x5 lateral supercell43. The root -mean -squared displacement of the atoms is determined by the Debye model with the Debye temperature 470 K. A 28x 28 k-mesh is used to sample the two -dimensional Brillouin zone and five different configurations of disorder have been calculated for each Fe thickness. The total Gilbert damping exhibits a linear dependence on the length of Fe and the intercept of the linear function can be extracted corresponding to the contribution of the spin pumping at the Pd/Fe interface44. The interfacial contribution i s converted to the effective spin mixing conductance, plotted in Fig. 6( b) as a function of the magnetization orientation. It can be seen that the effective spin mixing conductance across Pd/Fe interface 19 2=1.29 10effgm↑↓ −× is independent of the magnet ization direction , and is in very good agreement with the experimental value19 21.23 10 m−× . According to the Elliott- Yafet mechanism in a nonmagnetic metal , spins relax indiscriminately energy and momentum along all orientation in Pd-layer since a cubic metal is expected to possess a weak anisotropy of the Elliott -Yafet parameter46. Incidentally , the fitting error will mislead an aniso tropic Gilbert damping if ones use Eq. ( 9) to fit the entire curves H∆ vs f. Besides , an epitaxial magnetic film integrated into a pseudo spin valve could lead to an anisotropic absorption of spin current based on s pin transfer torque mechanism since it is demanding to drag magnetization parallel ing to the applied field11. I V. CONCLUSIONS In summary, a non-local Gilbert damping is induced by the spin pumping in Pd/Fe bilayers as spin currents transfer angular momentum into the Pd- layer . Due to strong magnetocrystalline anisotropy, the field -dragging effect makes the line width versus frequency deviate from the linear relationship except magnetic field along hard or easy ax es. Extrinsic relaxation , such as TMS and mosaicity, relies heavil y on magnetization orientation. Howeve r, an epitaxial interface could significantly decrease and minimize the extrinsic contributions, especially TMS and mosaicity. It is noteworthy that an isotropic non- local Gilb ert damping factor is clarified after ruling out the misleading field-dragging effect and the deceitful extrinsic contributions. Magnetization orientation has a negligible contribution to the non- local Gilbert damping based on both theoretical and experimental results , manifesting that the absorption of pure spin currents across interface Pd(100)[110]/Fe(001)[100] is independent of Fe magnetization orientation. Our works provide deeper i nsight into the non- local Gilbert damping mechanism. ACKNOWLEDGMENTS This work is supported by the National Key Research Program of China (Grant Nos. 2015CB921403, 2016YFA0300701, and 2017YFB0702702), the National Natural Sciences Foundation of China (Grant Nos. 51427801,1187411 ,51671212, and 11504413) and the Key Research Program of Frontier Sciences, CAS (Grant Nos. QYZDJ -SSW -JSC023, KJZD -SW-M01 and ZDYZ2012- 2). The work at Beijing Normal University is partly supported by the National Natural Sciences Foundation of China (Grant Nos. 61774017, 61704018, and 11734004), the R ecruitment Program of Global Youth Experts and the Fundamental Research Funds for the Central Universities (Grant No. 2018EYT03). REFERENCES 1. I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76 (2), 323 -410 (2004). 2. K. Ando, S . Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa and E. Saitoh, Phys. Rev. Lett. 101 (3), 036601 (2008). 3. J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back and T. Jungwirth, Rev. Mod. 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Condens. Matter 19 (18), 183201 (2007). 40. J. Foros, G. Woltersdorf, B. Heinrich and A. Brataas, J. Appl. Phys. 97 (10), 10A714 (2005). 41. M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rog alev and W. E. Bailey, Phys. Rev. B 94 (1) (2016). 42. J. M. Shaw, H. T. Nembach and T. J. Silva, Phys. Rev. B 85 (5) (2012). 43. Y . Liu, A. A. Starikov, Z. Yuan and P . J. Kelly, Phys. Rev. B 84 (1) (2011). 44. Y . Liu, Z. Yuan, R. J. Wesselink, A. A. Stari kov and P . J. Kelly, Phys. Rev. Lett. 113 (20), 207202 (2014). 45. A. A. Starikov, Y . Liu, Z. Yuan and P . J. Kelly, Phys. Rev. B 97 (21) (2018). 46. B. Zimmermann, P . Mavropoulos, S. Heers, N. H. Long, S. Blugel and Y . Mokrousov, Phys. Rev. Lett. 109 (23), 236603 (2012). FIGURE CAPTIONS Fig. 1 ( Color online) (a) Dark field scanning high -resolution transmission electron microscopy image and ( b) selected area electron diffraction pattern of Pd/Fe/MgO(001). The inset of Fig. 1(b) shows a schematic of the ep itaxial relationship . Fig. 2 ( Color online) (a) A schematic illustration of the stacked sample Pd/Fe/MgO(001). The sample is placed on the CPW for FMR measurement, and could be rotated in plane . (b) Typical real FMR spectra of Pd/Fe at fixed frequency 13.4 GHz at various magnetic field angle sHϕ. Magnetic field angle Hϕ dependen ce of the resonanc e field H r at a fixed frequency 13.4 GHz for Pd/Fe (c) and Cu /Fe (d) . The red curves are fit to Kittel’s formula (2). (In order to show clearly the tendency , we show the data at 45 225Hϕ−≤≤aa, the same below ) Fig. 3 ( Color online) The measured linewidth H∆ as a function of Hϕat 13.4 GHz for Pd /Fe (a) and Cu/Fe (b). The line width H∆ is superimposed by several terms, such as TMS, mosaicity and Gilbert contribution with field- dragging. Fig. 4 (Color online) Field -dragging effect for Pd/Fe. (a) The green line denotes the equilibrium direction of magnetization as a function of magnetic field angleHϕ at 13.4 GHz. T he red line indicates the misalignment between the magnetization and the applied magnetic field according ly. (b) The equilibrium direction of the magnetization in the frequency -swept mode at variousHϕ. Fig. 5 ( Color online) Frequency dependence of the resonance field Hr (a-e) and frequency dependence of the resonance line width H∆ (f-j) for Pd/Fe at variousHϕ. The blue solid squares and curves in (f) and (j) corresponding to frequency dependence of H∆ at 0Hϕ=a and 45Hϕ=a for Cu/Pd. Fig. 6 (Color online) Angular dependent Gilbert damping and first principles calculation. ( a) The opened and solid green squares represent the obtained Gilbert damping for Pd/Fe and Cu/Fe films, respectively. The red and blue lines are guide to the eyes. ( b) The experimental and calculated spin mixing conductance as a function of the orientation of the equilibrium magnetization. Table I The fitted magnetic anisotropy parameters and magnetic relaxation parameters in Pd /Fe and Cu/Fe films . Fig.1 Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Table I The fitted m agnetic anisotropy parameters and magnetic relaxation parameters in Pd /Fe and Cu/Fe films in Fig. 3. Sample 4H(Oe) 2H(Oe) 0 dHµ (T) effa 100γ<>Γ (710Hz) ϕ∆(deg.) Pd/Fe 625 0 2.0 0.0060 0 0.23 Cu/Fe 625 0 1.9 0.0042 58 1.26

2018-09-28

Although both theoretical predications and experimental observations demonstrated that the damping factor is anisotropic at ferromagnet/semiconductor interface with robust interfacial spin-orbit coupling, it is not well understood whether non-local Gilbert damping driven by spin currents in heavy metal/ferromagnetic metal (HM/FM) bilayers is anisotropic or not. Here, we investigated the in-plane angular- and frequency- dependence of magnetic relaxation of epitaxial Fe/MgO(001) films with different capping layers of Pd and Cu. After disentangling the parasitic contributions, such as two-magnon scattering (TMS), mosaicity, and field-dragging effect, we unambiguously observed that both local and non-local Gilbert damping are isotropic in Fe(001) plane, suggesting that the pure spin currents absorption is independent of Fe magnetization orientation in the epitaxial Pd/Fe heterostructure. First principles calculation reveals that the effective spin mixing conductance of Pd/Fe interface is nearly invariant for different magnetization directions in good agreement with the experimental observations. These results offer a valuable insight into the transmission and absorption of pure spin currents, and facilitate us to utilize next-generation spintronic devices.

Isotropic non-local Gilbert damping driven by spin currents in epitaxial Pd/Fe/MgO(001) films

1809.11020v1

Gradient expansion formalism for generic spin torques Atsuo Shitade RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan (Dated: August 23, 2021) We propose a new quantum-mechanical formalism to calculate spin torques based on the gradient expansion, which naturally involves spacetime gradients of the magnetization and electromagnetic elds. We have no assumption in the small-amplitude formalism or no diculty in the SU(2) gauge transformation formalism. As a representative, we calculate the spin renormalization, Gilbert damp- ing, spin-transfer torque, and -term in a three-dimensional ferromagnetic metal with nonmagnetic and magnetic impurities being taken into account within the self-consistent Born approximation. Our results serve as a rst-principles formalism for spin torques. I. INTRODUCTION Spin torques have been investigated both theoreti- cally and experimentally in the eld of magnetic spin- tronics since the celebrated discovery of the current- induced magnetization reversal by the spin transfer torque (STT)1{5. When an electric eld is applied to a ferromagnetic metal with magnetic structures such as domain walls and skyrmions, the spin-polarized current ows, and electron spin is transferred to the magnetiza- tion via the exchange interaction. Furthermore, the so- called-term arises from spin relaxation6{11. Electronic contributions to spin torques in a ferromagnetic metal without spin-orbit interactions (SOIs) are expressed by ~ =~s_~ n~~ n_~ n(~js
~@)~ n~ n(~js
~@)~ n;(1) in which~ nis the magnetization which is dynamical and nonuniform. sandare the spin renormalization and electronic contribution to the Gilbert damping, respec- tively. The third and fourth terms are the STT and - term driven by the spin-polarized current ~js. In the pres- ence of SOIs, another spin torque called the spin-orbit torque is allowed even without magnetic structures12{14. In real materials, both magnetic structures and SOIs do exist, and hence a systematic formalism to calculate these spin torques is desired15. To calculate spin torques quantitatively, a quantum- mechanical formalism is desirable. It is dicult to take into account spin relaxation systematically in the semi- classical Boltzmann theory8,10,13{15, and phenomenolog- ical treatment may even lead to incorrect results on the -term8. The small-amplitude formalism, in which small transverse uctuations around a uniform state are as- sumed, is quantum-mechanical but cannot be applied to the nite-amplitude dynamics except for simple cases without SOIs9. The SU(2) gauge transformation formal- ism, where a magnetic structure is transformed to a uni- form state, is also quantum-mechanical and correct5,11,12. However, we should be careful when we deal with mag- netic impurities11. Magnetic impurities become dynami- cal and nonuniform by the SU(2) gauge transformation, which yields the additional SU(2) gauge eld. If this con- tribution is not taken into account, the Gilbert damping vanishes.Here we propose a new quantum-mechanical formalism to calculate generic spin torques based on the gradient ex- pansion. As a representative, we calculate four terms in Eq. (1) in a three-dimensional (3d) ferromagnetic metal with nonmagnetic and magnetic impurities. The gradient expansion is a perturbation theory with respect to space- time gradients16,17as well as electromagnetic elds18{20 in terms of the Wigner representations of the Keldysh Green's functions. The former two terms in Eq. (1) are linear responses of electron spin to a temporal gradient of the magnetization, and the latter two are the second- order responses to a spatial gradient and an electric eld. As mentioned in Ref. 10, it is a natural extension of the semiclassical Boltzmann theory8,10,13{15. We do not have to pay any attention to the SU(2) gauge eld even in the presence of magnetic impurities, SOIs, and sublattice de- grees of freedom as in antiferromagnets. II. GRADIENT EXPANSION In this Section, we review the gradient expansion of the Keldysh Green's function with external gauge elds being taken into account. We do not rely on any spe- cic form of the Hamiltonian, which may be disordered or interacting. Furthermore, gauge elds may be abelian or nonabelian. The gradient expansion was already car- ried out up to the innite order in the absence of gauge elds16,17and in the abelian case18,19and up to the rst order in the nonabelian case20. Although we are inter- ested in the abelian case, we give rigorous derivation up to the fourth order in the nonabelian case with the help of the nonabelian Stokes theorem21,22. A. Locally covariant Keldysh Green's function When we carry out the gradient expansion, it is es- sential to keep the local gauge covariance. First, let us explain its meaning here. Under a gauge transformation 0(x) =V(x) (x) for a eld (x), gauge eldsA(x), a locally gauge-covariant quantity ~A(x), and the Keldysh Green's function ^G(x1;x2) transform as A0 (x) =V(x)A(x)Vy(x)i~[@V(x)]Vy(x);(2a)arXiv:1708.03424v1 [cond-mat.mes-hall] 11 Aug 20172 ~A0(x) =V(x)~A(x)Vy(x); (2b) ^G0(x1;x2) =V(x1)^G(x1;x2)Vy(x2): (2c) The Green's function ^G(x1;x2) with the hat symbol is gauge-covariant in the sense of Eq. (2c). However, in the Wigner representation dened later in Eq. (6), the center-of-mass coordinate X12(x1+x2)=2 is the only coordinate, and hence the Green's function should be de- ned as locally gauge-covariant with respect to X12. It can be achieved by introducing the Wilson line, W(x1;x2)Pexp 1 i~Zx1 x2dyA(y) ; (3) which transforms in the same way as the Green's func- tion, i.e.,W0(x1;x2) =V(x1)W(x1;x2)Vy(x2).Pis the path-ordered product. The locally gauge-covariant Green's function ~G(x1;x2) with the tilde symbol is then dened by18{20 ~G(x1;x2)W(X12;x1)^G(x1;x2)W(x2;X12);(4) which transforms as ~G0(x1;x2) =V(X12)~G(x1;x2)Vy(X12); (5) instead of Eq. (2c). Similarly to Eq. (4), all the two- point quantities with the hat symbol should be replaced by those with the tilde symbol. B. Gauge-covariant Wigner representation Next, we dene the Wigner representation of the lo- cally gauge-covariant Green's function18{20, ~G(X12;p12)Z dDx12ep12x 12=i~~G(x1;x2);(6)whereX12(x1+x2)=2 andx12x1x2are the center- of-mass and relative coordinates, respectively, and p12is the relative momentum. Dis the spacetime dimension. Dynamics of the Green's function is determined by the Dyson equation involving convolution, which is a two- point quantity dened by \AB(x1;x2)Z dDx3^A(x1;x3)^B(x3;x2);(7) for any two-point quantities ^Aand ^B. Therefore, we have to nd the Wigner representation of the locally gauge- covariant convolution, ~A(X12;p12)?~B(X12;p12)^AB(X12;p12):(8) Since the Wigner representation is just the Fourier trans- formation with respect to x12, convolution turns into the simple product ~A(p12)~B(p12) for a translationally invari- ant system in the absence of gauge elds; otherwise, it becomes noncommutative and is called the Moyal prod- uct. It is evaluated by expanding Eq. (8) with respect to the relative coordinates x13andx32as in Appendix A and is expressed by ~A?~B=~A~B+ (i~=2)PD(~A;~B) + (i~=2)PF(~A;~B) + (1=2!)(i~=2)2PD2(~A;~B) + (i~=2)2PDF(~A;~B) + (1=2!)(i~=2)2PF2(~A;~B); (9a) PD(~A;~B)DX~A@p~B@p~ADX~B; (9b) PF(~A;~B)(F@p~A@p~B+ 2@p~AF@p~B+@p~A@p~BF)=4; (9c) PD2(~A;~B)DX1DX2~A@p1@p2~B2DX1@p2~A@p1DX2~B+@p1@p2~ADX1DX2~B; (9d) PDF(~A;~B)[F(DX@p~A@p@p~B@p@p~ADX@p~B) + 2(DX@p~AF@p@p~B@p@p~AFDX@p~B) + (DX@p~A@p@p~B@p@p~ADX@p~B)F]=4; (9e) PF2(~A;~B)(F11F22@p1@p2~A@p1@p2~B+ 4@p1@p2~AF11F22@p1@p2~B +@p1@p2~A@p1@p2~BF11F22+ 4F11@p1@p2~AF22@p1@p2~B + 4@p1@p2~AF11@p1@p2~BF22+ 2F11@p1@p2~A@p1@p2~BF22)=42: (9f)3 Here a covariant derivative and a eld strength are de- ned by DX~A(X;p)@X~A(X;p) + [A(X);~A(X;p)]=i~; (10a) F(X)@XA(X)@XA(X) + [A(X);A(X)]=i~: (10b) For simplicity, the arguments X;p are omitted in Eq. (9) and below. After all, the Moyal product is regarded as a pertur- bation theory with respect to spacetime gradients as well as eld strengths, but not to gauge elds. Thus, the gauge covariance of the results is guaranteed. PDand PFdenote the rst-order contributions with respect to spacetime gradients Dand eld strengths F, respectively. PDFis the mixed second-order contribution involving D andF. We also write down the second order with respect toFin Eq. (9f), which may be useful for studying other nonlinear responses in the future. In order to derive PF2, we need the fourth order with respect to x13andx32and obtain many other terms. All the terms up to the fourth order are written in Eq. (A8). C. Gradient expansion up to the second order Here we derive the gradient expansion of the Keldysh Green's function. We focus on the abelian case and as- sume a static and uniform eld strength. Similarly to Eq. (9a), we expand the Green's function and self-energy as23,24 ~G=~G0+ (~=2)~GD+ (~=2)~GF+ (1=2!)(~=2)2~GD2 + (~=2)2~GDF+ (1=2!)(~=2)2~GF2; (11a) ~ =~0+ (~=2)~D+ (~=2)~F+ (1=2!)(~=2)2~D2 + (~=2)2~DF+ (1=2!)(~=2)2~F2: (11b) Note that ~G0is the unperturbed Green's function with disorder or interactions being taken into account. ~GP and ~GPQ(P;Q =D;F) are the rst and second orders with respect to spacetime gradients or eld strengths, respectively. By substituting these into the left Dyson equation, (~L~)?~G= 1; (12) in which ~Lis the Lagrangian, we get ~G0= (~L ~0)1 and ~G1 0~GP=~P~G0iPP(~G1 0;~G0); (13a)~G1 0~GPQ=~PQ~G0i2PPQ(~G1 0;~G0) + [~Q~GP+iPP(~Q;~G0)iPP(~G1 0;~GQ) + (P$Q)]: (13b) The self-energies are determined self-consistently. To calculate the expectation values, the lesser Green's function is necessary. In the real-time representation, the Green's function and self-energy are of matrix forms23,24, ~G= GR2G< 0GA ; (14a) ~ = R2< 0 A ; (14b) in which R, A, and <indicate the retarded, advanced, and lesser components, respectively. For the rst order, the lesser one can be written as G< P=[(GR PGA P)f(p0) +G<(1) Pf0(p0)];(15a) < P=[(R PA P)f(p0) + <(1) Pf0(p0)];(15b) in which the upper and lower signs indicate boson and fermion, respectively, and f() = (e=T1)1is the distribution function at temperature T. By introducing PP(~A;~B)IJ P@I~A@J~B(I;J=X;p) withXp D = pX D = andpp F =F, we obtain an equivalent form of Eq. (13a)23,24, (GR 0)1GR P=R PGR 0iIJ P@I(GR 0)1@JGR 0; (16a) (GA 0)1GA P=A PGA 0iIJ P@I(GA 0)1@JGA 0; (16b) (GR 0)1G<(1) P=<(1) PGA 0+iIp0 Pf@I(GR 0)1(GR 0GA 0) [(GR 0)1(GA 0)1]@IGA 0g: (16c) Similarly, for the second order, we obtain the lesser Green's function, G< PQ=[(GR PQGA PQ)f(p0) +G<(1) PQf0(p0) +G<(2) PQf00(p0)]; (17a) < PQ=[(R PQA PQ)f(p0) + <(1) PQf0(p0) + <(2) PQf00(p0)]; (17b) and an equivalent form of Eq. (13b), (GR 0)1GR PQ=R PQGR 0+ [R QGR P+iIJ P@IR Q@JGR 0iIJ P@I(GR 0)1@JGR Q+ (P$Q)] +IJ PKL Q@I@K(GR 0)1@J@LGR 0; (18a)4 (GA 0)1GA PQ=A PQGA 0+ [A QGA P+iIJ P@IA Q@JGA 0iIJ P@I(GA 0)1@JGA Q+ (P$Q)] +IJ PKL Q@I@K(GA 0)1@J@LGA 0; (18b) (GR 0)1G<(1) PQ=<(1) PQGA 0+ R QG<(1) P+ <(1) QGA P+iIJ P@I<(1) Q@JGA 0iIp0 P[@IR Q(GR 0GA 0)(R QA Q)@IGA 0] iIJ P@I(GR 0)1@JG<(1) Q+iIp0 Pf@I(GR 0)1(GR QGA Q)[(GR 0)1(GA 0)1]@IGA Qg Ip0 PKL Qf@I@K(GR 0)1@L(GR 0GA 0) +@L[(GR 0)1(GA 0)1]@I@KGA 0g+ (P$Q) ; (18c) (GR 0)1G<(2) PQ=<(2) PQGA 0+ [iIp0 P<(1) Q@IGA 0+iIp0 P@I(GR 0)1G<(1) Q+ (P$Q)] +Ip0 PKp0 Qf@I@K(GR 0)1(GR 0GA 0) + [(GR 0)1(GA 0)1]@I@KGA 0g: (18d) Note that the left and right Dyson equations are equiv- alent as explicitly proved in Appendix B. Generally, the nth-order lesser Green's function with respect to a tem- poral gradient or an electric eld involves the nth deriva- tive of the distribution function. D. Spin torques Spin torques are proportional to the spin expectation value. The spin expectation value is given by h~ i=i~ZdDp (2~)Dtr~ G< =i~ZdDp (2~)Dtr~ < 0+ (~=2)G< D+ (~=2)G< F + (1=2!)(~=2)2G< D2+ (~=2)2G< DF + (1=2!)(~=2)2G< F2]: (19) Among many terms in Eq. (19), G< Dyields the spin renor- malization and Gilbert damping. G< Fyields the spin- orbit torque in the presence of SOIs. In order to calculate the STT and -term driven by an electric eld, G< DFis necessary. Equations (15)-(19) are our central results for calculating spin torques in generic systems. III. APPLICATION TO A 3D FERROMAGNETIC METAL As an example, we explicitly calculate spin torques in a 3d ferromagnetic metal, H(X;~ p) =~ p2=2mJ~ n(X)
~ ; (20) with the chemical potential and [~ n(X)]2= 1. We take into account nonmagnetic and magnetic impurities, Vimp(~X) =NiX jvi(~X~Xij)+NsX jvs~ mj
~ (~X~Xsj): (21) Magnetic impurities are assumed to be isotropic, namely, ma i= 0;ma imb j=ab=3 after average over the magnetiza- tion directions. The same system was studied within the Born approximation in the literature9,11. Here we em- ploy the self-consistent Born approximation, but it does not change any results quantitatively. We only have to calculate the momentum integrals of G< 0,G< D,G< F, andG< DFwith Eqs. (15)-(18) in order. First, the unperturbed Green's function is given by (GR 0)1(X;;~ p ) =H(X;~ p)R 0(X;) =R()x+JR()~ n(X)
~ ; (22a) GR 0(X;;~ p ) =R 0(;~ p)[R()x JR()~ n(X)
~ ]; (22b) R 0(;~ p) =f[R()x]2[JR()]2g1;(22c) withR()+R 00(),JR()J R 03(), andx~ p2=2m. Note that the self-energy is expressed as R 0(X;) = R 00() + R 03()~ n(X)
~ due to the spin rotation symmetry. For conve- nience, we dene gas the momentum integral of Gmutiplied by 4 (~2=2m)3=2and introduce i niv2 i(2m=~2)3=2=4; snsv2 s(2m=~2)3=2=4. The mo- mentum integral and self-energy are obtained by self- consistently solving gR 0(X;)4~2 2m3=2Zd3p (2~)3GR 0(X;;~ p ) =IR 11()IR 01()~ n(X)
~ ; (23a) R 0(X;) = igR 0(X;) + s~ mi
~ gR 0(X;)~ mj
~  =( i+ s)gR 00() +( i s=3)gR 03()~ n(X)
~ ; (23b) in which we dene IR mn()Z 0pxdx [R()x]m[JR()]2nm1 f[R()x]2[JR()]2gn:(24)5 Second, the rst-order Green's functions are given by GR D(X;;~ p ) =GR 0(X;;~ p )R D(X;)GR 0(X;;~ p ) + 2JR()[R 0(;~ p)]2fJR()R0()[R()x]JR0()g~ n(X)_~ n(X)
~  + 2[JR()]2[R 0(;~ p)]2(pi=m)~ n(X)@Xi~ n(X)
; (25a) G<(1) D(X;;~ p ) =GR 0(X;;~ p )<(1) D(X;)GA 0(X;;~ p ) i[JR() +JA()]jR 0(;~ p)j2[jR()xj2jJR()j2]_~ n(X)
~  + [JR() +JA()]jR 0(;~ p)j2f[R()x]JA()[A()x]JR()g~ n(X)_~ n(X)
~  i@X0[GR 0(X;;~ p ) +GA 0(X;;~ p )]; (25b) GR F(X;;~ p ) =GR 0(X;;~ p )R F(X;)GR 0(X;;~ p ); (25c) G<(1) F(X;;~ p ) =GR 0(X;;~ p )<(1) F(X;)GA 0(X;;~ p ) +iFj0(pj=m)f2GR 0(X;;~ p )GA 0(X;;~ p )[GR 0(X;;~ p )]2[GA 0(X;;~ p )]2g: (25d) The self-energies are expressed as R D(X;)R D2()~ n(X)_~ n(X)
~ ;<(1) D(X;)<(1) D1()_~ n(X)
~ +<(1) D2()~ n(X) _~ n(X)
~ and obtained by solving the following sets of linear equations, gR D2() =IR 01()R D2()=JR() + 2[R0()IR 02()JR0()IR 12()]=JR(); (26a) R D2() =( i s=3)gR D2(); (26b)" g<(1) D1() g<(1) D2()# =" J<(1) 1()J<(1) 2() J<(1) 2()J<(1) 1()#" <(1) D1() <(1) D2()# 2i" [JR() +JA()]J<(1) 1()=2[IR 01() + c:c:]=2 [JR() +JA()]J<(1) 2()=2# ; (26c) " <(1) D1() <(1) D2()# =( i s=3)" g<(1) D1() g<(1) D2()# : (26d) Here we dene J<(1) 1()Z 0pxdx jR()xj2jJR()j2 j[R()x]2[JR()]2j2; (27a) J<(1) 2()Z 0pxdx i[R()x]JA()i[A()x]JR() j[R()x]2[JR()]2j2; (27b) J<(1) 3()J<(1) 2()2 3[iJR()iJA()]Z 0pxdx x j[R()x]2[JR()]2j2; (27c) J<(1) 4()Z 0pxdx 2 j[R()x]2[JR()]2j2 jJR()j2 +2 3xJR() [R()x]2[JR()]2+JA() [A()x]2JA2() f[R()x]JA() + [A()x]JR()g : (27d) These momentum integrals in Eqs. (24) and (27) are explicitly calculated in Appendix C. The other components vanish, i.e., R F(X;) =gR F(X;) = 0 and <(1) F(X;) =g<(1) F(X;) = 0. Third, the second-order Green's functions are given by GR DF(~X;;~ p ) =GR 0(~X;;~ p )R DF(~X;)GR 0(~X;;~ p ) 2[R 0(;~ p)]2 fJR()R0()[R()x]JR0()gij JR0()pipj=m
Fj0@Xi~ n(~X)
~ =m; (28a) G<(1) DF(~X;;~ p ) =GR 0(~X;;~ p )<(1) DF(~X;)GA 0(~X;;~ p ) 2jR 0(;~ p)j2 f[R()x]JA()[A()x]JR()gij [JR()JA()]pipj=m
Fj0@Xi~ n(~X)
~ =m + 4ijR 0(;~ p)j2 jJR()j2ij+ [R 0(;~ p)JR() + A 0(;~ p)JA()]6 f[R()x]JA() + [A()x]JR()gpipj=m
Fj0~ n(~X)@Xi~ n(~X)
~ =m; (28b) where we drop the X0dependence in ~ n(X) because we are interested in the STT and -term only. The self- energies are expressed as R DF(~X;)R DF1()Fi0@Xi~ n(~X)
~ =m; <(1) DF(~X;)<(1) DF1()Fi0@Xi~ n(~X)
~ =m + <(1) DF2()Fi0~ n(~X)@Xi~ n(~X)
~ =m and obtained by solving the following sets of linear equations, gR DF1() =IR 01()R DF1()=JR() 2f[3JR()R0()2R()JR0()]IR 02()JR()JR0()IR 12()g=3JR3(); (29a) R DF1() =( i s=3)gR DF1(); (29b)" g<(1) DF1() g<(1) DF2()# =" J<(1) 1()J<(1) 2() J<(1) 2()J<(1) 1()#" <(1) DF1() <(1) DF2()# + 2i" J<(1) 3() J<(1) 4()# ; (29c) " <(1) DF1() <(1) DF2()# =( i s=3)" g<(1) DF1() g<(1) DF2()# : (29d) The spin expectation value is given by h~ i(X) =i~Zd 2~Zd3p (2~)3tr~ G<(X;;~ p ) =h~ i0(X)~ 42J2mJ ~23=2 [ren~ n(X)]_~ n(X) 1 42J2mJ ~21=2 Fi0[STT~ n(~X)]@Xi~ n(~X); (30a) h~ i0(X)1 222m ~23=2 ~ n(X)Z df()[=gR 03()]; (30b) J1=2Z d[f0()][ig<(1) D1()=2]; (30c) renJ1=2Z dff()[=gR D2()][f0()][ig<(1) D2()=2]g; (30d) J1=2Z dff()[=gR DF1()] + [f0()][g<(1) DF1()=2i]g; (30e) STTJ1=2Z d[f0()][g<(1) DF2()=2i]; (30f) and spin torques by ~ (X) =~ n(X)Jh~ i(X) =~ 422mJ ~23=2 [ren+~ n(X)]_~ n(X) 1 422mJ ~21=2 Fi0[STT+~ n(~X)]@Xi~ n(~X): (31) renandare the dimensionless spin renormalization and Gilbert damping, while STT andare the STT and-term. To evaluate Eqs. (30c)-(30f), we carry out numerical integrals by putting the energy unit J= 1, the momentum cuto  = 103, and temperature T= 103 and dividing the energy interval jj<5 into 217subinter- vals. In Fig. 1, we show their chemical-potential depen-dences for dierent iand s. We also show the previous results obtained by the small-amplitude9and the SU(2) gauge transformation formalisms11at zero temperature, ren=(3=2 +3=2 )=3J3=2; (32a) = s(1=2 ++1=2 )2=3J3=2; (32b) STT=1 3J1=2X 3=2  i1=2 + s(21=2 +1=2 )=3 !2J1=2(3 i+ 5 s)=9( i+ s)2STT1;(32c) =2 s(1=2 ++1=2 )STT=3J; (32d) with+J. Note that STT1in Eq. (32c) is STT for the!1 limit. Our results completely coincide with these previous ones.7 0 1 2 3 4 5 6 -1-0.5 0 0.5 1 1.5 2 2.5 3(a) 3τren µγi=4e-3, γs=4e-3 γi=4e-3, γs=8e-3 γi=8e-3, γs=4e-3 γi=8e-3, γs=8e-3 0 2 4 6 8 10 12 -1-0.5 0 0.5 1 1.5 2 2.5 3(b) 3τα/γs µ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1-0.5 0 0.5 1 1.5 2 2.5 3(c) τSTT/τSTT∞ µ 0 0.5 1 1.5 2 2.5 3 3.5 -1-0.5 0 0.5 1 1.5 2 2.5 3(d) 3τβ/2γsτSTT µ FIG. 1. (Color online) Chemical-potential dependences of (a) 3 ren, (b) 3= s, (c)STT=STT1, and (d) 3 =2 sSTTfor dierent iand s. Points are obtained by the gradient expansion formalism and numerical calculation of Eqs. (30c)-(30f), while lines are obtained by the small-amplitude9and the SU(2) gauge transformation formalisms11. IV. DISCUSSION AND SUMMARY Let us clarify how the SU(2) gauge transformation formalism corresponds to the gradient expansion for- malism. In the former, the Green's function is diag- onalized by a unitary matrix U~ u
~ with~ u [sin=2 cos;sin=2 sin;cos=2], which transforms the magnetization ~ n[sincos;sinsin;cos] to~ zand yields the SU(2) gauge eld A iUy@XU=~ u @X~ u
~ 5,11,12. Thus, spacetime gradients of the magne- tization are described by this SU(2) gauge eld. In the latter,@X(GR 0)1/@X~ n
~ in Eqs. (16) and (18) is transformed to Uy@X(GR 0)1U/2~ z~A
~ and thus plays the same role as the SU(2) gauge eld. We emphasize that there is another source of the SU(2) gauge eld in the SU(2) gauge transformation formal- ism11. Magnetic impurities, which are quenched in the original frame, become dynamical and nonuniform in the adiabatic frame and yield the SU(2) gauge eld. If this contribution is not taken into account, the Gilbert damp- ing vanishes, and the -term is not fully reproduced when magnetic impurities are anisotropic. In the gradient ex- pansion, we do not rely on the SU(2) gauge transforma-tion and hence do not encounter such diculty. In summary, we demonstrated how the gradient ex- pansion works for calculating spin torques quantum- mechanically. We derived the gradient expansion up to the fourth order with respect to the relative coordinates with abelian or nonabelian gauge elds being taken into account, which enables us to investigate nonlinear re- sponses with respect to spacetime gradients as well as eld strengths. We applied this formalism to a 3d ferro- magnetic metal with nonmagnetic and magnetic impuri- ties and successfully reproduced the previous results on the spin renormalization, Gilbert damping, STT, and - term. The greatest advantage of our formalism is that we do not assume any assumptions such as small transverse uctuations or suer from the SU(2) gauge eld arising from magnetic impurities or SOIs. Our central results Eqs. (15)-(19) serve as rst-principles formulas of spin torques. ACKNOWLEDGMENTS We thank J. Fujimoto and G. Tatara for discussion and reading our manuscript. This work was supported by RIKEN Special Postdoctoral Researcher Program. Appendix A: Derivation of Eq. (9)and more In this Appendix, we evaluate the Wigner representation of convolution Eq. (8), ^AB(X12;p12) =Z dDx12Z dDx3ep12x 12=i~W(X12;x1)^A(x1;x3)^B(x3;x2)W(x2;X12) =Z dDx12Z dDx3ZdDp13 (2~)DZdDp32 (2~)Dep12x 12=i~ep13x 13=i~ep32x 32=i~ W(X12;x1)W(x1;X13)~A(X13;p13)W(X13;x3) W(x3;X32)~B(X32;p32)W(X32;x2)W(x2;X12) (A1) =Z dDx12Z dDx3ZdDp13 (2~)DZdDp32 (2~)Dep12x 12=i~ep13x 13=i~ep32x 32=i~ [W(X12;x1)W(x1;X13)W(X13;X12)][W(X12;X13)~A(X13;p13)W(X13;X12)]8 (a) X12 X32X13x1 x3 x2(b) X12 X32X13x1 x3 x2(c)v -1 -11 1 u S2S3S1 FIG. 2. Transformation from (a) the Wilson line corresponding to Eq. (A1) to (b) that to Eq. (A2). The plane spanned by x1;x2;x3is divided into three planes, S1,S2, andS3. (c) Three planes are parametrized by y=X12+ux32=2 +vx13=2. [W(X12;X13)W(X13;x3)W(x3;X32)W(X32;X12)][W(X12;X32)~B(X32;p32)W(X32;X12)] [W(X12;X32)W(X32;x2)W(x2;X12)]: (A2) Here we insert the identities W(X13;X12)W(X12;X13) =W(X32;X12)W(X12;X32) = 1, which corresponds to trans- formation of the Wilson line as shown in Fig. 2. We dene three planes S1,S2, andS3as in Fig. 2(b), whose origin isX12. To evaluate the rst factor of the integrand in Eq. (A2), we use the nonabelian Stokes theorem21,22, W(X12;x1)W(x1;X13)W(X13;X12) =Sexp 1 2i~Z S1dydyW(X12;y)F(y)W(y;X 12) =Sexp1 8i~Z1 0duZu 0dv(x 13x 32x 13x 32)W(X12;y)F(y)W(y;X 12) ;(A3) in whichSis the surface-ordered product21,22, andy=X12+ux32=2+vx13=2. By using the Taylor expansion around X12, we obtain F(y) =e(ux 32+vx 13)@X 12=2F =F+ (ux 32+vx 13)@X 12F=2 + (ux1 32+vx1 13)(ux2 32+vx2 13)@X1 12@X2 12F=8 +O(x3); (A4a) W(y;X 12) =11 2i~Z1 0dw(ux 32+vx 13)[A+w(ux 32+vx 13)@X 12A=2] +1 4(i~)2Z1 0dw1Zw1 0dw2(ux1 32+vx1 13)(ux2 32+vx2 13)A1A2+O(x3) =1(ux 32+vx 13)A=2i~(ux 32+vx 13)(ux 32+vx 13)@X 12A=8i~ + (ux1 32+vx1 13)(ux2 32+vx2 13)A1A2=8(i~)2+O(x3); (A4b) W(X12;y)F(y)W(y;X 12) =F+ (ux 32+vx 13)(@X 12F+ [A;F]=i~)=2 + (ux1 32+vx1 13)(ux2 32+vx2 13)[@X1 12@X2 12F+ [@X2 12A1;F]=i~ + 2[A1;@X2 12F]=i~+ [A1;[A2;F]]=(i~)2]=8 +O(x3) =F+ (ux 32+vx 13)DX 12F=2 + (ux1 32+vx1 13)(ux2 32+vx2 13)DX1 12DX2 12F=8 +O(x3) =e(ux 32+vx 13)DX 12=2F: (A4c) Here and below we omit the argument X12for simplicity. Then, we express Eq. (A3) as W(X12;x1)W(x1;X13)W(X13;X12) =1 +1 8i~Z1 0duZu 0dv(x 13x 32x 13x 32)[F+ (ux 32+vx 13)DX 12F=29 + (ux1 32+vx1 13)(ux2 32+vx2 13)DX1 12DX2 12F=8] +1 64(i~)2Z1 0du1Zu1 0dv1Zu1 0du2Zu2 0dv2 (x1 13x1 32x1 13x1 32)(x2 13x2 32x2 13x2 32)F11F22+O(x5) =1 + (x 13x 32x 13x 32)F=16i~+ (x 13x 32x 13x 32)(2x 32+x 13)DX 12F=96i~ + (x1 13x1 32x1 13x1 32)(x2 13x2 32x2 13x2 32)F11F22=512(i~)2 + (x 13x 32x 13x 32)[6x1 32x2 32+ 3(x1 32x2 13+x2 32x1 13) + 2x1 13x2 13] DX1 12DX2 12F=1536i~+O(x5): (A5) The other factors in Eq. (A2) are similarly obtained as W(X12;X13)~A(X13;p13)W(X13;X12) =ex 32DX 12=2~A(p13); (A6a) W(X12;X32)~B(X32;p32)W(X32;X12) =ex 13DX 12=2~B(p32); (A6b) W(X12;X13)W(X13;x3)W(x3;X32)W(X32;X12) =1 + (x 13x 32x 13x 32)F=8i~ + (x 13x 32x 13x 32)(x 32x 13)DX 12F=32i~ + (x1 13x1 32x1 13x1 32)(x2 13x2 32x2 13x2 32)F11F22=128(i~)2 + (x 13x 32x 13x 32)[4x1 32x2 323(x1 32x2 13+x2 32x1 13) + 4x1 13x2 13] DX1 12DX2 12F=768i~+O(x5); (A6c) W(X12;X32)W(X32;x2)W(x2;X12) =1 + (x 13x 32x 13x 32)F=16i~ (x 13x 32x 13x 32)(x 32+ 2x 13)DX 12F=96i~ + (x1 13x1 32x1 13x1 32)(x2 13x2 32x2 13x2 32)F11F22=512(i~)2 + (x 13x 32x 13x 32)[2x1 32x2 32+ 3(x1 32x2 13+x2 32x1 13) + 6x1 13x2 13] DX1 12DX2 12F=1536i~+O(x5): (A6d) Thus, the integrand in Eq. (A2) is given by O(1) = ~A(p13)~B(p32); (A7a) O(x) =[x 32DX 12~A(p13)~B(p32)x 13~A(p13)DX 12~B(p32)]=2; (A7b) O(x2) =(x 13x 32x 13x 32)[F~A(p13)~B(p32) + 2 ~A(p13)F~B(p32) +~A(p13)~B(p32)F]=16i~ (A7c) + [x1 32x2 32DX1 12DX2 12~A(p13)~B(p32)2x1 32x2 13DX1 12~A(p13)DX2 12~B(p32) +x1 13x2 13~A(p13)DX1 12DX2 12~B(p32)]=8; (A7d) O(x3) =(x 13x 32x 13x 32)fF[x 32DX 12~A(p13)~B(p32)x 13~A(p13)DX 12~B(p32)] + 2[x 32DX 12~A(p13)F~B(p32)x 13~A(p13)FDX 12~B(p32)] + [x 32DX 12~A(p13)~B(p32)x 13~A(p13)DX 12~B(p32)]Fg=32i~ (A7e) + (x 13x 32x 13x 32)[(2x 32+x 13)DX 12F~A(p13)~B(p32) + 3(x 32x 13)~A(p13)DX 12F~B(p32) (x 32+ 2x 13)~A(p13)~B(p32)DX 12F]=96i~ (A7f) + [x1 32x2 32x3 32DX1 12DX2 12DX3 12~A(p13)~B(p32)3x1 32x2 32x3 13DX1 12DX2 12~A(p13)DX3 12~B(p32) + 3x1 32x2 13x3 13DX1 12~A(p13)DX2 12DX3 12~B(p32)x1 13x2 13x3 13~A(p13)DX1 12DX2 12DX3 12~B(p32)]=48; (A7g) O(x4) =(x1 13x1 32x1 13x1 32)(x2 13x2 32x2 13x2 32) [F11F22~A(p13)~B(p32) + 4 ~A(p13)F11F22~B(p32) +~A(p13)~B(p32)F11F22 + 4F11~A(p13)F22~B(p32) + 4 ~A(p13)F11~B(p32)F22+ 2F11~A(p13)~B(p32)F22]=512(i~)2(A7h) + (x 13x 32x 13x 32)fF[x1 32x2 32DX1 12DX2 12~A(p13)~B(p32)2x1 32x2 13DX1 12~A(p13)DX2 12~B(p32)10 +x1 13x2 13~A(p13)DX1 12DX2 12~B(p32)] + 2[x1 32x2 32DX1 12DX2 12~A(p13)F~B(p32) 2x1 32x2 13DX1 12~A(p13)FDX2 12~B(p32) +x1 13x2 13~A(p13)FDX1 12DX2 12~B(p32)] + [x1 32x2 32DX1 12DX2 12~A(p13)~B(p32)2x1 32x2 13DX1 12~A(p13)DX2 12~B(p32) +x1 13x2 13~A(p13)DX1 12DX2 12~B(p32)]Fg=128i~ (A7i) + (x 13x 32x 13x 32)f(2x1 32+x1 13)DX1 12F[x2 32DX2 12~A(p13)~B(p32)x2 13~A(p13)DX2 12~B(p32)] + 3(x1 32x1 13)[x2 32DX2 12~A(p13)DX1 12F~B(p32)x2 13~A(p13)DX1 12FDX2 12~B(p32)] (x1 32+ 2x1 13)[x2 32DX2 12~A(p13)~B(p32)x2 13~A(p13)DX2 12~B(p32)]DX1 12Fg=96i~ (A7j) + (x 13x 32x 13x 32)f[6x1 32x2 32+ 3(x1 32x2 13+x2 32x1 13) + 2x1 13x2 13]DX1 12DX2 12F~A(p13)~B(p32) + [8x1 32x2 326(x1 32x2 13+x2 32x1 13) + 8x1 13x2 13]~A(p13)DX1 12DX2 12F~B(p32) + [2x1 32x2 32+ 3(x1 32x2 13+x2 32x1 13) + 6x1 13x2 13]~A(p13)~B(p32)DX1 12DX2 12Fg=1536i~ (A7k) + [x1 32x2 32x3 32x4 32DX1 12DX2 12DX3 12DX4 12~A(p13)~B(p32)4x1 32x2 32x3 32x4 13DX1 12DX2 12DX3 12~A(p13)DX4 12~B(p32) + 6x1 32x2 32x3 13x4 13DX1 12DX2 12~A(p13)DX3 12DX4 12~B(p32)4x1 32x2 13x3 13x4 13DX1 12~A(p13)DX2 12DX3 12DX4 12~B(p32) +x1 13x2 13x3 13x4 13~A(p13)DX1 12DX2 12DX3 12DX4 12~B(p32)]=384: (A7l) Now we can carry out the integrals by putting p13=p12+q13;p32=p12+q32. Sinceq13andq32appear in the forms of~A(p12+q13) and ~B(p12+q32),x 13andx 32can be replaced with i~@p12acting on ~Aand ~B, respectively. Finally, we obtain ~A?~B=~A~B+ (i~=2)PD(~A;~B) + (i~=2)PF(~A;~B) + (1=2!)(i~=2)2PD2(~A;~B) + (i~=2)2PDF(~A;~B) + (1=3)(i~=2)2PDF(~A;~B) + (1=3!)(i~=2)3PD3(~A;~B) + (1=2!)(i~=2)2PF2(~A;~B) + (1=2!)(i~=2)3PD2F(~A;~B) + (1=3)(i~=2)3PDDF(~A;~B) + (1=2!)(1=3)(i~=2)3PD2F(~A;~B) + (1=4!)(i~=2)4PD4(~A;~B); (A8a) PD(~A;~B)DX~A@p~B@p~ADX~B; (A8b) PF(~A;~B)(F@p~A@p~B+ 2@p~AF@p~B+@p~A@p~BF)=4; (A8c) PD2(~A;~B)DX1DX2~A@p1@p2~B2DX1@p2~A@p1DX2~B+@p1@p2~ADX1DX2~B; (A8d) PDF(~A;~B)[F(DX@p~A@p@p~B@p@p~ADX@p~B) + 2(DX@p~AF@p@p~B@p@p~AFDX@p~B) + (DX@p~A@p@p~B@p@p~ADX@p~B)F]=4; (A8e) PDF(~A;~B)[DXF(2@p~A@p@p~B+@p@p~A@p~B) + 3(@p~ADXF@p@p~B@p@p~ADXF@p~B) (@p~A@p@p~B+ 2@p@p~A@p~B)DXF]=4; (A8f) PD3(~A;~B)DX1DX2DX3~A@p1@p2@p3~B3DX1DX2@p3~A@p1@p2DX3~B + 3DX1@p2@p3~A@p1DX2DX3~B@p1@p2@p3~ADX1DX2DX3~B; (A8g) PF2(~A;~B)(F11F22@p1@p2~A@p1@p2~B+ 4@p1@p2~AF11F22@p1@p2~B +@p1@p2~A@p1@p2~BF11F22+ 4F11@p1@p2~AF22@p1@p2~B + 4@p1@p2~AF11@p1@p2~BF22+ 2F11@p1@p2~A@p1@p2~BF22)=42; (A8h) PD2F(~A;~B)[F(DX1DX2@p~A@p1@p2@p~B2DX1@p2@p~A@p1DX2@p~B +@p1@p2@p~ADX1DX2@p~B) + 2(DX1DX2@p~A@p1F@p2@p~B 2DX1@p2@p~AF@p1DX2@p~B+@p1@p2@p~AFDX1DX2@p~B) + (DX1DX2@p~A@p1@p2@p~B2DX1@p2@p~A@p1DX2@p~B11 +@p1@p2@p~ADX1DX2@p~B)F]=4; (A8i) PDDF(~A;~B)fDX1F[2(DX2@p~A@p1@p2@p~B@p2@p~ADX2@p1@p~B) + (DX2@p1@p~A@p2@p~B@p1@p2@p~ADX2@p~B)] + 3[(DX2@p~ADX1F@p1@p2@p~B@p2@p~ADX1FDX2@p1@p~B) (DX2@p1@p~ADX1F@p2@p~B@p1@p2@p~ADX1FDX2@p~B)] [(DX2@p~A@p1@p2@p~B@p2@p~ADX2@p1@p~B) + 2(DX2@p1@p~A@p2@p~B@p1@p2@p~ADX2@p~B)]DX1Fg=4; (A8j) PD2F(~A;~B)fDX1DX2F[6@p~A@p1@p2@p~B+ 3(@p2@p~A@p1@p~B+@p1@p~A@p2@p~B) + 2@p1@p2@p~A@p~B] + [8@p~ADX1DX2F@p1@p2@p~B6(@p2@p~ADX1DX2F@p1@p~B +@p1@p~ADX1DX2F@p2@p~B) + 8@p1@p2@p~ADX1DX2F@p~B] + [2@p~A@p1@p2@p~B + 3(@p2@p~A@p1@p~B+@p1@p~A@p2@p~B) + 6@p1@p2@p~A@p~B]DX1DX2Fg=16; (A8k) PD4(~A;~B)DX1DX2DX3DX4~A@p1@p2@p3@p4~B4DX1DX2DX3@p4~A@p1@p2@p4DX4~B + 6DX1DX2@p3@p4~A@p1@p2DX3DX4~B4DX1@p2@p3@p4~A@p1DX2DX3DX4~B +@p1@p2@p3@p4~ADX1DX2DX3DX4~B: (A8l) The arguments X;p are omitted. Equations (A8f), (A8g), and (A8i)-(A8l) are not written in Eq. (9). Appendix B: Equivalence of the left and right Dyson equations Equations (13), (16), and (18) are derived from the left Dyson equation. From the right Dyson equation, ~G?(~L~) = 1; (B1) we also obtain ~GP~G1 0=~G0~PiPP(~G0;~G1 0); (B2a) ~GPQ~G1 0=~G0~PQi2PPQ(~G0;~G1 0) + [ ~GP~Q+iPP(~G0;~Q)iPP(~GQ;~G1 0) + (P$Q)]; (B2b) GR P(GR 0)1=GR 0R PiIJ P@IGR 0@J(GR 0)1; (B3a) GA P(GA 0)1=GA 0A PiIJ P@IGA 0@J(GA 0)1; (B3b) G<(1) P(GA 0)1=GR 0<(1) P+iIp0 Pf@IGR 0[(GR 0)1(GA 0)1](GR 0GA 0)@I(GA 0)1g; (B3c) and GR PQ(GR 0)1=GR 0R PQ+ [GR PR Q+iIJ P@IGR 0@JR QiIJ P@IGR Q@J(GR 0)1+ (P$Q)] +IJ PKL Q@I@KGR 0@J@L(GR 0)1; (B4a) GA PQ(GA 0)1=GA 0A PQ+ [GA PA Q+iIJ P@IGA 0@JA QiIJ P@IGA Q@J(GA 0)1+ (P$Q)] +IJ PKL Q@I@KGA 0@J@L(GA 0)1; (B4b) G<(1) PQ(GA 0)1=GR 0<(1) PQ+ G<(1) PA Q+GR P<(1) Q+iIJ P@IGR 0@J<(1) QiIp0 P[@IGR 0(R QA Q)(GR 0GA 0)@IA Q] iIJ P@IG<(1) Q@J(GA 0)1+iIp0 Pf@IGR Q[(GR 0)1(GA 0)1](GR QGA Q)@I(GA 0)1g Ip0 PKL Qf@I@KGR 0@L[(GR 0)1(GA 0)1] +@L(GR 0GA 0)@I@K(GA 0)1g+ (P$Q) ; (B4c) G<(2) PQ(GA 0)1=GR 0<(2) PQ+ [iIp0 P@IGR 0<(1) QiIp0 PG<(1) Q@I(GA 0)1+ (P$Q)] +Ip0 PKp0 Qf@I@KGR 0[(GR 0)1(GA 0)1] + (GR 0GA 0)@I@K(GA 0)1g: (B4d)12 Thus, the left and right Dyson equations seem dierent from each other but in fact are equivalent. Both equations lead to GR P=GR 0R PGR 0+iIJ PGR 0@I(GR 0)1GR 0@J(GR 0)1GR 0; (B5a) GA P=GA 0A PGA 0+iIJ PGA 0@I(GA 0)1GA 0@J(GA 0)1GA 0; (B5b) G<(1) P=GR 0<(1) PGA 0iIp0 PfGR 0@I[(GR 0)1+ (GA 0)1]GA 0+@I(GR 0+GA 0)g: (B5c) and GR PQ=GR 0R PQGR 0+fGR 0R QGR 0R PGR 0+iIJ PGR 0[@IR QGR 0@J(GR 0)1@I(GR 0)1GR 0@JR Q + R QGR 0@I(GR 0)1GR 0@J(GR 0)1+@I(GR 0)1GR 0R QGR 0@JGR 0+GR 0@I(GR 0)1GR 0@J(GR 0)1GR 0R Q]GR 0 + (P$Q)g+IJ PKL QGR 0f@I@K(GR 0)1GR 0@J@L(GR 0)1 @J(GR 0)1GR 0@I@K(GR 0)1GR 0@L(GR 0)1@L(GR 0)1GR 0@I@K(GR 0)1GR 0@J(GR 0)1 +@I@K(GR 0)1GR 0[@J(GR 0)1GR 0@L(GR 0)1+@L(GR 0)1GR 0@J(GR 0)1] + [@J(GR 0)1GR 0@L(GR 0)1+@L(GR 0)1GR 0@J(GR 0)1]GR 0@I@K(GR 0)1 @I(GR 0)1GR 0@J(GR 0)1GR 0@K(GR 0)1GR 0@L(GR 0)1@I(GR 0)1GR 0@K(GR 0)1GR 0@J(GR 0)1GR 0@L(GR 0)1 @I(GR 0)1GR 0@K(GR 0)1GR 0@L(GR 0)1GR 0@J(GR 0)1@K(GR 0)1GR 0@L(GR 0)1GR 0@I(GR 0)1GR 0@J(GR 0)1 @K(GR 0)1GR 0@I(GR 0)1GR 0@L(GR 0)1GR 0@J(GR 0)1@K(GR 0)1GR 0@I(GR 0)1GR 0@J(GR 0)1GR 0@L(GR 0)1g GR 0; (B6a) GA PQ=GA 0A PQGA 0+fGA 0A QGA 0A PGA 0+iIJ PGA 0[@IA QGA 0@J(GA 0)1@I(GA 0)1GA 0@JA Q + A QGA 0@I(GA 0)1GA 0@J(GA 0)1+@I(GA 0)1GA 0A QGA 0@JGA 0+GA 0@I(GA 0)1GA 0@J(GA 0)1GA 0A Q]GA 0 + (P$Q)g+IJ PKL QGA 0f@I@K(GA 0)1GA 0@J@L(GA 0)1 @J(GA 0)1GA 0@I@K(GA 0)1GA 0@L(GA 0)1@L(GA 0)1GA 0@I@K(GA 0)1GA 0@J(GA 0)1 +@I@K(GA 0)1GA 0[@J(GA 0)1GA 0@L(GA 0)1+@L(GA 0)1GA 0@J(GA 0)1] + [@J(GA 0)1GA 0@L(GA 0)1+@L(GA 0)1GA 0@J(GA 0)1]GA 0@I@K(GA 0)1 @I(GA 0)1GA 0@J(GA 0)1GA 0@K(GA 0)1GA 0@L(GA 0)1@I(GA 0)1GA 0@K(GA 0)1GA 0@J(GA 0)1GA 0@L(GA 0)1 @I(GA 0)1GA 0@K(GA 0)1GA 0@L(GA 0)1GA 0@J(GA 0)1@K(GA 0)1GA 0@L(GA 0)1GA 0@I(GA 0)1GA 0@J(GA 0)1 @K(GA 0)1GA 0@I(GA 0)1GA 0@L(GA 0)1GA 0@J(GA 0)1@K(GA 0)1GA 0@I(GA 0)1GA 0@J(GA 0)1GA 0@L(GA 0)1g GA 0; (B6b) G<(1) PQ=GR 0<(1) PQGA 0+ GR 0(R QGR 0<(1) P+ <(1) QGA 0A P)GA 0+iIJ PGR 0[@I<(1) QGA 0@J(GA 0)1@I(GR 0)1GR 0@J<(1) Q + <(1) QGA 0@I(GA 0)1GA 0@J(GA 0)1+@I(GR 0)1GR 0@J(GR 0)1GR 0<(1) Q+@I(GR 0)1GR 0<(1) QGA 0@J(GA 0)1]GA 0 +iIp0 PGR 0f@I(R Q+ A Q)R QGR 0@I[(GR 0)1+ (GA 0)1]@I[(GR 0)1+ (GA 0)1]GA 0A QgGA 0 +Ip0 PKL QGR 0f@I@K[(GR 0)1+ (GA 0)1]GA 0@L(GA 0)1@K(GR 0)1GR 0@I@L[(GR 0)1+ (GA 0)1] +@K(GR 0)1GR 0@I[(GR 0)1+ (GA 0)1]GA 0@L(GA 0)1+@I[(GR 0)1+ (GA 0)1]GA 0@K(GA 0)1GA 0@L(GA 0)1 +@K(GR 0)1GR 0@L(GR 0)1GR 0@I[(GR 0)1+ (GA 0)1]gGA 0iIp0 P@I(GR Q+GA Q) + (P$Q) ; (B6c) G<(2) PQ=GR 0<(2) PQGA 0+fiIp0 PGR 0[@I(GR 0)1GR 0<(1) Q<(1) QGA 0@I(GA 0)1]GA 0+ (P$Q)g +Ip0 PKp0 Q GR 0f@I@K[(GR 0)1(GA 0)1]@I[(GR 0)1+ (GA 0)1]GA 0@K(GA 0)1 @K[(GR 0)1+ (GA 0)1]GA 0@I(GA 0)1+@I(GR 0)1GR 0@K[(GR 0)1+ (GA 0)1] +@K(GR 0)1GR 0@I[(GR 0)1+ (GA 0)1]gGA 0+@I@K(GR 0GA 0)
: (B6d)13 Appendix C: Momentum integrals in Eqs. (24)and (27) In this Appendix, we give the explicit forms of the momentum integrals dened above. Here we introduce D0(E)Z 0pxdx 1 Ex= 2[1=2E1=2tanh1(=E)1=2]=; (C1a) D1(E) =Z 0pxdx 1 (Ex)2= [1=2=(E)E1=2tanh1(=E)1=2]=; (C1b) andER ()R()JR(). Equation (24) is given by IR 01() =[D0(ER +())D0(ER ())]=2; (C2a) IR 11() =[D0(ER +()) +D0(ER ())]=2; (C2b) IR 02() =[D0(ER +())D0(ER ())JR()D1(ER +())JR()D1(ER ())]=4; (C2c) IR 12() =[JR()D1(ER +())JR()D1(ER ())]=4; (C2d) IR 22() =[D0(ER +())D0(ER ()) +JR()D1(ER +()) +JR()D1(ER ())]=4; (C2e) and Eq. (27) by J<(1) 1() =1 2D0(ER +()) ER +()EA ()+D0(ER ()) ER ()EA +() + c:c:; (C3a) J<(1) 2() =i 2D0(ER +()) ER +()EA ()D0(ER ()) ER ()EA +() + c:c:; (C3b) J<(1) 3() =J<(1) 2()1 3[iJR()iJA()] ER +()D0(ER +()) JR()[ER +()EA +()][ER +()EA ()]ER ()D0(ER ()) JR()[ER ()EA ()][ER ()EA +()]+ c:c: ;(C3c) J<(1) 4() =1 3[ER +()]2jR()j2+ 2jJR()j2 JR()[ER +()EA +()][ER +()EA ()]D0(ER +()) [ER ()]2jR()j2+ 2jJR()j2 JR()[ER ()EA ()][ER ()EA +()]D0(ER ()) ER +() ER +()EA +()D1(ER +())ER () ER ()EA ()D1(ER ()) + c:c: (C3d) 1L. Berger, J. Appl. Phys. 55, 1954 (1984). 2L. Berger, J. Appl. Phys. 71, 2721 (1992). 3J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). 4L. Berger, Phys. Rev. B 54, 9353 (1996). 5G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004). 6S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004). 7A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Eu- rophys. Lett. 69, 990 (2005). 8Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 74, 144405 (2006). 9H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006). 10F. Pi echon and A. Thiaville, Phys. Rev. B 75, 174414(2007). 11H. Kohno and J. Shibata, J. Phys. Soc. Jpn. 76, 063710 (2007). 12K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008). 13A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). 14A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009). 15E. van der Bijl and R. A. Duine, Phys. Rev. B 86, 094406 (2012). 16J. Rammer, Quantum Field Theory of Non-equilibrium States (Cambridge University Press, New York, 2007). 17A. Kamenev, Field Theory of Non-Equilibrium Systems (Cambridge University Press, New York, 2011). 18M. Levanda and V. Fleurov, J. Phys.: Condens. Matter 6, 7889 (1994).14 19M. Levanda and V. Fleurov, Ann. Phys. 292, 199 (2001). 20C. Gorini, P. Schwab, R. Raimondi, and A. L. Shelankov, Phys. Rev. B 82, 195316 (2010). 21I. Y. Aref'eva, Theor. Math. Phys. 43, 353 (1980).22N. E. Brali c, Phys. Rev. D 22, 3090 (1980). 23S. Onoda, N. Sugimoto, and N. Nagaosa, Prog. Theor. Phys. 116, 61 (2006). 24A. Shitade, Prog. Theor. Exp. Phys. 2014 , 123I01 (2014).

2017-08-11

We propose a new quantum-mechanical formalism to calculate spin torques based on the gradient expansion, which naturally involves spacetime gradients of the magnetization and electromagnetic fields. We have no assumption in the small-amplitude formalism or no difficulty in the SU($2$) gauge transformation formalism. As a representative, we calculate the spin renormalization, Gilbert damping, spin-transfer torque, and $\beta$-term in a three-dimensional ferromagnetic metal with nonmagnetic and magnetic impurities being taken into account within the self-consistent Born approximation. Our results serve as a first-principles formalism for spin torques.

Gradient expansion formalism for generic spin torques

1708.03424v1

arXiv:1904.10946v3 [math.AP] 9 Oct 2019ON THE ENERGY DECAY RATE OF THE FRACTIONAL WAVE EQUATION ON RWITH RELATIVELY DENSE DAMPING WALTON GREEN Abstract. We establish upper bounds for the decay rate of the energy of t he damped fractional wave equation when the averages of the dam ping coefficient on all intervals of a fixed length are bounded below. If the power of the fractional Laplacian, s, is between 0 and 2, the decay is polynomial. For s≥2, the decay is exponential. Our assumption is also necessary for energy de cay. Second, we prove that exponential decay cannot hold for s <2 if the damping vanishes at all. Consider the following damped fractional wave equation on Rfors >0 andγ:R→ R≥0: (1) wtt(x,t)+γ(x)wt(x,t)+(−∂xx+1)s/2w(x,t) = 0,(x,t)∈R×R+. Thedamping force is represented by γwt. Herein, we study the decay rate of the energy ofw, defined by E(t) =/bardbl(w(t),wt(t))/bardblHs/2×L2:=/parenleftbigg/integraldisplay R|(−∂xx+1)s/4w(x,t)|2+|wt(x,t)|2dx/parenrightbigg1/2 Standard analysis shows that if γ= 0, then the energy is conserved, i.e. there is no decay. On the other hand, for constant damping γ=c >0, it can be shown thatE(t) decays exponentially. Thus, the interest in this problem i s in interpolating between these cases. The fractional model (1) was introduce d recently by Malhi and Stanislavova in [9]. Our results are inspired by their paper , but we are even able to recover what is known in the classical case of s= 2 from a new perspective. In this classical setting, the problem was initially studie d on boundeddomains under theso-called Geometric Control Condition (GCC)[1, 14]whi chrequires γtobepositive (in some sense) on certain geodesic curves determined by the geometry of the domain. OnR, or more generally Rd, the GCC simplifies to the following: there exist Rand c >0 such that for all line segments ℓ∈Rdof length R, (2)/integraldisplay ℓγ(x)dx > c. Extending the results of Bardos, Lebeau, Rauch, Taylor, and Phillips [1, 14] to the non-compact case, Burq and Joly [3] proved exponential deca y of the energy under the GCC on Rdusing the semiclassical analysis of [15]. These methods, ut ilizing pseudodifferential calculus, require γto be sufficiently smooth. For the one-dimensional problem, this smoothness conditio n was relaxed by Malhi and Stanislavova in [10] to γwhich are continuous and bounded. Moreover, through intricate spectral analysis, it is shown that condition (2) is equivalent to exponential decay of the energy of solutions to (1). One special case of ou r results (Theorem 1 with 12 WALTON GREEN s= 2) recovers this result and moreover our proof of this equiv alence is of a different nature, applying ideas from the study of the harmonic analys t’s uncertainty principle, specifically the Paneah-Logvinenko-Sereda Theorem (see Th eorem 3 below). The initial investigation into the fractional case in [9] re quired the restrictive as- sumption that the set {x∈R:γ(x)≥ε}contains a periodic set. In this case, it was proved that the rate of decay is polynomial if s <2 and exponential if s≥2. Herein, we relax this periodic condition on {γ≥ε}to require that {γ≥ε}be relatively dense [5] which means there exists R >0 such that (3) inf a∈Rm({γ≥ε}∩[a−R,a+R])>0 wheremis the Lebesgue measure. Theorem 1. Let0≤γ∈L∞(R). There exists R >0such that (4) inf a∈R/integraldisplaya+R a−Rγ(x)dx >0 if and only if there exists C,ω >0such that E(t)≤ C(1+t)−s 4−2s/bardblw(0),wt(0)/bardblHs×Hs/2if0< s <2 Ce−ωtE(0) ifs≥2 for allt >0whenever the right-hand side is finite. Note that for γbounded, the condition (4) is equivalent to {x∈R:γ(x)≥ε}being a relatively dense set (3) for εsmall enough. However, if γis unbounded, then (4) is the weaker condition. The above result does not say anything about the optimality o f the rates. However, we can answer the question posed in [9] concerning the value o f the threshold between exponential and polynomial decay. In the final section, we sh ow that exponential decay neccesitates that sbe greater than 2 (as long as γis not bounded away from zero), thus establishing s= 2 as the threshold. Theorem 2. Let0≤γ∈L∞(R)ands >0. Suppose (i)m({γ= 0})>0. (ii)There exists C,ω >0such that E(t)≤Ce−ωtE(0) for allt >0andE(0)<∞. Thens≥2. The main ingredient in our proof is a resolvent estimate for t he fractional Laplacian (Proposition 1 below). In proving this, we will rely on the st udy of the uncertainty principle for the Fourier transform [5] which is defined by F(f)(ξ) :=ˆf(ξ) =1√ 2π/integraldisplay Rf(x)e−ixξdx forξ∈R,f∈L1(R)∩L2(R).Fthen uniquely extends to a unitary operator on L2(R). The manifestation of the uncertainty principle we will use i s a generalization due to O. Kovrijkine of the classical Paneah-Logvinenko-Sereda The orem [11, 8].DECAY OF FRACTIONAL WAVE EQUATION WITH DENSE DAMPING 3 Theorem 3 (Thm 2 from [7]) .Let{Jk}n k=1be intervals in Rwith|Jk|=b. LetE⊂R which is relatively dense. Then, there exists c >0such that /bardblf/bardblLp(E)≥c/bardblf/bardblLp(R) for allf∈Lp,p∈[1,∞]withsuppˆf⊂/uniontextn k=1Jk. Moreover, cdepends only on the number and size of the intervals, not on how they are placed. In the proof of the proposition, we will only need the case whe np= 2 and there are two intervals J1,J2. In order to conclude the polynomial or exponential decay in T heorem 1, we will use the following two results on semigroups which connect re solvent bounds for the generator to the decay of the semigroup. For exponential dec ay, there is the following characterization from [6, Theorem 3] (See also [4, 13]). Theorem 4 (Gearhart-Pruss Test) .LetetAbe aC0-semigroup in a Hilbert space H and assume there exists M >0such that /bardbletA/bardbl ≤Mfor allt≥0. Then, there exists C,ω >0such that /bardbletA/bardbl ≤Ce−ωt if and only if iR⊂ρ(A)andsupλ∈R/bardbl(A−iλ)−1/bardbl<∞. For the polynomial decay, we use the following result from [2 , Theorem 2.4]: Theorem 5 (Borichev-Tomilov) .LetetAbe aC0-semigroup on a Hilbert space H. Assume there exists M >0such that /bardbletA/bardbl ≤Mfor allt≥0andiR⊂ρ(A). Then for a fixed α >0, /bardbletAA−1/bardbl=O(t−1/α)ast→ ∞ if and only if /bardbl(A−iλ)−1/bardbl=O(λα)asλ→ ∞. 1.Resolvent Estimates Proposition 1. LetΩ⊂Rbe relatively dense, s >0. There exists c >0(depending onΩ,s) such that for all f∈L2(R),λ≥0. (5) c/bardblf/bardbl2 L2(R)≤(1+λ)2 s−2/bardbl((−∂xx+1)s/2−λ)f/bardbl2 L2(R)+/bardblf/bardbl2 L2(Ω). The operator ( −∂xx+1)s/2is understood as a strictly positive Fourier multiplier: (−∂xx+1)s/2f(x) :=1√ 2π/integraldisplay R(|ξ|2+1)s/2ˆf(ξ)eixξdξ. Throughout, wedenoteby /bardbl·/bardblthenorm /bardbl·/bardblL2(R). Webeginwith thefollowing algebraic lemma. Lemma 1. Lets >0. There exists cs>0such that |τs−λ| ≥cs(1+λ)1−1/s for allτ,λ≥0in the region |τ−λ1/s|>1. Proof.First, for any s >0, there exists ds>0 such that dsmax(x,y)s−1|x−y| ≤ |xs−ys| for allx,y∈R+. Next, consider two cases.4 WALTON GREEN (i) Ifτ≥λ1/s+1, then |τs−λ| ≥dsmax(τ,λ1/s)s−1|τ−λ1/s|=dsτs−1|τ−λ1/s|. The function x/ma√sto→xs−1(x−µ) is positive and increasing for x > µ+1, so we can bound the final term from below by its value at τ=λ1/s+1 which yields |τs−λ| ≥ds(λ1/s+1)s−1. (ii) Ifτ≤λ1/s−1, then |τs−λ| ≥dsmax(τ,λ1/s)s−1·1 =ds(λ1/s)s−1. Ifs <1, thens−1<0 so (λ1/s)s−1≥(λ1/s+1)s−1. Since 0 ≤τ≤λ1/s−1, λ≥1. So, for s≥1, (λ1/s)s−1=(λ1/s+λ1/s)s−1 2s−1≥(λ1/s+1)s−1 2s−1. Therefore, there exists cssuch that |τs−λ| ≥cs(λ1/s+1)s−1≥cs(λ+1)1−1 s where in the final step, we have used the fact that for p≤q, (xq+yq)1/q≤(xp+ yp)1/p. /square Proof of Proposition 1. Letλ≥0,s >0 andg∈L2(R) such that suppˆ g⊂Aλ:={ξ∈ R:/vextendsingle/vextendsingle(|ξ|2+1)1/2−λ1/s/vextendsingle/vextendsingle≤1}. Notice that Aλis the union of two intervals of length no more than 4. Therefore, for Ω ⊂Rwhich is relatively dense, by Theorem 3, there existsC >0 (independent of λandg) such that /bardblg/bardbl ≤C/bardblg/bardblL2(Ω). Denote by Pλthe projection Pλf=F−1( /BDAλF(f)). Then, for f∈L2(R), /bardblf/bardbl2=/bardblPλf/bardbl2+/bardbl(I−Pλ)f/bardbl2 ≤C/bardblPλf/bardbl2 L2(Ω)+/bardbl(I−Pλ)f/bardbl2 =C/bardblf−(I−Pλ)f/bardbl2 L2(Ω)+/bardbl(I−Pλ)f/bardbl2 ≤2C/bardblf/bardbl2 L2(Ω)+2C/bardbl(I−Pλ)f/bardbl2 L2(Ω)+/bardbl(I−Pλ)f/bardbl2 ≤2C/bardblf/bardbl2 L2(Ω)+(2C+1)/bardbl(I−Pλ)f/bardbl2. It remains to estimate the final term. Applying Lemma 1 with τ= (|ξ|2+ 1)1/2, we obtain /bardbl((−∂xx+1)s/2−λ)f/bardbl2=/integraldisplay [(|ξ|2+1)s/2−λ]2|ˆf(ξ)|2dξ ≥/integraldisplay Ac λ[(|ξ|2+1)s/2−λ]2|ˆf(ξ)|2dξ ≥cs(λ+1)2−2 s/integraldisplay Ac λ|ˆf(ξ)|2dξ =cs(λ+1)2−2 s/bardbl(I−Pλ)f/bardbl2.DECAY OF FRACTIONAL WAVE EQUATION WITH DENSE DAMPING 5 /square To apply (5) to the wave equation (1), we first represent the wa ve equation as a semigroup: Setting W(t) = (w(t),wt(t)), we see that (1) is equivalent to d dtW(t) =AγW(t) whereAγ:Hs×Hs/2→Hs/2×L2is densely defined by Aγ(u1,u2) = (u2,−(−∂xx+ 1)s/2u1−γu2). The Sobolev space Hrforr >0 is defined by the decay of the Fourier transform: Hr:=/braceleftbigg u∈L2:/bardblu/bardbl2 Hr=/integraldisplay R(|ξ|2+1)r|ˆu(ξ)|2dξ <∞/bracerightbigg . The definition above is more convenient for our setting so tha t/bardblu/bardblHs/2=/bardbl(−∂xx+ 1)s/4u/bardbl, but the multiplier is equivalent to the usual multiplier ( |ξ|+1)2r. It is standard thatA0is a closed skew-adjoint operator therefore etA0is a semigroup of unitary operators. Then, since γ≥0, forU= (u1,u2)∈Hs×Hs/2, Re/an}bracketle{tA∗ γU,U/an}bracketri}htHs/2×L2= Re/an}bracketle{tAγU,U/an}bracketri}htHs/2×L2 = Re/an}bracketle{tA0U,U/an}bracketri}htHs/2×L2−/an}bracketle{tγu2,u2/an}bracketri}htL2=−/an}bracketle{tγu2,u2/an}bracketri}htL2≤0. Moreover, since γ∈L∞(R), the domain of Aγis the same as A0. So, by classical semigrouptheory[12] etAγisaC0-semigroupofcontractions. WenowapplyProposition 1 toA0andAγ. The first step is an observability inequality for the undamp ed wave equation (1). Proposition 2. LetΩ⊂Rbe relatively dense, s >0. Then, there exists c >0such that c/bardblU/bardbl2 Hs/2×L2≤(|λ|+1)4 s−2/bardbl(A0−iλ)U/bardbl2 Hs/2×L2+/bardblu2/bardbl2 L2(Ω) for allU= (u1,u2)∈Hs×Hs/2andλ∈R. Proof.ForU= (u1,u2)∈Hs(R)×Hs/2(R), setw1= (−∂xx+ 1)s/4u1−iu2and w2= (−∂xx+1)s/4u1+iu2. First, by the parallelogram identity, /bardblw1/bardbl2 L2(R)+/bardblw2/bardbl2 L2(R)= 2/bardbl(−∂xx+1)s/4u1/bardbl2+2/bardblu2/bardbl2= 2/bardblU/bardbl2 Hs/2×L2. Second, /bardbl(A0−λI)U/bardbl2 Hs/2×L2=/bardbl(−∂xx+1)s/4(−λu1+u2)/bardbl2+/bardbl−(−∂xx+1)s/2u1−λu2/bardbl2 =/bardbl−λw1+w2 2+i(−∂xx+1)s/4w1−w2 2/bardbl2 +/bardbl−(−∂xx+1)s/4w1+w2 2−iλw1−w2 2/bardbl2 =/bardbl−iλw1−(−∂xx+1)s/4w1/bardbl2+/bardbl−iλw2+(−∂xx+1)s/4w2/bardbl2.6 WALTON GREEN So, applying Proposition 1 to w1withsreplaced by s/2, we have, for λ≥0, 2c/bardblU/bardbl2 Hs/2×L2=c(/bardblw1/bardbl2+/bardblw2/bardbl2) ≤(|λ|+1)4 s−2/bardbl((−∂xx+1)s/4−λ)w1/bardbl2+/bardblw1/bardbl2 L2(Ω)+c/bardblw2/bardbl2 ≤(|λ|+1)4 s−2/bardbl((−∂xx+1)s/4−λ)w1/bardbl2+2/bardblw1−w2/bardbl2 L2(Ω)+(c+2)/bardblw2/bardbl2 ≤(|λ|+1)4 s−2/bardbl((−∂xx+1)s/4−λ)w1/bardbl2+8/bardblu2/bardbl2 L2(Ω)+c+2 (|λ|+1)2/bardbl((−∂xx+1)s/4+λ)w2/bardbl2 ≤(c+2)(|λ|+1)4 s−2/bardbl(A0−iλI)U/bardbl2 Hs/2×L2+8/bardblu2/bardbl2 L2(Ω). We get the case λ <0 by exchanging the roles of w1andw2. /square Finally we extend this to Aγ−iλIand prove Theorem 1. 2.Proof of the Decay Rates in Theorem 1 First notice that for any R,ε >0,a∈R, /integraldisplaya+R a−Rγ(x)dx≤ /bardblγ/bardbl∞m({γ≥ε}∩[a−R,a+R])+2Rε. So, (4) implies that {γ > ε}is relatively dense for εsmall enough. Therefore, taking Ω ={γ≥ε}and applying Proposition 2, c/bardblU/bardbl2 Hs/2×L2≤(|λ|+1)4 s−2/bardbl(A−iλI)U/bardbl2 Hs/2×L2+/bardblu2/bardbl2 L2(Ω) ≤2(|λ|+1)4 s−2/bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2+/bracketleftbigg 2(|λ|+1)4 s−2+ε−2/bracketrightbigg /bardblγu2/bardbl2 L2(Ω). (6) We estimate the final term. Since A0is skew-adjoint, Re/an}bracketle{t(Aγ−iλI)U,U/an}bracketri}ht= Re/an}bracketle{t(A0−iλI)U,U/an}bracketri}ht−/an}bracketle{tγu2,u2/an}bracketri}ht=−/bardbl√γu2/bardbl2 which implies D/bardblγu2/bardbl2≤D/bardblγ/bardbl∞/bardbl√γu2/bardbl2≤D2/bardblγ/bardbl2 ∞/bardbl(Aγ−iλ)U/bardbl2 δ+δ/bardblU/bardbl2 for anyD,δ >0. Choosing D= 2(|λ|+1)4 s−2+ε−2andδ=c/2, from (6) we obtain c/bardblU/bardbl2 Hs/2×L2≤C/bracketleftbigg (|λ|+1)4 s−2+(|λ|+1)8 s−4+1/bracketrightbigg /bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2+c 2/bardblU/bardbl2 Hs/2×L2. Thus, we have proved the following estimate for ( Aγ−iλI)−1: (7) /bardbl(Aγ−iλI)−1/bardblHs/2×L2→Hs/2×L2≤/braceleftBigg C(|λ|+1)4 s−20< s <2 C s ≥2. Applying the Theorems 4 and 5 allows one to conclude the decay rates in Theorem 1 from (7).DECAY OF FRACTIONAL WAVE EQUATION WITH DENSE DAMPING 7 3.Neccessity of (4) and Threshold Value In this final section we prove the converse in Theorem 1 and sub sequently Theorem 2. By the Gearhart-Pruss Test (Theorem 4) and Borichev-Tomi lov (Theorem 5), the decay rates of the energy in Theorem 1 imply (8) c/bardblU/bardbl2 Hs/2×L2≤ /bardbl(Aγ−iλI)U/bardbl2 Hs/2×L2 for some c=c(s,λ)>0 and for all U∈Hs/2×L2and allλ∈R. Taking U= ((−∂xx+1)−s/4u,iu) foru∈L2(R), we have 2c/bardblu/bardbl2≤ /bardbl(−λ+(−∂xx+1)s/4)u/bardbl2+/bardbl(−(−∂xx+1)s/4−iγ+λ)u/bardbl2 (9) ≤3/bardbl((−∂xx+1)s/4−λ)u/bardbl2+2/bardblγu/bardbl2. 3.1.Converse in Theorem 1. Now, we only consider the special case λ= 1. Let u∈L2(R) such that suppˆ u⊂[−D,D] for some D >0 to be fixed later. For such u, /bardbl((−∂xx+1)s/4−1)u/bardbl2=/integraldisplayD −D[(|ξ|2+1)s/4−1]2|ˆu(ξ)|2dξ≤[(D2+1)s/4−1]2/bardblu/bardbl2. So, taking Dsmall enough, we obtain that there exists C >0 such that (10) /bardblu/bardbl2≤C/bardblγu/bardbl2 for allu∈L2(R) satisfying suppˆ u⊂[−D,D]. Setf(x) =sin(Dx) Dx. Then, supp ˆf⊂ [−D,D]. For each a∈R, setfa(x) =f(x−a). Of course, supp ˆfa⊂[−D,D] and /bardblfa/bardbl=/bardblf/bardbl. Thus, for any R >0, /bardblf/bardbl2=/bardblfa/bardbl2≤C/bardblγfa/bardbl2=C/integraldisplay [a−R,a+R]+/integraldisplay [a−R,a+R]c|γ(x)fa(x)|2dx The second integral goes to 0 (uniformly in a) asR→ ∞sinceγis bounded and f∈L2. The first integral becomes /integraldisplaya+R a−R|γ(x)fa(x)|2dx≤ /bardblγ/bardbl∞/integraldisplaya+R a−Rγ(x)dx sincefis bounded by 1. Thus there exists Rlarge such that (4) holds. We remark that to prove the neccessity of the condition (4), t he decay rates from Theorem 1 can be replaced by an a priori weaker condition, nam ely that there exists λ≥1 such that iλ∈ρ(Aγ) andAγ−iλhas closed range. Then, setting µ=/radicalbig λ2/s−1, we obtain (10) for suppˆ u⊂[µ−D,µ+D] (Dsmall enough). The proof is completed analogously by taking f(x) =eiµxsin(Dx) Dx. 3.2.Proof of Theorem 2. Now, to prove the threshold value (Theorem 2), we use the fact that exponential decay yields (8) with cindependent of λ, from which (9) follows. Suppose that s <2. We will derive a contradiction. In this case, we take suppˆu⊂ {ξ∈R:/vextendsingle/vextendsingle(|ξ|2+1)s/4−λ/vextendsingle/vextendsingle≤K}=:Aλ(K) forKto be chosen later. Then, we have /bardbl((−∂xx+1)s/4−λ)u/bardbl2=/integraldisplay Aλ(K)[(|ξ|2+1)s/4−λ]2|ˆu(ξ)|2dξ≤K2/bardblu/bardbl2.8 WALTON GREEN So taking Ksmall enough, we have, as above, (11) c/bardblu/bardbl ≤ /bardblγu/bardbl whenever suppˆ u⊂Aλ(K),λ∈R.Aλ(K) is the union of the two intervals ±/bracketleftbigg/radicalBig (λ−K)4/s−1,/radicalBig (λ+K)4/s−1/bracketrightbigg and we notice that the length of these intervals is increasin g ifs <2. Indeed, lim λ→∞/radicalBig (λ+K)4/s−1−/radicalBig (λ−K)4/s−1 = lim λ→∞λ4/s−1 λ2/s which is ∞ifs <2. Thus, (11) holds for suppˆ ucontained in any ball since (11) does not see modulation of u(translation of ˆ u). We demonstrate that this is a violation of the uncertainty pr inciple. Let f(x) =/BD{γ=0}(x)φ(x), where φis some positive L2function so that f∈L2andγf= 0. Then, ˆf∈L2so setting gR=F−1( /BDB(0,R)ˆf),gRconverges to fin theL2norm. Therefore, since suppˆ gR⊂B(0,R), by (11), c/bardblgR/bardbl ≤ /bardblγgR/bardbl ≤ /bardblγf/bardbl+/bardblγ(gR−f)/bardbl ≤ /bardblγ/bardbl∞/bardblgR−f/bardbl. The LHS goes to c/bardblf/bardbl>0 (fis nonzero since m({γ= 0})>0) while the RHS appoaches zero as R→ ∞which is a contradiction. Acknowledgements The author is thankful to Milena Stanislavova for introduci ng him to this problem as well as to Benjamin Jaye and Mishko Mitkovski for comments that greatly improved the presentation of this paper. References [1] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient condit ions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. , 30(5):1024–1065, 1992. [2] A. Borichev and Y. Tomilov. Optimal polynomial decay of f unctions and operator semigroups. Mathematische Annalen , 347(2):455–478, 2010. [3] N. Burq and R. Joly. Exponential decay for the damped wave equation in unbounded domains. Communications in Contemporary Mathematics , 18(06):1650012, 2016. [4] L. Gearhart. Spectral theory for contraction semigroup s on hilbert space. Transactions of the American Mathematical Society , 236:385–394, 1978. [5] V. P. Havin and B. J¨ oricke. The uncertainty principle in harmonic analysis , volume 28. Springer Science & Business Media, 2012. [6] F. Huang. Characteristic conditions for exponential st ability of linear dynamical systems in hilbert spaces.Ann. of Diff. Eqs. , 1:43–56, 1985. [7] O.Kovrijkine.SomeresultsrelatedtotheLogvinenko-S eredatheorem. Proceedings of the American Mathematical Society , 129(10):3037–3047, 2001. [8] V. N. Logvinenko and J. F. Sereda. Equivalent norms in spa ces of entire functions of exponential type.Teor. FunkciıFunkcional. Anal. i Prilozen. Vyp , 20:102–111, 1974. [9] S. Malhi and M. Stanislavova. On the energy decay rates fo r the 1d damped fractional Klein- Gordon equation. arXiv preprint arXiv:1809.09531 , 2018. [10] S. Malhi and M. Stanislavova. When is the energy of the 1d damped Klein-Gordon equation decaying? Mathematische Annalen , 372(3-4):1459–1479, 2018.DECAY OF FRACTIONAL WAVE EQUATION WITH DENSE DAMPING 9 [11] B. P. Paneah. Some theorems of Paley–Wiener type. In Doklady Akademii Nauk , volume 138, pages 47–50. Russian Academy of Sciences, 1961. [12] A. Pazy. Semigroups of linear operators and applications to partial differential equations , vol- ume 44. Springer-Verlag, 1983. [13] J. Pr¨ uss. On the spectrum of C0-semigroups. Transactions of the American Mathematical Society , 284(2):847–857, 1984. [14] J. Rauch, M. Taylor, and R. Phillips. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana university Mathematics journal , 24(1):79–86, 1974. [15] M. Zworski. Semiclassical analysis , volume 138. American Mathematical Soc., 2012. School of Mathematical and Statistical Sciences, Clemson U niversity, Clemson, South Carolina 29634 E-mail address :awgreen@clemson.edu

2019-04-24

We establish upper bounds for the decay rate of the energy of the damped fractional wave equation when the averages of the damping coefficient on all intervals of a fixed length are bounded below. If the power of the fractional Laplacian, $s$, is between 0 and 2, the decay is polynomial. For $s \ge 2$, the decay is exponential. Second, we show that our assumption on the damping is necessary for the energy to decay exponentially.

On the Energy Decay Rate of the Fractional Wave Equation on $\mathbb{R}$ with Relatively Dense Damping

1904.10946v3

arXiv:0805.3306v1 [cond-mat.mes-hall] 21 May 2008Non-equilibrium thermodynamic study of magnetization dyn amics in the presence of spin-transfer torque Kazuhiko Seki and Hiroshi Imamura Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan The dynamics of magnetization in the presence of spin-trans fer torque was studied. We derived the equation for the motion of magnetization in the presence of a spin current by using the local equilibrium assumption in non-equilibrium thermodynamic s. We show that, in the resultant equa- tion, the ratio of the Gilbert damping constant, α, and the coefficient, β, of the current-induced torque, called non-adiabatic torque, depends on the relaxa tion time of the fluctuating field τc. The equality α=βholds when τcis very short compared to the time scale of magnetization dyn amics. We apply our theory to current-induced magnetization rever sal in magnetic multilayers and show that the switching time is a decreasing function of τc. Spin-transfer torque-induced magnetization dynamics such as current-induced magnetization reversal [1, 2, 3], domain wall motion [4], and microwave generation [5] have attracted a great deal of attention because of their potential applications to future nano-spinelectronic de- vices. In the absence of spin-transfer torque, magnetiza- tion dynamics is described by either the Landau-Lifshitz (LL) equation [6] or the Landau-Lifshitz-Gilbert (LLG) equation [7]. It is known that the LL and LLG equations become equivalent through rescaling of the gyromagnetic ratio. However, this is not the case in the presence of spin- transfer torque. For domain wall dynamics, the following LLG-type equation has been studied by several groups [8, 9, 10]: ∂t/angbracketleftM/angbracketright+v·∇/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright +α M/angbracketleftM/angbracketright×∂t/angbracketleftM/angbracketright+β M/angbracketleftM/angbracketright×[(v·∇)/angbracketleftM/angbracketright],(1) whereMrepresents the magnetization, vis the velocity, γis the gyromagnetic ratio and αis the Gilbert damping constant. The second term on the left-hand side repre- sents the adiabatic contribution of spin-transfer torque. The first and the second terms on the right-hand side are the torque due to the effective magnetic field Hand the Gilbert damping. The last term on the right-hand side of Eq. (1) represents the current-induced torque, called “non-adiabatic torque” or simply the βterm. The direc- tions of the adiabaticcontribution of spin-transfertorque and non-adiabatic torque are shown in Fig. 1 (a). As shownbyThiaville et al., the value ofthe coefficient βstrongly influences the motion of the domain wall [8]. However, the value of the coefficient βis still controver- sial, and different conclusions have been drawn from dif- ferent approaches[9, 10, 11, 12, 13, 14, 15]. For example, Barnes and Maekawa showed that the value of βshould beequaltothatoftheGilbertdampingconstant αtosat- isfy the requirement that the relaxation should cease at the minimum of electrostatic energy, even under particle flow. Kohno et al.performed microscopic calculationsFIG. 1: (a) The direction of the magnetization M, the adia- batic contribution of spin-transfer torque, ( v·∇)M, and the βterm,M×[(v·∇)M], are shown. The direction of the ve- locityvis indicated by the dotted arrow. (b) The magnetic multilayers, in which the pinned and the free layers are sepa - rated by a nonmagnetic spacer layer are schematically shown . The magnetization vectors of the pinned and free layers are represented by S1andSs, respectively. The effective mag- netic field to which S2is subject is represented by H. (c) The direction of the magnetization of the free layer, S2, the spin-transfer torque ( S2×S1)×S2, and the non-adiabatic torque,S2×S1, are shown. The direction of S1is indicated by the dotted arrow. of spin torques in disordered ferromagnets and showed that theαandβterms arise from the spin relaxation processes and that α/negationslash=βin general [10]. Tserkovnyak et al.[11] derived the βterm using a quasiparticle approx- imation and showed that α=βwithin a self-consistent picture based on the local density approximation. In the current-induced magnetization dynamics in the magnetic multilayers shown in Fig. 1 (b) [16, 17, 18], the non-adiabatic torque exerts a strong effect, and therefore affectsthedirect-currentvoltageofthe spintorquediode, as shown in Refs. [17, 18]. The magnetization dynam- ics of the free layer, S2, has been studied by using the following LLG-type equation, ∂tS2−I eg/planckover2pi1(S2×S1)×S2=γH×S2+α S2S2×∂tS2 +ηIS2×S1, (2) whereIis the charge current density, gis the amplitude of the spin torque introduced by Slonczewski [1], /planckover2pi1is2 the Dirac constant and ηrepresents the magnitude of the “non-adiabatictorque” which is sometimes called the field-like torque [17, 18]. In this paper, we study the magnetization dynamics induced by spin-transfer torque in the framework of non- equilibrium thermodynamics. We derive the equation of motion of the magnetization in the presence of a spin currentby usingthe local equilibrium assumption. In the resultant equation, the Gilbert damping term and the β term are expressed as memory terms with the relaxation time of the fluctuating field τc. We show that the value of the coefficient βis not equal to that of the Gilbert damping constant αin general. However, we also show that the equality α=βholds ifτc≪1/(γH). We apply our theory to the current-induced magnetization reversal in magnetic multilayersand showthat the switching time is a decreasing function of τc. Let us first briefly introduce the non-equilibrium sta- tistical theory of magnetization dynamics in the absence of spin current [19]. The LLG equation describing the motion of magnetization Munder an effective magnetic fieldHis given by ∂tM=γH×M+α MM×∂tM.(3) The equivalent LL equation is expressed as ∂tM=γ 1+α2H×M−αγ M(1+α2)M×(M×H).(4) The Langevin equations leading to Eqs. (3) and (4) by taking the ensemble average of magnetization m, are ∂tm=γHtot×m (5) ∂tδH=−1 τc(δH−χsm)+R(t), (6) where the total magnetic field Htotis the sum of the effective magnetic field Hand the fluctuating magnetic fieldδHandχsis the susceptibility ofthe local magnetic fieldinducedatthepositionofthespin. AccordingtoEq. (6) the fluctuating magnetic field δHrelaxes toward the reaction field χsmwith the relaxation time τc. The ran- dom field R(t) satisfies /angbracketleftR(t)/angbracketright= 0 and the fluctuation- dissipation relation, /angbracketleftRi(t)Rj(t′)/angbracketright=2 τcχskBTδi,jδ(t−t′), wherekBis the Boltzmann constant, Tis the tempera- ture,/angbracketleft···/angbracketrightdenotestheensembleaverage,and i,j= 1,2,3 represents the Cartesian components. It was shown that Eqs. (5) and (6) lead to Kawabata’s extended Landau- Lifshitz equation [20] derived by the projection operator method [19]. In the Markovian limit, i.e.,τc≪1/(γH), we can obtain the LLG equation (3) and the correspond- ing LL equation (4) with α=γτcχsM[19]. In order to consider the flow of spins, i.e., spin cur- rent, we introduce the positional dependence. Since we are interested in the average motion, it is convenient to introducethemeanvelocityofthecarrier, v. Theaveragemagnetization, /angbracketleftm(x,t)/angbracketright, is obtained by introducing the positional dependence and taking the ensemble average of Eq. (5). In terms of the mean velocity, the ensemble average of the left-hand side of Eq. (5) leads to ∂t/angbracketleftm/angbracketright+(v·∇)/angbracketleftm/angbracketright. (7) Assuming /angbracketleftδH×m/angbracketright ≈ /angbracketleftδH/angbracketright×/angbracketleftm/angbracketright, which is applicable when the thermal fluctuation is small compared to the mean value, we obtain ∂t/angbracketleftm/angbracketright+(v·∇)/angbracketleftm/angbracketright=γ/angbracketleftHtot(x,t)/angbracketright×/angbracketleftm(x,t)/angbracketright.(8) The mean magnetization density is expressed as /angbracketleftM(x,t)/angbracketright=ρ(x,t)/angbracketleftm(x,t)/angbracketright,i.e., by the product of the scalar and vectorial components both of which depend on the position of the spin carrier at time t. The spin carrier density satisfies the continuity equation, ∂tρ(x,t)+∇·(vρ(x,t)) = 0. (9) By multiplying the left-hand side of Eq. (8) by ρ(x,t) and using the continuity equation (9), the closed expres- sion for the mean magnetization is obtained as [21] ρ(∂t/angbracketleftm/angbracketright+v·∇/angbracketleftm/angbracketright) =∂tρ/angbracketleftm/angbracketright+/angbracketleftm/angbracketright∇·vρ+ρv·∇/angbracketleftm/angbracketright =∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright, (10) where Div v/angbracketleftM/angbracketrightis defined by Divv/angbracketleftM/angbracketright=3/summationdisplay i=1∂vi/angbracketleftM/angbracketright ∂xi=/angbracketleftM/angbracketright(∇·v)+(v·∇)/angbracketleftM/angbracketright.(11) By multiplying the right-hand side of Eq. (8) by ρ(x,t) and using Eq. (10), we obtain ∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γ(H+/angbracketleftδH/angbracketright)×/angbracketleftM/angbracketright.(12) Equation (12) takes the standard form of a time- evolution equation for extensive thermodynamical vari- ables under flow [21]. The average of Eq. (6) with the positional dependence is given by ∂t/angbracketleftδH(x,t)/angbracketright=−1 τc[/angbracketleftδH(x,t)/angbracketright−χ/angbracketleftM(x(t),t)/angbracketright],(13) wherex(t) is the mean position at time tof the spin car- rier, which flows with velocity v=∂tx(t) andχ=χs/ρ is assumed to be a constant independent of the position. Equations (12) and (13) constitute the basis for the sub- sequent study of magnetization dynamics in the presence of spin-transfer torque. The formal solution of Eq. (13) is expressed as /angbracketleftδH(x,t)/angbracketright=χ τc/integraldisplayt −∞ψ(t−t′)/angbracketleftM(x(t′),t′)/angbracketrightdt′,(14) where the memory kernel is given by ψ(t) = exp[−t/τc]. Using partial integration, we obtain /angbracketleftδH(x,t)/angbracketright=χ/angbracketleftM/angbracketright−/integraldisplayt −∞ψ(t−t′)χ/angbracketleft˙M(t′)/angbracketrightdt′,(15)3 where the explicit expression for ˙M(t) =˙M(x(t),t) is given by the convective derivative, ˙M(t) =∂tM(x(t),t)+(v·∇)M(x(t),t).(16) Substituting Eq. (15) into Eq. (12), we obtain the equa- tion of motion for the mean magnetization density, ∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright +γ/integraldisplayt −∞dt′ψ(t−t′)χ/angbracketleftM(t)/angbracketright×/angbracketleft˙M(t′)/angbracketright.(17) Equation (17) supplemented by Eq. (16) is the principal resultof this paper. When the relaxation time of the fluctuating field, τc, is very short compared to the time scale of the magnetiza- tion dynamics, the memory kernel is decoupled and Eq. (17) can be written in the form of an LLG-type equation as ∂t/angbracketleftM/angbracketright+Divv/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright+α M/angbracketleftM/angbracketright×˙/angbracketleftM/angbracketright,(18) whereα=γτcχMis the Gilbert damping constant. Sub- stituting the explicit form of the convective derivative, Eq. (16), into Eq. (18) and using Eq.(11) we obtain the following LLG-type equation: ∂t/angbracketleftM/angbracketright+/angbracketleftM/angbracketright(∇·v)+(v·∇)/angbracketleftM/angbracketright=γH×/angbracketleftM/angbracketright +α M/angbracketleftM/angbracketright×∂t/angbracketleftM/angbracketright+α M/angbracketleftM/angbracketright×[(v·∇)/angbracketleftM/angbracketright].(19) If∇·v= 0, Eq. (19) reduces to Eq. (14) of Ref. [9], which is derived by replacing the time derivative of mag- netization∂tMon both sides of the LLG equation (3) by the convective derivative ∂tM+v·∇·M. The term /angbracketleftM/angbracketright(∇·v) appears not on the right-hand side ofEq. (19) but on the left-hand side, which means we cannot obtain Eq. (19) using the same procedure used in Ref. [9]. As shown in Refs. [9, 22], Eq. (19) with /angbracketleftM/angbracketright(∇·v) = 0 leads to a steady-state solution in the comoving frame, /angbracketleftM(t)/angbracketright=/angbracketleftM0(x−vt)/angbracketright, where /angbracketleftM0(x)/angbracketrightdenotes the stationary solution in the absence of domain wall mo- tion. However, if /angbracketleftM/angbracketright(∇·v)/negationslash= 0, the steady-state so- lution may break the Galilean invariance. The situa- tion/angbracketleftM/angbracketright(∇·v)/negationslash= 0 can be realized, for example, in magnetic semiconductors [23, 24], where the spin carrier density is spatially inhomogeneous, i.e.,∇ρ/negationslash= 0. The last term of Eq. (19) represents the non-adiabatic component of the current-induced torque, which is also known as the “ βterm”. By comparing Eq. (19) with Eq. (1), one can see that the coefficient of the last term isequaltotheGilbert dampingconstant α. However,Eq. (19) is valid when the relaxation time of the fluctuating field,τc, is very short compared to the time scale of the magnetization dynamics. It should be noted that the generalformoftheequationdescribingthemagnetization dynamics is given by Eq. (17) where the last term on theright-hand side is the origin of the αandβterms. It is possible to projectthe torque representedby the memory function onto the direction of the αandβterms. This projection leads to α/negationslash=βin general. In order to observe the effect of τcon the magneti- zation dynamics we applied our theory to the current- induced magnetization switching in the magnetic multi- layer shown in Fig.1 (b). We assumed that the fixed and free layers are single-domain magnetic layers acting as a large spin characterized by the total magnetization vec- tor defined as Si=/integraltext dV/angbracketleftMi/angbracketright, wherei= 1(2) for the fixed (free) layer and/integraltext dVdenotes the volume integra- tion over the fixed (free) layer. Both the magnetization vector of the fixed layer S1and the effective magnetic field,H, acting on the free layer lie in the plane. Integrating Eqs. (12) and (13) over the volume of the free layer, we obtain the equations, ∂tS2+/integraldisplay dSˆn·J=γ(H+/angbracketleftδH/angbracketright)×S2,(20) ∂t/angbracketleftδH/angbracketright=−1 τc(/angbracketleftδH/angbracketright−χVS2), (21) whereJ=v⊗/angbracketleftM/angbracketrightis the spin current tensor/integraltext dSrep- resents the surface integration over the free layer, ˆnis the unit normal vector of the surface, and χV=χ/Vis defined by the volume of the free layer V. The same procedure used to derive Eq. (17) yields ∂tS2+/integraldisplay dSˆn·J=γH×S2 +γ/integraldisplayt −∞dt′ψ(t−t′)χVS2(t)×∂t′S2(t′),(22) whereψ(t) = exp[−t/τc]. When the relaxation time of the fluctuating field is short compared to the time scale of magnetization dy- namics, the LLG-type equation in the presence of the spin-transfer torque is obtained as ∂tS2+/integraldisplay dSˆn·J=γH×S2+α S2S2×∂tS2,(23) whereα=γτcχVS2. By introducing the conventional form of the spin-transfer torque [1], we obtain the follow- ing LLG-type equation: ∂tS2−I eg/planckover2pi1(S2×S1)×S2=γH×S2+α S2S2×∂tS2.(24) However, Eq. (24) is valid only when τc<1/(γH). As mentioned before, the torque represented by using the memory function generally has a component parallel to the non-adiabatic torque. In order to observe the ef- fect of the non-adiabatic torque induced by the memory function on the magnetization dynamics, we performed numerical simulation using Eqs. (20) and (21).4 FIG. 2: The z-component of the magnetization S2is plotted as a function of time for various values of τc. The initial direction of the free layer is taken to lie in the direction of the effective magnetic field, which is aligned to the zaxis. The initial angle between S1andS2is taken to be 45◦. The Gilbert damping constant αis 0.01. For the simulation, we used the following conditions. At the initial time of t= 0, we assumed that the mag- netization of the free layer is aligned parallel to the ef- fective magnetic field Hand the angle between the mag- netizations of the fixed and the free layers is 45◦. This arrangement corresponds to the recent experiment on a magnetic tunnel junction system [18]. We also assumed that the fluctuation field has zero mean value at t= 0, i.e.,/angbracketleftδH(0)/angbracketright=0. In Fig. 2, we plot the time dependence of the zcom- ponent of the magnetization of the free layer, S2, under the large-enough spin current to flip the magnetization of the free layer, Ig/planckover2pi1S2 2S1/(eαγH) =−10. The value of τcis varied while the value of α= 0.01 is maintained. The solid, dotted, and dot-dashed lines correspond to γHτc= 0.1,1.0, and 10.0, respectively. As shown in Fig. 2, the time required for the magnetization of the free layer to flip decreases with increasing τc, which can be understood by considering the non-adiabatic torque induced by the spin current. The non-adiabatic torque induced by the spin current is obtained by projecting the torque given by the last term of Eq. (22) onto the direction of S2×S1, which results in the positive con- tribution to the spin-flip motion of S2. Since the last term of Eq. (22) includes a memory function, the non- adiabatic torque induced by the spin current increases with increasing τc. Therefore, the time required for S2 to flip decreases with increasing τc. ForγHτc>10 we observe no further decrease of the time required for S2 to flip because the memory function is an integral of the vectorS2(t)×∂t′S2(t′) and the contributions from the memory at t−t′≫1/(γH) is eliminated. In conclusion, we derived the equation for the motion of magnetization in the presence of a spin current by us- ing the local equilibrium assumption in non-equilibriumthermodynamics. We demonstrated that the value of the coefficientβis not equal to that of the Gilbert damping constantαin general. However, we also show that the equalityα=βholds ifτc≪1/(γH). We then applied our theory to current-induced magnetization reversal in magnetic multilayersand showed that the switching time is a decreasing function of τc. The authors would like to acknowledge the valuable discussions they had with S.E. Barnes, S. Maekawa, P. M. Levy, K. Kitahara, K. Matsushita, J. Sato and T. Taniguchi. This work was supported by NEDO. [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000). [4] M. Kl¨ aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, and L. J. Heyderman, Appl. Phys. Lett.83, 105 (2003). [5] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em- ley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature425, 380 (2003). [6] L. LandauandE. Lifshitz, Phys. Z.Sowjet. 8, 153(1935). [7] T. L. Gilbert, Armour Research Foundation Project No. A059, Supplementary Report, May 1, 1959; IEEE Trans. Magn.40, 3443 (2004). [8] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Eu- rophys. Lett. 69, 990 (2005). [9] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). [10] H. Kohno, G. Tatara, andJ. Shibata, J. Phys. Soc. Japan 75, 113706 (2006). [11] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W. Bauer, Phys. Rev. B 74, 144405 (2006). [12] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn. Magn. Mater 320, 1282 (2008). [13] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73, 054428 (2006). [14] M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zang- will, Phys. Rev. B 75, 214423 (2007). [15] R. A. Duine, A. S. Nunez, J. Sinova, and A. H. MacDon- ald, Phys. Rev. B 75, 214420 (2007). [16] J. Zhang, P. M. Levy, S. Zhang, and V. Antropov, Phys. Rev. Lett. 93, 256602 (2004). [17] A. A. Tulapurkar et al., Nature 438, 339 (2005). [18] H. Kubota et al., Nature Physics 4, 37 (2008). [19] K. Miyazaki and K. Seki, J. Chem. Phys. 108, 7052 (1998). [20] A. Kawabata, Prog. Theor. Phys 48, 2237 (1972). [21] S. R. de Groot and P. Mazur, Nonequilibrium thermody- namics(North-Holland, Amsterdam, 1962). [22] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, J. Magn. Magn. Mater 8, 153 (1935). [23] H. Ohno, Science 281, 951 (1998). [24] T. Dietl and H. Ohno, Materials Today 9, 18 (2006).

2008-05-21

The dynamics of magnetization in the presence of spin-transfer torque was studied. We derived the equation for the motion of magnetization in the presence of a spin current by using the local equilibrium assumption in non-equilibrium thermodynamics. We show that, in the resultant equation, the ratio of the Gilbert damping constant, $\alpha$, and the coefficient, $\beta$, of the current-induced torque, called non-adiabatic torque, depends on the relaxation time of the fluctuating field $\tau_{c}$. The equality $\alpha=\beta$ holds when $\tau_c$ is very short compared to the time scale of magnetization dynamics. We apply our theory to current-induced magnetization reversal in magnetic multilayers and show that the switching time is a decreasing function of $\tau_{c}$.

Non-equilibrium thermodynamic study of magnetization dynamics in the presence of spin-transfer torque

0805.3306v1

Controlling the nonlinear relaxation of quantized p ropagating magnons in nanodevices M. Mohseni, 1, * Q. Wang, 2 B. Heinz, 1,3 M. Kewenig, 1 M. Schneider, 1 F. Kohl, 1 B. Lägel, 4 C. Dubs, 5 A. V. Chumak, 2 and P. Pirro 1 1 Fachbereich Physik and Landesforschungszentrum OPT IMAS, Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany 2 Faculty of Physics, University of Vienna, Boltzman ngasse 5, A-1090 Vienna, Austria 3 Graduate School Materials Science in Mainz, Staudi ngerweg 9, 55128 Mainz, Germany 4 Nano Structuring Center, Technische Universität Kai serslautern, 67663 Kaiserslautern, Germany 5 INNOVENT e.V., Technologieentwicklung, Prüssingstr aße 27B, 07745 Jena, Germany Relaxation of linear magnetization dynamics is well described by the viscous Gilbert damping pro- cesses. However, for strong excitations, nonlinear damping processes such as the decay via magnon-mag- non interactions emerge and trigger additional rela xation channels. Here, we use space- and time-resol ved micro-focused Brillouin light scattering spectrosco py and micromagnetic simulations to investigate the nonlinear relaxation of strongly driven propagating spin-waves in yttrium iron garnet nanoconduits. We show that the nonlinear magnon relaxation in this h ighly quantized system possesses intermodal feature s, i.e. magnons scatter to higher-order quantized mode s through a cascade of scattering events. We furthe r show how to control such intermodal dissipation pro cesses by quantization of the magnon band in single - mode devices, where this phenomenon approaches its fundamental limit. Our study extends the knowledge about nonlinear propagating spin-waves in nanostruc tures which is essential for the construction of ad - vanced spin-wave elements as well as the realizatio n of Bose-Einstein condensates in scaled systems. Relaxation of magnons, the quanta of spin waves (SWs), due to magnetic damping is a complicated process and involves different (non)linear contribu - tions. Relaxation mechanisms which can be de- scribed by the phenomenological Gilbert damping drive the magnetization towards its equilibrium sta te by e.g. dissipating the energy to the lattice. It i s one of the key elements of performance in many practica l devices and fundamental phenomena [1–10]. Dissipation of the energy can be more intricate for strongly driven excitations, where nonlinear re - laxation mechanisms via magnon-magnon interac- tions open up additional dissipation channels [11– 17]. Unlike the Gilbert damping, these types of in- trinsic dissipation processes can redistribute the magnon energy within the magnon spectrum [18– 27]. The classical works of Suhl predicted that l arge amplitude uniform magnetization oscillations lead t o the onset of instability processes, allowing the no n- linear relaxation of strongly driven magnons by a d e- cay into secondary magnon modes [25]. In particu- lar, the common second-order Suhl instability pro- cess can be: ( i) a disadvantage since it comes along with detrimental influence on the magnon transport and decay characteristics, potentially dominating t he competing linear damping [17,22,28] , or, ( ii ) an ad- vantage by providing additional degrees of freedom of magnon transport for device architectures and quantum computing concepts [23,29,30]. So far, most of such investigations in scaled systems, whic h are of large interest for applications, have been c ar- ried out for standing SW modes with vanishing mo- mentum ( k = 0), e.g. the Ferromagnetic resonance (FMR) mode. However, SWs carrying a momentum are not only essential for applications, but they p os- sess an enriched physics behind their nonlinear ins ta- bilities due to the increased amount of potential s cat- tering channels. Nevertheless, little investigation s have been carried out in this direction yet. Recent development of ultra-low damping na- noscale systems based on YIG, the most promising hosts for SWs, provides access to quasi-1D systems with highly quantized magnon spectra [31,32] . By imposing limitations on the available relaxation channels due to the strong quantization of the mag- non band, and a drastically modified SW character- istics including the SW dispersion relation, mode profile and their ellipticity, nonlinear SW dynamic s in such devices can be different compared to contin - uous films and quasi-2D systems [33-34]. Further- more, recent experimental and theoretical studies o f SW dynamics and magnon condensates in nano- scopic systems [32, 35,36] enforce us to better un - derstand nonlinear SW dynamics and magnon ther- malization processes in nano-scaled 1D systems. Here, we use space- and time-resolved mi- cro-focused Brillouin light scattering (µBLS) to un - cover the mechanism of nonlinear relaxation of strongly driven propagating magnons via the second- order Suhl instability in YIG nanoconduits. We demonstrate how magnons nonlinearly relax to other quantized modes via four-magnon scattering pro- cesses, and such nonlinear processes can be con- trolled using quantization of the magnon band. To demonstrate the effect of quantization on the nonlinear dynamics, we use two exemplary mag- nonic nanoconduits structured from a Liquid Phase Epitaxial (LPE) YIG film grown on top of a Gado- linium Gallium Garnet (GGG) substrate [37]. The multi-mode nanoconduit with a lateral width of w = 400 nm (Fig. 1a) and a thickness of d = 85 nm was fabricated using a hard mask and ion beam milling process [31]. A comparative single-mode conduit with a smaller width of w = 100 nm and d= 44 nm was fabricated using a similar method (Fig. 1b). SW s in both devices are excited by a microwave antenna which is placed on top of the nanoconduits by elec- tron beam lithography and a lift-off process [31]. Ap- plying a microwave rf current to the antenna gener- ates a dynamic Oersted field which in return excite s SWs resonantly, see supplemental materials SM [40]. The detection of the generated SWs has been carried out using space- and time-resolved µBLS [38]. An incident laser light with an effecti ve spot size of 300 nm (focused by a ×100 microscope objective with a numerical aperture NA=0.85) is used to probe the SWs through the GGG substrate under the antenna. The inelastically scattered ligh t was analyzed using a tandem Fabry-Perot interfer- ometer to obtain the frequency and intensity of the magnons. FIG. 1. (a)-(b) SEM images of the w = 400 nm (multi- mode) and w =100 nm (single-mode) wide conduits (shaded in orange), respectively. (c)-(d) Magnon ba nd structures of the multi-mode and single-mode condui ts, respectively. Color plots are obtained by micromagn etic simulations and dashed lines from analytical calcul ations. Note the different scales of the frequencies. (e)-( f) Meas- ured spin-wave spectra of the multi-mode and single - mode conduits in the presence of different powers, respec- tively. The excited modes are represented by the ye llow dots in (c) and (d). A static external field (µ 0He = 60 mT) saturates the nanoconduits along their length. Thus, the wave vector of the propagating SWs is parallel to the ma g- netization vector, /g2193 ‖ /g2169 , and waveguide (WG) modes appears [32]. The width of the multi-mode waveguide is large enough to ensure dipolar pinning of the spins at the edges, while spins at the edges of the single-mode conduit are fully unpinned [32]. Moreover, due to the interplay between the contribu - tions of the dipolar and exchange energy to the SW dispersion, the different WG modes are well quan- tized on the frequency axis. The dispersion relatio n of the fundamental mode and the first two WG modes are shown in Fig 1c-d, in which the dashed lines are analytical results based on method discus ses in Ref [32], and the color plot is obtained by mic ro- magnetic simulations using the MuMax 3.0 pack- age [39, 40]. The fundamental mode and higher or- der WG modes are labeled as n = 0 and n = 1, 2 re- spectively. Please note that the spectrum is much more dilute in the 100 nm wide conduit due to the higher contribution of the exchange energy to the magnon band structure, which leads to a strong quan - tization and the absence of degenerate states among modes (single-mode system for wave vectors below approx. 40 rad/µm). We first set the rf frequency to f = 3.85 GHz where dipolar SWs having a wave vector of kx = 1.5 rad/µm are excited in the multi-mode device [40]. T o characterize the linear SW dynamics, we set the rf power to P = 10 dBm and measure the intensity of the generated magnons as displayed in Fig 1e (black circles). Up to P = 18 dBm, only the frequency of the resonantly driven SW mode is observed (red and green triangles). A further increase in the rf power up to P = 20 dBm (blue curve) leads to the appearance of two additional peaks in the SW frequency spec- trum labeled as f – and f + in Fig. 1e. We refer to these magnons as secondary magnons which are modes populated by nonlinear scattering processes. They have the lowest threshold for the observed instabil ity process and can fulfill the fundamental conservatio n laws to permit the scattering process [22]. The en- ergy and momentum conservation laws of these pro- cesses generally read [18,20,22,28], /g1858/g2869+ /g1858/g2870= /g1858/g2871+ /g1858/g2872, /g2193/g2869+ /g2193/g2870= /g2193/g2871+ /g2193/g2872 (1) where two magnons with the frequencies f 1 & f 2 and momenta k 1 & k 2 scatter to two magnons with the frequencies f 3 & f 4 and momenta k 3 & k 4. Note that the lateral component of the k vector is symmetric, and the out of plane component is zero in this fre-quency range due to the small thickness. In our ex- periments, two magnons with a frequency of f = 3.85 GHz scatter finally to two magnons with the frequen - cies of f 3 = f - = 3.25 GHz and f 4 = f + = 4.45 GHz. We note that this process is not a special peculiar ity of the chosen spectral position, see SM [40]. For comparison, we now investigate the same nonlinear process in the comparative single- mode waveguide. We set the f = 3.71 GHz and meas- ure the intensity of the driven mode as shown in Fi g. 1f. Clearly, even in the presence of high powers l ike P = 20 dBm, side peaks cannot be observed, evidenc- ing the absence of a similar nonlinear dissipation processes. Here, only the µBLS intensity drops at high powers which is caused by the nonlinear fre- quency shift of the dispersion relation and possibl e impacts of the higher temperature [31, 41]. In prin ci- ple, the absence of side peaks demonstrates that su ch scattering processes can be efficiently suppressed in narrower conduits where the magnon band structure is highly quantized and therefore, the fundamental conservation laws required for the scattering pro- cesses cannot be fulfilled. To understand the fundamental differences between the two waveguide types, let us investigate the observed nonlinear dynamics in the multi-mode conduit in more detail. A nonlinear scattering inst a- bility is characterized by a clear threshold of the ini- tial magnon intensity which is required for its on- set [18,22,24,42]. Neglecting SW radiation losses, the threshold magnon amplitude is defined by the ef - fective relaxation frequency of the secondary mag- nons divided by the four-magnon coupling strength [16,22]. To investigate the threshold beh av- ior in the multi-mode conduit in which the scatteri ng is observed, we sweep the rf power for a fixed fre- quency f = 3.85 GHz as shown in Fig. 2. Once the instability threshold is reached at P = 18 dBm (indi- cated by the black arrow), the growth rate of the d i- rectly excited magnon intensity as a function of mi - crowave power drops. Increasing the power to P = 19 dBm leads to an abrupt increase of the intensity of the secondary magnons labeled as f + and f – (indi- cated by the gray arrow). From this power ( P = 19 dBm) on, the intensity growth rate of the directly ex- cited mode with respect to the power is decreased, evidencing that the energy transfers to the seconda ry magnon modes. FIG. 2. Spin-wave amplitude in the multi-mode conduit as a function of microwave excitation power when f = 3.85 GHz. The secondary magnons created by the second or der Suhl instability are denoted as f + and f -. The back and gray arrows indicate the onset of instability and t he rise of the secondary magnons, respectively. A closer look on Fig. 2 near the instability threshold opens the question what happens when the instability threshold is approached at P = 18 dBm and the µBLS intensity of the initially excited mod e drops, while the amplitudes of the secondary mag- nons at f+ and f- are still at the thermal level, implying the absence of magnon scattering to these modes. We perform micromagnetic simulations to uncover the wave vector of the scattered magnons and address the discussed question. Figure 3a shows the frequency spectrum of the simulated multi-mode conduit ( f = 3.85 GHz) in which different amplitude of the rf currents are used to drive the system. For a small rf current equal to irf = 4 mA, only the reso- nantly excited SWs can be observed in the frequency spectrum (black curve). The corresponding popula- tion of the magnon band is depicted in Fig. 3b show - ing the wave vector of kx = 1.5 rad/µm of the directly excited mode. Increasing the rf current to a higher value of irf = 8 mA increases the amplitude and the linewidth of the resonant SWs (red curve in Fig 3.a) [28]. As shown in Fig. 3c, this is related to the onset of a first-level four magnon scattering proce ss in which the frequency of the magnons is conserved. Such a process cannot be observed in the measured frequency spectrum of the conduit, but it can mani- fest itself in the observed drop of the directly ex cited mode intensity with power. As evidenced by the sim- ulations, two incoming magnons from the resonantly driven mode with opposite momenta scatter to two outgoing magnons at the same frequency, but with different momenta. The scattered magnons populate the fundamental mode ( n = 0) at a higher wave num- ber of /g1863/g3051= 30 /g1870/g1853/g1856/µ/g1865 , and two spectral position at the first WG mode ( n = 1). These frequency-con- serving scattering processes which are similar to plane films [25] are indicated by the pink arrows in Fig. 3c, and can also be observed in the single-mod e conduit, see SM [40]. A further increase of the rf current to irf = 13 mA leads to the onset of the sideband peaks in the frequency spectrum (blue curve in Fig 3.a), similar to the experiments. As evidenced from the simulated band structure (Fig. 3d), this is due to the second level of the magnon scattering cascade. Once the magnons scattered by the first level process to the n=1 WG mode reach a critical amplitude, they un- dergo themselves another second order instability. In this process, two magnons with the frequency of f = 3.85 GHz and identical momentum of /g1863/g3051= 10.7 /g1870/g1853/g1856/µ/g1865 at the first WG mode ( n = 1), scatter to two outgoing magnons with the frequencies of f - = 3.46 GHz and f + = 4.24 GHz at the fundamental mode ( n = 0) and the second WG mode ( n = 2), re- spectively. The simulated values are in very good agreement with the experimentally obtained frequen- cies. In Fig. 3d, this type of frequency-noncon- serving scattering is represented by the red arrows. The scattered magnons feature /g1863/g3051/g2878= 14.3 /g1870/g1853/g1856/µ/g1865 and /g1863/g3051/g2879= 7.1 /g1870/g1853/g1856/µ/g1865 , assuring momentum conser- vation laws given by 2/g1863/g3051= /g1863 /g3051/g2878+ /g1863 /g3051/g2879. We note that the second scattering step clearly shows that the f i- nite momentum of the ingoing magnon opens the op- portunity to scatter to two new, different frequenc ies and thus, to redistribute the magnon energy towards the bottom of the spectrum and to higher frequencie s (modes). Unlike the first level process, it involve s only magnons of a single propagation direction (+ k or –k) and can only occur for propagating waves. This is evidenced by the momentum and energy con- servation laws which require a finite sum of the mo - menta of the two incoming magnons to allow for a frequency non-degenerated splitting. This is a sign if- icant difference to the nonlinear instabilities of the FMR mode without momentum ( kx = 0) in which magnon instabilities are always degenerated [43-44] . Thus, if the FMR undergoes a second-order instabil- ity, this process never leads to a redistribution o f the magnon energy across the spectrum. FIG. 3. Results of the micromagnetic simulations in the multi-mode conduit. (a) spin-wave frequency spectra when the microwave current varies. (b-d) Magnon ban d structures (linear scale) of the driven system corr espond to the black, red and blue curves in (a), respectiv ely. The scaling of b-d is independent from each other. The properties of the cascade-like magnon scattering events coupling different waveguide modes in the multi-mode waveguide and the absence of this effect in the single mode waveguide also im - plies that thermalization of magnons is significant ly changed in systems with strongly diluted spectra compared to earlier investigations in systems which quasi-continuous spectra. The simulations also explain the observed peculiarity in the threshold curve of the experimen ts as were discussed in the context of Fig. 2. Indeed, the magnons scattered to higher wave numbers via the first level frequency-conserving scattering process (Fig. 3c) cannot be detected experimentally due to the maximum detectable momentum using µBLS spectroscopy, which is approximately kx ~ 21 rad/µm in our experiments [38]. This explains at least par - tially the decrease of the measured magnon intensit y at the driving frequency. Since the different level s of the cascade process have different threshold powers , the nonlinear scattering to the secondary magnon modes at different frequencies is observed at a slightly higher power than the start of the drop of the intensity at the directly excited frequency. In add i- tion, the limited wave vector sensitivity of the BL S can pose inconsistency for the SW amplitude ob- served in the simulations compared to the experi- ments. FIG. 4. Time-resolved spin-wave amplitude measured by µBLS spectroscopy. (a) Beginning of the pulse. (b) End of the pulse. Black arrows indicate the onset and t he decay of the instability, respectively. Note that the dec ay rates correspond to the intensity of the magnons. To further characterize the impact of the nonlinear relaxation on the total relaxation of the sys- tem [16], we perform time-resolved µBLS measure- ments in the multi-mode conduit. The measured in- tensity of the driven and secondary magnons at the beginning and the end of a 1µs long microwave rf pulse ( f = 3.85 GHz and P = 24 dBm) at the meas- urement position are shown in Fig. 4a-b. Figure 4a illustrates that the resonantly driven SW mode (blu e curve) undergoes the second-level four-magnon scattering after t ~ 4 ns, evidenced by the rise of the secondary magnons (yellow and red curves). This is indicated by the black arrow in Fig. 4a. Note that the growth rate of the driven mode drops immediately when the rise of the secondary magnons sets in, evi - dencing the conservation of the energy in the nonli n- ear redistribution process. The decay of the magnons at the end of pulse is presented in Fig. 4b. In particular, the decay o f the secondary magnons begins once the intensity of the driven SW mode is decayed enough after t ~ 4 ns (indicated by the black arrow). More interestingly, the decay of the magnons at the resonantly driven frequency to the thermal level includes two steps manifesting the high nonlinearity of the dynamics. First, it decays with an exponential decay time of t1,d = 19 ns, which is accompanied by the decay of the secondary magnons at f+ and f-. Afterwards, it de- cays with a longer exponential decay time of t2,d = 24 ns suggesting a transition from a nonlinear relaxat ion to a linear relaxation with a lower decay rate. In other words, the first decay includes an energy flow to t he secondary magnons which acts as an additional dis- sipation channel for the driven magnons. After the secondary magnons decayed to the thermal level, thi s additional dissipation channel is switched off, whi ch leads to a slower decay time of the driven SWs. In summary, we explored the nonlinear re- laxation of strongly driven propagating spin waves in nanodevices. The finite momentum of the mag- nons investigated in our study provides an addition al playground for the nonlinear magnon instability pro - cesses. Furthermore, it was shown that such inter- modal dissipation process is strongly suppressed in systems with a strongly quantized magnon band (sin- gle-mode systems), suggesting the fundamental lim- itation of this process in nanodevices. This can o pen a new avenue for coherent nonlinear nano-mag- nonics. The nonlinear dynamics studied in this lett er are general and thus, can be applied to devices bas ed on other deposition techniques as well. Our study c an be used for several device architectures, namely, f re- quency mixers [45], squeezed states [46], signal a nd data processing units [29, 47-50], and quantum com- puting concepts [23], and further open doors to eng i- neered dissipation of magnons in nanodevices. Acknowledgments The authors thank Burkard Hillebrands for support and valuable discussions. This project is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 173 - 268565370 (“Spin+X”, Project B01) and by the pro- ject - 271741898, the European Research Council within the Starting Grant No. 678309 “MagnonCir- cuits” and by the Austrian Science Fund (FWF) through the project I 4696-N. We appreciate our col - leagues from the Nano Structuring Center of the TU Kaiserslautern for their assistance in sample prepa - ration. B.H. acknowledges support by the Graduate School Material Science in Mainz (MAINZ). Correspondence to: * mohseni@rhrk.uni-kl.de References [1] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D. Appl. Phys. 43 , 264001 (2010). [2] S. Mamica, M. Krawczyk, and D. Grundler, Phys. Rev. Applied 11 , 054011 (2019) [3] A. V. Sadovnikov, C. S. Davies, S. V. Grishin, V. V. Kruglyak, D. V. Romanenko, Y. P. Sharaevskii, and S. A. Nikitov, Appl. Phys. Lett. 106 , 192406 (2015). [4] A. V. Sadovnikov, E. N. Beginin, S. E. 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2020-06-05

Relaxation of linear magnetization dynamics is well described by the viscous Gilbert damping processes. However, for strong excitations, nonlinear damping processes such as the decay via magnon-magnon interactions emerge and trigger additional relaxation channels. Here, we use space- and time-resolved microfocused Brillouin light scattering spectroscopy and micromagnetic simulations to investigate the nonlinear relaxation of strongly driven propagating spin waves in yttrium iron garnet nanoconduits. We show that the nonlinear magnon relaxation in this highly quantized system possesses intermodal features, i.e., magnons scatter to higher-order quantized modes through a cascade of scattering events. We further show how to control such intermodal dissipation processes by quantization of the magnon band in single-mode devices, where this phenomenon approaches its fundamental limit. Our study extends the knowledge about nonlinear propagating spin waves in nanostructures which is essential for the construction of advanced spin-wave elements as well as the realization of Bose-Einstein condensates in scaled systems.

Controlling the nonlinear relaxation of quantized propagating magnons in nanodevices

2006.03400v2

1 Thermally excited spin waves in a nano-structure: thermal gradient vs. constant temperature Simone Borlenghi1, Mattero Franchin2, Hans Fangohr2, Lars Bergqvist1,4, and Anna Delin1,3,4, 1Department of Materials and Nanophysics, School of Information and Communication Technology, Electrum 229, Royal Institute of Technology, SE-16440 Kista, Sweden 2Department of Engineering and the Environment, University of Southampton, SO17 1BJ Southampton, United Kingdom 3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden 4SeRC (Swedish e-Science Research Center), KTH, SE-10044 Stockholm, Sweden. Using micromagnetic simulations, we have investigated spin dynamics in a nanostructure in the presence of thermal fluctuations. In particular, we have studied the effects of a uniform temperature and of a uniform thermal gradient. In both cases, the stochastic field leads to an increase of the precession angle of the magnetization, and to a mild decreas of the linewidth of the resonance peaks. Our results indicate that the Gilbert damping parameter plays the role of control parameter for the amplification of spin waves. I. I NTRODUCTION Recent experiments have shown that a temperature gradient across a magnetic material (conductor or insulator) generates a pure spin current. This phenomenon, known as the Spin- Seebeck Effect (SSE) [1], [2] has opened a new research field in which temperature, magnetism and electronic transport are considered simultaneusly [3]. In insulating ferromagnets, the spin current cannot be car- ried by electrons and therefore a spin wave spin current [2] associated with the magnetization dynamics in the sample is at the core of SSE. This effect is not limited to ferromagnets: in a recent experiment Padron-Hernandez et al. [4] suggest that Spin Waves (SW), excited by a radio frequency generator in an yttrium iron garnet sample, are amplified in presence of a uniform thermal gradient through the sample. In their paper, they suggest that heat flow acts as a thermal torque that opposes the damping, in a similar way as spin torque does in spin transfer nano-oscillators [5]. In conducting ferromagnets, pure spin currents are generated by itinerant electrons with opposite spins that flow in opposite directions as well as by magnons. Experiments and theoretical analysis performed so far have mainly focused on the effect of thermal gradient at the interface between different materials through the Inverse Spin-Hall Effect [6], [7], or on domain wall motion [8], [9]. More recently, Machado et. al. [10] investigated through micromagnetic simulations the role of thermal gradient on the vortex core magnetization dynamics in a Permalloy disk. So far, a micromagnetic study of the effect of heat flow on SW amplification in a ferromagnet, and a comparison with the uniform temperature case, is missing. The possibility to propagate SW using a thermal gradient suggests many possible applications [11], and an understanding of the direct effect of heat flow on the magnetization dynamics is highly desirable. In this paper we present detailed micromagnetic simulations of the SW spectrum of a nanoscructure in the presence Corresponding author: S. Borlenghi (email: simonebg@kth.se.)of a uniform and non uniform temperature distribution. For practical reasons we have chosen to use the parameters valid for Permalloy (Py). However, our results should not been interpreted as quantitatively specific for Py, but rather as a qualitative description of the effect of heat flow in a nanos- tructure. While the studies performed so far have focused on the local properties of Spin-Seebeck devices (i.e. spin current propagation), we have here chosen to investigate the effect of thermal fluctuations on SW. II. F ORMULATION OF THE PROBLEM The dynamics of the magnetization in a ferromagnet is described, at the length scale of the exchange length, by the classical LLG equation of motion [12], [13] @M @t=j 0jMHe+ MsM@M @t; (1) where 0=2:21105m/(As) is the gyromagnetic ratio, is the dimensionless Gilbert damping parameter and Ms is the saturation magnetization of the sample. The first term at the right-hand side of Eq.(1) is the adiabatic torque, which accounts for the precession of the magnetization Maround the effective field He. The effective field itself is the functional derivative of the Gibbs free energy of the system with respect to the magnetization [14]. The second term on the right-hand side, proportional to , accounts for energy dissipation. In general, Hecontains a Zeeman term, which describes the interaction of the precessing magnetization with the applied field, as well as exchange, shape anisotropy and demagnetizing field. [14]. Our numerical simulations have been performed with Nmag [15], a micromagnetic package based on finite-element dis- cretization of the sample, which is represented by a network of sites (mesh). In this package, the magnetization dynamics at each sitekof the mesh is described by the LLG equation, and the interactions with neighbouring sites are taken into account in the computation of the effective field.arXiv:1303.2895v1 [cond-mat.mes-hall] 12 Mar 20132 Figure 1. (Color online) a)Magnetization vector precessing around the field Hext, aligned with the zaxis. The transverse component mx+imy is the SW complex amplitude, which describes the magnetization precession at the Larmor frequency in the x-yplane. b)Cartoon of the Py nanopillar studied in this paper, showing the mesh used in the computations. A uniform thermal gradient @xTis applied in the xdirection, and a uniform magnetic fieldHextis applied in the zdirection. Temperature is introduced by adding to the effective field Hk eat sitek, a stochastic field Hk th, which is assumed to be a Gaussian random process with zero mean and amplitudeD Hk th;iHl th;jE = 2Dkijkl(tt0). Herei;j=x;y;z stand for the cartesian components of the field, while k;lrefers to the sites on the mesh. The fluctuation amplitude Dkis given by [16] Dk=2kBTk Ms 00Vk
t; (2) wherekBis the Boltzmann constant, 0is the vacuum magnetic permeability, Tkis the temperature at site k,Vkis the volume containing the magnetic moment at site kand
tis the integration time step. We have taken Vkas the average volume per site, given by the total volume of the sample divided by the number of sites of the mesh. In agreement with the results of Ref. [17], we have neglected the weak temperature dependence of . Notice that Eq. (2) is valid in a spin dynamics atomistic description, where each site corresponds to a single precessing spin. In a micromagnetic framework, where each site corresponds to a large number of precessing spins inside a finite volume, Dkhas to be multiplied by a scaling factor, which for Py is equal to 10 (see Ref. [18]). III. N UMERICAL SIMULATIONS The sample used in our simulations, shown in Fig. (1b) is a cuboid with dimensions of 1003030nm. The mesh contains 4100 nodes, giving a lattice distance smaller than 3 nm. This is of the order of the Py exchange length. The micromagnetic parameters are those of Py: the exchange stiffness of is J= 1:31011J/m, while for the saturation magnetization we have taken 0:86106A/m. In most of our simulations, the sample is saturated by an external field Hext of 10 kOe applied in the zdirection, which corresponds to the precession axis of the magnetization. The computations performed at different fields will be clearly indicated. The temperature Tis uniform along the yandzdirections, whilea uniform thermal gradient @xTis applied in the xdirection. This configuration, with the field orthogonal to the thermal gradient, is the one commonly used in SSE experiments [1], [2], [4]. The quantity of interest in our simulations is the normalized magnetization averaged over the volume Vof the sample: hm(t)i=1 VM sR VM(r;t)dV. In particular, the complex SW amplitude (t) =hMxi+ihMyip Ms(Ms+Mz); (3) describes the transverse magnetization precessing in the x-y plane [see Fig. (1a)]. for small oscillations considered here, MzMs, so thathmxi+ihmyi. The LLG equation, written in terms of this variable, reads [5] d dt=i!+  e: (4) For a single spin, the solution (t)exp [(i! e)t]is an oscillating signal with frequency !0= He. In an extended confined ferromagnet, the effective field is spatially dependent, so that the solution consists of a discrete set of SW modes with frequencies !`, whose spatial profile depends on the geometry of the system [14], [19]. In the cuboid geometry, they consist of sine and cosine. The time decay of the signal is controlled by the damping rate e, which in general is a function of and of the reso- nant frequency [5]. For small precession angles and damping parameter considered here, each SW mode has an effective damping  e!`. The SW power spectrum, given by the absolute value of the Fourier transform of the SW amplitude, consists of Lorentzians centered at the resonance frequencies, whose linewidths (full width at half height) correspond to  e[5], [14]. When the system absorbs energy, the heights of the resonance peaks in- crease, while their linewidths decrease [14]. This corresponds to and increase of the precession angle. In our simulations, we started from an initial condition where the magnetization is uniformly tilted 8in thexdirec- tion with respect to the precession axis z. We then computed the time evolution for 10 ns with a time step of 1 ps. Fig. (2a) shows the typical output mx(t)andmy(t)of our simulations at zero temperature, computed for =0.01, while Fig. (2b) shows the corresponding power spectrum, which is dominated by the two modes f`,`= 1;2with frequencies 31.2 and 40.1 GHz correspondingly. These low energy modes are the only ones visible in the linear regime, since their damping is relatively weak. Fig. (2d) shows their spatial profile. These modes have different linewidths, and consequently different effective dampings ;` e. We performed the computations have in a wide range of thermal gradients, spanning between 0 and 102K/nm. Fig. 2c) shows the magnetization as a function of temperature, in good agreement with Ref. [18]. T he gradients at which SW amplification is observed are comprised between 101and10 K/nm, which correspond to temperatures between 3 and 300 K in the hottest part of the system, well below the Curie point.3 Figure 2. (Color online) a)Time evolution of the transverse components of the magnetization mxandmy, computed for = 0:01.b)Power spectrum of the system, with the two dominating modes and the corresponding effective damping rate. c)Magnetization as a function of temperature. d)Spatial profile of the precessing component of the magnetization mx, which gives the profile of the standing SW modes in the system, the mode f1corresponds to the SW along thezdirection, while the degenerate mode f2to the SW along the x andydirections. The low-temperature side is set at 0 K, so that the system is studied in the simplest possible condition. The heights of the SW peaks fluctuate considerably from sample to sample, due to different realizations of the thermal field. Below follows the results pertaining to simulations performed on a single sample. An analysis of the signal averaged over many samples is performed in the subsequent section. Fig. (3) shows the power spectrum as a function of fre- quency and thermal gradient, for = 0:01. Both the color code and the gradient are in logarithmic scale. The two modes f1andf2are clearly visible. Their frequency is independent of@xT, while their amplitude grows up to a factor 3, and reaches its maximum around 10K/nm. For larger gradients, the temperature in the sample approaches the Curie point, so that the magnetization drops and the spectrum is dominated by noise. Other thermally excited SW modes are visible at frequencies around 70 GHz. Because of their higher damping, their height is almost one order of magnitude smaller than f1 andf2, so that they give a negligible contribution to the SW power. In our system, the damping parameter plays a key role, since it is responsible both for the dissipation and the strength of the thermal field. To obtain better insight regarding the effect of the thermal gradient, we computed the linewidth (full width at half maximum) of the modes f`, as a function of @xT, for different values of [Fig. (4) a) and b)]. Interestingly, we find that the linewidth of the two modes f`, shows a quadratic Figure 3. (Color online) Power spectrum (in color code) as a function of frequency and thermal gradient @xT. Two modes f1(31.2 GHz) and f2 (40.1 GHz) are visible. The frequency of both modes is independent of @xT, while their amplitude grows as a function of it, and reaches its maximum around 10K/nm. dependence on the damping parameter and on the thermal gradient. ` e(@xT) = ` +()b`()@xT(b`()@xT)2;(5) where ` +()is the positive damping rate (i.e. the linewidth at zero thermal gradient) of mode `. From our numerical simulations, we can extract ` +(), which fits the functions 1 +() = 166=(1 + 7:510325:51063);(6) 2 +() = 180=(1 + 4:810322:91053); and of the coefficients b`() b1() = 1:7109+ 4:51052; (7) b2() = 3:53108+ 7:691052: For low values of (up to 3103) the quadratic correction is negligible. Let us now turn to a discussion of the amplification of SW signal given by the temperature gradient. Fig. (4c) shows the gainA(i.e.the ratio between the SW power at a given @xTand the SW power at @xT= 0) as a function of @xT, calculated for different values of . Between 0 and 10 K/nm, the signal grows as a function of @xT. At gradients larger than 10 K/nm, corresponding to an average temperature of 1500 K (the Curie point of Py), the magnetization drops abruptly and the signal is destroyed. Fig. (4d) shows the maximum gain Amax(computed at @xT= 10 K/nm) as a function of , which fits the functionAmax() = 1 +c+ (c)2, withc= 129:9. Fig. (4e) shows the gain as a function of the thermal gradient, for= 102and an applied field between 1 and 9 KOe. The amplification starts at a critical gradient gc. This critical threshold, which is different for each mode, is plotted as a4 Figure 4. (Color online) a)andb)Linewidths of the modes f1andf2(in GHz ) vs@xT(in Log scale), computed for different values of . Their linewidths decrease quadratically with the thermal gradient (see text). c)Gain Avs@xT, computed for different values .Agrows exponentially with the gradient. d)Maximum gainAmax (at@xT= 10 K/nm) vs. The maximum gain increases quadratically with (see text). e)Gain Vs thermal gradient (in Log scale), computed for = 0:01and different values of the applied field. f)Critical gradient gc(in Log scale) vs , for both modes. function of in Fig. (4f). Remarkably, an increase in of a factor six corresponds to a decrease in gcof two orders of magnitude. Thus, both the critical threshold and the SW amplification are dramatically affected by the Gilbert damping parameter , but they do not depend on the applied field. A simple analysis can give a qualitative understanding of this phenomenon. The SW power spectrum for the SW amplitude of a single spin obeying Eq.(4), in presence of thermal fluctuation, reads [20] p(!) =D (!2!2 0)2+ +()2; (8) which is a Lorentzian centered around the resonance frequency !0 0Hext, with damping rate +!0. The strength of the fluctuations D/Tis given in Eq. (2). Thus, the height of the resonant peaks is proportional to . On the contrary, The applied field Hextcontrols the resonance frequency and the damping rate, but not the height of the peaks, so that it does not influence the amplification. However, this qualitative consideration is strictly valid only for a single spin in the linear regime, where the random field acts as an additive noise. In the case of many interacting spins, and in a non-linear regime, the noise acts in a multiplicative way [21]–[23], and the damping rate itself is not anymore linear function !0. Thus, the behaviour of the resonant peaks is expected to depend strongly on the geometry of the system, and on the intensity of the thermal gradient. The transition between linear and nonlinear regimes, and between additive and multiplicative noise, has not yet been investigated in within the SSE. Figure 5. (Color online) Computations of gain and noise/signal ratio for = 0:003 (black dots) and = 0:01(red squares), and an applied field of 10 KOe. Panels a)andb)show respectively the gain and the noise/signal ratio for a system with uniform thermal gradient (in Log scale). Panels c) andd)show respectively the gain and the noise/signal ratio for a system with uniform temperature. IV. C OMPARISON BETWEEN THERMAL GRADIENT AND CONSTANT TEMPERATURE The results shown in the previous section suggest that a thermal gradient is an effective means to amplify SW in a ferromagnet. However, a uniform temperature distribution might lead to a similar effect. In both the isothermal and the uniform gradient case, only a part of the thermal excitation contributes to SW amplification, the rest being wasted into thermal noise. In this section we analyse these issues, comparing the effects of thermal gradient with the ones of uniform temperature. For this purpose, we have averaged the SW spectrum over 24 samples with different realization of the random thermal field, and we have analyzed the average amplification and the noise/signal ratio for a system with thermal gradient and for an isothermal one. The computations were performed for an applied field of 10 KOe, and for two values of the Gilbert damping parameter, = 3103and= 101. The gradient in the non-isothermal system spans between 0 and 10 K/nm, while the temperature in the isothermal one spans between 0 and 1500 K. The panels a) and c) in Fig. (5) show the average gain of the SW signal respectively for the non-isothermal and for the isothermal system, with the error bars indicating the variance of the average signal. In fact, the amplification is similar in both systems, where SW are amplified up to a factor 2 (for = 102), with large error bars exceeding 2.4. Apart from a slight overestimation of the amplification effect, the study performed on a single sample agree with the sample-averaged results. In the non-isothermal system, the maximum amplification is between at 10 K/nm, which corresponds to an average temperature between of 150 K in the system, and a maximum temperature on the hotter side of 300 K. In the isothermal system, a similar amplification is reached at a temperature of 300 K. The panels b) and d)5 show the noise/signal ratios for the two systems, whose values are similar in the amplification region. Remarkably, the Gilbert damping parameter has a negligible influence on noise/signal ratio. V. C ONCLUSIONS We have performed a numerical study, which analyses the effect of thermal excitations on SW amplification in a nanostructure. Our simulations suggest that the system should behave in a similar way with a uniform temperature and a uniform gradient. The Gilbert damping parameter affects dramatically the SW amplification: a system with higher damping dissipates energy at a higher rate, but is also more effective in absorbing thermal energy. In this study we have not taken into account the spin transfer effect, since our objective consists in studying the effect of thermal gradient on the LLG equation in the simplest possible configuration, where the only source that affects the magnetization dynamics is the stochastic thermal field. A thorough simulation of the spin-Seebeck effect, which takes into account the effect of electronic transport, will be the subject of a future paper. Concerning possible experimental study of SW excitations through a temperature gradient, the paper of Naletov et al. [19] analyses the SW modes excited by various means (rf field and rf current) in a perpendicularly magnetized nanopillar, using a magnetic resonance force microscope. This is an effective means to investigate samples buried under several electrodes, and is sensitive to all the SW modes excited in the system. We believe that an experimental investigation performed with a setup similar to the one of Ref. [19] could elucidate the effect of thermal gradient on SW modes with different symmetries. VI. A CKNOWLEDGMENT We gratefully acknowledge the Carl Tryggers foundation and the Göran Gustafssons foundation for financial support. This work was financed through the EU project NexTec, VR (the Swedish Research Council), and SSF (Swedish Founda- tion for Strategic Research). The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Centre in LinkÃ˝ uping (NSC). We wish to thank M. Rasander and J. Chico for reviewing the manuscript. REFERENCES [1] K. Uchida et al. Observation of the spin seebeck effect. Nature , 455:778– 781, 2008. [2] K. Uchida et al. Spin seebeck insulator. Nat. Mater. , 9:894–897, 2010. [3] J. Sinova. Spin seebeck effect: Thinks globally but acts locally. Nat. Mater. , 9:880–881, 2010. [4] E. Padrón-Hernández et al. Amplification of spin waves by thermal spin-transfer torque. Phys. Rev. Lett. , 107:197203, 2011. [5] A. Slavin and V . Tiberkevich. Nonlinear auto-oscillator theory of microwave generation by spin-polarized current. IEEE Transactions on Magnetics , 45(4):1875 –1918, 2009. [6] H. Adachi et al. arXiv:1010.2325v2 , 2010. [7] Xiao et al. Theory of magnon-driven spin seebeck effect. Phys. Rev. B , 81:214418, 2010. [8] Hals et al. Thermopower and thermally induced domain wall motion in (ga, mn)as. Solid State Communications , 150(11â ˘A¸ S12):461 – 465, 2010. Spin Caloritronics.[9] D. Hinzke and U. Nowak. Domain wall motion by the magnonic spin seebeck effect. Phys. Rev. Lett. , 107:027205, 2011. [10] Tiago S. Machado, Tatiana G. Rappoport, and Luiz C. Sampaio. V ortex core magnetization dynamics induced by thermal excitation. Applied Physics Letters , 100(11):112404, 2012. [11] G. E. W. Bauer. arXiv:1107.4395v1 , 2011. [12] L. D. Landau and E. M. Lifshitz. To the theory of the dispersion of the ferromagnetic-body permeability. In Collected papers . Ed. Pergamon, 1965. [13] T.L. Gilbert. A phenomenological theory of damping in ferromagnetic materials. IEEE, Transaction on Magnetics , 40(6):3443 – 3449, 2004. [14] A. G. Gurevich and G. A. Melkov. Magnetization Oscillation and Waves . CRC Press, 1996. [15] T. Fischbacher et al. A systematic approach to multiphysics extensions of finite-element-based micromagnetic simulations: Nmag. Magnetics, IEEE Transactions on , 43(6):2896 –2898, 2007. [16] E. Martinez et al. Thermal effects in domain wall motion: Micromag- netic simulations and analytical model. Phys. Rev. B , 75:174409, 2007. [17] P. Lubitz, M. Rubinstein, J. J. Krebs, and S.-F. Cheng. Frequency and temperature dependence of ferromagnetic linewidth in exchange biased permalloy. Journal of Applied Physics , 89(11):6901–6903, 2001. [18] G. Grinstein and R. H. Koch. Coarse graining in micromagnetics. Phys. Rev. Lett. , 90:207201, 2003. [19] V . V . Naletov et al. Identification and selection rules of the spin- wave eigenmodes in a normally magnetized nanopillar. Phys. Rev. B , 84:224423, 2011. [20] N. Saito M. Toda, R. Kubo. Statistical physics . Springer-Verlag, 1983. [21] José Luis Garcia-Palacios and Francisco J. Lázaro. Langevin-dynamics study of the dynamical properties of small magnetic particles. Phys. Rev. B , 58:14937–14958, Dec 1998. [22] Werner Scholz, Thomas Schrefl, and Josef Fidler. Micromagnetic simulation of thermally activated switching in fine particles. Journal of Magnetism and Magnetic Materials , 233(3):296 – 304, 2001. [23] O. Chubykalo, R. Smirnov-Rueda, J.M. Gonzalez, M.A. Wongsam, R.W. Chantrell, and U. Nowak. Brownian dynamics approach to interacting magnetic moments. Journal of Magnetism and Magnetic Materials , 266(1â ˘A¸ S2):28 – 35, 2003. Proceedings of the 4th International Conference on Fine Particle Magnetism (ICFPM).

2013-03-12

Using micromagnetic simulations, we have investigated spin dynamics in a nanostructure in the presence of thermal fluctuations. In particular, we have studied the effects of a uniform temperature and of a uniform thermal gradient. In both cases, the stochastic field leads to an increase of the precession angle of the magnetization, and to a mild decreas of the linewidth of the resonance peaks. Our results indicate that the Gilbert damping parameter plays the role of control parameter for the amplification of spin waves.

Thermally excited spin waves in a nano-structure: thermal gradient vs. constant temperature

1303.2895v1

Spin dynamics and frequency dependence of magnetic damping study in soft ferromagnetic FeTaC lm with a stripe domain structure B. Samantaray1a), Akhilesh K. Singh2, A.Perumal2, R. Ranganathan1and P. Mandal1, 2 1)Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, Indiaa) 2)Department of Physics, Indian Institute of Technology Guwahati, Guwahati - 781039, India (Dated: 12 November 2021) Perpendicular magnetic anisotropy (PMA) and low magnetic damping are the key factors for the free layer magnetization switching by spin transfer torque technique in magnetic tunnel junction devices. The mag- netization precessional dynamics in soft ferromagnetic FeTaC thin lm with a stripe domain structure was explored in broad band frequency range by employing micro-strip ferromagnetic resonance technique. The polar angular variation of resonance eld and linewidth at dierent frequencies have been analyzed numeri- cally using Landau-Lifshitz-Gilbert equation by taking into account the total free energy density of the lm. The numerically estimated parameters Land e g-factor, PMA constant, and eective magnetization are found to be 2.1, 2105erg/cm3and 7145 Oe, respectively. The frequency dependence of Gilbert damping parameter () is evaluated by considering both intrinsic and extrinsic eects into the total linewidth analysis. The value ofis found to be 0.006 at 10 GHz and it increases with decreasing precessional frequency. Spin transfer torque (STT) has grater credibility com- pared to other techniques towards ultrafast spin dynam- ics in ferromagnet by electric current induced magneti- zation reversal of spin valves and magnetic tunnel junc- tions (MTJ).1The current researchers are more keen to focus on STT technology for its high density magnetic random access memories (MRAM),2,3STT-driven do- main wall devices4and perpendicular magnetic record- ing media5applications. In order to make this technol- ogy more ecient, lowering the critical current density is essential which requires the material specications with low saturation magnetization ( MS), high spin polariza- tion, large uniaxial perpendicular magnetic anisotropy (PMA) constant and low magnetic damping.6{8The mag- netic damping parameter ( ) can be described well by the phenomenological Landau-Lifshitz-Gilbert equation and is known as the Gilbert damping.9,10Several at- tempts have been made for understanding the origin of Gilbert damping in spin dynamics relaxation in single layer as well as multilayered magnetic alloys, which arises from both intrinsic and extrinsic parts of the material. The intrinsic contribution to the Gilbert damping pa- rameter has been studied by tuning the strength of the spin-orbit coupling.8,11,12Recently, Ikeda et al.13have re- ported that CoFeB-MgO based MTJ with PMA would be reliable for high-density non-volatile memory application due to its high thermal stability and eciency towards STT technology. The investigation on magnetic dynam- ics, PMA and the apparent magnetic damping have been studied extensively in CoFeB based soft ferromagnetic thin lm by ferromagnetic resonance (FMR) and time- resolved magneto-optical Kerr eect.14,15Malinowski et al.16have reported a large increase in Gilbert damping with applied magnetic eld in perpendicularly magne- tized CoFeB thin lm. a)Electronic mail: iitg.biswanath@gmail.comIn this letter, we focus on amorphous FeTaC layer due to its interesting soft ferromagnetic (FM) properties.17,18 The amorphous soft FM layer reduces the number of pin- ning centers which may lead to the STT-driven domain wall motion along with high tunneling magnetoresistance ratio (TMR). The transcritical loop along with the stripe domain structure, which are the manifestation of PMA component were reported on FeTaC thin lm with thick- ness of 200 nm.18,19To shed some more light onto its dy- namic magnetic properties, we have further studied this lm by using ferromagnetic resonance technique. Though the magnetic anisotropy and Gilbert damping have been studied by FMR technique in several magnetic thin lms like Heusler alloys, permalloy, soft magnetic ma- terials and multilayered (FM/antiferromagnetic or non- magnetic/FM) magnetic lms for magnetic recording, MTJ and TMR reader applications, most of the reports are limited to single frequency due to the measurements in X-band electron-spin-resonance spectrometer where the cavity resonates at particular frequency.6,8,15,20{23In this report, spin dynamics and magnetic relaxation are studied at dierent magnetization precessional frequen- cies. Soft ferromagnetic Fe 80Ta8C12single layer lm with thickness 200 nm was deposited by dc magnetron sput- tering technique and the details of growing environment were reported earlier.18The static and dynamic magnetic properties were explored by using a vector network ana- lyzer (VNA) based homemade micro-strip ferromagnetic resonance (MS-FMR) spectrometer. The micro-strip line which was coupled to VNA and Schottky diode detector (Agilent 8473D) through high frequency coaxial cables was mounted in between the pole pieces of the electro- magnet. The magnetic thin lm with lm side down- ward was mounted on the strip line. The frequency of the microwave signal was xed by using an Agilent Tech- nologies made VNA (Model PNA-X, N-5242A) with a constant microwave power of 5 dBm. The rst deriva-arXiv:1503.07043v5 [cond-mat.mtrl-sci] 14 May 20152 FIG. 1. Schematic diagram of MandHvectors in spherical polar coordinate system. 'Mand'Hare the in-plane angle of magnetization ( M) and external magnetic eld ( H) with respect toxaxis, while MandHare out-of-plane angles with respect to zaxis. tive of the absorption spectrum with respect to magnetic eld (H) was collected by eld modulation and lock-in detection technique. The FM thin lm was treated in- plane and out-of-plane orientations. The magnetic eld sweeping FMR spectra were recorded by varying two pa- rameters: precessional frequency and the angle between Hand normal of the lm. The frequency ( f) and po- lar angle (H) dependence of resonance eld ( Hr) and linewidth (
HPP) were extracted from each FMR spec- trum and the numerical calculations were carried out by mathematica program for dierent relaxation processes. The precession of magnetization ( M) in the sample plane under the in uence of microwave and external mag- netic eld is illustrated in Fig. 1 in a polar coordinate system.'H('M) is the in-plane angle between H(M) andxaxis andH(M) is the polar angle between zaxis andH(M). The uniform precession of magnetization can be described by the Landau-Lifshitz-Gilbert (LLG) equation of motion,9,10 @ !M @t=  !M !Heff +G M2 S" !M@ !M @t# (1) The rst term corresponds to the precessional torque in the eective magnetic eld and the second term is the Gilbert damping torque. =gB=~is denoted as gyro- magnetic ratio and written in terms of Land e gfactor, Bohr magneton B, and Planck constant ~.G= MS is related to the intrinsic relaxation rate of the material. is the dimensionless Gilbert damping parameter. The free energy density of a single magnetic thin lm can be written as, E=MSH[sinHsinMcos('H'M) + cosHcosM] 2M2 Ssin2M+K?sin2M (2) where the rst term is analogous to the Zeeman energy, the second term is dipolar demagnetization energy, the third term signies the anisotropy energy, MSis the sat- uration magnetization, K?is the PMA constant with corresponding anisotropic eld H?= 2K?=MS. The resonance frequency frof the uniform precession mode is FIG. 2. (a) shows typical FMR spectra at dierent frequencies in planar orientation, (b) shows fdependence of Hrfor in- plane applied magnetic eld. Experimentally and numerically calculatedHrvalues are shown as open circles and solid line respectively and (c) shows the plot of room temperature M Hloop and domain images reproduced from Ref. [19]. deduced from the energy density by using the following expression,24 f2 r= 22 1 M2 Ssin2M" @2E @2 M@2E @'2 M@2E @M@'M2# (3) where the derivatives are evaluated at equilibrium posi- tions ofMandH. For the in-plane orientation, the typical FMR spectra at dierent frequencies are shown in Fig. 2(a). The mea- surements were carried out by varying the frequency from 1 to 18 GHz with an interval of 0.5 GHz. In the lower frequency range 1-6 GHz, the FMR spectra show two res- onance peaks. The low-eld resonance peaks named as secondary mode and are marked by 2 and 2for 2.5 and 6 GHz, respectively in Fig. 2(a). This mode arises from the linear unsaturated zone of the transcritical M(H) loop19 and is usually observed in stripe-domain structure.25,26 The transcritical loop along with the domain structure are reproduced from earlier report19and is shown in Fig. 2(c). The dense stripe-domain structure observed in this lm conrms the presence of perpendicular magnetic anisotropy. The primary modes usually called uniform mode are marked as 1 and 1for 2.5 and 6 GHz, respec- tively. The value of Hrfor secondary resonance peak increases with the increase in fup to 4.5 GHz and then follows the reverse trend as depicted in Fig. 2(b). This3 FIG. 3. Equilibrium angle of the magnetization, M, as a function of the applied eld direction, H, in out-of-plane con- guration at dierent frequencies. could be explained on the basis that the value of Hrof the uniform mode above 4.5 GHz overcomes the parallel saturation eld, i.e., 280 Oe as observed in M(H) curve. Above 6 GHz, Hrexceeds the parallel saturation eld in large extent and this could be the reason for the strong attenuation of secondary phase. In planar conguration (M=H==2), the solution for the in-plane resonance frequency can be calculated by incorporating the total energy in Eq. 3 and is given by, fr= 2[(4M+Hcos ('H'M)) (Hcos ('H'M))]1 2 (4) The value of 'Mcan be calculated by using the solution ofHat equilibrium condition, i.e.,@E @'M=0. However, for the present thin lm, we could not nd any planar anisotropy from the 'Hdependence of Hrand hence conclude,'H='M. Thefdependence of Hris nu- merically calculated by using Eq. 4 and is shown as a solid line in Fig. 2(b). The numerically calculated values yielded a good t and the parameters are found to be re- liable with 4 MS= 779110 Oe and = 2.95 MHz/Oe with ag-factor of 2.1. The deduced value of saturation magnetization is very close to earlier reported value from M(H) loop measurement.18 In out-of-plane conguration, the solution for the res- onance frequency is deduced from Eq. 3 by employing the conditions, 'H='M=0and is represented in Eq. 5. f2 r= 2
2h Hcos (MH) 4MS2K? MS cos 2Mi h Hcos (MH) 4MS2K? MS cos2Mi (5) The equilibrium angle Mis numerically calculated for FIG. 4. Angular dependence of resonance eld Hrin out- of-plane conguration at dierent frequencies. ( ) shows the experimental points and the line (-) shows the modeled data. each value of Hby minimizing the energy, i.e.,@E @M= 0 and is depicted in Fig. 3 for dierent frequencies. Fig. 3 demonstrates that magnetization suddenly attempts to align in planar direction as the magnetic eld goes away from theH=0and 180. Fig. 4 shows one complete round ofHdependence of Hrat dierent frequencies. Uniaxial PMA is found to be observed along with singu- larity atH= 0andH= 180, which signies that innite magnetic eld is required to turn the Mvector parallel toHin perpendicular conguration. The depen- dence ofHronHis modeled at dierent frequencies starting from 4 to 10 GHz with 2 GHz intervals by using Eq. 5 and the interpolated values of Mfrom Fig. 3. The modeled values of Hrare plotted as a solid line in Fig. 4 and a very good agreement with experimental data is observed. The parameters deduced from this calculation are found to be K?=2105erg/cm3and 4Meff=7145 Oe. Finally, the damping of magnetization precession has been analyzed from linewidth of FMR spectra. The H dependence of
HPPas shown in Fig. 5 was extracted from the polar angle variation of FMR spectrum at dier- ent frequencies in the range of 4-10 GHz with an interval of 2 GHz. In order to get better clarity of Fig. 5, the data for the 4 GHz frequency are not shown. The total linewidth broadening due to the intrinsic and extrinsic parts of the material has been expressed in the following equation,8,27
HPP=
H() +
H(
4Meff) +
H(
H) =2p 31 j@! @Hrj MS @2E @2 M+1 sin2M@2E @'2 M + 1p 3@H @4Meff
4Meff +1p 3@H @H
H (6) where 4Meff= 4MS2K?=MS, is the eective mag-4 FIG. 5. Out-of-plane angular dependence of total linewidth,
Hppat dierent frequencies. ( ) shows the experimental points and the line ( ) shows the modeled data. FIG. 6.Hdependence of
Hpp,
H(),
H(
4Meff) and
H(
H) modeled data at 6 GHz frequency. netization.
H() arises from the intrinsic Gilbert type damping and has large contribution towards linewidth broadening. The parameter signies how fast the pre- cessional energy is dissipated into the lattice. The terms
H(4
Meff) and
H(
H) represent the linewidth broadening due to the spatial dispersion of the magni- tude and direction of Meff, respectively. The Hdepen- dence of
HPPwas modeled by using Eq. 6 and the interpolated values of Mfrom Fig. 3 at dierent fre- quencies. The numerically calculated values of
HPP are shown as solid lines in Fig. 5. The individual contri- butions towards the total linewidth (
Hpp) is also shown in Fig. 6. The curves are shown for a single frequency for better clarity. The linewidth broadening is observed mainly due to the intrinsic Gilbert damping. The ex-trinsic contribution is found to be negligible when H is away from 0and 180but it is large near the per- pendicular conguration. The Gilbert damping param- eter at dierent frequencies for the FeTaC thin lm is plotted in the inset of Fig. 6. It shows that the de- crease monotonically with the increase in frequency. The low value of damping parameter observed in the present thin lm can be more relevant towards the STT tech- nology or MTJ applications. Such an increase of by decreasing precessional frequency has been attributed to the inhomogeneous linewidth broadening due to the dis- persion of anisotropic eld.28,29The dispersion in mag- nitude and direction of eective magnetization are found to be
4Meff=0.1 KOe,
H1104degree. The value of Gilbert damping constant for the present Fe- TaC thin lm is found to be comparable to those re- ported in Fe-based magnetic thin lms, such as FePd ternary alloy11, permalloy30, NiFe/CoFeB/CoFe multi- layered sturucture15and (FeCo) 1xGdx8. However, the Mn- and Co- based thin lms6,7have larger damping pa- rameter as compared to the present lm which could be understood on the basis of spin-orbit coupling. In conclusion, PMA and Gilbert damping which are very important and crucial parameters for STT, STT-MRAM and TMR applications, have been analyzed in FeTaC soft ferromagnetic thin lm with a striped domain structure by using MS-FMR technique in broad band frequency range. The precise estimation of Land e g-factor, PMA constant and 4 Meffwere carried out by using total energy density function for magnetic thin lm. 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Miyazaki, \Eect of spin diusion on gilbert damping for a very thin permalloy layer in cu/permalloy/cu/pt lms," Physical Review B, 66, 104413 (2002). 28Z. Celinski and B. Heinrich, \Ferromagnetic resonance linewidth of fe ultrathin lms grown on a bcc cu substrate," Journal of Applied Physics, 70, 5935{5937 (1991). 29T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, \Inductive measurement of ultrafast magnetization dynamics in thin-lm permalloy," Journal of Applied Physics, 85, 7849{7862 (1999). 30G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, \Spin wave contributions to the high-frequency mag- netic response of thin lms obtained with inductive methods," Journal of Applied Physics, 95, 5646{5652 (2004).

2015-03-24

Perpendicular magnetic anisotropy (PMA) and low magnetic damping are the key factors for the free layer magnetization switching by spin transfer torque technique in magnetic tunnel junction devices. The magnetization precessional dynamics in soft ferromagnetic FeTaC thin film with a stripe domain structure was explored in broad band frequency range by employing micro-strip ferromagnetic resonance technique. The polar angular variation of resonance field and linewidth at different frequencies have been analyzed numerically using Landau-Lifshitz-Gilbert equation by taking into account the total free energy density of the film. The numerically estimated parameters Land\'{e} $g$-factor, PMA constant, and effective magnetization are found to be 2.1, 2$\times10^{5}$ erg/cm$^{3}$ and 7145 Oe, respectively. The frequency dependence of Gilbert damping parameter ($\alpha$) is evaluated by considering both intrinsic and extrinsic effects into the total linewidth analysis. The value of $\alpha$ is found to be 0.006 at 10 GHz and it increases with decreasing precessional frequency.

Spin dynamics and frequency dependence of magnetic damping study in soft ferromagnetic FeTaC film with a stripe domain structure

1503.07043v5

arXiv:1311.6305v1 [cond-mat.mes-hall] 25 Nov 2013Spin-wave excitation and propagation in microstructured w aveguides of yttrium iron garnet (YIG) /Pt bilayers P. Pirro,1T. Brächer,1, 2A. Chumak,1B. Lägel,1C. Dubs,3O. Surzhenko,3P. Görnet,3 B. Leven,1and B. Hillebrands1 1)Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, D-67663 Kaisersl autern, Germany 2)Graduate School Materials Science in Mainz, Gottlieb-Daim ler-Strasse 47, D-67663 Kaiserslautern, Germany 3)Innovent e.V., Prüssingstraße 27B, 07745 Jena, Germany (Dated: 7 May 2018) We present an experimental study of spin-wave excitation an d propagation in microstructured waveguides patterned from a 100 nm thick yt trium iron garnet (YIG)/platinum (Pt) bilayer. The life time of the spin waves is found to be more than an order of magnitude higher than in comparably sized metall ic structures despite the fact that the Pt capping enhances the Gilbert damping. Ut ilizing microfocus Brillouin light scattering spectroscopy, we reveal the spi n-wave mode structure for different excitation frequencies. An exponential spin-wav e amplitude decay length of 31µm is observed which is a significant step towards low damping , insulator based micro-magnonics. 1The concept of magnon spintronics, i.e., the transport and m anipulation of pure spin currents in the form of spin-wave quanta, called magnons, ha s attracted growing interest in the recent years1–12. One of the key advantages of magnon spin currents is their la rge decay length which can be several orders of magnitude higher than the spin diffusion length in conventional spintronic devices based on spin-polarize d electron currents. Considering possible applications, the miniaturization of magnonic ci rcuits is of paramount importance. Up to now, downscaling has been achieved using metallic ferr omagnets like NiFe or Heusler compounds2–7. But even the best metallic ferromagnets exhibit a damping w hich is two or- ders of magnitude larger than for Yttrium Iron Garnet (YIG), a ferrimagnetic insulator1,13,14. However, to the best of our knowledge, as high quality YIG film s could only be grown with thicknesses in the range of microns, no microstructured YIG devices have been fabricated so far. A big step forward has been taken with the recent introdu ction of methods to produce high quality, low damping YIG films with thicknesses down to s everal nanometers9,15–17. In this Letter, we show that microscaled waveguides (see Fig . 1) can be fabricated from liquid phase epitaxy (LPE) grown YIG films of 100nm thickness whose high quality has been confirmed by ferromagnetic resonance spectroscopy (FM R). Studying the excitation and propagation of spin-waves in these waveguides by microf ocus Brillouin light scattering, we demonstrate that the damping of the unstructured film can b e preserved during the structuring process. Another key feature of magnon spintronics is its close relat ionship to a multitude of phys- ical phenomena like spin-pumping, spin-transfer torque, s pin Seebeck effect, and (inverse) spin Hall effect, which allow for the amplification, generati on and transformation between charge currents and magnonic currents7–12,15–24. Hetero-structures of YIG covered with a thin layer of platinum (Pt) have proven to show these effects w hich opens a way to a new class of insulator based spintronics. Therefore, we directly stu dy bilayers of YIG/Pt, providing a basis for further studies utilizing the described effects. The used YIG film is prepared by liquid phase epitaxy from a PbO -B2O3-FeO3flux melt using a standard isothermal dipping technique with a gr owth rate of 20nm/min. The incorporation of Pb and Pt ions into the garnet lattice allow s for a low relative lattice mismatch of 3·10−4. We determine the magnetic properties of the film using FMR and compare the results to measurements performed after the deposition of a 9nmPt film onto YIG using plasma 2FIG. 1. Sample schematic: In a 5µm wide waveguide patterned from a bilayer of YIG/Pt (100nm /9nm), spin waves are excited using the dynamic Oersted fields of a microwave current flowing in a copper antenna. An external bias field Hextis applied along the short axis of the waveguide. The spin-wave intensity is detected using micro focus Brillouin light scattering spec- troscopy. cleaning and RF sputtering. From the resonance curve HFMR(fFMR), a saturation magne- tization of Ms= 144±2kA/mhas been determined for the pure YIG film. We find that the deposition of Pt slightly reduces the resonance field µ0HFMR(for example by 1mT for fFMR= 7.0GHz ) compared to the pure YIG film. This shift agrees with the rece nt findings of Ref. 15, where a proximity induced ferromagnetic orderin g of Pt combined with a static exchange coupling to YIG has been proposed as possible expla nation. Figure 2 shows the ferromagnetic resonance linewidth (FWHM )µ0∆Hwith and with- out Pt and the corresponding fits to evaluate the effective Gil bert damping parameter α according to22 µ0∆H=µ0∆H0+2αfFMR γ(1) with the gyromagnetic ratio γ= 28GHz /T. The Gilbert damping αincreases by almost a factor of 5 due to the deposition of Pt: from (2.8±0.3)×10−4to(13.0±1.0)×10−4. The inhomogeneous linewidth µ0∆H0is unchanged within the accuracy of the fit ( 0.16±0.02mT and0.14±0.04mT , respectively). Please note that the increase of the dampin g cannot be explained exclusively by spin pumping from YIG into Pt. Othe r interface effects, like the already mentioned induced ferromagnetic ordering of Pt in c ombination with a dynamic exchange coupling may play a role15,23. Using the spin mixing conductance for YIG/Pt (g↑↓≈1.2×1018m−2from22,24), we find that the expected increase in Gilbert damping due to spin pumping11,20–22isαsp= 1.25×10−4,i.e., it is by a factor of 8 smaller than the 3/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s89/s73/s71/s61/s50/s46/s56/s32/s120/s49/s48/s45/s52/s32/s89/s73/s71/s47/s80/s116/s32/s98/s105/s108/s97/s121/s101/s114 /s32/s112/s117/s114/s101/s32/s89/s73/s71/s181 /s48/s32 /s72 /s32/s40/s109/s84/s41 /s32/s102 /s70/s77/s82/s40/s71/s72/s122/s41/s89/s73/s71/s47/s80/s116/s61/s49/s51/s46/s48/s32/s120/s49/s48/s45/s52 FIG. 2. Linewidth µ0∆Has a function of the ferromagnetic resonance frequency fFMRfor the pure YIG film (blue circles) and the YIG/Pt bilayer (red squar es). The deviations from the linear increase of µ0∆HwithfFMR(fit according to Eqn. 1) are mainly due to parasitic modes cau sing a small systematical error in the measurement of the linewidt h. measured increase. This clearly demonstrates the importan ce of additional effects15,23. As shown recently in Ref.15, this additional damping can be strongly reduced by the intr oduction of a thin copper (Cu) layer in between YIG and Pt, which does no t significant influence the spin-pumping efficiency. The micro structuring of the YIG/Pt waveguide is achieved us ing a negative protective resist mask pattered by electron beam lithography and physi cal argon ion beam etching. As last production step, a microwave antenna (width 3.5µm,510nm thickness) made of copper is deposited on top of the waveguide (see Fig. 1). To experimentally detect the spin waves in the microstructu red waveguide, we employ microfocus Brillouin light scattering spectroscopy (BLS)3–8. This method allows us to study the spin-wave intensity as a function of magnetic field and sp in-wave frequency. In addition, it provides a spatial resolution of 250nm , which is not available in experiments using spin pumping and inverse spin Hall effect8–10as these methods integrate over the detection area (and also over the complete spin-wave spectrum12). To achieve an efficient spin-wave excitation, we apply a stati c magnetic field of 70mT perpendicular to the long axis of the waveguide. The dynamic Oersted field of a microwave current passing through the antenna exerts a torque on the st atic magnetization. This config- uration results in an efficient excitation of Damon-Eshbach l ike spin waves which propagate 4/s51/s46/s52 /s51/s46/s53 /s51/s46/s54 /s51/s46/s55 /s51/s46/s56/s48/s46/s49/s49/s32/s99/s101/s110/s116/s101/s114/s32/s114/s101/s103/s105/s111/s110 /s32/s101/s100/s103/s101/s32/s114/s101/s103/s105/s111/s110/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s66/s76/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121 /s102 /s77/s87/s32/s40/s71/s72/s122/s41 /s49 /s49 /s32/s181 /s109 /s97/s110/s116/s101/s110/s110/s97/s89/s73/s71/s47/s80/s116 /s72 /s101/s120/s116 FIG. 3. Normalized BLS intensity (log scale) as a function of t he applied microwave frequency fMW (external field µ0Hext= 70mT ). The blue line (circular dots) shows the spectrum measured at the edges of the waveguide (see inset). The red line (rectang ular dots) is an average of the spectra recorded in the center of the waveguide. perpendicular to the static magnetization. A microwave pow er of0dBm (pulsed, duration 3µs, repetition 5µs) in the quasi-linear regime, where nonlinearities are not s ignificantly influencing the spin-wave propagation, has been chosen. To o btain a first characterization of the excitation spectrum, BLS spectra as a function of the a pplied microwave frequency (fMW) have been taken at different positions across the width of th e waveguide at a distance of11µm from the antenna. Figure 3 shows the spectrum of the edge re gions (blue circles) and of the center of the waveguide (red squares, see sketch in the inset). The main excitation in the center of the waveguide takes place at frequencies bet weenfMW= 3.49−3.66GHz and we will refer to these spin-wave modes as the waveguide modes . At the borders of the waveguide, edge modes have their resonance around fMW= 3.44GHz . The reason for the appearance of these edge modes is the pronounced reduction o f the effective magnetic field Heffat the edges by the demagnetization field and the accompanyin g inhomogeneity of the z-component of the static magnetization. This situation has been analyzed experimentally and theoretically in detail for metallic systems25,26. To get a better understanding of the nature of the involved sp in-wave modes, mode profiles at different excitation frequency measured at a distance of 6µm from the antenna are shown in Fig. 4. The evolution of the modes can be seen clearly: for f requencies below fMW= 3.45GHz , the spin-wave intensity is completely confined to the edges of the waveguide. In 5/s48 /s49 /s50 /s51 /s52 /s53/s51/s46/s52/s48/s51/s46/s52/s53/s51/s46/s53/s48/s51/s46/s53/s53/s51/s46/s54/s48/s51/s46/s54/s53 /s80/s111/s115/s105/s116/s105/s111/s110/s32 /s122 /s32/s97/s108/s111/s110/s103/s32/s119/s105/s100/s116/s104/s32/s111/s102/s32/s119/s97/s118/s101/s103/s117/s105/s100/s101/s32/s40/s181/s109/s41/s102 /s77/s87/s40/s71/s72/s122/s41 /s48/s48/s46/s51/s48/s46/s53/s48/s46/s56/s49/s32 /s66/s76/s83/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41 FIG. 4. BLS intensity (linear scale) as a function of the posit ion along the width of the waveguide for different excitation frequencies fMW(µHext= 70mT ). Frequencies below 3.45GHz show strongly localized edge modes which start to extend into the center of the waveguide for frequencies between 3.45−3.50GHz . For higher fMW, waveguide modes appear which have their local intensity ma xima in the center of the waveguide. The dashed lines indicate the calculated minimal frequencies of the waveguide modes shown in Fig. 5. the range fMW= 3.45−3.50GHz , the maximum of the intensity is also located near the edges, but two additional local maxima closer to the center o f the waveguide appear. For frequencies in the range of 3.50−3.57GHz , three spin-wave intensity maxima symmetrically centered around the center of the waveguide are observed. Th is mode is commonly labeled as the third waveguide mode n= 3(ndenotes the number of maxima across the width of the waveguide). For higher fMW, only one intensity maximum is found in the center of the waveguide (first waveguide mode, n= 1). For the waveguide modes, we can compare the experimental res ults to theoretical consid- erations. The theory for spin waves in thin films27with the appropriate effective field from micromagnetic simulations and a wave-vector quantization over the waveguide’s short axis provides an accurate description of the spin-wave mode disp ersions4–6. Figure 5 shows the dispersion relations and the excitation efficiencies of the w aveguide modes n= 1,3,5. Only odd waveguide modes can be efficiently excited4,5(even modes have no net dynamic mag- netic moment averaged over the width of the waveguide) using direct antenna excitation. The minimal frequencies of these three modes are indicated a s dashed lines for comparison in Fig. 4. Comparing Fig. 4 and Fig. 5, we find a reasonable agre ement between theory and experiment for the first and the third waveguide mode. The n= 5and higher waveguide 6/s51/s46/s52/s51/s46/s53/s51/s46/s54/s51/s46/s55/s51/s46/s56/s51/s46/s57 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s49/s49/s102/s32/s40/s71/s72/s122/s41/s32/s49 /s32/s51 /s32/s53/s119/s97/s118/s101/s103/s117/s105/s100/s101/s32/s109/s111/s100/s101/s69/s120/s99/s46/s32/s101/s102/s102/s46/s32/s40/s97/s114/s98/s46/s117/s110/s105/s116/s41 /s32/s87/s97/s118/s101/s32/s118/s101/s99/s116/s111/s114 /s32/s107 /s120/s32/s40/s114/s97/s100/s47/s181/s109/s41/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s32 FIG. 5. (color online) Dispersion relations and amplitude e xcitation efficiencies for the first three odd waveguide modes of a transversally magnetized YIG waveg uide and an antenna width of 3.5µm (external field µHext= 70mT , width of waveguide 5µm, further parameters see28). modes are not visible in the experiment. Due to the fact that t he excitation efficiency and the group velocity of the spin-wave modes decreases with inc reasingn(see Ref. 4 and 5 for details), this can be attributed to a small amplitude of thes e modes. To visualize the influence of the spatial decay of the spin wav es on the mode composition, Fig. 6 (a) shows 2D spin-wave intensity maps for two exemplar y excitation frequencies. For fMW= 3.45GHz , edge modes can be detected for distances larger than 20µm. Different higher order waveguide modes are also excited, but they can o nly be detected within 5µm from the antenna. From this findings, we can conclude that the edge modes are dominating the propagation in this frequency range because of their hig h group velocities (proportional to the decay length) compared to the available waveguide mod es (n≥5). The situation is completely different for fMW= 3.60GHz . Here, the preferably excited n= 1waveguide mode is interfering with the weaker n= 3waveguide mode causing a periodic beating effect3–5of the measured spin-wave intensity. In this frequency rang e, no significant contribution of modes confined to the edges is vis ible. An important parameter for magnonic circuits and applicati ons is the exponential de- cay length δampof the spin-wave amplitude. To determine δexper ampforfMW= 3.60GHz , we integrate the spin-wave intensity over the width of the wave guide (Fig. 6 (b)) and obtain δexper amp= 31µm which is substantially larger than the reported decay len gths in metallic mi- crostructures made of Permalloy or Heusler compounds3,4. This value can be compared 7FIG. 6. (a) BLS intensity maps (linear scale) for two different excitation frequencies ( µ0Hext= 70mT ). (b) Integrated BLS intensity (logarithmic scale) over the width of the waveguide for fMW= 3.60GHz including a fit to determine the exponential amplitude decay length (δamp= 31µm). to the expected theoretical value δtheo amp=vgτwherevgis the group velocity and τis the life time of the spin wave. The Gilbert damping of the unpa tterned YIG/Pt bilayer α= 1.3·10−3measured by FMR corresponds to a life time τ≈28ns for our experi- mental parameters. The group velocity vgcan be deduced from the dispersion relations in Fig. 5 or from dynamic micromagnetic simulations yieldin gvg≈1.0−1.1µm/ns, thus δtheo amp= 28−31µm. The agreement with our experimental findings δexper amp= 31µm is excel- lent, especially if one considers that the plain film values o fαandMs, which might have been changed during the patterning process, have been used f or the calculation. This indi- cates that possible changes of the material properties due t o the patterning have only an negligible influence on the decay length of the waveguide mod es and that the damping of the spin waves due to the Pt capping is well described by the measu red increase of the Gilbert damping. To conclude, we presented the fabrication of micro-magnoni c waveguides based on high quality YIG thin films. Spin-wave excitation and propagatio n of different modes in a mi- crostructured YIG/Pt waveguide was demonstrated. As expec ted, the enhancement of the Gilbert damping due to the Pt deposition leads to a reduced li fe time of the spin waves com- pared to the pure YIG case. However, the life time of the spin w aves in the YIG/Pt bilayer is still more than an order of magnitude larger than in the usual ly used microstructured metallic 8systems. This leads to a high decay length reaching δexper amp= 31µm for the waveguide modes. 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Gubbiotti, M. Conti, G. Carlotti, P. Candeloro, E. D. Fabr izio, K.Y. Guslienko, A. André, C.Bayer, and A.N. Slavin, J. Phys.: Condens. Matter 16, 7709 (2004). 26C. Bayer, J.P. Park, H. Wang, M. Yan, C.E. Campbell, and P.A. C rowell, Phys. Ref. B 69, 134401 (2004). 1027B.A. Kalinikos and A.N. Slavin, Journal of Physics C: Solid S tate Physics, 19pp. 7013, (1986). 28Dispersion relations calculated according to Ref. 27 with p arameters Beff= 68mT and effective width = 4µm from a micromagnetic simulation29,Ms= 144kA /m,100nm thick- ness, exchange constant A= 3.5pA/m. 29M.J. Donahue and D.G. Porter, Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (Sept 1999). 11

2013-11-25

We present an experimental study of spin-wave excitation and propagation in microstructured waveguides patterned from a 100 nm thick yttrium iron garnet (YIG)/platinum (Pt) bilayer. The life time of the spin waves is found to be more than an order of magnitude higher than in comparably sized metallic structures despite the fact that the Pt capping enhances the Gilbert damping. Utilizing microfocus Brillouin light scattering spectroscopy, we reveal the spin-wave mode structure for different excitation frequencies. An exponential spin-wave amplitude decay length of 31 {\mu}m is observed which is a significant step towards low damping, insulator based micro-magnonics.

Spin-wave excitation and propagation in microstructured waveguides of yttrium iron garnet (YIG)/Pt bilayers

1311.6305v1

arXiv:1412.2479v1 [cond-mat.mes-hall] 8 Dec 2014Magnetization Dynamics driven by Non-equilibrium Spin-Or bit Coupled Electron Gas Yong Wang,1Wei-qiang Chen,2and Fu-Chun Zhang3,1,4 1Department of Physics, The University of Hong Kong, Hong Kon g SAR, China 2Department of Physics, South University of Science and Tech nology of China, China 3Department of Physics, Zhejiang University, China 4Collaborative Innovation Center of Advanced Microstructu res, Nanjing University, Nanjing, 210093, China The dynamics of magnetization coupled to an electron gas via s-d exchange interaction is investi- gated by using density matrix technique. Our theory shows th at non-equilibrium spin accumulation induces a spin torque and the electron bath leads to a damping of the magnetization. For the two-dimensional magnetization thin film coupled to the elec tron gas with Rashba spin-orbit cou- pling, the result for the spin-orbit torques is consistent w ith the previous semi-classical theory. Our theory predicts a damping of the magnetization, which is abs ent in the semi-classical theory. The magnitude of the damping due to the electron bath is comparab le to the intrinsic Gilbert damping and may be important in describing the magnetization dynami cs of the system. I. INTRODUCTION In study of spin transfer torque (STT), it has been proposed [1, 2] to manipulate magnetic order parameter dynamics by using non-equilibrium electron bath instead of external magnetic fields. The proposal has already led to commercial products in spintronics engineering. Re- cently, there has been much attention on the ”spin-orbit torque”(SOT), which was first proposed in theory[3, 4], and later confirmed in experiments[5–8] (see Ref. 9 and 10 for a comprehensive review). After applying an ex- ternal electric field to the electron gas with spin-orbit in- teraction(SOI), a component ofthe accumulated electron spin density mis-aligned with the ferromagnetic ordering can be created[3, 4], which then will induce a field-like torque. The SOT opens the possibility of manipulat- ing the magnetic order parameter in collinear magnetic structures and may efficiently reduce the critical current density for magnetization switching[3, 4]. In the theoret- ical side, a full quantum theory has been proposed and developed to describe the dynamics of a single domain magnetunderthecontinuousscatteringbyspin-polarized electrons. The quantum STT theory recovers the results of the semiclassical STT theory, and has revealed more detailsaboutthemagnetizationdynamicsintheSTT[11– 13]. Therefore, it will be natural to apply a full quantum theory to study the magnetization dynamics influenced by the SOI electron gas. This may be an extension of the quantum STT theory to SOT. In the full quantum theory, the quantum dynamics of the magnetization can be described by the evolution of its density matrix under the influence of the electron gas, which can be tuned by the external electric field. This treatment will not only give the mean-field effect on the magnetization dynamics by the electron bath, but also include the damping of the magnetization due to the fluctuation of the electron spin. The similarstrategyhas been exploited to investigatethe photo-excited dynamics of the order parameter in Peierls chain[14]. This paper is organized as follows. In section II, we apply density matrix technique to derive general formal-E FIG. 1. (Color online). Schematic diagram for the lattice of localized spins (orange) coupled to the conductions electr ons (blue) through s-d exchange interaction. An external elect ric fieldEcan be applied to tune the electron bath. ism for the magnetization dynamics driven by the elec- tron bath through s-d exchange interaction. In section III, we apply the general formalism to the special case where the spatially uniform magnetization is coupled to a two dimensional electron gas with Rashba SOI, and calculate the spin-orbit torque and the damping effect of the electron bath. The main results are summarized and discussed in section IV. II. GENERAL FORMALISM We apply density matrix technique to study dynamics of the magnetization driven by the electron bath via an s-d exchange interaction. The system is schematically illustratedinFig.1, wheretheelectronbathcanbe tuned byanexternalelectricfield. TheHamiltonianofthetotal system is formally written as H=HM+He+Hsd. (1) Here,HMis the Hamiltonian for the magnetization sub- system in terms of the local spin operators /hatwideSi,µat sitei with spin directions µ(=x,y,z);Heis the Hamiltonian oftheelectronsubsystem; Hsddescribesthes-dexchange interaction between the magnetization and the electron,2 where Hsd=J/summationdisplay i,µ/hatwideSi,µ/hatwideσi,µ. (2) Here,/hatwideσi,µrepresents the electron spin operator at site i without /planckover2pi1/2, andJis the exchange coupling strength. Note that we have not specified the forms of HMandHe yet, thus the results below will be quite general. The effect of the s-d exchange interaction Hsdis twofold. On one hand, the magnetization dynamics is driven by the electron bath via Hsd; on the other hand, the electron states are also affected by the magnetization configuration in turn due to Hsd. Since the time scale of the electron dynamics is usually much faster than that of the magnetization dynamics, we may assume that the electrons under the bias voltage establish a stationary non-equilibrium distribution in a very short time inter- val, during which the change of the magnetization con- figuration is negligible and the non-equilirium electron bath is approximated to be constant. The validity of this assumption only holds if the spin-lattice interaction is stronger than the s-d exchange interaction to relax the electron spin. Consider a short time interval [ t0,t], where the initial density matrices of the magnetization and the electron bath are ρM(t0) andρe(t0) respectively. Then the initial magnetization configuration at each site isSi,µ(t0) = Tr[/hatwideSi,µρM(t0)], and the initial electron den- sity matrix ρe(t0) is determined by the bath Hamiltonian HB=He+J/summationtext i,µSi,µ(t0)/hatwideσi,µand the open boundary conditions. In order to investigate the magnetization dynamics during the time interval [ t0,t] defined above, we rede- fine the local spin operators /hatwideSi,µ=Si,µ(t0) +/hatwidesi,µ, then the Hamiltonian Hin Eq. (1) can be rewritten as H=HM+HB+Vsd, (3) with the interaction term Vsd=J/summationdisplay i,µ/hatwidesi,µ/hatwideσi,µ. (4) During this time interval, the electron density matrix ρe may be approximated to be constant because of the neg- ligible change of the magnetization, and this can be jus- tified in the limit t→t0. Assuming the total density matrix as ρ(t) =ρM(t)⊗ρe(t0) and to the second or- der of interaction strength, the equation for the density matrix/tildewideρM(t) in the interaction picture is[15] d dt/tildewideρM(t) =J i/planckover2pi1/summationdisplay i,µσi,µ(t)[/tildewidesi,µ(t),/tildewideρM(t0)] +(J i/planckover2pi1)2/summationdisplay i,µ;j,ν/integraldisplayt t0dτ{Ci,µ;j,ν(t,τ)[/tildewidesi,µ(t),/tildewidesj,ν(τ)/tildewideρM(τ)] −Cj,ν;i,µ(τ,t)[/tildewidesi,µ(t),/tildewideρM(τ)/tildewidesj,ν(τ)]}. (5) Here,/tildewider···denotes the operators in the interaction picture; the electron spin polarization is σi,µ(t) =Tre[/tildewideσi,µ(t)/tildewideρe(t0)]; the electron spin-spin correlation func- tion isCi,µ;j,ν(t,τ) = Tr e[/tildewideσi,µ(t)/tildewideσj,ν(τ)/tildewideρe(t0)], which is a function of t−τonly and satisfies the relation Ci,µ;j,ν(t,τ) =C∗ j,ν;i,µ(τ,t). In priciple, the solution of Eq. (5) gives the density matrix of the magnetization in the time interval [ t0,t] under the influence of the electron bath, and can be applied to study the physical qualities that we are particularly interested in. Based on Eq. (5), the dynamical equation for Sl,λ(t) = TrM[/tildewideSl,λ(t)/tildewideρM(t)] is obtained as d dtSl,λ(t) =1 i/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J /planckover2pi1/summationdisplay µ,νǫλµνσl,µ(t)Sl,ν(t) +i(J i/planckover2pi1)2/summationdisplay j,µ,ν,ξǫλµν/integraldisplayt t0dτ{Cl,µ;j,ξ(t,τ)/an}b∇acketle{t/hatwideSl,ν(t)/hatwidesj,ξ(τ)/an}b∇acket∇i}htτ −Cj,ξ;l,µ(τ,t)/an}b∇acketle{t/hatwidesj,ξ(τ)/hatwideSl,ν(t)/an}b∇acket∇i}htτ}. (6) Here,/an}b∇acketle{t···/an}b∇acket∇i}htt≡TrM[···ρM(t)], and the spin commutation relation [/hatwideSl,λ,/hatwideSi,µ] =iδli/summationtext νǫλµν/hatwideSl,νhas been exploited. The first term in the r.h.s.of Eq. (6) gives the intrin- sic magnetization dynamics due to HM; the second term is the spin torque term due to the accumulation of the electron spin density; the third term gives the damping effect of the electron bath. If the operator /hatwideSl,ν(t) in the damping term is approximately replaced by its expecta- tion value Sl,ν(t), Eq. (6) becomes d dtSl,λ(t) =1 i/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J /planckover2pi1/summationdisplay µ,νǫλµνσl,µ(t)Sl,ν(t) +2J2 /planckover2pi12/summationdisplay j,µ,ν,ξǫλµνSl,ν(t)/integraldisplayt t0dτKl,µ;j,ξ(t−τ)sj,ξ(τ),(7) whereKl,µ;j,ξ(t−τ) is the imaginary part of Cl,µ;j,ξ(t,τ), andsj,ξ(τ) =/an}b∇acketle{t/hatwidesj,ξ/an}b∇acket∇i}htτ. We introduce the kernel func- tionγ(t) which satisfies the relation dγl,µ;j,ξ(t)/dt= Kl,µ;j,ξ(t). The integral in the last term in Eq. (7) is rewrittenas/integraltextt t0dτγl,µ;j,ξ(t−τ)˙Sj,ξ(τ) afterintegratingby parts and neglecting the boundary terms in the limiting caset→t0. It can be further simplified as Γ l,µ;j,ξ˙Sj,ξun- der the Markovian approximation ˙Sj,ξ(τ)≈˙Sj,ξ(t), with the coefficient Γ l,µ;j,ξ=/integraltextδt 0dτγl,µ;j,ξ(τ) forδt=t−t0. Based on the discussions above, Eq. (7) can be written in a compact form d dtSl(t) =1 i/planckover2pi1/an}b∇acketle{t[/hatwideSl,HM]/an}b∇acket∇i}htt+γeBl(t)×Sl(t),(8) whereγeis the gyromagnetic ratio; Blis the effective magnetic field on the the local spin Sloriginating from the electron bath. The µ-component of Blis expressed as Bl,µ(t) =J γe/planckover2pi1σl,µ(t)+2J2 γe/planckover2pi12/summationdisplay j,ξΓl,µ;j,ξ(t)˙Sj,ξ(t).(9) The first term in (9) will give the torque term due to the electron spin accumulation, which has been discussed ex- tensively in previous studies; the second term will give3 the damping effect of the electron bath on the magne- tization dynamics, which only emerges in the quantum treatment. The non-local feature of the damping term can be found here, which depends on the spatial correla- tion of Γ l,µ;j,ξ. So far we have established a general dynamical equa- tion for the magnetization when it is coupled to the elec- tron bath via s-d exchange interaction. Here, both the Hamiltonian for the magnetization subsystem HMand the Hamiltonian for the electron subsystem Hehave not been specified yet. The treatment is similar to the previ- ous work on the order parameter dynamics in the photo- excited Peierls chain[14]. In the next section, we apply this general formula to study the magnetization dynam- ics of a two-dimensional ferromagnetic thin film under the influence of an electron gas with Rashba SOI, i.e. a model system for “spin-orbit torque”. III. SPIN-ORBIT TORQUE A. Electron Bath with Rashba SOI We consider a special system studied by Manchon and Zhang[3] for the spin-orbit torque. The two-dimensional magnetization thin film in x-y plane consists of N= M×Nlattice sites with the lattice constant a, and we will use the discrete notations in both real and reciprocal space. The magnetization is assumed to be uniform due to strong exchange interaction. The lack of inversion symmetry in z-direction induces the Rashba spin-orbit interaction in the two-dimensional electron gas. In this case, the Hamiltonian for the electron bath is given as[3] HB=/hatwidep2 2m∗e+αR /planckover2pi1(/hatwidep×/hatwidez)·/hatwideσ+JS·/hatwideσ,(10) where/hatwidepis the electron momentum operator; m∗ eis the effective mass of electrons; αRis the Rashba interaction strength; S=Siis the localized spin at each site. For S=S(sinθcosφ,sinθsinφ,cosθ), the energy dispersion relation of the electron is Ek,±=/planckover2pi12k2 2m∗e±∆k. (11) Here, we have denoted the electron wavevector k= k(cosϕ,sinϕ), and ∆k=/radicalBig J2S2+α2 Rk2−2JSαRksinθsin(φ−ϕ). The corresponding electron eigenstates |k,±/an}b∇acket∇i}htare |k,±/an}b∇acket∇i}ht=1√ Neik·r/parenleftBigg cosΘk,± 2e−iΦk sinΘk,± 2/parenrightBigg ,(12)where the angles Θ k,±and Φ kare determined by cosΘk,± 2=/radicalbig ∆2 k−J2S2cos2θ/radicalbig 2∆2 k∓2JS∆kcosθ, sinΘk,± 2=±∆k−JScosθ/radicalbig 2∆2 k∓2JS∆kcosθ, cosΦk=JSsinθcosφ+αRksinϕ/radicalbig ∆2 k−J2S2cos2θ, sinΦk=JSsinθsinφ−αRkcosϕ/radicalbig ∆2 k−J2S2cos2θ. Thespinpolarizationvectorforthestate |k,±/an}b∇acket∇i}htisPk,±= (sinΘk,±cosΦk,sinΘk,±sinΦk,cosΦk,±). The statistical properties of the electron bath are de- termined by the probability distribution function fk,sfor the state |k,s=±/an}b∇acket∇i}ht, which can be tuned by the external field. If an electric field Eis applied, the non-equilibrium distribution of the electron states will be established due to the random scattering potential by impurities[3]. The distribution function fk,sis determined by the Boltz- mann equation −eE /planckover2pi1·∇kfk,s=Sc[fk,s]. (13) The collision integral Sc[fk,s] describes the relaxation of the occupied state |k,s/an}b∇acket∇i}htand can be treated by the relax- ation time approximation, namely, Sc[fk,s] =−fk,s−f0 k,s τ. (14) Here,f0 k,sis the equilibrium distribution function, and an isotropic relaxation time τhas been assumed[3]. To the first orderofthe electric field, the solution of Eq. (13) isfk,s=f0 k,s+gk,s, where the out of equilibrium part induced by the external electric field is gk,s=∂f0 k,s ∂EeE·vk,sτ, (15) with the electron velocity vk,s=1 /planckover2pi1∇kEk,s. Such a treat- ment of the non-equilirium electron distribution was also exploited in the previous semiclassical theory[3]. B. Electron Spin Polarization and Torque With the non-equilibrium distribution function fk,s given above, the electron spin polarization σl,µat sitel and the correlationfunction Cl,µ;j,ξ(t,τ) in Eq. (9) can be calculated, and the torque and damping effect due to the electron bath can be obtained. In the second quantiza- tion representation of the basis set {|k,s/an}b∇acket∇i}ht}, the operator /hatwideσl,µis expressed as /hatwideσl,µ=1 N/summationdisplay k,s;k′,s′χµ k,s;k′,s′ei(k′−k)·rl/hatwidec† k,s/hatwideck′,s′,4 where the matrix element χµ k,s;k′,s′= (cosΘk,s 2eiΦk,sinΘk,s 2)σµ/parenleftBigg cosΘk′,s′ 2e−iΦk′ sinΘk′,s′ 2/parenrightBigg . Then the electron spin polarization σl,µis σl,µ=1 N/summationdisplay k,sχµ k,s;k,sfk,s=1 N/summationdisplay k,sPµ k,sfk,s.(16) For the physically relevant case αRk≪JS, the approx- imate value of Pk,±to the first order ofαRk JSis Pk,±=± Sx+αR JSSxSykx+αR JS(1−S2 x)ky Sy−αR JS(1−S2 y)kx−αR JSSxSyky Sz+αR JSSySzkx−αR JSSxSzky . Here, the unit vector for the magnetization is denoted as S= (sinθcosφ,sinθsinφ,cosφ). For the electric current density je=je(cosϑ,sinϑ,0), the non-equilibrium spin polarization δσlwhich is per- pendicular to Sis calculated to be (Appendix A) δσl=−αRm∗ ejea3 e/planckover2pi1Ef cosϑSxSy+sinϑ(1−S2 x) −cosϑ(1−S2 y)−sinϑSxSy cosϑSySz−sinϑSxSz , whereEfdenotes the Fermi energy. The torque Tlis then obtained as Tl=JSαRm∗ ejea3 e/planckover2pi12Ef cosϑSz sinϑSz −cosϑSx−sinϑSy =JαRm∗ ea3 e/planckover2pi12Ef(/hatwidez×je)×Sl. This result reproduces the form of SOT obtained before[3], but the magnetization vector is not restricted in two-dimensional x-y plane in our derivations. It is easily understood from the effective Hamiltonian (10), where the non-equilibrium distribution of electron states will produce an extra electron spin polarizationalong the direction/hatwidez×je. C. Correlation Function and Damping We now calculate the correlation function Cl,µ;j,ξ(t,τ), which gives the damping term for the magnetization dynamics due to the electron bath. Since /hatwideck,s(t) = /hatwideck,se−iEk,st//planckover2pi1, the correlation function Cl,µ;j,ξ(t,τ) is for- mally written as Cl,µ;j,ξ(t,τ) =1 N2/summationdisplay k,s;k′,s′/summationdisplay k′′,s′′;k′′′,s′′′ei(k′−k)·rlei(k′′′−k′′)·rj ×ei(Ek,s−Ek′,s′)t//planckover2pi1ei(Ek′′,s′′−Ek′′′,s′′′)τ//planckover2pi1 ×χµ k,s;k′,s′χξ k′′,s′′;k′′′,s′′′/an}b∇acketle{t/hatwidec† k,s/hatwideck′,s′/hatwidec† k′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}ht.(17)We see that Cl,µ;j,ξ(t,τ) is the function of rl−rjand t−τ, due to the space and time translation invari- ance for the investigated system. For simplicity, we es- timateCl,µ;j,ξ(t,τ) with several approximations. Firstly, we assumethat the phase factor ei(k′−k)·(rl−rj)will cause the cancellation of the summations over kandk′if rl/ne}ationslash=rj, thusCl,µ;j,ξ=Cµξδlj. Secondly, χµ k,s;k′,s′are calculated to the zeroth order ofαRk JSfor the relevant caseαRk≪JS, where the electron spin states are k- independent, i.e. χ±±=±(sinθcosφ,sinθsinφ,cosθ), χ+−= (−cosθcosφ−isinφ,−cosθsinφ+icosφ,sinθ). Furthermore, we calculate the correlation function /an}b∇acketle{t/hatwidec† k,s/hatwideck′,s′/hatwidec† k′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}htwith the electron bath at equi- librium, where the effect of the non-equilibrium electric current induced by the external field will be neglected. This enable us to apply the Wick contraction[16] to sim- plify the calculations. The negligence of the dependence of the damping coefficient on the Rashba SOI and the non-equilibrium electric current is valid if the dynamical equation (8) is kept to the first order of these two factors. With the above approximations, we get Cµξ(t) =1 N2/summationdisplay k,sχµ ssχξ ssfk,s+1 N2/summationdisplay k,s;k′,s′χµ ssχξ s′s′fk,sfk′,s′ +1 N2/summationdisplay k,s;k′,s′ei(Ek,s−Ek′,s′)t//planckover2pi1χµ ss′χξ s′sfk,s(1−fk′,s′), where|k,s/an}b∇acket∇i}htand|k′,s′/an}b∇acket∇i}htare different states. Since the kernel function γl,µ;j,ξ(t) is given by the relation dγl,µ;j,ξ(t)/dt=Kl,µ;j,ξ(t), where Kl,µ;j,ξ(t) = ℑ[Cl,µ;j,ξ(t)], their Fourier transformations are related by γl,µ;j,ξ(ω) =i ωKl,µ;j,ξ(ω). The Fourier transformation of Kl,µ;j,ξ(t) is obtained as (Appendix B) Kl,µ;j,ξ(ω) =δlj(m∗ ea2 2π/planckover2pi12)2/planckover2pi1 2i/summationdisplay s,s′[χµ ss′χξ s′sgs(ω)−(χµ ss′χξ s′s)∗gs(−ω)], where the function gs(ω) is defined as gs(ω) = 0,/planckover2pi1ω <0; /planckover2pi1ω ,0</planckover2pi1ω < E f−sJS; Ef−sJS , /planckover2pi1ω > E f−sJS. Then the damping kernel function γl,µ;j,ξ(t) can be calculated by the inverse Fourier transformation from γl,µ;j,ξ(ω), which results in (Appendix B) γl,µ;j,ξ(t) =δlj(m∗ ea2 2π/planckover2pi1)21 2/summationdisplay s(δµξg− s(t)+is/summationdisplay νǫµξνSνg+ s(t)). (18) Here,g± s(t) =/integraltext+∞ −∞dωg± s(ω)e−iωtandg± s(ω) = 1 /planckover2pi1ω(gs(ω)±gs(−ω)), as schematically shown in Fig. 2.5 Then the coefficient Γ l,µ;j,ξin Eq. (9) is obtained as Γl,µ;j,ξ=δlj(m∗ ea2 2π/planckover2pi1)2(Γ(1)δµξ+Γ(2)/summationdisplay νǫµξνSν),(19) with Γ(1)=1 2/summationtext s/integraltextδt 0dτg− s(τ) and Γ(2)= i 2/summationtext ss/integraltextδt 0dτg+ s(τ). Then the damping part in Eq. (8) can be explicitly written as Dl= 2(Jm∗ ea2 2π/planckover2pi12)2(Γ(1)˙Sl×Sl+SΓ(2)˙Sl),(20) which is independent of the Rashba constant and the electric current due to our approximations above. 0−1−0.500.51 hωgs+(ω)(a) Ef+JSs = −s = + Ef−JS 000.51 hωgs−(ω)(b) s = + s = −Ef−JSEf+JS FIG. 2. (Color online). Schematic diagram for g± s(ω). Blue line fors= +, and red line for s=−. Notice that g+ sis an odd function of ωandg− s(ω) is an even function of ω, and they approach to 0 when |ω| → ∞. The first term in (20) will give the damping effect which drivesthe local spin towardsthe direction with the lower energy; while the second term in (20) will give a renormalized factor in Eq. (8). Assuming that J∼1 eV, m∗ e∼me,a∼1˚A, one gets the rough estimation of the magnitudeorderforthe factor(Jm∗ ea2 2π/planckover2pi12)2∼10−3, thusthe damping effect due to the electron bath is comparable to the intrinsic Gilbert damping of some ferromagnetic ma- terials. This damping effect can become important to understand the dissipative features of the magnetization dynamics driven by spin-orbit torque. IV. CONCLUSION In conclusion, we have applied density matrix tech- nique to formulate the magnetization dynamics of a sys- tem consisting of local magnetic moments influenced by an electron gas through s-d exchange interaction. In this approach, the magnetic subsystem is treated as an open quantum system and the electron gas acts as a non- equilibrium bath tuned by the external electricfield. The spin torque due to the non-equilibrium electron spin ac- cumulation and the damping effect of the electron bath have been taken into account simultaneously. We ap- ply the developed formula to the model system for spin- orbit torque, where the two-dimensional magnetization film is coupled to the Rashba electron gas through s-dexchange interaction. We have calculated the spin-orbit torque and the results are consistent with the previous study. However, our method does not require the mag- netization direction to be in the two-dimensional plane as in the previous study. Our approach enables us to ob- tain the damping effect due to the electron bath, which is a new feature absent in the semiclassical theory. The damping caused by the electron bath is estimated to be comparableto the intrinsic Gilbert damping, and may be importanttodescribethemagnetizationdynamicsdriven by spin-orbit torque. In brief, this work has extended the previous semiclassical theory for spin-orbit torque to a more complete description. Further applications of this approach are expected to understand and to manip- ulate the magnetization dynamics through electron gas in other complex cases. ACKNOWLEDGMENTS This work was supported in part by the Hong Kong’s University Grant Council via grant AoE/P-04/08. This work is also partially supported by National Basic Re- search Program of China (No. 2014CB921203), NSFC grant (No.11274269), and NSFC grant (No.11204186). Appendix A: Electron Spin Polarization We first assume that the electric field is applied along x-direction, then δσl=1 N/summationdisplay k,sgk,sPk,s=1 N/summationdisplay k(gk,+−gk,−)kxαR JSΣx, whereΣx= (SxSy,−(1−S2 y),SySz). The corresponding electric current density is je=−e Na3/summationdisplay k,sgk,s(vk,s)x≈ −e/planckover2pi1 m∗e1 N/summationdisplay k,sgk,skx, and the spin current density is js=/planckover2pi1 2Na3/summationdisplay k,sgk,s(vk,s)xPk,s ≈/planckover2pi12 2m∗e1 Na3/summationdisplay k(gk,+−gk,−)kxS. Thus a rough relation is obtained as δσl=−αRm∗ ejea3 e/planckover2pi1EfΣx, where the relation js≈ −/planckover2pi1JS 2eEfjeShas been used here. Similarly, if the electric field is applied along the y- direction, the non-equilibrium spin polarization will be δσl=−αRm∗ ejea3 e/planckover2pi1EfΣy,6 withΣy= (1−S2 x,−SxSy,−SxSz). Therefore, for the electric current density je=je(cosϑ,sinϑ,0), we get δσl=−αRm∗ ejea3 e/planckover2pi1Ef cosϑSxSy+sinϑ(1−S2 x) −cosϑ(1−S2 y)−sinϑSxSy cosϑSySz−sinϑSxSz . Appendix B: Correlation Function and Damping Kernel The imaginary part of Cµξ(t) is given as Kµξ(t) =ℑ[Cµξ(t)] = (m∗ ea2 2π/planckover2pi12)2/summationdisplay s,s′ℑ[χµ ss′χξ s′s/integraldisplayEf sJSdǫ/integraldisplay∞ Efdǫ′ei /planckover2pi1(ǫ−ǫ′)t] = (m∗ ea2 2π/planckover2pi12)2/summationdisplay s,s′/integraldisplayEf sJSdǫ/integraldisplay∞ Efdǫ′[−i 2χµ ss′χξ s′sei /planckover2pi1(ǫ−ǫ′)t+h.c.]. Here,fk,sis approximated as the zero-temperature Fermi distribution function, and the relation1 N/summationtext k→ a2 (2π)2/integraltext d2k=m∗ ea2 2π/planckover2pi12/integraltext dǫhas been used. Its Fourier trans-formation Kµξ(ω) is then Kµξ(ω) =1 2π/integraldisplay+∞ −∞dtKµξ(t)eiωt = (m∗ ea2 2π/planckover2pi12)2/summationdisplay s,s′/integraldisplayEf sJSdǫ/integraldisplay∞ Efdǫ′ ×[−i 2χµ ss′χξ s′sδ(ω+ǫ−ǫ′ /planckover2pi1)+i 2(χµ ss′ξξ s′s)∗δ(ω+ǫ′−ǫ /planckover2pi1)] =−(m∗ ea2 2π/planckover2pi12)2i/planckover2pi1 2/summationdisplay s,s′[χµ ss′χξ s′sgs(ω)−(χµ ss′χξ s′s)∗gs(−ω)], where the function g(ω) is defined as gs(ω) = 0,/planckover2pi1ω <0; /planckover2pi1ω ,0</planckover2pi1ω < E f−sJS; Ef−sJS , /planckover2pi1ω > E f−sJS. Therefore, γl,µ;j,ξ(ω) =δlj(m∗ ea2 2π/planckover2pi1)21 2/summationdisplay s,s′[ℜ(χµ ss′χξ s′s)g− s(ω)+iℑ(χµ ss′χξ s′s)g+ s(ω)], whereg± s(ω) =1 /planckover2pi1ω(gs(ω)±gs(−ω)), andγl,µ;j,ξ(t) is cal- culated as γl,µ;j,ξ(t) =/integraldisplay+∞ −∞dωγl,µ;j,ξ(ω)e−iωt =δlj(m∗ ea2 2π/planckover2pi1)21 2/summationdisplay s,s′[ℜ(χµ ss′χξ s′s)g− s(t)+iℑ(χµ ss′χξ s′s)g+ s(t)] =δlj(m∗ ea2 2π/planckover2pi1)21 2/summationdisplay s(δµξg− s(t)+is/summationdisplay νǫµξνSνg+ s(t)) ≈δljδµξ(m∗ ea2 2π/planckover2pi1)2g−(t), whereg± s(t) =/integraltext+∞ −∞dωg± s(ω)e−iωtand we have used the expressions χµ +,+χξ +,+=χµ −,−χξ −,−=SµSξ. χµ +,−χξ −,+= (χµ −,+χξ +,−)∗=δµξ−SµSξ+i/summationdisplay νǫµξνSν. [1] J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). [2] L. Berger, Phys. Rev. B 54, 9353 (1996). [3] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008). [4] A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009). [5] A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna, Y. Lyanda-Geller, and L. P. Rokhinson, Nat. Phys. 5, 656(2009). [6] I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Nat. Mater.9, 230 (2010). [7] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Bl¨ ugel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013). [8] Y.B. Fan, P. Upadhyaya,X.F. Kou, M. R.Lang, S.Takei, Z.X. Wang, J. S. Tang, L. He, L.-T. Chang, M. Montaz-7 eri, G.Q. Yu, W. J. Jiang, T. X. Nie, R. N. Schwartz, Y. Tserkovnyak, and K. L. Wang, Nat. Mater. 13, 699 (2014). [9] P. Gambardella and L. M. Miron, Phil. Trans. R. Soc. A 369, 3175 (2011). [10] A. Brataas and K. M. D. Hals, Nat. Nanotechnol. 9, 86 (2014). [11] Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403 (2012). [12] Y. Wang and L. J. Sham, Phys. Rev. B 87, 174433 (2013).[13] T. Tay and L. J. Sham, Phys. Rev. B 87, 174407 (2013). [14] Y. Wang, W.Q. Chen, and F.C. Zhang, Phys. Rev. B 90, 205110 (2014). [15] K. Blum, Density Matrix Theory and Applications (Springer-Verlag, Berlin Heidelberg, 2012). [16] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems , (McGraw-Hill, New York, 1971).

2014-12-08

The dynamics of magnetization coupled to an electron gas via s-d exchange interaction is investigated by using density matrix technique. Our theory shows that non-equilibrium spin accumulation induces a spin torque and the electron bath leads to a damping of the magnetization. For the two-dimensional magnetization thin film coupled to the electron gas with Rashba spin-orbit coupling, the result for the spin-orbit torques is consistent with the previous semi-classical theory. Our theory predicts a damping of the magnetization, which is absent in the semi-classical theory. The magnitude of the damping due to the electron bath is comparable to the intrinsic Gilbert damping and may be important in describing the magnetization dynamics of the system.

Magnetization Dynamics driven by Non-equilibrium Spin-Orbit Coupled Electron Gas

1412.2479v1

arXiv:1810.05956v1 [math.AP] 14 Oct 2018CRITICAL EXPONENT FOR NONLINEAR DAMPED WAVE EQUATIONS WITH NON-NEGATIVE POTENTIAL IN 3D VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA Abstract. We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impac t on the critical exponent of the corresponding nonlinear Cauchy pr oblem with small initial data. The main new phenomena is that certain relation between these coefficientsmaycauseverystrongjump ofthe crit icalStrauss exponent in 3D to the critical 5D Strauss exponent for the wave eq uation without damping coefficients. 1.Introduction We consider the Cauchy problem for the nonlinear damped wave equa tion with a non-negative potential: (∂2 t+2w(|x|)∂t−∆+V(|x|))U=λ|U|pin (0,T)×R3, (1.1) U(0,x) =εf0(|x|),(∂tU)(0,x) =εf1(|x|) forx∈R3, (1.2) wheref0,f1,w, andVare assumed to be radially symmetric functions in R3. The case without any damping term, i.e. the case when w=V= 0,has been intensively studied for few decades (see [16], [9], [6], [5], [2], [18], [11 ], or references in [3]) and in this case there is a critical nonlinear expone nt known asStrausscritical exponent thatseparatesglobalsmalldataso lutionsandblow - up of the small data solution for finite time. This critical exponent p0(n) is the positive root of p/parenleftbiggn−1 2p−n+1 2/parenrightbigg = 1. In case of presence the damping terms with w(r) =c1/r, V(r) =c2/r2, wherec1,c2>0, one can pose the question if the interaction between damping The first author was supported in part by Project 2017 ”Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari” of INDAM, GNAMPA - Gr uppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni, by Institut e of Mathemat- ics and Informatics, Bulgarian Academy of Sciences and Top Global U niversity Project, Waseda University, by the University of Pisa, Project PRA 2018 49 a nd project ”Dinamica di equazioni nonlineari dispersive”, ”Fondazione di Sardegna” , 2 016. The second author was partially supported by Grant-in-Aid for Scie nce Research (No.16H06339 and No. 26220702), JSPS. The third author was partially supported by Grant-in-Aid for Scient ific Research (No.18H01132), JSPS. 12 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA terms and nonlineare source term can produce appropriate shift o f the Strauss exponent. Forthis, wesupposethat w(r)isapositivedecreasingfunctionin C([0,∞))∩ C1(0,∞) satisfying w(r) =1 rforr≥r0 (1.3) with some positive number r0. First, we recall known results concerning the linear damped wave eq uations, i.e.,V≡0 andλ= 0. It was shown in Mochizuki [14] that if 0 < w(|x|)≤ C(1 +|x|)−αforx∈R3andα >1, then the scattering to the free wave equation occurs in the energy space, without assuming the radial s ymmetry. On the other hand, if w(x)≥C(1+|x|)−αforx∈R3andα≤1, then we see from the work of Matsumura [13] that the energy for the wave eq uation decays to zero as time goes to infinity. In this paper we focus on the border line case α= 1. For the semilinear wave equation with potential (∂2 t−∆+V(x))u=λ|u|pin (0,T)×R3, one can find blow up result in [15] or global existence part in [4]. In the case where the coefficient of the damping term is a function of time variable, D’Abbicco, Lucente and Reissig [1] derived the critical expo nent for the Cauchy problem to /parenleftbigg ∂2 t+2 1+t∂t−∆/parenrightbigg U=λ|U|pin (0,T)×R3, (1.4) byassumingtheradialsymmetry. Indeed, theyprovedthatthep roblemadmits a global solution for sufficiently small initial data if p > p 0(5), and that the solution blows up in finite time if 1 <p<p 0(5). This result can be interpreted as a result of the action of the dampe d term in (1.4) that shifts the critical exponent for small data solutions fr omp0(3) to p0(5). The assumption about the radial symmetry posed in [1] was removed by Ikeda, Sobajima [8] for the blow-up part (actually, they treated m ore general damping term µ(1+t)−1∂tuwithµ>0), and by Kato, Sakuraba [10] and Lai [12] for the existence part, independently. Now we turn back to the case where the coefficient of the damping te rm is a function of spatial variabletime variavles. Ikeda, Sobajima [7] cons idered the Cauchy problem for /parenleftbigg ∂2 t+V0 |x|∂t−∆/parenrightbigg U=λ|U|pin (0,T)×Rn, (1.5) and proved a blow-up result together with the upper bound of the lif espan, provided that 0 <V0<(n−1)2/(n+1),n/(n−1)<p≤p0(n+V0), and that p<(n−2)/(n−4) ifn≥5.CRITICAL EXPONENT FOR NONLINEAR DAMPED WAVE EQUATIONS 3 Weshallstudythecombinedeffectbetweenthedampingandpotent ialterms in this paper, provied the following relation V(r) =−w′(r)+w(r)2forr>0. (1.6) Since we assumed that wis a decreasing function, we see that Vis a non- negative function. Roughly speaking, our result is similar to [1] in the s ense that the critical exponent is shifted from p0(3) top0(5). Another important phenomena, closely related to the space shift o f the crit- ical exponent, is the behavior of the supercritical solution near th e light cone. Indeed, for p>p0(5) we shall see that the far field behavior of U(t,r) is given by U(t,r)/lessorsimilar1 t2, r∈(t/2,t), t→ ∞, so the decay rate for the solution Uto the 3D problem (1.1) is the same as the 5D linear wave equation. Thispaperisorganizedasfollows. Inthesection2,weformulatethe problem and state our results in Theorems 2.1 and 2.2. The section 3 is devote d to the proof of a blow-up result given in Theorem 2.1. In the section 4, we de rive a priori upper bounds and complete the proof of Theorem 2.2. 2.Formulation of the problem and Results Since the Cauchy problem (1.1)-(1.2) is rotationally invariant, we can make the substitution U(t,rω) =u(t,r) rwithr=|x|, ω=x/|x|, and obtain (∂2 t+2w(r)∂t−∂2 r+V(r))u=|u|p/rp−1in (0,T)×(0,∞), (2.1) u(0,r) =εr˜f0(r),(∂tu)(0,r) =εr˜f1(r) forr>0 (2.2) together with the boundary condition u(t,0) = 0 for all t∈(0,T). By the relation (1.6), we have the following factorization of the operator in (2.1): ∂2 t+2w(r)∂t−∂2 r+V(r) = (2.3) =(∂t−∂r+w(r))(∂t+∂r+w(r)) forr>0. This relation (2.3) suggests us to consider the following equations: (2.4) P+v+=g, P −v−=gin (0,∞)×(0,∞) with P±=∂t±∂r+w(r).4 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA SettingW(r) =/integraldisplayr 0w(τ)dτforr≥0, (2.4) can be rewitten as ∂s(eW(r−s)v+(t−s,r−s)) =−eW(r−s)g(t−s,r−s),0≤s≤min{t,r}, ∂s(e−W(r+s)v−(t−s,r+s)) =−e−W(r+s)g(t−s,r+s),0≤s≤t for a fixed ( t,r). Then a simple integration over (0 ,T) gives Lemma 2.1. Lett>0,r>0. Ifv±solves(2.4), then we have v+(t,r) = (2.5) =e−W(r)+W(r−T)v+(t−T,r−T)+/integraldisplayT 0e−W(r)+W(r−s)g(t−s,r−s)ds for0<T≤min{t,r}, and v−(t,r) = (2.6) =eW(r)−W(r+T)v−(t−T,r+T)+/integraldisplayT 0eW(r)−W(r+s)g(t−s,r+s)ds for0<T≤t. Therefore, the mixed initial-boundary valued problem P−P+u=Fin (0,∞)×(0,∞), (2.7) u(t,0) = 0,fort∈(0,∞), u(0,r) =ϕ(r),(∂tu)(0,r) =ψ(r) forr∈(0,∞), has a solution represented via integration over ∆−(t,r) ={(σ,y)∈(0,∞)×(0,∞);|t−r|<σ+y<t+r, σ−y<t−r} with an appropriate kernel E−(t,r,y) =e−W(r)e2W(2−1(y−t+r))e−W(y)fort,r≥0, y≥t−r. (2.8) More precisely, we have the following assertion. Proposition 2.2. Ifusolves(2.7), then for 0<r<twe have u(t,r) =1 2/integraldisplay/integraldisplay ∆−(t,r)E−(t−σ,r,y)F(σ,y)dydσ (2.9) +1 2/integraldisplayt+r t−rE−(t,r,y)(ψ(y)+ϕ′(y)+w(y)ϕ(y))dy. Furthermore, for 0<t<rwe have u(t,r) =1 2/integraldisplay/integraldisplay ∆−(t,r)E−(t−σ,r,y)F(σ,y)dydσ+E−(t,r,r−t)ϕ(r−t)(2.10) +1 2/integraldisplayt+r r−tE−(t,r,y)(ψ(y)+ϕ′(y)+w(y)ϕ(y))dy.CRITICAL EXPONENT FOR NONLINEAR DAMPED WAVE EQUATIONS 5 Proof.Settingu+=P+u, we find P−u+=F, (2.11) P+u=u+. Using (2.6) with T=t, we get u+(t,r) =eW(r)−W(r+t)u+(0,r+t) (2.12) +/integraldisplayt 0eW(r)−W(r+s)F(t−s,r+s)ds. Using (2.5) with T=rorT=t, we find u(t,r) =e−W(r)u(t−r,0)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright =0+/integraldisplayr 0e−W(r)+W(r−s)u+(t−s,r−s)ds for 0<r<t, and u(t,r) =e−W(r)+W(r−t)u(0,r−t)+ +/integraldisplayt 0e−W(r)+W(r−s)u+(t−s,r−s)ds for 0< t < r. Combining these relations, we get (2.9) and (2.10). Indeed, whent>r, we have u(t,r) =/integraldisplayr 0e−W(r)+2W(r−s)−W(r+t−2s)u+(0,r+t−2s)ds +/integraldisplayr 0/integraldisplayt−s 0e−W(r)+2W(r−s)−W(r−s+s′)F(t−s−s′,r−s+s′)ds′ds =1 2/integraldisplayt+r t−rE−(t,r,y)u+(0,y)dy +1 2/integraldisplay/integraldisplay ∆−(t,r)E−(t−σ,r,y)F(σ,y)dydσ. Sinceu+(0,y) =ψ(y) +ϕ′(y)+w(y)ϕ(y), we get (2.9). On the other hand, when 0<t<r, we have u(t,r) =e−W(r)+W(r−t)u(0,r−t) +/integraldisplayt 0e−W(r)+2W(r−s)−W(r+t−2s)u+(0,r+t−2s)ds +/integraldisplayt 0/integraldisplayt−s 0e−W(r)+2W(r−s)−W(r−s+s′)F(t−s−s′,r−s+s′)ds′ds =E−(t,r,r−t)u(0,r−t)+1 2/integraldisplayt+r r−tE−(t,r,y)u+(0,y)dy +1 2/integraldisplay/integraldisplay ∆−(t,r)E−(t−σ,r,y)F(σ,y)dydσ,6 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA which implies (2.10). This completes the proof. /square Remark 2.3. The assumption (1.3)implies eW(r)∼ /an}bracketle{tr/an}bracketri}ht, r>0. (2.13) Then the definition (2.8)ofE−implies E−(t,r,y)∼/an}bracketle{tr−t+y/an}bracketri}ht2 /an}bracketle{tr/an}bracketri}ht/an}bracketle{ty/an}bracketri}ht. (2.14) Remark 2.4. Suppose that (1.3)and(1.6)hold. Ifusolves (∂2 t+2w(r)∂t−∂2 r+V(r))u=|u|p/rp−1in(0,T)×(0,∞), (2.15) u(t,0) = 0 fort∈(0,T), u(0,r) = 0,(∂tu)(0,r) =ψ(r)forr∈(0,∞), then from (2.9)and(2.10)we find the following lower bound: u(t,r)/greaterorsimilar/tildewiderI−(|u|p/yp−1)(t,r)+/tildewiderJ−(ψ)(t,r) (2.16) fort>0,r>0, where we put /tildewiderI−(F)(t,r) =/integraldisplay/integraldisplay ∆−(t,r)/tildewiderE−(t−σ,r,y)F(σ,y)dydσ, (2.17) /tildewiderJ−(ψ)(t,r) =/integraldisplayt+r |t−r|/tildewiderE−(t,r,y)ψ(y)dy (2.18) with /tildewiderE−(t,r,y) =/an}bracketle{tr−t+y/an}bracketri}ht2 /an}bracketle{tr/an}bracketri}ht/an}bracketle{ty/an}bracketri}ht. Now we are in a position to state our blow-up result. Theorem 2.1. Suppose that (1.3)and(1.6)hold. Let ψ∈C0(R3)be a nonzero non-negative function. If 1< p <(3+√ 17)/4 =p0(5), then the classical solution to the problem (2.15)blows up in finite time. Moreover, there exists a positive constant C∗independent of εsuch that T∗(ε)≤/braceleftbiggexp(C∗ε−p(p−1))ifp=p0(5), C∗ε−p(p−1)/(1+3p−2p2)if1<p<p 0(5).(2.19) HereT∗(ε)denotes the lifespan of the problem (2.15). To show the counter part of Theorem 2.1, we introduce an integral equation associated with the problem (2.1)-(2.2): u(t,r) =εu0(t,r)+1 2/integraldisplay/integraldisplay ∆−(t,r)E−(t−σ,r,y)|u(σ,y)|p yp−1dydσ (2.20)CRITICAL EXPONENT FOR NONLINEAR DAMPED WAVE EQUATIONS 7 fort>0,r>0, where we have set u0(t,r) =1 2/integraldisplayt+r |t−r|E−(t,r,y)(ψ(y)+ϕ′(y)+w(y)ϕ(y))dy (2.21) +χ(r−t)E−(t,r,r−t)ϕ(r−t) withϕ(r) =r˜f0(r),ψ(r) =r˜f1(r), whereχ(s) = 1 fors≥0, andχ(s) = 0 for s<0. Theorem 2.2. Suppose that (1.3)and(1.6)hold. Assume p>(3+√ 17)/4 = p0(5). Let˜f0∈C1([0,∞)),˜f1∈C0([0,∞))satisfy |˜f0(r)| ≤ /an}bracketle{tr/an}bracketri}ht−κ−2,|˜f′ 0(r)|+|˜f1(r)| ≤ /an}bracketle{tr/an}bracketri}ht−κ−3forr≥0 (2.22) with some positive constant κ≥2p−3. Then there exists ε0>0so that the corresponding integral equation (2.20)to the problem (2.1)-(2.2)has a global solution satisfying |u(t,r)|/lessorsimilarεr/an}bracketle{tr/an}bracketri}ht−2/an}bracketle{tt−r/an}bracketri}ht−(2p−3), t>0, r>0 for anyε∈(0,ε0]. 3.Blow-up In this section we prove the blow-up result in an analogous manner to [9] (see also [17] and [11]). Our first step in this subsection is to obtain ba sic lower bounds of the solution to the problem (2.15). Lemma 3.1. LetR≥1andp>1. We assume ψ(r)>0,∀r∈(0,R), ψ(r) = 0,∀r≥R. (3.1) Then we have (3.2) /tildewiderJ−(ψ)(t,r)/greaterorsimilarc0 /an}bracketle{tr/an}bracketri}ht, c0:= inf R/2≤r≤2R/3ψ(r) for (3.3) t<r<t +(R/2), t+r>R. Moreover, if uis the solution to (2.15), then we have (3.4) /tildewiderI−(|u|p/yp−1)(t,r)/greaterorsimilarcp 0 /an}bracketle{tr/an}bracketri}ht(t−r)2p−3 for0<t<2randt−r>R. Proof.The estimate (3.2) follows from (2.18) and our choice of ψ.Indeed, if (t,r) are close to the light cone as in (3.3), then we have /tildewiderJ−(ψ)(t,r)/greaterorsimilar/integraldisplayt+r r−t/an}bracketle{tr−t+y/an}bracketri}ht2 /an}bracketle{tr/an}bracketri}ht/an}bracketle{ty/an}bracketri}htψ(y)dy /greaterorsimilarc0/integraldisplay2R/3 R/21 /an}bracketle{tr/an}bracketri}ht/an}bracketle{ty/an}bracketri}htdy/greaterorsimilarc0/an}bracketle{tr/an}bracketri}ht−1,8 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA because of (3.3). The estimate (3.4) follows from (2.17) and (3.2), provided 0 < t <2rand t−r>R. Indeed, with F(σ,y) =|u(σ,y)|p/yp−1we have /tildewiderI−(F)(t,r) =/integraldisplay/integraldisplay ∆−(t,r)/tildewiderE−(t−σ,r,y)|u(σ,y)|p yp−1dydσ. (3.5) Since the domain Σ ={(σ,y)∈(0,∞)×(0,∞);0≤y−σ≤R/2, t−r<σ+y<t+r} is a subset of the integration domain ∆ −(t,r) fort−r>R, we see from (2.16) and (3.2) that u(σ,y)/greaterorsimilarc0/an}bracketle{ty/an}bracketri}ht−1for (σ,y)∈Σ. Therefore, we get /tildewiderI−(F)(t,r)/greaterorsimilarcp 0/integraldisplay/integraldisplay Σ/an}bracketle{t−t+σ+r+y/an}bracketri}ht2 /an}bracketle{tr/an}bracketri}ht/an}bracketle{ty/an}bracketri}ht2pdydσ. Now, introducing the coordinates α=σ+y,β=σ−y, we obtain /tildewiderI−(F)(t,r)/greaterorsimilarcp 0/integraldisplayt+r t−r/an}bracketle{tα−t+r/an}bracketri}ht2 /an}bracketle{tr/an}bracketri}ht/an}bracketle{tα/an}bracketri}ht2pdα, (3.6) becauseβ∼1. Sincet<2r, we havet+r>3(t−r), so that /an}bracketle{tr/an}bracketri}ht/tildewiderI−(F)(t,r)/greaterorsimilarcp 0/integraldisplay3(t−r) t−r(α−t+r)2 /an}bracketle{tα/an}bracketri}ht2pdα /greaterorsimilarcp 0/an}bracketle{tt−r/an}bracketri}ht−2p/integraldisplay3(t−r) t−r(α−t+r)2dα /greaterorsimilarcp 0(t−r)−2p+3, fort−r>R. This completes the proof. /square Forη>0, we introduce the following quantity: /an}bracketle{tu/an}bracketri}ht(η) = inf{/an}bracketle{ty/an}bracketri}ht(σ−y)2p−3|u(σ,y)|: (σ,y)∈Σ(η)}, (3.7) Σ(η) ={(σ,y); 0≤σ≤2y, σ−y≥η}. (3.8) Since we may assume 0 <R≤1, (2.16) and (3.4) yield /an}bracketle{tu/an}bracketri}ht(y)≥C1εpfory≥1. (3.9) We shall show that there exists a constant C2>0 such that (3.10) /an}bracketle{tu/an}bracketri}ht(ξ)≥C2/integraldisplayξ 1/parenleftbigg 1−β ξ/parenrightbigg[/an}bracketle{tu/an}bracketri}ht(β)]p ηpp∗dβ, ξ≥1 for somep∗>0. Letξ≥1 and (t,r)∈Σ(ξ). Forη>0 we set ˜Σ(η,t−r) ={(σ,y);y≥t−r, σ+y≤3(t−r), σ−y≥η}.CRITICAL EXPONENT FOR NONLINEAR DAMPED WAVE EQUATIONS 9 It is easy to see that ˜Σ(η,t−r)⊂∆−(t,r) for anyη>0 and (t,r)∈Σ(ξ) and that (σ,y)∈˜Σ(1,t−r) implies (σ,y)∈Σ(σ−y). Therefore, we have /tildewiderI−(F)(t,r)/greaterorsimilar1 /an}bracketle{tr/an}bracketri}ht/integraldisplay/integraldisplay ˜Σ(1,t−r)(−t+σ+r+y)2 /an}bracketle{ty/an}bracketri}ht|u(σ,y)|p yp−1dydσ /greaterorsimilar(t−r)2 /an}bracketle{tr/an}bracketri}ht/integraldisplay/integraldisplay ˜Σ(1,t−r)[/an}bracketle{tu/an}bracketri}ht(σ−y)]p /an}bracketle{ty/an}bracketri}ht2p(σ−y)p(2p−3)dydσ, because−t+σ+r+y≥ −t+r+(σ−y)+2y≥1+(t−r) for (σ,y)∈˜Σ(1,t−r). Changing the variables by β=σ−y,z=y, we have u(t,r)/greaterorsimilar(t−r)2 /an}bracketle{tr/an}bracketri}ht/integraldisplayt−r 1/parenleftBigg/integraldisplay(3(t−r)−β)/2 t−r[/an}bracketle{tu/an}bracketri}ht(β)]p /an}bracketle{tz/an}bracketri}ht2pβp(2p−3)dz/parenrightBigg dβ /greaterorsimilar1 /an}bracketle{tr/an}bracketri}ht(t−r)2p−2/integraldisplayt−r 1t−r−β 2[/an}bracketle{tu/an}bracketri}ht(β)]p βp(2p−3)dβ /greaterorsimilar1 /an}bracketle{tr/an}bracketri}ht(t−r)2p−3/integraldisplayt−r 1/parenleftbigg 1−β t−r/parenrightbigg[/an}bracketle{tu/an}bracketri}ht(β)]p βp(2p−3)dβ. Since the function y/ma√sto→/integraldisplayy 1/parenleftbigg 1−β y/parenrightbigg[/an}bracketle{tu/an}bracketri}ht(β)]p βp(2p−3)dβ is non-decreasing, for any ( t,r)∈Σ(ξ), we have /an}bracketle{tr/an}bracketri}ht(t−r)p∗u(t,r)≥C2/integraldisplayξ 1/parenleftbigg 1−β ξ/parenrightbigg[/an}bracketle{tu/an}bracketri}ht(β)]p βp(2p−3)dβ, which implies (3.10) with p∗= 2p−3. Now we are in a position to employ Lemma 3.2 below with α=p,β= 0 and κ=p(2p−3). Then we see that /an}bracketle{tu/an}bracketri}ht(y) blows up in a finite time y=T∗(ε), providedpp∗=p(2p−3)≤1. Thelastconditionisequivalent to1 <p≤p0(5). Therefore, the solution of (2.15) blows up in a finite time T∗(ε)≤T∗(ε), if 1< p≤p0(5) and (3.1) hold. Moreover, we have the upper bound (2.19) of the life span T∗(ε). Therefore, we can conclude the proof of Theorem 2.1, provided the following lemma is valid. Although its proof has been given in [11], we shall present it in a compact way in the appendix, for the sake of completeness. Lemma 3.2. LetC1,C2>0,α,β≥0,κ≤1,ε∈(0,1], andp>1. Suppose thatf(y)satisfies f(y)≥C1εα, f(y)≥C2εβ/integraldisplayy 1/parenleftbigg 1−η y/parenrightbiggf(η)p ηκdη, y≥1.10 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA Then,f(y)blows up in a finite time T∗(ε). Moreover, there exists a constant C∗=C∗(C1,C2,p,κ)>0such that T∗(ε)≤/braceleftbigg exp(C∗ε−{(p−1)α+β})ifκ= 1, C∗ε−{(p−1)α+β}/(1−κ)ifκ<1. 4.Small data global existence Our first step is to obtain the following estimates for the homogeneo us part of the solution to the problem (2.20). Lemma 4.1. Suppose that (1.3)and(1.6)hold. Assume that ϕ∈C1([0,∞)), ψ∈C0([0,∞))satisfy |ϕ(r)| ≤C0r/an}bracketle{tr/an}bracketri}ht−(κ+2),|ϕ′(r)|+|ψ(r)| ≤C0/an}bracketle{tr/an}bracketri}ht−(κ+2)forr≥0 (4.1) with some positive constants C0andκ. Then we have (4.2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt+r |t−r|E−(t,r,y)(ψ(y)+ϕ′(y)+w(y)ϕ(y))dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/lessorsimilarC0r /an}bracketle{tr/an}bracketri}ht2/an}bracketle{tt−r/an}bracketri}htκ. fort>0,r>0. Moreover, for 0<t≤rwe have (4.3) |E−(t,r,r−t)ϕ(r−t)|/lessorsimilarC0r /an}bracketle{tr/an}bracketri}ht2/an}bracketle{tt−r/an}bracketri}htκ. Proof.We begin with the proof of (4.2). In the following, let t >0,r >0. Since 0≤r−t+y≤2yfory≥ |t−r|, from (2.14) we have |E−(t,r,y)|/lessorsimilar/an}bracketle{ty/an}bracketri}ht//an}bracketle{tr/an}bracketri}htfory≥ |t−r|. (4.4) Therefore, by using the assumptions on the data, the left hand sid e of (4.2) is estimated by /an}bracketle{tr/an}bracketri}ht−1/integraldisplayt+r |r−t|/an}bracketle{ty/an}bracketri}ht/parenleftbig |ψ(y)|+|ϕ′(y)|+/an}bracketle{ty/an}bracketri}ht−1|ϕ(y)|/parenrightbig dy/lessorsimilarC0/an}bracketle{tr/an}bracketri}ht−1/integraldisplayt+r |r−t|/an}bracketle{ty/an}bracketri}ht−κ−1dy, which leads to (4.2), because the last integral is estimated as follows : /integraldisplayt+r |r−t|/an}bracketle{ty/an}bracketri}ht−κ−1dy/lessorsimilar/an}bracketle{tt−r/an}bracketri}ht−κ,/integraldisplayt+r |r−t|/an}bracketle{ty/an}bracketri}ht−κ−1dy/lessorsimilarr/an}bracketle{tt−r/an}bracketri}ht−κ−1, and/an}bracketle{tr/an}bracketri}ht ∼rforr≥1. Next we prove (4.3), by assuming 0 <t≤r. From (4.4) we have |E−(t,r,y)ϕ(y)|/lessorsimilarC0y /an}bracketle{tr/an}bracketri}ht/an}bracketle{ty/an}bracketri}htκ+1fory≥ |t−r|. Therefore, we have |E−(t,r,r−t)ϕ(r−t)|/lessorsimilarC0(r−t) /an}bracketle{tr/an}bracketri}ht/an}bracketle{tr−t/an}bracketri}htκ+1, which implies (4.3). This completes the proof. /squareCRITICAL EXPONENT FOR NONLINEAR DAMPED WAVE EQUATIONS 11 It follows from (2.21) and Lemma 4.1 that |u0(t,r)|/lessorsimilarεr/an}bracketle{tr/an}bracketri}ht−2/an}bracketle{tr−t/an}bracketri}ht−κfort>0, r>0, (4.5) provided (2.22) holds, because we have set ϕ(r) =r˜f0(r),ψ(r) =r˜f1(r). This estimate suggests us to define the following weighted L∞-norm: (4.6) /bardblu/bardbl= sup (r,t)∈[0,∞)×[0,T]{r−1/an}bracketle{tr/an}bracketri}ht2/an}bracketle{tt−r/an}bracketri}ht2p−3|u(t,r)|}. Our next step is to consider the integral operator appeared in (2.2 0): I−(F)(t,r) =1 2/integraldisplay/integraldisplay ∆−(t,r)E−(t−σ,r,y)F(σ,y)dydσ. (4.7) Lemma 4.2. Ifp>p0(5) = (3+√ 17)/4, then we have (4.8) /bardblI−(F)/bardbl/lessorsimilar/bardblu/bardblp withF(t,r) =|u(t,r)|p/rp−1, and (4.9) /bardblI−(G)/bardbl/lessorsimilar/bardblu−v/bardbl(/bardblu/bardbl+/bardblv/bardbl)p−1 withG(t,r) = (|u(t,r)|p−|v(t,r)|p)/rp−1. Proof.We begin with the proof of (4.8). For ( y,σ)∈∆−(t,r) we havey≥ |t−r−σ|, so that (4.4) yields E−(t−σ,r,y)/lessorsimilar/an}bracketle{tr/an}bracketri}ht−1/an}bracketle{ty/an}bracketri}htfor (y,σ)∈∆−(t,r). Therefore, using the following estimate /an}bracketle{tr/an}bracketri}ht2p/an}bracketle{tt−r/an}bracketri}htp(2p−3)|F(t,r)| ≤r/bardblu/bardblp in (4.7), we get |I−(F)|/lessorsimilar/bardblu/bardblpI(t,r), where we put I(t,r) =/integraldisplay/integraldisplay ∆−(t,r)y /an}bracketle{tr/an}bracketri}ht/an}bracketle{ty/an}bracketri}ht2p−1/an}bracketle{tσ−y/an}bracketri}htp(2p−3)dydσ. First, suppose t≥r. To evaluate the integral, we pass to the coordinates β=σ−y, z=y (4.10) and deduce I(t,r)/lessorsimilar/integraldisplayt−r r−t/integraldisplay(t+r−β)/2 (t−r−β)/21 /an}bracketle{tr/an}bracketri}ht/an}bracketle{tβ/an}bracketri}htp(2p−3)/an}bracketle{tz/an}bracketri}ht2p−2dzdβ (4.11) +/integraldisplayr−t −t−r/integraldisplay(t+r−β)/2 −β1 /an}bracketle{tr/an}bracketri}ht/an}bracketle{tβ/an}bracketri}htp(2p−3)/an}bracketle{tz/an}bracketri}ht2p−2dzdβ.12 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA Letr≥1. Noting that t−r−β >0 forβ <t−r, and−β >0 forβ <r−t, we get /an}bracketle{tr/an}bracketri}htI(t,r)/lessorsimilar/integraldisplayt−r r−t1 /an}bracketle{tβ/an}bracketri}htp(2p−3)/an}bracketle{tt−r−β/an}bracketri}ht2p−3dβ +/integraldisplayr−t −t−r1 /an}bracketle{tβ/an}bracketri}htp(2p−3)/an}bracketle{tβ/an}bracketri}ht2p−3dβ, sincep>p0(5)>3/2. Splitting the integral at β= (t−r)/2 in the first term of the right hand side, we obtain /an}bracketle{tr/an}bracketri}htI(t,r)/lessorsimilar1 /an}bracketle{tt−r/an}bracketri}htp(2p−3)/integraldisplayt−r (t−r)/21 /an}bracketle{tt−r−β/an}bracketri}ht2p−3dβ +1 /an}bracketle{tt−r/an}bracketri}ht2p−3/integraldisplay(t−r)/2 −t−r1 /an}bracketle{tβ/an}bracketri}htp(2p−3)dβ ≡I1+I2. Sincep>p0(5) is equivalent to p(2p−3)>1, it is clear that /an}bracketle{tt−r/an}bracketri}ht(2p−3)I2/lessorsimilar1. (4.12) On the other hand, it follows that /an}bracketle{tt−r/an}bracketri}htp(2p−3)I1/lessorsimilar 1 if p>2, log(2+t−r) ifp= 2, /an}bracketle{tt−r/an}bracketri}ht4−2pifp<2, so that /an}bracketle{tt−r/an}bracketri}ht(2p−3)I1/lessorsimilar1, (4.13) because 4 −2p−(p−1)(2p−3) = 1−p(2p−3)<0 forp>p0(5). Combining (4.12) with (4.13), we obtain for r≥1 I(t,r)/lessorsimilarr/an}bracketle{tr/an}bracketri}ht−2/an}bracketle{tt−r/an}bracketri}ht−(2p−3). (4.14) When 0<r≤1, we see from (4.11) that /an}bracketle{tr/an}bracketri}htI(t,r)/lessorsimilarr/integraldisplayt−r r−t1 /an}bracketle{tβ/an}bracketri}htp(2p−3)/an}bracketle{tt−r−β/an}bracketri}ht2p−2dβ+/integraldisplayr−t −t−r1 /an}bracketle{tβ/an}bracketri}ht2p−3dβ /lessorsimilarr(/an}bracketle{tt−r/an}bracketri}ht−p(2p−3)+/an}bracketle{tt−r/an}bracketri}ht−(2p−2))+r/an}bracketle{tt−r/an}bracketri}ht−(2p−3) forp>p0(5)>3/2. Thus we get (4.14) for 0 <r≤1. Next, suppose 0 <t<r. Then the change of variables (4.10) gives I(t,r)/lessorsimilar/integraldisplayt−r −t−r/integraldisplay(t+r−β)/2 −β1 /an}bracketle{tr/an}bracketri}ht/an}bracketle{tβ/an}bracketri}htp(2p−3)/an}bracketle{tz/an}bracketri}ht2p−2dzdβ /lessorsimilar/integraldisplayt−r −t−r1 /an}bracketle{tr/an}bracketri}ht/an}bracketle{tβ/an}bracketri}htp(2p−3)/an}bracketle{tβ/an}bracketri}ht2p−3dβ,CRITICAL EXPONENT FOR NONLINEAR DAMPED WAVE EQUATIONS 13 since−β >0 forβ <t−r, andp>3/2. Whenr≥1, we get /an}bracketle{tr/an}bracketri}htI(t,r)/lessorsimilar/an}bracketle{tt−r/an}bracketri}ht−(2p−3), sincep > p 0(5)>3/2. On the other hand, when 0 < r≤1, we use the following estimate: /an}bracketle{tr/an}bracketri}htI(t,r)/lessorsimilart/an}bracketle{tt−r/an}bracketri}ht−p(2p−3)−(2p−3)/lessorsimilarr/an}bracketle{tt−r/an}bracketri}ht−(2p−3). These estimates leads to (4.14), and hence (4.8) holds. In order to prove (4.9), it suffices to notice the following estimate: /an}bracketle{tr/an}bracketri}ht2p/an}bracketle{tt−r/an}bracketri}htp(2p−3)|G(r,t)| ≤pr/bardblu−v/bardbl(/bardblu/bardbl+/bardblv/bardbl)p−1, because the remaining part of the proof is the same as before. This completes the proof. /square Proof of Theorem 2.2. If we define a sequence {un}∞ n=−1by un+1(t,r) =εu0(t,r)+I−(|un|p/rp−1) fort>0, r>0, (4.15) withu−1≡0, then (4.5) and Lemma 4.2 shows that it is a Cauchy sequence in X={u∈C([0,∞)×[0,∞));/bardblu/bardbl<∞} for sufficiently small ε. Thus, we get a solution to the integral equation (2.20). This completes the proof. /square Appendix A: Proof of Lemma 3.2 First, we consider the case κ= 1. We put F(z) = (C1εα)−1f(exp(ε−µz)), µ= (p−1)α+β. Then we have F(z)≥1, F(z)≥Cp−1 1C2/integraldisplayz 0/parenleftBig 1−e−ε−µ(z−ζ)/parenrightBig F(ζ)pdζ, z≥0. Since the function z/ma√sto→(1−e−z) is increasing on [0 ,∞), for 0< ε≤1, we obtain (A.1)F(z)≥1, F(z)≥Cp−1 1C2/integraldisplayz 0/parenleftbig 1−e−(z−ζ)/parenrightbig F(ζ)pdζ, z≥0. By Lemma A.3 below, we can conclude that F(z) blows up in a finite time and the desired estimates of the lifespan hold for the case κ= 1. Next, we consider the case κ<1. If we put G(z) = (C1εα)−1f(ε−νez), ν=(p−1)α+β 1−κ,14 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA then we have G(z)≥1, G(z)≥Cp−1 1C2ε(p−1)α+β/integraldisplayz 0/parenleftbig 1−e−(z−ζ)/parenrightbigG(ζ)p (ε−νeζ)κ−1dζ forz≥0 and 0< ε≤1, which implies (A.1), because eζ≥1 forζ≥0. Therefore, what we have to do is to show the following lemma. Lemma A.3. LetC >0andp>1. Suppose that f(t)satisfies (A.2) f(t)≥1, f(t)≥C/integraldisplayt 0(1−e−(t−τ))f(τ)pdτ for anyt≥0. Then,f(t)blows up in a finite time. Proof.By (A.2), for t≥1, we have f(t)≥C/integraldisplayt 0(1−e−(t−τ))dτ=C(t−1+e−t) ≥C(t−1). We put (A.3)A1= exp/parenleftBigg 1+logγ p−1+2∞/summationdisplay j=1logj pj/parenrightBigg , γ= max{2 C(1−e−1),1}. Then, there exists T1>1 such that f(t)≥A1for anyt≥T1. Now, we define sequences {Ak}and{Tk}by (A.4) Ak+1=Ap k γk2, Tk+1=Tk+2 k2, k∈N. Then, for any k∈N, we see that f(t)≥Akfort≥Tk. Indeed, by (A.2), for t≥Tk+2k−2, we have f(t)≥C/integraldisplayt t−k−2(1−e−(t−τ))f(τ)pdτ ≥CAp k/integraldisplayk−2 0(1−e−σ)dσ≥C(1−e−1) 2Ap kk−2, because 1 −e−σ≥(1−e−1)σfor 0≤σ≤1. 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Georgiev, Semilinear hyperbolic equations , second ed., MSJ Memoirs, vol. 7, Mathe- matical Society of Japan, Tokyo, 2005, With a preface by Y. Shibat a. MR 2145150 4. V. Georgiev, Ch. Heiming, and H. Kubo, Supercritical semilinear wave equation with non-negative potential , Comm.PartialDifferentialEquations 26(2001),no.11-12,2267– 2303. MR 1876418 5. V. Georgiev, H. Lindblad, and Ch. Sogge, Weighted Strichartz estimates and global existence for semilinear wave equations , Amer. J. Math. 119(1997), no. 6, 1291–1319. MR 1481816 6. R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equati ons, Math. Z. 177(1981), no. 3, 323–340. MR 618199 7. M. Ikeda and M. Sobajima, Life-span of blowup solutions to semilinear wave equation with space-dependent critical damping , arXiv:1709.04401 (2017). 8. ,Life-span of solutions to semilinear wave equation with tim e-dependent critical damping for specially localized initial data , Mathematische Annalen (2018), 1–24. 9. F. John, Blow-up of solutions of nonlinear wave equations in three sp ace dimensions , Manuscripta Math. 28(1979), 235–268. 10. M. Kato and M. Sakuraba, Global existence and blow-up for semilinear damped wave equations in three space dimensions , arXiv:1807.04327 (2018). 11. H. Kubo and M. Ohta, On the global behavior of classical solutions to coupled sys tems of semilinear wave equations , OperatorTheory Adv. and Appl., vol. 159, Birkhuser Verlag, 2005. 12. N. A. Lai, Weighted L2-L2estimate for wave equation and its applications , arXiv:1807.05109 (2018). 13. A. Matsumura, Energy decay of solutions of dissipative wave equations , Proc. Japan Acad., Ser. A 53(1977), 232–236. 14. K. Mochizuki, Scattering theory for wave equations with dissipative term s, Publ. Res. Inst. Math. Sci. 12(1976), 383–390. 15. W. Strauss and K. Tsutaya, Existence and blow up of small amplitude nonlinear waves with a negative potential , Discrete Contin. Dynam. Systems 3(1997), no. 2, 175–188. MR 1432072 16. W. A. Strauss, Nonlinear wave equations , CBMS Regional Conference Series in Mathe- matics, vol. 73, American Math. Soc., Providence, RI, 1989. 17. H. Takamura, An elementary proof of the exponential blow-up for semi-lin ear wave equa- tions, Mathematical Methods in the Applied Sciences 17(1994), 239–249. 18. Borislav T. Yordanov and Qi S. Zhang, Finite time blow up for critical wave equations in high dimensions , J. Funct. Anal. 231(2006), no. 2, 361–374. MR 219533616 VLADIMIR GEORGIEV, HIDEO KUBO, AND KYOUHEI WAKASA V. Georgiev Dipartimento di Matematica Universit `a di Pisa Largo B. Pontecorvo 5, 56100 Pisa, Italy, and, Faculty of Science and Engineering, Wased a University, 3- 4-1, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan and IMI–BAS , Acad. Georgi Bonchev Str., Block 8, 1113 Sofia, Bulgaria E-mail address :georgiev@dm.unipi.it H. Kubo Department of Mathematics, Faculty of Science, Hokkaido Un iversity, Sap- poro 060-0810, Japan E-mail address :kubo@math.sci.hokudai.ac.jp K.Wakasa Department of Mathematics, Faculty of Science and Technolo gy, Tokyo Uni- versity of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Ja pan. E-mail address :wakasakyouhei@ma.noda.tus.ac.jp

2018-10-14

We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impact on the critical exponent of the corresponding nonlinear Cauchy problem with small initial data. The main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical Strauss exponent in 3D to the critical 5D Strauss exponent for the wave equation without damping coefficients.

Critical exponent for nonlinear damped wave equations with non-negative potential in 3D

1810.05956v1

Large enhancement of spin pumping due to the surface bound states in normal metal/superconductor structures M.A. Silaev1, 2, 3 1Department of Physics and Nanoscience Center, University of Jyv askyl a, P.O. Box 35 (YFL), FI-40014 University of Jyv askyl a, Finland 2Moscow Institute of Physics and Technology, Dolgoprudny, 141700 Russia 3Institute for Physics of Microstructures, Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia We show that the spin pumping from ferromagnetic insulator into the adjacent metallic spin sink can be strongly stimulated by the superconducting correlations. The key physical mechanism responsible for this eect is the presence of quasiparticle surface states at the ferromagnetic insu- lator/superconductor interface. We consider the minimal model when these states appear because of the suppressed pairing constant within the interfacial normal layer. For thin normal layers we obtain a strongly peaked temperature dependence of the Gilbert damping coecient which has been recently observed in such systems. For thicker normal layers the Gilbert damping monotonically increases down to the temperatures much smaller than the critical one. The suggested model paves the way to controlling the temperature dependence of the spin pumping by fabricating hybrid normal metal/superconductor spin sinks. PACS numbers: Introduction Spin transport and spin dynamics in superconductors have attracted signicant attention recently1{7. Quite interesting experimental results have been obtained for the spin pumping eects8{18which in general play the central role in spintronics19{21. It was found that superconducting correlations can lead ei- ther to the signicant suppression8or to the signicant enhancement9{13,17of Gilbert damping (GD) coecient FIG. 1: Schematic setup of the ferromagnetic insulator (FI) lm with the adjacent metallic spin sink consisting of of nor- mal (N) and superconducting (S) layers. The constant ex- ternal magnetic eld is H0x. The magnetization precession m(t) is driven by the external magnetic eld H ei ty. It generates spin current i pumped from F to the spin sink. Upper panel shows the coordinate dependencies of the order parameter
( x) and local density of states N(x) at the energy "= 0:5
0fordN= 0:20,dS= 30,T= 0:7Tc.in systems consisting of superconducting and ferromag- netic layers, such as in the generic example shown in Fig.1. The basic mechanism for changing GD in such systems is the spin pumping eect. This mechanism is based on the spin angular momentum transfer from the ferromagnet into the the adjacent metallic lm via the pumped spin current i(t) generated by the time- dependent magnetization m(t). The spin relaxation in the metallic spin sink leads to the damping-like spin torque and modies the eective GD coecient of the system. In this way the suppression of GD with decreasing temperature T < Tcin systems with superconducting spin sink8can be qualitatively understood as result- ing from the the freezing out of quasiparticles in the superconductor22. However, the strong increase of GD with lowering temperature9{13,17seems to be counter- intuitive and its understanding requires further theoret- ical eorts. In ferromagnetic insulator (FI) /superconductor (S) bi- layers GdN/NbN the peaked behaviour of GD as a func- tion of temperature has been observed13. The maximal GD reached at about T0:7Tcis several times larger than in the normal state =N23, whereis the spin-pumping related change of GD. Because of the several reasons such behaviour cannot be explained23by the coherence peak of spin susceptibility in homogeneous superconductors24. First, this peak occurs at T0:9Tc and for the realistic values of the Dynes parameter25 0:1Tcin NbN its magnitude is23=N0:20:3. Such behaviour is typical for the line widths of nu- clear magnetic resonance26,27and electronic paramag- netic resonance28in superconductors. It is clearly dier- ent from the observed behaviour of GD in FI/S systems13 which has an order of magnitude larger peak =N 23 at signicantly lower temperatures T0:7Tc. In this Letter we suggest a minimal theoretical modelarXiv:2008.06253v1 [cond-mat.supr-con] 14 Aug 20202 which explains the large enhancement of GD in FI/S structures. The key physical mechanism responsible for this eect is the existence of quasiparticle states localized at the FI/S interface. Such states appear due to the sup- pressed pairing within the interfacial normal layer29{32 (N) as illustrated in Fig.1. Shown on top of the Fig.1 are the spatial proles of the order parameter
( x) and the local density of states (DOS) N(x) at the subgap energy"= 0:5
0, where
0is the bulk energy gap at T= 0. The overall N/S lm thickness is dS= 30, where 0=p DS=Tc0is the coherence length, DSis the diu- sion constant in S, Tc0is the bulk critical temperature. Near the interface at x= 0 the DOS is enhanced due to the subgap quasiparticle states which are formed in the N/S structure33{36and occupy the certain energy interval between the bulk gap and Thouless energy DN=d2 Nwhere DNis the diusion coecient and dNis the thickness of N. The existence of surface bound states in N/S struc- tures is demonstrated37in Fig.2a,c where the N(x;") pro- les are shown to have a maximum at x= 0 and energies which depend on dN. The order parameter and DOS in Figs.1,2 are calculated within the Usadel theory38as explained below. In Fig.1 we choose identical diusion coecient in N and S layers DN=DS=Dwhile in Fig.2DN= 0:05DS. At low frequencies 
0the DOS enhancement leads to the increased probability of the magnon ab- sorption by conductivity electrons in the N/S layer. Qualitatively, at a given energy level this probability is determined by number of available states for transi- tionN(")N("+ )N2(") and the dierence of oc- cupation numbers n0("+ )n0(") @"n0where n0(") = tanh("=2T) is the equilibrium distribution func- tion. The product of these factors leads to the energy- resolved magnon absorption probability Pm= N2@"n0. In Fig.2b,d one can that of Pm(") atT= 0:7Tc0is en- hanced at the boundary of N layer x= 0 (red curves) as compared to x=dS(blue curves). Besides that, the localization of surface states is qualitatively equivalent to the decrease of the spin sink volume which and the corresponding increase of the non-equilibrium spin po- larization. As we show by an exact calculation below these mechanisms lead to the large enhancement of spin pumping in the N/S lms. Interestingly, besides explaining the large peak of the spin pumping for dN0the model described above yields also the qualitatively dierent regime with almost monotonic increase of GD down to the temperatures TTc. This behaviour is obtained for dN0when the bound states are pushed down to lower energies as shown in Fig.2c and the absorption probability us en- hanced for quasiparticles with "
0which are not frozen out down to the signicantly low temperatures determined by the Thouless energy TthDN=d2 N. Sim- ilar behaviour of GD has been observed experimentally in Py/Nb/Pt superconducting heterostructures12,17, al- though its physical origin can be dierent. Model of spin pumping To quantify the spin (a) (b) (c) (d) FIG. 2: (a,c) Density of states prole N(";x) in the N/S structure. The position of N/S boundary shown by the dashed line is at (a) dN= 0:20and (c)dN= 0:80.T= 0:7Tc0, = 0:1Tc0,dS= 50,DN= 0:05DS. Plots for other dSare shown in Appendix37. (b,d) Magnon absorption probability Pm(") = @"n0N2for the frequency = 0 :02Tc0, Red and blue curves are taken at x= 0 andx=dS, respectively. Parameters are the same as in (a,b). pumping eect we consider the microscopic model of the spin-dependent scattering of electrons at the FI interface37,39,40. As we show below, it formally yields the spin current identical to the one given by the inter- facial exchange interaction between the localized spins in FI and conduction elections in the adjacent metal41. Within this model the local spin polarization close to the interfaceS(t) acts as eective eld for the localized mag- netic moments. This process can be taken into account by introducing the additional term i(t) into the Landau- Lifshitz-Gilber equation (1 +m)@tm+ mHeff=i=SF0dF (1) i(t) =JsdS(t)m(t) (2) HereSF0is the equilibrium spin density in F, dFis the F lm thickness, Heffis the eective eld and is the intrinsic Gilbert damping coecient. The term i(t) can be interpreted as the spin current between FI and metal. To calculateS(t) we need to nd the spin response of the superconductor to the interfacial exchange eld. In the linear regime it is given by S =heffmm (3) where we introduce the eective exchange eld heff= Jsd=dS, normal metal DOS at the Fermi level and the local spin susceptibility m.3 The spin-pumping related change of the GD is deter- mined by the dissipative part of the susceptibility =CTc0Imm= (4) where the dimensionless coecient determining the cou- pling strength between the FI and metallic lms is23 C=heff Tc0heff SF0dS dF(5) From there one can see that since heff/1=d2 Sthe cou- pling coecient is C/1=dS. Localization of surface states provides the eective decrease of dSwhich leads to the increase of Cand the spin response. Calculation of the time-dependent spin re- sponse. What is left is to calculate the local spin susceptibility min the Eq.4 for the FI/N/S structure in Fig.1. We do so by developing the microscopic ki- netic theory of spin pumping generalizing the quasiclas- sical approach2,40,42,43to the time-dependent situation. The magnetization of conduction electrons is deter- mined by spin accumulation and can be written in terms of the Keldysh quasiclassical Green's function (GF) as S(t) =Tr [^3^gK(t;t)]=8 (6) gKis the (22 matrix) Keldysh component of the qua- siclassical GF matrix  g= ^gR^gK 0 ^gA which depends on two times and a single spatial coordinate variable g= g(t1;t2;r). GF gobeys the Usadel equation f^3@t;ggt+r(Dgrg) =
[^1;g]+[;g][so;g]t:(7) where ^k;^k,k= 0;1;2;3 are Pauli matrices, Dis the diusion coecient. The commutator operator is dened as [X;g]t=X(t1)g(t1;t2)g(t1;t2)X(t2), similarly for anticommutatorf;gt. The symbolic product operator is given by (AB)(t1;t2) =R dtA(t1;t)B(t;t2). Spin relaxation is determined by the spin-orbital scat- tering self energy ^so=
^g=(6so) (8) The self-consistency equation for the gap function is
=Tr[^1^gK]=4 (9) whereis the pairing coecient. In our model we assume the pairing constant to be suppressed in the N region (x<dN) = 0:05(x>dN) as compared to its value in S. We scan over the values of the diusion coecient in the N layer DNwhile keeping it xed in S layer DS. The inelastic scattering is described by the Dynes44param- eter which enters to the Eq.7 as the matrix in Nambu- Keldysh space with ^R;A=^3which described both the DOS singularity broadening and the relaxation of non-equilibrium distribution functions as described be- low. Note that this terms conserves the total spin in ac- cordance with the general property of spin-independent electron-phonon scattering.Eq.7 is supplemented by the dynamical boundary con- ditions atx= 0 describing the spin splitting and pump- ing induced by the electron scattering at the FI inter- face with time-dependent magnetization. These bound- ary conditions are derived37from the spin-dependent scattering matrix ^Sconnecting the incident ^ iand re- ected ^ relectronic waves ^ r=^S(t)^ i. For frequen- cies small compared to the exchange eld in FI we use the adiabatic approximation which yields the expression ^S=ei(m^)^3=2, where  is the time-independent spin- mixing angle. Then, assuming that jj1 and Dg@xg(x= 0) =iJsd[m^3;^g]t (10) wherem=m(t) is the time-dependent magnetiza- tion. Within the minimal band model of the FI39,40 the interfacial exchange constant is expressed through the spin-mixing angle as Jsd=vF 4R1 1d^pxj^pxj(^px), where ^pxis the electron momentum projection on the interface normal. Eq.10 generalizes the static bound- ary condition at the spin-active interface39,40,43,45to the case of time-pendent magnetization. The induced spin current is obtained using the general expression i(t) = DTr[^g@xg](t;t). With the help of Eqs.(10,6) it yields the phenomenological Eq.(2). Introducing the usual parametrization of quasiclassical Keldysh function in terms of the distribution function ^gK= ^gR^f^f^gAwe can identify the terms which are essential to calculate linear response in the low-frequency limit. Expanding the energy representation of ^ gKto the rst order in we obtain the non-equilibrium correction ^gK= (^m ) (^gR 0^gA 0)fh+ @"n0 2(gR h+gA h) (11) where we parametrise the spin-dependent corrections as follows ^f= (^m )fhandgR;A= (^m )gR;A h. In contrast to stationary non-equilibrium situations42 when only the rst term in (11) is important the time- dependent case requires taking into account also the sec- ond term with the corrections of spectral functions23. In the low-frequency limit the calculation is simplies by ne- glecting the frequency dependence of the perturbed spec- tral GF in (11). Using (11) we write the time-dependent spin polarization in the metallic lm as follows S =i m Z1 1d"[2Nfh+ (gR 3h+gA 3h)@"n0] (12) whereN= Tr(^3^gR)=2 is the local DOS and gR;A 3h= Tr(^3^gR;A h)=2 . Equations for zero-order spectral func- tion ^gR;A 0(";x), corrections ^ gR;A h(";x) and the distribu- tion function fh("; ;x) are obtained straightforwardly37 from Eqs.(7, 10). The zero-order GF ^ gR;A 0(";x) are calcu- lated in the N/S structure self-consistently together with the order parameter 9. This gives in particular the
( x) andN(";x) proles shown in Fig.1,2. The corrections fh and ^gR;A hare determined by the linear equation37.4 DN=DS, = 0:1Tc0 dN 0= (a) DN= 0:05DS, = 0:1Tc0 (b) DN=DS, = 0:01Tc0 (c) DN= 0:05DS, = 0:01Tc0 (d) (e) (f) (g) (h) FIG. 3: Upper row: temperature dependencies of the GD (T) in FI/N/S systems. The three curves in each plot correspond todN=0= 0:8; 0:2; 0. Lower row: color plots of the functions (dN;T)=N. Horizontal lines in each panel are positioned as guide for eyes at dN=0= 0:8; 0:2; 0 corresponding to the curves in the upper plot. The four columns correspond to various Dynes parameters =Tc0= 0:1; 0:01 and ratios of diusion coecients in N and S layers DN=DS= 1; 0:05 specied on top of the panels. Common parameters are dS= 30,snTc0= 1, = 0:02Tc0. Results and discussion Using the described formal- ism we calculate the non-equilibrium spin polarization (12) in the N/S structure shown in Fig.1. This gives us the local susceptibility (3) and the excess GD (4 ). The resulting temperature dependencies of (T) are shown in Fig. 3 for various parameters. The rst column in Fig.3 corresponds to = 0 :1Tc0and identical diusion coecients in N ans S layers. In the absence of N layer dN= 0 there is a usual coherence peak at T0:9Tc with the small amplitude =N1:4. Adding the thin N layer with dN>0:10leads to the increase of the peak amplitude to =N1:9 and shifting to lower temperatures. The peak is enhanced by decreasing the diusion coef- cientDNin the normal layer. Qualitatively, this leads to better localization of surface bound states and hence to the increase of surface DOS. As shown in the second column of Fig.3 for DN= 0:05DSand = 0:1Tc0the peak is enhanced to =N2:5 reached at T0:7Tc withdN= 0:20. This behaviour is quite similar to the experimental observation13. For larger dN>0:50the temperature dependence is qualitatively changed to the monotonic increase down to the low temperatures. As shown by the yellow curve with dN= 0:80the increase continues to T0:1Tc. Even larger increase is obtained for smaller Dynes pa- rameters = 0 :01Tc0as shown in the third and fourth columns of the Fig. 3. For DN=DSwe obtain the max-imal value=N= 3. ForDN= 0:05DSwe obtain the maximal value =N= 4:8. For all values of we note that forDNDSthe monotonically increasing (T) is obtained down to the threshold temperature of the order of Thouless ennergy TthDN=d2 N. As one can see in the color plots Fig.3f,h for increasing dNit can be rather smallTthTc. The introduced model can explain the observed spin- pumping enhancement in GdN/NbN system13assuming that there is a naturally formed thin normal layer at the FI/S interface. The pairing suppression at the inter- face can result from various reasons, including magnetic disorder46,47, strong usual disorder48or the band struc- ture modication49. It is straightforward to check our prediction of the enhanced GD by fabricating articial FI/N/S structures with various parameters. The behaviour of (T) obtained in Figs.3b,d with dN= 0:80is qualitatively similar to the one observed ex- perimentally in Py/Nb/Pt heterostructures12,17. In the equilibrium state of our model the spin-triplet supercon- ductivity is absent. Therefore the monotonic increase of GD due to the supercondducting correlations is not in principle an exclusive feature of the system with spin super-currents. However, the spin-triplet correlations are generated in the non-equilibrium case (11) providing23 signicant contribution to the spin response (12). The developed quasiclassical theory of spin pumping can be generalized to the case of metallic ferromagnets5 by introducing the nite spin-dependent tunnelling prob- ability through the F/S interface43,50,51to the boundary condition (10). This provides the way to study charge and heat transport induced by the magnetic precession as well as spin torques induced by voltage and tempera- ture biases52{56. Conclusions We have developed the general formalism to calculate spin-pumping in spatially- inhomogeneous metallic lms with spin-active interfaces. As an example we have considered the FI/N/S structure and found that the the presence of quasiparticle bound states localized near the spin-active interface provides strong enhancement of spin pumping which shows up in the strong increase of the GD coecient with decreasing temperature below Tc. The model explains large peak of GD in Gd/NbN structures and shows the way to controlling spin pumping properties in superconducting systems. Acknowledgements This work was supported by the Academy of Finland (Project No. 297439) and Russian Science Foundation, Grant No. 19-19-00594. I thank Yakov Fominov for comments. Appendix A: Stationary spin-mixing scattering matrix Near the at FI/M surface we write wave functions in the form kkeikkrwherekk=kzz+kyyis the conserved momentum parallel to the interface. Along zcoordinate we have 1D Shrodinger equations i@t = (^H"F?) (A1) ^H=@2 x=2m+ ["F+V+ (m^)Vs](x) (A2) wherem=m(t). Let us rst nd the frozen scattering matrix which de- pends adiabatically on time. In this case the energy of incoming and scattered electrons coincide so that writing /ei"twe get stationary 1D Shrodinger equation ^H = ("+"F?) (A3) ^H=@2 x=2m+ ["F+V0+ (m^)Vs](x) (A4) where"F?="Fk2 k=2m. For the energy we have "= k2=2m"Fwherek2=k2 x+k2 k. First, we nd the scattering matrix writing solutions kk=A+eikxx+Aeikxx(A5) kk=Bex=(A6) where2 = 2mVk2 xandV"(#)=V0+ ()Vsare the spin-up (down) band energies in FI. The re ection coecientS=A+=Ais then S=ei'ei=2=1 +ikx 1ikx(A7)Since we are interested in spin-dependent re ection phase we get the spin-mixing angle ei=1 +k2 x++ikx(+) 1 +k2x+ikx(+)(A8) which yields =2 = arcsin kx(+)p (1 +k2x+)2+k2x(+)2! (A9) Finally, the spin-dependent part of the scattering ma- trix connecting the incident ^ iand re ected ^ relectronic waves written in the basis-independent form ^ r=^S^ i (A10) ^S=ei(m^)^3=2(A11) Appendix B: Time-dependent boundary conditions at the FI/metal interface Here we derive boundary conditions () starting from the scattering theory of the interface between FI and metal, either normal or superconducting one. The main dierence from the previous works deriving boundary conditions at FI/M interface is that the magnetization of FI depends on time m=m(t). We consider matrix GF dened in a Keldysh-Nambu- spin space G(r1;r2;t1;t2) =^GR^GK 0^GA (B1) where retarded, advanced and Keldysh parts are dened in a standard way as follows ^GR(r1;r2;t1;t2) =(t1t2) (B2)h h^ (r1;t1)^ +(r2;t2)i+h^ (r1;t1)^ +(r2;t2)ii ^GA(r1;r2;t1;t2) =(t2t1) (B3)h h^ (r1;t1)^ +(r2;t2)i+h^ (r1;t1)^ +(r2;t2)ii ^GK(r1;r2;t1;t2) = (B4) h^ (r1;t1)^ +(r2;t2)i+h^ (r1;t1)^ +(r2;t2)i where the eld operators ^ = ( ^ ";^ #;^ + #;^ + ") satisfy the equations of motion i@t^ = ^H(t)^ (B5) and the Hamiltonian has time-dependent order parame- ter
=
( t), boundary potential V=V(t) ^H(t) = ^3(k2=2m"F) + ^2
(t) +^V(t) (B6)6 The GF satises Gor'kov equations [i@t1^H(t1;r1)]^G=(t12)(r12) (B7) ^G[i@t2^H(t2;r2)] =(t12)(r12) (B8) wherer12=r1r2andt12=t1t2. Assuming the at FI/M interface we consider transverse momentum com- ponentskz;yas conserved quantities. The perpendicular component kxchanges to the opposite one upon elec- tron re ection. We are interested in the components of GF which are slowly varying as function of the center of mass coordinate r= (r1+r2)=2 and thus can be written as follows Gkk(x1;x2;t1;t2) =Z dr12eikkr12G(r1;r2;t1;t2) The GF satises Gor'kov equations [i@t1^H(t1;z1)]Gkk=(t12)(x12) (B9) Gkk[i@t2^H(t2;x2)] =(t12)(x12) (B10) ^H(t;x) =(@2 x=2m+"?)^3+ ^2
(t) +^V(t) (B11) where"?="Fk2 k=2m. Let's consider the Fourier expansion Gkk(x1;x2) =X k1;2ei(k1x1k2x2)Gkk(k1;k2) (B12) Near the M/FI interface z= 0 we can establish the con- nection between amplitudes Gkk(k1;k2) =^S(t1)Gkk(k1<0;k2) (B13) Gkk(k1;k2) =Gkk(k1;k2<0)^S+(t2) (B14) From these two relations we get Gkk(k1;k2) =^S(t1)Gkk(k1<0;k2<0)^S+(t2) (B15) Relations (B13,B14) can be obtained as follows. First, consider the vicinity of interface jx1;2jwhere= vx=
. In this case we can use the simplied equation for GF neglecting the time derivative and order parameter [(@2 x1=2m+"?)^3^V(t1;x1)]G(x1<x2) = 0 (B16) (@2 x2=2m+"?)G(x2<x1)^3G(x2<x1)^V(t2;x2)] = 0 (B17) These are two independent equations identical to the Shrodinger equation (A3) at "= 0. Thus we can write the solution Gkk(x1<x2) =X k1>0[eik1x1+eik1x1^S(t1)]F2(x2) (B18) Gkk(x2<x1) =X k2>0F1(x1)[eik2x2+eik2x2^S+(t2)] (B19)where ^F1;2(x) in principle can be arbitrary functions. Comparing these relations with the general Fourier ex- pansion (B12) we get Eqs. (B13,B14). The quasiclassical GF in general is introduced accord- ing to the following general procedure gp(r) =1 Z1 1dpZ dr12eipr12^3G(r1;r2) Near the at surface we have only the z-dependence ^gp(x) =1 Z1 1dqeiqxZ1 1dp^3^Gkk(kx+q;kxq) where we denote r12=r1r2,p= (k2 z+k2 k)=2m"F. Then atx= 0 we have gp(x= 0) =1 ZZ1 1dqdp^3Gkk(kxq;kx+q) gp(x= 0) =1 ZZ1 1dqdp^3Gkk(kx+q;kxq) (B20) Then using relations B15 gp(x= 0) =1 ZZ1 1dqdp^3Gkk(kx+q;kxq) = 1 ZZ1 1dqdp^S(t1)^3Gkk(kxq;kx+q)^S+(t2) ^S(t1)^gp(z= 0) ^S+(t2) where in the last relation we assume that ^Sdoes not de- pend onq. Finally we get the time-dependent boundary condition for quasiclassical functions gp(x= 0) = ^S(t1)gp(x= 0) ^S+(t2) (B21) Expanding ^S(t)1 +i^m(t)=2 in Eq.B21 we get the matrix current at the M/FI boundary ^I(t1;t2) =Zd p 4(n
vF)gp(t1;t2) = (B22) vFZ1 0d^px^pz[gp(t1;t2)gp(t1;t2)] ivF 2Z1 0d^px^px(^px)[^m^3;g]t where we denote [ ^X;g]t=^X(t1)g(t1;t2)g(t1;t2)^X(t2), n=zis the normal to FI interface and denote the inci- dent ^p
n<0 and re ected ^p
n>0 momenta. This expression can be simplied even more if we as- sume that due to the impurity scattering the anisotropic parts of GF are small. Then we can use two lowest order terms in the spherical harmonics expansion gp= g+p
ga=p (B23)7 Keeping only the s-wave term we get for the matrix cur- rent (B22) I(t1;t2) =i1Jsd[^3^m;g]t (B24) where the conductance is given by Jsd=vF 4Z1 1d^pxj^pxj(^px) (B25) We can nd the spin current using the general expres- sion i(t) =Tr4[^^IK(t;t)]: (B26) Taking into account the denition of the spin density S(t) =Tr [^3^gK(t;t)]=8 (B27) the spin current B26 owing from FI to the spin sink can be written as i(t) =JsdS(t)m(t) (B28) Appendix C: Equation for the spectral and distribution functions Kinetic equation From the Keldysh-Usadel equation in the main text we obtain the nite-frequency kinetic equation r(Drfh) = [1 so+ 2(2 +i )N]fh (C1) D@xfh(x= 0) =2iheffN@"n0 (C2) @xfh(x=dS) = 0 (C3) whereD=DTr(1^gR^gA)=2 and1 so= 4D=3Dsn. The system (C1, C2, C3) is linear with the coecients de- termined by the zero-order spectral function. Solving it we nd the spin-dependent non-equilibrium distribution function generated by the dynamical spin-active inter- face.Spectral functions In the adiabatic approximation we nd the spectral functions from the stationary Usadel equation i[("+i)^3;^g] +@x(D^g@x^g) =
[^1;^g][^so;^g] (C4) with the boundary conditions D^g@x^g=iJsd[^3^m(t);g] (C5) Using the normalization condition (^ gR)2= 1 we use the following parametrization for equilibrium GF and correc- tions in the low-frequency adiabatic approximation ^gR 0= cos0^3+ sin0^1 (C6) ^gR h= (sin0^3+ cos0^1)h (C7) Then we get the following equations for the parameters 0,h i("+i) sin0+
cos0+@xD 2@x0 = 0 (C8) @x0(x= 0;dS) = 0 (C9) h i("+i) cos02 3so
sin0 +@xD 2@xh = 0 (C10) DN@xh(x= 0) = 2iheffsin0; (C11) @xh(x=dS) = 0 (C12) Solving the nonlinear Eq.(C8,C9) together with the self- consistency equation for
we obtain the zero-order spec- tral functions in the N/S structure. The corresponding DOS proles are shown in Fig.2 and in more detail in Fig.4. Using them we nd the coecients in the linear Eq.(C10,C11,C12) for the correction hwhich yields the perturbation of spectral functions by the spin-active in- terface. 1J. Linder and J. W. A. Robinson, Nat Phys 11, 307 (2015), ISSN 1745-2473, URL http://dx.doi.org/10. 1038/nphys3242 . 2F. S. Bergeret, M. Silaev, P. Virtanen, and T. T. Heikkil a, Rev. Mod. Phys. 90, 041001 (2018), URL https://link. aps.org/doi/10.1103/RevModPhys.90.041001 . 3W. Han, S. Maekawa, and X.-C. Xie, Nature materials pp. 1{14 (2019). 4C. Quay and M. Aprili, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, 20150342 (2018). 5K. Ohnishi, S. Komori, G. Yang, K.-R. Jeon, L. Olde Olthof, X. Montiel, M. Blamire, and J. Robinson, Applied Physics Letters 116, 130501 (2020).6D. Beckmann, Journal of Physics: Condensed Matter 28, 163001 (2016). 7M. Eschrig, Reports on Progress in Physics 78, 104501 (2015), ISSN 0034-4885, URL http://stacks.iop.org/ 0034-4885/78/i=10/a=104501 . 8C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Physical review letters 100, 047002 (2008). 9K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen, S. Komori, J. W. Robinson, and M. G. Blamire, Physical Review B 99, 144503 (2019). 10K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Co- hen, X. Montiel, M. Eschrig, S. Komori, J. W. A. Robinson, and M. G. Blamire, Phys. Rev. B 99, 024507 (2019), URL https://link.aps.org/doi/10.8 FIG. 4: Density of states prole N(";x) in the N/S structures of various lengths. The position of N/S boundary shown by the dashed line is at dN= 0:20, = 0:01Tc0,DN=DS. 1103/PhysRevB.99.024507 . 11K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen, X. Montiel, M. Eschrig, T. Wagner, S. Komori, A. Sri- vastava, J. W. Robinson, et al., Phys. Rev. Applied 11, 014061 (2019), URL https://link.aps.org/doi/10. 1103/PhysRevApplied.11.014061 . 12K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kurebayashi, L. F. Cohen, X. Montiel, M. Eschrig, J. W. A. Robin- son, and M. G. Blamire, Enhanced spin pumping into su- perconductors provides evidence for superconducting pure spin currents (2018), URL https://doi.org/10.1038/ s41563-018-0058-9 . 13Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan, Y. Ma, Y. Yun, X. Xie, J. S. Moodera, and W. Han, Phys- ical Review B 97, 224414 (2018). 14L.-L. Li, Y.-L. Zhao, X.-X. Zhang, and Y. Sun, Chinese Physics Letters 35, 077401 (2018). 15Y. Zhao, Y. Yuan, K. Fan, and Y. Zhou, Applied PhysicsExpress 13, 033002 (2020). 16I. Golovchanskiy, N. Abramov, V. Stolyarov, V. Chichkov, M. Silayev, I. Shchetinin, A. Golubov, V. Ryazanov, A. Ustinov, and M. Y. Kupriyanov, arXiv preprint arXiv:2006.00348 (2020). 17K.-R. Jeon, X. Montiel, S. Komori, C. Ciccarelli, J. Haigh, H. Kurebayashi, L. F. Cohen, A. K. 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2020-08-14

We show that the spin pumping from ferromagnetic insulator into the adjacent metallic spin sink can be strongly stimulated by the superconducting correlations. The key physical mechanism responsible for this effect is the presence of quasiparticle surface states at the ferromagnetic insulator/superconductor interface. We consider the minimal model when these states appear because of the suppressed pairing constant within the interfacial normal layer. For thin normal layers we obtain a strongly peaked temperature dependence of the Gilbert damping coefficient which has been recently observed in such systems. For thicker normal layers the Gilbert damping monotonically increases down to the temperatures much smaller than the critical one. The suggested model paves the way to controlling the temperature dependence of the spin pumping by fabricating hybrid normal metal/superconductor spin sinks.

Large enhancement of spin pumping due to the surface bound states in normal metal/superconductor structures

2008.06253v1

arXiv:2107.04966v1 [math.AP] 11 Jul 2021SPACE-TIME ARITHMETIC QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS TOMOYUKI OKA Abstract. Thispaperisconcernedwithspace-timehomogenizationproblemsfo rdamped wave equations with spatially periodic oscillating elliptic coefficients and t emporally (arithmetic) quasi-periodic oscillating viscosity coefficients. Main res ults consist of a homogenization theorem, qualitative properties of homogenized ma trices which appear in homogenized equations and a corrector result for gradients of s olutions. In particular, homogenized equations and cell problems will turn out to deeply depe nd on the quasi- periodicity as well as the log ratio of spatial and temporal periods of the coefficients. Even types of equations will change depending on the log ratio and qu asi-periodicity. Proofs of the main results are based on a (very weak) space-time t wo-scale convergence theory. 1.Introduction and main results Space-time homogenization problems forhyperbolic equations were first studied byBen- soussan, Lions and Papanicolaou. In [4], based on a method of asymptotic expansion , the following wave equation is treated: (1.1) ∂2 ttuε−div(aε∇uε) =fin Ω×(0,T), where Ωisabounded domainin RNwithsmoothboundary ∂Ω,N≥1,T >0,f=f(x,t) is a given data, a:TN×T→RN×Nis anN×Nsymmetric matrix field satisfying a uniform ellipticity and 1-periodicity and aε:=a(x ε,t εr) forr>0 (i.e.,aεisε×εr-periodic). The homogenization problem concerns asymptotic behavior as ε→0+of (weak) solutions uε=uε(x,t)aswellasarigorousderivationoflimitingequations, oftencalled homogenized equation. In [4], it is assumed that (weak) solutions uε=uε(x,t) can be expanded as a series: (1.2) uε(x,t) =∞/summationdisplay j=0εjuj(x,t,x ε,t εr), whereuj=uj(x,t,y,s) : Ω×(0,T)×TN×T→Rforj= 0,1,2,...are some periodic functions, andthen, bysubstituting (1.2)to(1.1), ataformallev el,u0=u0(x,t)turnsout to be independent of microscopic variable (y,s) and to solve the following homogenized equation: ∂2 ttu0−div(ahom∇u0) =fin Ω×(0,T), whereahomis the so-called homogenized matrix and represented as (1.3)ahomek=/integraldisplay1 0/integraldisplay /squarea(y,s)/parenleftbig ∇yΦk(y,s)+ek/parenrightbig dydsfork= 1,2,...,N. Date: July 13, 2021. 2010Mathematics Subject Classification. Primary : 35B27; Secondary : 80M40, 47J35. Key words and phrases. quasi-periodic space-time homogenization, two-scale convergenc e, very weak two-scale convergence, damped wave equation, hyperbolic-para bolic equation. 12 TOMOYUKI OKA Here/square:= (0,1)Nis a unit cell, ∇ystands for the gradient operator with respect to the third variable y,{ek}={[δjk]j=1,2,...,N}stands for a canonical basis of RNand Φ k: TN×T→R(fork= 1,2,...,N) is thecorrector which will be explained latter (see Remark 1.12 below). Moreover, Φ kis determined by the so-called cell problems . In particular, if the log-ratio of the spatial and temporal periods of t he coefficients is the hyperbolic scale ratio (i.e., r= 1), then the cell problem is also a wave equation. ∂2 ssΦk−divy/bracketleftbig a(y,s)(∇yΦk+ek)/bracketrightbig = 0 in TN×T (otherwise, cell problems are always elliptic equations, e.g., (1.17) be low). In [4], the following heat equation is also treated: (1.4) ∂tuε−div(aε∇uε) =fin Ω×(0,T). By substituting (1.2) to (1.4), u0=u0(x,t) is a (weak) solution to the following homoge- nized equation: (1.5) ∂tu0−div(ahom∇u0) =fin Ω×(0,T), whereahomis defined by (1.3). Furthermore, if r= 2 (i.e.,aε=a(x ε,t ε2)), then the corrector Φ kis the unique solution to the following cell problem: (1.6) ∂sΦk−divy/bracketleftbig a(y,s)(∇yΦk+ek)/bracketrightbig = 0 in TN×T (asin(1.1), cell problems arealways ellipticequations forany r/\e}atio\slash= 2). Thus thetypeofthe cellproblemdependsonthelog-ratioofthespatialandtemporalp eriodsofthecoefficients. Moreover, these formal arguments based on the asymptotic exp ansion for (the Cauchy- Dirichlet problem for) (1.4) are justified via two-scale convergence theory by A. Holmbom in [19]. The notion of two-scale convergence was first proposed by G . Nguetseng [24], and then, developedbyG.Allaire[2,3](seealso,e.g.,[22,33,36]). Iten ablesustoanalyzehow strong compactness of bounded sequences in Sobolev spaces fails due to their oscillatory behaviors (see (i) and (ii) of Remark 2.5 below). A. Holmbom extended the two-scale convergence theory to space-time homogenization and derived (1 .5) and (1.6) rigorously. Moreover, thenotionof very weak two-scale convergence isintroduced, andthen, itplays a crucial role for characterizing homogenized matrices (see Corollar y 2.8 below for details). Besides, homogenization problems for various parabolic equations h ave been studied not only for linear ones but also for nonlinear ones (e.g. [1, 13, 17, 20, 2 7, 34]). In particular, for p-Laplace type [13, 34] and porous medium type [1], it has been pr oved that cell problems are given as parabolic equations at the critical scale (i.e., aε=a(x ε,t ε2) in (1.4)). On the other hand, the following more general hyperbolic-parabolic equation is treated (e.g. [5, 6, 7, 12, 14, 23, 31, 32]). (1.7) hε∂2 ttuε−div(aε∇uε)+gε∂tuε=fin Ω×(0,T). Herehεandgεareε×εr-periodic functions rapidly oscillating. Furthermore, [26, 29, 30, 35] deal with nonlinear wave equations, and in particular, in [26, 2 9], almost periodic settings are studied via Σ-convergence theory developed in [25]. He re (1.7) is called damped wave equations forhε≡1 andgε>0 and it is noteworthy that asymptotic expansions of solutions to damped wave equations are performed w ith the aid of solutions to diffusion equations (e.g. (1.4)), and moreover, asymptotic beha viors of solutions to dampedwave equations aresimilar tothose ofdiffusion equations as t→+∞(see e.g. [18, 28]). Therefore, it is expected that cell problems for (1.7) will chan ge at the critical scale for (1.4) (i.e., aε=a(x ε,t ε2)). However, at least to our knowledge, it does not seem toSPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 3 occur under the periodic homogenization in the fixed domain except f orhε=−ε2and gε≡1 (see [4, Chapter 2, Section 4.5] for details). 1.1.Setting of the problem. One of main purposes of the present paper is to find conditions under which the cell problems of (1.7) will be different from elliptic ones (see (1.17) below). As a consequence, we emphasize that the (arithmet ic) quasi-periodicity of the time-dependent coefficient gεin (1.7) is crucial and it is defined as follows. Definition 1.1 (Quasi-periodic functions) .The function ϕ∈C(R)is said to be (arith- metic) quasi-periodic if it satisfies ϕ(s+1) =ϕ(s)+C∗for alls∈[0,1)andC∗∈R (i.e.,ϕis(0,1)-periodic if C∗= 0). Remark 1.2. The notion of quasi-periodicity has been defined in several different ways (see e.g., [9, 11]). We stress that quasi-periodic functions in the sen se of Definition 1.1 do not satisfy the almost-periodicity in the sense of Besicovitch, wh ich is known as a generalization of periodicity. Indeed, if ϕ∈C(R) is quasi-periodic, there exists a (0 ,1)- periodic function ϕper∈Cper(/square)1such that ϕ(s) =ϕper(s)+C∗s, which implies that /parenleftbigg limsup R→+∞1 |2R|/integraldisplayR −R|ϕ(s)|rds/parenrightbigg1/r = +∞,for allr∈[1,+∞). Thusϕdoes not belong to the generalized Besicovitch space Br(R) (see e.g., [8, 21]). Moreover, we shall consider both the effect of the periodic homoge nization and the effect of the singular limit due to ϕ(t εr) =ϕper(t εr)+C∗t εrandC∗t εr→+∞fort>0 asε→0+. In this paper, we shall consider the Cauchy-Dirichlet problem for th e following damped wave equation: (1.8)/braceleftBigg ∂2 ttuε−div/bracketleftbig a/parenleftbig t,x ε/parenrightbig ∇uε/bracketrightbig +g/parenleftbigt εr/parenrightbig ∂tuε=fεin Ω×(0,T), uε|∂Ω= 0, uε|t=0=v0 ε, ∂tuε|t=0=v1 ε. Here we make the following Assumption (A). Let Ω be a bounded domain in RNwith smooth boundary ∂Ω,N≥1. (i) LetT >0,ε>0 andr>0. Letv0 ε∈H1 0(Ω) andv1 ε∈L2(Ω) be such that v0 ε→v0weakly inH1 0(Ω) and v1 ε→v1weakly inL2(Ω). Letfε,f∈L2(Ω×(0,T)) be such that fε→fweakly inL2(Ω×(0,T)). (ii) TheN×Nsymmetric matrix a∈[C1(0,T;L∞(RN))]N×Nsatisfies a uniform ellipticity, i.e., there exists λ>0 such that (1.9) λ|ξ|2≤a(t,y)ξ·ξ≤ |ξ|2for anyξ∈RNand a.e. (t,y)∈(0,T)×RN, and (0,1)N-periodicity: a(t,y+ej) =a(t,y) a.e. in ( t,y)∈(0,T)×RN. 1Indeed, setting Φ(s) :=ϕ(s)−C∗s, we see that Φ(s) is (0,1)-periodic.4 TOMOYUKI OKA (iii) Setg∈C(R;R+) as follows: g(s) =gper(s)+C∗s>0 for alls∈R+. Heregperis a (0,1)-periodic function and C∗≥0 is a constant. In addition, if r= 2, we further assume C∗≤2λ C/square, whereC/square=N/π2is the best constant of the Poincar´ e inequality on the unit cell, that is, /ba∇dblw/ba∇dblL2(/square)≤C/square/ba∇dbl∇w/ba∇dblL2(/square)for allw∈H1 per(/square) (see Notation below). (iv) In addition, if C∗/\e}atio\slash= 0 and 2< r <+∞, thena=a(y),v0 ε,v1 ε,a(y),g(s) andfε are smooth, ( −div(a(x ε)∇v0 ε)), (v1 ε), (fε) and (∂tfε) are bounded in L2(Ω),H1 0(Ω), L∞(0,T;L2(Ω)) andL2(Ω×(0,T)), respectively. In this paper, we shall consider the convergence of solutions ( uε) to (1.8) and the homogenized equation as ε→0+. We also discuss how the homogenized matrix can be represented for each r>0. 1.2.Main results. We start with the following definition of weak solutions to (1.8): Definition 1.3 (Weak solution of (1.8)) .A function uε∈L∞(0,T;H1 0(Ω))is said to be a weak solution to (1.8), if the following (i)-(iii)are all satisfied : (i) (Regularity )uε∈W2,2(0,T;H−1(Ω))∩W1,∞(0,T;L2(Ω)). (ii) (Initial condition )uε(t)→v0 εstrongly in L2(Ω)ast→0+and∂tuε(t)→v1 εin H−1(Ω)ast→0+. (iii) (Weak form )It holds that, for all φ∈H1 0(Ω), (1.10)/angbracketleftbig ∂2 ttuε(t),φ/angbracketrightbig H1 0(Ω)+At ε(uε(t),φ)+/a\}b∇acketle{tg(t εr)∂tuε(t),φ/a\}b∇acket∇i}htH1 0(Ω)=/a\}b∇acketle{tfε(t),φ/a\}b∇acket∇i}htH1 0(Ω) for a.e. int∈(0,T), whereAt ε(v,w)is a bilinear form in H1 0(Ω)defined by At ε(v,w) =/integraldisplay Ωa/parenleftbig t,x ε/parenrightbig ∇v(x)·∇w(x)dxforv,w∈H1 0(Ω). By Galerkin’s method (cf. [10, Theorem 12.2]), we have Theorem 1.4 (Existence and uniqueness of weak solutions to (1.8)) .Suppose that a(t,x ε)∈[C1(0,T;L∞(Ω))]N×N sym, g(t εr)∈C(0,T), fε∈L2(Ω×(0,T)), v0 ε∈H1 0(Ω), v1 ε∈L2(Ω). Then for every ε>0there exists a unique weak solution uεto(1.8). Then we first obtain the following homogenization theorem: Theorem 1.5 (Homogenization theorem) .Suppose that (A)is satisfied. Let uε∈ L∞(0,T;H1 0(Ω))be a unique weak solution to (1.8). There exist u0∈L∞(0,T;H1 0(Ω)) andh∈L2 loc((0,T];H−1(Ω))such that, for any σ>0, uε→u0 weakly-∗inL∞(0,T;H1 0(Ω)), (1.11) uε→u0 strongly in C([0,T];L2(Ω)), (1.12) g(t εr)∂tuε→ /a\}b∇acketle{tgper/a\}b∇acket∇i}hts∂tu0+C∗hweakly in/braceleftBigg L2(0,T;H−1(Ω))ifC∗= 0, L2(σ,T;H−1(Ω))ifC∗/\e}atio\slash= 0,(1.13)SPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 5 ∂2 ttuε→∂2 ttu0 weakly in/braceleftBigg L2(0,T;H−1(Ω))ifC∗= 0, L2(σ,T;H−1(Ω))ifC∗/\e}atio\slash= 0,(1.14) a(t,x ε)∇uε→ /a\}b∇acketle{ta(t)(∇u0+∇yu1)/a\}b∇acket∇i}htyweakly in [L2(Ω×(0,T))]N, (1.15) where/a\}b∇acketle{tw/a\}b∇acket∇i}hts=/integraltext1 0w(s)ds,/a\}b∇acketle{tˆw/a\}b∇acket∇i}hty=/integraltext /squareˆw(y)dyandu1is written by u1(x,t,y) :=N/summationdisplay k=1∂xku0(x,t)Φk(t,y). (1.16) HereΦkis a corrector for each k= 1,...,Nand it is characterized as follows : (i)In caser∈(0,+∞)\ {2},Φk∈H1 per(TN)/R(see Notation below )is the unique solution to (1.17) −divy[a(t,y)(∇yΦk+ek)] = 0inTN×(0,T), whereekis thek-th vector of the canonical basis of RN. (ii)In caser= 2,Φk∈L2(0,T;H1 per(TN)/R)is the unique solution to (1.18) C∗t∂tΦk−divy[a(t,y)(∇yΦk+ek)] = 0inTN×(0,T). In particular, if either C∗= 0ora=a(y), thenΦk∈H1 per(TN)/Ris the unique solution to (1.17). Furthermore, for any C∗≥0,u0is the unique weak solution to (1.19)/braceleftBigg ∂2 ttu0−div[ahom(t)∇u0]+/a\}b∇acketle{tgper/a\}b∇acket∇i}hts∂tu0+C∗h=finΩ×(0,T), u0|∂Ω= 0, u0|t=0=v0, ∂tu0|t=0= ˜v1. Hereu0≡v0wheneverC∗/\e}atio\slash= 0, and moreover, ˜v1=/braceleftBigg v1ifC∗= 0, 0ifC∗/\e}atio\slash= 0. Moreover,ahom(t)is the homogenized matrix given by (1.20) ahom(t)ek=/integraldisplay /squarea(t,y)/parenleftbig ∇yΦk(t,y)+ek/parenrightbig dy, k= 1,2,...,N. Remark 1.6. It is noteworthy that, due to the loss of the time periodicity, the fo llowing facts hold: (i)(Homogenized equation). Thehomogenized equation(1.19)isofthesametype as the original equation (1.8) for the periodic case (i.e., C∗= 0). On the other hand, for the quasi-periodic case (i.e., C∗/\e}atio\slash= 0), by the effect of the singular limit ofg, (1.19) is represented as the following elliptic equation: −div(ahom∇u0) =f−C∗hin Ω×(0,T), u0∈H1 0(Ω). Furthermore, the limit of the solution to (1.8) coincides with the limit of the initial datav0 ε. (ii)(Cell problem). For the periodic case C∗= 0, the corrector Φ kis always de- scribed as the solution to the elliptic equation (1.17). On the other ha nd, for the quasi-periodic case, at the critical case r= 2, the cell problem (1.18) is different from (1.17) and it is given as the parabolic equation by the effect of th e singular limit ofg. Thus Φ kdepends on t∈(0,T), and then, qualitative properties of the homogenized matrix ahomwill change due to (1.20) (see Proposition 1.7 below).6 TOMOYUKI OKA Moreover, as for the homogenized matrix, we next have the followin g Proposition 1.7 (Qualitative properties of the homogenized matrix ahom).Under the same assumption as in Theorem 1.5, let 0< r <+∞andahom(t)be the homogenized matrices defined by (1.20). Then the following (i)and(ii)hold: (i) (Uniform ellipticity )It holds that λ|ξ|2+λ/ba∇dbl∇yΦξ(t)/ba∇dbl2 L2(/square)+C∗t 2d dt/ba∇dblΦξ(t)/ba∇dbl2 L2(/square) ≤ahom(t)ξ·ξ≤ |ξ|2+/ba∇dbl∇yΦξ(t)/ba∇dbl2 L2(/square)+C∗t 2d dt/ba∇dblΦξ(t)/ba∇dbl2 L2(/square) for anyξ∈RNand a.e.t∈(0,T), whereλ>0is the ellipticity constant of a(t,y) defined by (1.9)andΦξis the corrector given by either (1.17)or(1.18)withek replaced by ξ∈RN. (ii) (Symmetry and asymmetry )Ifa(t,y)is the symmetric matrix, then ahom(t)is the asymmetric matrix for r= 2andC∗/\e}atio\slash= 0. Otherwise, ahom(t)is also the symmetric matrix. Remark 1.8. We stress that, in the critical case (i.e., r= 2 andC∗/\e}atio\slash= 0), even though the elliptic constant of a(t,y) is independent of t∈(0,T), that ofahom(t) depends on t. Furthermore, the symmetry breaking of ahom(t) occurs but it makes no contribution to the divergence (see Remark 5.1 below). We finally get the following corrector result. Theorem 1.9 (Corrector result for time independent coefficients) .Suppose that (A)is fulfilled and assume that a=a(y),v0 ε,v1 ε,a(y),g(s)andfεare smooth, (−div(a(x ε)∇v0 ε)), (v1 ε),(fε)and(∂tfε)are bounded in L2(Ω),H1 0(Ω),L∞(0,T;L2(Ω))andL2(Ω×(0,T)), respectively. Let uεandu0be the unique solutions to (1.8)and(1.19), respectively. Then it holds that (1.21) lim ε→0+/integraldisplayT 0/integraldisplay Ω/vextendsingle/vextendsingle∇uε(x,t)−/parenleftbig ∇u0(x,t)+∇yu1(x,t,x ε)/parenrightbig/vextendsingle/vextendsingle2dxdt= 0 for allr∈(0,+∞), whereu1=/summationtextN k=1∂xku0ΦkandΦk∈L2(0,T;H1 per(TN)/R)is the corrector for r∈(0,+∞). As for the time dependent case a=a(t,y), we have the following corrector result for more specific settings: Corollary 1.10 (Corrector result for time dependent coefficients) .Suppose that C∗/\e}atio\slash= 0. In addition, assume that a(t,y)is smooth and the following (1.22)-(1.25)hold: ∂ta(t,y)ξ·ξ≤0for allξ∈RNand all(t,y)∈(0,T)×RN, (1.22) −div(a(0,x ε)∇v0 ε)→ −div(ahom(0)∇v0)strongly in H−1(Ω), (1.23) lim ε→0+/ba∇dblv1 ε/ba∇dblL2(Ω)= 0, (1.24) fε→fstrongly in L2(Ω×(0,T))or(fε/√ t)is bounded in L2(Ω×(0,T)), (1.25) In addition, if r= 2andC∗/\e}atio\slash= 0, assume that (1.26) ∂ta(t,y) =−a(t,y)for all(t,y)∈(0,T)×RN.SPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 7 Hereahom(t)is the homogenized matrix defined by (1.20). Letuεandu0be the unique solutions to (1.8)and(1.19), respectively. Then (1.21)holds. Remark 1.11. Initial data v0 ε∈H1 0(Ω) satisfying (1.23) can actually be constructed (see e.g. [10, pp. 236]). Remark 1.12. From Theorem 1.9, it holds that uε/\e}atio\slash→u0strongly in L2(0,T;H1 0(Ω)) in general due to the oscillation of the third term u1(x,t,x ε) asε→0+. Thusu1(x,t,x ε) plays a role as the corrector term recovering the strong compact ness in this topology. For this reason, Φ kis often called a corrector. 1.3.Plan of the paper and notation. This paper is organized as follows. In the next section, we summarize relevant material on space-time two-scale c onvergence. Section 3 is devoted to proving uniform estimates for solutions uεto (1.8) asε→0+. Furthermore, we shall prove their weak(- ∗) and strong convergences. In Section 4, we shall prove Theorem 1.5. To prove Proposition 1.7, we shall discuss qualitative properties of the homogenized matrixahom(t) in Section 5. The final section is devoted to proofs of Theorem 1.9 a nd Corollary 1.10. Notation. Throughout this paper, C >0 denotes a non-negative constant which may vary from line to line. In addition, the subscript A of CAmeans dependence of CAon A. Letδijbe the Kronecker delta, ei= (δij)1≤j≤Nbe thei-th vector of the basis of RN,/ba∇dbl · /ba∇dblH1 0(A)be defined by /ba∇dbl · /ba∇dblH1 0(A):=/ba∇dbl∇ · /ba∇dbl L2(A)for domains A⊂RN,∇and∇y denote gradient operators with respect to xandy, respectively, and div and div ydenote divergence operators with respect to xandy, respectively. Furthermore, we shall use the following notation: •/square= (0,1)N,I= (0,T),J= (0,1),dZ=dydsdxdt . •Define the set of smooth /square-periodic functions by C∞ per(/square) ={w∈C∞(/square):w(·+ek) =w(·) inRNfor 1≤k≤N}. •We also define W1,q per(/square) andLq per(/square) as closed subspaces of W1,q(/square) andLq(/square) by W1,q per(/square) =C∞per(/square)/bardbl·/bardblW1,q(/square), Lq per(/square) =C∞per(/square)/bardbl·/bardblLq(/square), respectively, for 1 ≤q <+∞. In particular, set H1 per(/square) :=W1,2 per(/square). We shall simply write Lq(/square) instead of Lq per(/square), unless any confusion may arise. •We often write Lq(Ω×/square) instead by Lq(Ω;Lq per(/square)) sinceLq per(/square) is reflexive Banach space for 1 <q <+∞. •Define the mean /a\}b∇acketle{tw/a\}b∇acket∇i}hty:=/integraltext /squarew(y)dyinyofw∈L1(/square). •We setW1,q per(/square)/R={w∈W1,q per(/square):/a\}b∇acketle{tw/a\}b∇acket∇i}hty=/integraltext /squarew(y)dy= 0}. •Furthermore, let Xbe a normed space with a norm /ba∇dbl·/ba∇dblXand a duality pairing /a\}b∇acketle{t·,·/a\}b∇acket∇i}htXbetweenXanditsdualspace X∗. Moreover, wewrite XN=X×X×···×X (N-product space), e.g., [ L2(Ω)]N=L2(Ω;RN). In order to clarify variables of integration, we shall often write, e.g .,/ba∇dblu(x ε)/ba∇dblLq(Ω)and /ba∇dblu(x,x ε)/ba∇dblLq(Ω)instead of /ba∇dblu(· ε)/ba∇dblLq(Ω)and/ba∇dblu(x,· ε)/ba∇dblLq(Ω), respectively. We often write u(t) instead ofu(·,t) for eacht∈Iandu: Ω×I→R.8 TOMOYUKI OKA 2.Space-time two-scale convergence theory In this section, we introduce the notion of space-time two-scale convergence and briefly summarize its crucial properties (see, e.g., [24], [2, 3], [22], [36] and [19 ] for more details). Throughout this section, let q∈[1,+∞] andq′denote for the H¨ older conjugate of q(i.e., 1/q+1/q′= 1 if 1<q<+∞;q′= 1 ifq= +∞;q′= +∞ifq= 1) unless any confusion may arise. Furthermore, let I= (0,T),/square= (0,1)NandJ= (0,1). We first define a class of test functions for the space-time two-sc ale convergence, called admissible test functions . Definition 2.1 (Admissible test function) .Letq′∈[1,+∞]and letX⊂Lq′(Ω×I× /square×J)be a separable normed space equipped with norm /ba∇dbl · /ba∇dblX. Then (X,/ba∇dbl · /ba∇dblX)is called an admissible test function space ( for the weak space-time two-scale convergence inLq(Ω×I×/square×J)), if it holds that, for all Ψ∈X,(x,t)/ma√sto→Ψ(x,t,x ε,t εr)is Lebesgue measurable in Ω×Iforε>0, and lim ε→0+/vextenddouble/vextenddoubleΨ/parenleftbig x,t,x ε,t εr/parenrightbig/vextenddouble/vextenddouble Lq′(Ω×I)=/ba∇dblΨ(x,t,y,s)/ba∇dblLq′(Ω×I×/square×J), /vextenddouble/vextenddoubleΨ/parenleftbig x,t,x ε,t εr/parenrightbig/vextenddouble/vextenddouble Lq′(Ω×I)≤C/ba∇dblΨ(x,t,y,s)/ba∇dblXforε>0. Moreover, Ψ∈Xis called an admissible test function ( for the weak space-time two-scale convergence in Lq(Ω×I×/square×J)). The following fact is well known and often used, in particular, to discu ss weak conver- gence of periodic test functions. Proposition 2.2 (Mean-value property) .Letw∈Lq(/square×J)and setwε(x,t) =w(x ε,t εr) forε>0and0<r <+∞. For any bounded domain Ω⊂RNand any bounded interval I⊂R, it holds that/braceleftBigg wε→ /a\}b∇acketle{tw/a\}b∇acket∇i}hty,sweakly inLq(Ω×I) ifq∈[1,+∞), wε→ /a\}b∇acketle{tw/a\}b∇acket∇i}hty,sweakly-∗inL∞(Ω×I)ifq= +∞ asε→0+. Here/a\}b∇acketle{tw/a\}b∇acket∇i}hty,sdenotes the mean of w, i.e., /a\}b∇acketle{tw/a\}b∇acket∇i}hty,s=/integraldisplay1 0/integraldisplay /squarew(y,s)dyds. Proof.See [10, Theorem 2.6]. /square Now, we are in a position to define the notion of space-time two-scale convergence in the following. Definition 2.3 (Weak space-time two-scale convergence and very weak two-sca le conver- gence). (i)A bounded sequence (vε)inLq(Ω×I)is said to weakly space-time two-scale con- vergeto a limitv∈Lq(Ω×I×/square×J)if it holds that lim ε→0+/integraldisplayT 0/integraldisplay Ωvε(x,t)Ψ(x,t,x ε,t εr)dxdt=/integraldisplayT 0/integraldisplay Ω/integraldisplay1 0/integraldisplay /squarev(x,t,y,s)Ψ(x,t,y,s)dZ for any admissible function Ψ∈X⊂Lq(Ω×I×/square×J)and it is denoted by vε2,2⇀vinLq(Ω×I×/square×J).SPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 9 (ii)A bounded sequence (vε)inLq(Ω×I)is said to very weakly two-scale converge to a limitvinLq(Ω×I×/square×J)when we choose Ψ(x,y,t,s) =φ(x)b(y)ψ(t)c(s) for anyφ∈C∞ c(Ω),b∈C∞ per(/square)/R,ψ∈C∞ c(I)andc∈C∞ per(J), and then, it is written by vε2,2⇀ vwvinLq(Ω×I×/square×J). Remark 2.4. Due to the extension of the original definition in [24, 2] of the test fu nction to Ψ∈Lq′(Ω×I;Cper(/square×J)) the boundedness of ( vε) is essential. Indeed, some coun- terexamples that the (weak space-time) two-scale limit does not co incide with the weak limit are known in [22, Examples 11 and 12]. Remark 2.5. As for the relation between weak or strong convergence and weak space- time two-scale convergence, the following holds: (i) Ifvε2,2⇀vinLq(Ω×I×/square×J), thenvε→ /a\}b∇acketle{tv/a\}b∇acket∇i}hty,sweakly inLq(Ω×I). (ii) Ifvε→ˆvstrongly in L1(Ω×I), thenvε2,2⇀ˆvinLq(Ω×I×/square×J). The following theorem is concerned with weak space-time two-scale c ompactness of bounded sequences in Lq(Ω×I). Theorem 2.6 (Weak space-time two-scale compactness) .Letq∈(1,∞]. Then, for any bounded sequence (vε)inLq(Ω×I), there exist a subsequence (εk)of(ε)such thatεk→0+ and a limit v∈Lq(Ω×I×/square×J)such that vεk2,2⇀vinLq(Ω×I×/square×J). Proof.See [19, Theorem 2.3]. /square As for weak space-time two-scale compactness of gradients, we o btain Theorem 2.7 (Weak space-time two-scale compactness for gradients) .Letq∈(1,+∞) and let(vε)be a bounded sequence in W1,q(Ω×I). Then there exist a subsequence εk→0+, a limitv∈Lq(Ω×I)and a function v1∈Lq(Ω×I;W1,q per(/square×J)/R)such that ∇t,xvεk2,2⇀∇t,xv+∇s,yv1in[Lq(Ω×I×/square×J)]N+1. Here and henceforth, ∇t,x= (∂t,∂x1,...,∂ xN)and∇s,y= (∂s,∂y1,...,∂ yN). Proof.See [22, Theorem 20]. /square As a corollary of Theorem 2.7, the following is obtained. Corollary 2.8 (cf. [15, 16, 19]) .Under the same assumptions as in Theorem 2.7, it holds that vεk εk2,2⇀ vwv1inLq(Ω×I×/square×J). Proof.See [19, Corollary 3.3]. /square10 TOMOYUKI OKA 3.Uniform estimates To discuss convergence of solutions uεfor (1.8), we shall first verify their uniform boundedness, and then, we shall prove their weak or strong conv ergences. Lemma 3.1 (Uniform estimates) .Letuε∈L∞(I;H1 0(Ω))be the unique weak solution of (1.8)under the same assumptions as in Theorem 1.5 and let Iσ= (σ,T)for anyσ >0. Then the following (i)-(vi)hold: (i) (uε) is bounded in L∞(I;H1 0(Ω)), (ii) (∂tuε) is bounded in L∞(I;L2(Ω)), (iii) (√ tε−r∂tuε) is bounded in L2(Ω×I), provided that C∗/\e}atio\slash= 0, (iv) (∂2 ttuε+g(t εr)∂tuε) is bounded in L2(I;H−1(Ω)), (v) (∂2 ttuε) is bounded in/braceleftBigg L2(I;H−1(Ω)) ifC∗= 0, L2(Iσ;H−1(Ω)) ifC∗/\e}atio\slash= 0, (vi) (tε−r∂tuε) is bounded in L2(Iσ;H−1(Ω)), provided that C∗/\e}atio\slash= 0. Proof.Recall (1.10), i.e., /a\}b∇acketle{t∂2 ttuε(t),φ/a\}b∇acket∇i}htH1 0(Ω)+At ε(uε(t),φ)+/a\}b∇acketle{tg(t εr)∂tuε(t),φ/a\}b∇acket∇i}htH1 0(Ω)=/a\}b∇acketle{tfε(t),φ/a\}b∇acket∇i}htH1 0(Ω) for allφ∈H1 0(Ω). Testing it by ∂tuε(see Remark 3.2 below), we deduce by the symmetry ofa(t,y) that/integraldisplay Ωa(t,x ε)∇uε(x,t)·∇∂tuε(x,t)dx (3.1) =1 2d dt/integraldisplay Ωa(t,x ε)∇uε(x,t)·∇uε(x,t)dx−1 2/integraldisplay Ω∂ta(t,x ε)∇uε(x,t)·∇uε(x,t)dx a.e. inI. Thus we have 1 2/integraldisplays 0d dt/integraldisplay Ω/bracketleftBig |∂tuε(x,t)|2+a/parenleftbig t,x ε/parenrightbig ∇uε(x,t)·∇uε(x,t)/bracketrightBig dxdt (3.2) (3.1)=1 2/integraldisplays 0/integraldisplay Ω∂ta(t,x ε)∇uε(x,t)·∇uε(x,t)dxdt +/integraldisplays 0/integraldisplay Ωfε(x,t)∂tuε(x,t)dxdt−/integraldisplays 0/parenleftBig gper(t εr)+C∗t εr/parenrightBig /ba∇dbl∂tuε(t)/ba∇dbl2 L2(Ω)dt for alls∈I. Then we observe from the uniform ellipticity (1.9), (3.2) and (A)that /ba∇dbl∂tuε(s)/ba∇dbl2 L2(Ω)+λ/ba∇dbluε(s)/ba∇dbl2 H1 0(Ω) (1.9) ≤ /ba∇dblv1 ε/ba∇dbl2 L2(Ω)+/ba∇dblv0 ε/ba∇dbl2 H1 0(Ω)+/integraldisplays 0d dt/parenleftbigg /ba∇dbl∂tuε(t)/ba∇dbl2 L2(Ω)+/integraldisplay Ωa(t,x ε)∇uε(x,t)·∇uε(x,t)dx/parenrightbigg dt (3.2)=/ba∇dblv1 ε/ba∇dbl2 L2(Ω)+/ba∇dblv0 ε/ba∇dbl2 H1 0(Ω)+/integraldisplays 0/integraldisplay Ω∂ta(t,x ε)∇uε(x,t)·∇uε(x,t)dxdt +2/parenleftbigg/integraldisplays 0/integraldisplay Ωfε(x,t)∂tuε(x,t)dxdt−/integraldisplays 0/parenleftBig gper(t εr)+C∗t εr/parenrightBig /ba∇dbl∂tuε(t)/ba∇dbl2 L2(Ω)dt/parenrightbigg (A) ≤ /ba∇dblv1 ε/ba∇dbl2 L2(Ω)+/ba∇dblv0 ε/ba∇dbl2 H1 0(Ω)+sup t∈I/ba∇dbl∂ta(t)/ba∇dblL∞(/square)/integraldisplays 0/ba∇dbluε(t)/ba∇dbl2 H1 0(Ω)dt +2/integraldisplays 0/bracketleftBig /ba∇dblfε(t)/ba∇dblL2(Ω)/ba∇dbl∂tuε(t)/ba∇dblL2(Ω)+/parenleftBig β−C∗t εr/parenrightBig /ba∇dbl∂tuε(t)/ba∇dbl2 L2(Ω)/bracketrightBig dtSPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 11 ≤/parenleftBig /ba∇dblv1 ε/ba∇dbl2 L2(Ω)+/ba∇dblv0 ε/ba∇dbl2 H1 0(Ω)+/ba∇dblfε/ba∇dbl2 L2(Ω×I)/parenrightBig +Cβ/integraldisplays 0/parenleftBig /ba∇dbl∂tuε(t)/ba∇dbl2 L2(Ω)+/ba∇dbluε(t)/ba∇dbl2 H1 0(Ω)/parenrightBig dt−C∗/integraldisplays 0/ba∇dbl√ tε−r∂tuε(t)/ba∇dbl2 L2(Ω)dt. Hereβ= max s∈[0,1]|gper(s)|. From the boundedness of ( fε) inL2(Ω×I), we get /ba∇dbl∂tuε(s)/ba∇dbl2 L2(Ω)+λ/ba∇dbluε(s)/ba∇dbl2 H1 0(Ω)+C∗/integraldisplays 0/ba∇dbl√ tε−r∂tuε(t)/ba∇dbl2 L2(Ω)dt (3.3) ≤C+Cβ/integraldisplays 0/parenleftBig /ba∇dbl∂tuε(t)/ba∇dbl2 L2(Ω)+/ba∇dbluε(t)/ba∇dbl2 H1 0(Ω)/parenrightBig dt, which together with Gronwall’s inequality yields (i) and (ii). Moreover, ( iii) also follows from (i), (ii) and (3.3). We next prove (iv). For any φ∈H1 0(Ω), the weak form (1.10) yields |/a\}b∇acketle{t∂2 ttuε(t)+g(t εr)∂tuε(t),φ/a\}b∇acket∇i}htH1 0(Ω)| ≤ /ba∇dblφ/ba∇dblH1 0(Ω)/parenleftBig /ba∇dblfε(t)/ba∇dblH−1(Ω)+/ba∇dbluε(t)/ba∇dblH1 0(Ω)/parenrightBig . (3.4) Here we used the fact that |a(t,y)ξ·ζ| ≤ |ξ||ζ|for allξ,ζ∈RNand a.e. (t,y)∈I×RN, which follows from the Rayleigh-Ritz variational principle. By the boun dedness of ( fε) in L2(I;H−1(Ω)) together with (i) and (3.4), we deduce that /integraldisplayT 0/ba∇dbl∂2 ttuε(t)+g(t εr)∂tuε(t)/ba∇dbl2 H−1(Ω)dt (3.4) ≤/integraldisplayT 0/parenleftBig /ba∇dblfε(t)/ba∇dblH−1(Ω)+/ba∇dbluε(t)/ba∇dblH1 0(Ω)/parenrightBig2 dt≤2/parenleftBig /ba∇dblfε/ba∇dbl2 L2(I;H−1(Ω))+/ba∇dbluε/ba∇dbl2 L2(I;H1 0(Ω))/parenrightBig , which implies that (iv) holds true. Here noting that /ba∇dbl∂2 ttuε+C∗tε−r∂tuε/ba∇dblL2(I;H−1(Ω)) ≤ /ba∇dbl∂2 ttuε+g(t εr)∂tuε/ba∇dblL2(I;H−1(Ω))+β/ba∇dbl∂tuε/ba∇dblL2(I;H−1(Ω))<C, we get (v) if C∗= 0. As for C∗/\e}atio\slash= 0, we infer that /integraldisplayT 0t/parenleftBig /ba∇dbl∂2 ttuε(t)/ba∇dbl2 H−1(Ω)+/ba∇dblC∗t εr∂tuε(t)/ba∇dbl2 H−1(Ω)/parenrightBig dt (3.5) =/integraldisplayT 0t/parenleftBig /ba∇dbl∂2 ttuε(t)+C∗t εr∂tuε(t)/ba∇dbl2 H−1(Ω)−2/parenleftbig ∂2 ttuε(t),C∗t εr∂tuε(t)/parenrightbig H−1(Ω)/parenrightBig dt ≤CT−/integraldisplayT 0C∗t2 εrd dt/ba∇dbl∂tuε(t)/ba∇dbl2 H−1(Ω)dt =CT−C∗T2/ba∇dbl√ ε−r∂tuε(T)/ba∇dbl2 H−1(Ω)+2C∗/integraldisplayT 0/ba∇dbl√ tε−r∂tuε(t)/ba∇dbl2 H−1(Ω)dt≤C. Hence we have/integraldisplayT σ/parenleftBig /ba∇dbl∂2 ttuε(t)/ba∇dbl2 H−1(Ω)+/ba∇dblt εrC∗∂tuε(t)/ba∇dbl2 H−1(Ω)/parenrightBig dt (3.6) ≤1 σ/integraldisplayT 0t/parenleftBig /ba∇dbl∂2 ttuε(t)/ba∇dbl2 H−1(Ω)+/ba∇dblt εrC∗∂tuε(t)/ba∇dbl2 H−1(Ω)/parenrightBig dt≤C σ for all 0<σ<T, which implies (v) and (vi). /square12 TOMOYUKI OKA Remark 3.2. Theargumentmentionedaboveisnotfullyrigorousinviewoftheregu larity of weak solutions. However, an approximate solution constructed by the Galerkin method inTheorem1.4satisfies alltheestimatesand(weak) lower semicontin uity ofnormsassures the assertions. Moreover, for smooth data, we have Lemma 3.3 (Uniform estimates with smooth data) .Suppose that a=a(y),v0 ε,v1 ε,a(y), g(s)andfεare smooth, (−div(a(x ε)∇v0 ε)),(v1 ε),(fε)and(∂tfε)are bounded in L2(Ω), H1 0(Ω),L∞(I;L2(Ω))andL2(Ω×I), respectively. Let uε∈L∞(I;H1 0(Ω))be the unique weak solution to (1.8). Assume that (A)holds. LetIσ= (σ,T)for anyσ>0. Then the following (i)-(v)hold: (i) (−div(a(x ε)∇uε))is bounded in L∞(I;L2(Ω)), (ii) (∂tuε)is bounded in L∞(I;H1 0(Ω)), (iii) (∂2 ttuε+g(t εr)∂tuε)is bounded in L∞(I;L2(Ω)), (iv) (∂2 ttuε)is bounded in L2(Ω×Iσ), (v) (tε−r∂tuε)is bounded in L2(Ω×Iσ), provided that C∗/\e}atio\slash= 0. Proof.Test (1.10) by −div(a(x ε)∇∂tuε). Then we observe that /integraldisplays 0/integraldisplay Ω∂2 ttuε(x,t)/parenleftBig −div/parenleftbig a(x ε)∇∂tuε(x,t)/parenrightbig/parenrightBig dxdt =/integraldisplays 0/integraldisplay Ω∂t/parenleftbig ∇∂tuε(x,t)/parenrightbig ·a(x ε)∇∂tuε(x,t)dxdt =1 2/integraldisplays 0d dt/parenleftbigg/integraldisplay Ωa(x ε)∇∂tuε(x,t)·∇∂tuε(x,t)dx/parenrightbigg dt(1.9) ≥λ 2/ba∇dbl∂tuε(s)/ba∇dbl2 H1 0(Ω)−1 2/ba∇dblv1 ε/ba∇dbl2 H1 0(Ω) and/integraldisplays 0/integraldisplay Ω/parenleftbig −div/parenleftbig a(x ε)∇uε(x,t)/parenrightbig/parenrightbig/parenleftbig −div/parenleftbig a(x ε)∇∂tuε(x,t)/parenrightbig/parenrightbig dxdt =1 2/ba∇dbl−div(a(x ε)∇uε(s))/ba∇dbl2 L2(Ω)−1 2/ba∇dbl−div(a(x ε)∇v0 ε)/ba∇dbl2 L2(Ω) for alls∈I. Hence we derive that λ 2/ba∇dbl∂tuε(s)/ba∇dbl2 H1 0(Ω)+1 2/ba∇dbl−div(a(x ε)∇uε(s))/ba∇dbl2 L2(Ω) ≤1 2/ba∇dblv1 ε/ba∇dbl2 H1 0(Ω)+1 2/ba∇dbl−div(a(x ε)∇v0 ε)/ba∇dbl2 L2(Ω) +/integraldisplay Ωfε(x,s)/parenleftBig −div(a(x ε)∇uε(x,s))/parenrightBig dx−/integraldisplay Ωfε(x,0)/parenleftBig −div(a(x ε)∇v0 ε(x))/parenrightBig dx −/integraldisplays 0/integraldisplay Ω∂tfε(x,t)/parenleftBig −div(a(x ε)∇uε(x,t))/parenrightBig dxdt −/integraldisplays 0/integraldisplay Ωg(t εr)∇∂tuε(x,t)·a(x ε)∇∂tuε(x,t)dxdt ≤1 2/ba∇dblv1 ε/ba∇dbl2 H1 0(Ω)+/ba∇dbl−div(a(x ε)∇v0 ε)/ba∇dbl2 L2(Ω) +/ba∇dblfε(s)/ba∇dbl2 L2(Ω)+1 4/ba∇dbl−div(a(x ε)∇uε(s))/ba∇dbl2 L2(Ω)+1 2/ba∇dblfε(0)/ba∇dbl2 L2(Ω)SPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 13 +1 2/ba∇dbl∂tfε/ba∇dbl2 L2(Ω×I)+1 2/integraldisplays 0/ba∇dbl−div(a(x ε)∇uε(t))/ba∇dbl2 L2(Ω)dt−/integraldisplays 0λg(t εr)/ba∇dbl∇∂tuε(t)/ba∇dbl2 L2(Ω)dt /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright ≥0 ≤C+1 4/ba∇dbl−div(a(x ε)∇uε(s))/ba∇dbl2 L2(Ω)+1 2/integraldisplays 0/ba∇dbl−div(a(x ε)∇uε(t))/ba∇dbl2 L2(Ω)dt, which together with Gronwall’s inequality yields (i) and (ii). Hence one ca n derive that (3.7)/ba∇dbl∂2 ttuε+g(t εr)∂tuε/ba∇dblL∞(I;L2(Ω))≤ /ba∇dblfε/ba∇dblL∞(I;L2(Ω))+/ba∇dbl−div(a(x ε)∇uε)/ba∇dblL∞(I;L2(Ω)), which implies (iii). As in the proofs of (3.5) and (3.6), (iv) and (v) follow f rom (3.7). /square Applying Lemma 3.1, we next get the following Lemma 3.4 (Weak(-star) and strong convergences) .Letuε∈L∞(I;H1 0(Ω))be the unique weak solution of (1.8)under the same assumption as in Lemma 3.1. Then there exist a subsequence (εn)of(ε),u0∈L∞(I;H1 0(Ω)),w∈L2(I;H−1(Ω))andh∈ L2 loc((0,T];H−1(Ω))such that, for any σ∈I, uεn→u0 weakly-∗inL∞(I;H1 0(Ω)), (3.8) ∂tuεn→∂tu0weakly-∗inL∞(I;L2(Ω)), (3.9) ∂2 ttuεn+g(t εrn)∂tuεn→w weakly inL2(I;H−1(Ω)), (3.10) ∂2 ttuεn→∂2 ttu0weakly in/braceleftBigg L2(I;H−1(Ω))ifC∗= 0, L2(Iσ;H−1(Ω))ifC∗/\e}atio\slash= 0,(3.11) tε−r n∂tuεn→h weakly inL2(Iσ;H−1(Ω)) ifC∗/\e}atio\slash= 0, (3.12) uεn→u0 strongly in C(I;L2(Ω)), (3.13) ∂tuεn→∂tu0strongly in/braceleftBigg C(I;H−1(Ω))ifC∗= 0, C(Iσ;H−1(Ω))ifC∗/\e}atio\slash= 0,(3.14) √ t∂tuεn→0 strongly in L2(Ω×I) ifC∗/\e}atio\slash= 0. (3.15) In particular, if C∗/\e}atio\slash= 0, then∂tu0(·,t)≡0for a.e.t∈I, and hence, u0is independent oft∈I, i.e.,u0=u0(x). Furthermore, there exists w1∈L2(Ω×I;H1 per(/square×J)/R)such that ∂tuεn2,2⇀∂tu0+∂sw1 inL2(Ω×I×/square×J), (3.16) a(t,x εn)∇uεn2,2⇀a(t,y)(∇u0+∇yw1) in[L2(Ω×I×/square×J)]N. (3.17) Thus it holds that a(t,x εn)∇uεn→ /a\}b∇acketle{ta(t,·)(∇u0+∇yw1)/a\}b∇acket∇i}hty,sweakly in [L2(Ω×I)]N, (3.18) where /angbracketleftbig a(t,·)/parenleftbig ∇u0(x,t)+∇yw1(x,t,·,·)/parenrightbig/angbracketrightbig y,s=/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x,t)+∇y/a\}b∇acketle{tw1(x,t,y,·)/a\}b∇acket∇i}hts/parenrightbig dy. Proof.Thanks to Lemma 3.1, we readily obtain (3.8)-(3.12). Furthermore, from (i) and (ii) of Lemma 3.1, the Asocili-Arzel´ a theorem yields (3.13). In the sam e way, (3.14) also14 TOMOYUKI OKA holds true by (ii) and (v) of Lemma 3.1. As for (3.15), noting by (iii) of L emma 3.1 that limsup εn→0+/ba∇dbl√ t∂tuεn/ba∇dbl2 L2(Ω×I)≤limsup εn→0+Cεr n= 0, we obtain (3.15). Thus u0=u0(x), provided that C∗/\e}atio\slash= 0. We finally show (3.16), (3.17) and (3.18). All the assumptions of Theorem 2.7 can be checked by (i) and (ii) of Lemma 3.1. Hence (3.16) holds true. Moreover, note that, for any Ψ ∈[L2 per(/square×J;Cc(Ω×I))]N, Ψ anda(t,y)Ψ are admissible test functions in [ L2(Ω×I×/square×J)]N(see [22, Theorems 2 and 4] for details) and define Ξ ∈[L2(Ω×I×/square×J)]Nby a(t,x εn)∇uεn2,2⇀Ξ in [L2(Ω×I×/square×J)]N. Then, Theorem 2.7 yields that /integraldisplayT 0/integraldisplay Ω/integraldisplay1 0/integraldisplay /squareΞ(x,t,y,s)·Ψ(x,t,y,s)dZ = lim εn→0+/integraldisplayT 0/integraldisplay Ω∇uεn(x,t)·ta(t,x εn)Ψ(x,t,x εn,t εrn)dxdt =/integraldisplayT 0/integraldisplay Ω/integraldisplay1 0/integraldisplay /square/parenleftbig ∇u0(x,t)+∇yw1(x,t,y,s)/parenrightbig ·ta(t,y)Ψ(x,t,y,s)dZ, which implies (3.17), and hence, (i) of Remark 2.5 yields (3.18). This com pletes the proof. /square 4.Proof of Theorem 1.5 We first derive the homogenized equation by setting jhom(x,t) :=/angbracketleftBig a(t,·)/parenleftbig ∇u0(x,t)+∇yw1(x,t,·,·)/parenrightbig/angbracketrightBig y,s. Recalling (3.10) and (3.18), we observe that, for all φ∈H1 0(Ω) andψ∈C∞ c(I), /integraldisplayT 0/a\}b∇acketle{tf(t),φ/a\}b∇acket∇i}htH1 0(Ω)ψ(t)dt = lim εn→0+/integraldisplayT 0/a\}b∇acketle{tfεn(t),φ/a\}b∇acket∇i}htH1 0(Ω)ψ(t)dt (1.10)= lim εn→0+/integraldisplayT 0/bracketleftBig /a\}b∇acketle{t∂2 ttuεn(t)+g(t εrn)∂tuεn(t),φ/a\}b∇acket∇i}htH1 0(Ω)+/parenleftbig a(t,x εn)∇uεn(t),∇φ/parenrightbig L2(Ω)/bracketrightBig ψ(t)dt (3.10),(3.18)=/integraldisplayT 0/bracketleftBig /a\}b∇acketle{tw,φ/a\}b∇acket∇i}htH1 0(Ω)+/parenleftbig jhom(t),∇φ/parenrightbig L2(Ω)/bracketrightBig ψ(t)dt. Herewcan be regarded as (4.1) w=∂2 ttu0+/a\}b∇acketle{tgper/a\}b∇acket∇i}hts∂tu0+C∗h. Actually, due to ψ∈C∞ c(Ω), this follows from (3.12), (3.14) and Proposition 2.2. Hence, by the arbitrariness of ψ∈C∞ c(I),u0turns out to be a weak solution to (4.2)/braceleftBigg ∂2 ttu0−divjhom+/a\}b∇acketle{tgper/a\}b∇acket∇i}hts∂tu0+C∗h=fin Ω×I, u0|∂Ω= 0, u0|t=0=v0, ∂tu0|t=0= ˜v1,SPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 15 where ˜v1=/braceleftBigg v1ifC∗= 0, 0 ifC∗/\e}atio\slash= 0. Indeed, noting that /ba∇dblu0(0)−v0/ba∇dblL2(Ω)≤ /ba∇dblu0(0)−uεn(0)/ba∇dblL2(Ω)+/ba∇dbluεn(0)−v0/ba∇dblL2(Ω) ≤ /ba∇dblu0−uεn/ba∇dblC(I;L2(Ω))+/ba∇dblv0 εn−v0/ba∇dblL2(Ω), we see by (3.13) and (A)that /ba∇dblu0(0)−v0/ba∇dblL2(Ω)≤limsup εn→0+/ba∇dblu0−uεn/ba∇dblC(I;L2(Ω))+limsup εn→0+/ba∇dblv0 εn−v0/ba∇dblL2(Ω)= 0, which implies that u0(x,0) =v0. Thusu0≡v0by∂tu0≡0, provided that for C∗/\e}atio\slash= 0. To check∂tu0(x,0) =v1(x) a.e. in Ω for C∗= 0, letψ∈C∞(I) be such that ψ(T) = 0 andψ(0) = 1. Then we infer that, for all φ∈C∞ c(Ω), /integraldisplay Ωv1(x)φ(x)dx(A),(1.10)= lim εn→0+/integraldisplay Ωv1 εn(x)φ(x)dx + lim εn→0+/integraldisplayT 0/angbracketleftBig ∂2 ttuεn(t)+g(t εrn)∂tuεn(t),φ/angbracketrightBig H1 0(Ω)ψ(t)dt + lim εn→0+/integraldisplayT 0/integraldisplay Ω/bracketleftBig a(t,x εn)∇uεn(x,t)·∇φ(x)ψ(t)−fεn(x,t)φ(x)ψ(t)/bracketrightBig dxdt = lim εn→0+/integraldisplayT 0/integraldisplay Ω/bracketleftBig −∂tuεn(x,t)φ(x)∂tψ(t)+g(t εrn)∂tuεn(x,t)φ(x)ψ(t) +a(t,x εn)∇uεn(x,t)·∇φ(x)ψ(t)−fεn(x,t)φ(x)ψ(t)/bracketrightBig dxdt =/integraldisplayT 0/integraldisplay Ω/bracketleftBig −∂tu0(x,t)φ(x)∂tψ(t)+/a\}b∇acketle{tgper/a\}b∇acket∇i}hts∂tu0(x,t)φ(x)ψ(t) +jhom(x,t)·∇φ(x)ψ(t)−f(x,t)φ(x)ψ(t)/bracketrightBig dxdt (4.2)=/integraldisplay Ω∂tu0(x,0)φ(x)dx, which together with the arbitrariness of φ∈C∞ c(Ω) yields that ∂tu0(x,0) =v1(x) a.e. in Ω forC∗= 0. The rest of the proof is to show that (4.3) jhom=ahom(t)∇u0(x,t). Hereahom(t) is the homogenized matrix defined by (1.20). Thus it suffices to prov e (1.16), that is, (4.4) /a\}b∇acketle{tw1/a\}b∇acket∇i}hts=u1:=N/summationdisplay k=1∂xku0(x,t)Φk(t,y), where Φ kis the corrector defined by either (1.17) or (1.18). Indeed, if (4.4) holds, then we derive that jhom(x,t) =/angbracketleftbig a(t,·)/parenleftbig ∇u0(x,t)+∇yw1(x,t,·,·)/parenrightbig/angbracketrightbig y,s16 TOMOYUKI OKA (4.4)=/integraldisplay /squarea(t,y)/parenleftBig ∇u0(x,t)+N/summationdisplay k=1∂xku0(x,t)∇yΦk(t,y)/parenrightBig dy =N/summationdisplay k=1/parenleftBig/integraldisplay /squarea(t,y)(∇yΦk(t,y)+ek)dy/parenrightBig /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright =ahom(t)ekby (1.20)∂xku0(x,t) =ahom(t)∇u0(x,t), whichimplies(4.3). Hence u0turnsouttobeauniqueweaksolutionto(1.19). Indeed, this follows from the uniqueness of the corrector Φ kand the similar argument as in Theorem 1.4 ifC∗= 0 andu0≡v0wheneverC∗/\e}atio\slash= 0. Thus we have uε→u0asε→0+ without taking any subsequence ( εn). Therefore, (1.11)–(1.15) hold by Lemma 3.4 and (4.1). Thus we get all the assertions. In the rest of this section, we shall prove (4.4) for all 0 < r <+∞. To this end, we show the following Lemma 4.1. Under the same assumption as in Theorem 1.5, it holds that lim εn→0+ε1−r n/integraldisplayT 0/integraldisplay Ω/bracketleftBig −∂tuεn(x,t)∂sc(t εrn)+C∗t∂tuεn(x,t)c(t εrn)/bracketrightBig φ(x)b(x εn)ψ(t)dxdt (4.5) + lim εn→0+/integraldisplayT 0/integraldisplay Ωa(t,x εn)∇uεn(x,t)·φ(x)∇yb(x εn)ψ(t)c(t εrn)dxdt= 0 for allφ∈C∞ c(Ω),b∈C∞ per(/square),ψ∈C∞ c(I)andc∈C∞ per(J). Proof.Taking a difference of the weak forms for (1.8) and (4.2) and recalling win (4.1), we observe that 0 =/integraldisplayT 0/angbracketleftBig ∂2 ttuεn(t)+g(t εrn)∂tuεn(t)−w(t),φb(· εn)/angbracketrightBig H1 0(Ω)ψ(t)c(t εrn)dt +/integraldisplayT 0/integraldisplay Ω/parenleftbig a(t,x εn)∇uεn(x,t)−jhom(x,t)/parenrightbig ·∇/parenleftbig φ(x)b(x εn)/parenrightbig ψ(t)c(t εrn)dxdt −/integraldisplayT 0/integraldisplay Ω(fεn−f)(x,t)φ(x)b(x εn)ψ(t)c(t εrn)dxdt =−/integraldisplayT 0/integraldisplay Ω∂tuεn(x,t)φ(x)b(x εn)/parenleftbig ∂tψ(t)c(t εrn)+ψ(t)ε−r n∂sc(t εrn)/parenrightbig dxdt +/integraldisplayT 0/integraldisplay Ω/parenleftbig gper(t εrn)+C∗t εrn/parenrightbig ∂tuεn(x,t)φ(x)b(x εn)ψ(t)c(t εrn)dxdt −/integraldisplayT 0/angbracketleftBig w(t),φb(· εn)/angbracketrightBig H1 0(Ω)ψ(t)c(t εrn)dt +/integraldisplayT 0/integraldisplay Ω/parenleftbig a(t,x εn)∇uεn(x,t)−jhom(x,t)/parenrightbig ·/parenleftbig ∇φ(x)b(x εn)+ε−1 nφ(x)∇yb(x εn)/parenrightbig ψ(t)c(t εrn)dxdt −/integraldisplayT 0/integraldisplay Ω(fεn−f)(x,t)φ(x)b(x εn)ψ(t)c(t εrn)dxdt.SPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 17 Multiplying both sides by εn, we conclude that −ε1−r n/integraldisplayT 0/integraldisplay Ω∂tuεn(x,t)φ(x)b(x εn)ψ(t)∂sc(t εrn)dxdt (4.6) +/integraldisplayT 0/integraldisplay Ωa(t,x εn)∇uεn(x,t)·φ(x)∇yb(x εn)ψ(t)c(t εrn)dxdt +ε1−r n/integraldisplayT 0/integraldisplay ΩC∗t∂tuεn(x,t)φ(x)b(x εn)ψ(t)c(t εrn)dxdt =εn/integraldisplayT 0/integraldisplay Ω∂tuεn(x,t)φ(x)b(x εn)∂tψ(t)c(t εrn)dxdt −εn/integraldisplayT 0/integraldisplay Ωgper(t εrn)∂tuεn(x,t)φ(x)b(x εn)ψ(t)c(t εrn)dxdt +εn/integraldisplayT 0/angbracketleftBig w(t),φb(· εn)/angbracketrightbig H1 0(Ω)ψ(t)c(t εrn)dt −εn/integraldisplayT 0/integraldisplay Ω/parenleftbig a(t,x εn)∇uεn(x,t)−jhom(x,t)/parenrightbig ·∇φ(x)b(x εn)ψ(t)c(t εrn)dxdt +/integraldisplayT 0/integraldisplay Ωjhom(x,t)·φ(x)∇yb(x εn)ψ(t)c(t εrn)dxdt +εn/integraldisplayT 0/integraldisplay Ω(fεn−f)(x,t)φ(x)b(x εn)ψ(t)c(t εrn)dxdt→0 asεn→0+. Here we used (A), Lemmas 3.1 and 3.4, Proposition 2.2 and /a\}b∇acketle{t∇yb/a\}b∇acket∇i}hty= 0. /square Employing Lemma 3.4 and Corollary 2.8, we shall apply (4.5) for any φ∈C∞ c(Ω), b∈C∞ per(/square)/R,ψ∈C∞ c(I) andc∈C∞ per(J) to show (4.4). Lemma 4.2. For any0<r≤1,(4.4)holds. Proof.Setc(s)≡1 in (4.5). By (3.15), the first term in (4.5) is zero. Thanks to (3.17), we deduce by (4.5) that /integraldisplayT 0/integraldisplay Ω/integraldisplay1 0/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x,t)+∇yw1(x,t,y,s)/parenrightbig ·φ(x)∇yb(y)ψ(t)c(s)dZ= 0. From the arbitrariness of φ∈C∞ c(Ω) andψ∈C∞ c(I), we get (4.7)/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x,t)+∇y/a\}b∇acketle{tw1(x,t,y,·)/a\}b∇acket∇i}hts/parenrightbig ·∇yb(y)dy= 0 a.e. in Ω ×I. Recalling that u1=N/summationdisplay k=1∂xku0(x,t)Φk(y), where Φ kis the solution to (1.17), we check that /integraldisplay /squarea(t,y)/parenleftbig ∇u0(x,t)+∇yu1(x,t,y)/parenrightbig ·∇yb(y)dy =N/summationdisplay k=1∂xku0(x,t)/integraldisplay /squarea(t,y)/parenleftbig ∇yΦk(y)+ek/parenrightbig ·∇yb(y)dy(1.17)= 0.18 TOMOYUKI OKA Hence (4.7) with /a\}b∇acketle{tw1/a\}b∇acket∇i}htsreplaced by u1(x,t,y) holds. Setting b= (/a\}b∇acketle{tw1/a\}b∇acket∇i}hts−u1)(x,t,·) and subtracting (4.7) for /a\}b∇acketle{tw1/a\}b∇acket∇i}htyandu1, we deduce by the Poincar´ e-Wirtinger inequality that 0 =/integraldisplay /squarea(t,y)∇y(/a\}b∇acketle{tw1/a\}b∇acket∇i}hts−u1)(x,t,y)·∇y(/a\}b∇acketle{tw1/a\}b∇acket∇i}hts−u1)(x,t,y)dy (1.9) ≥λ/ba∇dbl∇y(/a\}b∇acketle{tw1/a\}b∇acket∇i}hts−u1)(x,t)/ba∇dbl2 L2(/square)≥λ C/square/ba∇dbl(/a\}b∇acketle{tw1/a\}b∇acket∇i}hts−u1)(x,t)/ba∇dbl2 L2(/square), which implies that /a\}b∇acketle{tw1/a\}b∇acket∇i}hts=u1. This completes the proof. /square Before discussing the case r>1, we claim that w1=w1(x,t,y) for allr∈(1,+∞). Indeed, multiplying both sides by ε−2(1−r) nin (4.6), we see that the third term in (4.6) is zero asεn→0+due to (3.15), and then, one can derive by Lemma 3.1 and Corollary 2.8 that 0 =−lim εn→0+εr−1 n/integraldisplayT 0/integraldisplay Ω∂tuεn(x,t)φ(x)b(x εn)ψ(t)∂sc(t εrn)dxdt = lim εn→0+εr−1 n/integraldisplayT 0/integraldisplay Ωuεn(x,t)φ(x)b(x εn)∂tψ(t)∂sc(t εrn)dxdt /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright =0 + lim εn→0+/integraldisplayT 0/integraldisplay Ωuεn εn(x,t)φ(x)b(x εn)ψ(t)∂2 ssc(t εrn)dxdt =/integraldisplayT 0/integraldisplay Ω/integraldisplay1 0/integraldisplay /squarew1(x,t,y,s)φ(x)b(y)ψ(t)∂2 ssc(s)dZ, which implies that ∂sw1is independent of s∈Jand so isw1byJ-periodicity. Thus w1∈L2(Ω×I;H1 per(/square)/R) for allr>1. We choose c(s)≡1 in (4.5) below. Then one can get the following Lemma 4.3. For any1<r≤2,(4.4)holds. Proof.As for the periodic case, (4.7) follows from (4.5) and (3.17) with c(s)≡1 and C∗= 0. Thus the assertion is obtained as in the proof of Lemma 4.2. We ne xt consider the quasi-periodic case. Applying Corollary 2.8 to (4.5) with c(s)≡1, we deduce that lim εn→0+ε1−r n/integraldisplayT 0/integraldisplay ΩC∗t∂tuεn(x,t)φ(x)b(x εn)ψ(t)dxdt (4.8) =−lim εn→0+ε2−r n/integraldisplayT 0/integraldisplay ΩC∗uεn εn(x,t)φ(x)b(x εn)∂t/parenleftbig tψ(t)/parenrightbig dxdt = 0 if 1<r<2, −/integraldisplayT 0/integraldisplay Ω/integraldisplay /squareC∗w1(x,t,y)φ(x)b(y)∂t/parenleftbig tψ(t)/parenrightbig dydxdt ifr= 2. Thus (3.17) and (4.5) yield (4.7) for the case 1 <r<2. On the other hand, for r= 2, we find by (3.17), (4.5) and (4.8) that /integraldisplayT 0/integraldisplay Ω/angbracketleftbig C∗t∂tw1(x,t,·),b/angbracketrightbig H1per(/square)/Rφ(x)ψ(t)dxdtSPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 19 +/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yw1(x,t,y)/parenrightbig ·φ(x)∇yb(y)ψ(t)dydxdt= 0. Here we used the fact that −C∗/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarew1(x,t,y)φ(x)b(y)∂t/parenleftbig tψ(t)/parenrightbig dydxdt (4.9) =C∗/integraldisplayT 0/integraldisplay Ω/angbracketleftbig t∂tw1(x,t,·),b/angbracketrightbig H1per(/square)/Rφ(x)ψ(t)dxdt. Indeed, define ξ(x,t,·)∈(H1 per(/square)/R)∗by /integraldisplayT 0/angbracketleftbig ξ(x,t,·),ζ(t,·)/angbracketrightbig H1per(/square)/Rdt=/integraldisplayT 0/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yw1(x,t,y)/parenrightbig ·∇yζ(t,y)dydt forζ∈L2(I;H1 per(/square)/R). Here (H1 per(/square)/R)∗is the dual space of H1 per(/square)/R. By Pettis’s theorem, we see that ξ: Ω→L2(I;(H1 per(/square)/R)∗) is measurable, and moreover, ξ∈ L2(Ω;L2(I;(H1 per(/square)/R)∗)) due tou0∈H1 0(Ω) and∇yw1∈[L2(Ω×I×/square)]N. Hence one can verify by (4.5) and (4.8) that C∗/integraldisplayT 0w1(x,t,·)∂t/parenleftbig tψ(t)/parenrightbig dt=/integraldisplayT 0ξ(x,t,·)ψ(t)dtin (H1 per(/square)/R)∗. Sinceψ∈C∞ c(I) is arbitrary, we have C∗t∂tw1(x,t,·) =−ξ(x,t,·) in (H1 per(/square)/R))∗ in the distributional sense for a.e. ( x,t)∈Ω×I. Thus (4.9) follows, and then, the arbitrariness of φ∈C∞ c(Ω) yields that /integraldisplayT 0/angbracketleftbig C∗t∂tw1(x,t,·),b/angbracketrightbig H1per(/square)/Rψ(t)dt (4.10) +/integraldisplayT 0/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yw1(x,t,y)/parenrightbig ·∇yb(y)ψ(t)dydt= 0 a.e. in Ω. Now, recall (1.16), where Φ kis the solution to (1.18). Then (4.10) with w1replaced by u1(x,t,y) holds. Hence choosing bψ= (w1−u1)(x,·,·) and subtracting (4.10) for w1and u1, we derive that 0 =/integraldisplayT 0C∗t 2d dt/ba∇dbl(w1−u1)(x,t)/ba∇dbl2 L2(/square)dt (4.11) +/integraldisplayT 0/integraldisplay /squarea(t,y)∇y(w1−u1)(x,t,y)·∇y(w1−u1)(x,t,y)dydt (1.9) ≥C∗T 2/ba∇dbl(w1−u1)(x,T)/ba∇dbl2 L2(/square)−C∗ 2/integraldisplayT 0/ba∇dbl(w1−u1)(x,t)/ba∇dbl2 L2(/square)dt +λ/integraldisplayT 0/ba∇dbl∇y(w1−u1)(x,t)/ba∇dbl2 L2(/square)dt ≥C∗T 2/ba∇dbl(w1−u1)(x,T)/ba∇dbl2 L2(/square)+/parenleftbigg −C∗ 2+λ C/square/parenrightbigg /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright ≥0 by(A)/integraldisplayT 0/ba∇dbl(w1−u1)(x,t)/ba∇dbl2 L2(/square)dt,20 TOMOYUKI OKA which implies w1=u1. Furthermore, (4.11) yields the uniqueness of (1.18), andmoreov er, ifa=a(y), then Φ kis the solution to (1.17) due to the uniqueness of (1.18). This completes the proof. /square We finally discuss the case where 2 <r<+∞. Lemma 4.4. For any2<r<+∞,(4.4)holds. Proof.Due to (3.17) and (4.5), it suffices to check lim εn→0+ε1−r n/integraldisplayT 0/integraldisplay ΩC∗t∂tuεn(x,t)φ(x)b(x εn)ψ(t)c(t εrn)dxdt= 0 withc(s)≡1. It is clear if C∗= 0. IfC∗/\e}atio\slash= 0, since ( tε−r∂tuε) is bounded in L2(Ω×Iσ) by (v) of Lemma 3.3, we find by ψ∈C∞ c(I) that ε1−r n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayT 0/integraldisplay ΩC∗t∂tuεn(x,t)φ(x)b(x εn)ψ(t)dxdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cεn/ba∇dbltε−r n∂tuεnψ/ba∇dblL1(Ω×I)→0 asεn→0+, which completes the proof. /square By Lemmas 4.2, 4.3 and 4.4, we obtain (4.4) for all r∈(0,+∞). Therefore, Theorem 1.5 is proved. 5.Proof of Proposition 1.7 We consider the case of C∗/\e}atio\slash= 0 andr= 2 only (see [1, Proposition 1.8] for the proof of the other case). We first prove (i). For each ξ∈RN, there exists a unique solution Φξ∈L2(I;H1 per(/square)/R) to C∗t∂tΦξ−divy/bracketleftbig a(t,y)(∇yΦξ+ξ)/bracketrightbig = 0 inI×/square. (5.1) Using (1.20) and (1.9), we derive that, for a.e. t∈I, ahom(t)ξ·ξ(1.20)=/integraldisplay /squarea(t,y)/parenleftbig ∇yΦξ(t,y)+ξ/parenrightbig ·ξdy (5.1)=/integraldisplay /squarea(t,y)(∇yΦξ(t,y)+ξ)·(∇yΦξ(t,y)+ξ)dy+C∗t 2d dt/ba∇dblΦξ(t)/ba∇dbl2 L2(/square) (1.9) ≥λ/integraldisplay /square|ξ+∇yΦξ(t,y)|2dy+C∗t 2d dt/ba∇dblΦξ(t)/ba∇dbl2 L2(/square) =λ/parenleftBig |ξ|2+/ba∇dbl∇yΦξ(t)/ba∇dbl2 L2(/square)/parenrightBig +C∗t 2d dt/ba∇dblΦξ(t)/ba∇dbl2 L2(/square). Here we used the fact that /a\}b∇acketle{t∇yΦξ(t,·)/a\}b∇acket∇i}hty= 0. In an analogous way, we get ahom(t)ξ·ξ≤/parenleftBig |ξ|2+/ba∇dbl∇yΦξ(t)/ba∇dbl2 L2(/square)/parenrightBig +C∗t 2d dt/ba∇dblΦξ(t)/ba∇dbl2 L2(/square). We next prove (ii). Let Φ jbe the unique solution to (5.1) with ξreplaced by ej. Then we observe from the symmetry of a(t,y) that, for a.e. t∈I, tahom(t)ej·ek=ahom(t)ek·ej =/integraldisplay /squarea(t,y)(∇yΦk(t,y)+ek)·ejdySPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 21 +/integraldisplay /squarea(t,y)(∇yΦk(t,y)+ek)·∇yΦj(t,y)dy+/angbracketleftbig C∗t∂tΦk(t),Φj(t)/angbracketrightbig H1per(/square)/R /bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright =0 by (1.18) =/integraldisplay /square(∇yΦk(t,y)+ek)·ta(t,y)(∇yΦj(t,y)+ej)dy+/angbracketleftbig C∗t∂tΦk(t),Φj(t)/angbracketrightbig H1per(/square)/R =ahom(t)ej·ek +/bracketleftbigg/integraldisplay /squarea(t,y)(∇yΦj(t,y)+ej)·∇yΦk(t,y)dy+/angbracketleftbig C∗t∂tΦk(t),Φj(t)/angbracketrightbig H1per(/square)/R/bracketrightbigg . However, for the second term in the last line, it follows that/integraldisplay /squarea(t,y)(∇yΦj(t,y)+ej)·∇yΦk(t,y)dy+/angbracketleftbig C∗t∂tΦk(t),Φj(t)/angbracketrightbig H1per(/square)/R =C∗t/parenleftBig −/angbracketleftbig ∂tΦj(t),Φk(t)/angbracketrightbig H1per(/square)/R+/angbracketleftbig ∂tΦk(t),Φj(t)/angbracketrightbig H1per(/square)/R/parenrightBig /\e}atio\slash= 0 forj/\e}atio\slash=k, which completes the proof. Remark 5.1. The skew symmetric part of ahom(t) is defined by askew hom(t)ej·ek=/parenleftbiggahom(t)−tahom(t) 2/parenrightbigg ej·ek =−1 2C∗t/parenleftBig −/angbracketleftbig ∂tΦj(t),Φk(t)/angbracketrightbig H1per(/square)/R+/angbracketleftbig ∂tΦk(t),Φj(t)/angbracketrightbig H1per(/square)/R/parenrightBig . Then we note that the skew-symmetric part of ahom(t) makes no contribution to the divergence for a.e. t∈I. Assume that u0and Φ kare smooth enough. Then we find by the symmetry of the Hessian that, for a.e. t∈I, div(askew hom(t)∇u0) =1 2div(askew hom(t)∇u0) −1 4N/summationdisplay j,k=1/bracketleftbigg C∗t/integraldisplay /square/parenleftBig −∂tΦj(t,y)Φk(t,y)+∂tΦk(t,y)Φj(t,y)/parenrightBig dy/bracketrightbigg ∂2 xjxku0 =1 2div(askew hom(t)∇u0) +1 4N/summationdisplay j,k=1/bracketleftbigg C∗t/integraldisplay /square/parenleftBig −∂tΦk(t,y)Φj(t,y)+∂tΦj(t,y)Φk(t,y)/parenrightBig dy/bracketrightbigg ∂2 xkxju0 =1 2div(askew hom(t)∇u0)−1 2div(askew hom(t)∇u0) = 0, which yields the assertion. 6.Proof of a corrector result This section is devoted to proving Theorem 1.9 and Corollary 1.10. 6.1.Proof of Theorem 1.9. Letaε=a(x ε) for the sake of simplicity. To show Theorem 1.9, we observe from (1.9) that λ/integraldisplayT 0/integraldisplay Ω/vextendsingle/vextendsingle∇uε−/parenleftbig ∇u0+∇yu1(x,t,x ε)/parenrightbig/vextendsingle/vextendsingle2dxdt22 TOMOYUKI OKA (1.9) ≤/integraldisplayT 0/integraldisplay Ωaε/parenleftbig ∇uε−/parenleftbig ∇u0+∇yu1(x,t,x ε)/parenrightbig/parenrightbig ·/parenleftbig ∇uε−/parenleftbig ∇u0+∇yu1(x,t,x ε)/parenrightbig/parenrightbig dxdt =/integraldisplayT 0/integraldisplay Ωaε∇uε·∇uεdxdt−2/integraldisplayT 0/integraldisplay Ωaε∇uε·/parenleftbig ∇u0+∇yu1(x,t,x ε)/parenrightbig dxdt +/integraldisplayT 0/integraldisplay Ωaε/parenleftbig ∇u0+∇yu1(x,t,x ε)/parenrightbig ·/parenleftbig ∇u0+∇yu1(x,t,x ε)/parenrightbig dxdt=:Iε 1−2Iε 2+Iε 3. In what follows, we shall estimate these three terms, Iε 1,Iε 2andIε 3for allr∈(0,+∞). We first estimate Iε 1. Lemma 6.1. Under the same assumption as in Theorem 1.9, it holds that lim ε→0+/integraldisplayT 0/integraldisplay Ωaε∇uε(x,t)·∇uε(x,t)dxdt =/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(y)/parenleftbig ∇u0(x,t)+∇yu1(x,t,y)/parenrightbig ·/parenleftbig ∇u0(x,t)+∇yu1(x,t,y)/parenrightbig dydxdt. Proof.From(A), (3.13) and (iii) of Lemma 3.3, it follows that /integraldisplayT 0/integraldisplay Ωaε∇uε(x,t)·∇uε(x,t)dxdt (1.10)=/integraldisplayT 0/integraldisplay Ωfε(x,t)uε(x,t)dxdt−/integraldisplayT 0/integraldisplay Ω/parenleftbig ∂2 ttuε(x,t)+g(t εr)∂tuε(x,t)/parenrightbig uε(x,t)dxdt →/integraldisplayT 0/integraldisplay Ωf(x,t)u0(x,t)dxdt−/integraldisplayT 0/integraldisplay Ωw(x,t)u0(x,t)dxdt =/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(y)/parenleftbig ∇u0(x,t)+∇yu1(x,t,y)/parenrightbig ·/parenleftbig ∇u0(x,t)+∇yu1(x,t,y)/parenrightbig dydxdt asε→0+. Here we used the fact that Φ kis the unique solution to (1.17) due to a=a(y). This completes the proof. /square Before discussing the limit of Iε 2, recall that u1is written by u1=/summationtextN k=1∂xku0Φk. Due to the smoothness of a(y), noting that a(y)(∇u0+∇yu1) belongs to [ L2(Ω×I;Cper(/square))]N, we see by [22, Theorem 4] that it is an admissible test function in [ L2(Ω×I×/square)]N. Hence (3.17) yields that Iε 2→/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(y)/parenleftbig ∇u0(x,t)+∇yu1(x,t,y)/parenrightbig ·(∇u0(x,t)+∇yu1(x,t,y))dydxdt (6.1) asε→0+. We finally estimate Iε 3. Thanks to ∇yΦk∈Cper(/square), one can derive by Proposition 2.2 that Iε 3=/integraldisplayT 0/integraldisplay Ω/bracketleftBig aε∇u0·∇u0+2aε∇yu1(x,t,x ε)·∇u0+aε∇yu1·∇yu1(x,t,x ε)/bracketrightBig dxdt (6.2) →/integraldisplayT 0/integraldisplay Ω/integraldisplay /square[a(y)∇u0·∇u0+2a(y)∇yu1·∇u0+a(y)∇yu1·∇yu1]dydxdt =/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(y)(∇u0+∇yu1)·(∇u0+∇yu1)dydxdt asε→0+.SPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 23 Consequently, with the aid of Lemma 6.1, (6.1) and (6.2), we obtain lim ε→0+(Iε 1−2Iε 2+Iε 3) = 0, which completes the proof. 6.2.Proof of Corollary 1.10. The strategy of the proof of Corollary 1.10 is the same as Theorem 1.9 and it suffices to show the following Lemma 6.2. Under the same assumption as in Corollary 1.10, it holds that limsup ε→0+/integraldisplayT 0/integraldisplay Ωa(t,x ε)∇uε(x,t)·∇uε(x,t)dxdt ≤/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig ·/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig dydxdt. Proof.Letaε=a(t,x ε) for simplicity. Define Eε(uε(t)) by Eε(uε(t)) =1 2/ba∇dbl∂ρuε(t)/ba∇dbl2 L2(Ω)+1 2/integraldisplay Ωaε∇uε(x,t)·∇uε(x,t)dx +/integraldisplayt 0g(ρ εr)/ba∇dbl∂ρuε(ρ)/ba∇dbl2 L2(Ω)dρ−1 2/integraldisplayt 0/integraldisplay Ω∂ρa(ρ,x ε)∇uε(x,ρ)·∇uε(x,ρ)dxdρ fort∈I. From (i), (ii) and (iii) of Lemma 3.1 and (A), we have |Eε(uε(t))| ≤Cfor all t∈I. Moreover, we derive by Remark 3.2 that, for any t∈Iandh∈(0,T−t), |Eε(uε(t+h))−Eε(uε(t))| ≤/integraldisplayt+h t/vextendsingle/vextendsingle/vextendsingle/a\}b∇acketle{t∂2 ρρuε(ρ),∂ρuε(ρ)/a\}b∇acket∇i}htH1 0(Ω)+/parenleftbig aε∇uε(ρ),∇∂ρuε(ρ)/parenrightbig L2(Ω)+g(ρ εr)/ba∇dbl∂ρuε(ρ)/ba∇dbl2 L2(Ω)/vextendsingle/vextendsingle/vextendsingledρ (1.10) ≤/integraldisplayt+h t/ba∇dblfε(ρ)/ba∇dblL2(Ω)/ba∇dbl∂ρuε(ρ)/ba∇dblL2(Ω)dρ≤ /ba∇dblfε/ba∇dblL2(Ω×I)/ba∇dbl∂ρuε/ba∇dblL∞(I;L2(Ω))|h|1/2, which along with (ii) of Lemma 3.1 and the boundedness of ( fε) inL2(Ω×I) yields the equicontinuous of t/ma√sto→Eε(uε(t)) onI. Then Ascoli-Arzel´ a’s theorem ensures that there existsξ∈C(I) such that Eε(uε)→ξstrongly in C(I). (6.3) Furthermore, it holds that, for any t∈I, Eε(uε(t)) =/integraldisplayt 0/vextendsingle/vextendsingle/vextendsingle/a\}b∇acketle{t∂2 ρρuε(ρ),∂ρuε(ρ)/a\}b∇acket∇i}htH1 0(Ω)+/parenleftbig aε∇uε(ρ),∇∂ρuε(ρ)/parenrightbig L2(Ω)+g(ρ εr)/ba∇dbl∂ρuε(ρ)/ba∇dbl2 L2(Ω)/vextendsingle/vextendsingle/vextendsingledρ +1 2/ba∇dblv1 ε/ba∇dbl2 L2(Ω)+1 2/integraldisplay Ωa(0,x ε)∇v0 ε(x)·∇v0 ε(x)dx (1.10)=/integraldisplayt 0/integraldisplay Ωfε(x,ρ)∂ρuε(x,ρ)dxdρ+1 2/ba∇dblv1 ε/ba∇dbl2 L2(Ω)+1 2/integraldisplay Ωa(0,x ε)∇v0 ε(x)·∇v0 ε(x)dx. Then, by (1.23), (1.24) and (1.25), ξ(t) is identified with ξ(t) =1 2/integraldisplay Ωahom(0)∇v0(x)·∇v0(x)dx24 TOMOYUKI OKA =1 2/integraldisplay Ω/integraldisplay /squarea(0,y)/parenleftbig ∇u0(x)+∇yu1(x,0,y)/parenrightbig ·∇u0(x)dydx. Hence defining J(t) by J(t) :=/integraldisplayt 0/integraldisplay Ω/integraldisplay /square∂ρa(ρ,y)/parenleftbig ∇u0(x)+∇yu1(x,ρ,y)/parenrightbig ·/parenleftbig ∇u0(x)+∇yu1(x,ρ,y)/parenrightbig dydxdρ and noting by (1.22) and [36, Example 1 in Section 7] that liminf ε→0+/integraldisplayt 0/integraldisplay Ω(−∂ρaε)∇uε(x,ρ)·∇uε(x,ρ)dxdρ≥ −J(t), one can derive by (6.3) that limsup ε→0+/integraldisplayT 0/integraldisplay Ωaε∇uε(x,t)·∇uε(x,t)dxdt ≤lim ε→0+/integraldisplayT 02Eε(uε)(t)dt+limsup ε→0+/integraldisplayT 0/integraldisplayt 0/integraldisplay Ω∂ρa(ρ,x ε)∇uε(x,ρ)·∇uε(x,ρ)dxdρdt ≤T/integraldisplay Ω/integraldisplay /squarea(0,y)/parenleftbig ∇u0(x)+∇yu1(x,0,y)/parenrightbig ·∇u0(x)dydx+/integraldisplayT 0J(t)dt. We estimate the second term below. Since u1(x,·,·) is smooth in /square×(η,T) for allη∈I due to the smoothness of a(t,y), it holds that /integraldisplayT η/integraldisplayt η/integraldisplay Ω/integraldisplay /square∂ρa(ρ,y)/parenleftbig ∇u0(x)+∇yu1(x,ρ,y)/parenrightbig ·/parenleftbig ∇u0(x)+∇yu1(x,ρ,y)/parenrightbig dydxdρdt =/integraldisplayT η/integraldisplay Ω/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig ·/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig dydxdt −(T−η)/integraldisplay Ω/integraldisplay /squarea(η,y)/parenleftbig ∇u0(x)+∇yu1(x,η,y)/parenrightbig ·/parenleftbig ∇u0(x)+∇yu1(x,η,y)/parenrightbig dydx −2/integraldisplayT η/integraldisplayt η/integraldisplay Ω/integraldisplay /squarea(ρ,y)/parenleftbig ∇u0(x)+∇yu1(x,ρ,y)/parenrightbig ·∇y∂ρu1(x,ρ,y)dydxdρdt =:Jη 1+Jη 2+Jη 3. Then one can verify that Jη 1→/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig ·/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig dydxdt and (6.4) Jη 2→ −T/integraldisplay Ω/integraldisplay /squarea(0,y)/parenleftbig ∇u0(x)+∇yu1(x,0,y)/parenrightbig ·∇u0(x)dydx asη→0+. Here we used the fact (1.18) at t= 0 in (6.4). Furthermore, if r/\e}atio\slash= 2,Jη 3= 0 readily follows due to u1=u1(x,y). On the other hand, if r= 2, (1.18) and (1.26) yield that Jη 3(1.26)= 2/integraldisplayT η/integraldisplayt η/integraldisplay Ω/integraldisplay /square∂ρa(ρ,y)/parenleftbig ∇u0(x)+∇yu1(x,ρ,y)/parenrightbig ·∇y∂ρu1(x,ρ,y)dydxdρdt ≤2/integraldisplayT η/integraldisplay Ω/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig ·∇y∂ρu1(x,t,y)dydxdtSPACE-TIME QUASI-PERIODIC HOMOGENIZATION FOR DAMPED WAVE EQUATIONS 25 −2(T−η)/integraldisplay Ω/integraldisplay /squarea(η,y)/parenleftbig ∇u0(x)+∇yu1(x,η,y)/parenrightbig ·∇y∂ρu1(x,η,y)dydx −2/integraldisplayT η/integraldisplayt η/integraldisplay Ω/integraldisplay /squarea(ρ,y)/parenleftbig ∇u0(x)+∇yu1(x,ρ,y)/parenrightbig ·∇y∂2 ρρu1(x,ρ,y)dydxdρdt (1.18)=−2/integraldisplayT ηC∗t/ba∇dbl∂ρu1(t)/ba∇dbl2 L2(Ω×/square)dt+2(T−η)C∗η/ba∇dbl∂ρu1(η)/ba∇dbl2 L2(Ω×/square) +/integraldisplayT η/integraldisplayt ηC∗ρd dρ/ba∇dbl∂ρu1(ρ)/ba∇dbl2 L2(Ω×/square)dρdt ≤ −/integraldisplayT ηC∗t/ba∇dbl∂ρu1(t)/ba∇dbl2 L2(Ω×/square)dt+2(T−η)C∗η/ba∇dbl∂ρu1(η)/ba∇dbl2 L2(Ω×/square)≤0 asη→0+. Hence we conclude that /integraldisplayT 0J(t)dt= lim η→0+Jη 1+ lim η→0+Jη 2+ lim η→0+Jη 3 ≤/integraldisplayT 0/integraldisplay Ω/integraldisplay /squarea(t,y)/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig ·/parenleftbig ∇u0(x)+∇yu1(x,t,y)/parenrightbig dydxdt −T/integraldisplay Ω/integraldisplay /squarea(0,y)/parenleftbig ∇u0(x)+∇yu1(x,0,y)/parenrightbig ·∇u0(x)dydx, which completes the proof. /square Acknowledgment The author is partially supported by Division for Interdisciplinary Adv anced Research and Education, Tohoku University and Grant-in-Aid for JSPS Fellows (No. 20J10143). He would like to thank Professor Goro Akagi (Tohoku University) wh o is his supervisor, for many stimulating discussions. References [1] G. Akagi, T. Oka, Space-time homogenization for nonlinear diffusio n, preprint, arXiv:2007.09977 (2020), 1–58. [2] G. Allaire, Homogenization and two-scaleconvergence, SIAM J. M ath. Anal. 23(1992), 1482–1518. [3] G. Allaire, M. 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Corduneanu, Almost periodic oscillations and waves , Springer, New York, 2009. [10] D. Cioranescu, P. Donato, An Introduction to Homogenization , Oxford Lecture Series in Mathe- matics and Its Applications, vol. 17, Oxford University Press, New Y ork, 1999.26 TOMOYUKI OKA [11] S. Cooper, Quasi-periodictwo-scalehomogenisationand effect ive spatial dispersion in high-contrast media, Calc. Var. Partial Differential Equations 57(2018), no. 3, 1–33. [12] H. Douanla, E. Tetsadjio, Reiterated homogenization of hyper bolic-parabolic equations in domains with tiny holes, Electron. J. Differential Equations 59(2017), 1–22. [13] Y. Efendiev, A. Pankov,Homogenizationofnonlinearrandompa rabolicoperators,Adv. Differential Equations 10(2005), 1235–1260. [14] G. A. Francfort, F. Murat, Oscillations and energy densities in t he wave equation, Comm. Partial Differential Equations 17(1992), 1785–1865. [15] L. Flod´ en, A. Holmbom, M. Olsson Lindberg, J. Persson, Two-s cale convergence. 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Wall, Two-scale convergence , Int. J. Pure Appl. Math. 2(2002), 35–86. [23] S. Migorski, Homogenization of hyperbolic-parabolic equations in perforated domains, Univ. Iagel. Acta Math. 33(1996), 59–72. [24] G. Nguetseng, Ageneralconvergenceresult forafunctiona l relatedto the theoryofhomogenization, SIAM J. Math. Anal. 20(1989), 608–623. [25] G. Nguetseng, Homogenization structures and applications I, Z. Anal. Anwend. 22(2003), 73–107. [26] G. Nguetseng, H. Nnang, N. Svanstedt, Deterministic homoge nization of quasilinear damped hy- perbolic equations, Acta Math. Sci. Ser. B (Engl. Ed.) 31(2011), no. 5, 1823–1850. [27] G. Nguetseng, J.-L. Woukeng, Deterministic homogenization of parabolic monotone operators with time dependent coefficients, Electron. J. Differential Equations. 82(2004), 1–23. [28] K. Nishihara, Lp-Lqestimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z. 244(2003), 631–649. [29] H. Nnang, Deterministic homogenization of weakly damped nonline ar hyperbolic-parabolic equa- tions, NoDEA Nonlinear Differential Equations Appl. 19(2012), 539–574. [30] C. Timofte, Homogenization results for a nonlinear wave equatio n in a perforated domain, Po- litehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72(2010), no. 2, 85–92. [31] C. Timofte, Homogenization results for hyperbolic-parabolic eq uations, Romanian Rep. Phys. 62 (2010), 229–238. [32] H.T. To, Homogenization of dynamic laminates, J. Math. Anal. App l.354(2009), 518–538. [33] A. Visintin, Towards a two-scale calculus, ESAIM Control Optim. Calc. Var. 12(2006), 371–397. [34] J.-L. Woukeng, Periodic homogenization of nonlinear non-monot one parabolic operators with three time scales, Ann. Mat. Pura Appl. 189(2010), no. 3, 357–379. [35] J.L. Woukeng, D. Dongo, Multiscale homogenization of nonlinear h yperbolic equations with several time scales, Acta Math. Sci. Ser. B (Engl. Ed.) 31(2011), no. 3, 843–856. [36] V.V. Zhikov, On two-scale convergence, J. Math. Sci. 120(2004), 1328–1352. (TomoyukiOka) Graduate School of Sciences, Tohoku University, Sendai 980 -8579 Japan Email address :tomoyuki.oka.q3@dc.tohoku.ac.jp

2021-07-11

This paper is concerned with space-time homogenization problems for damped wave equations with spatially periodic oscillating elliptic coefficients and temporally (arithmetic) quasi-periodic oscillating viscosity coefficients. Main results consist of a homogenization theorem, qualitative properties of homogenized matrices which appear in homogenized equations and a corrector result for gradients of solutions. In particular, homogenized equations and cell problems will turn out to deeply depend on the quasi-periodicity as well as the log ratio of spatial and temporal periods of the coefficients. Even types of equations will change depending on the log ratio and quasi-periodicity. Proofs of the main results are based on a (very weak) space-time two-scale convergence theory.

Space-time arithmetic quasi-periodic homogenization for damped wave equations

2107.04966v1

1 The following article has been accepted by Journal of Applied Physics . After it is published, it will be found at Link . Spin -torque nano -oscillator based on two in-plane magnetized synthetic ferrimagnets E. Monteblanco1,a), F. Garcia -Sanchez1, M. Romera1,2, D. Gusakova1, L. D. Buda -Prejbeanu1, U. Ebels1 1Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP *, INAC, SPINTEC, F -38000 Grenoble, France 2GFMC, Departamento de Física de Materiales, Universidad Complutense, Madrid, Spain. * Institute of Engineering Univ. Grenoble Alpes a) Electronic email: nmonteblanco@gmail.com We report the dynamic characterization of the spin -torque -driven in- plane precession modes of a spin -torque nano- oscillator based on two different synthetic ferrimagnets: a pinned one characterized by a strong RKKY interaction which is exchange coupled to an antiferromagnetic layer; and a second one, non -pinned characterized by weak RKKY coupling. The microwave properties associated with the steady -state precession of both SyFs are characterized by high spectral purity and power spectral density. However, f requency dispersion diagrams of the damped and spin transfer torque modes reveal drastically different dynamical behavior and microwave emission properties in both SyFs. In particular, the weak coupling between the magnetic layers of the non- pinned SyF raises discontinuous dispersion diagrams suggesting a strong influence of mode crossing. An interpretation of the different dynamical features observed in the damped and spin torque modes of both SyF systems was obtained by solving simultaneously, in a macros pin approach, a linearized version of the Landau -Lifshitz -Gilbert equation including the spin transfer torque term. I. INTRODUCTION The exceptional and multi -functional properties of spin- torque1-2 nano- oscillators (STOs) made them promising candidates for a wide range of emerging technologies which span from microwave emitters3 and detectors4-5 to neuromorphic computing systems6-10. These devices use the transfer of angular momentum from a spin-polarized current to the local magnetization of a ferromagnetic layer11,12 to induce self -sustained oscillations of the magnetization, which translate into a microwave signal whose frequency can be finely tuned with the applied direct current13-17. A widely studied structure of the STO is of the type (AF/SyF/MgO/SL) with in -plane magnetization , including a synthetic ferrimagnet structure (called SyF- Polarizer) pinned via exchange bias to an antiferromagnetic layer (AF) and a sin gle ferromagnetic layer (SL)18-20. A standard SyF layer is composed of two ferromagnetic layers, a top (TL) and bottom (BL) layer coupled antiferromagnetically through a thin metallic layer via the Ruderman- Kittel- Kasuya- Yosida (RKKY) interaction.21,22 In STO s, spin-polarized electrons affect damped oscillations by modifying damping and thus linewidth and amplitudes and overcoming a critical current density (j c) polarized electrons can induce the steady -state oscillations (STT excitations) . Steady -state oscillations can be obtained in both parts, SyF or SL, by changing the polarity of the current. The STT excitations of a SyF structure present some advantages in comparison with the SL excitations, as the higher spectral purity (smaller linewidth), zero field excitations23,24, frequency tuning as a function of the current with the possibility to change from redshift (df/dj app<0) to blueshift (df/dj app>0) applying an in-plane field and achieving large thermal stability25-29. However, since the SyF is pinned by an antiferromagnet, achieving steady -state excitations require s a relatively large critical current. The use of exchange -coupled layers with perpendicular anisotropy30,31 or in- plane magnetized magnetic layers32-34 have been shown to increase the magnetic stiffness and can potentially improve the performance of the oscillator . Replacing the standard free layer SL by an unpinned SyF layer is thus of potential interest towards improving the microwave properties of spin torque oscillators . In this article, we report the main static and dynamic features of a spin torque oscillator based on two SyFs with the following structure : IrMn(6.1)/SyF -Polarizer/MgO(0.9)/SyF -FL, w here numbers 2 represent thickness in nanometers. The composition of the SyF- Polarizer and SyF -FL are CoFe(1.8)/Ru(0.4)/CoFe B(2) and CoFe(0.5)/CoFeB(3.4)/Ru/CoFe(3.6) respectively. The structure of this manuscript is the following: section II introduces the numerical techniques used to predict some features of the STO dispersion diagrams and it is devoted to the macrospin analysis of the STO structure, section III presents the experi mental characterization of the STO dynamics. Section IV presents the discussion and conclusions. In the following, the nano- oscillator based on a double SyF will be called D-SyF for simplicity. II. NUMERICAL ANALYSIS Two types of numerical studies were performed in the framework of the macrospin approximation: (1) computation of the hysteresis loops (MH) and magnetoresistance loops (MR) using the minimization of the energy and (2) stability analysis based on the linearization of the Landau- Lifshitz -Gilbert (LLG) equation enhanced with the spin transfer torque term (Slonczewski term) around the equilibrium positions, in order to find instabilities due to the applied current and fiel d. The D -SyF under study present s the following structure: AF/SyF -Polarizer/insulator barrier/SyF -Free layer, wh ere the bottom layer of the SyF -Polarizer is pinned into the positive x direction by an exchange bias field (H eb) induced by an antiferromagnetic layer, see Figure 1. Fig. 1. (a) Schematics of the double SyF oscillator structure with the labels used in this article. Negative applied current corresponds to electrons flowing from SyF -FL to SyF- Polarizer. (b) Schematic of the standard RF setup. II.1. Simulation of static hysteresis loops In order to understand the complex frequency dispersion diagrams of the D -SyF oscillator, its hysteresis loop has been simulated as the first step considering the different coupling between the ferromagnetic layers. The total free energy density of the coupled system is written as 𝜎𝜎 𝑡𝑡𝑡𝑡𝑡𝑡=𝛴𝛴 𝑖𝑖𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖=𝛴𝛴 𝑖𝑖(𝜎𝜎𝑖𝑖𝑖𝑖𝑡𝑡,𝑖𝑖+ 𝜎𝜎𝑒𝑒𝑒𝑒𝑡𝑡,𝑖𝑖) where the internal and external component for each layer are defined as follow, 𝜎𝜎𝑖𝑖𝑖𝑖𝑡𝑡,𝑖𝑖=𝜎𝜎𝑧𝑧𝑒𝑒𝑒𝑒𝑧𝑧 ,𝑖𝑖+𝜎𝜎𝑎𝑎𝑖𝑖,𝑖𝑖+𝜎𝜎𝑑𝑑,𝑖𝑖+𝜎𝜎𝑒𝑒𝑒𝑒,𝑖𝑖 (1) 𝜎𝜎𝑒𝑒𝑒𝑒𝑡𝑡,𝑖𝑖=𝜎𝜎𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ,𝑖𝑖+𝜎𝜎𝑑𝑑𝑖𝑖𝑑𝑑,𝑖𝑖𝑖𝑖 (2) The magnetic layers are labelled as i, j =1 to 4 and i≠ j, corresponding to the structure on Figure 1. The model includes the different internal energies contributions such as the demagnetizing σ d and magnetocrystalline anisotropies σan, exchange bias σeb (only for the BL of the SyF pinned) and the Zeeman terms σzeem. Also, we include the Ruderman- Kittel- Kasuya -Yosida (RKKY) interlayer exchange interaction σRKKY (internal to each SyF) and the dipolar stray field35-36 σdip, see Eq. (2). The RKKY coupling is taken into account only between the layers 1- 2 and 3- 4 (σRKKY,1(3) = σRKKY,2(4) ) and the 3 dipolar field between the four layers ( σdip,ij= σdip,ji). More details about each term of the equations are found in Ref. 34 and 38. Two different RKKY coupling constants were considered: J RKKY =-0.1mJ/m2 and -1.5mJ/m2 for the SyF -FL (weak coupling) and SyF -Polarizer (strong coupling) layer respectively . The influence of a high or weak RKKY coupling in the hysteresis loops (MH) and magnetoresistance curves (MR) has been shown in previous studies34. The structure considered is not compensated, i.e., the product (M S*t*S) for the TL and BL of each SyF are not close , (M S*t*S)TL,SyF- FL= 35.12 μA nm2, (M S*t*S)BL,SyF- FL=28.79 μA nm2, (M S*t*S)TL,SyF -Polarizer =14.06 μA nm2, and (M S*t*S)BL,SyF -Polarizer =17.56 μA nm2. The net magnetic moment of the SyF -FL is 6.33 μA nm2 and for the SyF -Polarizer is 3.5 μA nm2 so we should consider the stray magnetic field (dipolar field) between both SyFs. This is fundamental to understanding the magnetization dynamics of this structure. SyF Pinned Layer SyF Free Layer Parameters 1 2 3 4 MS(kA/m) 1470 1060 1112.5 1470 K (J/m3) 7957.75 t (nm) 1.8 2.0 3.9 3.6 MS*t *S (µA.nm2) 17.56 14.06 28.79 35.12 Nx Ny Nz 0.024952 0.046343 0.928705 0.027061 0.050280 0.922658 0.044620 0.083119 0.872260 0.042085 0.078372 0.879543 p - -m3 m2 - η 0.3 α 0.02 S (nm2) 6636.63 Heb (kA/m) 79.5 0 0 0 JRKKY (mJ/m2) -1.5 -0.1 Table I. Parameters used in the numerical simulations . Here M S is the saturation magnetization, K u is a uniaxial magneto crystalline anisotropy constant (//ox axis in the plane), t is the film thickness, S is the surface, Nx, Ny, and Nz are demagnetization factors, α is the damping constant. H eb is the exchange bias field that acts on the BL of the SyF- Polarizer and η is the spin efficiency. Numerical simulation of the hysteresis loop and the magnetoresistance of the D -SyF oscillator are shown in Figure 2(a). These curves provide information about the relative orientation of the magnetizations of the different layers, as a function of the applied field. The MR was calculated with the scalar product of the magnetizations adjacent to the insulator barrier i.e. layers 2 and 3. The parameters are listed in Table I. In our samples, the anisotropy axis corresponds to the longer axis of the ellipse, and it is parallel to the X -axis. Arrows on Figure 2 represent the SyF -FL and SyF- Polarizer magnetization s respectively following the color convention of the layers in Figure 1. For relatively large positive values of the applied field [100mT, 400mT] the magnetization of both layers of the SyF -FL are parallel to the external field while the magnetization of the TL of the SyF -pinned is pointing in opposite direction, corresponding to the antiparallel state in the magnetoresistance curve (Figure 2b). Upon sweeping the field from positive to negative values, a first magnetization switching is observed at μ oHsw ≈ +8.4 mT (Hsat+SyF- FLin Figure 2(a)). It corresponds to the switching of the magnetization of the BL of the SyF -FL, which presents a lower net magnetic moment in comparison with the TL. The magnetization switching of the BL of the SyF -FL is accompanied by a change from the antiparallel (high resistance) to the parallel (low resistance) state in the magnetoresistance curve, see Figure 2(b). Upon sweeping further the applied field towards negative values, the magnetization of the TL layer of the SyF -FL is inverted at Hsat-SyF- FL (Figure 2(a)), which does not affect the magnetoresistance (Figure 2(b)). From now on, we refer to th e region of around 100mT between the two character istic saturati on field values ( Hsat-SyF-FL and Hsat+SyF-FL) as “plateau”. It corresponds to the P state in the RH curve. Sweeping the field in the opposite direction (from negative to positive values ) the bottom layer of the SyF -FL is reversed first again leading to a 4 similar scenario except because the plateau region is shifted towards positive field value s and now corresponds to the antiparallel state (Figure 2(b)) . II.2. Dynamics features, the LLG equation The magnetization dynamics of the D -SyF structure is described in a macrospin approach solving the generalized Landau- Lifshitz -Gilbert (LLG) equations enhanced by the spin torque term34,38. The equation for each ferromagnetic layer is written as follow s, 𝑑𝑑𝒎𝒎𝑖𝑖 𝑑𝑑𝑡𝑡=−𝛾𝛾0(𝒎𝒎𝑖𝑖×𝑯𝑯𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒)+𝛼𝛼𝑖𝑖�𝒎𝒎𝑖𝑖×𝑑𝑑𝒎𝒎𝑖𝑖 𝑑𝑑𝑡𝑡�+�𝑑𝑑𝒎𝒎𝑖𝑖 𝑑𝑑𝑡𝑡� 𝑆𝑆𝑆𝑆𝑆𝑆 (3) �𝑑𝑑𝒎𝒎𝑖𝑖 𝑑𝑑𝑡𝑡� 𝑆𝑆𝑆𝑆𝑆𝑆=𝛾𝛾0𝑗𝑗𝑎𝑎𝑑𝑑𝑑𝑑𝐺𝐺(𝜂𝜂)𝒎𝒎𝑖𝑖×(𝒎𝒎𝑖𝑖×𝐩𝐩𝑖𝑖) (4) where the layers are identified by the number i =1 to 4, see structure in Figure 1(a). The vector m i=Mi/MSi is the normalized magnetization vector, M Si is the corresponding saturation magnetization, γ 0 is the gyromagnetic ratio, α i is the Gilbert damping constant. All the parameters are listed in Table I. Heffi=Hinti+Hcouplingi is the effective field, composed of the internal field Hinti and the coupling field Hcouplingi of the ith layer. The effective field is derived from the energy term of equations (1) and (2) for each layer. The last term is the spin torque term that affects the damping (second term in (3)). For this model we have not taken into account the effects of a mutual spin through the Ru spacer During simulations, three types of dynamic couplings are considered : dynamic RKKY interaction between the two magnetic layers of each SyF, dynamic dipolar interaction between the four magnetic layers of the oscillator , and the mutual spin transfer torque (MSTT) only between the BL of the SyF -FL and the TL of the SyF -Polarizer , see Eq (4) . Dynamic RKKY interaction and dynamic dipolar interaction are conservative couplings included in the precession term of the LLG equation (first term in Eq. (3)) , without an energy loss . The Gilbert term and the MSTT , are dissipative couplings considered by the damping and the spin-transfer torque term (second and last term in Eq. (3)) . Here j app is the applied current density, the pre -factor G (𝜂𝜂) is given by Eq. 2 in Ref. 38, with the spin polarization efficiency 𝜂𝜂 and the unit vector p i represents the direction of the spin polarization vector of electrons. We are only considering the STT between layers 2 and 3 with the polarizer of layer 2 is being layer 3 and the opposite as shown in Table I. The equation (3) should be read as follow s: the spin-polarized electrons from the mj layer reach the layer mi where the intrinsic damping starts to be counteracted (japp<jc) (only for the correct sign of current ) due to the spin transfer torque. These damped excitations produced by the applied current are called the STT damped modes in this manuscript . There are four STT damped modes such as the result of the hybridization of the isolated SyFs modes, due to the dipolar coupling present between the layers in the structure. Overcoming a critical current value (j c) the damping is completely counteracted, and the magnetization state is destabilized, leading to switching to another stable state or into steady st ate oscillations (STT excitations). All excitations involve all four layers of the oscillator. In the case of positive applied current (j app>jc>0) the BL of the SyF -FL magnetization is driven by the spin transfer torque of spin -polarized current and this leads to excitations dominated by the SyF -FL, in which the SyF -Polarizer participate following the dynamics due to the dipolar coupling. For the case of the SyF -Polarizer dominant precession (j app<jc<0), it is the TL magnetization of the SyF -Polarizer which is controlled by the spin transfer torque. As this structure is coupled by the dipolar field, the four layers move together at the same frequency. In the absence of current (j app=0) the spin torque term disappears, and we can simulate the excitation modes in the linear regime. 5 Fig. 2. (a) -(b) Hysteresis and magnetoresistance curves of the D -SyF oscillator. (c) Dispersion diagram of hybridized damped modes f1,f2, f3, and f4 for j app=0. The magnetic field has been swept from positive to negative values. The regions (i) -(iv) are defined by the crossing of modes. To calculate the STT damped modes (j app≠0), the set of equations (3) are rewritten in the form of a matrix, giving a periodic solution for mi and linearizing around an equilibrium position, mi=(M eq,0,0)+(0,n y,i,nz,i)ept. The corresponding equilibrium positions were extracted from the hysteresis loop. The eigenvalue of the characteristic matrix will be the complex number p=λ+iω, with a real part λ corresponding to the decay rate of the excitation modes and the imaginary part ω corresponding to the 2π frequency of the excitations. The eigenvectors (0,n y,i,nz,i) define the character of the oscillation mode and the amplitude of oscillation for each layer. Two types of instabilities can be described using the linearization. The first one corresp onds to spin wave mode softening (f>0) and will coincide with the magnetization switching32. In the second type the real part of the characteristic eigenvalue can cross the zero axis. To monitor such instability, we have calculated the decay rates of the system as a function of the current density and the applied field λ(j app,Happ). This provides us with the main criteria to find the critical current density of the STT regime. A negative value means that the system relaxes into a stable state (damped regime), while a positive value indicates that the system becomes unstable, where one of the possibilities is the STT regime (auto oscillations). The passage from negative to positive defi nes j c, i.e. the current where the decay rate is zero: λ(j app,Happ)=0→ japp=jc. This method does not consider temperature. Parameters used for the simulations were extracted from the experimental devices and correspond to SyF structure s, which are not magnetically compensated. The dipolar field was calculated numerically for both SyFs in the AP magnetic configuration (H app= 0), finding for the SyF -FL values around ±0.2 mT, for the BL and TL respectively. In the case of the SyF - Polarizer, the dipolar field calculated in the same X direction was - 2.1 mT and - 1.2 mT, for TL and BL respectively. Moreover, when the SyF -FL is already saturated (positive X direction) the dipolar field in the SyF -Polarizer increases up to -21.9 mT and -18.9mT. Therefore, due to the non- compensated SyF structures, we always expect the influence of the dynamical dipolar coupling in the magnetization dynamics. Figure 2(c) shows the calculated STT damped mode frequencies at j app=0 obtained by sweeping the applied field from positive to negative values. These modes are labeled f1, f2, f3, and f4 from low to high frequency. The dispersion diagram show s a well -defined mode splitting between modes 1 and 2, as well as two mode s anti-crossings between modes 1 and 2, and 2 and 3 respectively. These effects are reminiscent of the splitting between the acoustic and optical -like modes provoked by the conservative RKKY inter action on single SyFs structures37. Here, the dipolar field between the four layers is 6 responsible for the splitting and anti -crossings of the frequency dispersion diagram ,35 see Figure 2(c). Due to the crossing of modes, we define four regions, indicated in Figure 2(c) as (i) -(iv). The region (i) is located for negative fields in the P state (low resistance state), and the three regions (ii) -(iv) on the AP state (high resistance state of the structure and interesting region to study the STT modes), see MR curve in Figure 2(b). In the following section, we study the evolution of the damped modes upon increasing the applied current and their stability . II.3. Magnetization stability analysis We start the study of the stability of the magnetization precession around the equilibrium positions. Using the decay rate λ criteria, λ(j app,Happ)<0→STT damped modes, λ(j app,Happ)>0→STT modes and λ(japp,Happ)=0→ japp=jc introduced before, we can distinguish between stable and unstable regimes, by computing the critical current for each value of the applied field . Figure 3(a) shows the corresponding decay rate s λ(0,H app) of the damped modes f1,f2,f3, and f4 respectively. We will respect the color of modes defined in Figure 2(c). As it was expected for j app=0, the decay rate of the four modes remain negative s, corresponding to the stability for the damped modes. Since the frequency dispersion vs. field was divided into several regions (i)-(iv), due to the crossing of the damped modes, we notice that the decay rates suffer inflections or in some cases abrupt jumps from one of these regions to another one. It is important to remark that the mode f4 is less affected by the crossing of modes, showing the larger decay rate and without distortions in the whole range of applied field. Fig. 3. Decay rate versus applied field dispersion diagrams, for the corresponding hybridized damped modes f1, f2, f3, and f4. Colors correspond to the modes defined in Figure 2(c). In (a) below the critical current j app=0, in (b) for the SyF -FL dominant precession, j app=1x1012 A/m2 and in (c) for the SyF -Polarizer dominant precession, j app=- 1x1012 A/m2. (d) State diagram H app vs j app. Positive current corresponds to an electron flow from the TL of the SyF- Polarizer (red arrow) to the SyF -FL (green arrows). The light grey and black regions correspond to the 7 unstable region of excitations, and the dark grey to the stable region. The black region corresponds to the switching of the SyF -FL. When the positive current density is increased up to japp=1x1012 A/m2, the decay rate tendencies of the system change, see Figure 3( b). It is noticed that the decay rate for the modes f1 and f2 (dominated by the SyF -FL) are already positive λ>0, which is an indication that these modes evolve into the STT regime due to the damping compensation by the increase of the applied current. As this method is a linearization of the LLG equation, small magnetization precession, it is not possible to study the tendency of the decay rate when the system is already in the steady state regime, large magnetization precession. Applying negative current density, j app=-1x1012 A/m2, the system reaches the SyF -Polarizer dominant precession; see Figure 3( c). The decay rate is already positive for the modes f1, f2 and f3 (dominated by the SyF -Polarizer), generating STT modes in the frequency field diagram. For both senses of current density, it is observed a crossing of the decay rate in the (i) region. In conclusion, it is possible to obtain the critical current of the STT modes using the criteria λ(j app,Happ)=0 ( for positive and negative current density) and we can predict which of the modes will be stable or unstable. Sweeping the current density for each value of magnetic field we obtain the critical lines to build the state diagram H app vs j app, shown in Fig ure 3(d). The magnetic field was swept from positive to negative values. i.e . from the positive saturation magnetization of the SyF -FL until its plateau region in the P state (low resistance). The arrows correspond to the orientation of the magnetization of the four layers of the structure. The stable and unstable dynamical states are indicated by the dark and light grey regions, respectively. Excitations dominated by the SyF -FL (SyF- Polarizer) are observed in the region of positive (negative) current. Yellow lines indicate the critical current densities, j c,SyF -FL and j c,SyF- Polari zer. The critical current values obtained and shown in Figure 3(d) are referential due to the simplification of the model and to the parameters used in performing simulations (exchange bias, RKKY coupling, thicknesses, M S etc). The critical currents should be taken as an approximation when compared to experimental results . The dashed yellow lines represent the border with the switching of the SyF -FL magnetizations, represented by the dark black region. In the SyF -FL dominant precession region, two small def lections can be observed, which corresponds to the splitting shown in Figure 1(c). The critical current for the SyF -FL in the saturated state is around 0.9x1012 A/m2. This numerical simulation framework allows to predict the magnetization dynamic behavior of the coupled SyFs of the STO structure and the critical current as a function of the applied field. As w e will verify in the next section, this framework provides useful information to understand the complex frequency dispersion diagrams studied experimentally, a fundamental issue in designing STO devices. III. EXPERIMENTAL SECTION In this section, we present the static and dynamic features of the D-SyF nano- oscillator . Measurements were carried out using a standard microwave measurement setup. III.1. Static measurements The experimental results are obtained for magnetic tunnel junctions with the following structure: IrMn(6.1)/SyF -Polarizer/MgO(0.9)/SyF -FL, where numbers represent thickness in nanometers. The composition of the SyF- Polarizer and SyF -FL are CoFe(1.8)/Ru(0.4)/CoFe B(2) and CoFe(0.5)/CoFeB(3.4)/Ru/CoFe(3.6) respectively. The thickness of Ru in SyF -FL was selected to achieve a weak and negative RKKY coupling. Samples were grown by sputter -deposition and patterned into elliptical nanopillars (130nm x 65nm ) with and area of 6636.63nm 2. The uniaxial shape anisotropy stabilizes the magnetization in the direction of the longest axis. We have characterized tens of devices which we classified in two categories depending on whether the TMR is above or below 50% (High-TMR and Low -TMR devices respectively) 16. The MgO barrier of the latter is considered to have inhomogeneities and/or pin holes either caused during deposition or electrical characterization. In this 8 work, we mainly focus on HTMR devices. However, due to the slightly high critical current of STT excitations dominated by SyF -Polarizer, those cannot be achieved easily in HTMR devices applying voltages below 400 mV (MgO barrier degradation) . Thus, LTMR devices characterized by a smaller resistance are used to characterize STT excitations dominated by the SyF -Polarizer . Figure 4(a) shows the MR curve s of a High -TMR (HTMR, red curve) device and a Low -TMR (LTMR, black curve) device respectively. The MR curves are in good agreement with the numerical simulations (Figure 2(b) ), which provide information about the magnetic orientations of the layers as a function of the applied field. It has been found during the optimization of D -SyF structures, (not shown in this article) that the roughness in a multilayer structure gives less control of the thicknesses of the layers on top, thus the RKKY coupling of the SyF -FL becomes weak, reducing the size of the plateau. As it was shown in previous studies34, the size of the plateau region of a SyF is directly related to the strength of its RKKY coupling and increased by the exchange bias field. The SyF -FL plateau is located between two characteris tic fields, the Hsat-SyF- FL and Hsat+SyF-FL=H sw, however , it is not evident in the MR curve due to the weak RKKY coupling. The plateau size will be estimated in the next section measuring the STT damped modes. Both magnetizations of the SyF -Polarizer remain in its AP configuration ( plateau region) for a quite large range of applied field (>500mT) thus we can consider this static configuration for all our measurements, until the spin flop field (Hsf-+SyF- Polarizer ). III.2. Dynamic Measurements In this section, the dynamical features of the D -SyF oscillator device are presented . Section ( A) shows the study of the STT damped modes on the plateau region of the SyF -FL. Section (B) corresponds to the study of the STT damped modes on the AP region (high resistance state) . Section (C) is focused on the analysis of the STT modes for SyF -FL and SyF- Polarizer dominant precession. The same sign convention of the numerical simulations is considered: electrons flow from the SyF -Polarizer towards the SyF -FL for positive applied current, promoting STT excitations in the SyF -FL. It is worth noting that during the STT measurements, the STT damped modes are also excited. A. STT damped modes, positive current First, the STT damped modes were measured for a HTMR device (TMR= 60%) around zero applied field. The corresponding excitation frequency dispersion as a function of the applied field for positive applied current I app=1 mA is shown in Figure 4(b). The power spectral density (PSD) of the STT damped modes is shown on a logarithmic scale. The magnetic field was swept from positive to negative values. We identify three different regimes in Figure 4(b) corresponding to the three different magnetic configurations of the SyF -FL (plateau (i) and both saturation regions ) described by the numerical simulations (Figure 2). Regions (i) and (ii) in Figure 4(b) correspond to regions (i) and (ii) in Figure 2(c). Experimentally, we identified region (i) between -27 mT<μ 0Happ<10 mT. Out of this range, the SyF- FL is saturated. 9 Fig. 4. (a) MR curves of an elliptical device (130x65nm2) for HTMR device in red (TMR 60%) and LTMR in blue (TMR 28%) respectively. (b) Frequency vs. applied field of the dB ( 10 log of power spectral density (nV2/Hz)) for positive applied current (I app=1 mA). The regions (i) and (ii) are identified in agreement with numerical simulations and the arrows correspond to the magnetic direction of magnetizations. The STT damped fundamental modes f1, f2 and f3 are identified, and the higher order damped modes f 11, f21, and the harmonics 2f1, 2f2. The dashed lines are included to identify the modes only as a visual reference. Linewidth as a function of the applied field of the damped f1 and f 2 STT damped modes for ( c) positive and ( d) negative applied currents (I app =±1 mA). By comparing with the numerical simulations (Figure 2(c)), we can identify the STT damped modes f1, f2, f3 in regions (i) and (ii) in Figure 4(b), as well as other harmonics (2f1 is the 2nd harmonic of the f1 mode) . In region (i) we observe the splitting and curvature of modes, which indicates weak conservative RKKY coupling of the SyF -FL, according to simulations. In region (ii) we observe other higher damped modes such as f11, f21, and the harmonics (2f1, 2f2) , in agreement with results given in Ref. 39, where the finite size of the device generates quantized spin waves. As we will see in the next section, the appearance of higher -order modes has negative consequences on the microwave properties of the D - SyF oscillator. As can be seen in Figure 4(b), the intensi ty of the different modes change s with the applied field (and applied current). As previously discussed , the amplitude of the magnetization precession and the linewidth should be proportional to the absolute value of the decay rate λ(japp,Happ) of the STT damped modes in the linear regime. The tendency of the linewidth observed experimentally ( Figure s 4(c) and 4(d)) agrees well with the evolution of modes 1 and 2 obtained by numerical simulations in the region (i) (Figure s 3(b) and 3(c) ), indicating a good correspondence between our model and experiments . Large values of the linewidth (≈400- 800 MHz ), for a quality factor Q=∆f/f ≈50-100, corresponding to STT damped modes (linear regime) were measured. 10 Fig. 5. Frequency vs. applied field diagram for regions (i) -(iv) for the device with TMR=60%, for I app=-1 mA. Dashed lines correspond to the damped f1, f2, and f3 modes only as a visual reference. Arrows indicate the magnetization directions on the magnetic layers for the different regions. B. STT damped modes negative current STT damped modes in both SyFs structures were measured by applying -1mA and they are introduced in the frequency dispersion diagram in Figure 5. For applied magnetic fields >10mT the SyF -FL is completely saturated (green arrows) while the SyF pinned remains in the AP magnetic configuration. By comparing with numerical simulations in Figure 2(c), we can also identify the regions (ii)-(iv) and modes f1, f2, and f3. The experimental STT damped modes follow the frequency- field dependence of the simulated modes, and they are accompanied by additional higher -order modes 27 (f11, f31, f22). At low positive fields, modes f1 and f2 are separated 2.4GHz, as a result of RKKY and dipolar interactions within the SyF -FL (conservative couplings). Two additional mode splits are observed around 100mT, between modes f1 and f2 and between modes f2 and f3 respectively , in agreement with the numerical analysis, see region (iii) in Figure 2( c). In the next section, we discuss the evolution of the different STT damped modes into STT excitations in the SyF -FL or the SyF- Polarizer upon increasing the applied current for positive and negative values respectively . C. STT excitation modes C.1 HTMR device : SyF -FL dominantes STT mode Steady -state oscillations (STT modes) were first analyzed by applying positive current, which favors STT modes dominated by the SyF -FL. Figure 6(a) shows the frequency -field dispersion curve with an applied current of I app=0.92 mA. The observed SyF -FL dominated STT modes correspond to the evolution of the STT damped modes f1 and f2 on region (ii) and (iii), as predicted by the stability analysis (Figure 3) . The fundamental STT mode f1 shows several discontinuities around 40, 58 and 82 mT, in agreement with the macro -spin simulations (Figure 2(c)). These discontinuities are interpreted to be due to interactions between the steady state mode and other damped modes of the system through higher harmonics29-35. Indeed, modes f 3, f11 and f 31 cross the second harmonic of the STT mode f1 (lines 11 in Figure 5 and 6(a)). Non -linear mode interactions through higher harmonics can produce discontinuities and kinks in the fundamental STT mode f140-41. Figure 6(b) displays t he linewidth of the STT mode f1. The linewidth of STT mode 1 increases each time there is a discontinuity in the f -H dispersion, as expected from the interaction with other damped modes via higher harmonics26. The STT mode f2 is characterized by a much lower linewidth, reaching a minimum of 42 MHz , Q≈4.94, around 100 mT . At low fields (region (i) ), the STT damped mode in the SyF -FL plateau shows a continuous frequency- field dispersion, without jumps or kinks, since t here are no mode crossings in this range of field. Exciting STT modes in this region would potentially offer excitations at zero fields without linewidth enhancements due to undesired mode interactions. Due to the switching of the SyF -FL for an applied current below the critical current, it was not possible to obtain STT in the region (i) . Fig. 6. (a) Frequency dispersion as a function of the field for the SyF -FL dominant precession, I app=0.92 mA. The circles indicate the kink and jumps in the STT f1 mode. (b) Frequency (black) and linewidth (blue) versus applied field. Deviations in frequency correspond to an abrupt increase of ∆f. (c) Frequency dispersion as a function of the applied field for the SyF -FL dominant precession at 0. 6, 0.92 and 0.94 mA . Inset: peaks of modes interactions around the bi -stable region ( around 82 mT) . (d) Frequency dispersion as a function of the applied current for three values of applie d field. Device HTMR ( TMR= 60%). To study the discontinuities of the STT modes , the frequency field dispersion is plotted for three different values of the applied current (Figure 6(c)). The evolution from a continuous STT mode at 0.6mA ( black curve ), into a discontinuous STT mode with jumps and kinks at 0.94mA ( blue curve). The PSD spectra at different fields around 82 mT are shown in the inset. The PSD spectra shows a region of bi-stability where two modes co -exist . This behavior has been reported to come from the interaction between a STT mode with other damped modes of the system through higher harmonics37,39,41,42, where the apparent mode co -existence is indeed thermally activated mode hopping. This kind of discontinuities 12 and jumps are observed only for large values of current, which implies a large amplitude of the magnetization precession and thus large dipolar interaction . An interesting feature of the magnetization dynamic on SyF pinned structures is the possibility to tune the frequency -current dependence from redshift (df/dj app<0) to blueshift (df/dj app>0) by applying an in- plane field . Figure 6 (d) shows th e frequency as a function of the applied current for μ0Happ=20, 70 and 85 mT respectively. A transition from redshift to blueshift is observed upon increasing field from 20 to 85 mT. Interestingly both curves are continuous with n o abrupt discontinuities . However, at μ 0Happ= 70 mT, a bi -stable region characterized by mode is observed around I app=0.82 mA (see dashed line). C.2. LTMR device Achieving SyF- Polarizer dominant precession requires increasing the applied current above the breakdown voltage in HTMR devices. Thus, an LTMR device (TMR= 28%) is used to pursue the study of the STT excitations dominated by the SyF -Polarizer. Figure 7 displays the STT modes of a LTMR device for positive (Figure 7(a)) and negative (Figure 7(b)) currents ( ±1.4 mA). The SyF- FL dominant excitations (Figure 7(a)) exhibit several discontinuities and kinks in the STT modes f1 and f2 mode s and higher harmonic s (2f1). Indeed, more discontinuities than the equivalent frequency- field dispersion of a HTMR device ( Figure 6(a)) are observed . This is expected since the existence of pinholes in the tunnel barrier increases interlayer interaction. The frequency -field dependence becomes flat in the field region around 100mT because of the interaction between higher harmonics f31 and 2f1. For negative current , the SyF -PL dominant precession shows a gap between the STT f3 and STT f1 mode s (Figure 7 (b)). Fig. 7. (a) SyF -FL dominant precession, I app=+1.4 mA and (b) SyF -PL dominant precession, for I app=-1.4 mA. Interactions between the harmonics of the STT f1 and f2 mode s with the damped or higher order modes will be 13 transmitted as a form of kinks, deviations or jumps of the normal tendency of the frequency and linewidth versus applied field dispersion, generating discontinuities in the linewidth. Device with LTMR ( TMR= 28%). The general tendency of the linewidth not shown is to decrease upon increasing the applied field , although several discontinuities (regions of linewidth enhancement) are observed corresponding to the interaction of the S TT f1, f2 , and f3 modes with damped and higher order modes, as in the HTMR device (Figure 6(b)). The local minimum linewidths measured for the SyF -FL dominant precession at 1.4 mA, were 20.5 MHz (Q≈1.73) at 112 mT, 13.6 MHz (Q≈1.1) at 194 mT and 12.9 MHz (Q≈1.17) at 214 mT. In the case of the SyF- Polarizer dominant precession at -1.4 mA , the minimum linewidths were 88MHz (Q≈8) at 46 mT, 53 MHz (Q≈5.57) at 150 mT and 46 MHz (Q≈5.41) at 214 mT. Interestingly, t he minimum values of the linewidth associated to the SyF-FL dominant precession are smaller than the minimum linewidth associated to the precession of a single free layer o n standard in-plane magnetized spin torque oscillators 22. For the case of the SyF -Polarizer, the minimal linewidth was around ≈46 MHz. We note that continuous excitation branches can be found for specific conditions in the case of the SyF - Polarizer. Finally, we analyze the redshift and blueshift regimes associated to SyF-FL dominant excitations. We note that the redshift and blueshift regimes have been studied only on SyF pinned structures23,25 so far. Figure 8 displays the frequency- current dependence at different magnetic fields for the SyF -FL dominant regime. In (a) -(b) (20 mT -plateau region and 54 mT -P state respectively) , the applied current remains below the critical current, therefore STT damped mode shown in (a) the standard redshift behavior with large linewidth Δf≈395- 550 MHz (Q≈79-110). In (b) t he STT damped mode shows the blueshift regime with a reduction of the linewidth by the current from Δf≈480 MHz (Q≈80) for low current for Iapp <1 mA to 125 MHz (Q≈17.8) for Iapp>1.2 mA. Upon increasing the field, the critical current gets smaller. Thus, at μ 0Happ=70.7 mT in (c) the system evolves into the STT excitations. The frequency dispersion shows discontinuities and a transition from a blueshift regime to a redshift regime upon increasing current , around 1.1 mA . Looking at Figures 7 and 5, we observe that this discontinuity is a product of the crossing of the STT mode f1 with the damped mode f3, however , this transition is opposite to the previous observed (redshift into blueshift) , and due to the complexity of these coupled systems. The mode t ransition into the redshift regime is accompanied by a reduction of the linewidth into Δf≈75 MHz (Q≈10) around 1.25 mA. At large fields (Figure 8(d) ) a redshift to blueshift transition is observed upon increasing current, and a very low linewidth of Δf≈28 MHz (Q≈2.54) is obtained in the blueshift regime . This region of STT excitations at large fields and applied current in LTMR devices is of potential interest for applications as it offers the possibility of selecting excitations in the redshift or in the blueshift regime. 14 Fig. 8. Frequency as a function of the current density for I app>0 for different field values (SyF -FL dominant precession). (a) STT damped mode f1 on the SyF -FL plateau region . (b) In the P state close to the switching field. The system shows a blueshift regime and large linewidth . In (c) the system shows a blueshift regime until Iapp=1.1mA. Overcoming this value of current , the regime changes towards a redshift regime. In (d) the STT f1 mode shows first a redshift regime and after the splitting a blueshift regime with a minimum local linewidth of 28 MHz at 1.25 mA. IV. C ONCLUSIONS In this manuscript, we conducted a comprehensive investigation into the spin- transfer torque damped modes and steady- state oscillations of a spintronic nano -oscillator employing two SyF structures. The study involved both numerical simulations and experimental analyses. Numerical simulation s were carried out using two different values of R KKY coupling, J RKKY ≈-0.1mJ/m2 and J RKKY ≈-1.5 mJ/m2 for the SyF -FL and SyF- Polarizer respectively. The small RKKY coupling of a SyF -FL eliminates the spin flop region, producing an abrupt switching of the magnetization layers and introducing a small plateau region (≈40 mT), in comparison with the corresponding of the pinned SyF- Polarizer (>600 mT). Static and dynamical experimental measurement s confirmed the weak RKKY coupling of the SyF -FL on top of the structure. The experimental STT damped hybridized modes have been identified using numerical modeling. W e found that it is possible to find the tendency of the linewidth and the PSD of the STT damped hybridized modes following the tendency of the decay rate (λ), also we can obtain the critical current of the STT modes. This study shows that the analysis of the stability predicts the state diagram of different structures, with an arbitrary number of layers. The frequency versus field dispersion diagrams for b oth devices (TMR 60% and 28%) show several discontinuities, attributed to the crossing of the STT mode or its harmonics with other damped hybridized or higher order modes, as was reported 15 in Ref. 2 4, 36, 37 and 39. The introduction of a SyF -FL instead of a single layer in the STO based will generate more damped modes which produce more discontinuities in the STT mode s. This last fact is also attributed to the dipolar coupling between the magnetic layers. The frequency current tuning shows two regimes for the SyF FL, the redshift (df/dI app<0) and a blueshift (df/dI app>0) which can be selected for different values of applied field. In the case of the SyF -Polarizer STT modes, a flat regi on (df/dI app≈0) with a small redshift and blueshift regimes was found. We found the minimum linewidth on a LTMR device, around ≈12.9 MHz (Q≈1.17) for SyF- FL and around ≈46 MHz (Q≈5.41) for the SyF -Polarizer dominant precession, evidence of the stability of the system. V. A CKNOWLEDGE M.R. acknowledges financial support from Spanish MIC, AEI and FEDER through Grant No. PID2020- 116181RB -C33 (MCI/AEI/FEDER, UE) and from Comunidad de Madrid (Atracción de Talento Grant No. 2018- T1/IND -11935). VI. REFERENCES 1 N. Locatelli, V. Cros, and J. Grollier, Nat Mater 13, 11 (2014). 2 Kent, A., Worledge, D. A new spin on magnetic memories. Nature Nanotech 10, 187–191 (2015) 3 A. Dussaux et al., Nat. Comm . 1, 8 (2010). 4 S. Miwa et al., Nature Materials 13, 50 (2014). 5 A. Jenkins et al., Nature Nanotechnology 11, 360 (2016). 6 Zeng, Z., Finocchio, G., Zhang, B. et al. Sci Rep 3, 1426 (2013).63 7 AV. Silva et al. Eur. Phys. J. Appl. Phys. (2015) 72: 10601 8 Torrejon, J., Riou, M., Araujo, F. et al. 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2023-12-16

We report the dynamic characterization of the spin-torque-driven in-plane precession modes of a spin-torque nano-oscillator based on two different synthetic ferrimagnets: a pinned one characterized by a strong RKKY interaction which is exchange coupled to an antiferromagnetic layer; and a second one, non-pinned characterized by weak RKKY coupling. The microwave properties associated with the steady-state precession of both SyFs are characterized by high spectral purity and power spectral density. However, frequency dispersion diagrams of the damped and spin transfer torque modes reveal drastically different dynamical behavior and microwave emission properties in both SyFs. In particular, the weak coupling between the magnetic layers of the non-pinned SyF raises discontinuous dispersion diagrams suggesting a strong influence of mode crossing. An interpretation of the different dynamical features observed in the damped and spin torque modes of both SyF systems was obtained by solving simultaneously, in a macrospin approach, a linearized version of the Landau-Lifshitz-Gilbert equation including the spin transfer torque term.

Spin-torque nano-oscillator based on two in-plane magnetized synthetic ferrimagnets

2312.10451v2

Coarse-graining in micromagnetic simulations of dynamic hysteresis loops R Behbahani1;2, M L Plumer1and I Saika-Voivod1;2 1 Department of Physics and Physical Oceanography, Memorial University of Newfoundland, Canada 2 Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada, N6A 3K7 E-mail: saika@mun.ca Abstract. We use micromagnetic simulations based on the stochastic Landau- Lifshitz-Gilbert equation to calculate dynamic magnetic hysteresis loops at nite temperature that are invariant with simulation cell size. As a test case, we simulate a magnetite nanorod, the building block of magnetic nanoparticles that have been employed in preclinical studies of hyperthermia. With the goal to eectively simulate loops for large iron-oxide-based systems at relatively slow sweep rates on the order of 1 Oe/ns or less, we modify and employ a previously derived renormalization group approach for coarse-graining (Grinstein and Koch, Phys. Rev. Lett. 20, 207201, 2003). The scaling algorithm is shown to produce nearly identical loops over several decades in the model cell volume. We also demonstrate sweep-rate scaling involving the Gilbert damping parameter that allows orders of magnitude speed-up of the loop calculations. Keywords : Landau-Lifshitz-Gilbert equation, micromagnetics, coarse-graining, mag- netic hyperthermia, nanorods The fundamental premise of micromagnetics is that the physics of interest can be modeled by a macrospin representing a collection of atomic spins within a small nite volume, or cell. The approximation that all spins within a cell point in the same direction is valid at temperature T= 0, so long as cells remain smaller than the exchange length [1]. A limiting factor for micromagnetic computer simulations is the number of cells used to model the system; using larger cells is computationally advantageous. At niteT, a few schemes have been proposed to account for how parameters used for modelling the magnetic properties of the material must vary with cell size in order to keep system properties invariant with cell size. For example, Kirschner et al. [2, 3] suggested an approximate scaling of saturation magnetization Msbased on the average magnetization of blocks of spins in atomistic Monte Carlo simulations, and subsequently scaling the exchange and uniaxial anisotropy constants AandKto preserve the exchange length and anisotropy eld. Feng and Visscher [4] proposed that the damping parameter , which models the dynamics of magnetic energy loss [5], should scale with cell size, arguing that using larger cells is analogous to having more degrees of freedom for energy absorption; see also [6] for eorts related to . The renormalizationarXiv:1912.00310v3 [cond-mat.mtrl-sci] 8 Nov 2021Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 2 group (RG) approach of Grinstein and Koch [7], based on mapping a Fourier space analysis of the non-linear sigma model to ferromagnets in order to scale A,K, eldH and magnetization M, has garnered signicant attention. However, to the best of our knowledge, no scaling theory has been applied to the calculation of magnetization-eld (MH) hysteresis loops [8], which are the foundation of experimental characterization of magnetic systems. In this Letter, we modify and employ the approach proposed by Grinstein and Koch [7] to the test case of calculating MH loops for magnetite nanorods at sweep rates relevant to magnetic hyperthermia, allowing us to make estimates of specic loss power that would otherwise be computationally impractical. The magnetite nanorods we simulate are the building-blocks of the nanoparticles that were shown by Dennis et al to successfully treat cancerous tumours in mice via hyperthermia [9]. It is reasonable to choose the smallest micromagnetic cell to be the cubic unit cell, which is of length a0= 0:839 nm and contains 24 magnetic Fe ions. We set the exchange stiness constant to A0= 0:981011J/m, which for cell length a0yields an eective exchange constant between neighbouring cells of Je=a0A0= 8:2221021J, which in turn yields a bulk critical temperature of Tc= 1:44Je=kB= 858 K for the bulk 3D-Heisenberg-model version of our system. This value of A0is close to what can be theoretically determined by considering the atomic-level exchange interactions across the faces of neighbouring unit cells [10], and is in reasonable agreement with experimental values [11, 12, 13, 14, 15, 16, 17]. The nanorod dimensions are approximately 6.7 nm 20 nm47 nm (8a024a056a0), with its length along the z-axis. We set Ms= 480 kA/m [11, 18, 19], the bulk value for magnetite. We do not consider magnetostatic interactions explicitly, but rather implicitly through an eective uniaxial anisotropy. For the purposes of this study, we choose a strength of K0= 10 kJ/m3, which is consistent with other studies of iron oxide nanoparticles [20, 21], and for which a more precise estimate can be obtained by considering the nanorod's demagnetization tensor [19, 22, 23, 24, 25, 26], maghemite content [9], and the eect of neighbouring nanorods within a nanoparticle. We omit cubic crystalline anisotropy as it has negligible eects on the hysteresis loops of magnetite nanoparticles with even modest aspect ratios, as discussed in Refs. [19, 26] (we have also veried that adding cubic anisotropy of strength 10 kJ/m3has no impact on the loops presented here). Anisotropy is set along the z-axis with a 5dispersion to mimic lattice disorder [21]. For convenience we set = 0:1, a choice consistent with previous studies [21, 27] and with magnetite thin lms [28]. While hysteretic heating is at the heart of magnetic nanoparticle hyperthermia, preventing eddy current heating of healthy tissue limits the frequency fand amplitude Hmaxof the external eld such that the sweep rate SR = 4 Hmaxfis less than a target value of 0:25 Oe/ns [29, 18]. For our simulation, we set Hmax= 500 Oe, which for the target SR implies a target value of f= 125 kHz, a value large enough to restrict unwanted Brownian relaxation [18]. To model the dynamics of the magnetization of a cell Mof xed magnitude Ms,Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 3 𝑎"𝝁𝑎" Nanoparticle SingleBlock𝑎$=2𝑎"(b=2)1344 Cells 𝑎=𝑎"(b=1) 10752 cells𝑎'=4𝑎"(b=4)168 Cells 𝑎)=8𝑎"(b=8)21 Cells Figure 1. Coarse-grained modelling of a magnetite nanorod. The smallest micromagnetic cell models the atomic spins within a cubic unit cell of length a0= 0:839 nm with a single magnetic moment. Our goal is to model the system using a smaller number of larger cells (of length ab=ba0forb > 1) with appropriately scaled parameters. The number of cells drawn and their sizes are only approximate. Illustrative spins for half of the tetrahedral Fe3+sites (FCC sites) are drawn over a spinel unit cell taken from Ref. [30]. we solve the Landau-Lifshitz-Gilbert (LLG) equation [22, 5, 31], dM dt= 1MHe 1 MsM(MHe) (1) wheretis time, 1=0 e=(1 +2), e= 1:761011rad/(s.T) is the gyromagnetic ratio for an electron, 0is the vacuum permeability, and Heis due to the combination of an external eld, uniaxial anisotropy, exchange interactions and a thermal eld. We perform our simulations using OOMMF (Object Oriented Micromagnetic Framework) software [32]. In particular, we include the Theta Evolve module [33] used for simulations at niteTvia a stochastic thermal eld [31]. We simulate the rod using cubic cells of length ba0, withbtaking on values 1, 2, 4 and 8. See Fig. 1. For b= 1, 10752 cells make up the rod. For b= 2, there are 10752 =23= 1344 cells. The volume of the rod is xed for all simulations at 10752a3 0(22a0)3. Additionally, we simulate the rod as a single cell { a single rectangular prism, or block. While there is some ambiguity in assigning a single length scale to represent a rectangular prism, we choose b= 22 from the geometrical mean, i.e., the side length of the cube of the same volume as the rod.Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 4 The goal of coarse-graining is to determine A(b) andK(b), i.e., how the exchange and anisotropy parameters should change with bto keep system properties invariant with b. Theb= 22 case is a practical limit where all the atomic spins are represented by a single macrospin, where exchange interactions are no longer required in the simulations, and which provides for an interesting test of a coarse-graining procedure in predicting K(b). In calculating hysteresis loops for a system with cell length ba0, we apply an external eld along the zaxis ofH(b) =Hmaxsin (2ft), and report the z-component of the average (over cells) magnetization unit vector mH=Mz=Ms, averaged over 88 to 100 independent simulations for b>1. Forb= 1 we use 250 simulations. In Fig. 2a we plot hysteresis loops at T= 310 K using dierent cell sizes (varying b) while keeping the exchange and anisotropy parameters xed at A0andK0. A value of SR = 2:5 Oe/ns is chosen to make the simulations computationally feasible at b= 1. Both the coercivity Hcand the remanence increase with increasing b. The increasing loop area is consistent with the stronger exchange coupling ( Je=ba0A0) between magnetization vectors of adjacent cells. For b4, it appears that the exchange is strong enough for the system to be nearly uniformly magnetized, and so Hcremains largely unchanged for b4 sinceKis constant. This means that for b= 1, at this T and for our rod size, exchange is not strong enough to be able to treat the nanorod as a single macrospin in a trivial way. Clearly, varying cell size changes the loops and a coarse-graining procedure is required. In their coarse-graining procedure, Grinstein and Koch introduced a reduced temperature T, which for a three dimensional system is given by, T=kBT A: (2) where  = 2 =ba 0is a high wave-number cut-o that re ects the level of coarse-graining. Similarly, the reduced parameters for eld and anisotropy constants are dened as, h=0MsH A21000 4; g =K A2; (3) withHgiven in Oe. Introducing the parameter l= ln(b), they gave the following set of equations for calculating the reduced parameters as functions of cell size, dT(l) dl= [1 +F(T(l);h(l);g(l))]T(l) dh(l) dl= 2h(l) dg(l) dl= [22F(T(l);h(l);g(l))]g(l)(4) where F(T;h;g) =T 2(1 +h+g): (5) Additionally, the magnetization of the coarse-grained system is scaled via, M(T;h) =(l)M(T(l);h(l)) (6)Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 5 where (l) =eRl 0F(T(l0);h(l0);g(l0))dl0: (7) For our system parameters and range of H, bothg1 andh1, and so F'T=2, which makes the numerical solution of Eq. 4 practically indistinguishable from the approximate analytic solution, which we nd to be, A(b) =(b)A0 (8) K(b) =(b)3K0 (9) H(b) =(b)H0 (10) M0=(b)M(b) (11) wheret=T=Tcand(b) =t=b+ 1t. AtT= 310 K,t= 0:3613,(2) = 0:8193, (4) = 0:7290,(8) = 0:6839, and(22) = 0:6551. Eqs. 8 and 9 provide a prescription for changing material parameters with b, while Eqs. 10 and 11 provide the prescription for scaling HandMafter a loop calculation. However, we nd that the prescription does not yield loops that are invariant with b, on account of Eq. 11; the correction of the coarse-grained values of Mback to those corresponding to the unscaled system is too large (the corrected remanance is too small), as we show in Fig. 2b. In Fig. 2c, we apply a correction to Eq. 11 and obtain good agreement between the reference ( b= 1) and coarse-grained ( b>1) loops. To motivate our correction to the rescaling of the magnetization, we begin by noting that the same value of Tin Eq. 2 can be achieved by either having a rescaled temperature T(b) or having a rescaled A(b). Combining this idea with Eq. 8 yields, T(b) =T0 b(b;T0); (12) which together with Eq. 11 [after solving for M(b)] predicts an overly simple dependence ofMonT, parametrically through b: a line passing through M0andT0atb= 1 and throughM= 0 andT=Tcasb!0. To obtain a model that better matches the data, we introduce a phenomenolgical correction to Eq. 11, one in which M0is a weighted average of M(b) and the RG expression for M0, M0=(b;T0)M(b) + (1)M(b): (13) We useas a free parameter to t the M(T) data for the nanorod. This yields a value of= 0:511, which we use in rescaling mHin Fig. 2c. The t reasonably recovers M(T) in theTrange corresponding to values of bbetween 1 and 22, as shown in Fig. 3. The collapse of the data in Fig. 2c is remarkable, with the biggest discrepancy arising between b= 1, corresponding to the most ne-grained simulation, and b= 2, the rst step in coarse-graining. The dierence lies most noticeably in the shoulder region where magnetization begins to change, where the microscopic details likely matter most. Loss of some detail is expected with coarse-graining and consistent with previous studies involving atomic-level magnetization switching in a grain [34]. The magnetization inCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 6 400 200 0 200 400 H(b) (Oe)1.00 0.75 0.50 0.25 0.000.250.500.751.00 mH(a) b=1 b=2 b=4 b=8 block 400 200 0 200 400 H(b)/ (Oe) 1.00 0.75 0.50 0.25 0.000.250.500.751.00×mH (b) b=1 b=2 b=4 b=8 block 400 200 0 200 400 H(b)/ (Oe) 1.00 0.75 0.50 0.25 0.000.250.500.751.00((1 )+)×mH (c) b=1 b=2 b=4 b=8 block Figure 2. Application of RG coarse graining to nanorod MH loops at T= 310 K and SR= 2 :5 Oe/ns. (a) Changing cell length ( a=ba0) without changing magnetic parameters. (b) AandKare scaled according to Eqs. 8 and 9, respectively, and mHandHare scaled according to Eqs. 11 and 10, respectively. (c) As in panel (b), exceptmHis scaled according to Eq. 13 with =0.511.
t= 1 fs for all simulations. Horizontal error bars shown for Hcrepresent one standard error and are vertically displaced to avoid overlap. Uncertainty in Hcis approximately 7 to 13%. the shoulder areas appears to diminish with increasing b. The behavior of b= 22 runs counter to this trend, but at this level of coarse-graining, there is only a single cell. It is signicant, however, that scaling seems to hold even in this limit. (We note that in this limit, even though there are no exchange interactions in the simulations, the value of the eective anisotropy still depends on exchange through the dependence of TconA0.) The loop areas for b= 1, 2 , 4, 8 and 22 are 495, 488, 443, 432 and 472 Oe, respectively. The smallest loop area (for b= 8) is 13% smaller than the area for b= 1. We note that the unrenormalized exchange length for our simulated material is lex;0=q 2A0 0M2s= 8:23 nm, which is longer than a8= 6:712 nm, and so only our b= 22 single block simulations scale the cell size beyond lex;0. Under renormalization, however, the exchange length becomes lex;b=q 2(b)A0 0M2s, which decreases with increasing b, and takes on values 7.45, 7.02, 6.80 and 6.66 nm for b= 2;4;8;and 22, respectively. Thus forb= 8, the cell length and the exchange length are approximately the same. We now turn our attention to speeding up simulations by considering the relationship between SR and . A larger value of signies a faster loss of energy and a shorter relaxation time for alignment of the magnetic moments to the eld, and results in a smaller hysteresis loop. Likewise, a slower SR is equivalent to a longer measurement time and consequently a smaller hysteresis loop. To build on these ideas, we recall Sharrock's equation for Hcas a function of T[35], Hc=HK" 1s kBT KVlnf0 ln 2# : (14) Sharrock derived this equation by calculating the time required for half of the magnetization vectors in the system, which are initially anti-aligned with the eld, to overcome an energy barrier that grows with KV and align with a eld of strength Hc. In this context, is the relaxation time. In the context of hysteresis loops, Hc is the eld required to ip half of the magentization vectors in an observation time ,Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 7 0.0 0.2 0.4 0.6 0.8 1.0 T/Tc0.00.20.40.60.81.0m b=1b=2b=4b=8b=22Simulations m(b), =0.511 Figure 3. Determining a scaling function for M(b) from the Tdependence of the nanorod magnetization. is used as a tting parameter to match nanorod data, yielding a value of 0.511. Vertical dot-dash lines indicate reduced temperatures corresponding to dierent values of b. which is related to SR via /1=SR.f0is the so-called attempt frequency, for which Brown [31, 36, 37, 38, 39] derived an expression in the high-barrier limit. At small , f0/, and so the product f0/=SR, implying that so long as SR == constant, Hc should remain the same. In Fig. 4 we show loops calculated for SR == 2:5 (Hmax = 500 Oe, and f= 125 kHz), the ratio obtained using a clinically relevant SR = 0 :25 Oe/ns and the estimate of = 0:1. Data for b= 4 and 8 and for various SR- pairs show good agreement. At 0 :25 Oe/ns, simulations using b= 1 are prohibitively long, taking several months on available computing resources. The results shown here combine the RG approach to reduce the number of cells, the ability to use a larger time step
tfor larger cells in solving the LLG equation [6], and the SR =scaling to employ a faster SR, all to dramatically reduce simulation time { by a factor of 43to 83for reducing the number of cells, a factor of at least 5 for the time step, and a factor of up to 1000 when using the fastest SR. The average area of the ve loops for b= 4 in Fig. 4 is S= 171:32:8 Oe, translating to a specic loss power of f01000 4MsS== 207 W/g 10% (using = 5:17 g/cm3), which is consistent with clinical expectations [40]. The loop area for b= 8 is 13% lower at 149 :4 Oe. In summary, we show that our modication to the RG approach of Grinstein and Koch [7] yields a scaling of exchange and anisotropy parameters and nite temperature nanorod hysteresis loops that are, to approximately 10-15%, invariant with cell size. WeCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 8 400 200 0 200 400 H(b)/ (Oe) 1.00 0.75 0.50 0.25 0.000.250.500.751.00((1 )+)×mH SR=0.25,=0.1,b=8 SR=0.25,=0.1 SR=2.5 ,=1 SR=25 ,=10 SR=50 ,=20 SR=250 ,=100 Figure 4. Invariance of MH loops. We combine RG scaling of magnetic quantities, larger time step with block size, and SR =scaling to predict the behaviour of prohibitively long ne-grain ( b= 1) simulations. b= 4 unless otherwise noted. note that the coarse-graining of magnetostatic interactions is beyond the framework of Ref. [7]. We are currently investigating magnetostatic scaling, and intend to report on it in future work. Scaling results hold even to the point where the nanorod is represented by a single magnetization vector that experiences anisotropy only. Whether this limit holds for systems with weaker exchange remains to be studied. This reduction to an eective Stoner-Wohlfarth (SW) model [41] should facilitate comparison with experiments on nanorods, since an analytic solution to the SW model at nite Tand SR exists [27]. It should also simplify computational studies of nanoparticles (nanorod composites) and collections of nanoparticles used in a wide variety of applications and hence facilitate comparison with experimental MH loops and quantication of system properties through simulations. In addition to the computational speedup resulting from the use of fewer micromagnetic cells, the invariance of loops when SR =is xed provides another avenue for computational speedup by allowing one to use a larger SR than the target value. We caution, however, that the theoretical motivation for this invariance stems from considering the Sharrock equation for only small . While both SR and set time scales, we have not provided any reasoning for why the invariance should hold as well as it does for larger . The data that support the ndings of this study are available from theCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 9 corresponding author upon reasonable request. Acknowledgments We thank Johan van Lierop, Rachel Nickel and Mikko Karttunen for enlightening discussions, and Martin D. Leblanc for guidance in using OOMMF. R.B. and I.S.-V. thank Mikko Karttunen and Styliani Consta for hosting our stay at Western University. We acknowledge the nancial support from the Natural Sciences and Engineering Research Council (Canada). Computational resources were provided by ACENET and Compute Canada. References [1] Abo G S, Hong Y K, Park J, Lee J, Lee W and Choi B C 2013 IEEE Trans. Magn. 494937{4939 [2] Kirschner M, Schre T, Hrkac G, Dorfbauer F, Suess D and Fidler J 2006 Physica B 372277{281 [3] Kirschner M, Schre T, Dorfbauer F, Hrkac G, Suess D and Fidler J 2005 J. Appl. Phys. 9710E301 [4] Feng X and Visscher P B 2001 J. Appl. Phys. 896988{6990 [5] Gilbert T L 2004 IEEE Trans. Magn. 403443{3449 [6] Wang X, Gao K and Seigler M 2011 IEEE Transactions on Magnetics 472676{2679 [7] Grinstein G and Koch R H 2003 Phys. Rev. Lett. 90207201 [8] Westmoreland S, Evans R, Hrkac G, Schre T, Zimanyi G, Winklhofer M, Sakuma N, Yano M, Kato A, Shoji T, Manabe A, Ito M and Chantrell R 2018 Scripta Materialia 154266{272 [9] Dennis C, Jackson A, Borchers J, Hoopes P, Strawbridge R, Foreman A, Van Lierop J, Gr uttner C and Ivkov R 2009 Nanotechnology 20395103 [10] Victora R, Willoughby S, MacLaren J and Xue J 2003 IEEE Trans. Magn. 39710{715 [11] Heider F and Williams W 1988 Geophys. Res. Lett. 15184{187 [12] Kouvel J 1956 Phys. Rev. 1021489 [13] Moskowitz B M and Halgedahl S L 1987 J. Geophys. Res. Solid Earth 9210667{10682 [14] Glasser M L and Milford F J 1963 Phys. Rev. 1301783 [15] Srivastava C M, Srinivasan G and Nanadikar N G 1979 Phys. Rev. B 19499 [16] Srivastava C and Aiyar R 1987 J. Phys. C: Solid St. Phys. 201119 [17] Uhl M and Siberchicot B 1995 J. Phys. Condens. Matter 74227 [18] Dutz S and Hergt R 2013 Int. J. Hyperth. 29790{800 [19] Usov N, Gudoshnikov S, Serebryakova O, Fdez-Gubieda M, Muela A and Barandiar an J 2013 J. Supercond. Nov. Magn. 261079{1083 [20] Usov N, Serebryakova O and Tarasov V 2017 Nanoscale research letters 121{8 [21] Plumer M, van Lierop J, Southern B and Whitehead J 2010 J. Phys. Condens. Matter 22296007 [22] Cullity B D and Graham C D 2011 Introduction to magnetic materials (John Wiley & Sons) [23] Newell A J, Williams W and Dunlop D J 1993 Journal of Geophysical Research: Solid Earth 98 9551{9555 [24] Aharoni A 1998 Journal of Applied Physics 833432{3434 [25] Fukushima H, Nakatani Y and Hayashi N 1998 IEEE Trans. Magn. 34193{198 [26] Usov N and Barandiar an J 2012 Journal of Applied Physics 112053915 [27] Usov N 2010 J. Appl. Phys. 107123909 [28] Serrano-Guisan S, Wu H C, Boothman C, Abid M, Chun B, Shvets I and Schumacher H 2011 J. Appl. Phys. 109013907 [29] Hergt R and Dutz S 2007 J. Magn. Magn. Mater. 311187{192 [30] 2019 Introduction to Inorganic Chemistry (Wikibooks) chap 8.6, see Creative CommonsCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 10 licence http://creativecommons.org/licenses/by-nc-sa/3.0/us/ URL https://en.wikibooks. org/wiki/Introduction_to_Inorganic_Chemistry [31] Brown Jr W F 1963 Phys. Rev. 1301677 [32] Donahue M J and Porter D G 1999 OOMMF User's Guide, Version 1.0, Interagency Report NISTIR 6376 National Institute of Standards and Technology Gaithersburg, MD URL https: //math.nist.gov/oommf/ [33] Lemcke O 2004 ThetaEvolve for OOMMF releases: 1.2a3, see https://math.nist.gov/oommf/contrib/oxsext/oxsext.html URL http://www.nanoscience. de/group_r/stm-spstm/projects/temperature/download.shtml [34] Mercer J, Plumer M, Whitehead J and Van Ek J 2011 Appl. Phys. Lett. 98192508 [35] Sharrock M and McKinney J 1981 IEEE Trans. Magn. 173020{3022 ISSN 0018-9464 [36] Garc a-Palacios J L and L azaro F J 1998 Phys. Rev. B 5814937 [37] Breth L, Suess D, Vogler C, Bergmair B, Fuger M, Heer R and Brueckl H 2012 J. Appl. Phys. 112 023903 [38] Taniguchi T and Imamura H 2012 Phys. Rev. B 85184403 [39] Leliaert J, Vansteenkiste A, Coene A, Dupr e L and Van Waeyenberge B 2015 Med. Biol. Eng. Com- put.53309{317 [40] Das P, Colombo M and Prosperi D 2019 Colloids and Surfaces B: Biointerfaces 17442{55 [41] Stoner E C and Wohlfarth E 1948 Philos. Trans. Royal Soc. A 240599{642

2019-12-01

Micromagnetic simulations based on the stochastic Landau-Lifshitz-Gilbert equation are used to calculate dynamic magnetic hysteresis loops relevant to magnetic hyperthermia. With the goal to effectively simulate room-temperature loops for large iron-oxide-based systems at relatively slow sweep rates on the order of 1 Oe/ns or less, a previously derived renormalization group approach for coarse-graining (Grinstein and Koch, Phys. Rev. Lett. 20, 207201, 2003) is modified and applied to calculating loops for a magnetite nanorod. The nanorod modelled is the building block for larger nanoparticles that were employed in preclinical studies (Dennis et al., Nanotechnology 20, 395103, 2009). The scaling algorithm is shown to produce nearly identical loops over several decades in the model grain size. Sweep-rate scaling involving the Gilbert damping parameter is also demonstrated to allow orders of magnitude speed-up of the loop calculations.

Coarse-graining in micromagnetic simulations of dynamic hysteresis loops

1912.00310v3

Landau-Lifshitz theory of the longitudinal spin Seebeck eect Silas Homan, Koji Sato, and Yaroslav Tserkovnyak Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA Thermal-bias-induced spin angular momentum transfer between a paramagnetic metal and ferro- magnetic insulator is studied theoretically based on the stochastic Landau-Lifshitz-Gilbert (LLG) phenomenology. Magnons in the ferromagnet establish a nonequilibrium steady state by equilibrat- ing with phonons via bulk Gilbert damping and electrons in the paramagnet via spin pumping, according to the uctuation-dissipation theorem. Subthermal magnons and the associated spin cur- rents are treated classically, while the appropriate quantum crossover is imposed on high-frequency magnetic uctuations. We identify several length scales in the ferromagnet, which govern qualitative changes in the dependence of the thermally-induced spin current on the magnetic lm thickness. PACS numbers: 85.75.-d,72.25.Mk,73.50.Lw,75.30.Ds I. INTRODUCTION Over the past three decades, spintronics has evolved from a focus on equilibrium phenomena in magnetic het- erostructures, such as giant magnetoresistance1and in- terlayer exchange interactions,2to dynamic processes, such as spin-transfer torque3,4and spin pumping,5,6and, more recently, nonequilibrium thermodynamics, heralded by the spin Seebeck eect7{9and thermally-induced motion of domain walls.10,11From a practical stand- point, magnetic nanostructures are useful for eld sens- ing and nonvolatile information storage,12where magne- toresistance is paramount for the readout, while current- induced spin torques are useful for fast and scalable bit switching.13 One rapidly-developing avenue of research concerns out-of-equilibrium spin phenomena in insulating systems, where spin is carried by collective excitations, such as spin waves (magnons), rather than electronic quasipar- ticles. To this end, spin waves in the ferrimagnetic in- sulator yttrium iron garnet (YIG) appear particularly promising as they suer from a remarkably low Gilbert damping (at microwave frequencies), 104, and the host material has Curie temperature of 500 K, thus remaining magnetic at room temperature.14Spin waves in YIG have recently been shown to undergo room- temperature Bose-Einstein condensation under nonlinear microwave pumping,15exhibit large spin pumping into adjacent conductors,16{18manifest the longitudinal spin Seebeck eect,19and eciently move domain walls un- der small thermal gradients.11These phenomena hold promise for integrated circuits based on nonvolatile mag- netic elements20with essentially no Ohmic losses and thus very low dissipation. Furthermore, thermal control of magnetic dynamics and spin currents21provides an attractive alternative to voltage control, especially since magnons, which are neutral objects, can respond more directly to tempera- ture gradients. The spin Seebeck eect, i.e., the gen- eration of thermal spin current between magnetic insu- lators and normal metals, is the basic phenomenon of central interest in this context. The purpose of this pa-per is to develop a systematic semi-phenomenological ap- proach to this problem, based on the Landau-Lifshitz- Gilbert (LLG) theory of ferromagnetic dynamics,22de- parting from the spin-pumping6perspective on the inter- action between electrons and magnons at ferromagnetic- insulatorjnormal-metal interfaces put forward in Ref. 8. In Sec. VIII, we comment on how the theory could be ex- panded to account for magnon and phonon kinetics when the standard LLG phenomenology fails. II. FERROMAGNETIC BULK DYNAMICS In the ferromagnetic bulk, away from the Curie temperature, magnetic dynamics are described by the stochastic LLG equation22 @tm= m(He+hl) +m@tm; (1) where m=M=Msis the unit-vector magnetization di- rection (Ms=jMjbeing the saturated magnetization magnitude), (minus) the gyromagnetic ratio ( > 0 for free electrons), dimensionless Gilbert damping con- stant, HeMF=Haz+Axr2m+Hr (2) the eective eld (consisting of applied eld Hain the zdirection, exchange eld /Ax, and relativistic cor- rections Hrthat include dipolar interactions and crys- talline anisotropies), and hlrandom Langevin eld with correlator23 hhl;i(r;t)hl;j(r0;t0)i=2 MskBT(r)ij(rr0)(tt0); (3) in accordance with the uctuation-dissipation theorem. We are interested at intermediate temperatures: much lower than the Curie temperature, such that the Landau- Lifshitz phenomenology based on the directional magne- tization dynamics [SO(3) nonlinear model] is appro- priate, while not too low such that the classical theory can be used as a starting point. We will, furthermore, ne- glectHrin Eq. (2), for simplicity, which is justied whenarXiv:1304.7295v2 [cond-mat.mes-hall] 12 Aug 20132 xyz0dT1T2With a little effort the layout of this diagram can actually be improvedby enlarging the inner box, see page 29 below.Here is the resonants-channel contribution toe+e−→4f. (From nowon, we do no longer display the\begin{fmfgraph}(40,25)\fmfpen{thick}...\end{fmfgraph}environment surrounding all pictures.)\fmfleftn{i}{2}\fmfrightn{o}{4}\fmf{fermion}{i1,v1,i2}\fmf{photon}{v1,v2}\fmfblob{.15w}{v2}\fmf{photon}{v2,v3}\fmf{fermion}{o1,v3,o2}\fmf{photon}{v2,v4}\fmf{fermion}{o4,v4,o3} e−e+µ+νµs¯cAnd the resonantt-channel contribution:\fmfleftn{i}{2}\fmfrightn{o}{4}\fmf{fermion}{i1,v1,v2,i2}\fmf{photon}{v1,v3}\fmf{fermion}{o1,v3,o2}\fmf{photon}{v2,v4}\fmf{fermion}{o4,v4,o3} e−e+µ+νµs¯cTwo point loop diagrams pose another set of problems. We must havea way of specifying that one or more of the lines connecting the twovertices arenotconnected by a straight line. The optionsleft,rightandstraightoffer the possibility to connect two vertices by a semicircledetour, either on the left or on the right. Since by default all lines con-tribute to the tension between two vertices, thetensionoption allows usto reduce this tension. The next examples shows both options in action.The lower fermion line is given an tension of 1/3 to make is symmetricalwith the upper line with consists of three parts. The loop photon is usinga detour on the right and does not contribute any tension.\fmfleft{i1,i2}\fmfright{o1}\fmf{fermion,tension=1/3}{i1,v1}\fmf{plain}{v1,v2}\fmf{fermion}{v2,v3}\fmf{photon,right,tension=0}{v2,v3}21nHa⊗N1N2Fjs1js2 FIG. 1. Schematic of an N 1jFjN2sandwich structure studied in this paper. The normal-metal layer N 1is treated as a poor spin sink, which blocks spin current, js10. The normal- metal layer N 2, on the other hand, is a perfect spin sink, thus establishing a thermal contact between its itinerant electrons and magnons in the ferromagnetic insulator (F), which results in spin current js2js. We assume the phonons in the F layer follow a linear temperature prole from T1atx= 0 (N 1jF interface) to T2atx=d(FjN2interface), corresponding to electron temperatures in N 1and N 2, respectively. kBT~ Ms. The Langevin correlator (3) is white at frequencies !kBT=~, corresponding to classical be- havior. In Sec. VI, we will adapt our theory to account for quantum uctuations at !&kBT=~, by matching with the fully quantum treatment of Ref. 24. In order to streamline discussion of the spin transfer, let us switch from the magnetization to the spin density: ssn=Ms m; (4) wheres=Ms= is the saturated spin density and n= mits direction. The LLG equation then becomes s(1 +n)@tn+n(Hz+h) +@ijs;i= 0;(5) where js;i=An@in (6) is identied as the magnetic spin current and hhi(r;t)hj(r0;t0)i= 2skBT(r)ij(rr0)(tt0);(7) whereiandjstand for the Cartesian coordinates. Here, HMsHaandAMsAx. In equilibrium, n=z, assuming the applied eld Ha>0. In this paper, we focus on the trilayer heterostructure depicted in Fig. 1. The temperature T(r) entering Eq. (3) is taken to correspond to the x-dependent phonon tem- perature inside of the ferromagnetic lm, T(x) =T1+x d(T2T1); (8) assuming Gilbert damping stems from the local magnon- phonon scattering. (We will revisit this assumption in Sec. VIII.)III. BOUNDARY CONDITIONS The boundary conditions for a ferromagnet sand- wiched between two normal metals need to be simi- larly constructed to account both for deterministic6and stochastic25spin-transfer torques. We will start with the former and then include the latter according to the uctuation-dissipation theorem. We assume the spin current is blocked by the N 1layer atx= 0, due to its weak spin-relaxation rate: js;x= 0 (x= 0): (9) In other words, the spin current pumped across the F jN1 interface is balanced by an equal back ow.6For our pur- poses, N 1can thus be replaced by an insulator, as long as it makes a good thermal contact with phonons in the ferromagnet. A net spin current across the F jN2inter- face, on the other hand, is allowed, if we treat N 2as a perfect spin sink:6 js;x=~g"# 4ndn dt(x=d); (10) whereg"#is the real part of the dimensionless interfa- cial spin-mixing conductance (per unit area).26We disre- gard the imaginary part of the spin-mixing conductance, since it governs the typically smaller6nondissipative spin- current component /dn=dt, which vanishes over a cycle of precession. A specic realization for such an N 1jFjN2 trilayer could be provided by the Cu jYIGjPt combina- tion, where Cu (Pt) is a light (heavy) element with weak (strong) spin-orbit interaction. (We will generalize our ndings to arbitrary N 1jFjN2trilayers, such as symmetric PtjYIGjPt type structures or general asymmetric struc- tures, in Sec. VII.) For the two N 1jFjN2interfaces, we correspondingly have the following (deterministic) boundary conditions for magnetic dynamics (as T!0): 8 < :@xn= 0; x = 0 A@xn+~g"# 4@tn= 0; x=d; (11) re ecting continuity of spin current, which is given by Eq. (6) inside the ferromagnet and Eqs. (9) and (10) in N1and N 2, respectively, across the corresponding inter- faces. Since spin pumping (10) aects magnetic dynamics similarly to Gilbert damping,6it is accompanied with a similar stochastic term.25The latter can be accounted for by modifying the boundary condition at x=d: A@xn+~g"# 4@tn+h0= 0; (12) where h0 i(;t)h0 j(0;t0) =~g"# 2kBT2ij(0)(tt0) (13)3 and= (y;z) is the two-dimensional position along the interface at x=d. The Langevin correlator strength is proportional to the electron temperature T2at the FjN2 interface, since the noise originates in the thermal uc- tuations of electronic spin currents in N 2. The spin Seebeck eect is embodied in the thermal- averaged spin current owing through the F jN2 interface:8 js;x=An@xn=n~g"# 4@tn+h0 : (14) Since our system is axially symmetric with respect to the zaxis, it is convenient to switch to complex notation: nnxiny. Thermal spin-current density, hjs;xi=jsz, can thus be written for small-angle dynamics (relevant at temperatures well below the Curie temperature) as js=AImhn@xnijx=d: (15) Exploiting, furthermore, translational invariance in the yzplane, we nd in the steady state: js=AImZd2qd! (2)3hn(q;!)@xn(q0;!0)i (2)3(qq0)(!!0);(16) where n(q;!) =Z d2dtei(!tq
)n(;d;t) (17) is the Fourier transform over and timet. The delta functions in the denominator of Eq. (16) cancel delta functions that factor out of the numerator when evaluat- ing the averageh:::i(with the remaining integrand inde- pendent of q0and!0). Similarly transforming Langevin correlators, Eqs. (7) and (13), we have: hh(x;q;!)h(x0;q0;!0)i=4(2)3skBT(x) (xx0)(qq0)(!!0);(18) for the bulk and hh0(q;!)h0(q0;!0)i= 4(2)30skBT2(qq0)(!!0) (19) for the FjN2interface, dening 0~g"# 4s; (20) which has dimensions of length. 0=dis the enhanced Gilbert damping for a monodomain precession of the fer- romagnetic lm.6 IV. SPIN SEEBECK COEFFICIENT We now have all the necessary ingredients in order to evaluate the (longitudinal) spin Seebeck coecient (which has units of inverse length squared)27 Sjs kB(T1T2)(21)of the N 1jFjN2structure shown in Fig. 1. To simplify our subsequent analysis, let us optimize the notation, as follows. The stochastic LLG equation (5) in the lm bulk is written as A(@2 x2)n(x;q;!) =h(x;q;!); (22) after linearizing transverse dynamics and Fourier trans- forming it in the yzplane and time. Here, 2q2+H(1 +i)s! A: (23) The stochastic boundary condition at x=d, Eq. (12), in this notation is A(@x0)n(x;q;!) =h0(q;!) (x=d); (24) while@xn= 0 atx= 0. Here, 0i0s! A: (25) Eqs. (22)-(25) now form a closed system of inhomo- geneous linear dierential equations, with source terms given by stochastic elds handh0. These are straight- forward to solve for nusing Green's functions. Substitut- ing the solution for ninto Eq. (16), we nd, after some algebra, the spin Seebeck coecient (21): S=0s2 23A2dZ1 1d2qZ1 1d!! Zd 0dxxcosh[(xd)] sinh(d)0cosh(d)2 :(26) Integrating over the longitudinal coordinate x, this nally becomes S=0s2 83A2dZ1 1d2qZ1 1d!! jsinh(d)0cosh(d)j2 sin2(id) 2 i+sinh2(rd) 2r ; (27) wherer(i) is the real (imaginary) part of . Eq. (27) is our central results and the main departure point for the subsequent analysis. Let us, for convenience, dene the following length scales: r A H(28) is the magnetic exchange length, l00 (29) is the spin-pumping length (i.e., the F thickness at which the monodomain Gilbert damping enhancement due to spin pumping6equals the intrinsic damping), r ~A skBT(30)4 is the thermal de Broglie wavelength (in the absence of applied eld), where Tis the ambient temperature, and l (31) is the decay length for thermal magnons in the bulk. In this notation, 2=q2+1 21 +i 2~! kBT: (32) V. QUASIPARTICLE APPROXIMATION In the following, we are primarily interested in the thickness,d, dependence of the spin Seebeck coecient, S, assuming the the length-scale hierarchy l0l. In YIG, for example, taking144Ms2 kG,A 1=2106erg/cm, and 104, we nd the follow- ing lengths: (1) .1 nm, at room temperature, (2) l0100 nm, taking g"#1014cm2from Ref. 17 and proportionately larger l0withg"#51014cm2from Ref. 18 (g"#is very sensitive to the preparation and qual- ity of the YIGjmetal interfaces), (3) l.10m, and (4) 10 nm at 1 kG (corresponding to typical magneto- static elds). We start by performing integration over frequency ! in Eq. (27) in the limit of low damping (both intrinsic and spin pumping). In this case, the integrand is peaked at sinh(d)0, corresponding to !n(q) =A s q2+n22 d2+1 2 ; (33) wheren= 0;1;2;3;::: labels magnon subbands [not to be confused with unit vector nintroduced in Eq. (4)]. These resonances are well separated when their width is much smaller than their spacing, allowing for a quasipar- ticle treatment of the energy integral. For the bulk damp- ing, this condition is 1=dp 1=21=2, for thermal magnons. Additionally, the occupation of magnons is ex- ponentially suppressed when the temperature is smaller than the gap, i.e., kBT=~.!0H=s. Therefore, we are interested in the opposite regime, when the thermal de Broglie wavelength is smaller than the magnetic exchange length, <  . Thus in the regime dl, these reso- nances are well-dened quasiparticle peaks corresponding to monodomain precession ( n= 0) and standing waves (n>0) along the longitudinal direction. See Fig. 2. This allows us to evaluate the spin Seebeck coecient by sum- ming the contributions from individual magnon modes, whendl. Expanding around !0, the contribution from the lowest energy resonance is S0=0 43dZ1 1d2qZ1 1d!! (!!0)2+ (+0=d)2!2; (34) 1.01.21.41.61.82.011.52!/!0(q)FIG. 2. Integrand of the spin Seebeck coecient, Eq. (27), in arbitrary units, illustrating the rst four quasiparticle res- onances,n= 0;1;2;3, according to Eq. (33), for a xed q. We set= 104,0= 102nm,q2+2= 1 nm2, and d= 10 nm in the main plot. The inset shows the essentially continuum spectrum when d= 100m, keeping other param- eters unmodied. which can be readily integrated over frequency: S0=0 42dZ1 1d2q (+0=d) [1 + (+0=d)2] 0=d 1 +l0=dZ1 1d2q (2)2; (35) where we have assumed small damping, +0=d1. Similarly, we nd for the n>0 subbands:28 Sn20=d 1 + 2l0=dZ1 1d2q (2)2: (36) The total Seebeck coecient in the quasiparticle approx- imation is thus S=S0+X n>0Sn: (37) The wave vector qin integrals (35) and (36) should be bounded by requiring that ~!n(q)<kBT, at which point our classical treatment breaks down, as discussed in the next section. The spurious ultraviolet divergence is elim- inated by cutting o at a total three-dimensional wave number,qc=p 1=21=2, corresponding to energy kBT. While a more careful quantum treatment for large wave numbers is constructed in the next section, we ex- pect a crude cuto to adequately capture the behavior of the Seebeck coecient, up to an overall factor of order one. Allow us to momentarily focus on the regime when d , where the quasiparticle peaks are dense (recovering three-dimensional behavior), so that SX n>0Sn20 1 + 2l0=dZ qcd3q (2)3=0 1 + 2l0=dq3 c 32(38)5 and, therefore, S=1 21 23=2 8 >>< >>:d 62; dl0 0 32; dl0: (39) That is, when the thickness of the ferromagnet is much smaller or larger than spin-pumping length, the spin cur- rent scales linearly with dor isdindependent, respec- tively. If the quasiparticle peaks are not dense, d, nite- size eects are important, re ecting individual magnon subbands. In the extreme low-temperature case when d, only monodomain precession along the longi- tudinal direction contributes to the Seebeck coecient: SS0. If the transverse dimensions are also much smaller than , the full volume of the (nano)magnet un- dergoes stochastic monodomain precession, and S=0 V1 1 +l0=d; (40) where we have retained only one mode associated with the transverse momentum in Eq. (35). Vhere is the vol- ume of the F layer. This coincides with the spin Seebeck coecient for a monodomain obtained in Ref. 8 [dening (T1+T2)=2!TF,T2!TN, and0=d!0, to match their notation]. Finally, for largest thicknesses dl, the quasiparticles are no longer well-dened (see inset of Fig. 2) and the above analysis cannot be applied. Because the thickness is beyond the magnon propagation length, only magnons within a distance lfrom the FjN2interface contribute to the spin current, which should, therefore, be independent of thickness, d, for a xed thermal gradient, ( T1T2)=d. Since, in this regime, the magnon propagation length is the largest length scale in the problem, we can send d! 1in Eq. (27), which gives S=0s2 83A2dZ1 1d2qZ1 1d!! 2rj0j2: (41) The integrand, which can be evaluated numerically, is independent of thickness, and, therefore, S/1=d, as expected. VI. QUANTUM CROSSOVER Our classical Langevin theory needs to be appropri- ately modied when approaching magnon frequencies of ~!kBT. On the one hand, this is an important limit, as the spin transport is dominated by thermal magnons in our model. On the other hand, the classical theory is inadequate for the treatment of quantum uctuations that dominate at high (on the scale of the ambient tem- perature) frequencies.To this end, we use a quantum-mechanical result24for the thermal spin current, which is exact for a tunneling spin-exchange Hamiltonian at an F jN interface:29 js=4kBT0Z1 0dD ()2@nBE() @; (42) whereD() is the magnon density of states, nBE(x) (ex1)1is the Bose-Einstein distribution, 1=kBT, andTis the temperature drop across the F jN inter- face (assuming magnons are equilibrated to a uniform temperature T, such that d). This limit can be di- rectly compared to our Eqs. (21) and (27), by rst send- ing0!0 in the integrand (thus reproducing the weak FjN contact and allowing for the magnons in F to equili- brate with phonons) and then sending !0 [such that the magnon spectral properties are unaected by Gilbert damping, as assumed in the derivation of Eq. (42)]. The magnons are correspondingly equilibrated to the average phonon temperature ( T1+T2)=2, such that we identify T= (T1T2)=2, in the present notation. According to Eq. (38), our semiclassical spin current becomes in this limit js!2kB(T1T2)0Zd3q (2)3: (43) This agrees with Eq. (42) in the limit kBT, where Z1 0dD ()2@nBE() @!Z1 0dD()Zd3q (2)3: (44) We conclude that the classical-to-quantum crossover can be accounted for by inserting the factor 2@nBE() @=~!=2 sinh(~!=2)2 F(~!) (45) in the energy integrand of Eq. (27), which eectively cuts o the contribution from magnons with energy ~! kBT. VII. RESULTS We summarize the spin Seebeck coecient dependence on the ferromagnetic layer thickness, when l0l: S(d)1 228 >< >:; d d=l; dl0 l0=l; l0dl l0=d; ld; (46) assuming(or else the magnon transport is ex- ponentially frozen out). The four regimes correspond respectively to the following physical situations: (1) Only the lowest magnon subband is thermally active (d), (2) quasi-3D subband structure is activated, but damping is still dominated by interfacial spin pumping6 0.01110010410610-510-410-310-210-1l0l d/S2 FIG. 3. Plot of the spin Seebeck coecient, Eq. (27), as a function of ferromagnet thickness dfor= 1 nm,= 10 nm, l0= 100 nm, and l= 10m. We use Eq. (37) (solid curve on the left) and Eq. (41) (solid line on the right), which are valid when d.landd&l, respectively, when N 1is a poor spin sink and N 2a perfect spin sink. To account for the classical-to-quantum crossover, we have inserted factor (45) in the integrands of Eqs. (35) and (36) [with !!!n(q)] and Eq. (41). The dotted curve shows the enhanced spin Seebeck coecient when also N 1is a perfect spin sink (which increases the eective thermal bias between magnons and electrons at the FjN2interface). (dl0), (3) bulk damping overtakes spin pumping, but the magnetic lm is still thinner than the thermal magnon decay length, such that magnons probe the full lm width ( l0dl), and (4) bulk regime is nally established when the lm is thicker than the magnon de- cay length ( ld). To illustrate these crossovers, we plot the spin Seebeck coecient as a function of din Fig. 3, using lengths characteristic of YIG, which are consistent with the above length-scale hierarchy. Notice that even thoughdetermines the magnon energy gap and the as- sociated Ginzburg-Landau correlation length in the clas- sical theory, it does not govern any prominent crossover in the function S(d). We conclude that S(d) has nonmonotonic thickness de- pendence, with the maximum value S(max)0 23; (47) attained at l0.d.l, i.e., below the magnon decay length but above l0, such that magnons equilibrate fully to the average phonon temperature T= (T1+T2)=2 (d&l0) but still remain coherent on the scale of d (d.l). This agrees with the result obtained in Ref. 30. S(max)is proportional to the spin-mixing conductance [see Eq. (20)] but is independent of the bulk Gilbert damping (as the magnon quasiparticle structure is still well resolved). According to Eq. (21), S(max)determines the largest spin current emitted thermally by a lm of magnetic insulator, as a function of d, when subjectedto a certain temperature dierence (for example, in a wedged magnetic insulator coated by metallic contacts). If, on the other hand, a well-dened temperature gradi- entis supplied (corresponding, for example, to a certain phonon-dominated heat- ux density), while thickness d is varied, the spin-current density js/Sdincreases with dsaturating at d&l(the magnon decay length): j(max) s xed@xTl0 22kB@xT; (48) which corresponds to the bulk regime. j(max) s vanishes when0!0 (no spin pumping) or !1 (no magnetic dynamics). It is interesting to ask how the above results would modify if both N 1and N 2in our model (see Fig. 1) were perfect spin sinks. For an inversion-symmetric struc- ture (e.g., PtjYIGjPt), spin currents at the two inter- faces must be equal, js1=js2js. Whendl, we should recover the bulk limit (41), since the magnons de- cay before traversing the full width of the lm (and thus the spin current at one interface should not be sensitive to the boundary condition at the other). When dl, however,S0andSnentering Eq. (37) need to be modi- ed. To that end, we notice that the factor (1+ l0=d)1in Eq. (35) re ects the dierence between Tm;0, the eective temperature of the magnons, and T2, the temperature of the electrons in N 2:8 Tm;0T2=T+ (0=d)T2 +0=d=T1T2 21 1 +l0=d:(49) Similarly for the n>0 subbands, the factor (1+2 l0=d)1 in Eq. (36) stems from the eective magnon-electron tem- perature dierence across the F jN2interface of Tm;nT2=T1T2 21 1 + 2l0=d: (50) When the bulk damping dominates over the interfa- cial spin pumping, i.e., l0d(while still dl), Tm;n!(T1+T2)=2, while in the opposite limit, i.e., dl0,Tm;n!T2. The magnon temperature thus becomes strongly skewed toward the F jN2interface for thinner ferromagnetic lms, when N 1is a poor spin sink (with electrons and magnons, therefore, being essentially decoupled at the N 1jF interface, for our purposes). In the case when both N 1and N 2are perfect spin sinks, on the other hand, the eective magnon-electron tempera- ture dierence driving spin current is given simply by TT2= (T1T2)=2 for all subbands (when dl). We account for this increased thermal gradient by dropping the factor (1 + l0=d)1on the right-hand side of Eq. (35) and likewise (1+2 l0=d)1in Eq. (36). The corresponding enhancement of the spin Seebeck coecient reverses the trend in Eq. (46) at d.l0to give S(d)1 228 < :0=d; d l0=l; dl l0=d; ld; (51)7 0.01110010410610-510-410-310-210-1l0l d/S2 FIG. 4. Plot of the spin Seebeck coecient using magnetic length scales as in Fig. (3) but calculated for a non-inversion- symmetric N 1jFjN2structure with spin-pumping parameters 0 1=0=10,0 2=0, respectively, at the two interfaces (upper trace) and0 1=0,0 2=0=10 (lower trace). Note that the former case is intermediate between the two curves plotted in Fig. 3 (where the solid curve corresponds to 0 1= 0,0 2=0 and the dotted curve to 0 1=0 2=0). which is now monotonically decreasing with d, as plotted by the dotted line in Fig. 3. When the structure N 1jFjN2is not mirror symmetric (either because the spin-mixing conductances or the spin- sink characteristics are dierent), which we characterize by dierent 0 1and0 2spin-pumping parameters at the two interfaces, we can repeat the above analysis for dl, nding Tm;0T2=T+ (0 1=d)T1+ (0 2=d)T2 +0 1=d+0 2=d =T1T2 21 + 2l0 1=d 1 + (l0 1+l0 2)=d(52) and Tm;nT2=T1T2 21 + 4l0 1=d 1 + 2(l0 1+l0 2)=d; (53) where we dened the spin-pumping lengths l0 i0 i=as- sociated with the left, i= 1, and right, i= 2, interfaces. We thus generalize the spin Seebeck contributions (at the FjN2interface) from dierent magnon subbands to S00 2 d1 + 2l0 1=d 1 + (l0 1+l0 2)=dZ1 1d2q (2)2F[~!0(q)] (54) and Sn>020 2 d1 + 4l0 1=d 1 + 2(l0 1+l0 2)=dZ1 1d2q (2)2F[~!n(q)] (55) in lieu of Eqs. (35) and (36), respectively. The asymptotic and crossover trends are now given by (assuming l0 1;l0 2l): S(d)1 228 >>< >>:2=(0 11+0 21)d; d 2=(l0 11+l0 21)l; dl0 1 l0 2=l; l0 1;l0 2dl l0 2=d; l d; (56) which coincides with Eq. (51) when 0 1=0 2but has an additional shoulder-like feature at d(l0 1+l0 2)=2 . This feature makes S(d) nonmonotonic when 0 1< 0 2. In Fig. 4, we plot the spin Seebeck coecient for two asym- metric cases: (1) 0 1=0=10,0 2=0and (2)0 1=0, 0 2=0=10 (physically corresponding to spin currents on two sides of a non-inversion-symmetric N 1jFjN2struc- ture, such as, PdjYIGjPt). VIII. DISCUSSION Our theory provides a minimalistic application of the LLG phenomenology to the problem of the spin See- beck eect, yet disregards magnon-magnon interactions. These become important at high temperatures, especially approaching the Curie temperature. Magnon-phonon in- teractions are included only insofar as a contribution to the total Gilbert damping. Elastic magnon scattering on impurities, which would manifest as an inhomogeneous broadening of ferromagnetic-resonance linewidth, may be an important impediment to the thermally-induced spin currents in disordered lms. The bulk limit of spin current, Eq. (48), is reduced by disorder, as well as the magnon decay length describing the crossover to the bulk regime. When the magnon mean free path lis shorter than our Gilbert damping decay length l, in particular, we expect the eective decay length to be lep ll(the spin diusion length) and j(max) s in Eq. (48) to be reduced by a factor of l=lep l=l(assuming that l0le, such that our length-scale hierarchy is unchanged). The Seebeck coecient behavior (46) [as well as Eq. (47)], however, remain essentially intact up to the thickness dle. Finally, we want to comment on a possibility of nonlo- cal magnetic relaxation. In this paper, we have assumed that Gilbert damping is a local and isotropic tensor. The locality would be a reasonable approximation if the damping bottleneck was due to some local dynamic de- fects. In the case of YIG,31which is known for its highly coherent elastic properties, nonlocality of the bulk mag- netic relaxation could signicantly modify our ndings. First of all, this could introduce new phonon-dependent length scales into the problem, which would show in the S(d) dependence. Standard long-wavelength ferromag- netic resonance on thick lms would reveal damping  that could be very dierent from that of short-wavelength thermal magnons relevant here. An eective damping parameter ~ of thermal magnons (which may itself be thickness dependent) would result in the bulk crossover thickness of ~l=~6=l. Whend.~l, we may still invoke8 the rst three regimes of our ndings, Eq. (46) (with  andlcorresponding to the thermal-magnon inverse qual- ity factor and decay length, respectively, due to magnon- phonon scattering), which should, furthermore, be indif- ferent to the fact that the local temperature, Eq. (8), is not well dened for highly-coherent phonons. The rea- son for this is that, in these regimes, when dis below the magnon decay length ~l, only the average phonon tem- perature Tis relevant for our theory. 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2013-04-26

Thermal-bias-induced spin angular momentum transfer between a paramagnetic metal and ferromagnetic insulator is studied theoretically based on the stochastic Landau-Lifshitz-Gilbert (LLG) phenomenology. Magnons in the ferromagnet establish a nonequilibrium steady state by equilibrating with phonons via bulk Gilbert damping and electrons in the paramagnet via spin pumping, according to the fluctuation-dissipation theorem. Subthermal magnons and the associated spin currents are treated classically, while the appropriate quantum crossover is imposed on high-frequency magnetic fluctuations. We identify several length scales in the ferromagnet, which govern qualitative changes in the dependence of the thermally-induced spin current on the magnetic film thickness.

Landau-Lifshitz theory of the longitudinal spin Seebeck effect

1304.7295v2

Magnon decay theory of Gilbert damping in metallic antiferromagnets Haakon T. Simensen, Akashdeep Kamra, Roberto E. Troncoso, and Arne Brataas Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Dated: July 3, 2019) Gilbert damping is a key property governing magnetization dynamics in ordered magnets. We present a theoretical study of intrinsic Gilbert damping induced by magnon decay in antiferromagnetic metals through s-dexchange interaction. Our theory delineates the qualitative features of damping in metallic antiferromagnets owing to their bipartite nature, in addition to providing analytic expressions for the damping parameters. Magnon- induced intraband electron scattering is found to predominantly cause magnetization damping, whereas the Néel field is found to be damped via disorder. Depending on the conduction electron band structure, we predict that magnon-induced interband electron scattering around band crossings may be exploited to engineer a strong Néel field damping. Introduction.— The dynamical properties of a harmonic mode are captured by its frequency and lifetime [ 1,2]. While the eigenfrequency is typically determined by the linearized equations of motion, or equivalently by a non-interacting de- scription of the corresponding quantum excitation, the lifetime embodies rich physics stemming from its interaction with one or more dissipative baths [ 1,3]. Dissipation plays a central role in the system response time. In the context of magnetic systems employed as memories, the switching times decrease with increasing damping thereby requiring a stronger dissi- pation for fast operation [ 4–6]. The dissipative properties of the system also result in rich phenomena such as quantum phase transitions [ 7–10]. Furthermore, the formation of hybrid excitations, such as magnon-polarons [ 11–18] and magnon- polaritons [ 19–24], requires the dissipation to be weak with respect to the coupling strengths between the two participating excitations [ 25]. Therefore, in several physical phenomena that have emerged into focus in the recent years [ 12,16,26–30], damping not only determines the system response but also the very nature of the eigenmodes themselves. Understanding, exploiting and controlling the damping in magnets is thus a foundational pillar of the field. The success of Landau-Lifshitz-Gilbert (LLG) phenomenol- ogy [ 31,32] in describing ferromagnetic dynamics has inspired vigorous e orts towards obtaining the Gilbert damping param- eter using a wide range of microscopic theories. The quantum particles corresponding to magnetization dynamics - magnons - provide one such avenue for microscopic theories and form the central theme in the field of magnonics [ 33,34]. While a vast amount of fruitful research has provided a good under- standing of ferromagnets (FMs) [ 35–54], analogous studies on antiferromagnets (AFMs) are relatively scarce and have just started appearing [ 55,56] due to the recently invigorated field of antiferromagnetic spintronics [ 57–62]. Among the ongoing discoveries of niches borne by AFMs, from electrically and rapidly switchable memories [ 63], topological spintronics [ 60], long range magnonic transport [ 64] to quantum fluctuations [65], an unexpected surprise has been encountered in the first principles evaluation of damping in metallic AFMs. Liu and coworkers [ 56] and another more recent first-principles study [66] both found the magnetization dissipation parameter to bemuch larger than the corresponding Néel damping constant, in stark contrast with previous assumptions, exhibiting richer features than in FMs. An understanding of this qualitative dierence as well as the general AFM dissipation is crucial for the rapidly growing applications and fundamental novel phenomena based on AFMs. Here, we accomplish an intuitive and general understanding of the Gilbert damping in metallic AFMs based on the magnon picture of AFM dynamics. Employing the s-d, two-sublattice model for a metallic AFM, in which the dandselectrons constitute the magnetic and conduction subsystems, we derive analytic expressions for the Gilbert damping parameters as a function of the conduction electron density of states at the Fermi energy and s-dexchange strength. The presence of spin- degenerate conduction bands in AFMs is found to be the key in their qualitatively di erent damping properties as compared to FMs. This allows for absorption of AFM magnons via s- dexchange-mediated intraband conduction electron spin-flip processes leading to strong damping of the magnetization as compared to the Néel field [ 67]. We also show that interband spin-flip processes, which are forbidden in our simple AFM model but possible in AFMs with band crossings in the conduc- tion electron dispersion, result in a strong Néel field damping. Thus, the general qualitative features of damping in metallic AFMs demonstrated herein allow us to understand the Gilbert damping given the conduction electron band structure. These insights provide guidance for engineering AFMs with desired damping properties, which depend on the exact role of the AFM in a device. Model.— We consider two-sublattice metallic AFMs within thes-dmodel [ 35,36,44]. The delectrons localized at lat- tices sites constitute the magnetic subsystem responsible for antiferromagnetism, while the itinerant selectrons form the conduction subsystem that accounts for the metallic traits. The two subsystems interact via s-dexchange [Eq. (3)]. For ease of depiction and enabling an understanding of qualitative trends, we here consider a one-dimensional AFM (Fig. 1). The re- sults within this simple model are generalized to AFMs with any dimensionality in a straightforward manner. Furthermore, we primarily focus on the uniform magnetization dynamics modes.arXiv:1907.01045v1 [cond-mat.mes-hall] 1 Jul 20192 FIG. 1: Schematic depiction of our model for a metallic AFM. The red and blue arrows represent the localized delectrons with spin up and down, respectively, thereby constituting the Néel ordered magnetic subsystem. The green cloud illustrates the delocalized, itinerant selectrons that forms the conduction subsystem. At each lattice site i, there is a localized delectron with spin Si. The ensuing magnetic subsystem is antiferromagnetically ordered (Fig. 1), and the quantized excitations are magnons [68,69]. Disregarding applied fields for simplicity and as- suming an easy-axis anisotropy along the z-axis, the magnetic Hamiltonian, Hm=˜JP hi;jiSi
SjKP i(Sz i)2, wherehi;ji denotes summation over nearest neighbor lattice sites, is quan- tized and mapped to the sublattice-magnon basis [69] Hm=X qh Aq ay qaq+by qbq +By qay qby q+Bqaqbqi ; (1) where we substitute ~=1,Aq=(2˜J+2K)SandBq= ˜JS eiq
aP hieiq
, where S=jSij,ais the displacement between the two atoms in the basis, and hidenotes sum- ming over nearest neighbor displacement vectors. aqandbq are bosonic annihilation operators for plane wave magnons on the A and B sublattices, respectively. We diagonalize the Hamiltonian [Eq. 1] through a Bogoliubov transforma- tion [ 69] toHm=P q!q y qq+y qq ;with eigenenergies !q=q A2qjBqj2. In the absence of an applied field, the magnon modes are degenerate. Theselectron conduction subsystem is described by a tight- binding Hamiltonian that includes the “static” contribution from the s-dexchange interaction [Eq. (3)] discussed below: He=tX hi;jiX cy icjJX i(1)i cy i"ci"cy i#ci# :(2) Here ciis the annihilation operator for an selectron at site iwith spin.t(>0)is the hopping parameter, and J(>0) accounts for s-dexchange interaction [Eq. (3)]. The (1)i factor in the exchange term reflects the two-sublattice nature of the AFM. The conduction subsystem unit cell consists of two basis atoms, similar to the magnetic subsystem. As a result, there are four distinct electron bands: two due to there being two basis atoms per unit cell, and twice this due to the two possible spin polarizations per electron. Disregarding applied fields, these bands constitute two spin-degenerate bands. We label these bands 1 and 2, where the latter is higher in energy.The itinerant electron Hamiltonian [Eq. (2)] is diagonalized into an eigenbasis (c1k;c2k)with eigenenergies 1k=k and2k= +k, wherek=p J2S2+t2j kj2, where k=P hieik
. The itinerant electron dispersion is depicted in Fig. 2. The magnetic and conduction subsystems interact through s-dexchange interaction, parametrized by J: HI=JX iSi
si; (3) where si=P 0cy i0ci0is the spin of the itinerant elec- trons at site i, where is the vector of Pauli matrices. The term which is zeroth order in the magnon operators, and thus ac- counts for the static magnetic texture, is already included in He [Eq. (2)]. To first order in magnon operators, the interaction Hamiltonian can be compactly written as Hem=X X kk0qcy k"ck0# WA; kk0qay q+WB; kk0qbq +h.c.;(4) whereandare summed over the electron band indices. As detailed in the Supplemental material, WA; kk0qandWB; kk0q, both linear in J, are coe cients determining the amplitudes for scattering between the itinerant electrons and the aqandbq magnons, respectively. Specifically, when considering plane wave states, WA=B; kk0qbecomes a delta function, thereby enforc- ing the conservation of crystal momentum in a translationally invariant lattice. Inclusion of disorder or other many-body eects results in deviation of the eigenstates from ideal plane waves causing a wave vector spread around its mean value [ 2]. The delta function, associated with an exact crystal momentum conservation, is thus transformed to a peaked function with finite width (
k). Thecombinations 11and22describe intraband electron scattering, while 12and21describe in- terband scattering. Intraband scattering is illustrated in Fig. 2. Interband scattering is prohibited within our model due to energy conservation, since the uniform q=0magnon energy is much smaller than the band gap. The scattering described by Hem[Eq. (4)] transfers spin angular momentum between the magnetic and conduction sub- systems. The itinerant electrons are assumed to maintain a thermal distribution thereby acting as a perfect spin sink. This is consistent with a strong conduction electron spin relaxation observed in metallic AFMs [ 70,71]. As a result, the magnetic subsystem spin is e ectively damped through the s-dexchange interaction. Gilbert damping.— In the Landau-Lifshitz-Gilbert (LLG) phenomenology for two-sublattice AFMs, dissipation is ac- counted via a 22 Gilbert damping matrix [ 72]. Our goal here is to determine the elements of this matrix in terms of the parameters and physical observables within our microscopic model. To this end, we evaluate the spin current “pumped” by the magnetic subsystem into the sconduction electrons, which dissipate it immediately within our model. The angu- lar momentum thus lost by the magnetic subsystem appears as Gilbert damping in its dynamical equations [ 72,73]. The3 - /2 - /4 0 /2 /4 e e kF,1a/epsilon1=µ1 m e e m /epsilon1=µ2 kF,2a FIG. 2: The selectron dispersion in metallic AFM model, where the red and blue dispersions depict electron bands 1 and 2, respectively. Illustrations of intraband electron-magnon scattering at two di erent Fermi levels, 1and2, are added. The depicted momentum transfer is exaggerated for clarity. second essential ingredient in identifying the Gilbert damping matrix from our microscopic theory is the idea of coherent states [ 74,75]. The classical LLG description of the magne- tization is necessarily equivalent to our quantum formalism, when the magnetic eigenmode is in a coherent state [ 74–76]. Driving the magnetization dynamics via a microwave field, such as in the case of ferromagnetic resonance experiments, achieves such a coherent magnetization dynamics [73, 77]. The spin current pumped by a two-sublattice magnetic sys- tem into an electronic bath may be expressed as [78] Iz=Gmm(m˙m)z+Gnn(n˙n)z +Gmn(m˙n)z+(n˙m)z;(5) where mand nare the magnetization and Néel field nor- malized by the sublattice magnetization, respectively. Here, Gi j=i j(M=j j), wherei jare the Gilbert damping co- ecients, is the gyromagnetic ratio of the delectrons andMis the sublattice magnetization. Considering the uni- form magnetization mode, Izis the spin current operator Iz=i[Hem;Sz][79], where Sz=P iSz i. We get Iz=iX X kk0qcy k"ck0# WA; kk0qay q+WB; kk0qbq h.c.:(6) The expectation value of this operator assuming the uniform magnetization mode to be in a coherent state corresponds to the spin pumping current [Eq. (5)]. In order to evaluate the spin pumping current from Eq. (6), we follow the method employed to calculate interfacial spin pumping current into normal metals in Refs. [ 73,77,78], and the procedure is described in detail therein. Briefly, this method entails assuming the magnetic and conduction subsystems to be independent and in equilibrium at t=1, when the mu- tual interaction [Eq. (4)] is turned on. The subsequent timeevolution of the coupled system allows evaluating its physical observables in steady state. The resulting coherent spin-current corresponds to the classical spin current Izthat can be related to the motion of the magnetization and the Néel field [Eq. (5)]. As a last step, we identify expressions for (m˙m)z,(m˙n)z and(n˙n)zin terms of coherent magnon states, which enables us to identify the Gilbert damping coe cientsmm,nnand mn. Results.— Relegating the detailed evaluation to Supplemen- tal Material, we now present the analytic expression obtained for the various coe cients [Eq. (5)]. A key assumption that allows these simple expressions is that the electronic density of states in the conduction subsystem does not vary significantly over the magnon energy scale. Furthermore, we account for a weak disorder phenomenologically via a finite scattering length lassociated with the conduction electrons. This results in an eective broadening of the electron wavevectors determined by the inverse electron scattering length, (
k)=2=l. As a result, the crystal momentum conservation in the system is enforced only within the wavevector broadening. By weak disorder we mean that the electron scattering length is much larger than the lattice parameter a. Ifkandk0are the wave vectors of the incoming and outgoing electrons, respectively, we then have (kk0)a=(
k)a1. This justifies an expansion in the wave vector broadening (
k)a. The Gilbert damping coe cients stemming from intraband electron scattering are found to be mm=0(J)0(J) 40BBBBBBBB@1+2 J 2 J+84 cos2(kFa)  2 J+4 cos2(kFa)21CCCCCCCCA[(
k)a]2; nn=0(J) 40BBBBBB@1+sin2(kFa) cos2(kFa)2 J 2 J+4 cos2(kFa)1CCCCCCA[(
k)a]2: (7) whereJ=JS=t,kFis the Fermi momentum and ais the lattice parameter, and where 0(J)=v2J2 8g2()j˜Vj24 cos2(kFa) 2 J+4 cos2(kFa): (8) Here, vis the unit cell volume, g()is the density of states per unit volume, is the Fermi level, and !0is the energy of theq=0magnon mode. ˜Vis a dimensionless and generally complex function introduced to account for the momentum broadening dependency of the scattering amplitudes. It satisfies ˜V(0)=1and0j˜V(
k)j1within our model. These analytic expressions for the Gilbert damping parameters constitute one of the main results of this letter. Discussion.– We straightaway note that nn=mm [(
k)a]21.nnis strictly dependent upon (
k)a, and is non- zero only if there is disorder and a finite electron momentum broadening. mmis large even when considering a perfectly ordered crystal. This latter result is in good accordance with recent first-principles calculations in metallic AFMs [ 56,66]. We moreover observe that both mmandnnare quadratic inJandg(). This result is shared by Gilbert damping ow- ing to spin-pumping in insulating ferrimagnet |normal metal4 e e m kFa/epsilon1=µ FIG. 3: A schematic depiction of magnon-induced interband scattering in a band crossing at the Fermi level. (NM) and AFM |NM bilayers with interfacial exchange cou- pling [ 78]. Metallic AFMs bear a close resemblance to these bilayer structures. There are however two main di erences: Thes-dexchange coupling exists in the bulk of metallic AFMs, whereas it is localized at the interface in the bilayer structures. Additionally, the itinerant electron wave functions are qual- itatively di erent in metallic AFMs and NMs, owing to the magnetic unit cell of the AFM. Indeed, these di erences turn out to leave prominent signatures in the Gilbert damping in metallic AFMs. The uniform mode magnon energy is much smaller than the electron band gap within our simple model. Interband scat- tering is thus prohibited by energy conservation. However, in real AFM metals, the electron band structure is more com- plex. There may for instance exist band crossings [ 80–82]. In such materials, magnon-induced interband electron scatter- ing should also contribute to Gilbert damping, depending on the position of the Fermi surface. Motivated by this, we now consider Gilbert damping stemming from interband scattering, while disregarding the energy conservation for the moment, labeling the coe cientsI mmandI nn. We then find the same expressions as in Eq. (7) with the roles of I mm;nninterchanged with respect to mm;nn. This implies that I nnis large and inde- pendent of electron momentum broadening, whereas I mmis proportional to the electron momentum broadening squared. Although arriving at this result required disregarding the en- ergy conservation constraint, the qualitative e ect in itself is not an artifact of this assumption. In materials with a band crossing, as depicted in Fig. 3, I nn=I mm>  nn=mmis a gen- eral result. This generic principle derived within our simple model provides valuable guidance for designing materials with an engineered Gilbert damping matrix. We now provide a rough intuitive picture for the damping dependencies obtained above followed by a more mathemati- cal discussion. Consider a conventional di raction experiment where an incident probing wave is able to resolve the two slits only when the wavelength is comparable to the physical separation between the two slits. In the case at hand, the wave- functions of electrons and magnon participating in a scatteringprocess combine in a way that the wavenumber by which the conservation of crystal momentum is violated becomes the probing wavenumber within a di raction picture. Therefore, the processes conserving crystal momentum have vanishing probing wavenumber and are not able to resolve the opposite spins localized at adjacent lattice sites. Therefore, only the aver- age magnetization is damped leaving the Néel field una ected. With disorder, the probing wavenumber becomes non-zero and thus also couples to the Néel field. The interband scattering, on the other hand, is reminiscent of Umklapp scattering in a single-sublattice model and the probing wavenumber matches with the inverse lattice spacing. Therefore, the coupling with the Néel field is strong. The Gilbert damping in metallic AFMs here considered is caused by spin pumping from the magnetic subsystem into thesband. This spin pumping induces electron transitions between spin"/#states among the selectrons. The Gilbert damping coe cients depend thus on transition amplitudes pro- portional to products of itinerant electron wave functions such as y k"(x) k0#(x). The damping e ect on sublattice A depends on this transition amplitude evaluated on the A sublattice, and equivalently for the B sublattice. Assuming without loss of gen- erality that site i=0belongs to sublattice A, we find in the one- dimensional model that the damping on sublattice A is a func- tion ofP jcos2xj 2a y k"(xj) k0#(xj), whereas the damping on sublattice B is a function ofP jsin2xj 2a y k"(xj) k0#(xj). Equivalently, by straightforwardly using that m=(mA+mB)=2 andn=(mAmB)=2, this analysis predicts that mmis a function ofP j y k"(xj) k0#(x), whereasnnis a function of P jcosxj a y k"(xj) k0#(x). Assuming plane wave solutions of the electron wave functions, and if we consider intraband scattering only, we more concretely find that mmis a function of(1i(
k)a), where iis the imaginary unit, whereas nnis a function of (
k)a. This coincides well with Eq. (7). Above, we presented a discussion of interband scattering in the minimal model where the band gap artificially was set to zero. In this limit, the upper electron band is a continuation of the lower band with a =amomentum shift. We may then write 2k= 1;k+=a;. Under the assumption of a disappear- ing band gap, momentum-conserving interband scattering at momentum kis therefore equivalent to intraband scattering be- tween kandk=a. This is the exact phase shift which results in a smallmmand a large nnconsistent with the discussion above. In real metallic AFMs with complex band structures, the exact wave function relations unveiled above do not apply. However, interband transition amplitudes will undoubtedly carry a position dependent phase. This position dependence results in a dephasing of transition amplitudes at neighboring lattice sites, which gives rise to a non-negligible nn. The pre- cise damping coe cients in real metallic AFMs depend on the detailed electron wave functions. We may however generally conclude that I nn=I mm> nn=mm. Conclusion.— We have provided a microscopic derivation of Gilbert damping resulting from magnon decay through s-d exchange interaction in metallic antiferromagnets. Analytic5 expressions for Gilbert damping coe cients resulting from in- traband electron scattering are presented, while Gilbert damp- ing resulting from interband electron scattering is discussed on a conceptual level. We find that intraband electron scattering gives rise to a large magnetization damping and a negligible Néel field damping. The intraband Néel field damping is pro- portional to the inverse electron scattering length squared, and disappears exactly if there is no crystal disorder. 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2019-07-01

Gilbert damping is a key property governing magnetization dynamics in ordered magnets. We present a theoretical study of intrinsic Gilbert damping induced by magnon decay in antiferromagnetic metals through $s$-$d$ exchange interaction. Our theory delineates the qualitative features of damping in metallic antiferromagnets owing to their bipartite nature, in addition to providing analytic expressions for the damping parameters. Magnon-induced intraband electron scattering is found to predominantly cause magnetization damping, whereas the N\'eel field is found to be damped via disorder. Depending on the conduction electron band structure, we predict that magnon-induced interband electron scattering around band crossings may be exploited to engineer a strong N\'eel field damping.

Magnon decay theory of Gilbert damping in metallic antiferromagnets

1907.01045v1

Spin Pumping into Carbon Nanotubes K. Fukuzawa1, T. Kato1, M. Matsuo2,3,4,5, T. Jonckheere6, J. Rech6, and T. Martin6 1Institute for Solid State Physics, The University of Tokyo, Kashiwa, 277-8581, Japan 2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China 3CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing, 100190, China 4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan 5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan 6Aix Marseille Univ, Universit´ e de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France (Dated: October 31, 2023) We theoretically study spin pumping from a ferromagnetic insulator (FI) into a carbon nanotube (CNT) . By employing the bosonization method, we formulate the Gilbert damping induced by the FI/CNT junction, which can be measured by ferromagnetic resonance. We show that the increase in the Gilbert damping has a temperature dependence characteristic of a Luttinger liquid and is highly sensitive to the Luttinger parameter of the spin sector for a clean interface. We also discuss the experimental relevance of our findings based on numerical estimates, using realistic parameters. I. INTRODUCTION Spin pumping induced by ferromagnetic resonance (FMR) [1, 2] is a fundamental technique in spintronics for generating spin current from a ferromagnet to an ad- jacent material [3, 4]. While spin pumping has been used for injecting spin into various materials, it can also be uti- lized for detecting spin excitations in various systems [5– 17]. Compared with bulk measurement techniques, such as nuclear magnetic resonance (NMR) and neutron scat- tering experiments, spin pumping has an advantage in sensitivity for nanostructured systems such as surfaces, thin films and atomic-layer compounds [5]. The study of exotic spin excitations which emerge in specific materials is one of the forefront topics of con- densed matter physics. A typical example is spin exci- tation in quasi-one-dimensional interacting electron sys- tems, whose low-energy excitation can be described by the Tomonaga-Luttinger liquid [18–20]. Spin excitations inherent to the Tomonaga-Luttinger liquid have been studied in carbon nanotubes (CNTs) by using NMR [21– 23]. While NMR can detect the local spin susceptibility in CNTs, the use of spin pumping to detect spin excita- tions is expected to provide useful information reflecting the exotic character of the Luttinger liquid, which can- not be captured by NMR. It is thus important to clarify what kind of information about the Luttinger liquid can be obtained from a spin pumping experiment. In this work, we theoretically formulate the increase in the Gilbert damping due to spin pumping in a setup in which spin is injected into CNTs. We consider a magnetic junction composed of a ferromagnetic insula- tor (FI) and a single-wall CNT (see Fig. 1) and take in- terfacial randomness into account with a simple model. We derive an analytic expression for the increase in the Gilbert damping by utilizing the bosonization method and second-order perturbation with respect to the inter- facial exchange coupling. We will focus on the two limiting cases, i.e., a clean in- terface and a dirty interface. We show that for both cases FIG. 1. Magnetic junction composed of a ferromagnetic in- sulator (FI) and a single-wall carbon nanotube (CNT). The dimension of the FI is W×W′×d′. the temperature dependence of the increase of the Gilbert damping shows a power-law behavior, with an exponent reflecting the Luttinger parameters. For a clean inter- face, the exponent includes information on the Luttinger parameters in the spin sector and is shown to be sensitive to small deviations from unity (which is the value of the SU(2) symmetric model in the spin sector). For a dirty interface, the exponent depends on the Luttinger param- eters of both the spin and charge sectors as in an NMR measurement. We estimate the increase of the Gilbert damping using realistic parameters and discuss the ex- perimental feasibility. Our paper is organized as follows. We introduce the microscopic model of the FI/CNT magnetic junction in Sec. II. We analytically calculate the increase in the Gilbert damping in Sec. III and subsequently estimate it with realistic parameters in Sec. IV. Finally, we briefly discuss the experimental relevance of our findings in Sec. V and summarize our results in Sec. VI. A detailedarXiv:2303.01343v2 [cond-mat.mes-hall] 28 Oct 20232 derivation of the analytic expressions is given in the two Appendices. II. MODEL Let us consider a junction composed of a CNT and FI, whose Hamiltonian is given by H=HCNT+HFI+ Hint. Here, HCNTandHFIdescribe electrons in the CNT and FI, respectively, and Hintrepresents the interfacial exchange interaction between the CNT and FI. We will give their explicit forms in the subsections that follow. A. Carbon nanotube The low-energy Hamiltonian of electrons in CNTs is given by HCNT=HK+HC, (1) where HKandHCrepresent the kinetic energy and the forward scattering potential due to the screened Coulomb interaction, respectively. Using standard con- ventions [24], the Hamiltonians describing these energies of electrons in CNTs are given by HK=−ivFZ dxX rασrψ+ rασ(x)∂xψrασ(x), (2) HC=1 2Z dx dy ρ (x)V(x−y)ρ(y), (3) where ψrασ(x) is the slowly varying part of the field operator of electrons, vFis the Fermi velocity, V(x) is the screened Coulomb potential, and ρ(x) =P rασψ† rασ(x)ψrασ(x) is the electron density operator. The subscripts, r(=±),α(=±), and σ(=±), repre- sent the direction of propagation, the nanotube branch (the valley), and the spin orientation, respectively. Using the bosonization method [19, 24], the annihilation oper- ator describing fermions in the CNT can be expressed in terms of bosonic fields, θασ(x) and ϕασ(x), as ψrασ(x) =ηrασ√ 2πaei(−rθασ(x)+ϕασ(x)), (4) where ηrασis the Klein factor, and ais a short-length cutoff which can be identified with the lattice constant of the CNT. To diagonalize the Hamiltonian, we introduce new bosonic fields for the charge and spin sectors, θjδ(x) andϕjδ(x) as θασ(x) =1 2X jδhjδ(α, σ)θjδ(x), (5) ϕασ(x) =1 2X jδhjδ(α, σ)ϕjδ(x), (6) where δ(=±) represents symmetric/antisymmetric modes, j(=c, s) indicates the charge/spin mode, hc+=1,hc−=α,hs+=σ, and hs−=ασ. The Hamiltonian of the CNTs can be written as HCNT=X j,δvjδ 2πZ dx[K−1 jδ(∂xθjδ)2+Kjδ(∂xϕjδ)2],(7) where Kjδis the Luttinger parameter and vjδ=vF/Kjδ. B. Ferromagnetic insulator We consider a bulk FI described by the quantum Heisenberg model and employ the spin-wave approxi- mation assuming that the temperature is much lower than the magnetic transition temperature and the mag- nitude of the localized spin, S0, is much larger than one [8, 9, 11, 15–17, 25]. In this situation, the Hamil- tonian for the FI is approximately written as a superpo- sition of magnon modes: HFI=X kℏωkb† kbk, (8) where bkis the annihilation operator of magnons, ℏωk= Dk2+ℏγghdcis the magnon dispersion, Dis spin stiffness, γgis the gyromagnetic ratio, and hdcis the static mag- netic field. We will only focus on uniform spin precession induced by external microwaves. For this purpose, it is sufficient to consider the magnon mode of k=0with the simplified Hamiltonian HFI=ℏω0b† 0b0. (9) Microwave absorption in FMR can be related to the imaginary part of the retarded spin correlation function, which is defined as GR(ω) =−i ℏZ∞ 0dt ei(ω+iδ)t⟨[S+ 0(t), S− 0]⟩, (10) where S+ 0=√2S0b0and S− 0=√2S0b† 0are spin ladder operators of the FI for k=0andS+ 0(t) = eiHt/ℏS+ 0e−iHt/ℏ. For an isolated bulk FI, the spin sus- ceptibility is calculated as: GR 0(ω) =2S0/ℏ ω−ω0+iδ. (11) In real experiments, the FMR linewidth is finite due to the the Gilbert damping. To represent this finite spin relaxation in the bulk FI, we introduce a phenomeno- logical dimensionless parameter αGand express the spin correlation function as GR 0(ω) =2S0/ℏ ω−ω0+iαGω. (12)3 C. Interfacial exchange interaction Now let us consider the interfacial exchange interaction between the FI and the CNT with the Hamiltonian, Hint=S+ 0s−+S− 0s+, (13) where s±is the spin ladder operator of the CNT, defined as s−=r 1 NFIX r,r′X α,α′ZW 0dx J(x) ×e−i(α−α′)kFx−i(r−r′)qFxψ† rα−(x)ψr′α′+(x),(14) ands+= (s−)†. Here, Wis the length of the interface, J(x) is the interfacial exchange coupling, NFIis the num- ber of unit cells in the FI, kFis the Fermi wavenumber, andqF(≪kF) is the momentum mismatch associated with the two modes. Because the interfacial exchange coupling J(x), which is induced by quantum mechanical mixing between CNT and FI, is sensitive to distances of atoms across the junction, we assumed that it depends on the position xdue to random atomic configuration near the interface. A simplified model for randomness in J(x) will be accounted for in the next section. III. FORMULATION A. Gilbert damping Using second-order perturbation with respect to the interfacial exchange coupling, the spin susceptibility is calculated as G(iωn) =1 G0(iωn)−1−Σ(iωn)(15) Σ(iωn) =−1 ℏZℏβ 0dτeiωnτ⟨Tτs+(τ)s−(0)⟩ (16) where s±(τ) = eHCNTτ/ℏs±e−HCNTτ/ℏ. The retarded spin correlation function is obtained by analytic contin- uation iωn→ω+iδas GR(ω) =2S0/ℏ ω−(ω0+δω0) +i(αG+δαG)ω0,(17) δω0 ω0≃2S0 ℏω0Re ΣR(ω0), (18) δαG≃ −2S0 ℏω0Im ΣR(ω0), (19) where ΣR(ω) is the retarded self-energy defined by ΣR(ω) =Z dt eiωtΣR(t), (20) ΣR(t) =−iθ(t) ℏ⟨[s+(t), s−(0)]⟩, (21)θ(t) is the step function, and αG+δαG≪1 has been assumed. In our work, we focus on the increase in the Gilbert damping due to the junction, δαG, which is writ- ten in terms of the dynamic spin susceptibility of CNTs. B. Self-energy of electrons in CNTs By substituting Eq. (14) into Eq. (21), we obtain ΣR(t) =−i ℏθ(t)2S0 NFIX r,r′X α,α′ZW 0dxZW 0dy⟨J(x)J(y)⟩imp ×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(22) Crαr′α′(x, y, t ) =⟨[ψ† rα,+(x, t)ψr′α′,−(x, t), ψ† r′α′,−(y,0)ψrα,+(y,0)]⟩0. (23) Here, ⟨···⟩ impindicates a random average for the in- terfacial exchange coupling. For simplicity, we assume that the exchange coupling follows a Gaussian distribu- tion whose average and variance are given by ⟨J(x)⟩imp=J1, (24) ⟨δJ(x)δJ(y)⟩imp=J2 2aδ(x−y), (25) where δJ(x) =J(x)− ⟨J(x)⟩imp. Here, J1andJ2repre- sent respectively the average and the standard deviation of the distribution. The ratio J2/J1reflects the random- ness of the interfacial exchange coupling. In particular, the case of J2/J1= 0 corresponds to a clean interface without randomness. Accordingly, the self-energy is calculated as ΣR(t) = ΣR 1(t) + ΣR 2(t), (26) ΣR 1(t) =−iθ(t)2S0J2 1 ℏNFIX r,r′,α,α′ZW 0dxZW 0dy ×e−i(kF(α−α′)+qF(r−r′))(x−y)Crαr′α′(x, y, t ),(27) ΣR 2(t) =−iθ(t)2S0J2 2a ℏNFIX r,r′,α,α′ZW 0dx C rαr′α′(x, x, t ). (28) Since the integrand of ΣR 1(t) includes a rapidly oscillating part as a function of ( x−y), the integral is negligibly small except for the case of α=α′andr=r′. There, we obtain ΣR 1(t) =−iθ(t)2S0J2 1 ℏNFIX r,αZW 0dxZW 0dy C rαrα(x, y, t ). (29) We should note that ΣR 1(t) corresponds to the process of electron creation and annihilation in the same branch and represents momentum-conserving spin relaxation for a clean junction. In contrast, ΣR 2(t) represents spin re- laxation for a “dirty” junction that is independent of the4 electron momentum. Here, the word “dirty” means that during spin exchange process the momentum of electrons in the CNT is not conserved and transitions between dif- ferent branches of valleys and propagation directions are allowed. The following discussion will consider two lim- iting cases for the interface. For the clean interface limit (J1≫J2), the magnon self-energy is represented with ΣR 1(t), while in the dirty interface limit ( J1≪J2), it is represented with ΣR 2(t). C. Clean interface Since the correlation function Crαr′α′(x, y, t ) can be calculated using the bosonization method (see Ap- pendix A), the self-energy ΣR 1(t) can be obtained ana- lytically. Therefore, the corresponding increase in the Gilbert damping is obtained as δαG,1=−2S0 ℏω0Im ΣR 1(ω0) =−4S0J2 1 ℏ2ω0(2πa)2NFIZW 0dxZW 0dyZ∞ 0dtsinω0t ×Im"sinh(iπa/β ℏvF) sinh(π(ia−(x−y)−vFt)/βℏvF)γ−1 ×sinh(iπa/β ℏvF) sinh(π(ia+ (x−y)−vFt)/βℏvF)γ+1# ,(30) γ≡Ks+ 4+Ks− 4+1 4Ks++1 4Ks−. (31) After analytic integration with respect to t(see Ap- pendix B for details), we obtain δαG,1=2 πΓ(γ)2 Γ(2γ)S0J2 1Wa ℏ2v2 FNFI2πa βℏvF2γ−3 ×I(πW/β ℏvF, γ), (32) I(w, γ) =1 wZw 0dz′Zz′ 0dz e−2(γ−1)z ×F(γ−1, γ,2γ; 1−e−4z), (33) where F(a, b, c ;x) is the hypergeometric function. D. Dirty interface The self-energy ΣR 2(t) can be obtained in a similar way as above. The corresponding increase in the Gilbertdamping is given by δαG,2=−2S0 ℏω0Im ΣR 2(ω0) =−S0J2 2aW ℏ2ω0(πa)2NFIX r,r′,α,α′Z∞ 0dtsinω0t ×Im"sinh(iπa/β ℏvF) sinh(π(ia−vFt)/βℏvF)2γrαr ′α′# ,(34) γrαr′α′=γ1δr,r′δα,α′+γ2δr,−r′δα,α′ +γ3δr,r′δα,−α′+γ4δr,−r′δα,−α′,(35) γ1= (Ks++Ks−+ 1/Ks++ 1/Ks−)/4, (36) γ2= (Kc++Kc−+ 1/Ks++ 1/Ks−)/4, (37) γ3= (Ks++Kc−+ 1/Kc++ 1/Ks−)/4, (38) γ4= (Kc++Ks−+ 1/Ks++ 1/Kc−)/4. (39) We should note that δαG,2is proportional to W, since the spin relaxation rate is determined through spatially-local spin exchange in the dirty interface and is proportional to the number of spin-exchange channels. After analytic integration with respect to t(see Appendix B for details), we obtain δαG,2=1 2πS0J2 2aW ℏ2v2 FNFI ×X r,r′,α,α′Γ(γrαr′α′)2 Γ(2γrαr′α′)2πa βℏvF2γrαr ′α′−2 .(40) IV. NUMERICAL ESTIMATE Next, we evaluate numerically the increase in the Gilbert damping by using realistic experimental parame- ters. While the increase was formulated for a single CNT in the previous section, to increase the signal, it would be more useful if we considered a junction with a bundle of CNTs. Thus, in the following, we will consider a junction composed of a FI and a bundle of CNTs with an area of W×W′(see Fig. 1) and multiply δαG,1andδαG,2by the number of CNTs in the junction, NCNT=W′/d(d: the diameter of CNTs). The parameters are given in Table I. The Fermi ve- locity vF, lattice constant a, diameter d, Luttinger pa- rameters of CNTs, Kc+,Kc−, and Ks−, are taken from Refs. [20, 24, 26]. The value of Ks+is an experiment result [22] under a magnetic field of 3 .6 T[27]. The spin amplitude S0and the lattice constant a′are determined by assuming that the FI is made from yttrium iron gar- net (YIG). The interfacial exchange coupling ( J1orJ2) is roughly estimated to be 2 K [28]. The number of unit cells is estimated as NFI=WW′d′/a′3, where d′is the thickness of the FI.5 TABLE I. Parameters used for the numerical estimate. Microwave frequency ω0 1 GHz Fermi velocity of CNT vF 106m/s Lattice constant of CNT a 2.46˚A Diameter of CNT d 1.5 nm Amplitude of spins of FI S0 10 Lattice constant of FI a′12.376˚A Thickness of FI d′10 nm Interfacial exchange couplings J1,J2 clean interface J1= 2 K, J2= 0 dirty interface J1= 0 K, J2= 1,2,3 K Luttinger parameters Kc+ 0.20 Ks+ 1.07 Kc−,Ks−1 100 1010-2 10-3 10-4 3 300 FIG. 2. Temperature dependence of increase in the Gilbert damping, δαG,1, for a clean interface ( J1≫J2). A. Clean interface The estimated increase in the Gilbert damping for a clean interface ( J1= 2 K ≫J2) is shown in Fig. 2 as a function of temperature. While δαG,1is proportional to 1/Tat high temperatures, it is almost constant at low temperatures. The crossover temperature for a fixed length Wis given by T∗=g(γ)ℏvF/(kBW) (kB: Boltz- mann constant), which is proportional to 1 /W. The fac- torg(γ), which depends only on γ, is explicitly shown later. The increase in the Gilbert damping is shown as a function of the junction length Win Fig. 3. While δαG,1 is proportional to Wfor a short junction, it is almost constant for a long junction. The crossover length for a fixed temperature Tis given by W∗=g(γ)ℏvF/(kBT). In the present estimate, the condition Lth≪vF/ω0 always holds, where Lth=ℏvF/kBTis a thermal length. Under this condition, the increase in the Gilbert damping becomes independent of ω0and is approximately given 3K 10K 30K 100K 300K 10-310-410-510-610-2 10-510-3 10-4 FIG. 3. Junction-length dependence of increase in the Gilbert damping, δαG,1, for a clean interface ( J1≫J2). by δαG,1=Γ(γ)2 Γ(2γ)S0J2 1a′3a (ℏvF)2dd′2πa Lth2γ−3 f(γ, πW/L th), (41) f(γ, w) =( w/π, (w/π≪g(γ)), g(γ),(w/π≫g(γ)),(42) g(γ) =2 πZ∞ 0dz e−2(γ−1)zF(γ−1, γ,2γ; 1−e−4z). (43) From this analytic expression, we obtain δαG,1∝( T2γ−2W, (W≪g(γ)Lth), T2γ−3g(γ),(W≫g(γ)Lth).(44) The exponent γ= (Ks++Ks−+K−1 s++K−1 s−)/4 cor- responds to unity when Ks+=Ks−= 1. Even in the present estimate employing Ks+= 1.07, the exponent is almost unity ( γ= 1.00114). By setting γ= 1, we can reproduce the power in the temperature and junction- length dependence of δαG,1shown in Figs. 2 and 3. Finally, let us discuss the factor g(γ). If γis slightly larger than 1 as in the present estimate, the geometric function is approximated as F(γ−1, γ,2γ;x)≃1. Then, the factor g(γ) is approximately given as g(γ) =1 π(γ−1). (45) This expression indicates that the increase in the Gilbert damping in the high-temperature limit ( T≫T∗) or the long-junction limit ( W≫W∗) is highly sensitive to the deviation of γfrom unity. The crossover temperature T∗and the crossover length W∗also include the factor6 100 1010-5 10-810-6 10-7 3 300 FIG. 4. Temperature dependence of increase in the Gilbert damping, δαG,2, for a dirty interface ( J2≫J1). The three lines correspond to J2= 1, 2, and 3 K, respectively. g(γ)∝(γ−1)−1. Thus, the increase in the Gilbert damp- ing can be used to investigate small deviations of γfrom unity. Then, the Luttinger parameter Ks,+in the spin sector can also be determined from Eq. (31) if we know whether it is greater or less than unity. We note that in the NMR measurement [22] Ks,+decreases as the mag- netic field increases. Using this experimental tendency, we expect that Ks,+can be determined uniquely. B. Dirty interface Next, we consider a dirty interface ( J2≫J1). Figure 4 shows the increase of the Gilbert damping, δαG,2, as a function of the temperature for J2= 1, 2, and 3 K. In this case, δαG,2is proportional to T−0.43in the whole temperature range and shows a nontrivial exponent in- herent to the Tomonaga-Luttinger liquid. The condition Lth≪vF/ω0also holds for a dirty in- terface. Therefore, δαG,2can be approximated as δαG,2=1 2πS0J2 2aa′3 (ℏvF)2dd′ ×X r,r′,α,α′Γ(γrαr′α′)2 Γ(2γrαr′α′)2πa Lth2γrαr ′α′−2 .(46) Noting that a≪Lth, the factor (2 πa/L th)2γrαr ′α′in Eq. (46) is largely reduced as γrαr′α′increases. There- fore, in the sum of Eq. (46), it is sufficient to keep the terms in which γrαr′α′takes a minimum value. In the present estimate, γrαr′α′is given by Eq. (35) with (γ1, γ2, γ3, γ4) = (1 .001,0.784,1.001.0.784). (47) Upon setting the minimum exponent to be γmin= 0.784, we obtain δαG,2∝T2γmin−2=T−0.432, which is consis-tent with the numerical results shown in Fig. 4. There- fore, the nontrivial exponent inherent to the Tomonaga- Luttinger liquid appears in spin pumping through a dirty junction. Note that the approximate expression is inde- pendent of the junction length Wfor a fixed thickness, since the W-linear factor in NFI=WW′d/a′3cancels out the factor of Win Eq. (34). The equation for the increase in the Gilbert damping for the dirty interface has almost the same form as that for 1/T1Tin NMR experiments where T1is the longitu- dinal relaxation time of nuclear spins[21–23]. Therefore, the power law of the temperature dependence for the dirty interface is the same as in NMR experiments. This is because the spin transfer occurs at a spatially localized point due to the impurity average at the dirty interface, leading to the same situation as the NMR experiment in which 1 /T1Tis related to the local dynamic spin suscep- tibility. V. EXPERIMENTAL RELEVANCE We estimated the increase in the Gilbert damping δαG in two limiting situations, i.e., clean and dirty interfaces. If we choose YIG as the ferromagnet, δαGshould be roughly in the range 10−5–10−2, because it should be comparable to the Gilbert damping of bulk YIG, αG, which is of order of 10−5–10−3. For a clean interface, δαGis large enough to be measured in FMR experiments (see Figs. 2 and 3). Note that δαGcan be reduced by in- creasing the thickness of YIG (denoted by d′). On the other hand, for a dirty interface, δαGis too small for it to be observable by spin pumping (see Fig. 4). However, we will moderate judgement on the possibility of observing δαGfor a dirty interface, because detailed information on the interfacial exchange coupling is still lacking. We should note that in the present modeling of randomness, the increase in the Gilbert damping is given by a sum of these two contributions, i.e., δαG=δαG,1+δαG,2, for an arbitrary strength of interfacial randomness. Our calculation can be applied straightforwardly to other one-dimensional electron systems such as quasi- one-dimensional magnets, whose low-energy states are also described by the Tomonaga-Luttinger liquid model. In particular, the low-energy states of spin systems with in-plane anisotropy are characterized by a Luttinger pa- rameter Kssmaller than 1. If Ksis sufficiently smaller than 1, δαGshould show nontrivial power-law behavior with respect to the temperature even for a clean inter- face. VI. SUMMARY We theoretically studied spin pumping from a ferro- magnetic insulator into carbon nanotubes. First, we for- mulated the increase in the Gilbert damping in terms of the spin susceptibility and described the interfacial7 exchange coupling with a simple model, in which two types of spin-flip process, i.e., momentum-conserving and momentum-nonconserving processes, coexist. Then, we analytically calculated the increase in the Gilbert damp- ing by treating electrons in carbon nanotubes in the framework of the Luttinger liquid. For a clean interface, the increase in damping is proportional to the inverse of the temperature at high temperatures while it is almost constant at low temperatures. The crossover tempera- ture includes information on the Fermi velocity in carbon nanotubes. We also found that the increase in damping is highly sensitive to the deviation of the Luttinger param- eter in the spin sector from unity. For a dirty interface, the increase in damping shows a power-law dependence on the temperature with a nontrivial exponent reflecting the nature of the Tomonaga-Luttinger liquid. We also estimated the increase of the Gilbert damping using re- alistic parameters. Our results indicate a possible appli- cation of spin pumping for detecting power-law behavior of spin excitation in low-dimensional systems. Detection of other types of spin excitation in exotic many-body states will be left as a future study. ACKNOWLEDGMENTS This French-Japanese collaboration is supported by the CNRS International Research Project “Excitations in Correlated Electron Systems driven in the Giga- Hertz range” (ESEC). This work received support from the French government under the France 2030 invest- ment plan, as part of the Initiative d’Excellence d’Aix- Marseille Universit´ e - A*MIDEX. We acknowledge sup- port from the institutes IPhU (AMX-19-IET-008) and AMUtech (AMX-19-IET-01X). T. K. acknowledges sup- port from the Japan Society for the Promotion of Sci- ence (JSPS KAKENHI Grants No. 20K03831). M. M. acknowledges support by a Grant-in-Aid for Sci- entific Research B (23H01839 and 21H01800) and A (21H04565) from MEXT, Japan, and by the Priority Pro- gram of Chinese Academy of Sciences, under Grant No. XDB28000000. Appendix A: Correlation functions Here, we briefly summarize the calculation of the corre- lation function Crαr′α′(x, y, t ) defined in Eq. (23). Using the bosonic fields, the correlation function is written as Crαr′α′(x, y, t ) =1 (2πa)2 ⟨eAeBeCeD⟩0− ⟨eCeDeAeB⟩0 , (A1) A=−i(−rθα+(x, t) +ϕα+(x, t)), (A2) B=i(−r′θα′−(x, t) +ϕα′−(x, t)), (A3) C=−i(−r′θα′−(y,0) +ϕα′−(y,0)), (A4) D=i(−rθα+(y,0) +ϕα+(y,0)), (A5)where we set r= +1 ( r=−1) for the left-going (right- going) branch. Using the formula, ⟨eA1eA2···eAN⟩= exp 1 2X i⟨A2 i⟩+X i<j⟨AiAj⟩ , (A6) which holds when [ Ai, Aj] is a c-number, we obtain ⟨eAeBeCeD⟩0≡eFrαr ′α′(x−y,t) = exph1 2⟨(A2+B2+C2+D2)⟩+⟨AB⟩+⟨CD⟩ +⟨AC⟩+⟨AD⟩+⟨BC⟩+⟨BD⟩i , (A7) ⟨eCeDeAeB⟩0=eFr′α′rα(y−x,−t). (A8) The correlation functions of the bosonic fields, which are defined as GXY jδ=⟨X(x, t)Y(y,0)⟩, are calculated as [19] Gθθ jδ(x, t) =Kjδ 4(I(x, t) +I(−x, t)), (A9) Gϕϕ jδ(x, t) =1 4Kjδ(I(x, t) +I(−x, t)), (A10) Gθϕ jδ(x, t) =Gϕθ jδ(x, t) =1 4(I(x, t)−I(−x, t)),(A11) I(x, t) =−log2iβℏvF Lsinhπ(ia−x−vFt) βℏvF . (A12) Using these correlation functions, we obtain Frαr′α′(x, t) =F1δr,r′δα,α′+F2δr,−r′δα,α′ +F3δr,r′δα,−α′+F4δr,−r′δα,−α′,(A13) F1(x, t) =˜Gθθ s++˜Gθθ s−+˜Gϕϕ s++˜Gϕϕ s− −r(˜Gθϕ s++˜Gθϕ s−+˜Gϕθ s++˜Gϕθ s−),(A14) F2(x, t) =˜Gθθ c++˜Gθθ c−+˜Gϕϕ s++˜Gϕϕ s−, (A15) F3(x, t) =˜Gθθ s++˜Gθθ c−+˜Gϕϕ s++˜Gϕϕ c− −r(˜Gθϕ s++˜Gθϕ c−+˜Gϕθ s++˜Gϕθ c−),(A16) F4(x, t) =˜Gθθ c++˜Gθθ s−+˜Gϕϕ s++˜Gϕϕ c−, (A17) where ˜GXY jδ(x, t)≡GXY jδ(x, t)−GXY jδ(0,0). Com- bining these results enables the correlation function Crαr′α′(x, y, t ) to be obtained analytically.8 Appendix B: Analytic expressions of integrals For a clean interface, the increase in damping is given as δαG,1=−4S0J2 1 ℏ2ω0(2πa)2NFIIγ, (B1) Iγ=v2 Fβℏ π3Zw 0dx′Zw 0dy′Z∞ 0dusin(˜ω0u) ×Im(sinh(iα) sinh(iα+x−y−u)γ+1 ×sinh(iα) sinh(iα−x+y−u)γ−1) , (B2) where ˜ ω0=βℏω0/π,u=πt/β ℏ,w=πW/β ℏvF,x′= πx/β ℏvF,y′=πy/β ℏvF, and α=πa/β ℏvF. Changing variables from x′andy′with Z= (x+y)/2 and z=x−y, the integral is modified as Iγ=v2 Fβℏ π3Z∞ 0dusin(˜ω0u) ×"Zw/2 0dZZ2Z −2Zdz+Zw w/2dZZ2(w−Z) −2(w−Z)dz# ×Im(sinh(iα) sinh(iα+z−u)γ+1 ×sinh(iα) sinh(iα−z−u)γ−1) =−v2 F 4βℏ π3Zw/2 0dZZ−2z 2zdzZ∞ −∞du ×(ei˜ω0u−e−i˜ω0u) ×sinh(iα) sinh(iα+z−u)γ+1sinh(iα) sinh(iα−z−u)γ−1 . (B3) In the last equation, we have used the relation sinh(iα) sinh(iα±z−u)∗ =sinh(iα) sinh(iα∓z+u)(B4) and the symmetry of the integrand with respect to Z↔ w−Zandz↔ −z. At this stage, it is useful to introduce Bγ(ζ, z) =Z∞ −∞du eiζusinh(iα) sinh(iα+z−u)γ+1 ×sinh(iα) sinh(iα−z−u)γ−1 , (B5) so that Iγ=−v2 F 2βℏ π3Zw/2 0dZZ2Z −2Zdz ×[Bγ(˜ω0, z)− Bγ(−˜ω0, z)]. (B6)Setting v= 2uand rearranging the hyperbolic sine func- tion, we obtain Bγ(ζ, z) =1 2(1−e−2iα)2γe−2zZ∞ −∞dv ×e−v(γ−iζ/2) [e−v+e−2z+i(π−2α)]γ+1[e−v+e−2z+i(π−2α)]γ−1, (B7) which can be computed analytically, invoking the formula 3.315.1 in Ref. [29] as Bγ(ζ, z) =1 2(1−e−2iα)2γe−i(π−2α)(γ+iζ/2)e−2z(γ−1−iζ/2) ×|Γ(γ+iζ/2)|2 Γ(2γ)F(γ−1, γ,2γ; 1−e−4z), (B8) where F(a, b, c ;x) is the Gauss hypergeometric function. To leading order in the small parameter α(≪1), this reduces to Bγ(ζ, z) =1 2(2α)2γeπζ/2+iζze−2z(γ−1) ×|Γ(γ+iζ/2)|2 Γ(2γ)F(γ−1, γ,2γ; 1−e−4z).(B9) In practice, we are mostly interested in the regime where ˜ω0=βℏω0≪1 so we can focus on values of ζsuch that|ζ| ≪1. This allows us to expand Bγ(ζ, z) for small values of ζ, which yields, after substituting back into the expression for Iγ, Iγ=−π 2v2 Fω0βℏ π4 (2α)2γΓ(γ)2 Γ(2γ)Zw/2 0dZ ×Z2Z 0dze−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z),(B10) where we have used the symmetry of the integrand with respect to z↔ −z. Combining this result with Eq. (B1), Eqs. (32) and (33) can be derived, after rewriting the integral variable as z′= 2Z. In the limiting case, the integral I(w, γ) given in Eq. (33) can be approximated into a simple form. For the short-junction limit ( w/π=W/β ℏvF=W/L th≪g(γ)), we obtain I(w, γ)≃1 wZw 0dz′Zz′ 0dz,e−2z(γ−1)=w 2.(B11) For the long-junction limit ( w/π≫g(γ)), I(w, γ) ≃1 wZw 0dz′Z∞ 0dz,e−2z(γ−1)F(γ−1, γ,2γ; 1−e−4z) =π 2g(γ), (B12)9 where g(γ) is defined by Eq. (43). These analytical ex- pressions lead to Eqs. (41) and (42) in the main text. For a dirty interface, the increase in damping is ex- pressed as δαG,2=−S0J2 2aW ℏ2ω0(πa)2NFIX r,r′,α,α′I′ γrαr ′α′, (B13) I′ γ=βℏ πZ∞ 0dusin(˜ω0u)Im(sinh(iα) sinh(iα−u)2γ) , (B14) with the same dimensionless variables as for a clean in- terface. By a similar way as the clean case, the integral I′ γis modified as I′ γ=−βℏ 4π[Aγ(−˜ω0)−Aγ(−˜ω0)], (B15) Aγ(ζ) =Z∞ −∞du e−iζusinh(iα) sinh(iα−u)2γ . (B16) Setting v= 2uand rearranging the hyperbolic sine func-tion, we obtain Aγ(ζ) =1 2(1−e−2iα)2γZ∞ −∞dve−(γ+iζ/2)v (e−v+ei(π−2α))2γ. (B17) Invoking the formula 3.314 in Ref. 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Fujimoto, and M. Matsuo, Valley- dependent spin transport in monolayer transition-metaldichalcogenides, Phys. Rev. Lett. 124, 166803 (2020). [11] M. Yama, M. Tatsuno, T. Kato, and M. Matsuo, Spin pumping of two-dimensional electron gas with rashba and dresselhaus spin-orbit interactions, Phys. Rev. B 104, 054410 (2021). [12] M. Inoue, M. Ichioka, and H. Adachi, Spin pumping into superconductors: A new probe of spin dynamics in a su- perconducting thin film, Phys. Rev. B 96, 024414 (2017). [13] M. A. Silaev, Finite-frequency spin susceptibility and spin pumping in superconductors with spin-orbit relax- ation, Phys. Rev. B 102, 144521 (2020). [14] M. A. Silaev, Large enhancement of spin pumping due to the surface bound states in normal metal–superconductor structures, Phys. Rev. B 102, 180502(R) (2020). [15] Y. Ominato, A. Yamakage, T. Kato, and M. Matsuo, Ferromagnetic resonance modulation in d-wave super- conductor/ferromagnetic insulator bilayer systems, Phys. Rev. B 105, 205406 (2022). [16] Y. Ominato, A. Yamakage, and M. Matsuo, Anisotropic superconducting spin transport at magnetic interfaces, Phys. Rev. B 106, L161406 (2022). [17] T. Funato, T. Kato, and M. Matsuo, Spin pumping into anisotropic dirac electrons, Phys. Rev. B 106, 144418 (2022). [18] J. Voit, One-dimensional fermi liquids, Rep. Prog. Phys. 58, 977 (1995). [19] J. von Delft and H. Schoeller, Bosonization for beginners – refermionization for experts, Ann. Phys. (Leipzig) 510, 225 (1998). [20] T. Giamarchi, Quantum Physics in One Dimension (Ox- ford University Press, 2003). [21] P. M. Singer, P. Wzietek, H. Alloul, F. Simon, and H. Kuzmany, Nmr evidence for gapped spin excitations in metallic carbon nanotubes, Phys. Rev. Lett. 95, 236403 (2005).10 [22] B. D´ ora, M. Gul´ acsi, F. Simon, and H. Kuzmany, Spin gap and luttinger liquid description of the nmr relaxation in carbon nanotubes, Phys. Rev. Lett. 99, 166402 (2007). [23] Y. Ihara, P. Wzietek, H. Alloul, M. H. R¨ ummeli, T. Pich- ler, and F. Simon, Incidence of the tomonaga-luttinger liquid state on the nmr spin-lattice relaxation in carbon nanotubes, Europhys. Lett. 90, 17004 (2010). [24] R. Egger and A. O. Gogolin, Correlated transport and non-fermi-liquid behavior in single-wall carbon nan- otubes, Eur. Phys. J. B 3, 281 (1998). [25] T. Kato, Y. Ohnuma, M. Matsuo, J. Rech, T. Jonck- heere, and T. Martin, Microscopic theory of spin trans-port at the interface between a superconductor and a ferromagnetic insulator, Phys. Rev. B 99, 144411 (2019). [26] H. Yoshioka and A. A. Odintsov, Electronic properties of armchair carbon nanotubes: Bosonization approach, Phys. Rev. Lett. 82, 374 (1999). [27] We treated effect of the external magnetic field and spin- orbit interaction phenomenologically through a change in Ks+. [28] J. Nogu´ es and I. K. Schuller, Exchange bias, J. Magn. Magn. Mater. 192, 203 (1999). [29] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products (Elsevier Academic Press, 2007).

2023-03-02

We theoretically study spin pumping from a ferromagnetic insulator (FI) into a carbon nanotube (CNT). By employing the bosonization method, we formulate the Gilbert damping induced by the FI/CNT junction, which can be measured by ferromagnetic resonance. We show that the increase in the Gilbert damping has a temperature dependence characteristic of a Luttinger liquid and is highly sensitive to the Luttinger parameter of the spin sector for a clean interface. We also discuss the experimental relevance of our findings based on numerical estimates, using realistic parameters.

Spin Pumping into Carbon Nanotubes

2303.01343v2

arXiv:1011.5054v1 [cond-mat.mtrl-sci] 23 Nov 2010Ultra-fastmagnetisationrates withintheLandau-Lifshit z-Bloch model. U. Atxitia and O. Chubykalo-Fesenko Instituto de Ciencia de Materiales de Madrid, CSIC, Cantobl anco, 28049 Madrid, Spain The ultra-fast magnetisation relaxation rates during the l aser-induced magnetisation process are analyzed in terms of the Landau-Lifshitz-Bloch (LLB) equation for di fferent values of spin S. The LLB equation is equivalentinthelimit S→∞totheatomisticLandau-Lifshitz-Gilbert(LLG)Langevind ynamicsandfor S=1/2 to the M3TM model [B. Koopmans, et al.Nature Mat. 9(2010) 259]. Within the LLB model the ultra-fast demagnetisation time( τM)and the transverse damping ( α⊥) are parameterizedby theintrinsic coupling-to-the- bath parameter λ, defined by microscopic spin-flip rate. We show that for the ph onon-mediated Elliott-Yafet mechanism, λis proportional to the ratio between the non-equilibrium ph onon and electron temperatures. We investigate the influence of the finite spin number and the sca ttering rate parameter λon the magnetisation relaxation rates. The relation between the fs demagnetisat ion rate and the LLG damping, provided by the LLB theory, is checked basing on the available experimental dat a. A good agreement is obtained for Ni, Co and Gd favoring the idea that the same intrinsic scattering proc ess is acting on the femtosecond and nanosecond timescale. PACS numbers: 75.40Gb,78.47.+p, 75.70.-i I. INTRODUCTION Magnetisation precession and the spin-phonon relaxation ratesat picosecondtimescale wereconsideredto bethe limi t- ingfactorforthespeedofthemagnetisationswitching1,2, un- til using optical excitation with fs pulsed lasers the possi bil- ity to influence the magnetisation on femtosecond timescale wasdemonstrated3–6. Theultra-fastlaser-induceddemagneti- sation immediately became a hot topic of solid state physics due to an appealing possibility to push further the limits of operation of magnetic devices. This ultra-fast process has now been shown to proceed with several important charac- teristictimescales6: (i)thefemtoseconddemagnetisationwith timescale τM(ii) the picosecond recovery with timescale τE and (iii) the hundredpicoseconds-nanosecondmagnetisati on precession, traditionally characterized by the ferromagn etic resonance frequency ωFMRand the Landau-Lifshitz-Gilbert dampingparameter αLLG(see Fig.1). Thephysicsofthemagnetisationchangesonfemto-second timescales is obviously not-trivial and will require novel the- orieswithintherelativisticquantumelectrodynamicsofm any electronsystems. From theoretical pointof view, the exist ing models try to answer an open question of the role of differ- entsubsystems(photons,phonons,electronsandspins)int he ultra-fast angular momentumtransfer7. This common goal is stimulated by experimental findings provided by the XMCD measurements showing the important role of the spin-orbit interactions8,9. For the present state of art quantummechani- cal descriptions8,10–13of ultra-fast demagnetisationprocesses involveunavoidablesimplificationsandsometimesevensom e ad-hoc assumptions necessary to explain experimental find- ings, such as reduced exchange interactions, enhanced spin - orbit coupling or a Gaussian distribution of occupied state s aroundthe Fermi level. While some degreeof agreementhas been achieved in modelling of the ultra-fast demagnetisati on (τM)scale14,themodellingofallthreeultra-fastdemagnetisa- tionrateswithinthesameapproachisoutsidethepossibili ties ofthequantummechanicalapproaches. The three-temperature (3T) phenomenological model in-volves the rate equations for the electron, phonon and spin temperatures (energies)10,15–17. Recently, it has been shown thattheintroductionofthespintemperatureisnotadequat e18 since the spin system is not in the equilibrium on the fem- tosecond timescale. It has been suggested to couple the spin dynamics to the two-temperature (2T) model for phonon and electrontemperatures18–22. Thesemodelsarebasedontheen- ergy flow picture and leave unidentified the angular momen- tum transfer mechanism and the underlying quantum mech- anism responsible for the spin flip22. They essentially inter- prettheultra-fastdemagnetisationas"thermal"processe s,un- derstanding the temperature as energy input from photon to electron and then to the spin system. By using these mod- els the important role of the linear reversal path in the femt o- seconddemagnetisationhasbeenidentified23,24. Thecompar- ison with experiment seems to indicate that in order to have magnetisation switching in the ultra-fast timescale, a com - binedactionof"heat"andlargefieldcomingfromtheinverse Faradayeffectisnecessary24. The most successful recent phenomenological models de- scribing the ultra-fast magnetisation dynamics are (i) the Langevin dynamics based on the Landau-Lifshitz-Gilbert (LLG) equationand classical HeisenbergHamiltonian for lo - calized atomic spin moments18,19, (ii) the Landau-Lifshitz- Bloch (LLB) micromagnetics21,22and (iii) the Koopmans’s magnetisation dynamics model (M3TM)25. The spin dy- namicscouldbe coupledtotheelectrontemperaturefromthe 2T model, underlying the electronic origin of the spin-flip process18,19,21,22,24or to both electron and phonon temper- atures, underlying the Elliott-Yafet mechanism mediated b y phonons25. When the 2T model was carefully parameterised fromthemeasuredreflectivity,it gaveanexcellentagreeme nt with the experiment in Ni22using the former approach or in Ni,Co andGdusingthe latterapproach25. In the classical derivation of the LLB equation the ther- mal averaging has been performed analytically within the mean field (MFA) approximation26. Thus, the LLB equa- tion for classical spins ( S→∞) is equivalent to an ensem- bleofexchange-coupledatomisticspinsmodelledbystocha s-2 tic LLG equations20,27. At the same time, in some cases the LLBequationmaybepreferablewith respectto theatomistic Heisenbergmodel,sincebeingmicromagneticit canincorpo - rate quantum nature of magnetism and the quantum deriva- tion of LLB also exists28. In particular the limits of validity forthestatisticalmechanicsbasedontheclassicalHeisen berg model for the description of materials with delocalized mag - netism of d-electrons in transition metals or magnetism of f- electronsin rare earthsare not clear. An alternativestati stical simplified descriptionof d-metalsconsists of a two level sys- tem with spin-up and spin-down bands (i.e. S=±1/2), as has been done by B. Koopmans et al.25. Their model, as we show in the present article, is also equivalent to the quantu m LLB equation with spin S=1/2. An additional advantage in the use of the LLB equation is the possibility to model larger spatial scales20,21. Therefore the LLB micromagnet- icsisanimportantparadigmwithinthemultiscalemagnetis a- tiondynamicsdescription. TheLLBequationhasbeenshown to describe correctly the three stages of the ultra-fast dem ag- netisationprocesses: thesub-picoseconddemagnetisatio n,the picosecondmagnetisationrecoveryand the nanosecondmag- netisationprecession20–22,see Fig.1. The intrinsic quantum mechanical mechanisms responsi- ble for the ultra-fast demagnetisation in the LLB model are included in the intrinsic coupling-to-the-bath parame - terλ22,28. The coupling process is defined by the rate of the spin flip. Several possible underlying quantum mech- anisms are currently under debate: the Elliott-Yafet (EY) electron scattering mediated by phonons or impurities13,25, or other electrons14and electron-electron inelastic exchange scattering29,30. By combining the macroscopic demagnetisa- tion equation (M3TM model) with the rate of spin flip calcu- latedonthebasisoffullHamiltonian,Koopmans etal.25have beenabletorelatetheultra-fastdemagnetisationtime τMwith thespinfliprateofthephonon-mediatedElliott-Yafetscat ter- ing. The authors fitted experimentaldemagnetisation rates in Ni,Co,GdtothephenomenologicalM3TMmodelandfound themtobeconsistentwiththevaluesestimatedonthebasiso f ab-initiotheory13. Thecoupling-to-the-bathparameter λ(mi- croscopicdampingparameterinatomisticLLGmodel)should be distinguished from that of the macroscopic damping αLLG (α⊥in the LLB model), a more complicated quantity which includesthe magnon-magnonprocesses. Thefirst attempttorelatethesub-picoseconddemagnetisa- tion time with the macroscopicdampingprocesses was given byKoopmans etal.6whosuggestedtherelation τM∼1/αLLG. Subsequently and with the aim to check this relation several experiments in doped permalloy were performed32–34. The permalloythinfilmsweredopedwithrareearthimpurities,a l- lowingtoincreaseinacontrolledwaythedampingparameter αLLG. The effect on the demagnetisation time τMwas shown to be opposite34or null32, in contrast to the above relation. However, it should be noted that the analysis leading to this expression was performed in terms of the Landau-Lifshitz- Gilbert equation, relating the ultra-fast demagnetisatio n time τMtothetransversedampingwithouttakingintoaccounttheir temperature dependence. Moreover, one should take into ac- count that the rare-earth impurities may introduce a differ ent Figure 1. Characteristic time scales in ultrafast laser-in duced mag- netisation dynamics experiments. The curve is obtained by t he in- tegration of the Landau-Lifshitz-Bloch equation coupled t o the two- temperature model with the parameters from Ref.21. For the m od- elling of precession the applied field Hap=1T at 30 degrees was used. scatteringmechanismwith aslowertimescale33. Partially basing on the above mentioned experimental re- sults andfroma generalpointof view,the longitudinalrela x- ation (the ultra-fast demagnetisation rate τM) and the trans- verse relaxation (the LLG damping αLLG) may be thought to be independent quantities. Indeed, different intrinsic and extrinsic mechanisms can contribute to the demagnetisatio n rates at different timescales. One can, for example, men- tionthatduringthefemtoseconddemagnetisationtheelect ron temperature is often raised up to the Curie temperature22,24. At this moment, the high frequency THz spinwaves35,36in- cludingtheStonerexcitations30contribute. At thesame time, the transverse relaxation is related to the homogeneous pre - cessional mode. The LLBequationtakescare of thedifferent naturesoflongitudinalandtransverserelaxation,arisin gfrom the spin disordering. The LLB model calculates them inde- pendently but basing on the same intrinsic scattering mecha - nismparameterizedbythe parameter λ. Theincrementof the number of scattering events is mimicked by the increases of the electron temperature. Consequently, the relation betw een the ultra-fast demagnetisation and precession remains val id butwithatemperature-dependentcorrection. Ifthisrelat ionis confirmedexperimentally,a uniqueintrinsic couplingpara m- etermeansthatthesamemainmicroscopicmechanismisact- ingonbothtimescales. Inthepresentarticlewewillshowth at the analysis of the available experimental data seems to ind i- cate towards this possibility, at least in pure transition m etals such as Ni or Co and in rare earth metal Gd. We did not find validityofthecorrespondingrelationin Fe. Up to now only classical version ( S→∞) of the LLB equation was used to model the ultra-fast demagnetisation processes20,21,24. In the present article we show the impor- tant role of the choice of the quantum spin value, resulting inthedifferencesinthecorrespondinglongitudinalrelax ation times. The article is organized as follows. In section II we presentdifferentformulationsofthequantumLLBmodeland its main features for different spin values S. In section III3 we present results on the modelling of the demagnetisation processes within LLB models with different choices of the quantumspinsnumber Sandofthe intrinsic scatteringmech- anisms. In section IV we present our attempts to link the ultra-fast demagnetisation rates in transition metals and Gd and comparison with available experimental data. Section V concludes the article. In the Appendix to the article we demonstrate the equivalence of the LLB model with S=1/2 andtheM3TMmodelbyB.Koopmans et al.25. II. THE LANDAU-LIFSHITZ-BLOCHMODELWITH QUANTUMSPINNUMBER S. TheLLBequationforaquantumspinwasderivedfromthe density matrix approach28. Although the model Hamiltonian was rather the simplest form of the spin-phonon interaction , thegeneralizationoftheapproachshouldbepossibleto mor e complex situations. The macroscopic equation for the mag- netisationdynamics,validatalltemperatures,iswritten inthe followingform: ˙n=γ[n×H]+γα/bardbl n²[n·Heff]n−γα⊥ n2[n×[n×Heff]](1) wheren=M/Me(T) =m/meis the reduced magnetisation, normalizedtotheequilibriumvalue Meatgiventemperature T andm=M/Me(T=0K). Theeffectivefield Heff,containsall usualmicromagneticcontributions,denotedby Hint(Zeeman, anisotropy,exchangeandmagnetostatic)andisaugmentedb y thecontributioncomingfromthetemperature Heff=Hint+me 2/tildewideχ/bardbl/parenleftbig 1−n2/parenrightbig n, (2) where/tildewideχ/bardbl(T) = (∂m/∂H)H→0is the longitudinal susceptibil- ity . The LLB equation contains two relaxational parame- ters: transverse α⊥andlongitudinal α/bardbl,relatedtotheintrinsic coupling-to-the-bathparameter λ. Inthe quantumdescription the couplingparameter λcontains the matrix elements repre- senting the scattering events and, thus, is proportional to the spin-fliprateduetotheinteractionwiththeenvironment. T his parameter,inturn,couldbetemperaturedependentand,ino ur opinion, it is this microscopic parameter which should be re - latedtotheGilbertparametercalculatedthroughab-initi ocal- culations as in Refs.38,39, since the contribution coming from thespindisorderingisnotproperlytakenintoaccountinth ese models. In the quantum case the temperature dependence of the LLB damping parameters is given by the following ex- pressions: α/bardbl=λ me2T 3TC2qS sinh(2qS)=⇒ S→∞λ me2T 3TC, (3) α⊥=λ me/bracketleftbiggtanh(qS) qS−T 3TC/bracketrightbigg =⇒ S→∞λ me/bracketleftbigg 1−T 3TC/bracketrightbigg ,(4)withqS=3TCme/[2(S+1)T], whereSis the quantum spin number and TCis the Curie temperature. In the case S→∞ thedampingcoefficientshavetheformsusedinseveralprevi - ouslypublishedworks40,suitableforthecomparisonwiththe LangevindynamicssimulationsbasedontheclassicalHeise n- bergHamiltonianandinagreementwiththem20,27. Eq.(1) is singular for T>TC, in this case it is more con- venient to use the LLB equation in terms of the variable m=M/Me(T=0K)27. The corresponding LLB equation is indistinguishable from Eq.(1) but with different relaxati onal parameters/tildewideα/bardbl=meα/bardbl,/tildewideα⊥=meα⊥and/tildewideα⊥=/tildewideα/bardblforT>TC, in this case the contribution of temperature to Heff[the sec- ondterminEq.(2)]is (−1//tildewideχ/bardbl)[1−3Tcm2/5(T−Tc)m]m. Al- though this formulation is more suitable for the modelling o f the laser-induced demagnetisation process, during which t he electronic temperature is usually raised higher than TC, it is the expression (4) which should be compared with the trans- verse relaxation parameter αLLGdue to the similarity of the formulationoftheEq.(1)withthemacromagneticLLGequa- tion. In the classical case and far from the Curie temperatur e T≪TC,λ=α⊥=/tildewideα⊥(αLLG). S→∞S=7/2S=3/2S=1/2 T/T CαLLG 10.90.80.70.60.50.40.09 0.06 0.03 S→∞S=7/2S=3/2S=1/2 T/T Cτ/bardbl[ps] 1.2 1 0.8 0.6 0.46 4 2 0.1 Figure 2. (Up) The transverse damping parameter α⊥(αLLG) as a function of temperature within the LLB model for different spin valuesS. The intrinsic coupling parameter was set to λ=0.03. (Down) The longitudinal relaxation time τ/bardblas a function of tem- perature within the LLB model for different spin values S. The temperature-dependent magnetisation and the longitudina l suscepti- bility/tildewideχ/bardblwereevaluatedinbothcasesintheMFAapproachusingthe Brillouinfunction. In the "thermal" model the nature of the longitudinal and4 the transverse relaxation differs from the point of view of characteristicspinwavefrequencies. Thetransverserela xation (knownastheLLGdamping)isbasicallytherelaxationofthe FMR mode. The contributionof other spinwave modes is re- ducedtothethermalaveragingofthemicromagneticparame- ters and the main effect comes fromthe decrease of the mag- netisation at high temperature. Consequently, the transve rse damping parameter increases with temperature (see Fig.2), consistentwithatomisticmodellingresults27andwell-known FMRexperiments37,41. On the contrary, the main contribution to the longitudinal relaxation comes from the high-frequency spin waves. This processoccursinastrongexchangefield. Asaresult,thelon - gitudinal relaxation time (the inverse longitudinal relax ation) is much faster and increases with temperature,knownas crit - ical slowing down, see Fig.2. This slowing down has been shown to be responsible for the slowing down of the femto- second demagnetisation time τMas a function of laser pump fluency18,22. The characteristic longitudinal timescale is not only defined by the longitudinal damping parameter (3) but also by the temperature-dependentlongitudinal susceptib ility /tildewideχ/bardbl(T)27, accordingtothefollowingequation: τ/bardbl(T)=/tildewideχ/bardbl(T) γ/tildewideα/bardbl(T). (5) As it can be observed in Fig. 2 the transverse relaxation parameter α⊥(αLLG) and the longitudinal relaxation time τ/bardbl haveastrongdependenceonthequantumspinnumber Scho- sen to describe system’s statistics. We conclude here about the occurrence of quite different relaxation rates for the t wo extremecases S=1/2andS=∞. B. Koopmans et al.recently used a different equation to describe the ultrafast demagnetisation dynamics25, called M3TMmodel: dm dt=RmTp TC/parenleftbigg 1−mcoth/parenleftbiggmTC Te/parenrightbigg/parenrightbigg . (6) Eq.(6)hasbeenobtainedthroughthegeneralMasterequatio n approach for the dynamics of the populations of a two level system (spin S=1/2 was used) with the switching probabil- ityevaluatedquantum-mechanicallyforthephonon-mediat ed EY spin-flips. Here TpandTeare phonon and electron tem- peratures,respectively,and Risa materialspecificparameter, relatedtothespin-flipprobabilityinthephonon-mediated EY scatteringevents asf, as R=8asfGepµBkBVaT2 C µatE2 D, (7) whereVaandµataretheatomicvolumeandmagneticmoment, respectively, Gepistheelectron-phononcouplingconstant, kB istheBoltzmannconstant, µBistheBohrmagnetonand EDis the Debye energy. This equation has allowed to fit the ultra- fast demagnetisation time ( τM) obtaining the values of Rin Ni, Co and Gd25and relating them to the phonon-mediated EYscatteringrates asf. As we show in the Appendix, the M3TM equation (6) cor- responds to the longitudinal part of the LLB equation withthermal field only ( Hint=0) and with spin S=1/2, i.e. it is equivalentto dm dt=γ/tildewideα/bardblHeff. (8) Thisgivesarelationbetweentheintrinsiccouplingparame ter λand the material specific parameter Rand finally with the phonon-mediatedEYspin-flipprobability asfviatheformula: λ=3R 2γµat kBTCTp Te=λ0Tp Te. (9) Thus the two approaches are reconciled, provided that the temperature-dependent coupling rate (9) is used in the LLB equation,incontrasttootherworks18,21,22wherethecoupling λis considered to be temperature-independent. Combining expressions (5) (7) and (9), one can immediately see that in thecaseofthephonon-mediatedEYprocess,thelongitudina l relaxationtimeisdeterminedby τ/bardbl∝/tildewideχ/bardbl asfE2 D GepVaTp. (10) InRef.25andbasingonthephonon-mediatedEYpicture,the classification of materials on the basis of the "magnetic in- teraction strength" parameter µat/Jwas proposed, where Jis thematerialexchangeparameter. Accordingtotheexpressi on above,thedemagnetisationratedependsonmoreparameters , among which the important one is also the electron-phonon coupling Gepdefining how fast the electron system can pass theenergytothephononone. Anotherimportantparameteris the microscopicspin-fliprate asf. Comparingto theB. Koop- manset al.25materialsclassification,thelongitudinalsuscep- tibilityinEq.(10) isindeeddefinedbythevalueoftheatomi c momentµatandbythefactthatthisfunctionrapidlyincreases withtemperatureanddivergescloseto TC∝J. AtT≈TCone obtainsasimplelinearrelation27/tildewideχ/bardbl∝µat/J,thusshowingthe dependenceof the demagnetisation rate on this parameter, a s suggestedinRef.25. In the case of the phonon-mediated EY process the tem- perature dependence of the longitudinal relaxation is comi ng from the longitudinalsusceptibiliy only (cf. Eq. (10)), as op- posed to the case λ=const (cf. Eq.(5)). (We do not discuss herethepossibilitythatthephonon-mediatedEYspin-flipr ate asfmay be also temperature dependent.) However, the tem- perature dependence of the susceptibility is characterize d by itsexponentialdivergencecloseto TC. Inthesecircumstances an additional linear temperature dependence provided by th e longitudinal damping is difficult to distinguish in the fitti ng procedureofexperimentaldata. III. MODELLINGOF THELASER-INDUCED ULTRA-FASTDEMAGNETISATIONWITHINTHE LLB MODELS. InthespiritofRefs.18,20–22,25forthemodellingofultra-fast demagnetisationdynamics,theLLBequationmaybecoupled5 to the electron temperature Teonly, understanding the elec- trons as the main source for the spin-flip mechanism18,20–22 or to both phonon and electron temperatures in the spirit of the phonon-mediatedElliott-Yafet process25. In both cases it is the electrontemperature T=Tewhich couplesto the mag- netisation in the LLB formalism, since the phonon tempera- ture could only enter into the temperature dependence of the coupling-to-the bath parameter λvia Eq.(9) . Note that the temperature Tis not the spin temperature, since the resulting dynamicsistakingplaceout-of-equilibrium. Theelectron Teandphonon Tptemperaturesaretakenfrom the two-temperature (2T) model15,45,46. Within this model theirdynamicsisdescribedbytwodifferentialequations: CedTe dt=−Gep(Te−Tp)+P(t), CpdTp dt=Gep(Te−Tp). (11) HereCe=γeTe(γe=const) and Cpare the specific heats of theelectronsandthelattice. TheGaussiansourceterm P(t)is a function which describes the laser power density absorbed in the material. The function P(t)is assumed to be propor- tional to the laser fluence Fwith the proportionality coeffi- cient which could be obtained from the long time scale de- magnetization data (for which Te=Tp)22. The dynamics of the electron temperature can be also measured directly in th e time-resolvedphotoemissionexperiment47. The first of Eqs.(11) may also include a diffusion term ∇z(κ∇zTe)taking into account a final penetration depth of the deposited energy into the film thickness25and a term, Ce(Te−300K)/τthdescribing the heat diffusion to the exter- nal space22. In the present article, the parameters for the 2T- model were taken either from Koopmans et al.25or from U. Atxitiaet al.22(for Ni only), where they were carefully pa- rameterized through the reflectivity measurements. The Ni (Co, Gd etc) parameters, such as magnetisation as a function of temperature were taken assuming the Brilloiun (Langevin forS→∞)function. The coupling of the 2T model to the LLB equation ade- quately describes all three stages of the ultra-fast demagn eti- sation rates: sub-ps demagnetisation, ps recovery and sub- ns precession21,22, see Fig.1. As a consequence of the temper- ature dependenceof both longitudinal dampingand suscepti - bility, and since the temperature is dynamically changed ac - cording to Eqs.(11), the longitudinal relaxation time is ti me- dependent via Eq.(5). It is also strongly dependent on the parameters of the 2T model and its dynamics is not simple. Consequently,thesub-psultra-fastdemagnetisationgene rally speaking is not exponentialand cannot be described in terms of one relaxation time τM. Simple analytical expression is possible to obtain with the supposition of a square-shaped temperaturepulse23. The two-exponentialfitting is also often used22,36. In our approachthe fs demagnetisation is fitted di- rectlytothesolutionoftheLLBequationwithoutassumptio n of the one- or two-exponential decay. However, to comply with the existing approaches, we still discuss the demagnet i- sationratein termsofauniqueparameter τM. In the experiment performedin the same material the onlyremainingfittingparameterfortheLLBmodelisthecoupling parameter λ. The choice of λtogether with the parameters of the 2T model defines all magnetisation rates. In Fig.3 we present modelling of the ultra-fast demagnetisation and re - magnetisationforvariousvaluesofthecouplingparameter λ, chosen to be independent on temperature, as in Ref. 22. If for some reason the scattering channel was suppressed, this wouldleadtoasmallscatteringrateandconsequentlyasmal l demagnetisation and a slow recovery. Indeed, the value of λ forGdwasfoundtobe60timessmallerthanforNi(seeTable I).Thissmallvalueof λassuresalargedelayinthemagnetis- arionrelaxationtowardstheequilibriumelectrontempera ture. Thus this parameter defines the diversity of the demagnetisa - tion rates in larger extend than the ratio µat/J, suggested in Ref.25anddiscussedintheprevioussubsection. λ=0.001 λ=0.01 λ=0.1 t[ps]∆m/m 0 20 10 00 -0.1 -0.2 -0.3 Figure 3. The result of integration of the LLB model ( S→∞) with different parameters λ(increasing from top to the bottom). In this case the the 2T model parameters were taken from Ref.22withl aser fluenceF=30 mJ/cm2 Another parameter strongly influencing the demagnetisa- tion rates is the phonon-electron coupling Gepdefining the rate of the electron temperature equilibration time. This i s the main parameter governing the magnetisation recovering timeτE. Indeed, in Ref.25the phonon-electroncoupling Gep was chosen to be 20 times smaller for Gd than for Ni. By adjustingthis parameter,the ultra-slowdemagnetisation rates observed in TbFe alloy48, Gd49and in half-metals50as well as the two time-scales demagnetisation25,49are also well- reproduced (see, as an example, Fig.4). Within this model thetwo-timescaleprocessconsistsofarelativelyfastdem ag- netisation (however much slower than in Ni), defined by the electrontemperatureandsmallvalueof λ,followedbyamuch slowerprocessduetoaslowenergytransferfromtheelectro n tothelattice system. As it was mentioned in the previous subsection, the phonon-mediatedEY mechanismpredictsthe couplingto the bath parameter λto be dependent on the ratio between the phonon and electron temperature through the relation (9). A decrease of λup to two times at high fluencies is observed for Ni and Co. The analysis of the data presented in Ref. 25 and47forGdhasshownthatduringthedemagnetisationpro- cess the ratio Te/Tphas increased almost 6 times. In Fig.5 we present the magnetisation dynamics for Ni evaluated for6 t[ps]400300200100m/m0 32101 0.8 0.6 0.4 Figure 4. The result of integration of the LLB model ( S→∞) with constantλ0=0.0015 (seeTableI).Inthiscase the2Tmodel param- eters were takenfrom Ref.25corresponding toGd. two laser pulse fluencies, assumingvariousvaluesof the spi n Sandtemperature-dependentandindependent λvalues. Note quite different demagnetisation rates at high fluency for tw o limiting cases S=1/2, used in Ref.25 and S=∞, used in Ref.22. The differences in the choice of λare pronounced at high pump fluency but are not seen at low fluency. One can also hope that in the fitting procedure of experimental data it would be possible to distinguish the two situations. Unfortunately, the fitting to experimental data procedure i s complicated and the changes coming from the two cases de- scribed above are competing with several different possibi l- ities such as an additional temperature dependency in elec- tron or phonon specific heats51. Additionally, we would like to mention different electron-phonon coupling constants Gep used in Refs. 22 and 25. Fitting to experimental data from Ref.25 for Ni for high fluence, we have found that the case of the temperature-dependent λ=λ0(Tp/Te)can be equally fitted with the temperature-independent λ≈λ0/2. To answer definitely which fitting is better, more experimental data pr o- moting one or another intrinsic mechanism and varying laser fluencyisnecessary. IV. LINKING DIFFERENTTIMESCALES Since the longitudinal relaxation occurs under strong ex- change field and the transverse relaxation - under external applied field, their characteristic timescales are quite di ffer- ent. However, the LLB equation provides a relation be- tween the ultra-fast demagnetisation(longitudinalrelax ation) and the transverse relaxation (ordinaryLLG dampingparam- eter) via the parameter λ0(λ=λ0orλ=λ0(Tp/Te)for Tp=Te). The two demagnetisation rates could be measured independently by means of the ultra-fast laser pump-probe technique52. It has been recently demonstrated53that the damping of the laser-induced precession coincides with the measuredby FMR intransitionmetals. By separatemeasure- mentsofthetwomagnetisationrates,therelations(4)and( 5) given by the LLB theory could be checked. This can pro- videthevalidationoftheLLBmodel,aswellastheanswerto the question if the same microscopic mechanism is acting onM3TMLLB S→∞S→1/2S→∞S→1/2 t[ps]∆m/m 0 3 2 1 00.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 Figure 5. Magnetisation dynamics during laser-induced dem agneti- sation process calculated within the LLB model with differe nt spin numbers and for two laser-fluencies F=10 mJ/cm2(upper curves) andF=40 mJ/cm2(bottom curves). Ni parameters from Ref.22 were used. The symbols are calculated with the LLB equation w ith the intrinsic damping parameter using a constant λ0=0.003 value, and the solid lines with the LLB equation and the intrinsic co upling withthe temperature dependent λ=λ0/parenleftbig Tp/Te/parenrightbig . femtosecond and picosecond timescales. Unfortunately, th e damping problem in ferromagnetic materials is very compli- cated and the literature reveals the diversity of measured v al- uesinthesamematerial,dependingonthepreparationcondi - tions. Thus, to have a definite answer the measurement on the same sample is highly desired. The measurements of bothα⊥andτMare available for Ni22where an excel- lent agreement between ultra-fast magnetisation rates via a uniquetemperature-independentparameter λ=0.04hasbeen reported22. The resultsof thesystematic measurementsof τM are also available for Ni, Co, Gd in Ref. 25, as well as for Fe55. The next problem which we encounter here is that the demagnetisationratesstronglydependon the spin value S, as is indicatedin Figs. 2 and 5. The fitting of experimentaldata using LLB model with different Svalues results in different values of the coupling parameter λ0. The use of S=1/2 value25orS=∞value22is quite arbitrary and these values do not coincide with the atomic spin numbers of Ni,Co, Gd. Generally speaking, for metals the spin value is not a good quantum number. The measured temperature dependence of magnetisation, however, is well fitted by the Brillouin func - tionwith S=1/2forNi andCoand S=7/2forGd54. These arethevaluesof Swhichwe usein TableI. Consequently in Table I we present data for the coupling parameter λ0extractedfromRef.25. Differentlytothisarticle, for Gd we corrected the value of the parameter Rto account foradifferentspinvaluebytheratioof thefactors,i.e. RS1= (fS2/fS1)RS2with fS=2qS sinh(2qS)1 m2 e,SχS /bardbl, (12) where the parameters are evaluated at 120 Kusing the MFA expressions for each spin value S. The data are evaluated7 Material S R25λ0α⊥ αLLG Ni 1/2 17.2 0.0974 0 .032 0 .01942-0.02841 Co 1/2 25.3 0.179 0.025 0.003641-0.00643-0.01144 Gd 7/2 0.009 0.0015 0.00036 0 .000533 Table I. The data for ultra-fast demagnetisation rate param eters for three different metals from ultrafast demagnetization rat es and from FMR mesurements. The third column presents the demagnetisa tion parameter Rfrom Ref. 25, corrected in the case of Gd for spin S=7/2. The fourth column presents the value of the λ0parame- ter, as estimated from the M3TM model25and the formula Eq.(9). The fifth column presents the data for α⊥estimated via the LLB model Eq.(4) and the λ0value from the third column, at room tem- peratureT=300Kfor Co and Ni and at T=120Kfor Gd . The last column presents the experimentally measured Gilbert d amping collectedfrom different references. for the phonon-mediated EY process with the temperature- dependent parameter λvia the expression (9). The value of theGilbertdampingparameter α⊥wasthenestimatedthrough formula(4)at300 K(forNiandCo)andat120 KforGd. Note that for temperature-independent λ=λ0the resulting λ0and α⊥valuesareapproximatelytwotimessmallerforNiandCo. Thelastcolumnpresentsexperimentalvaluesforthesamepa - rameterfoundinliteratureforcomparisonwiththeonesint he fifth column, estimated through measurements of the ultra- fast demagnetisation times τMand the relation provided by theLLBequation. Given the complexityof the problem, the results presented inTableIdemonstratequiteasatisfactoryagreementbetwe en the values, extractedfrom the ultra-fast demagnetisation time τMand the Gilbert damping parameter α⊥via one unique coupling-to-the-bath parameter λ. The agreement is particu- larlygoodforNi,indicatingthatthesamespinflipmechanis m isactingonbothtimescales. Thisistrueforbothexperimen ts in Refs.22 and 25. For Co the value is some larger. For the temperature-independent λ, the resulting value is two times smaller and the agreement is again satisfactory. We would like to note that no goodagreementwas obtainedforFe. The reporteddampingvalues41are5-10timessmallerasestimated fromthedemagnetisationratesmeasuredinRef. 55. V. CONCLUSIONS The Landau-Lifshitz-Bloch(LLB) equation providesa mi- cromagnetic tool for the phenomenological modelling of the ultra-fast demagnetisation processes. Within this model o ne can describe the temperature-dependent magnetisation dy- namics at arbitrary temperature, including close and above the Curie temperature. The micromagnetic formulation can take into account the quantum spin number. The LLB model includes the dynamics governed by both the atomistic LLG model and the M3TM model by Koopmans et al.25. In the future it represents a real possibility for the multisca le modelling20. We have shown that within this model the ultra-fast de- magnetisation rates could be parameterized through a uniqu etemperature dependent or independent parameter λ, defined by the intrinsic spin-flip rate. The magnetisation dynamics is coupled to the electron temperature throughthis paramet er and is always delayed in time. The observed delay is higher forhigherelectrontemperature. Thisisinagreementwitht he experimentalobservationthatdifferentmaterialsdemagn etize atdifferentrates25,50andthattheprocessissloweddownwith the increase of laser fluency. We have shown that for the phonon-mediatedEY mechanism the intrinsic parameter λis dependent on the ratio between phonon and electron temper- atures and therefore is temperature dependent on the femto second-severalpicosecondtimescale. TheLLBequationcan reproduce slow demagnetizing rates observed in several ma- terials such as Gd, TbFe and half metals. This is in agree- ment with both phonon-mediated EY picture since in Gd a lowerspin-flipprobabilitywaspredictedandalsowiththei n- elastic electron scattering picture, since the electron di ffusive processes are suppressed in insulators and half-metals31,50. However, we also stress the importance of other parameters determining the ultra-fast demagnetisation rates, such as the electron-latticecoupling. The macroscopic damping parameters (longitudinal and transverse) have different natures in terms of the involved spinwaves and in terms of the timescales. Their temperature dependenceisdifferent,however,theyarerelatedbythesp in- flip rate. We have tried to check this relation in several tran - sition metals such as Ni, Co, Fe and the rare-earth metal Gd. A good agreement is obtained in Co and Gd and an excel- lent agreement in Ni. This indicates that on both timescales the same main microscopic mechanism is acting. In Ni the agreement is good both within the assumptions λ=λ0and λ=λ0Tp/Te. InCotheagreementseemstobebetterwiththe temperature-independentparameter λ=λ0whichdoesnotin- dicate towards the phonon-mediated EY mechanism. How- ever, given a small discrepancy and the complexity of the damping problem, this conclusion cannot be considered defi- nite. Wecanneitherexcludeanadditionaltemperaturedepe n- dence of the intrinsic scattering probability (i.e. the par ame- terλ0)forbothphonon-mediatedEYandexchangescattering mechanismswhichwasnottakenintoaccount. Anopenquestionistheproblemofdopedpermalloywhere anattempttosystematicallychangethedampingparameterb y dopingwithrare-earthimpuritieswasundertaken33inorderto clarify the relation between the LLG damping and the ultra- fast demagnetisation rate32,34. The results are not in agree- ment with the LLB model. However in this case we think that the hypothesis of the slow relaxing impurities present ed in Ref.34 might be a plausible explanation. Indeed, if the relaxation time of the rare earth impurities is high, the sta n- dardLLB modelis not valid since it assumes an uncorrelated thermal bath. The correlation time could be introduced in theclassicalspindynamicsviatheLandau-Lifshitz-Miyas aki- Seki approach56. It has been shown that the correlation time of the order of 10 fs slows down the longitudinal relaxation independentlyon the transverse relaxation. Thus in this ca se, themodificationoftheoriginalLLBmodeltoaccountforthe colorednoiseisnecessary.8 VI. ACKNOWLEDGEMENT This work was supported by the Spanish projects MAT2007-66719-C03-01,CS2008-023. AppendixA To show the equivalence between the LLB model with S=1/2 and the M3TM model25, we compare the relaxation rates resulting fromboth equations. We start with the M3TM equation dm dt=−RTp TC/parenleftbigg 1−mcoth/bracketleftbigg/parenleftbiggTC Te/parenrightbigg m/bracketrightbigg/parenrightbigg m(A1) whereweidentifytheBrillouinfunctionforthecase S=1/2, B1/2=tanh(q)withq=q1/2=(TC/Te)m. Now, we use the identityB1/2=2/B′ 1/2sinh(2q)towrite dm dt=−RTp TC/bracketleftbigg2 sinh(2q)/bracketrightbigg/parenleftBigg 1−B1/2 m B′ 1/2/parenrightBigg m2(A2) we multiplyanddivideby qµatβto obtain dm dt=−RTp TCµat kBTC/bracketleftbigg2q sinh(2q)/bracketrightbigg/parenleftBigg 1−B1/2 m µatβB′ 1/2/parenrightBigg m(A3) M3TMLLB t[ps]∆m/m 0 6 4 2 00 -0.1 -0.2 -0.3 Figure 6. Longitudinal relaxation calculated with M3TM and LLB (S=1/2) models for Nickel parameters22andT/Tc=0.8.We expand around equilibrium me=B1/2(qe)the small quantity1 −B1/2/m 1−B1/2(q) m∼=δm me/parenleftbigg 1−/parenleftbiggTC Te/parenrightbigg B′ 1/2(qe)/parenrightbigg (A4) whereδm=m−me. Next,weexpand maroundm2 e m=me+1 2(m2−m2 e) me=⇒δm me=(m2−m2 e) 2m2e(A5) and, 1−B1/2/m βµatB′ 1/2≈1 2/tildewideχ/bardbl(m2−m2 e) m2e(A6) Finally,collectingthe equations(A3)and(A6)altogether : dm dt=/parenleftbigg3R 2µat kBTC/parenrightbigg2Tp 3TC2q sinh(2q)/parenleftbigg1 2/tildewideχ/bardbl(1−m2 m2e)m/parenrightbigg (A7) Comparing this to the LLB equation with longitudinal re- laxation only and without anisotropy and external fields, we canwriteEq. 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2010-11-23

The ultra-fast magnetisation relaxation rates during the laser-induced magnetisation process are analyzed in terms of the Landau-Lifshitz-Bloch (LLB) equation for different values of spin $S$. The LLB equation is equivalent in the limit $S \rightarrow \infty$ to the atomistic Landau-Lifshitz-Gilbert (LLG) Langevin dynamics and for $S=1/2$ to the M3TM model [B. Koopmans, {\em et al.} Nature Mat. \textbf{9} (2010) 259]. Within the LLB model the ultra-fast demagnetisation time ($\tau_{M}$) and the transverse damping ($\alpha_{\perp}$) are parameterized by the intrinsic coupling-to-the-bath parameter $\lambda$, defined by microscopic spin-flip rate. We show that for the phonon-mediated Elliott-Yafet mechanism, $\lambda$ is proportional to the ratio between the non-equilibrium phonon and electron temperatures. We investigate the influence of the finite spin number and the scattering rate parameter $\lambda$ on the magnetisation relaxation rates. The relation between the fs demagnetisation rate and the LLG damping, provided by the LLB theory, is checked basing on the available experimental data. A good agreement is obtained for Ni, Co and Gd favoring the idea that the same intrinsic scattering process is acting on the femtosecond and nanosecond timescale.

Ultra-fast magnetisation rates within the Landau-Lifshitz-Bloch model

1011.5054v1

Phonon anharmonic damping enhances the Tc of BCS-type superconductors Chandan Setty Department of Physics, University of Florida, Gainesville, Florida, USA Matteo Baggioliy Instituto de Fisica Teorica UAM/CSIC, c/Nicolas Cabrera 13-15, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain. Alessio Zacconez Department of Physics "A. Pontremoli", University of Milan, via Celoria 16, 20133 Milan, Italy. Department of Chemical Engineering and Biotechnology, University of Cambridge, Philippa Fawcett Drive, CB30AS Cambridge, U.K. Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, U.K. A theory of superconductivity is presented where the eect of anharmonicity, as entailed in the acoustic, or optical, phonon damping, is explicitly considered in the pairing mechanism. The gap equation is solved including diusive Akhiezer damping for longitudinal acoustic phonons or Klemens damping for optical phonons, with a damping coecient which, in either case, can be directly related to the Gr uneisen parameter and hence to the anharmonic coecients in the interatomic potential. The results show that the increase of anharmonicity has a strikingly non-monotonic eect on the critical temperature Tc. The optimal damping coecient yielding maximum Tcis set by the velocity of the bosonic mediator. This theory may open up unprecedented opportunities for material design whereTcmay be tuned via the anharmonicity of the interatomic potential, and presents implications for the superconductivity in the recently discovered hydrides, where anharmonicity is very strong and for which the anharmonic damping is especially relevant. I. INTRODUCTION Atomic vibrations in solids are inevitably aected by the shape of the interatomic potential. For all real ma- terials, the shape of the interatomic potential is far from being quadratic, i.e. harmonic. The intrinsic anhar- monicity of solids has many well known consequences such as thermal expansion, soft modes and instabili- ties, sound absorption, identication of stable crystalline phases etc. [1] A well established approach to anhar- monicity is the self-consistent method introduced by Born and Hooton [2], leading to the concept of renormal- ization of phonon frequencies in the quasiharmonic or self-consistent phonon approximation, where the renor- malized phonon frequencies arise from an eective vibra- tional dynamics within a region about equilibrium, which takes anharmonic terms of the potential into account via adjustable parameters obtained from a self-consistent so- lution to the many-body problem [3]. However, the eect of anharmonicity extends to far greater areas, including electron-phonon coupling, where traditionally the eect of anharmonic damping has al- ways been neglected, and where instead recent rst- principle calculations demonstrate an important eect of anharmonicity on band-structure [4, 5]. settychandan@gmail.com ymatteo.baggioli@uam.es zalessio.zaccone@unimi.itIn the context of high Tcsuperconductors, the eect of anharmonic enhancement on Tchas been studied in the early days following the discovery of high- Tcsuper- conductivity in the cuprates. In particular, several works by Plakida and others have studied the eect of anhar- monicity on Tcfor the case of structurally unstable lat- tices or deformed lattice potentials [6{8]. Even more re- cent works on the high- Tchydrides [9{19] only take into consideration phonon energy renormalizations due to an- harmonicity but neglect anharmonic damping. However, a fundamental understanding of the eect of anharmonic damping on phonon-mediated supercon- ductivity and e.g. on Tcis absent due to the lack of analytical approaches to this problem. Yet, this is a fun- damental issue in the context of high-T superconductors where anharmonicity becomes important due to the sig- nicant temperature values, since in general anharmonic- ity in solids grows roughly linear in T[1]. Even more urgent is the problem of the eect of anharmonicity in hydrogen-based materials, which have recorded the high- estTcvalues so far: in these systems the presence of a light element such as hydrogen induces a huge anhar- monicity due to the large oscillation amplitudes of the hydrogen atoms [13, 18, 20{23]. Numerical studies and rst-principle calculations can assess the eect of anharmonicity in an empirical way for a specic material by benchmarking against harmonic calculations, but a systematic fundamental understand- ing of the role of anharmonic damping on conventional superconductivity is missing. This would be highly bene-arXiv:2003.06220v2 [cond-mat.supr-con] 20 Nov 20202 cial to obtain system-independent guidelines to not only estimate the eect of anharmonic damping in general cases, but also to develop generic guidelines for material design. For example, by relating anharmonic damping to the interatomic potential it could become possible to design materials with ad-hoc or tunable electron-phonon coupling and superconducting properties. Here we take a rst step in this direction by studying the eect of phonon Akhiezer and Klemens damp- ing on superconductivity beyond the quasi-harmonic approximation. We do this by explicitly taking into account the phonon damping due to anharmonicity in the mediator for the electron pairing. The theory shows that, unexpectedly, the eect of the anharmonicity (as represented by the damping coecient) on Tcis non-monotonic, i.e. Tcrst increases then goes through a maximum and then decreases upon increasing the anharmonic damping. This occurs because electron- phonon scattering processes involving energy-loss and energy-gain (Stokes and anti-Stokes) act constructively to increase the eective attraction driving the formation of Cooper pairs. The enhancement is most ecient for a window of critical damping parameter ( Dmax) set by the bosonic velocity and correlated with the Ioe-Regel scale. Outside this window, the strength of pairing deteriorates leading a reduction in Tc. These results are valid for both cases of acoustic and optical phonons, as shown in in the Appendix A.3 below. II. THE THEORETICAL FRAMEWORK The displacement eld of an anharmonic solid obeys the following dynamical equation [24]: @2ui @t2=CT ijkl@2uk @xj@xlCT ijklkl@
T @xj+ijkl@2_uk @xj@xl (1) which is coupled to Fourier's law for heat transfer and to the energy balance equation for the thermal gradient
T. In Eq. (1) , uidenotes the i-th Cartesian component of the atomic displacement eld, CT ijklis the isothermal elastic constant tensor, klis the thermal expansion ten- sor, andijklis the viscosity tensor. The dot indicates derivative with respect to time of the elastic eld ukin the last dissipative term. For solids, where acoustic excitations can be split into longitudinal (LA) and transverse (TA), Eq. (1) can be split into two decoupled equations for LA and TA dis- placements, leading to the following Green's function in Fourier space [25]: G(!;q) =1 !2 2 (q) +i!(q)(2) where=TA;LA is the branch label, and (q) =Dq2 represents the Akhiezer damping, which coincides with the acoustic absorption coecient [24], while (q) =vqis the acoustic eigenfrequency, already renormalized to account for the shift induced by anharmonicity [26], withvthe speed of sound for branch . The quadratic dependence (q) =Dq2of the damp- ing stems directly from the viscous term in Eq. (1) and is typical of Akhiezer damping [24, 27]. In particular, it has been shown [27] that takes the following general form for longitudinal excitations (see also [28]): L=q2 2"4 3+ +T22v2 L C2p 14v2 T 3v2 L2# :(3) wherexyxy is the shear viscosity, is the bulk vis- cosity,is the solid density, is the thermal conductivity, is the longitudinal thermal expansion coecient, and Cpis the specic heat at constant pressure. The second term in Eq. (3), 2, represents the phonon damping due to heat exchange between the compressed and the rareed regions of the longitudinal wave. This second contribution, in practice, represents only a few percent of the rst viscous contribution in Eq. (3) and is there- fore negligible. The above derivation follows a hydrodynamic approach [29]; by comparing with the result of a microscopic ap- proach based on the Boltzmann transport equation for phonons, it has been shown that [24] DL=CvT 24 3h 2 xyih xyi2 CvT 2h 2 xyi(4) where we neglected the contribution from bulk viscosity , since normally . Furthermore,h:::iindicates averaging with respect to the Bose-Einstein distribution as a weight, while xyis thexycomponent of the tensor of Gr uneisen constants. Also, Cvis the specic heat at constant volume, while is the phonon life-time. Since T1(which is an experimental observation for most solids [24, 30]), the diusion constant DLis independent of temperature, i.e. a well-known experimental fact [30]. A substantially equivalent expression for the damping of longitudinal phonons, in terms of an average Gr uneisen constant of the material av, was derived by Boemmel and Dransfeld [30] DLCvT 2 2 av (5) and provides a good description of the Akhiezer damping measured experimentally in quartz at T >60K[30]. In turn, the Gr uneisen constant , or at least the lead- ing term [31] of avor xyabove, can be directly related to the anharmonicity of the interatomic potential. For perfect crystals with pairwise nearest-neighbour interac- tion, the following relation holds [31] =1 6V000(a)a2+ 2[V00(a)aV0(a)] V00(a)a+ 2V0(a)(6) whereais the equilibrium lattice spacing between nearest-neighbours, and V000(a) denotes the third deriva- tive of the interatomic potential V(r) evaluated in r=a.3 Hence, the phonon damping coecient DLcan be directly related to the anharmonicity of the interatomic potential via the Gr uneisen coecient and Eq. (6). III. RESULTS Because in crystals momentum is always conserved during electron-phonon scattering events, only longitu- dinal phonons contribute to pairing [32, 33], therefore we will focus on the LA phonon, =LA, and we will drop theindex in the following. According to Eq. (2) we thus choose a phonon propagator written in Matsubara frequency of the form (i n;q) =1 v2q2+ 2n+ (q) n; (7) with ( q) =Dq2being the Akhiezer damping discussed above, and vis the phonon velocity. We dene the Bosonic Matsubara frequency n= 2nT wherenis an integer number and Tthe temperature. The super- conducting gap equation for a generic gap at momentum kand Fermionic Matsubara frequency !n= (2n+ 1)T takes the form (see Ref. [34] or [35])
(i!n;k) =g2 VX q;!m
(i!m;k+q)(q;i!ni!m) !2m+2 k+q+
(i!m;k+q)2; (8) for a constant attractive interaction gand volume V. Herekis the free electron dispersion which we choose to be quadratic with a chemical potential . The inverse temperature is denoted by and we work in simplied units where twice the electron mass is set to unity. For analytical tractability, we also choose an isotropic gap function independent of frequency, i.e.,
( i!m;k+q)
. Converting the momentum summation into energy integral with variable and assuming a constant density of states, the gap equation reduces to 1 =X !mZ1 Td [(v2D!m)(+) +!2m] [!2m+2+
2] (9) where=N(0)g2andN(0) is the density of states at the Fermi level. To begin the discussion, we conne our- selves to small Dso that we can ignore DTTc even though the chemical potential is allowed to be large compared to Tc. This implies that the linear term in !m can be neglected. The remaining constant v2acts like a mass term and reduces Tcfor allD[36]. As this eect is only quantitative, this term can also be ignored, as a rst approximation, without aecting the central claims of the paper. The full eect of the chemical potential term will be included in the upcoming paragraphs. With these assumptions and using the energy integral identityR1 1d (z+s)(2+r2)=s r(s2+z2r2), we obtain 1 =X !mT!2 mp !2m+
2 !4m+ (!2m+
2)(v2D!m)2: (10) To determine the condition for Tc, we set the supercon- 0.5 1.4 v 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5D 0.10.20.30.40.5Tc 0.9 1.0 1.1 1.2 1.3 1.4v0.51.01.5Dmax Figure 1. Top: The dimensionless critical temperature Tc as a function of the damping constant Dat dierent dimen- sionless speeds  v2[0:5;1:4].Bottom: The position of the maximum temperature as a function of the dimensionless longitudinal sound speed. In both plots we xed  = 0:1. ducting gap
= 0. We can then perform the innite sum over Matsubara frequencies (see the Appendix A.1 for more details) to obtain the simplied gap equation 1 =1 v4" 1 2 +i(i+D) 4 1 2v2 2Tc(i+D) +i(i+D) 4 1 2+v2 2Tc(i+D) +c:c# ; (11) where, henceforth, the barred quantities are normalized byp , i.e., v=v=p and (x) is the digamma function. A solution for Tccan be obtained from Eq. (11) and is plotted in Fig. 1 (Top) as a function of the anharmonic damping parameter D. The plot shows that Tcis enhanced quadratically for small D, reaches a maximum at an optimal anharmonicity parameter Dmax (set by the dimensionless phonon velocity  v), and falls o as a power law for larger D. The optimal parameter Dmax increases with  vas shown in Fig. 1 (Bottom). In the4 0.01 1.5μ0.0 0.5 1.0 1.5 2.0 2.5 3.0D 0.100.150.200.250.300.350.40Tc 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4μ1.301.351.401.451.50Dmax Figure 2. Top: The dimensionless critical temperature Tcin function of the diusion constant Dat dierent dimensionless chemical potentials  2[0:01;1:5].Bottom: The position of the maximum temperature in function of the dimensionless chemical potential. In both plots we xed  v= 1:2. Appendix A.2 (see also Refs. [37{42] quoted therein), we discuss the behavior of Dmax for larger values of vwhere it saturates to a value Dmaxv2=Tc(not shown in Fig. 1). This condition for resonance can be obtained from the denominator in Eq. 10. Note that the enhancement of the transition temperature occurs only above a critical value of the phonon velocity that is set by the interaction parameterp . The reason forthe non-monotonic behavior of Tccan be understood from Eq.(10) and the anti-symmetry in !of the phonon damping term. Because of this property, Stokes and anti-Stokes processes ( !m<and>0, respectively) add up constructively to increase the eective attraction driving the formation of Cooper pairs. This constructive interference grows with Dwhich gets to the numerator upon adding the two processes. Eventually, however, for suciently large anharmonic damping Dv2=!m, the quadratic termD2!2 min the denominator of Eq.(10) becomes the dominant contribution, the Stokes and anti-Stokes processes now add up in a destructive way and superconductivity gets suppressed. In the regime wherevis very small, the last term in the denominator of Eq.(10) can be approximated as ( v2D!m)2D2!2 m and the non-monotonicity is absent even at small D values (see dark lines in Fig.1). IV. CHEMICAL POTENTIAL EFFECTS In the following paragraphs, we relax the assumptions made previously on the chemical potential. We restrict ourselves to the BCS/quasi-BCS regime where the chem- ical potential is positive and not below the band bottom. This assumption ignores eects where the pairing scale becomes comparable to the band-width and hence keep- ing the BCS-BEC cross-over regime inaccessible. Follow- ing the same steps of the previous section, we obtain the simplied formula 1 =X !mTc !2 mc+ (v2D!mc)
j!mcj1 (!2mc+ (v2D!mc))2+!2mc(v2D!mc)2 (12) where!mcis the Fermionic Matsubara frequency at T= Tc. After algebraic manipulations of the Matsubara sum, as shown in the Appendix A.1, the nal equation for Tc with a nite chemical potential reduces to 1 =1 2v2" 1 2 +( b+a 2(b+b) 1 2b+ +ab 2(b+b) 1 2b +c:c)# + [D$D]; (13) where we have the denitions az D+i, bzr z2+4v2 (2Tc)2 2(Di)andzv2 2Tc+iD 2Tc. A plot of the numerical solution for TcversusDis shown in Fig. 2 (Top). Many of the features appearing in Fig. 1 (Top) are reproduced when the chemical po- tential is introduced { a non-monotonic dependence on the anharmonicity parameter, a quadratic rise and power-law fall o for small and large Drespectively. This rearms the assumptions made on the chemicalpotential in deriving Eq. (11). However, the chemical potential has an additional non-trivial eect of reducing Tcat small and large D, but enhances its peak value at optimalD. Furthermore, the Tcpeak position ( Dmax) changes substantially for small  and remains virtually unchanged for larger  . A plot of Dmaxas a function of is shown in Fig. 2 (Bottom).5 V. DISCUSSION Much attention has been devoted to the role of dis- order induced damping on superconducting Tc(see [43] and references therein); however, only a few theoretical works have examined directly the eects of damping on the superconducting properties, mostly in terms of glassi- ness [36, 44{46]. Ref. [45] nds an enhancement of super- conducting transition driven by a spin-glass phase formed from paramagnetic spins interacting through Ruderman- Kittel-Kasuya-Yosida exchange couplings. On the other hand, Ref. [44] nds that a glassy phase leads to mono- tonically decreasing Tcbut does not take into account the role of anharmonic phonon damping explicitly. The dissipative aspect of the glass phase was considered at a phenomenological level in Ref. [36] in the context of the under-doped high- Tccuprates. While a similar non- monotonic behavior in Tcis found, its mechanism does not arise from the time-reversal symmetry breaking in the dissipation term. This is re ected in the linear rise ofTcfor small damping as opposed to the quadratic rise as found in this work. Furthermore, as alluded to ear- lier, the parameter Dis a characteristic of anharmonic damping and originates from the viscous damping term in Eq. (1) describing anharmonic phonons. It can be di- rectly related to the Gr uneisen constant, which, in turn, can be determined via rst-principle calculations of the inter-atomic potential through Eq. (6); therefore, this re- lation provides a microscopic handle for tuning Dgiving one signicant control in designing real materials. VI. CONCLUSION To conclude, we have developed superconducting gap equations which account for the eect of anharmonic damping of phonons. The phonon viscosity parameter Dcan be related directly to the Gr uneisen coecient and to the shape of the interatomic potential. Upon solving the gap equation, it is found that the Tcdepends non-monotonically upon the anharmonic damping parameterDand features a maximum as a function of D. The value of the critical damping parameter ( Dmax) around which Cooper pairing is the strongest is set by the velocity vof the phonon. Within this optimal range of damping, Stokes and anti-Stokes electron-phonon scattering processes act constructively to increase the eective coupling constant. Outside this window, the strength of pairing deteriorates leading to a reduction in Tc. The prominence of the peak is enhanced when the Fermi energy is large compared to the electron-phonon coupling. Since the phonon damping corresponds to the phonon linewidth, these predictions may be further tested and investigated experimentally. The same results (anharmonic enhancement of Tcand non-monotonicity with damping) and the same resonance mechanism (this time due to Klemens damping [47]) apply in the case of pairing mediated by optical phonons, as shown in theAppendix A.3 below. Hence, the presented framework may lead to new guidelines for material design to optimizeTcin conventional superconductors, including high-T hydrides. Acknowledgements { Useful discussions with Boris Shapiro are gratefully acknowledged. M.B. acknowledges the support of the Spanish MINECO's \Centro de Exce- lencia Severo Ochoa" Programme under grant SEV-2012- 0249. CS is supported by the U.S. DOE grant number DE-FG02-05ER46236. Appendix A: Details of the derivations 1. Theoretical framework To obtain Eq. 9 from the gap equation (Eq. 8; see Fig. 3 for the associated self-energy diagram) we make the assumption of an isotropic gap function independent of frequency, i.e.,
( i!m;k+q)
. This allows us to cancel the order parameter in the numerator on both sides of Eq. 8 and eliminate the !ndependence to yield 1 =g2 VX q;!m1 ((v2D!m)q2+!2m) (!2m+2q+
2): (A1) We can now convert the qmomentum sum into an in- tegral by replacing1 VP q!1 (2)dR ddq!R N()d, whereN() is the density of states at energy . For quadratic bands with chemical potential , we have q=q2written in units stated in the main text. We now further assume a featureless density of states and approximate N()'N(0) as in a BCS supercon- ductor. This is exact in two dimensions and works well when the chemical potential is far away from the band bottom in three dimensions. Dening =g2N(0), we nally obtain Eq. 9. To obtain Eq. 11 from Eq. 10, we can simplify the Matsubara sum by summing over only positive frequen- cies and writing the equation for Tcas 1 = 2(2Tc)21X m=0" 1 x(x2+ (v20+Dx)2) +1 x(x2+ (v20Dx)2)# (A2) wherexm+1 2and the primed quantities are di- mensionless variables normalized by 2 Tc(i.e,v20= v2=2Tc). One can then use partial fractions to sim- plify the denominators and use the identity (z) = limk!1n Pk1 n=01 n+z+ lnko . The logarithmic terms cancel to yield Eq. 11. Similarly, one can obtain Eq. 13 from Eq. 12 by shifting the summation over positive fre- quencies and writing the equation for Tcas6 1 =Tc (2Tc)31X m=0" x2+ (v20Dx)0 x1 h (x2+ (v20Dx)0)2+x2(v20Dx)2i+ x2+ (v20+Dx)0 x1 h (x2+ (v20+Dx)0)2+x2(v20+Dx)2i# :(A3) (q;i!ni!m) g(k+q;i!m) Figure 3. Feynman diagram for the anomalous self-energy. In the weak coupling BCS limit, the anomalous self-energy reduces to the gap function. The solid (zig-zag) line is the electron (boson) Green function in the superconducting state. We again expand the summand above in partial frac- tions by factoring the denominators. Performing the remaining integer summations using the identity for (x) dened above, we obtain Eq. 13. 2. The resonance condition In this paragraph, we provide more details about the resonance condition discussed in the main text. The idea is that at a specic frequency, sometimes referred to as theIoe-Regel frequency [37], the boson mediator for the phonons undergoes a crossover from a ballistic propaga- tion to a diusive incoherent motion. More precisely, this happens at: !IRv2 D(A4) This value is of fundamental importance in the realm of amorphous systems, because of its correlation with the boson peak frequency, where the vibrational density of states (VDOS), normalized by the Debye law !2, dis- plays a maximum value [38{40]. The same boson peak phenomenology, however, is also at play in strongly an- harmonic crystals [41, 42]. Physically, this means that the density of the boson mediators is maximal around the boson peak frequency. As a consequence, one would expect the eects of the mediators to be enhanced at such energy scale. By esti- mating that: !IRTc (A5) we arrive at the following phenomenological resonance 1.0 1.2 1.4 1.6v0.20.40.60.81.0Dmax v2πTcFigure 4. A validation of the resonance condition (A6) using the data of g.1. condition: Tcv2 Dmax(A6) which is quoted in the main text. Here Dmaxis the value of the phonon viscosity at which Tcis maximized. In order to validate this expression, we plot the ratio DTc=v2in gure 4 for the same curves shown in the main text in g.1. We observe, that, especially for large values of the sound speed (compared to the phonon vis- cosityD), the resonance condition (A6) holds to good accuracy. This observation provides a useful correlation between the energy scale of the boson peak (induced by anharmonicity) and the maximum critical temperature that can be reached. 3. Pairing mediated by anharmonic optical phonons In the main text we focused our attention on the case of pairing mediated by acoustic phonons, where the an- harmonic damping is diusive, q2, according to the Akhiezer mechanism. In this section, we consider the case of pairing mediated by optical phonons. In the case of optical phonons, the anharmonic damping is mainly related to the decay process of the optical phonon into two acoustic phonons. The damping coecient is in- dependent of q, in this case, and was famously calculated by Klemens using perturbation theory [47]. As shown by Klemens, the damping parameter for optical phonons is proportional to the square of the Gr uneisen constant of the material. Hence, also in this case the Tc-enhancement could be tuned via the interatomic potential of the pa- rameter through , in a material-by-design perspective. Hence, we take a typical dispersion relation for optical7 1 2 3 4Γ0.81.01.21.41.6T˜ c 1 2 3 4 5Γ0.10.20.30.40.50.60.70.8T˜ c Figure 5. The dimensionless critical temperature ~Tc 2Tc=p in function of the constant damping . Top: In- creasing the mass gap of the optical mode !2 0from orange to purple. Bottom: Increasing the curvature of the optical dispersion relation from yellow to black. phonons, opt(q) =!0+q2(A7)with Klemens damping given a constant . We imple- ment this model of optical phonons into the Green's func- tion Eq. (2) of the main article, this time with damping =const independent of q[47], leading to the following form of the Bosonic propagator: (i n;q) =1 [!2 0+ 2!0q2+O(q4) ] + 2n n: (A8) Upon implementing this propagator in the theoretical framework above, we obtain the theoretical predictions forTcas a function of anharmonic damping constant for pairing mediated by optical phonons, reported in Fig.5 above. These predictions align well with the eect of Tc- enhancement due to anharmonic damping at low damp- ing, followed by a peak and subsequent decrease of Tc, that was shown in the main article for acoustic phonons. Also, in this case, clearly, the anharmonic damping can lead to a substantial increase of Tc, by at least a factor three. 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2020-03-13

A theory of superconductivity is presented where the effect of anharmonicity, as entailed in the acoustic, or optical, phonon damping, is explicitly considered in the pairing mechanism. The gap equation is solved including diffusive Akhiezer damping for longitudinal acoustic phonons or Klemens damping for optical phonons, with a damping coefficient which, in either case, can be directly related to the Gruneisen parameter and hence to the anharmonic coefficients in the interatomic potential. The results show that the increase of anharmonicity has a strikingly non-monotonic effect on the critical temperature $T_{c}$. The optimal damping coefficient yielding maximum $T_c$ is set by the velocity of the bosonic mediator. This theory may open up unprecedented opportunities for material design where $T_{c}$ may be tuned via the anharmonicity of the interatomic potential, and presents implications for the superconductivity in the recently discovered hydrides, where anharmonicity is very strong and for which the anharmonic damping is especially relevant.

Anharmonic phonon damping enhances the $T_c$ of BCS-type superconductors

2003.06220v2

Giant anisotropy of Gilbert damping in a Rashba honeycomb antiferromagnet M. Baglai,1R. J. Sokolewicz,2A. Pervishko,1, 3M. I. Katsnelson,2O. Eriksson,1, 4D. Yudin,5, 1and M. Titov2, 3 1Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 20, Uppsala, Sweden 2Institute for Molecules and Materials, Radboud University Nijmegen, NL-6525 AJ Nijmegen, the Netherlands 3ITMO University, Saint Petersburg 197101, Russia 4School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden 5Skolkovo Institute of Science and Technology, Moscow 121205, Russia (Dated: November 11, 2019) Giant Gilbert damping anisotropy is identied as a signature of strong Rashba spin-orbit coupling in a two-dimensional antiferromagnet on a honeycomb lattice. The phenomenon originates in spin- orbit induced splitting of conduction electron subbands that strongly suppresses certain spin- ip processes. As a result, the spin-orbit interaction is shown to support an undamped non-equilibrium dynamical mode that corresponds to an ultrafast in-plane N eel vector precession and a constant perpendicular-to-the-plane magnetization. The phenomenon is illustrated on the basis of a two dimensional s-dlike model. Spin-orbit torques and conductivity are also computed microscopically for this model. Unlike Gilbert damping these quantities are shown to reveal only a weak anisotropy that is limited to the semiconductor regime corresponding to the Fermi energy staying in a close vicinity of antiferromagnetic gap. I. INTRODUCTION A gapless character of the spin-wave spectrum in isotropic Heisenberg magnets in two dimensions re- sults in the homogeneity of magnetic ordering being destroyed by thermal uctuations at any nite tem- peratures. In contrast, in van der Waals magnets, characterized by intrinsic magnetocrystalline anisotropy that stems from spin-orbit coupling1, an ordered mag- netic state can be retained down to a monolayer limit. Two-dimensional (2D) van der Waals magnets are currently experiencing a revived attention2{8driven by the prospects of gateable magnetism9{12, a con- tinuing search for Kitaev materials13,14and Majorana fermions15, topologically driven phenomena16as well as various applications3,4,7. The trade-o between quan- tum connement, nontrivial topology and long-range magnetic correlations determines their unique magneto- electronic properties, in particular a tunable tunneling conductance17and magnetoresistance18{20depending on the number of layers in the sample, as well as long- distance magnon transport21. Ferromagnetic thin lms have already entered commer- cial use in hard drives, magnetic eld and rotation angle sensors and in similar devices7,22,23, while keeping high promises for technologically competitive ultrafast mem- ory elements24and neuromorphic chips25. Moreover, it has recently been suggested that current technology may have a lot to gain from antiferromagnet (AFM) materi- als. Indeed, manipulating AFM domains does not induce stray elds and has no fundamental speed limitations up to THz frequencies26. Despite their ubiquitousness, AFM materials have, however, avoided much attention from technology due to an apparent lack of control over the AFM order parameter { the N eel vector. Switching the N eel vector orientation by short electric pulses has been put forward only recently as the basis for AFM spintronics27{29. The proposed phenomenon has beensoon observed in non-centrosymmetric crystals such as CuMnAs30{33and Mn 2Au34{36. It should be noted that in most cases AFMs are characterized by insulating type behavior37, limiting the range of their potential appli- cations, e.g., for spin injection38. Interestingly, antifer- romagnetic Mn 2Au possesses a typical metal properties, inheriting strong spin-orbit coupling and high conductiv- ity, and is characterized by collective modes excitations in THz range36. Despite a lack of clarity concerning the microscopic mechanisms of the N eel vector switching, these experi- ments have been widely regarded as a breakthrough in the emerging eld of THz spintronics26,30,36,39{43. It has been suggested that current-induced N eel vector dynam- ics in an AFM is driven primarily by the so-called N eel spin-orbit torques29,32,44{56. The N eel spin-orbit torque originates in a non-equilibrium staggered polarization of conduction electrons on AFM sublattices29,32,48,50. Characteristic magnitude of the non-equilibrium stag- gered polarization and its relevance for the experiments with CuMnAs and Mn 2Au remain, however, debated. The N eel vector dynamics in an AFM is also strongly aected by an interplay between dierent types of Gilbert dampings. Unlike in a simple single-domain ferromagnet with a single sublattice, the Gilbert damping in an AFM is generally dierent on dierent sublattices and includes spin pumping from one sublattice to another. A proper understanding of Gilbert damping is of key importance for addressing not only the mechanism of spin pumping but also domain wall motion, magnon lifetime, AFM res- onance width and many other related phenomena57{61. It is also worth noting that spin pumping between two thin ferromagnetic layers with antiparallel magnetic ori- entations share many similarities with Gilbert damping in a bipartite AFM62,63. A conduction electron mechanism for Gilbert damp- ing in collinear ferromagnet requires some spin-orbit in- teraction to be present. It is, therefore, commonly as-arXiv:1911.03408v1 [cond-mat.str-el] 8 Nov 20192 FIG. 1. A model of Rashba honeycomb antiferromagnet with two sublattices, AandB, and on-site exchange interaction between localized momenta and conduction electrons (see Eq. 1). The large blue arrow represents the N eel vector vector, n, that is in general, characterized by non-vanishing in-plane, nk, and perpendicular-to-the-plane, n?, components. We re- fer to a specic coordinate system with ^xaxis chosen to be in the direction of nk. sumed that spin-orbit interaction of electrons naturally enhances the Gilbert damping. Contrary to this in- tuition, we show that Rashba spin-orbit coupling does generally suppress one of the Gilbert damping coe- cients and leads to the appearance of undamped non- equilibrium N eel vector precession modes in the AFM. Spin dynamics in a bipartite AFM is described in terms of two mutually orthogonal vector elds, namely the vec- torn(t) that is proportional to the N eel vector (dierence between sublattice moments) and the vector m(t) that is proportional to the net magnetization (sum of sublattice moments) of a sample. Even though the AFM ground state corresponds to m= 0, it is widely understood that no N eel dynamics is possible without formation of a small but nite nonequilibrium magnetization m. It appears, however, that Gilbert damping terms associated with the time dynamics of m(t) andn(t) are essentially dierent from a microscopic point of view. Indeed, the Gilbert damping that is proportional to @tnis characterized by a coecient n, which is van- ishing in the absence of spin-orbit interaction, much like it is the case in the ferromagnets. This behavior can be traced back to a spin-rotational symmetry of the collinear AFM. Indeed, the absolute value of nis conserved up to the orderm2. Thus, the dynamics of the N eel vector is essentially a rotation that does not change the conduction electron spectrum as far as the spin-rotation invariance is present. Breaking the spin-rotation symmetry by spin- orbit interaction induces, therefore, a nite n, which is quadratic with respect to spin-orbit interaction strength. In contrast, the Gilbert damping that is proportional to@tmoriginates directly in the conduction electron scattering even in the absence of any spin-orbit inter-action. The strength of the damping in a simple sym- metric AFM is characterized by a coecient m, which is typically much larger than n. As a rule, the spin- orbit interaction tends to suppress the coecient mby restricting the ways in which electrons can damp their magnetic moments. The condition mnhas been indeed well documented in a metallic AFM57,59. In this paper, we uncover the microscopic mechanism of strong and anisotropic Gilbert damping suppression due to the in uence of spin-orbit interaction in a 2D AFM model on a honeycomb lattice. Below we focus mainly on the AFM in the regime of good metallic behavior, such that the Fermi energy of electrons exceeds by order of magnitude that of an ef- fectives-dexchange coupling between electron spins and localized AFM magnetic momenta. In this case, the tran- sition to the highly anisotropic regime takes place pro- vided the characteristic spin-orbit energy exceeds the scale ~=, whereis the electron scattering time. Alter- natively, one may think of characteristic spin-orbit length becoming smaller than the mean free path of conduction electrons. We show here that the splitting of 2D Fermi surfaces by spin-orbit interaction leads to a dramatic sup- pression of electron spin ips in certain directions. This results in a strong anisotropy of both Gilbert damping tensors ^nand ^m, that get some of their principal com- ponents vanishing. This extreme anisotropy in the damp- ing leads to essentially undamped N eel vector dynamics for certain nonequilibrium modes. In particular, we identify a specic undamped mode that corresponds to perpendicular-to-the-plane magneti- zationm/^zand in-plane N eel vector n(t)?^z. The N eel vector corresponding to the mode has a precission aroundmwith the frequency Jexm=~, whereJexis the value of the isotropic AFM exchange. The presence of the undamped mode identied here, illustrates how lowering the symmetry of the electronic bath (by spin-orbit interaction) may induce a conserva- tion law in the localized spin subsystem. Based on this microscopic mechanism we provide qualitative arguments in favor of a generality of the giant Gilbert damping anisotropy in a 2D metalic AFM with spin-orbit cou- pling. Even though the undamped mode cannot be asso- ciated with a single spin-wave or a magnon, its presence has a strong impact on the nonequilibrium N eel vector dynamics in 2D Rashba AFMs. Apart from the Gilbert damping our results extend to cover conductivity and spin-orbit torques in the Rashba honeycomb AFM model. We also demonstrate how weak anisotropy of all these quantities emerge with Fermi en- ergies approaching the AFM band gap. II. PHENOMENOLOGY OF AFM DYNAMICS In this paper, we choose to describe the AFM with a classical Heisenberg model for localized spins SX=SnX on two sublattices X=A;B. The spins have the same3 modulusSand antiparallel directions nA=nBin the ground state. The AFM Heisenberg model is coupled to an eective tight-binding model of conduction electrons (see Appendix A) by means of exchange interaction, Hsd=JX iX 0Si
0cy ici0; (1) whereJstands for an s-d-like exchange energy that is the same onAandBsublattices, the operators cy i(ci) are the standard creation (annihilation) operators for an elec- tron on the lattice site iwith the spin index , and the no- tation= (x;y;z) represents the three-dimensional vector of Pauli matrices. The real-time dynamics of AFM is, then, dened by two coupled dierential equations (Landau-Lifshitz- Gilbert equations) on the unit vectors nAandnB, _nA=HAnA+ (JA=~)nAsA; (2a) _nB=HBnB+ (JA=~)nBsB; (2b) where dot stands for the time derivative, sXis the spin density of conduction electrons on the sublattice X, sA,B(r) =1 2X i0D cy i0ci0E2 A; (3) andAis the area of the unit cell in the AFM. The no- tationsHA,Brefer to eective elds on the sublattices A andBthat are dened by the Heisenberg model. For an isotropic antiferromagnet, one nds an eec- tive eld28HA+HB=Jexm=~+ 2H, whereHis an external magnetic eld in frequency units and Jexis a direct antiferromagnetic exchange energy that is one of the largest energies in the problem. In turn, the combi- nationHAHBis proportional to magnetic anisotropy that we do not specify in this paper. Magnetization dynamics in AFM is conveniently for- mulated in terms of the N eel and magnetization vectors, n= nAnB
=2;m= nA+nB
=2;(4) that remain mutually perpendicular n
m= 0 and yield the constraint n2+m2= 1. The dynamics necessarily induces a nite nonequilibrium magnetization vector m, while the condition m1 remains to be fullled. From Eqs. (2) we obtain _n= nm+Hn+ns++ms;(5a) _m=Hm+ms++ns; (5b) where = 2 JexS=~ands=JA(sAsB)=2~. In Eqs. (5) we have deliberately skipped terms that are in- duced by anisotropy of AFM exchange since the latter depend on particularities of the AFM Heisenberg model that we do not discuss here. The vectors+is proportional to average polariza- tion of conduction electrons, while the vector sis pro- portional to the staggered polarization. The quantities −2.50.02.5 vp/∆−2024ε/∆θ=π/2 −2.50.02.5 vp/∆θ=π/4K-valley K/prime-valley −2.50.02.5 vp/∆θ= 0FIG. 2. Electronic band structure of the honeycomb AFM model of Eq. (9) for dierent orientations of the N eel vec- tor (nz= cos). Two-dimensional momenta pare measured with respect to the wave-vectors KandK0that specify two nonequivalent valleys. Deviation of the N eel vector from the perpendicular-to-the plane conguration ( = 0) lifts the val- ley degeneracy. We restrict our analysis to the metallic regime with Fermi energies corresponding to two Fermi surfaces per valley (an example is shown by a black dotted line). The energy scale
characterizes the strength of s-dexchange in- teraction. s=s 0+scontain equilibrium contributions s 0that characterize various interactions induced by conduction electrons. These contributions do renormalize the pa- rameters of the AFM Heisenberg model and are not the subject of the present paper. The nonequilibrium contributions soriginate from various forces applied to conduction electrons. One nat- ural example is the electric eld that not only induces an electric current in the sample but also contributes to s. The electric eld can be further related to electric current by the resistivity tensor. The response of spin densities to electric current denes the so-called spin-orbit torques in Eqs. (5) that we also compute. Similarly, the response of sto the time derivatives _n and _mdescribe various types of Gilbert damping induced by conduction electrons. Quite generally, such a response can be written in the form of a tensor  s+ s = ^m^mn ^nm ^n _m _n ; (6) where all tensor components may themselves depend on the vectorsnandm. Gilbert dampings, in their original meaning, corre- spond to the contributions to sthat are symmet- ric under the time reversion. The terms that change sign should, more appropriately, be referred to as ef- fective spin renormalizations. Both types of terms are, however, obtained from the microscopic analysis of the Gilbert damping tensors in Eq. (6) similarly to the case of ferromagnets64.4 Time reversion, mentioned above, applies exclusively to the Heisenberg model, while keeping the tight-binding model (a bath) non-reversed. In other words we do not reverse the electron scattering time . Such a denition helps to identify the dissipative (even with respect to the time reversion) contributions to sthat describe Gilbert dampings. These contributions must, however, change sign under the transformation !, because spin densities sare always odd with respect to complete time reversion (the one which also includes that of the electron bath). We will see below, indeed, that all Gilbert dampings are proportional to the scattering time in the same way as the longitudinal conductivity does. Before we proceed with the microscopic analysis of s for a particular model, it is instructive to draw some gen- eral consequences for Eqs. (5) based on symmetry argu- ments in the case of collinear AFM with sublattice sym- metry and spin-rotational invariance (i. e. for vanishing spin-orbit interaction). Assuming that deviations from the AFM ground state remain small we shall limit ourselves to the linear order inmin Eq. (7a) and to the quadratic order in min Eq. (7b). Thus, we shall retain terms up to linear order inmin the tensors ^ m, ^nm, and ^mnand terms up to quadratic order in min ^n. Mixing tensors ^ mnand ^nmmust be odd in m, which implies, for our precision, a linear in mapproximation. As a result, the sublattice symmetry (the symmetry with respect to renaming AandB) prescribes that the mix- ing tensors must also be linear in n. In the absence of spin-orbit coupling we are also restricted by spin-rotation invariance that (together with the sublattice and time- reversion symmetries) dictates the following form of the Gilbert damping contributions to the non-equilibrium spin densities s+=m_m+0 mn(n_m)+mnm(n_n);(7a) s=n_n+0 nm(m_n) +nmn(m_m);(7b) where all coecients are assumed to be constants. It is easy to see that the vector forms n(m_n) and m(n_m), which could have respectively entered the spin densities s+ands, do not contribute to Eqs. (5) in the precision explained above. Substitution of Eqs. (7) into Eqs. (5) gives _n= nm+Hn+ mn_m+nm_n;(8a) _m=Hm+nn_n + mm_m+ (nm)(n
_m)0 nm2n_n;(8b) where m=m0 mand =mn+nm+0 m0 n. Discarding the three last terms in Eq. (8b), which are all of the second order in m, we indeed arrive at a set of Gilbert damping terms that is widely used in the AFM literature57,58,60. The symmetry consideration behind Eqs. (8) has es- sentially relied upon the spin-rotation invariance. This also implies n= 0 as has been pointed out in the in- troductory section. The coecient mcan, in turn, benite and large, even in the absence of spin-orbit inter- action. As we will show below, the presence of spin- orbit interaction does not only provide us with a nite nbut also drastically change the symmetry structure of Eqs. (8). We will demonstrate that the onset of spin-orbit interaction strongly aects the coupling of the localized spin subsystem to the electron bath (described by the tight-binding model) resulting in a strong reduction in the ability of conduction electrons to ip spins in certain directions and, therefore, to impose a friction on magne- tization dynamics. In the following, we turn to the microscopic analysis of the conductivity (Sec. IV), spin-orbit torques (Sec. V) and Gilbert dampings (Sec. VI) in a particular model of Rashba honeycomb AFM that has been put forward recently by some of the authors65. Rashba spin-orbit in- teraction breaks spin-rotational invariance of the model by singling out the direction ^zperpendicular to the 2D plane. We, therefore, investigate how such spin-rotation breaking manifests itself in the anisotropy of the above- mentioned quantities. III. MICROSCOPIC MODEL For the sake of a microscopic analysis we adopt a sub- lattice symmetric s-d-like model of a 2D honeycomb an- tiferromagnet with Rashba spin-orbit coupling, that was introduced in Ref. 65. The energy dispersion of this model is illustrated schematically in Fig. 2. The low en- ergy model for conduction electrons responsible for the dispersion in Fig. 2, is described by an eective Hamilto- nian (see Appendix A) that in a valley-symmetric rep- resentation reads He=vp
+1 2[]^z
n
zz+V(r):(9) Here,, andare the vectors of Pauli matrices in sublattice, valley and spin space, respectively, vis the characteristic Fermi velocity, while and
=JSare the energy scales characterizing the strength of Rashba spin-orbit coupling and s-d-like exchange energy, corre- spondingly. The termV(r) stands for a scalar Gaussian white-noise disorder potential, which is proportional to the unit ma- trix in sublattice, valley and spin space. The potential has a zero mean value hV(r)i= 0 and is fully character- ized by the pair correlator, hV(r)V(r0)i= 2(~v)2d(rr0); (10) where the angular brackets denote the averaging over dis- order realizations. The dimensionless parameter d1 quanties the disorder strength. The disorder potential is responsible for a momentum relaxation of conduction electrons. Exchange interaction and spin-orbit scattering (or the scattering on a non- collinear congurations with m6= 0) enable coupling be- tween localized angular momenta and kinetic momenta5 of electrons. Together these mechanisms form a channel to dissipate angular momentum of localized spins into the lattice. Thus, our model provides us with a micro- scopic framework to study dissipative quantities such as Gilbert dampings, anti-damping spin-orbit torques and conductivity that we compute below. We also note that the computation of spin-relaxation time can be directly related to our analysis of Gilbert damping66,67. The spectrum of the model (9) with V(r) = 0 consists of two electron and two hole branches for each of the valleys as illustrated in Fig. 2, e ;&(p) =p v2p2+
2&
nz+2=4=2;(11a) h ;&(p) =p v2p2+
2&
nz+2=4=2;(11b) where&=is the valley index. All spectral branches are manifestly isotropic with respect to the direction of the electron momentum pirrespective of the N eel vector orientation (as far as m= 0). In order to limit the complexity of our microscopic analysis we restrict ourselves to the metallic regime that corresponds to the Fermi energy "F>
+above the minimum of the top electron branches, e +;&(p), as shown schematically in Fig. 2. Note that the Fermi en- ergy"Fis counted in the model from the center of the AFM gap. We also focus on the limit of weak disorder "F=~1 where=~=(d"F) stands for the electron scattering time. Also, in order to describe spin-orbit in- duced anisotropy we nd it convenient to decompose the N eel vector (as well as the magnetization vector) to the in-plane and perpendicular-to-the-plane components as n=nk+n?, wheren?=nz^z. IV. CONDUCTIVITY The electric conductivity in the metallic regime is dom- inated by electron diusion. Despite the fact that the Fermi surface (line) is isotropic and does not depend on the direction of nk, the conductivity appears to be weakly anisotropic with respect to in-plane rotations of the N eel vector due to the onset of spin-orbit interaction. In par- ticular, fornz= 0 we nd the diagonal conductivity com- ponents xx=4e2 h"F ~"2 F
2 "2 F+ 3
2; (12a) yy=xx+4e2 h"F ~"2 F "2 F+
22
2 "4 F+"2 F
2+ 2
4; (12b) where the principal axes correspond to choosing ^xdirec- tion alongnk(see Fig. 1). In the deep metal regime, and for a general direction of n, this anisotropy is evidently small xxyy xx+yy=2
2 "4 F(1n2 z); "F+
; (13)whereaa= 1=aais the corresponding resistivity tensor component. We note that the anomalous Hall conduc- tivity is identically vanishing in the model of Eq. (9). The results of Eq. (12) and all subsequent results of our paper are technically obtained from linear response Kubo formulas evaluated in the diusive approximation (ladder diagram summation). The details of these calculations can be found in Appendixes B, C, and D. V. SPIN-ORBIT TORQUE Before proceeding with the microscopic analysis of Gilbert damping we shall discuss the eects of spin-orbit induced anisotropy for spin-orbit torques in the model of Eq. (9). Since this anisotropy appears to be weak in the metal regime, we shall touch on it only brie y. As was already mentioned, the spin-orbit torques origi- nate in the response of nonequilibrium spin polarizations sto electric current. Technically, we compute rst the response of sto electric eld and, then, express the electric eld in terms of 2D electric current density j using the conductivity tensor of Eq. (12). A straightforward computation of such a response gives s= 0 (see Appendixes B and C for more detail) and s+=a(n2 z)^zj+b(n2 z)nk(nk(^zj)) +c(n2 z)nk(n?(^zj)); (14) where the coecients a,bandcdo generally depend on n2 z= 1n2 xn2 yand are shown in Fig. 3. It is appropriate to recall here that the computation of the responses from the model of Eq. (9) refers to the case when m= 0. The symmetry form of Eq. (14) in this case has been also established recently from numerical simulations65. Importantly, the rst term in the right-hand side of Eq. (14) represents the well-known Rashba-Edelstein eect68, while the other two terms represent higher har- monics of the same eld-like eect that arise due to spin- rotation symmetry breaking. Anti-damping like torques (that are even under time-reversal) are vanishing in the model due to the valley symmetry constraint. This sym- metry reads  xH[n]x=H[n], from which it follows that the response of s+to charge current must be an even function of n. The behavior of the coecients a,bandcas a function ofnzis illustrated in Fig. 3 for two dierent choices of the Fermi energy. For in-plane N eel vector orientations (nz= 0) we nd a=a01 + 32 1 + 2 22+426; (15a) b= 2a212242+4 1 + 2234; (15b) c=2a21 + 2 22223446 1 + 42+ 54+ 66; (15c)6 FIG. 3. The coecients a,b, andcin Eq. (14) as a function of the direction of the N eel vector, nz= cos, for two dierent Fermi energies: "F= 4
(left panel) and "F= 16
(right panel). We use = 1:5
. Fornz= 0 the results correspond to Eqs. (15). where a0=AJ e~v "F; = "F;  =
"F: (16) In the metal regime, "F+
, the results of Eqs. (15) are reduced to a=AJ e~v "F; b =c= 2AJ e~v "F
"F2 :(17) One can, therefore, see that the high harmonics terms (proportional to bandc) become irrelevant in the metal regime. Vanishing response of the staggered polarization, s= 0, for the model of Eq. (9) is a simple consequence of the sublattice symmetry. As shown below the presence of a nite, though small, mbreaks such a symmetry and leads to a nite s. Taking into account a linear in m term in the Hamiltonian is also necessary to obtain nite mixed Gilbert damping tensors ^ nmand ^mnin Eq. (6). A low-energy model that takes into account nite mag- netization vector reads (see also Appendix D) H=He
m
; (18) whereHeis given by Eq. (9). The conductivity tensor does not acquire a linear in mterms in the leading order with respect to the large metal parameter "F=~, because the anomalous Hall eect remains subleading with respet to the metal parameter. Similarly, the result of Eq. (14) is not aected by the linear in mcorrections. However, the direct computation of the staggered po- larization response (in the linear order with respect to m) gives rise to a nite result. In the limit of large Fermi energy"F+
, we nd s=AJ e~v "F
"F2h 2n?(m?(^zj)) (19) + 2mk(n?(^zj))3nk(m?(^zj))i ;where the overall strength of the eect is again of the order of the coecients bandc. This makes the eects of nonequilibrium staggered polarization (including the celebrated N eel spin-orbit torque) irrelevant in the metal regime. Indeed, staggered polarization can hardly be in- duced by electrons with wavelengths that strongly exceed the distance between AandBsublattices. The results of Eqs. (14), (15) clearly suggest that the only torques surviving in the large energy limit are those related to non-equilibrium polarization s+=a0^zj, which is nothing but the standard Rashba-Edelstein eect68. These torques have a form Tn=a0n(^zj) in the right-hand side of Eq. (5a) and Tm=a0m(^zj) in the right-hand side of Eq. (5b). The anisotropy of torques is, however, irrelevant in this limit. VI. GILBERT DAMPING Surprisingly, the situation is dierent when we consider Gilbert damping terms. In this case we nd that the gi- ant anisotropy of Gilbert damping persists to arbitrarily large Fermi energy as soon as spin orbit energy exceeds ~=. The latter condition ensures that the scattering be- tween spin-split subbands is suppressed. The direct computation of the Gilbert damping tensors for~=gives s+=k m_mk+ mm+ mn; (20a) s=? n_n?+ nm+ nn; (20b) where the terms abcontain various vector forms. Far in the metal regime, "F+
, we nd k m= 2"F ~AJ2S ~2v2 1
2 "2 F(2 +n2 z) +::: ;(21a) ? n="F ~AJ2S ~2v2" "F2 +:::# ; (21b) = 2"F ~AJ2S ~2v2"
"F2 +:::# ; (21c) while the vectors forms abcan be written as mm=n(n_m) +nk(nk_mk) 2nk(nk_m?); (22a) mn=n(mk_n?)m?(nk_n?) +n?(m?_nk)nk(mk_nk) 3m?(n?_nk); (22b) nm= 2nk(mk_m?) + 2mk(n?_mk) n?(m?_m) + 2m?(n_m) +mk(n_m?)m?(n?_mk);(22c) nn=m(n_nk): (22d) Thus, we see from Eqs. (21) that the coecients ? nand are vanishingly small in the metal regime. Moreover,7 in the limit "F
the only non-vanishing contribu- tions to Gilbert dampings are given by the rst terms on the right-hand sides of Eqs. (20) that are manifestly anisotropic. The onset of spin-orbit interactions therefore makes Gilbert dampings ultimately anisotropic, also in the deep metal regime. This is in contrast to conductivity and spin-orbit torques that are quickly becoming isotropic in the metal limit. For "F+
, we nd the well known Landau-Lifshitz-Gilbert equations _n= nm+Hn+ k mn_mk+? nm_n?; _m=Hm+? nn_n?+ k mm_mk; (23) where we again omit terms that originate e. g. from mag- netic anisotropy of the AFM. Eqs. (23) are clearly dif- ferent from Eqs. (8) derived on the basis of symmetry analysis in the absence of spin-orbit interaction. The very pronounced, highly anisotropic Gilbert damping terms in the Landau-Lifshitz-Gilbert equations of Eqs. (23) represent the main result of our paper. The phenomenon of the giant Gilbert damping anisotropy in the 2D AFM clearly calls for a qualitative understanding that we provide in Sec. VII. VII. QUALITATIVE CONSIDERATION The results of Eqs. (20), (21) suggest that the anisotropy of Gilbert damping is most pronounced in the metal limit, "F
+as far as=~1. In partic- ular, certain spin density responses are vanishing in this limit. One of them is the response of the average spin densitys+ zto _mzthat is dened by the tensor com- ponentzz min Eq. (6). The other four vanishing tensor components xx n,xy n,yx nandyy ncorrespond to the re- sponses of the in-plane staggered spin densities s xand s yto _nxand _ny. It is important to stress that the component zz mis not only nite but also quite large in the absence of spin-orbit interaction, i. e. for = 0. It is, therefore, instructive to understand how the onset of spin-orbit interaction may cancelzz mresponse and lead to the conservation of z projection of magnetization vector. Such a qualitative understanding can be achieved by considering the Kubo-Greenwood formula for zz mfor the model of Eq. (18) in the limit
!0 and!1 , zz m/X pX s;s0=jh p;sjzj p;s0ij2("Fe p;s)("Fe p;s0); (24) wheree p;=p v2p2+2=4=2 correspond to the two electronic branches of Eq. (11a) that are evidently valley degenerate in the limit
!0. The states p;sare simply the eigenstates of theHamiltonian H0=vp
+ (=2) []z, H0=0 B@0 0 v(pxipy) 0 0 0 i v (pxipy) v(px+ipy)i 0 0 0v(px+ipy) 0 01 CA; (25) that can be explicitly written as p;=1 2p v2p2e =20 BB@vpei ie  e  ivpei1 CCA; (26) where we have used px=pcos,py=psin. One may notice that h p;sjzj p;si= 0 for any value ofsuggesting that the response function zz min Eq. (24) is vanishing. This is, however, not the case for = 0. In- deed, in the absence of spin-orbit interaction the electron branches become degenerate e p;=vpsuch that the in- plane spin- ip processes contribute to the Kubo formula, h p;+jzj p;ij=0=h p;jzj p;+ij=0= 1:(27) These processes are exactly the ones responsible for a nite Gilbert damping component zz min the absence of spin-orbit interaction. The spin-orbit induced splitting of the subbands forbids these spin- ip processes as soon as=~1 and leads to a giant anisotropy of Gilbert damping in the metal limit. Indeed, the other elements of the Gilbert damping tensor xx mandyy mremain nite irrespective of the subband splitting, h p;j(x+iy)j p;i=ivpei p v2p2+2=4: (28) One can further show that for = 0 the entire Gilbert damping tensor ^ mbecomes isotropic ^ xx m= ^yy m= ^zz m as it have been expected on the basis of the symmetry analysis. Very similar physics is also responsible for the anisotropy of the tensor ^ n. It is worth noting that the same type of anisotropy is known to take place in the limit of large spin-orbit interaction in 2D Rashba ferromagnets64. Spin-orbit induced anisotropy of Gilbert damping plays, however, a lesser role in 2D ferromagnets due to the much stricter constraint on the single mag- netization vector. A less restricted dynamics of mand nvectors make the Gilbert damping anisotropy play a bigger role in 2D AFMs. Indeed, it can be directly seen from Eqs. (23) that a nonequilibrium state with m=m^zandn=nkbecomes undamped in the absence of external eld H= 0. Such a state corresponds to the undamped N eel vector pre- cession around ^zaxis with a frequency given by Jexm. The state clearly survives in the presence of easy plane magnetic anisotropy in the AFM. We believe that such a phenomenon remains to be rather generic for a vari- ety of 2D or layered AFM systems with strong spin-orbit coupling of Rashba type.8 VIII. CONCLUSIONS In this paper, we demonstrate that the presence of suf- ciently strong spin-orbit coupling =~1 results in the ultimate anisotropy of the Gilbert damping tensor in the metal regime, "F
+. One can trace the phenomenon to the spin-orbit induced splitting of Fermi surfaces that forbids scattering processes that are respon- sible for the relaxation of certain magnetization and N eel vector components. We also demonstrate that a nite in-plane projection nkof the N eel vector is responsible for a weak anisotropy of conductivity and spin-orbit torques for Fermi energies approaching the band edge, "F
+. This anisotropy is, however, absent in the metallic regime. Gilbert damping is, however, in the absence of spin- orbit interaction as it is required by symmetry consider- ations. Thus, we demonstrate that the onset of Rashba spin-orbit interaction in 2D or layered AFM systems leads to a giant anisotropy of Gilbert damping in the metallic regime. The physics of this phenomenon origi- nates in spin-orbit induced splitting of the electron sub- bands that destroys a particular decay channel for mag- netization and leads to undamped precession of the N eel vector. The phenomenon is based on the assumption that other Gilbert damping channels (e. g. due to phonons) remain suppressed in the long magnon wavelength limit that we consider. The predicted giant Gilbert damp- ing anisotropy may have a profound impact on the N eel vector dynamics in a variety of 2D and layered AFM ma- terials. ACKNOWLEDGMENTS We are thankful to I. Ado, H. Gomonay and J. Sinova for fruitful discussions. This research was supported by the JTC-FLAGERA Project GRANSPORT. D.Y. and M.T. acknowledge the support from the Russian Science Foundation Project No. 17-12-01359. A.P. acknowledges support from the Russian Science Foundation Project 18-72-00058. The work of D.Y. was also supported by the Swedish Research Council (Vetenskapsr adet, 2018- 04383). M.T. is especially thankful to the KITP visitor program \Spintronics Meets Topology in Quantum Ma- terials". O.E. acknowledges support from the Swedish Research Council (Vetenskapsr adet) and the Knut and Alice Wallenberg foundation. Appendix A: Model system In this Appendix, we shall brie y justify Eqs. (9) and (18) of the main text. We start from an s-d-like model for two-dimensional antiferromagnet on a honeycomb lattice65. The model includes a local exchange interac- tion between localized magnetic moments and conduction electron spins as given by Eq. (1). Itinerant electrons inthe model are, therefore, governed by the tight-binding Hamiltonian H0=tX iX 0cy ici0JX iX 0Si
0cy ici0 +i 3aX hi;jiX 0^z
(dij)0cy icj0; (A1) where we do ignore disorder for a moment. The model is characterized by the nearest neighbor hopping energy tand the Rashba spin-orbit coupling energy ,z-axis is aligned perpendicular to the two-dimensional plane, the in-plane vectors dijconnect the neighboring sites iandj on a honeycomb lattice. For any site ion the sublattice Awe choose d1=a 0 1 ;d2=a 2p 3 1 ;d3=a 2p 3 1 ; (A2) whereais the length of the bond between AandB. By projecting the tight-binding model of Eq. (A1) on states in a vicinity of the valley wave-vectors, K=4 3p 3a 1 0 ;andK0=K; (A3) we nd, in the valley symmetric approximation, the ef- fective Hamiltonian of Eq. (9) with the assumption that SA=SB, wherev= 3ta=2~. By relaxing the assump- tion we obtain the model of Eq. (18). Appendix B: Linear Response tensors In order to keep technical expressions compact we let ~= 1 and"F="below. Our technical analysis is based on linear response of electron spin density to various per- turbations at zero frequency ( dc) limit. In particular, we consider three types of responses: the one with respect to electric current (via electric eld and inverse conduc- tivity tensor), the one with respect to the time derivative of the N eel vector and the other one with respect to the time derivative of magnetization vector. These responses are summed up as s+=^SSOT +j+^SGD mn_n+^SGD m_m; (B1a) s=^SSOT j+^SGD nm_m+^SGD n_n; (B1b) where we dene the response tensors ^SSOT  that are describing spin-orbit torques (both eld-like and anti- damping) and various ^SGDtensors that are describing various contributions to Gilbert dampings (and to eec- tive spin renormalizations)64. In order to compute the linear response tensors in Eqs. (B1) we apply the standard Kubo formula s =J2Sv2A 2VX cTrD ^GR^s ^GA^FE@X @t; (B2)9 whereVis the system area, cTr is an operator trace, ^GR(A)= ("Hi0) are retarded (advanced) Green func- tion operators, ^ s+ =, ^s = zzare the operators corresponding to the average spin-polarization s+and staggered spin-polarization s, the product ^F
X(t) rep- resents the time-dependent perturbation in the Hamil- tonian, while the angular brackets denote the disorder averaging that we consider in diusive (ladder) approxi- mation. The linear-response formula Eq. (B2) assumes zero temperature and zero frequency limit that corresponds to taking both Green's functions at the same energy "="F. We also neglect the Fermi-sea contribution (also known as St reda contribution) since such a contribution appears to be either zero or subleading in the metal parameter "1 with respect to our results. Thus, in order to compute Gilbert dampings and spin- orbit torque tensors we consider linear response of s to the three perturbations mentioned above. Each per- turbation is parameterized by the term H=^F
X(t) with _X=_n; ^F=
zz; (B3a) _X=_m; ^F=
; (B3b) _X= (v=e )^1j;^F=; (B3c) where ^is the conductivity tensor (this is computed from the standard Kubo formula which is analogous to the one in Eq. (B2) but for the response of current density to electric eld). The disorder averaging amounts to replac- ing Green's functions in Eq. (B2) with the corresponding disorder-averaged Green's functions and to replacing one of the operators, ^ sor^F, with the corresponding vertex- corrected operator. Disorder-averaged Green's functions become diagonal in the momentum space due to restored translational in- variance and take the form GR(A) p = ["HR(A)]1, where the Hamiltonian His dened in Eq. (9) of the main text, while the self-energy R(A)is evaluated in the Born-approximation depicted schematically in Fig. 4a. We nd that the real part of the self-energy does renor- malize the Fermi energy "and thes-dexchange coupling strength
, while the imaginary part reads Im R(A)=d 2("
zzn
): (B4) In order to evaluate linear response tensors in the lead- ing order with respect to the metal parameter "1 one also needs to sum up the ladder diagrams as shown in Fig. 4b-c. To do that one denes the vertex corrected operator ^Fvc=^F+^F(1)+^F(2)+^F(3)+


; (B5) where we denote by ^F(i)the operator ^Fthat is dressed by the number of idisorder lines, ^F(i)= 2dZd2p (2)2GR p^F(i1)GA p: (B6) FIG. 4. Diagrammatic illustration. a) Born-approximation. b) Ladder-approximation. c) Disorder-averaged polarization bubble. d) Perturbative expansion of the disorder-averaged polarization bubble. It appears that the summation in Eq. (B5) can be re- duced to geometric series in a nite operator space. In- deed, let us dene the operator space that is spanned by 16 operators in each of the valleys Bi=1 2; i=f;;g; (B7) whereiis a cumulative index with = 0;za valley parity index and;taking on the four values f0;x;y;zgeach. ForB= (B1;B2;:::;B 32) we dene the vertex cor- rected operator vector as Bvc=B+FB+F2B+F3B+


=1 1FB;(B8) whereFstands for a matrix of vertex corrections. Using the normalization condition Tr BiBj= 2ijwe nd Fij=dZd2p (2)2Tr GA pBiGR pBj ; (B9) where Tr stands for the usual matrix trace in the valley, spin and sublattice spaces. It easy to imagine that the matrix inversion in Eq. (B8) might be a daunting analytical task. We note, however, that the matrix Fis evidently diagonal in the valley space, and it can also become block-diagonal in sublattice and spin spaces by choosing a more convenient basis. A particularly useful choice of basis corresponds to in- plane rotation of both spin and sublattice Pauli matrices to the frame associated with the in-plane projection nk of the N eel vector. For spin Pauli matrices this transfor- mation is given by x!znxx+nyyq n2x+n2y; y!znyxnxyq n2x+n2y;(B10) where we took advantage of the fact that the direction of nis opposite in the two valleys. The same transformation (B10) has to be applied to  xand y. The matrixFis instrumental for the analysis of all linear response tensors in Eq. (B1). Indeed, using the10 denition of Eq. (B9) in Eq. (B2) and summing up the diusion ladders we nd s =J2Sv2A 2dX X ijTr[^s Bi]RijTr[^FBj]@X @t; (B11) whereR=F(1F)1. Thus, the computation of all response tensors is reduced in the diusive approximation to the computation of the vertex correction matrix Fand subsequent matrix inversion. Appendix C: Vertex correction Still, nding an inverse matrix (1 F)1is not that straightforward due to a pair of eigenvalues (one per val- ley) that equal exactly 1. The presence of such eigenval- ues roots in the particle conservation and is, therefore, not an articial problem. The unit eigenvalues do evi- dently prevent the matrix inversion in Eq. (B8). Nev- ertheless, it can be shown that the corresponding eigen- vectors do not enter the nal equations of motion for localized spins. In the next section, we brie y illustrate how one can formally avoid the particle conservation di- vergence in the computation of vertex corrections. Let us dene by athe eigenvectors of Fthat corre- spond to two unit eigenvalues, Fa=a, with= 0;z. For the normalized vector awe dene special operators B=a
B="
zn
 2p "2+
2; (C1) which are conserved with respect to impurity dressing B=B(i) for any order i. This means that the vertex corrected operator Bvc is formally diverging in the dc limit. In what follows, we formally write Bvc =R1B, where the limit R1! 1 is taken at the end of the calculation. The response tensors dened by Eqs. (6) consist of dierent correlators of the operators  ,s+ =a, and s = zz. It is evident that most of these operators are already orthogonal to B, Tr B = Tr s+ B = Tr s B0 = 0; (C2) while the only dangerous sector is related to the projec- tion Tr s Bz =4
np "2+
2; (C3) which is evidently nite. The result of Eq. (C3) leads to formally diverging contribution s divthat is generally present in all components of s, s div;/R1X Tr[^s Bz] Tr[^FBz]@n @t: (C4)One can immediately see, however, that such a diverging contribution corresponds to a particular vector form, s div;/R1nn
@n @t= 0; (C5) that manifestly vanishes due to the constraint jnj= 1 which is exact in the limit m= 0. Thus, the diver- gency inBvc div(which originates in the diusion pole of the density-density response) is, in fact, harmless for the response tensors we are discussing. It is interesting to note that the irrelevance of the di- vergency in Bvc divoperator extends to higher orders in m, even though it becomes much harder to see. We touch on this problem in Appendix D. Appendix D: Finite magnetization The deviation from a collinear antiferromagnetic order can be accounted by considering a nite net magnetiza- tion term in the Hamiltonain perturbatively, H=He+U; U =
m
: (D1) In the paper, we build the rst order perturbation theory with respect to U. First of all, it can be shown that the self-energy ac- quires the linear in mcontribution as Im R(A)=d 2("
zzn
+
m
):(D2) Second, the Dyson expansion of the disorder-averaged Green's functions GR(A)with respect to mreads GR(A)!GR(A)+GR(A)UR(A)GR(A); (D3) whereUR(A)=U(1id=2) and we disregarded terms starting from quadratic order in m. Note, that we have kept the notations GR(A)for the disorder averaged Green's functions of the unperturbed system. The computation of linear response tensors amounts to considering an additional contribution to the response tensor represented by a complex class of diagrams de- picted schematically in Fig. 4d. Before ladder summa- tion is applied the diagrams of Fig. 4d correspond to a contribution to the correlator of two operators BiandBj of the type Uij= 2dZd2p (2)2Tr GAUAGABiGRBj +GABiGRURGRBj ; (D4) which has yet be dressed. The dressing amounts to re- placing both BiandBjoperators with the corresponding vertex corrected operators Bvc iandBvc j, respectively. The nal result for the response of spin density is still given by Eq. (B11), where the matrix R=F(1F)1 is, however, replaced with R=F 1F+1 1FU1 1F; (D5)11 which corresponds to diagrams Fig. 4c-d. It is again con- venient to consider a particular basis for the matrix Fas dened in Eq. (B10) to simplify analytical computation. The problem of divergence in the operators Bdoes now become less trivial. Careful analysis shows that the linear terms in mincluded in Eq. (D5) lead to additional diverging contributions to sof the form s;(1) div;/R1nm
@m @t; (D6) that is analogous to the one in Eq. (C5) for a nite m. (We remind that the constraint n2+m2= 1 provides a relation between these terms). The contribution in Eq. (D6) is, however, of too high order in min Eq. (5a) and cancels out completely in Eq. (5b). The terms linear in mare also responsible for diverging contributions in s+ of the type s+ div;/R1mn
@n @t=R1mm
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2019-11-08

Giant Gilbert damping anisotropy is identified as a signature of strong Rashba spin-orbit coupling in a two-dimensional antiferromagnet on a honeycomb lattice. The phenomenon originates in spin-orbit induced splitting of conduction electron subbands that strongly suppresses certain spin-flip processes. As a result, the spin-orbit interaction is shown to support an undamped non-equilibrium dynamical mode that corresponds to an ultrafast in-plane N\'eel vector precession and a constant perpendicular-to-the-plane magnetization. The phenomenon is illustrated on the basis of a two dimensional $s$-$d$ like model. Spin-orbit torques and conductivity are also computed microscopically for this model. Unlike Gilbert damping these quantities are shown to reveal only a weak anisotropy that is limited to the semiconductor regime corresponding to the Fermi energy staying in a close vicinity of antiferromagnetic gap.

Giant anisotropy of Gilbert damping in a Rashba honeycomb antiferromagnet

1911.03408v1

Spin injection eciency at metallic interfaces probed by THz emission spectroscopy Jacques Hawecker1, T. H. Dang2, Enzo Rongione2, James Boust2, Sophie Collin2, Jean-Marie George2, Henri-Jean Drouhin3, Yannis Laplace3, Romain Grasset3, Jingwei Dong3, Juliette Mangeney1, Jerome Tignon1, Henri Jar es2, Luca Perfetti3and Sukhdeep Dhillon1 1Laboratoire de Physique de l'Ecole normale sup rieure, ENS, Universit e PSL, CNRS, Sorbonne Universit e, Universit e de Paris, F-75005 Paris, France 2Unit e Mixte de Physique, CNRS, Thales, Universit e Paris-Sud, Universit e Paris-Saclay, F-91767 Palaiseau, France and 3Laboratoire des Solides Irradi es, CEA/DRF/lRAMIS, Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau, France Terahertz (THz) spin-to-charge conversion has become an increasingly important process for THz pulse generation and as a tool to probe ultrafast spin interactions at magnetic interfaces. However, its relation to traditional, steady state, ferromagnetic resonance techniques is poorly understood. Here we investigate nanometric trilayers of Co/X/Pt (X=Ti, Au or Au0:85W0:15) as a function of the 'X' layer thickness, where THz emission generated by the inverse spin Hall eect is compared to the Gilbert damping of the ferromagnetic resonance. Through the insertion of the 'X' layer we show that the ultrafast spin current injected in the non-magnetic layer denes a direct spin conductance, whereas the Gilbert damping leads to an eective spin mixing-conductance of the trilayer. Importantly, we show that these two parameters are connected to each other and that spin- memory losses can be modeled via an eective Hamiltonian with Rashba elds. This work highlights that magneto-circuits concepts can be successfully extended to ultrafast spintronic devices, as well as enhancing the understanding of spin-to-charge conversion processes through the complementarity between ultrafast THz spectroscopy and steady state techniques. PACS numbers: I. INTRODUCTION When a pure spin current pass through materials with large spin-orbit coupling, it can generate a transverse charge current1,2by means of the Inverse Spin-Hall-Eect (ISHE). A urry of activity on this topic has been mo- tivated by the intimate relation between ISHE and the direct SHE3. The latter can be very eciently employed to generate a spin transfer torque capable of switching the magnetization of ferromagnetic thin lms4,5. Most experiments in this eld have been performed by spin- pumping viaferromagnetic resonance while some works have investigated the ultrafast regime6,7. More recently, other authors have proven that ISHE can be employed to generate an intense THz radiation. This breakthrough highlighted that interfaces leading to large spin trans- fer torque are also excellent emitters of electromagnetic waves8{12. Theoretical simulations based on superdiu- sive transport equations have successfully reproduced the observed emission13,14. However, these frameworks do not cover the impact of the electronic transmission at in- terfaces, neither the discussion of the particular role of the interfacial spin-orbit elds originating from charge transfer and symmetry breaking15. To this end, a tighter connection with steady state spintronics is highly desir- able. For example, the magnetocircuits analogies are The authors to whom the correspondence show be addressed are luca.perfetti@polytechnique.edu, henri.jares@cnrs-thales.fr and sukhdeep.dhillon@phys.ens.frwidely employed to dene the eciency of the spin-to- charge conversion16,17. An extension of such formalism to impulsive excitations has been discussed in the case of ultrafast spin-Seebeck eect18but not yet for the spin current injected from ferromagnetic transition metals. In the eld of spintronics, the optimal eciency of spin orbit torque (SOT) requires the engineering of metallic interfaces favoring a higher spin-current generation. Re- cently, many authors have tackled this issue by inserting transion metals19,20or noble metals21between cobalt (or CoFe, CoFeB) and platinum. Experiments with dier- ent interlayers have shown clear correlations between the spin-transfer-torque, magnetoresistance19, perpendicular anisotropy15,19and spin memory loss22. Here, we in- vestigate this topic by comparing the THz emission ef- ciency detected by Time Domain Spectroscopy (TDS) with Ferro-Magnetic Resonance (FMR) spectroscopy of trilayers Co/X/Pt. The insertion of an atomically thick interlayer of X=Ti, Au or Au 0:85W0:15modies the ca- pability of the interface to generate spin currents. This property aects, on the same footing, the emission of electromagnetic radiation as well as the Gilbert damping of the multilayer. Our result show that the THz-TDS emission spectroscopy is contactless and non-destructive method that can give an accurate and reliable esti- mate of the spin-injection eciency at spintronic in- terfaces. We discuss the data in the magnetocircuit formalism16{18, by assuming that spin-currents follow the evolution of magnetic uctuations. The average spin- conductance ( g"+g#)=2 characterizes the ultrafast cur- rents in platinum18whereas an eective spin conductancearXiv:2103.09557v1 [cond-mat.mes-hall] 17 Mar 20212 g"# edescribes the damping torque in the ferromagnetic material ( in our case is cobalt )16. These two quantities would be proportional to each other if the spin ow was conserved at the interface17. In reality, strong spin-orbit assisted scattering processes generate a sink of angular momentum and limit the spin ow that can propagate in platinum23{26. The comparison between ( g"+g#)=2 andg"# eshows that the fraction of spin current lost at the interface25is proportional to the spin conductance and may exceed 40% in the Co/Pt bilayer. Our measure- ments highlight that passivation of the interface by dif- ferent compounds follows a common trend and suggests the existence of a general relation between spin memory loss and spin-conductance. We prove this claim by choos- ing inter-layer materials with very dierent properties: Ti is more chemically reactive and has small spin-orbit coupling whereas Au and Au 0:85W0:15are less chemically reactive and hosts a larger spin-orbit interaction (espe- cially the Au:W alloy). FIG. 1: A) Detection of spin-to-charge conversion in a spin- tronic emitter. The cobalt layer has magnetization ~Mpar- allel to the external magnetic eld ~Hand is in contact with the platinum layer. An ultrashort laser pulse photoexcites the sample generates, in the Pt side of the interface, a spin current~Jsthat is proportional to the direct spin conductance (g"+g#)=2. The inverse spin Hall eect of Pt leads to a trans- verse charge current ~Jc. Being shorter than one picosecond, the~Jcpulse emits radiation in the THz spectral range. B) Eect of the platinum layer on the ferromagnetic resonance of the underlying cobalt. The magnetization precession driven by a radiofrequency eld ~hrfinduces a spin current. The in- crease of Gilbert damping due to the ~Jsinjection in the Pt layer is proportional to the eective spin-mixing conductance. II. GENERAL FRAMEWORK OF SPINTRONIC THZ EMISSION A framework building on few hypothesis connects the spin conductance to the emitted THz radiation.In the thin lm limit, the THz electric eld of a plane wave at the surface of the sample is given by~ET(!) =eZR~Jc(!;z)dz. This expression links ~ET(!) to the charge current density ~Jcvia an eective impedance8,27,28: Z=Z0 1 +n+Z0R (z)dz; (1) wherenis the refractive index of the substrate, zis the coordinate perpendicular to the interface, Z0= 377 is the vacuum impedance andR (z)dzis the local conduc- tivity integrated over the total thickness of the multi- layer. The charge current ~Jc(z) arises in platinum be- cause of the inverse-spin-Hall-eect acting on the spin current ow ~Js(z) along the normal direction to the lm plane. The latter decreases exponentially over a distance equal to the spin diusion length. It follows that: ZdPt 0+~Jc(z)dz=Js(0+)(~ en~ es)Pt sstanhdPt 2Pts;(2) where~ enis a unitary vector normal to the interface, ~ es is the polarization direction of the spin current, dPtis the thickness of platinum layer, Pt sis the spin diusion length in platinum,  sis the spin-Hall-angle of platinum andJs(0+) is the magnitude of spin current density gen- erated in the ferromagnet, penetrating into the heavy metal, and thus responsible for the charge current os- cillations at the platinum side of the interface. At this stage, it is important to recall that the magnitude of theJspropagating in platinum can be smaller than the one generated in the ferromagnet. The discontinuity of spin current between the two sides of the active interface is generally ascribed to the spin-decoherence induced by local spin-orbit elds (also known as spin-memory-loss)23 and has been recently proved viarened spin-orbit torque experiments15. Emission over an ultrabroad spectral range8,18and theoretical modeling13,14,18show thatJsevolves on a timescale comparable to the energy and momentum re- laxation of hot electrons. We make use of the magneto- circuit formalism to write to the spin current in terms of spin conductance parameters17. The ultrafast generation of a spin accumulation on the ferromagnetic side leads to a longitudinal component whereas the spin accumulation on the Pt side induces a transverse component18. The resulting expression reads: ~Js=~ 4g"+g# 2h@tM M^mi+g"#h^m@t^mi ;(3) whereMis the magnetization magnitude in the very proximity of the interface, ^mis the local magnetization direction,g"(g#) is the spin conductance parallel (an- tiparallel) to the magnetization and g"#is the spin mix- ing conductance. The longitudinal component is propor- tional to (g"+g#)=2 and to the relative demagnetiza- tion@tM=M . This term is driven by the quasi-ballistic3 FIG. 2: A) THz waveforms emitted by a set of dierent tri- layers Co/Au 0:85W0:15(d)/Pt. B) Spin current generated in Co/Ti(d)/Pt, Co/Au( d)/Pt and Co/Au 0:85W0:15(d)/Pt tri- layers of dierent thickness. The Co and Pt layer have xed thickness of 2 nm and 5 nm, respectively. The thickness dof the X=Ti,Au,Au 0:85W0:15layer is instead varied between 0 nm and 2 nm. The parameter (d) has been extracted from the THz signal via Eqn. [5] and can be considered as a nor- malized spin current density in the platinum layer. transport of highly excited electrons from the Cobalt to Platinum and it represents the dominant contribution in the case of the spin current that are generated by ultra- fast laser pulse14. Accordingly, the THz emission from Co/Pt is many orders of magnitude more intense8than the one observed from an interface where the longitudinal component is inactive18. Owing to the quasi-ballistic nature of the injection, the spin current arises the spin accumulation taking place on a length scale 1 :4 nm18,27,28. As a consequence, the strength of the emitted THz radiation scales as the en- ergy density injected by the pump pulse8,27,28 j@tM Mj/ABFI d+dPt+dCo; (4) whereABis the absorbed fraction of pump pulse in the multilayer, FIis the incident uence of the pump pulse, dPt= 5 nm is the thickness of platinum layer, dCo= 2 nm is the thickness of cobalt layer, dis the thickness of the X = Ti, Au or Au 0:85W0:15layer. To investigate this, we have prepared Co/X( d)/Pt trilayers on glass and highly resistive Si(111) substrates by sputtering deposi- tion at room temperature with standard experimental conditions. The Au 0:85W0:15material has been obtained via the evaporation of a rod containing 85% of gold and 15% of tungsten. The thickness dof the interlayer is typ- ically varied between 0 and 2 nm. Within this range of d, theABcoecient can be considered constant8,27,28. Morever the incident laser uence FIhas been kept xed and stable.The THz TDS system is placed in a re ection geom- etry where the generated THz pulses are collected from the same surface of the spin-emitter as the excitation (i.e. no beam passes through the substrate). The emit- ters are mounted with small magnetic eld parallel ( =10 mT) to the spin interface. We veried that a switch- ing of the ~Morientation reverses the direction of the emitted THz eld, thereby conrming that charges cur- rents arise from the ISHE. Fig. 2A) displays a set of THz traces emitted from Co/Au 0:85W0:15(d)/Pt multilayers with dierent values of the Au 0:85W0:15thicknessd. The THz traces recorded for dierent values of dhold nearly identical waveforms (see also supplementary information le29). SinceET(t;d)=ET(d)f(t) (and equivalently ET(!;d)=ET(d)f(!)), we assume that spin uctua- tions, spin mixing conductance and spin-Hall-angle have negligible frequency dependence within the bandwidth of the detected THz. As observed experimentally, the drop of THz signal as a function of dis mainly due to a de- creasing spin-conductance. The latter is related to the detection of the THz eld via Eqs. [1-4]. By solving for the spin conductance we obtain: (d) =g"(d) +g#(d) g"(0) +g#(0)=ET(d) ET(0)Z(0) Z(d)d+dPt+dCo dPt+dCo;(5) where the impedance Z(d) has been calculated by as- suming the THz conductivity in thin lms27,28,30Co= 3106S/m,Pt= 4106S/m,Au= 4106S/m, Au:W= 1:2106S/m andTi= 0:5106S/m. Dier- ences of these conductivities with respect to bulk values are due to strong charge scattering at the landscape of the interface and to the formation of small grains30. As a matter of facts, the factor Z(0)=Z(d) remains close to unity, owing to the fact that metallic interlayers with nanometric thickness have small parallel conductivity. III. DATA ANALYSIS AND DISCUSSION The parameter (d) of Eqn. [5] re ects the relative reduction of the spin-injection eciency in Pt if an in- terlayer of thickness dis grown between Co and Pt. As shown by Fig. 2B), (d) follows nearly an exponential de- cay exp(d=lX), with characteristic length lAu= 4 nm for X=Au or X=Au 0:85W0:15andlTi= 1:5 nm for X=Ti. As can be observed, Ti aects the spin mixing conduc- tance much more eectively than Au or Au 0:85W0:15do. Recent experiments have shown that a submonolayer of Ti can indeed substantially modify the spin-transfer torque of the CoFeB/Pt20and Co/Pt31interfaces. The insertion of the chemically reactive Ti alters the spin dependent transmission/re ection probabilities that fa- vor the transport of one spin avor with respect to the other. Furthermore, the surface passivation by Ti atoms may modify the spin- ip scattering potential at the in- terface. Although the microscopic mechanisms leading to the large reduction of spin conductance is still debated,4 FIG. 3: A) Derivative of the spin susceptibility vs inten- sity of the static magnetic eld Hin the reference bi- layer Co/Pt. The dierent curves correspond to hrffre- quencies of 4-18 GHz, with step of 2 GHz. B) Varia- tion of resonance frequency as a function the static mag- netic eldHin the Co/Au 0:85W0:15(d)/Pt trilayers. C) Full width at half maximum of the ferromagnetic resonance in Co/Au 0:85W0:15(d)/Pt trilayers. The Co layer has thick- ness of 15 nm, the Pt layer has thickness of 5 nm and the Au0:85W0:15layer has thickness dvarying between 0 nm and 1.5 nm. a systematic investigation of spin orbit torque with dif- ferent transition metals concluded that the d-orbital ll- ing has a stronger in uence on charge-to-spin conversion than the atomic number19. Our measurements corrob- orate this nding: the passivation of Co/Pt interface is more eective in the case of a transition metal with in- complete 3d-shell like titanium than in the case of an alloy with larger atomic number but closed 5 dshell like Au. Moreover, the larger spin-orbit interaction of W in the Au 0:85W0:15does not seem to make any appreciable dierence with respect to pure gold. Our model in the last section of this article will further clarify this, some- how surprising, result. Next, we discuss the eective spin mixing conduc- tance that is measured by means of FerroMagnetic Res- onance (FMR)12. Samples made with 5 nm of Pt and thicker Co lms (15nm) were deposited on highly resis- tive Si/SiO2(111) substrates before lithography pattern- ing. The thicker ferromagnetic layer provides a clearer FIG. 4: A) Gilbert damping and spin conductance in Co/Ti(d)/Pt, Co/Au( d)/Pt and Co/Au 0:85W0:15(d)/Pt tri- layers as a function of thickness d. B) Gilbert damping and spin conductance of the two trilayer set plot against the pa- rameter extracted from the emitted THz. The green dot cor- responding to vanishing THz emission is the intrinsic Gilbert damping measured on cobalt capped by 2 nm of alumina. A model that includes the spin memory loss is calculated via Eq. [9] and superimposed (solid line) to the experimental data. resonance spectrum compared to a 2 nm layer. Fig. 3A) displays the dierential susceptibility of the Co/Pt bi- layer as a function of the external magnetic eld H. Curves of dierent colors stand for increasing frequency of radiofrequency eld hrf. We show in Fig. 3B) the reso- nance frequency !ras a function of Hfor the multilayers Co/Au 0:85W0:15(d)/Pt. The FMR theory predicts: !r= 0p H(H+M); (6) where is the gyromagnetic ratio and 0vacuum perme- ability and Mis the saturation magnetization. By tting the data with Eq. [6], it is possible to extract the satura- tion magnetization M= 150050 emu/cm3. The damp- ing term can be quantied by measuring the half width at half maximum
Hof FMR linewidth. As shown by Fig. 3C) the linear regression
H=
H0+!r 0; (7)5 provides the Gilbert damping (d) for the Co/Au 0:85W0:15(d)/Pt series. Likewise, this proce- dure is applied to extract the Gilbert damping of Co/Ti(d)/Pt trilayers. Moroever, the larger thickness of cobalt layer ( dCo= 15 nm in FMR experiments instead ofdCo= 2 nm chosen for the THz emission experiment) minimize the extra contribution of two-magnons scatter- ing to thevalue. Since two-magnon scattering scales as 1=d2 Co, the associated damping term26should not exceed 8104and it has been neglected. Therefore, diers from the intrinsic 0only by a term arising from the injected spin current. The eective spin mixing conductance g"# eis obtained via23,25:
=0=gB 4Md Cog"# e; (8) wheregstands for Land e factor of the electron and B is the Bohr magnetron. The value 0= 5103is obtained by measuring the Gilbert damping of a 15 nm cobalt capped by 2 nm of alumina. Fig. 4A) shows andg"# efor the two trilayer series as a function of interlayer thickness d. Similarly to THz mea- surements, the drop of spin mixing conductance is faster in Co/Ti(d)/Pt than in Co/Au 0:85W0:15(d)/Pt samples. This nding highlights the rst important outcome of this work: an intimate connection between ( g"+g#)=2 obtained by ultrafast currents in the THz spectral range, withg"# eextracted from the FMR damping linewidth. We nd phenomenologically the universal relation: g"# e/(d) 1(d): (9) The solid line of Fig. 4B is calculated from Eq. [9] with parameters g"# e(0) = 75nm2and(d) = 0:4(d). From their dependence on the transmission coecient at the interface17, we evince that g",g#and (g"+g#)=2< g"#=g"should scale as (d) upon the insertion of the interlayer. Namely, we assume that g"(d)=g"(0) = g#(d)=g#(0) =g"#(d)=g"#(0) =(d). Moreover, we set g"#= (1)g"# e, where the parameter  <1 arises from the spin-memory-loss22,23,25. Due the spin scattering at the interface, the spin-current leading to THz emission in platinum is 1times smaller than the spin current aecting the ~Mprecession. The second important result of our work is that is proportional to the spin conduc- tance at the interface. The more ecient the generation of spin current, the higher the spin memory loss. When expressed in terms of relative variation of spin conduc- tance, the spin memory loss appears to be insensitive to the compound and thickness that has been employed to perform the passivation of the interface. We now turn on to the modeling of the spin memory loss through Rashba elds at the interface.IV. MODELING OF SPIN MEMORY LOSS THROUGH RASHBA SPIN-ORBIT INTERACTION AT THE INTERFACE. A. Electronic quantum transmission with spin-orbit interaction The insertion of an interlayer X at the Co/Pt interface has two mains eects: i)the formation of a thin potential barrier is accompanied by smaller the spin-transmission vsCo/Pt. Indeed Co/Pt is known to build an excellent matching for the majority spin channel near the Fermi level whereas a larger chemical mismatch may take place in the case of Co/X/Pt with X=Ti, Au or Au 0:85W0:15 and; ii)since the Ti or pure Au lack the open 5 dshell of Pt, the presence of an interlayer has to reduce spin orbit interaction (SOI) at the interface32. In the following, we consider a simplied SOI assisted quantum transmission model that has been recently im- plemented with success for the description of SOT33{37. This model will rst highlight the role of i)andii)in the description of our data. The interface is treated as an ideal trilayer structure Co/X/Pt with a spin current Js propagating along the ~ endirection, normal to the layers (CPP geometry). Jsis computed from the propagation of selected plane waves with in-plane conserved wavevec- torkk, and normal wavector kzalong~ en. The quantum transmission is summed hereafter over the Fermi surface, as it is required within an extended Landauer treatment. We obtain theJs(z) prole across the interface viaa rened model involving a Rashba-like term33{36. We re- strict the electronic states to two electron bands with spin polarized states. The partitioned Hamiltonian in Co and Pt reads: ^H=^p2 2m+
e^m
^+^V (10) where ^p=i~rzis the impulsion operator, mis the eective mass, ^mis the magnetization direction,
e' 2 eV is the exchange coupling for Co, and ^V=^VCo= 0 represents the energy position of the bottom of the spin- averaged 3 dCo bands. Along the same lines, we set for Pt an exchange coupling
e= 0 and ^V=^VPt'1 eV. The potential dierence ^VCo^VPtis representative of the workfunction oset between the two metals. The addition of an interlayer is simulated by an inter- facial potential ^VSthat is expressed by34,36,37: tI^VS(z) =tIh VX+R ~ ^~ p~ en
^i (z); (11) wherezis the coordinate along the direction ~ en, the func- tion(z) is Dirac delta function and tIis the eective interface thickness. The operator ^VSis dened via:VX is the average interface of an unpolarized potential bar- rier andRis the strength of Rashba interaction. We introduce the two parameters having the dimension of inverse length. The quantity kX=VXtIm=~2tunes the transmission trough the barrier and kso=RkFtIm=~26 FIG. 5: A) Prole of spin current Jsin the Co/X/Pt surface at the vicinity of the Co/Pt interface for 3 dierent cases: no scattering potential (black curve), Rashba scattering only withkso= 2A1(red curve) and Rashba scattering plus a potential barrier ( kX= 3A1). The spin memory loss = (Js(0)Js(0+))=Js(0) is the relative discontinuity of Js at the interface. B) Spin memory loss as a function of potential barrier kXfor three dierent strengths of the Rashba scattering. C) Spin memory loss as a function of eective spin-conductance geobtained by varying kXand with spin orbit parameter equal to kso= 2A1(blue circles). As a term of comparison we also show the relation extracted from the experimental data (green dashed line). rule the strength of the spin-orbit scattering (see also supplementary information le29). B. Results of the model Our model provides the prole of a normalized spin- polarized current originating from Co (where it is normal- ized to unity) and propagating through a Co/X/Pt tri- layer. Figure 5A) depicts three specic cases, correspond- ing to: no interfacial potentials ( kX= 0 andkso= 0), a pure Rashba interaction ( kX= 0 andkso= 2A1) and, both a potential barrier and a Rashba interaction (kX= 3A1andkso= 2A1). The spin current is always maximal in the bulk of Co, while it goes towards zero when penetrating in the non-magnetic Pt layer and mov-ing away from the interface. In the absence of the scat- tering potential ^VS(black curve in Fig. 5A)) the Js(0) value at the Co/Pt interface results from an equilibrium condition between bulk spin- ip rates in the two regions. The spin-current is continuous everywhere (no spin-orbit scattering) and its value Js(0)0:6 coincides with the prediction of a pure diusive spin-model. This agreement corroborates the validity of our quantum transmission model in the absence of any ^VSscattering. Adding a Rashba interaction kso= 2A1(red curve in Fig. 5A)) leads to the spin-memory loss. Indeed the Rashba elds are not collinear to the incoming spin and induce a local spin-precession. Only a fraction of spin current coming from the Co reservoir is injected into the Pt layer so that Js(z) displays a sizable dis- continuity at the interface22. In order to quantify this eect, we introduce the memory loss parameter = (Js(0)Js(0+))=Js(0), where 0and 0 +are the lim- iting values reached by approaching the interface from the Co and Pt side, respectively. From the chosen pa- rameters we extract = 0:6, which is only 50% higher than our experimental value and in agreement with pre- vious FRM estimates23. The presence of an additional unpolarized scattering potential with kX= 3A1(Blue curve in Fig. 5A) has two main eects. On one hand, the larger back ow of Js in the Co layer leads to a smaller ejection of spin-current from the ferromagnet. On the other hand, an unchanged strength of the Rashba eld results in a smaller jump of theJscurrent at the interface. As shown in Fig. 5B), the monotonic reduction of spin memory loss as a function ofkXtakes place for two representatives values of the inverse spin length kso. We extract the eective spin conductance from the rescaled ratio between the spin current Js(0) obtained in the presence of an interlayer (i.e. for kX>0) and theJs(0) obtained for the bare Co/Pt interface (i.e. forkX= 0). Figure 5B) shows the calculated vs.ge when the potential barrier kXis increased linearly to 5A1while the value ksois kept xed to 2 A1. Note that the spin memory loss display the same trend of the curve that is extracted by combining FMR-spin-pumping and THz methods (green dashed line). This shows that an interposition of Ti, Au or Au 0:85W0:15introduces a chemical barrier at the interface. The enhanced back- ward diusion of electrons has the eect of decreasing both the spin mixing conductance and the spin memory loss. This eect takes place even if the spin dependent scatteringksoremains equal to the pristine value. V. CONCLUSIONS AND ACKNOWLEDGMENTS. In conclusion, we report that the spin-conductance can be extracted from broadband THz spectroscopy. The in- vestigation of Co/X( d)/Pt trilayers with X=Ti, Au and Au0:85W0:15show that in all cases, an interlayer reduces7 the spin-to-charge conversion. THz experiments have been bench-marked with the eective spin-mixing con- ductance extracted by FerroMagnetic Resonance mea- surements. A model including spin memory loss show that the relative drop of spin current at the interface is proportional to the spin conductance and attains = 0:4 at the Co/Pt interface. The simulations indicate that modied spin transmission probabilities at the interface can explain this correlation. Our ndings are very gen- eral and show that a combination of THz emission with FMR spectroscopy can bring accurate characterizations and provide new insights into spintronic multilayers. We acknowledge E. Jacquet for his contribution in the thin lm growth and M. Cosset-Cheneau for hishelp in the FMR experiments. We are very thank- ful to Tobias Kampfrath and Marco Battiato for the enlightening discussions on the interpretation of THz emission mechanism. Synchrotron Soleil hosts a THz setup where some transmission measurements have been done. Financial support has been provided by the DGA project ITEHR (No. 2018600074) as well as ANR Project TOPRISE No. ANR-16-CE24-0017. We acknowledge the Horizon2020 Framework Programme of the European Commission under FET-Proactive Grant agreement No. 824123 (SKYTOP). This project has received funding from the H2020 research and innovation programme s- Nebula under grant agreement No.0863155. 1E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, \Con- version of spin current into charge current at room tem- perature: Inverse spin-Hall eect," Appl. Phys. Lett. 88, 182509 (2006). 2S. O. Valenzuela and M. Tinkham, \Direct electronic mea- surement of the spin Hall eect," Nature 442, 176 (2006). 3J.Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, T. 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2021-03-17

Terahertz (THz) spin-to-charge conversion has become an increasingly important process for THz pulse generation and as a tool to probe ultrafast spin interactions at magnetic interfaces. However, its relation to traditional, steady state, ferromagnetic resonance techniques is poorly understood. Here we investigate nanometric trilayers of Co/X/Pt (X=Ti, Au or Au0:85W0:15) as a function of the 'X' layer thickness, where THz emission generated by the inverse spin Hall effect is compared to the Gilbert damping of the ferromagnetic resonance. Through the insertion of the 'X' layer we show that the ultrafast spin current injected in the non-magnetic layer defines a direct spin conductance, whereas the Gilbert damping leads to an effective spin mixing-conductance of the trilayer. Importantly, we show that these two parameters are connected to each other and that spin-memory losses can be modeled via an effective Hamiltonian with Rashba fields. This work highlights that magneto-circuits concepts can be successfully extended to ultrafast spintronic devices, as well as enhancing the understanding of spin-to-charge conversion processes through the complementarity between ultrafast THz spectroscopy and steady state techniques.

Spin injection efficiency at metallic interfaces probed by THz emission spectroscopy

2103.09557v1

Exploring Damping Effect of Inner Control Loops for Grid-Forming VSCs Liang Zhao, Student Member , IEEE , Xiongfei Wang, Fellow , IEEE , Zheming Jin, Member , IEEE Abstract—This paper presents an analytical approach to explore the damping effect of inner loops on grid -forming converters. First, an impedance model is proposed to characterize the behaviors of inner loops, thereby illustrating their influence on output impedance shaping. Then, based on the impedance represe ntation, the complex torque coefficient method is employed to assess the contribution of inner loops to system damping. The interactions among inner loops, outer loops, and the ac grid are analyzed . It reveal s that i nner loops shape the electrical damping torque coefficient and consequently influence both synchronous and sub -synchronous oscillation modes. The virtual admittance and current control -based inner -loop scheme is employed to illustrate the proposed analytical approach. The case study comprises the analysis of impedance profiles, the analysis of damping torque contributed by inner loops under various grid strengths, and the comparison between dq-frame and αβ-frame realizations of inner loops. Finally, simulation and experimental tests c ollaborate with theoretical approaches and findings. Index Terms —Voltage -source converters, grid -forming control, inner loops, virtual admittance, current control, active damping. I. INTRODUCTION Voltage -source converters (VSCs) are extensively employed with renewable energy sources , serving as an efficient interface to power grids. The g rid-forming (GFM) control emerges as a favorable approach for VSC -based resources [1]. Differing from traditional grid -following control that operates VSCs to follow the frequency of grid voltage [2], GFM -VSCs behave as a voltage source behind an impedance, and they autonomously regulate the frequency and magnitude of the voltage source [3]. The control system of a GFM -VSC typically comprises outer and inner loops. The outer loop employs synchronization and voltage magnitude control to generate the electromotive force (EMF) reference vector. The synchronization control is attained by regulating their active power output or their dc-link voltage [4]-[17]. A basic approach is the active power -frequency ( P-f) droop control [4], also known as the power -synchronization control (PSC) [5]. This approach can provid e damping and droop characteristics but lacks the inertia l response . To furnish the inertia l respo nse, the virtual synchronous generator control is developed [6], [7], which is mathematically equivalent to the P-f droop control with a low-pass filter (LPF) [8]. Besides the LPF, various forms of active power -frequency control lers are reported [9]-[11], such as the proportional -integral (PI), the P- derivative control, and the lead -lag controller, where the lead - lag controller can achieve flexible regulation of damping, inertia, and droop ch aracteristics [11]. When dealing with regulated dc -link dynamics in GFM -VSCs, the dc -link voltage control can synthesize the frequency directl y. Such dc -link voltage -synchronization control was developed for voltage angle control in wind turbines [12]. Then , similar control architectures are reported in [13]-[15]. While simple , this method may be subject to low -frequency instability issues when connected to dc -link constant -power loads [16]. To provide an additional degree of control for active damping, the dc-link voltage control and the PSC can be configured in cascade and parallel [5], [16], creating a flexible synchronization control [17]. Moreover, t he magnitude of EM F reference is comm only implemented through the reactive power -voltage ( Q-V) droop control [5], or the ac -bus voltage control (AVC) at the point of common coupling (PCC) [18]. Besides the outer loops, the inner loops of GFM -VSCs are of important concern. Some GFM control schemes omit the use of inner loops [5], [19], where the EMF reference vect or directly serves as the modulation -voltage vector. GFM -VSCs with this open -loop voltage control have a simpler controller design, while the overload ride -through issues under large disturbances may arise [20], [21]. Virtual impedance control is a straightforward method to regulate the output impedance under both steady state and transient operations [5], [22]-[24], which is a breed of current control. The virtual inductance or the quasi - static virtual reactance can be adopted [23], while the virtual inductance emulation involves derivative calcula tions. An alternate inner -loop scheme is the dual -loop vector voltage and current control [25]-[30]. This method has capabilities of reference tracking, current limiting, and harmonics rejection, while it poses a risk of underdamping or even instability when interacting with synchronization con trol and stiff ac grid [25], [26]. Instead , employing the grid -side current disturbance feedforward can enhance the damping effect for the stability robustness [27], [28]. Moreover, an asymmetrical vector voltage control is proposed in [29] for the damping enhancement, which introduces an integral control loop from d- axis voltage ( vd) to q-axis current reference ( iqref) to the vector voltage controller [30]. Yet, a sophisticated parameter design is required, and the damping enhancement effect is not significant under a weak grid [30]. Moreover, the virtual admittance and current control serve as another variant of inner loops [31]. This method can emulate an inductance without derivative calculations. Yet, the voltage drop on the virtual admittance reduces the PCC voltage , and the outer -loop AVC is thus required to maintain the PCC voltage magnitude around 1p.u. [32]. A recent development involves a unified voltage control [33], which synthesizes those different voltage control methods through a universal structure with varying virtual impedance specifications. However, despite the diversity of reported inner - loop schemes, there is little discussion concernin g the characterization of their dynamic behaviors. 2 The interaction s between inner and outer loops mainly result in two types of oscillations : synchronous oscillation (SO) and sub-synchronous oscillation (SSO) [2], [34]. The SO issue originates from the dynamics of the power -angle control plant [5]. The inductive power transfer impedance introduces a pair of conjugate poles at the synchronous frequency, i.e., the fundamental frequency of t he system. The basic principle of mitigating the SO issue is to increase the resistive part of the power transfer impedance between the internal -voltage phasor and the grid -voltage phasor. The inner control loops shape the output impedance of GFM -VSCs, the reby affecting the SO mode. The specific control loops for increasing the resistive effect include the virtual impedance control [5], the virtual admittance control [34], and the current control [34]. The SSO is another oscillatory mode, manifesting as oscillations below the fundamental frequency [24], [25]. For the virtual impedance control -based inner loop, a large virtual renaissance with a high -pass filter (HPF) may jeopardize the SSO, while utilizing a virtual reactance can mitigate it [24]. As for the dual -loop vector voltage and current control , the interaction between inner -loop vector voltage control and outer - loop PSC under a stiff grid may induce the SSO [25]. Increasing the voltage control bandwidth and employing an a symmetrical voltage controller can alleviate such adverse sub -synchronous control interactions [30], [35]. Moreover, when the virtual admittance and current control serve as the inner loop, the effects of inner -loop parameters on the SSO -mode damping ratio are investigated in [34] and [36] via sensitivity analysis. However, those analysis results are based on numerical methods, which cannot provide analytical or physical insight into interactions and instability issues. In [35], the effect of dual-loop vector voltage and current control is analyzed from the damping torque perspective. Yet, a quantitative assessment of damp ing is still missing. Therefore, there is still a lack of analytical assessment for the damping contribu tion of inner loops of GFM -VSCs. In this paper , an analytical approach is developed to investigate the damping effect of inner loops . It compris es the impedance -based dynamics characterization and the complex torque coefficient -based control interaction analysis. First, a n impedance model is proposed to uniformly represent the dynamics of various inner -loop schemes . The impedance - shaping effect of i nner-loop configurations and parameters is analyzed. Then , based on the impedance representation of inner loops, the complex torque coefficient method is utilized to reveal the interactions among outer loops, inner loops, and the grid impedance. The electrical damping torque quantifies the contributions of inner loops to the damping effect on both SO and SSO modes . Following this, the proposed analytical method is exemplified using the virtual admittance and current control -based inner loops. Parameter tuning guidelines are subsequently developed to enhance the system stability robustness. Moreover, the dq- and αβ-frame realizations of inner loops are com pared in terms of the damping effect. The theoretical method and analytical findings are finally validated by simulation and experimental results. II. SYSTEM DESCRIPTION Fig. 1 shows the system and control block diagram of the GFM -VSC . The single -converter infinite bus system is employed for the study , where the ac grid is represented by an ideal voltage source , gV ggV = , in series with the grid impedance Zg [37]. Such a setting is dedicated to analyzing the stability of a single converter, while it cannot cover the interaction issues of multiple converters. The VSC is connected to the PCC through the inductor filter Lf. Vpcc and ig denote the PCC voltage and grid current, respectively. The control scheme of GFM -VSC comprise s of outer and inner loops. The outer loop s generate the EMF -vector reference, including the phase ang le and magnitude. The PSC is employed for the synchronization control, where GPSC denotes the control ler, P PCC ig VpccZg ggV Outer LoopsInner Control Loops Voltage Reference GeneratorErefVpccigLf ErefVmag PP Q V mag calculationInner Loops VrefVmag V0 P PrefGPSC-+ -+++ ++ ivKs1/s w1 PSCAVC Fig. 1. System and control block diagrams of GFM -VSC s. (a) (b) (c) Fig. 2. Three types of inner -loop configurations. (a) Virtual impedance control. (b) Vector voltage and current control. (c) Virtual admittance and current control. Virtual impedance (VI)PWM Eref ig Vector voltage control (VVC)Current control (CC)PWM Vpcc igEref Virtual admittance (VA)Current control (CC)PWMEref ig Vpcc 3 and Pref denote the active power and its reference, respectively. The magnitude of the EMF vector is regulated by the AVC , which is intended to regulate the PCC voltage magnitude with zero steady -state error [5]. Vmag and Vref denote the PCC voltage magnitude and its r eference, respectively. Kiv denotes the integral gain of AVC. Fig. 2 shows three types of inner -loop configurations that are widely used in GFM -VSCs. 1) Virtual impedance control, as shown in Fig. 2(a) . 2) Dual -loop vector voltage and current control, as shown in Fig. 2( b). 3) Virtual admittance and current control, as shown in Fig. 2(c). III. IMPEDANCE -BASED DYNAMIC CHARACTERIZATION This Section employs an impedance model to characterize the dynamics of inner loops. The impedance shaping effect of inner loops is analyzed, which utilizes the virtual admittance and current control as a typical case. A. Basic Principle and Derivation Fig. 3 illustrates the basic principle of the impedance -based meth odology. Fig. 3(a) shows the GFM -VSC with studied inner loops. The outer lops generate the EMF -vector reference, Eref, via the voltage reference generator . Afterward, inner loops process Eref and generate the modulation -voltage reference. In contrast, Fig. 3 (b) shows the GFM -VSC with impedance -based representation of inner loops. The inner loops are omitted, while an impedance Zeq is utilized to represent the dynamics of inner loops. It can be observed from the comparison that inner loop s result in the shaping of Lf into Zeq. Fig. 4 shows the derivation process of Zeq. The control block diagram and derivation are performed in the αβ-frame. Stage 1 shows the i nitial control block diagram of inner loops. It is a uniform representation of different inner -loop configurations , as shown in Fig. 2 , including the voltage control, current control, and voltage decoupling control. Fvc denotes the flag of the voltage control, which can be zero or one. Gv denotes the voltage controller, where different forms can denote the vector voltage control and virtual admittance control . Fcc denotes the flag of current control, which can be unity gain or the virtual impedance. Gi denotes the current controller . Fv denotes the flag of voltage decoupling control, which can be zero or the LPF employed in voltage decoupling control . Gd denotes the time delay . Stage 2: According to the Block diagram algebra and the Superposition theorem [38], [39], the controlled voltage source can be decomposed into three series controller voltage sources, generated by Eref, Vpcc, and ig, respectively. ( ) ( ) ( )i i i i i i pcc igU U U U UE UV Uid i v d v d i v vc d i ccG G G G F G G G F G G F= + + = = - = - 1 2 3 1 ref 2 3 (1) Stage 3: According to Ohm ’s law [38], the current -controlled voltage source Ui3 can be represented with a virtual impedance, given by Z d i ccG G F= 1 (2) Stage 4: According to Norton’s theorem [38], a voltage source in series with impedance s can be equivalently represented with a current source parallel with admittance s. i i1 i i2 pccUiEZZ UiVZZff ffd i v d v d i v vcG G G sL sL G F G G G F sL sL = = ++ - = = ++ 1 ref 11 2 11 (3) Stage 5: According to Ohm’s law [38], the voltage - controlled current source ii2 can be represented with a virtual admittance , given by ( ) 1YZ fd i v vc vG G G F F sL -=+1 . (4) Stage 6: According to Thevenin’s theorem [38], a current source in parallel with admittance s can be equivalently represented with a voltage source in series with impedance s. The final representation of Zeq is given by ( )eqZf 1d i cc d i v vc vG G F sL G G G F F += - + . (5) The loop gain, from Eref to the modulation voltage, is given by ( )1d i v eq d i v vc vG G GGG G G F F= - + . (6) ig Lg Lf PCC+ -Voltage Reference Generat or Vpcc ErefVgOuter LoopsErefInner Loops Vpccig (a) ig Lg PCC+ -Voltage Reference Generat orZeq Outer Loops ErefErefVpcc Vg (b) Fig. 3. Basic principle of the impedance -based method . (a) GFM -VSC with studied inner loops. (b) GFM -VSC with impedance representation of inner -loop dynamics. 4 B. Case Analysis with Virtual Admittance -Current Control The virtual admittance - and current control -based inner loop is used for analysis, which is implemented in the αβ-frame . In such configuration , the parameters of Fig. 4 are settled as: Fvc=1 and Fcc=1. The voltage controller denotes the virtual admittance control, given by 1 v v R vGsL G R=+ . (7) Lv and Rv denote the virtual inductance and virtual resistance, respectively. In order to maintain a low R/X ratio of power transfer impedance, Rv is employed with a notch filter GR at the fundamental frequency [5]. Such a notch filter becomes a n HPF when the virtual admittance control is implemented in the dq- frame [24]. The current control employs an αβ-frame quasi -PR controller [40], given by 22 1 2ri i pi rksGkss ww=+++ . (8) kpi denotes the P gain. kri denotes the R gain and is equal to 0.4p.u. wri denotes the resonant cut -off frequency, which is utilized to mitigate the sensitivity of the R controller against the variation of fundamental frequency and is equal to 2/rad s . Note that the controller gain at fundamental frequency is reduced by wri, while it is still large enough for reference tracking. Moreover, the timed delay can be neglected when analyzing low -frequency dynamics, i.e., Gd=1. Under his assumption , the loop gain Geq is equal to 1. In the frequency domain, the impedance can be represented as the real and imaginary parts, denot ing the resistance and the reactance, respectively. Figs. 5 -7 present s the impedance profiles shaped by inner -loop virtual admittance and current control , where different parameters are tested. It is known that the resistance component can dampen the dc components (or ultra-low frequency ac components) in ac voltage and current [38]. Such issue typically manifest s as fundamental -frequency oscillations in the active and reactive power, namely the SO [5]. Thus, the low -frequency resistance part of Zeq is utilized to reflect the damping on the SO mode. Fig. 5 shows the impedance profiles with different P gain of current control . Rv=0.1p.u. and Lv=0.3p.u. are adopted , and the voltage decoupling loop is disabled ( Fv=0). It is shown that the low-frequency Req increases as kpi increases from 0.5p.u. to 3p.u., which, however, remains below the value of Rv. Further, Zeq is equal to the value of virtual admittance if kpi is infinite, given by piv v RksL R G→+= + eqZ , (9) The results indicate that the current control interacts with virtual admittance control , resulting in the negative resistance effect in the Zeq. Increasing the proportional gain of current control can partially alleviate such a negative resistance effect. Fig. 6 shows the impedance profiles with and without the voltage decoupling control. The voltage decoupling control is denoted by Fv=0 (disabled) and Fv=1 (enabled). The results reveal that enabling voltage decoupling control leads to an increase in the low -frequency Req, approaching the value of Rv. Consequently, voltage decoupling control effectively mitigates the low -frequency negative resistance effect resulting from the interaction between virtual admittance and cu rrent control . Fig. 7(a) shows the impedance profiles with different Rv. An increase in Rv from 0.08 p.u. to 0.12 p.u. is associated with an augmentation of the low -frequency Req, indicating a positive Gi +-+-Gv ++ FvGd PCC+ - Stage 1 Ui Vpccig Lf Eref Vpcc Vpcc igirefCurrent ControlVoltage Decoupling Fcc FvcVoltage Control GdGiGv+ - PCC+ - + -Gd(Fv-GiGvFvc) -GdGiFccUi1 Ui2 Ui3Eref Vpcc igLf Vpccig Stage 2 GdGiGv PCCGd(Fv-GiGvFvc)Z1 = GdGiFcc + - + -Ui1 Ui2Eref Vpccig VpccLf Stage 3 Gd(Fv-GiGvFvc)GdGiGv1 (Z1+sLf) 1 (Z1+sLf) PCCZ1+sLfii1 ii2Eref Vpccig Vpcc Stage 4 PCCZ1+sLfii1ErefVpccig GdGiGv (Z1+sLf) Stage 5 (Z1+sLf) Gd(GiGvFvc-Fv) Geq+ - PCCZeq ErefVpccig Stage 6 Fig. 4. Derivation of the impedance model . 5 damping effect. The low -frequency Req is increased as Rv increases from 0.08p.u. to 0.12p.u., indicating the positive damping effect. Fig. 7(b) shows the impedance profiles with different Lv. The Lv has minimal impact on the low -frequency resistance and little effect on the SO -mode dynamics . IV. COMPL EX TORQUE COEFFICIENT -BASED INTERACTION ANALYSIS This Section uses the complex torque coefficient method to reveal the interaction among inner loops, outer loops, and the ac gri d. The complex torque profiles are analyzed to quantify the damping effect of inner loops on SO and SO modes. A. Basic Principle and Stability Criteria The complex torque coefficient method was initially developed to investigate torsional interactions in synchronous - machine infinite -bus system s [41]. The mechanical and electrical torques characteri ze the dynamics of the electromechanical (rotor) and the electrical (stator) subsystems, respectively. Fig. 8 illustrates the mechanical and electrical torques i n the frequency domain . The real component , aligned with the phase angle θ, represents the synchronous torque . The imaginary component , aligned with frequency ω, denotes the damping torque. Mathematical expression s of the complex torque are given by ( ) ( ) ( ) ( ) ( ) ( )mm eej K j D j K j Dw w w w w w w w = + = + m eT T . (10) Tm and Te denote the mechanical and electrical torque with complex form. Km and Ke represent the mechanical and electrical synchronizing torque coefficient. Dm and De denote the mechanical and electrical damping torque coefficient. Given the power/ energy balance between mechanical and electrical subsystems , the equation to capture the interaction is given by ( ) ( ) ( ) ( ) 0m e m eK K j D Dw w w w w+ + + = . (11) The real part is the sum of synchronizing torques, i.e., ( ) ( ) 0meKKww+= , which determines the oscillation frequencies [42]. The imaginary part is the net damping ; and its value at the oscillatory frequencies determines the stability [43], i.e. 0 Stable 0 Unstableme meDD DD+ + (12) The advantage of this stability analysis method lies in that the contribution of mechanical and electrical subsystems to the system damping can be quantified clearly. B. Application of Complex Torque in GFM -VSCs Fig. 9 shows the dq-frame circuit diagram with impedance representation of inner loops , which is equivalent to the αβ- frame circuit diagram . The dq-frame impedance can be derived according to the frequency translation [44], given by 0.060.080.10.120.14 0.250.30.35 Frequency (Hz)10110010-1102Leq (p.u.) Req (p.u.) kpi = 1.0 p.u.kpi = 3 p.u. kpi = 0.5 p.u. Fig. 5. Impedance profiles with different P gains of current control . Lv=0.3p.u., Rv=0.1p.u., Fv=0. 0.060.080.10.12 0.250.30.35 Frequency (Hz)10110010-1102Leq (p.u.) Req (p.u.)With VD control ( Fv = 1) Without VD control ( Fv = 0) 0.05 Fig. 6. Impedance profiles with and without the voltage decoupling control. Lv=0.3p.u., Rv=0.1p.u., kpi=2p.u . 0.050.10.15 0.250.30.35 Frequency (Hz)10110010-1102Leq (p.u.) Req (p.u.)Rv = 0.12 p.u. Rv = 0.10 p.u. Rv = 0.08 p.u. (a) 0.040.060.080.10.12 0.10.20.30.40.5 Frequency (Hz)10110010-1102Leq (p.u.) Req (p.u.) Lv = 0.4 p.u. Lv = 0.3 p.u. Lv = 0.2 p.u. (b) Fig. 7. Impedance profiles with different parameters of virtual admittance . kpi=2p.u., Fv=1. (a) with different Rv. (b) with different Lv. 6 () ( ) dq eq_ αβ ZZ 1 s s j w =+ (13) () () ()() () () ()dqZdq dq qdZ s Z ss Z s jZ sZ s Z s -= + (14) The active power flowing through the PCC can be calculated according to the instantaneous power theory , given by ( )( )( ) ( )( )2 q gq d gd 22 d gd q gqsin cosggEV Z Z E EV Z Z P Z Z Z Z + + - + = + + + (15) The active power versus phase angle , under small -signal perturbation s, is expressed as ( ) ( ) ( )( )q gq 0 gd d 0 0 g022 d gd q gqcos sin Z Z Z ZP E V Z Z Z Z+ + - = + + + PθG (16) E0, Vg0, θ0 denote the operating point of EMF magnitude, grid voltage magnitude, and the power angle. It is clear from the equation that the power -angle dynamics are affected by the inner loops, the grid impedance, and the operating point. Fig. 10 shows the closed -loop control block diagram of the GFM -VSC in the frequency domain. GPSC represents the synchronization dynamics . GP includes the inner -loop dynamics and the grid impedance, as illustrated in (16). It is clear that 1 PSCG and GP correspond to mechanical and electrical torque coefficients in (10), respectively. The power - angle relationship is rewritten as () ()PSC PθG G1( ) ( ) ( ) ( )m m m e e eP K j Dj P j K j D w w w w w w w w = = + = = + (17) The control interaction can thus be analyzed through the frequency response of 1 PSCG and GP. Moreover, w hen integrating the dynamics of AVC into the complex torque coefficient , the control plant of active power and voltage magnitude is shown in Fig. 11. The transfer functions GP, GPE, GV, and GVE can be calculated through the circuit diagram Fig. 9. The transfer function from phase angle to active power can be reformulated by selecting the phase angle as the input signal and active power as the output signal , given by Pθ PE Vθ VEG G GG 1iv ivKsPKs = - + (18) (a) Tm w Synchronizing torque Damp ing torque KmDm (b) Te w Synchronizing torque Damp ing torque KeDe Fig. 8. Decomposition of complex torque. (a) Mechanical torque. (b) Electrical torque. Zgdq PCC+ -Zdq Vdq Vgdqidq dqEcos sinE E = Fig. 9. Circuit diagram with impedance representation of inner loops in the dq- frame. Pref P+- PmSynchronization control dynamics Inner -loop dynamics and grid impedance () PSCG jw () PθG jw Fig. 10. Closed -loop control block diagram of the GFM -VSC. Vref+- E++++ Vmag P VθG PθG PEG VEG ivKs Vmag Fig. 11. Active power and voltage magnitude control plants. TABLE I. CIRCUIT AND CONTROL PARAMETERS FOR CASE STUDIES . Description Values Grid inductance Lg Stiff grid: Lg=0.05 p.u. (SCR=20) Weak grid: Lg=0.8 p.u. (SCR=1.25) Outer -loop PSC: ( ) PSC psc p pG K s ww = + Case 1: Kpsc=0.1p.u., wp=3Hz Case 2: Kpsc=0.1p.u., wp=100Hz Virtual resistance Rv 0.05p.u. to 0.1p.u. (With a 5Hz HPF in dq-frame) Virtual inductance Lv 0.1p.u. to 0.3p.u. Current control proportional gain kpi 0.5p.u. to 2p.u. 7 C. Case Analysis with V irtual Admittance -Current Control Table I shows the circuit and control parameters for the case study. It is noted that the 100Hz LPF in the PSC loop is employed to mitigate noises in power calculation, while 3Hz LPF is the case of inertia emulation. Additionally, the αβ-frame notch filter employ ed with Rv is replaced with an HPF in the dq-frame [5], [24]. Fig. 12 shows complex torque profiles with different grid impedances , where Lv=0.1p.u., Rv=0.1p.u., kpi=2p.u., and PSC employs a 3Hz LPF. The damping torque De with Lg=0.8p.u. is larger than that with Lg=0.05p.u. It indicates that the net damping torque emDD+ is prone to be negative under stiff grids. The results provide an analytical interpretation that the GFM -VSC exhibits a higher instability risk under stiff grids . Therefore, the following case st udies are conducted under the grid condition Lg=0.05p.u. Fig. 13 shows complex torque profiles with different virtual resistance Rv, with Lv=0.1p.u. and kpi=2p.u. In Fig. 13(a), the synchronizing torques Ke and Km intersect in the frequency range of the SSO mode. When Rv=0.05p.u., the damping torque relationship at the interaction frequency is emDD- , thereby leading to a positive net damping and a stable operation. In contrast, when Rv change s to 0.1p.u., the damping torque relationship becomes em 0 DD+ , indicating the SSO issue. The comparative results show that increasing Rv jeopardizes the damping torque of the SSO mode. In Fig. 13(b), the synchr onizing torques Ke and Km intersect at the fundamental -Km -DmKe (Lg=0.05p.u. ) Ke (Lg=0.8p.u. ) De (Lg=0.05p.u. )De (Lg=0.8p.u. ) Frequency (Hz)101100102-5 -0.05051015 0Synchronizing torque (p.u.)Damping torque (p.u.) Fig. 12. Complex torque profiles with different grid impedances . Frequency (Hz)101100102-KmKe (Rv=0.05p.u. ) Ke (Rv=0.1p.u.) De (Rv=0.05p.u. ) De (Rv=0.1p.u.) -Dm-5 -0.05051015 0Synchronizing torque (p.u.)Damping torque (p.u.) (a) -Km Ke (Rv=0.05p.u. ) Ke (Rv=0.1p.u.) De (Rv=0.05p.u. )De (Rv=0.1p.u.)-Dm Frequency (Hz)101100102-5 -0.05051015 0Synchronizing torque (p.u.)Damping torque (p.u.) (b) Fig. 13. Complex torque profiles with different Rv. (a) Outer -loop PSC with 3Hz LPF. (b) Outer -loop PSC with 100Hz LPF. -Km -DmKe (Lv=0.2p.u. ) Ke (Lv=0.1p.u.) De (Lv=0.2p.u. )De (Lv=0.1p.u.) Frequency (Hz)101100102-5 -0.05051015 0Synchronizing torque (p.u.)Damping torque (p.u.) Fig. 14. Complex torque profiles with different Lv. -Km -DmKe (kpi=2p.u. )Ke (kpi=0.5p.u.) De (kpi=2p.u. ) De (kpi=0.5p.u.) Frequency (Hz)101100102-5 -0.05051015 0Synchronizing torque (p.u.)Damping torque (p.u.) (a) Frequency (Hz)101100102-5 -0.05051015 0Synchronizing torque (p.u.)Damping torque (p.u.)-Dm-Km De ( Fv = 1 ) De ( Fv = 0 )Ke ( Fv = 1 )Ke ( Fv = 0 ) (b) Fig. 15. Complex torque profiles with different parameters of current control and voltage decoupling control. (a) with different P gains of current control . (b) with and without the voltage decoupling control. 8 frequency, refer ring to the SO mode. As Rv increases from 0.05p.u. to 0.1p.u., the net damping torque at synchronous frequency changes from negative to positive, thereby enhancing the damping effect on the SO mode. The results in Fig. 13 demonstrate the trade -off of Rv between SSO -mode and SO - mode damping torque. Fig. 14 illustrate s complex torque profiles with different virtual inductance Lv, where Rv=0.1p.u., kpi=2p.u., and PSC adopt s a 3Hz LPF. It is shown that a s Lv transitions from 0. 1p.u. to 0.2p.u., the net damping torque at the synchronous frequency shifts from negative to positive, implying the advantageous damping effect of Lv on the SSO mode. Consequently , the Lv can be utilized to mitigate the adverse damping effect of Rv on the SSO mode. Moreover, Lv has little effect on the SO -mode damping torque, which changes a little under the variation of Lv. Fig. 15 shows complex torque profiles with different parameters o f current control and voltage decoupling control. Lv=0.1p.u., Rv=0.1p.u., and PSC adopts a 3Hz LPF. Increasing current control P gain kpi and employing the voltage decoupling control ( Fv=1) can enhance the damping torque of the SO mode , while slightly reducing the damping torque for the SSO mode. Yet, c ompared with virtual admittance control, the effect of current control and voltage decoupling control are insignificant . In light of the impedance -based dynamic characterization and the complex torque coefficient -based interaction analysis, the damping effects of the inner -loop virtual admittance and current control , along with their design guidelines, are formulated as follows: 1) Current co ntrol with high bandwidth is recommended to alleviate its adverse interaction with virtual admittance control , thereby enhancing damping on the SO mode. Moreover, the voltage decoupling control should be employed to mitigate the adverse impact of interacti ons between virtual admittance control and current control . 2) Virtual resistance Rv enhances the SO -mode damping but compromises the SSO -mode damping. Consequently, the design of Rv should prioritize the damping effects on the SO mode, namely, by increasing the value of Rv to mitigate the SO issues. 3) Virtual inductance Lv enhances the SSO -mode damping and little impacts the SO -mode damping. Hence, the design of Lv is dedicated to attenuating the SSO issue, through which the adverse effect of Rv on SSO -mode damping can also be mitigated. When Rv=0.1p.u. , Lv≥0.2p.u., and kpi=2p.u. , the damping for both SO and SSO modes can be guaranteed, as shown in the blue line in Fig. 14 . Moreover, it is noted that parameter tuning of inner loops needs to be coordinated with the outer loop. Inner loops shape the output impedance and the electrical damping torque, providing a stable range for outer -loop parameters to be adjusted. V. COMPARISON OF DIFFERENT REFERENCE -FRAM E REALIZATION S OF INNER LOOPS The different reference -frame re alizations of inner loops are also a critical concern to affect the inner -outer loop interactions and the system damping . This Section compares the control interactions with αβ-frame and dq-frame realizations of inner loops. The different coupling dynamics of those inner -loop realizations are illustrated via small -signal model ; and their damping effects are characterized through the closed -loop poles . Gv_ab Gi_ab +-+++- ababc ab abcFv ab abcab abcVabdqab Edq Eab αβrefi Vpcc VpccUi iab igGv_dq Gi_dq +-+++- Fv VpccVdqdqabc abcdq abcdq abcdqidq ig VpccUi dqrefi Edq (a) (b) Fig. 16. Different reference frame realizations of inner loops . (a) αβ-frame realization . (b) dq-frame realization. Vref+++-+-Gv_dq EG 1T- ++ ( )1 fgZZ-+ gZ gZ vF gZ u VpccGAVCInner Loops ˆwPref GPSC+- 1/sPSC GP dG ˆs dqE ˆc dqE ˆ +-Gi_dq ivKs ˆs dqi ˆs dqi ˆs dqi ˆs dqi ˆs dqi ˆs dqu ˆs dqu ˆs dqu s idqUˆ ˆP magˆV s dqrefiˆ Fig. 17. Small-signal model of the GFM -VSC with αβ-frame realization of inner loops . 9 A. Small -Single Model with αβ-Frame and dq -Frame Realizations of Inner Loop s Fig. 16(a) shows the αβ-frame realization of inner loops. dqE ref0TE= that is generated by outer loops is processed by dq-αβ transformation to generate the αβ-frame reference Eab. The inner -loop control is implemented with αβ-frame variables Eab , Vab , and iab. Fig. 16(b) shows the dq-frame realization of inner loops. Differently, the inner -loop control is implemented with αβ-frame variables Edq, Vdq, and idq. It can be observed from the comparison that the phase angle used in abc-dq and dq-abc transformations result s in different coupling dynamics with the outer -loop synchronization control. Fig. 17 show s the small -signal model of the GFM -VSC with αβ-frame realization of inner loops . The detailed derivation s of the matrix are presented in the Appendix [17], [34], [45]. The coupling loop ① denotes that the phase angle dynamic participates in the dq-αβ transformation of the EMF vector Eab . Fig. 18 show s the small -signal model of the GFM -VSC with dq-frame realization of inner loops . The detailed derivations of the matrix are presented in the Appendix [17], [34], [45]. The phase angle dynamic is coupled with the inner loop through four loops . 1) Loop ② denotes the abc-dq transformation of the PCC- voltage vector Vpcc. 2) Loop ③ denotes the abc-dq transformation of the grid- current vector ig. 3) Loop ④ denotes the abc-dq transformation of the PCC- Vref+-+- ˆ ˆwPref ++GPSC+- ++ ( )1 fgZZ-+ vF1/s u VpccG dG - - +PSC AVC GP 1T-Inner Loops uG iG EcG ˆc dqE Gv_dq +- vuFGGi_dq ivKs gZ gZ gZ ˆs dqu ˆs dqu ˆs dqu ˆs dqi ˆs dqi ˆs dqi ˆs dqi ˆs dqi s idqUˆ ˆP magˆV c dqrefiˆ Fig. 18. Small-signal model of the GFM -VSC with dq-frame realization of inner loops . -400-300-200-1000100200300400Imag. axis (seconds-1) -140 -120 -100 -80 -60 -40 -20 0 Real axis (seconds-1)dq-frame inner -loop ab-frame inner -loop -400-300-200-1000100200300400Imag. axis (seconds-1) -50 -40 -30 -20 -10 0 Real axis (seconds-1)dq-frame inner -loop ab-frame inner -loop (a) (b) -140 -120 -100 -80 -60 -40 -20 0-400-300-200-1000100200300400Imag. axis (seconds-1) Real axis (seconds-1)dq-frame inner -loop ab-frame inner -loop -400-300-200-1000100200300400Imag. axis (seconds-1) -50 -40 -30 -20 -10 0 Real axis (seconds-1)dq-frame inner -loop ab-frame inner -loop (c) (d) Fig. 19. Closed -loop poles of transfer function model. (a) Inverter mode with Lg=0.05 p.u . (b) Inverter mode with Lg=0.8 p.u. (c) Rectifier mode with Lg=0.05 p.u . (d) Rectifier mode with Lg=0.8 p.u. 10 voltage vector Vpcc. 4) Loop ⑤ denotes the dq-abc transformation of the modulation -voltage vector Ui. B. Comparative Analysis of Coupling Dynamics The difference between αβ-frame and dq-frame realizations of inner loops lies in the coupling dynamics as depicted in loops ①-⑤. The matrix of loop ① in Fig. 17 and the matrix of loop ② in Fig. 18 are rewritten as 00 00ss qq Euss ddEVGG EV --= - = , (19) where 0s dE and 0s qE denote the steady -state operating point of the EMF vector. 0s dV and 0s qV represent the steady -state operating point of the PCC -voltage vector. It can be observed from the electrical circuit that the EMF vector and PCC -voltage vector are comparable, near to 1p.u. Therefore , loop s ① and ② demonstrate comparable dynamics. The loops ④ and ⑤ in Fig. 18 can be combined when the voltage decoupling control is enabled ( Fv=1), given by 00 00ss q iq u Uiss id dVUGG UV -+= - , (20) where 0s idU and 0s iqU denote the steady -state operating point of the modulation -voltage vector. Since the modulation -voltage vector an d the PCC -voltage vector are comparable in the electrical circuit, near 1p.u., the dynamics of loops ④ and ⑤ can be cance lled. Therefore, the difference between αβ-frame and dq-frame realizations of inner loops is primarily attributed to the coupling dynamics represented by loop ③, introduced by the abc-dq transformation of the grid-current vector ig. The matrix of loop ③, iG , is represented by (41), which compromise s the operating point of dq-frame current . The d - axis current 0s dI is a positive value under invert er mode and a negative value under rectifier mode. Therefore, the dynamic effect of the loop ③ is different under inverter and rectifier modes. Fig. 1 9 shows the closed -loop poles with αβ- and dq-frame realizations of the inner loops. The results show that in the inverter operation mode, dq-frame realization demonstrates better damping for the SSO mode, while poorer damping for the SO mode. Conversely, in the rectifier operation mode, dq-frame realization exhibits poorer damping for the SSO mode, while better damping for the SO mode. However , this disparity is less apparent compared to the impact of control parameter variations. The inner loops can achieve favorable damping effects in both αβ- and dq-frame realizations by employing the robust parameter design presented in Section I V. VI. SIMULATI ON AND EXPERIMENTAL VERIFICATION Fig. 20 shows the experimental setup. The voltage source of ac grid is generated by the grid simulator. The VSC is connected to the grid through inductance filters and grid impedances. The control strategy is performed in the dSPACE -1007 platform . The main system constants and controller parameters are presented in Table II. Fig. 21 shows the measured impedance profiles of inner loops, which are conducted via the frequency scan approach [46]. The impedance profiles are measured by disabling outer loops , thereby focusing only on the inner -loop dynamics. Fig. 21(a) compares the impedance profiles with and without the voltage decoupling control. The results demonstrate that the voltage decoupling control leads to an increase in the low - frequency resistance component. Fig. 21(b) presents impedance profiles with different Rv. It is shown that a large Rv results in an increased resistance component in the low -frequency range while having little impact on the inductance component. Fig. 21(c) compares impedance profiles with different Lv. It is observed that a large Lv corresponds to an increased inductance component with little impact on the resistance component. The measured impedance profiles validate the effectiveness of the proposed impedance models prese nted in Section III. Zg Vg igPWMGrid simulator Converter VpccLf dSPACE -1007 OscilloscopePWM Vg Vpcc ig DS2102 D/A DS2004 A/D Fig. 20. Experimental setup. TABLE II. PARAMETERS FOR SIMULATION AND EXPERIMENTAL TEST Symbol Description Value (p.u.) Pn Rated power 3kW (1 p.u.) Vg Rated voltage (L -G, RMS) 110V/50Hz (1 p.u.) w1 Nominal angular frequency 100 (1 p.u.) Lg Grid impedance 2mH (0.05 p.u.) Lf Filter inductance 4mH (0.1 p.u.) V1 Rated EMF magnitude 110V (1 p.u.) KPSC PSC proportional gain 0.1 w1/Pn (0.1 p.u.) Kiv A VC integral gain 50 (50 p.u.) Lv Virtual inductance 0.1-0.3 p.u. Rv Virtual resistance 0.05-0.12 p.u. kpi Current control proportional gain 0.5-3 p.u. kii Current control integral gain 0.2 p.u. kri Current control resonant gain 0.4 p.u. Td Control time delay 150 s 11 Fig. 22 shows the measured complex torque coefficient of inner -loop virtual admittance and current control, corresponding to the transfer function from p hase ang le to active power GP. The frequency scan approach is utilized for the measurement. The results indicate that increasing Rv enhances damping torque for the SO mode but diminishes it for the SSO model. Conversely, a larger Lv enhances the damping torque for the SSO mode but exhibits little influence on the damping torque of the SO mode. Th ose measured complex torque profiles confirm the validity of the theoretical complex torque models outlined in Section I V. Fig. 23 shows experimental wave forms of SO -mode dynamics with virtual admittance and current control -based t1 t2 Vpcca [1 p.u./div] iga [1 p.u./div] P [0.5 p.u./div] Q [0.5 p.u./div]Kp = 3p.u. Kp = 0.5p.u. Kp = 3p.u. 20ms (a) t1 Fv = 1 Fv = 0 [200 ms/div] Vpcca [1 p.u./div] iga [1 p.u./div] P [1 p.u./div] Q [1 p.u./div] (b) t1 t2Rv = 0.1p.u. Rv = 0.05p.u. Rv = 0.1p.u. Vpcca [1 p.u./div] iga [1 p.u./div] P [1 p.u./div] Q [1 p.u./div][400 ms/div] (c) Fig. 23. Experimental waveform of SO -mode dynamics . (a) With different P gains of current control . (b) Comparison with and without the voltage decoupling control. (c) With different Rv. 0.040.060.080.10.12 0.20.30.4 Frequency (Hz)101100102Leq (p.u.) Req (p.u.)w/o VD control ( Fv = 0) w/ VD control ( Fv = 1) (a) 00.050.10.15 0.250.30.35 Frequency (Hz)101100102Leq (p.u.) Req (p.u.) Rv = 0.08p.u. Rv = 0.10p.u. Rv = 0.12p.u. (b) 0.040.060.080.10.12 0.10.20.30.40.5 Frequency (Hz)101100102Leq (p.u.) Req (p.u.)Lv = 0.2 p.u. Lv = 0.3 p.u. Lv = 0.4 p.u. (c) Fig. 21. Measured impedance profiles of inner -loop virtual admittance and current control . (a) Comparison with and without the voltage decoupling control. (b) With different Rv. (c) With different Lv. Frequency (Hz)101100102-5051015 -0.08-0.06-0.04-0.0200.02Synchronizing torque (p.u.)Damping torque (p.u.)Rv=0.10p.u.; Lv=0.1p.u. Rv=0.05p.u.; Lv=0.1p.u. Rv=0.10p.u.; Lv=0.2p.u. Fig. 22. Measured synchronizing and damping torques of inner -loop virtual admittance and current control . 12 inner loops. Wavefor ms with different current control P gain kpi are compared in Fig. 23(a) . It is shown that the system becomes unstable as kpi switches from 3p.u. to 0.5p.u. at t1. A dc component is exhibited in the ac current waveform , and SO issues arise in active and reactive power waveforms. However , such issues are mitigated as kpi is switched back from 0.5p.u. to 3p.u. at t2. It can be concluded that a large current control P gain has a beneficial damping effect on the SO mode , aligning with the theoretical analysis in Fig. 5 . Fig. 23(b) shows the SO -mode dynamics with different virtual resistances Rv. Switching Rv from 0.1p.u. to 0.05p.u. at t1 results in the instability of the GFM -VSC, which is characterized by a dc component in the ac current waveform and SO issues in active and reactive power waveforms. Such SO issues are mitigated as Rv is switched back from 0. 05p.u. to 0.1p.u. at t2. The results validate the beneficial damping effect of Rv on the SO mode , which is consistent with the theoretical analysis in Fig. 7 . Fig. 23(c) compares the SO -mode dynamics with and without the voltage decoupling control. In this validation, the PSC proportional gain KPSC is selected as 0.17p.u., leading to a worse damping effect on the SO mode [34]. It can be observed that the GFM -VSC becomes unstable when disabling the voltage decoupling control, where a dc component is exhibited in the ac current waveform , and synchronous -frequency oscillations arise in active and reactive power waveforms. The results validate the benef icial damping effect of the voltage decoupling control on the SO mode. , which is aligned with the theoretical analysis in Fig. 6 . Fig. 24 shows experimental waveforms of SSO -mode dynamics, where the GFM -VSC operates under the inverter mode with Lg=0.1p.u. It is shown that the system becomes unstable as Rv changes from 0.1p.u. to 0.15p.u. at t1. This instability manifests as low -frequency os cillations in voltage, current, and both active and reactive power waveforms. Subsequently, Lv switches from 0.1p.u. to 0.2p.u. at t2, mitigating those SSO issues . The results validate the adverse damping effect of Rv and the beneficial damping effect of Lv on the SSO -mode dynamics , aligning with the theoretical analysis in Fig. 13 and Fig. 14. Fig. 25 shows the simulated step response of the active power . The αβ-frame and dq-frame realizations of inner loops are compared under both rectifier and inverter modes. When the active power reference is subject to a step change from 1p.u. to 1.1p.u., the GFM -VSC with dq-frame inner loops demonstra tes a slightly smaller overshoot compared to αβ-frame inner loops. In contrast , during the rectifier mode, αβ-frame inner loops result in a slightly smaller active power overshoot than dq- frame inner loops. The comparative results validate the theoretical damping analysis for the SSO mode presented in Fig. 19. VII. CONCLUSION This paper has developed an analytical method to assess the dynamics behaviour and damping effects of inner loops for GFM -VSCs. 1) An impedance model at the ac output of the GFM -VSC can uniformly characterize the behavior of various inner loops . The configuration and parameters of inner loops can shape the output impedance, subsequently influencing system dynamics. 2) The complex torque coefficient -based interaction analysis reveals that the net damping , contributed by outer loops, inner loops, and gri d impedance, determines the system stability . Inner loops shape the electrical damping torque and subsequently influenc e the dynamics of both SO and SSO modes . 3) A study of virtual admittance and current control reveals that increasing the current control bandwidth and employing the voltage decoupling control are beneficial to Vpcca [1 p.u./div] iga [1 p.u./div] P [1 p.u./div] Q [1 p.u./div] [400 ms/div]t1 t2Rv: 0.1p.u. → 0.15p.u. Lv: 0.1p.u. → 0.2p.u. Fig. 24. Experimental waveform of SSO -mode dynamics. 0.9511.051.11.151.2 0 0.2 0.4 0.6 0.8 1 1.2Active power (p.u.) Time (s)dq-frame inner -loop ab-frame inner -loopReference1.165 1.162 (a) 0 0.2 0.4 0.6 0.8 1 1.2-1.2-1.15-1.1-1.05-1-0.95Active power (p.u.) Time (s)-1.158 -1.165dq-frame inner -loop ab-frame inner -loopReference (b) Fig. 25. Simulation s tep response of active power with αβ-frame and dq-frame realization s of inner loops . (a) Inverter mode. (b) Rectifier mode. 13 enhanc ing low-frequency resistance. Rv enhances the SO- mode damping but reduces the SSO-mode dampin g. In contrast, Lv enhances the SSO-mode damping with little impact on the SO mode. Additionally, in the inverter mode, the dq-frame realization of inner loops shows slightly better damping for SSO mode and slightly reduced damping for SO mode compared to the αβ-frame realization, but this trend reverses in the rectifier mode. APPENDIX A. Modeling of Outer -Loop Dynamics The synchron ization dynamics result in two dq-frames, namely system and controller dq-frames. The PCC voltage aligns with the system dq-frame, with variables denoted by superscript “s”. The EMF vector aligns with the controller dq- frame, with variables denoted by a su perscript “c”. Steady -state operating points are denoted by a subscript “0”. Small-signal perturbations are represented using a superscript “ ˆ”. The ac circuit dynamic is represented by ( ) ( )ˆ ˆgZ + - = +s s s s s dq0 dq gdq0 dq0 dqV u V I i (21) ˆˆgZ=ss dq dqui (22) The grid impedance is given by 1 1gg g ggsL LZL sLw w- = (23) The active power dynamics are represented by ss dq dqui 0 0 0 0ˆ ˆ ˆ ui PPs s s s d q d q GGP I I V V = + (24) s dqiˆˆ ,iu P P P P g P G G G G Z= = + (25) The synchronization dynamics are given by ( ) refˆ ˆPSCGPPs= - . (26) The PCC voltage magnitude dynamics are given by ()()s dqu00 mag22 00ˆ ˆ,ss dquu Vpcc Vpcc ss dqVV V G G VV= = + (27) The dynamic of AVC is given by ( ) ref magˆˆivKE V Vs= - . (28) B. Modeling of αβ-Frame Realizat ion of Inner Loops For the αβ-frame realization of inner loops, the EMF vector is first generated through the dq-αβ transformation, given by ( )( )s s c c dq0 dq dq0 dqE E E E0ˆˆˆje++ = + , (29) sc dq dqEE0 1 0ˆˆ ˆ,s q EEs dET G G E- -= + = , (30) 000 1 00cos sin sin cosjTe - - == (31) The control laws of inner loops are given by ( )s s s dqref dq dqi E u _ˆ ˆˆv dqG= - , (32) ( )s s s s idq dqref dq dqU i i u _ˆˆ ˆ ˆi dq vGF= - + (33) () ( ) () ( )_ _ 1 _ _ 1v dq v i dq iG s G s j G s G s jab abw w=+ =+ (34) The open -loop and closed -loop transfer function s, from the reference to the output of active power, are given by ( )1 1T T1 _ f _ _ 1 _ _ _ 1 __PSC open P g d i dq v dq E d v g d i dq d i dq v dq g u d i dq v dq iv Vpcc gGT G Z Z G G G Gs G F Z G G G G G Z G G G T K G Z s ab- - -= + --- = - (35) ()pcc _ _ _ refˆ ˆ 1open closed openPTTsT Pab ab ab==+ (36) C. Modeling of dq -Frame Realization of Inner Loops For the dq-frame realization of inner loops, the abc-dq transformation of PCC voltage is represented by ( )( )c c s s dq0 dq dq0 dqU u U u0ˆˆˆje-++ = + (37) ( )cs dq dquu0 0ˆ ˆˆ ,s q uus dVT G G V= + = - (38) 000 00cos sin sin cosjTe - == - (39) The abc-dq transformation of grid current is represented by ( )( )c c s s dq0 dq dq0 dqI i I i0ˆˆˆje-++ = + (40) ( )cs dq dqii0 0ˆˆ ˆ,s q iis dIT G G I= + = - (41) The small -signal representation s of control laws are given by ( )c c c dqref dq dqi E u _ˆ ˆˆv dqG= - , (42) ( )c c c c idq dqref dq dqU i i u _ˆˆ ˆ ˆi dq vGF= - + (43) The dq-abc transformation of modulation voltage is represented by ( )( )s s c c idq0 idq idq0 idqU U U U0ˆˆˆje++ = + (44) 14 sc idq idqUU0 1 0ˆˆ ˆ,s iq Ui Uis idUT G G U- -= + = (45) The open -loop and closed -loop transfer function s, from the reference to the output of active power, are given by ( )23 2 3TT T T1 _f 1 _ _ _ 1 __ _ _ _open P g d PSC d v g d i dq d i dq v dq g u d i dq v dq iv Vpcc g Ui v u i dq i i dq v dq uT G Z Z G G s G F Z G G G G G Z G G G T K G Z s G F G G G G G Gab - - - = + - -- = - = + - - (46) ()pcc _ _ _ refˆ ˆ 1open dq closed dq open dqPTTsT P==+ (47) REFERENCES [1] M. 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2023-10-14

This paper presents an analytical approach to explore the damping effect of inner loops on grid-forming converters. First, an impedance model is proposed to characterize the behaviors of inner loops, thereby illustrating their influence on output impedance shaping. Then, based on the impedance representation, the complex torque coefficient method is employed to assess the contribution of inner loops to system damping. The interactions among inner loops, outer loops, and the ac grid are analyzed. It reveals that inner loops shape the electrical damping torque coefficient and consequently influence both synchronous and sub-synchronous oscillation modes. The virtual admittance and current control-based inner-loop scheme is employed to illustrate the proposed analytical approach. The case study comprises the analysis of impedance profiles, the analysis of damping torque contributed by inner loops under various grid strengths, and the comparison between dq-frame and {\alpha}\b{eta}-frame realizations of inner loops. Finally, simulation and experimental tests collaborate with theoretical approaches and findings.

Exploring Damping Effect of Inner Control Loops for Grid-Forming VSCs

2310.09660v1

1 A Frequency -Domain Beamforming Procedure for Extracting Rayleigh Wave Attenuation Coefficients and Small -Strain Damping Ratio from 2D Ambient Noise Array Measurements Aser Abbas a, *, Mauro Aimar b, Brady R. Cox a, Sebastiano Fotib a Utah State University, Department of Civil and Environmental Engineering, Logan, UT, USA, 84322. b Politecnico di Torino, Department of Structural, Building and Geotechnical Engineering (DISEG), Torino, Italy. Abstract The small -strain damping ratio plays a crucial role in assessing the response of soil deposits to earthquake -induced ground motions and general dynamic loading. The damping ratio can theoretically be inverted for after extracting frequency -dependent Rayleigh wave attenuation coefficient s from wavefields collected during surface wave testing . However, determining reliable estimat es of in-situ attenuation coefficients is much more challenging than achieving robust phase velocity dispersion data , which are commonly measured using both active -source and ambient - wavefield surface wave methods . This paper introduces a new methodology for estimating frequency -dependent attenuation coefficient s through the analysis of ambient noise wavefield data recorded by two-dimensional (2D) arrays of surface seismic sensors for the subsequent evaluation of the small -strain damping ratio . The approach relies on the application of a n attenuation -specific wavefield conversion and frequency -domain beamforming. Numerical simulations are employed to verify the proposed approach and inform best practices for its application . Finally, the practical efficacy of the proposed approach is showcased through its application to field data collected at a deep, soft soil site in Logan, Utah, USA , where phase velocity and attenuation coefficients are extract ed from surface wave data and then simultaneously inverted to develop deep shear wave velocity and damping ratio profiles . Keywords: attenuation; damping; surface wave testing; inversion; in situ; noninvasive; ambient noise; vibrations *Corresponding author . E-mail: aser.abbas@usu.edu (A. Abbas) 2 Introduction The small -strain shear modulus ( Gmax) and small -strain damping ratio ( D) form the starting point for many soil constitutive models and play a crucial role in assessing the response of soil deposits to earthquake -induced ground motions and general dynamic loading. Gmax is directly related to the in-situ shear wave velocity ( Vs), and it represents the soil stiffness and its resistance to deformation under applied shear stress . D characterizes the energy dissipation properties of the material . The influence of D on the amplitude and frequency content of seismic waves has been recognized since at least 1940 (Ricker, 1940), with subsequent research establishing it as a pivotal parameter for seismic site response studies and for modeling ground -borne vibrations (e.g., Anderson et al., 1996; Tao and Rathje, 2019; Papadopoulos et al., 2019 ; Foti et al., 2021 ). Despite its significance, the in-situ estimation of D has received far less attention when compared to measurements of Vs (Parolai , 2014 ). D can theoretically be inverted for after extracting frequency -dependent Rayleigh wave phase velocity and attenuation coefficients (α) from wavefields collected during surface wave testing (Lai, 1998 ; Foti, 2004 ). However, in-situ α values are generally much more difficult to reliably measure than phase velocities (Haendel et al., 2016 ; Parolai et al., 2022 ), which are commonly measured using both active -source and ambient -wavefield surface wave methods. This paper introduces a new noninvasive method to estimate frequency -dependent Rayleigh wave α using ambient noise wavefield data collected with two -dimensional (2D) array s of surface seismic sensors for the subsequent evaluation of D. The approach relies on frequency -domain beamforming (FDBF) and applies an attenuation -specific wavefield conversion, known as the FDBFa approach. While Aimar et al. (2024a) previously used this approach for active -source surface wave testing, it has not be en applied to ambient noise surface wave testin g. In this paper, we introduce a new method called the noise FDBFa (NFDBFa) approach and document its development and application. The subsequent sections of this paper are organized as follows: first, we cover important background information on attenuation and damping. Second, we present a concise overview of the FDBF technique introduced by Lacoss et al. (1969) and the FDBFa wavefield conversion methodology proposed by Aimar et al. (2024a) , along with the integration of these methods within our proposed NFDBFa approach. Then, synthetic studies are presented to showcase the capabilities of the proposed NFDBFa approach and inform best practices for its application . The synthetic 3 studies offer valuable insights into the influence of 2D array size and proximity to noise sources on attenuation estimates. For example, it is demonstrated that the optimal 2D ambient noise array design principles for attenuation estimation differ from th e principles governing 2D array design for phase velocity estimation. Finally, we demonstrate the practical utility of our proposed NFDBFa technique through a field application at a deep, soft soil site in Logan, Utah, USA . In this field application, phase velocity and attenuation coefficients are extracted from surface wave data and then simultaneously inverted to develop deep Vs and D profiles. The good agreement observed between the attenuation estimates derived from our new NFDBFa technique and those obtained through the standard FDBFa analysis of active -source data collected using the multichannel analysis of surface waves (MASW) provides compelling evidence of the effectiveness of our new ambient noise approach. Background The attenuation of seismic waves in a continuum is related to the damping ratios of both compression waves ( Dp) and shear waves ( Ds). Surprisingly, little is known about the relative relationship between Dp and Ds, and one can find instances in the literature where researchers have assumed Dp = Ds (Badsar et al., 2010; Verachtert et al., 2017; Aimar et al., 2024b) , Dp > Ds (Bergamo et al., 2023) , and Dp < Ds (Xia et al., 2002 ). In this paper, when ' D' is used without a subscript, it implies that the statement or equation is valid for both Dp and Ds. The damping ratio (D) is commonly used in engineering, while its inverse, the quality factor ( Q), where 𝑄 −1= 2𝐷, is more prevalent in seismological and geophysical literature (Foti, 2004). Consequently, Q, being the inverse of D, also varies for compressional waves ( Qp) and shear waves ( Qs). Seismic wave attenuation is commonly attributed to three mechanisms: material damping, geometric spreading, and apparent attenuation (Zywicki, 1999). Material damping, or anelastic attenuation, arises from the collective interaction of diverse factors (Johnston et al., 1979). These factors encompass frictional losses among solid particles and fluid flow losses due to the relative motion between solid and fluid phases, a phenomenon particularly notable in coarse -grained soils (Biot, 1956; Walsh, 1966 and 196 8; Stoll, 1974). Fine -grained soils, however, showcase more intricate phenomena influenced by electromagnetic interactions between water dipoles and microscopic solid particles ( Lai, 1998 ). This intrinsic material damping is typically approximated as frequency -independent, especially within the seismic frequency band , primarily spanning 0.1 4 to 10 Hz (Aki and Richards , 1980; Shibuya et al. , 1995) . Material damping gives rise to a cyclic stress -strain curve exhibiting a hysteretic loop and is commonly referred to as hysteretic damping (Rix et al., 2000; Parolai et al ., 2022). Geometric or radiation damping involves the spread of a fixed amount of energy over a broader area or volume as the wavefront moves away from the source. Take, for instance, a harmonic unit point load applied along the normal direction to the surface of a homogeneous and isotropic half - space ; this perturbation generates both body waves and Rayleigh wave s. The body waves propagate radially from the source, forming a hemispherical wave front, while Rayleigh wave s travel outward along a cylindrical wave front. As these waves travel, they traverse an expanding volume of material, leading to a decrease in energy density as the distance from the source increases. The amplitude of the body waves attenuate s in proportion to the ratio of 𝑟−1 (where r is the radial distance from the source), except when along the surface of the half -space. In that case, the amplitude attenuates proportionally to 𝑟−2. Conversely, the amplitude of the Rayleigh wave s attenuate s as 𝑟−0.5 (Lamb, 1904; Ewing et al., 1957; Richart et al., 1970). Consequently, at substantial distances from the surface source, the dominant influence on overall particle motion stems from the surface wavefield (Lai, 1998) . It is worth mentioning that these geometric spreading rules do not hold with transient waveforms (Keilis -Borok, 1989) or non-homogeneous media (Lai, 1998 ). Apparent attenuation includes wave scattering, which arises from the interaction of waves with heterogeneities along the seismic path (O’Doherty and Anstey, 1971; Spencer et al., 1977), as well as the reflection and transmission of seismic waves at interfaces and mode conversions (Rix et al., 2000). Therefore, apparent attenuation is highly site -specific and difficult to generalize . Both laboratory tests and in-situ methods have been proposed to estimate D. Laboratory tests , such as the resonant column (ASTM D4015 -21), are valuable for parametrically studying the material/intrinsic damping ratio, but they cannot capture the other two mechanisms contributing to the attenuation of seismic waves in situ. Conversely, the damping ratio estimates obtained using in-situ methods are influenced by all the seismic wave damping mechanisms mentioned above (Parolai et al., 2022). In-situ methods also have the advantage of assessing soil characteristics in their natural and undisturbed state (Rix et al., 2000). Additionally, in -situ tests encompass a greater soil volume, effectively reducing result biases that might arise from localized variations in soil 5 properties (Badsar et al., 2010). Furthermore, they provide parameter estimates on a spatial scale relevant to common engineering applications (e.g., Comina et al. 201 1). In the scope of estimating D, in-situ methods can be dissected into two categories: invasive and noninvasive methods. Invasive methods encompass techniques such as cross -hole testing (Jongmans, 1989; Hall and Bodare, 2000) and downhole testing (Michaels, 1998; Crow et al., 2011 ). Noninvasive methods, particularly surface wave techniques, offer numerous advantages. By situating sensors at the ground surface, surface wave methods accelerate data acquisition, minimize costs, streamline validation of soil -receiver coupling, and encompass a frequency range closely aligned with those pertinent to earthquake engineering applications (Rix et al., 2000; Verachtert et al., 2017; and Parolai et al., 2022). Surface wave test ing became popular in the 1980’s as an effective way to non -invasively develop 1D layering and Vs profiles for both soil deposits and pavement systems (e.g., Nazarian et al. , 1983 ; Stokoe et al. , 1989). Typically, the use of surface wave methods involves acquiring experimental phase velocity dispersion data through active -source methods, ambient -noise methods, or a combination of both (Tokimatsu, 1995) . These dispersion data are then inverted to obtain layered subsurface models, with the primary goal of re solving changes in Vs. The combined use of active -source and ambient -noise methods facilitates the generation of dispersion data across a wide frequency range , which enables resolution of both near -surface and deeper layers . Active sources predominantly produce energy concentrated at higher frequencies, typically ranging from several Hertz to perhaps 100 Hertz , with limited energy generation below 5-10 Hz for small sources like sledgehammers and drop weights . Consequently, the effective profiling depth using active -source methods is often constrained to approximately 15 to 40 m, contingent on the subsurface velocity and source mass (Foti et al., 2018) . The primary hindrance to achieving increased penetration depths lies in gen erating lower frequency (i.e., longer wavelength) waves with affordable and highly -portable sources. This difficulty is circumvented by ambient -noise methods, which do not involve the active generation of wave energy. Instead, they rely on ground motion s induced by cultural noise and microtremors (i.e., ambient noise) , encompassing an abundance of low -frequency components (Lai, 1998). Consequently, ambient -noise surveys offer valuable insights for deep characterization, extending to depths of hundreds of meters or more (Foti et al., 2014 ; Teague et al., 2018 a). Nevertheless, the spectral power of microtremors is generally low at higher frequencies (Peterson , 1993) , which limits their ability to resol ve changes 6 in stiffness near the ground surface (Tokimatsu, 1995 ; Foti et al., 2014 ). Combining both active and ambient -noise measurements offers a solution to overcome this limitation. Ambient -noise surveys typically employ 2D arrays of surface seismic sensors due to the a -priori unknown location of the ambient noise source s. Unlike linear arrays, 2D arrays allow for the determination of wave propagation direction, which is necessary for resolving the true phase velocity (Cox and Beekman, 2011). While 2D ambient noise array measurements have been referred to using several names, in this paper we will refer to them as microtremor array measurements (MAM; Ohrnberger et al., 2004 ; Teague et al., 2018b ). A schematic representation of a typical survey utilizing both a ctive and ambient -noise arrays is presented in Figure 1a. The active -source array in Figure 1a is in accordance with the MASW method (Park et al., 1999), utilizing a linear array of receivers to capture the wavefield generated by active source s off each end of the array. Example waveforms recorded by 24 receivers placed in -line with one of the active sources to the left of the array are depicted in Figure 1b. The ambient wavefield array depicted in Figure 1a is in accordance with MAM testing, where surface sensors are deployed in a 2D circular pattern (note that other 2D geometries are also permissible). Example ambient noise waveforms recorded by nine sensors in the circular array are depicted in Figure 1d. Figure 1c schematically illustrates phase velocity dispersion data that are commonly extracted from active -source MASW waveforms and ambient noise MAM waveforms using various well -known wavefield transformation techniques (Vantassel and Cox, 2022). Examples of these techniques include frequency -domain beamforming (FDBF; Lacoss et al., 1969), high -resolution frequency - wavenumber (f -k) spectrum analysis (Capon 1969), cylindrical FDBF (Zywicki 1999; Zywicki and Rix, 2005), and Rayleigh three -component beamforming (Wathelet et al., 2018). The combined dispersion data from MASW and MAM spans a wide frequency range , encompassing both low frequencies obtained from the MAM testing and high frequencies obtained from the MASW testing, with some overlap in between . The phase velocity dispersion data are then typically used to solve the parameter identification problem (i.e., inversion) and obtain 1D Vs profile s of the subsurface (Foti et al., 2018; Vantassel and Cox, 2021) . Note that the inversion step and resulting Vs profiles are not illustrated schematically in Figure 1. 7 Figure 1. Schematic illustrating the data acquisition and processing stages of active -source and ambient - wavefield surface wave testing used to extract phase velocity and phase attenuation data . Panel (a) presents a typical acquisition setup consisting of concentric MASW and MAM arrays, featuring active sources for the MASW array and an ambient wavefield for the MAM array. Panel (b) shows waveforms from a single active -source location collected using the MASW array, while Panel (c) presents the combined phase velocity dispersion data resulting from MASW and MAM Frequency Domain Beamforming (FDBF) processing. Panel (d) depicts the ambient n oise waveforms collected from the MAM array. In Panel (e), phase attenuation data processed through active -source FDBFa and ambient -wavefield NFDBFa techniques are illustrated. 8 As noted above, much more effort has been devoted to extracting phase velocity information from surface wave approaches than to extracting attenuation information. Nonetheless, m ultiple active - source methods have been developed to estimate the attenuation of surface waves. The methods introduced by Lai (1998), Lai et al. (2002), Rix et al. (2000), Xia et al. (2002), and Foti (2004) are founded on assessing the spatial decay of Rayleigh waves , a phenomenon that is influenced by both Dp and Ds, as described b y Aki and Richards (1980) . These approaches assume the dominance of a single Rayleigh wave mode of propagation. Consequently, they might yield inaccurate results in soil profiles where multiple surface wave modes significantly contribute to the wavefield propagation (Rix et al. 2001). Badsar et al. (2010) introduced the half -power bandwidth method, originally developed in the field of mechanical and structural dynamics to determine the modal damping ratio of a structure , to assess Rayleigh modal attenuation by analyzing the width o f the Rayleigh peaks in the f -k domain. Verachtert et al. (2017) employed the circle fit method, originally developed to determine eigenfrequencies and modal damping ratios in structural dynamics (Ewins 1984), to estimate multimodal Rayleigh dispersion and attenuation curves. Both the half -power bandwidth and circle fit methods facilitated the determination of modal attenuation curves from multimode wavefields (Verachtert et al., 2017). Recently, Aimar et al. (202 4a) introduced an innovative technique that combines a novel wavefield conversion approach coupled with FDBF (Lacoss et al., 1969) for processing active -source data collected using MASW to estimate the frequency -dependent α values . They called this the FDBF attenuation (FDBF a) method. Notably, the wavefield conversion proposed by Aimar et al. (2024a) to extract differs from the conventional wavefield transformation s commonly used to go from the time -distance domain to the f -k domain , as detailed in the following section . To avoid confusion, we will refer to the wavefield transformation proposed by Aimar et al. (2024a) as ‘wavefield conversion, ’ while reserving the term ‘wavefield transformation ’ specifically for the more common f -k domain transformation s used to extract phase velocity data . While important research on extracting phase attenuation coefficients using active -source methods is ongoing, similar to phase velocity data, combining active -source and ambient noise methods is desirable for resolving attenuation data over a broader frequency band. The majority of ambient noise techniques aimed at estimating the attenuation of surface waves were developed for regional - scale estimation (Haendel et al., 2016; Parolai et al., 2022) . Only a limited number of approaches have considered local scales that hold relevance for engineering purposes, like site-specific seismic 9 ground response analyses or dynamic vibration studies . These local -scale approaches are predominantly based on retrieving attenuation properties from the cross -correlation of seismic noise (e.g., Albarello and Baliva , 2009; Parolai, 2014; Haendel et al., 2016). Albarello and Baliva (2009) proposed a methodology that reconstructs the Green’s function based on the temporal derivative of averaged cross -correlations from noise recordings obtained by pairs of geophones , thereby incorporating attenuation effects into the process. They further validated this approach by demonstrating its potential in estimating attenuation coefficients at two distinct sites. Parolai (2014) estimated the Rayleigh phase velocity and attenuation coefficient s by fitting a damped zero - order Bessel function , introduced by Prieto et al. (2009), using data generated from the space correlation function introduced by Aki (1957). To mitigate the impact of uneven source distribution on cross -correlations, Haendel et al. (2016) employed a higher -order noise cross -correlation technique to extract the phase velocity and attenuation coefficient of Love waves. They illustrated that their approach yields correlation functions with higher signal -to-noise ratios compared to simple noise cross -correlatio ns. The importance of seismic noise cross -correlation methods cannot be underestimated . Nonetheless, in theory, the reconstruction of the full Green’s function requires the noise wavefield energy to be equal ly partitioned in all directions (Sánchez -Sesma and Campillo, 2006; Snieder et al., 2007) . This is a highly specific condition that rarely met rigorously by ambient noise on Earth (Cupillard and Capdeville, 2010; Tsai, 2011; Haendel et al., 2016). Furthermore, w hile travel time measurements from cross -correl ation of ambient noise are theoretically understood, amplitude measurements lack a corresponding theoretical background, except when the noise is equipartitioned (Snieder et al. , 2007; Tsai, 2011). Studies by Cupillard and Capdeville (2010) and Tsai (2011) have shown that attenuation estimates using cross -correlations are significantly influenced by the distribution of the noise sources. In light of the challenges posed by the equipartitioning condition for the reconstruction of the full Green’s function in ambient noise studies (Sánchez -Sesma and Campillo, 2006; Snieder et al., 2007), and considering the limitations highlighted by Cupillard and Capdeville (2010) and Tsai (2011) regarding the influence of noise source distribution on attenuation estimates, w e introduce a paradigm -shifting approach herein for calculating attenuation coefficients from ambient noise . This novel method not only eliminates the need for an equipartitioned noise wavefield , but also remains robust in the face of uneven noise source distribution, marking a departure from existing methodologies. 10 This paper builds upon Aimar et al.'s (202 4a) work on developing a n FDBFa technique for estimating α from active -source MASW testing and expands the FDBFa approach to ambient noise data recorded using MAM . Importantly, using an FDBF approach enables the actual direction of ambient noise propagation to be determined for each noise window and frequency, and does not require equipartitioning of ambient noise energy. Furthermore, using a n FDBF approach enables the phase attenuation data generated from MASW and that from MAM to be combined in order to generate phase attenuation data spanning a broader frequency range , as illustrated schematically in Figure 1e. The experimental dispersion and attenuation data can then be combined and inverted to determine not only the Vs profile but also the D profile of the subsurface to greater depths . This inversion of dispersion and attenuation data to obtain Vs and D profiles can be carried out either sequentially, as demonstrated in the work of Rix et al. (2000), or simultaneously, as shown by both Lai (1998) and Aimar et al. (2024b ). Wavefield conversion Proposed by Aimar et al., (2024a ) The method introduced by Aimar et al. ( 2024a ) to estimate Rayleigh wave attenuation ( α) assumes that the recorded wavefield is dominated by planar surface waves, specifically Rayleigh waves observed in the far field , with a dominant propagation mode. Several techniques have been developed to estimate the wavenumber ( k) and therefore the phase velocity from such wavefields (e.g., Lacoss et al., 1969; Capon, 1969; Zywicki and Rix, 2005; Wathelet et al., 2018). Aimar et al. (2024a ) harnessed this concept and introduced a novel wavefield conversion approach that provides a pathway for calculating α by utilizing methods from existing literature originally developed for estimating k. The methodology involves converting the recorded wavefield into a function interpreted as a pseudo -wave. This pseudo -wave exhibits dispersion characteristics reflecting the phase attenuation of the original wave. The determination of α then becomes straightforward through the application of existing techniques for estimating k. Consider the harmonic, exponentially decaying displacement wavefield, U(r), depicted in Figure 2a and expressed by Equation 1. This wavefield is observed at many discrete distances at a specific moment in time and is induced by the passage of a monochromatic plane wave. Within this wavefield, α governs the amplitude decay resulting from material damping in accordance with a viscoelastic constitutive model . When the wavefield is plotted as log amplitude versus radial distance ( r) from the source, the slope of the amplitude decay is α, as illustrated in Figure 2b. When 11 the wavefield is plotted as phase angle versus r, k denotes the slope of the unwrapped phase (i.e., the linear phase shift) , as shown in Figure 2c . Aimar et al. ( 2024a ) proposed raising the recorded wavefield, U(r), to the power of the imaginary number , i (see Equation 2) . Consequently, a pseudo displacement wavefield, 𝑣(𝑟), is generated , wherein the wavenumber is modulated by α, signifying that when the unwrapped phase of the converted wavefield is graphed against radial distance, the slope of that phase corresponds to the value of α (refer to Figure 2d) . Conversely, when the log amplitude of the converted wavefield is plotted against distance, the slope manifests as k, with an inverted sign (refer to Figure 2e). This wavefield conversion allows for estimating α using any of the already established and common wavefield transformation techn iques for calculating k (e.g., f-k or FDBF methods) . 𝑈(𝑟)= 𝑒−𝛼𝑟𝑒−𝑖𝑘𝑟 (1) 𝑣(𝑟)=⌈𝑈(𝑟)⌉𝑖= 𝑒−𝑖𝛼𝑟𝑒𝑘𝑟 (2) This wavefield conversion can also be extend ed to a broadband wavefield , comprised of a superposition of monochromatic plane waves by exponentiating the wavefield in the frequency domain with the power of the imaginary number . To address numerical artifacts introduced by the wrapped phase on the pseudo wavefield, Aimar et al. ( 2024a ) recommended normalizing v(r) by its amplitude on a frequency -by-frequency basis . Aimar et al. ( 2024a ) showed that this wavefield conversion can be successfully applied to active -source wavefields recorded using MASW as a means to estimate α. In this paper, we extend this approach to estimate α from ambient noise wavefields recorded using MAM arrays, employing the FDBF technique introduced by Lacoss et al. (1969) . 12 Figure 2. Schematic illustrating the wavefield conversion approach proposed by Aimar et al. (2024a) to extract attenuation coefficients ( ). Panel (a) displays the particle displacement of a monochromatic wave experiencing exponential amplitude decay with distance, indicative of material damping in a viscoelastic constitutive model. Panel (b) depicts linear amplitude decay in log amplitude v ersus linear distance space, where the slope represents the phase attenuation coefficient. In Panel (c), the modulation of the unwrapped phase slope with distance by the wavenumber ( k) is demonstrated. Panel (d) illustrates the modulation of the unwrapped phase slope by the phase attenuation coefficient in the converted wavefield. Panel (e) showcases the control of the slope of the log amplitude decay with linear distance by the wavenumber, albeit with an inverted sign. Noise Frequency Domain Beam Forming - attenuation ( NFDBFa) The inherent challenge in ambient noise measurements stems from the lack of a priori information about the source location or the direction of wave propagation, necessitating the use of spatial 2D 13 array s to determine the noise propagation directions during post -processing (Zywicki, 1999). As ambient noise wavefields operate in two spatial dimensions (e.g., x and y) , it is necessary to represent the wavenumber using 2D vectors (Johnson and Dudgeon, 1993; Zywicki, 1999) , where 𝑘⃗ = 𝑘𝑥𝑖̂+𝑘𝑦𝑗̂, and 𝑖̂ and 𝑗̂ are unit vectors in the x and y directions, respectively . Similarly, 𝛼 is also expressed as a 2D vector (i.e., 𝛼 = 𝛼𝑥𝑖̂+𝛼𝑦𝑗̂) in this paper. Beamforming refers to a diverse set of array processing algorithms that concentrate the signal -capturing capabilities of the array in a specific direction. The fundamental concept behind beamforming is straightforward: when a propagating signal exists within an array's aperture, the outputs of the sensors, delayed by appropriate amounts and added together, enhance the coherent signal while mitigating the incoherent signal from waves propagating in different directions. The delays that enhance the signal are directly linked to the time it takes for the signal to travel between sensors (Johnson and Dudgeon, 1993). Delays in the time domain correspond to linear phase shifts in the frequency domain , providing information about the wavenumber . FDBF calculations are exclusively performed within the frequency domain . Applying FDBF to the original wavefield, 𝑈(𝑟 ), provides information about 𝑘⃗ , which informs the estimation of the phase velocity. This paper aims to demonstrate that applying FDBF to the converted, normalized pseudo wavefield , 𝑣(𝑟 ), informs the estimation of 𝛼 . Henceforth , in this paper, we will denote FDBF applied to the converted noise wavefield as NFDBFa, emphasizing its role in estimating the phase attenuation from ambient noise . In the NFDBFa approach, the first step is to partition the noise data collected by a 2D array of m sensors into B time windows. The m sensors are located at the ground surface at coordinates ( 𝑥𝑖,𝑦𝑖) denoted by the vector 𝑟𝑖⃗⃗ , where i varies from 1 to m. For each time window, Fourier spectra are calculated. Following this, the complex number at each frequency in the spectra is exponentiated to the imaginary power. Then, each exponentiated complex number is normalized by dividing it by its absolute amplitude . This process is conducted to obtain the normalized spectra of the pseudo wavefield (Aimar et al., 2024a ). These spectra are then used to compute the Hermitian symmetric spatio -spectral correlation matrix, 𝑅𝑖𝑗, with i and j representing indices of the m sensors in the 2D array, using Equation 3 : 𝑅𝑖𝑗(𝜔)= 1 𝐵∑𝑣𝑖,𝑛(𝜔)𝐵 𝑛=1 𝑣𝑗,𝑛∗(𝜔) (3) 14 where 𝑅𝑖𝑗(𝜔) is the averaged pseudo cross -power spectrum between the ith and jth sensors in the array across all windows, 𝑣𝑖,𝑛(𝜔) is the normalized pseudo spectra of the ith sensor’s data in the nth window , * indicates complex conjugation , and 𝜔 is the angular frequency . Despite being frequency -dependent, the spatio -spectral correlation matrix conveys spatial wavefield properties. Power within specific frequency -phase attenuation ( f-𝛼 ) pairs is determined by steering the array towards various directions and potential phase attenuation values . Array steering involves exponential phase shift vectors determined by trial 𝛼 values in pseudo space , as given by Equation 4: 𝑒(𝛼 )= [exp(−𝑖𝛼 .𝑟1⃗⃗⃗ ),…,exp(−𝑖𝛼 .𝑟𝑚⃗⃗⃗⃗ )]𝑇 (4) where 𝑒(𝛼 ) is a steering vector associated with a trial 𝛼 and T denotes the transpose of the vector. The power in a particular f-𝛼 pair, 𝑃𝑁𝐹𝐷𝐵𝐹𝑎(𝛼 ,𝜔), is estimated by multiplying 𝑅𝑖𝑗(𝜔) by 𝑒(𝛼 ) and summing the total power over all sensors , as given by Equation 5 : 𝑃𝑁𝐹𝐷𝐵𝐹𝑎(𝛼 ,𝜔)= 𝑒𝐻(𝛼 )𝑅𝑖𝑗(𝜔)𝑒(𝛼 ) (5) where H indicates the Hermitian transpose. The steering vectors aim to align the array with plane waves propagating from a specified direction and phase attenuation for each frequency . The successful alignment results in a peak within the 𝑃𝑁𝐹𝐷𝐵𝐹𝑎(𝛼 ,𝜔) pseudo -spectrum estimate. Thus, the NFDBFa technique presented herein allows for estimating α from ambient noise data without requiring an equipartitioned wavefield. Even though there are similarities between the FDFBa method proposed by Aimar et al. (2024a) for estimating α using an MASW test setup and the NFDBFa method introduced in this study, there are notable differences between the two. Part of the difference is a consequence of the inherent dissimilarities between MASW and MAM. In the FDBFa method, the source location is predetermined and the array is aligned with the source, simplifying the problem and enabling the use of wavefield transformations like cylindrical frequency domain beamforming (Zywicki and Rix, 2005) . Moreover, the signal -to-noise ratio can be readily enhanced by time -domain or frequency -domain stacking, as advocated by Vantassel and Cox (2022) and Foti et al., (2018) . Additionally, dispersion and attenuation uncertainties can be quantified using the multiple source offset approach proposed by Cox and Wood (201 1). In co ntrast, the NFDBFa approach developed in this study encounters distinct challenges, primarily arising from the a priori unknown location 15 of the source(s). This necessitates the utilization of 2D arrays and involves azimuthally scanning the 2D space to ascertain the direction of the most coherent source of energy at each frequency for each window . Furthermore, the enhancement of the coherent noise -to-incoherent noise ratio involves averaging multiple time windows, while uncertainty quantification involves analyzing various time blocks, each composed of different windows. Thus, in the NFDBFa approach, data are recorded for significantly longer durations (i.e., hours) compared to FDBFa (i.e., seconds). Additionally, the NFDBFa approach relies on measurement s of ambient noise, which is typically assumed to be generated by sources located in the far field. If this assumption holds true, it helps to mitigate the impact of geometric spreading, which plays a significant role on attenuation estimates near an active source (Badsar, 2012). Nearfield noise sources lead to complications in extracting accurate attenuation estimates , as discussed in greater detail below . Figure 3 presents examples of the FDBF and NFDBFa responses obtained from a synthetic wavefield recorded by a ten -receiver circular MAM array for a single frequency and single time window . The array comprises nine sensors equally spaced on the perimeter of the circle and one sensor in the middle . The FDBF method is utilized to estimate 𝑘⃗ and the NFDBFa method is utilized to estimate 𝛼 . In Figures 3a and 3b, the results of applying the FDBF technique to the original noise wavefield recorded by the array are depicted. Figure 3a illustrates the f-𝑘⃗ spectrum at the considered frequency in a 2D wave number space ( 𝑘𝑥-𝑘𝑦). Stronger powers are represented by a darker purple color. This spectrum offers insights into the power and vector velocities of propagating waves. In this example, a wave propagates along the x -axis with a velocity represented by a vector wave number 𝑘⃗ at the chosen frequency. Consequently, a spectrum peak emerges on the positive 𝑘𝑥 axis at a distance of |𝑘⃗ | from the origin. The associated phase velocity can be calculated as 𝑉𝑟=2𝜋𝑓/|𝑘⃗ |, and the wavelength, λ, can be determined as 𝜆=2𝜋/|𝑘⃗ |. Figure 3b illustrates the cross -section a -a from Figure 3a, revealing the main and side lobes. Generally, the narrower the main lobe and the shorter the side lobes the better the array and processing algorithm are at accurately identifying the correct 𝑘⃗ values for a given frequency . 16 Figure 3 . Schematic illustrat ing the FDBF and NFDBFa responses obtained from a n ambient noise wavefield recorded by a ten-receiver circular MAM array for a single frequency and single time window. Panel (a) presents the f-𝑘⃗ spectrum resulting from applying the FDBF method to the original wavefield , displaying the beamforming peak powers in kx-ky space . Panel (b) shows the cross -section a -a from Figure 3a, revealing the main and side lobes. Panel (c) presents the f-𝛼 spectrum resulting from applying the NFDBFa technique to the pseudo wavefield, presenting the beamforming peak powers in 𝛼𝑥-𝛼𝑦 space. Panel (d) illustrates the cross -section x -x from Figure 3c , showing the main and side lobes along the direction of wave propagation. Figures 3c and 3d display the f-𝛼 spectrum obtained from applying the NFDBFa method to the converted noise wavefield for the same time window used to develop Figure 3a . In this case , instead of presenting the beamforming peak powers in the 𝑘𝑥-𝑘𝑦 space, as seen in Figure 3a, they are now depicted in the 𝛼𝑥-𝛼𝑦 space. This transition occurs because the phase in the pseudo wavefield is modulated by 𝛼 (refer to Figure 2), rather than k. Figure 3c employs a different color scheme, where stronger powers are represented by darker blue colors. The f-𝛼 spectrum shown in Figure 3c illustrates wave propagation for a single frequency along the x -axis with a phase 17 attenuation represented by the vector 𝛼 . Figure 3d illustrates the cross -section x -x from Figure 3c , revealing the main and side lobes along the positive x -axis (i.e., direction of wave propagation). Similar to estimating 𝑘⃗ , the narrower the main lobe and the shorter the side lobes the better the array and processing algorithm are at accurately identifying the correct 𝛼 values for a given frequency. The ability of the NFDBFa approach to develop phase attenuation estimates from ambient -noise recorded using MAM arrays is investigated in the following section using synthetic data. NFDBFa evaluation with synthetic wavefields This section uses synthetic data to validate the effectiveness of the NFDBFa approach in estimating phase attenuation from ambient noise recorded using MAM arrays. Specifically, the approach is tested on two soil models: a half -space model and a single layer above a half -space model. All numerical simulations discussed in this section were executed using Salvus (Afanasiev et al., 2019), a comprehensive 2D and 3D full-waveform modeling software suite based on the spectral element method. The simulations were performed on the Texas Advanced Computing Center’s (TACCs) high -performance cluster Lonestar6 using two compute nodes. Half -space model This subsection presents a simple wave propagation simulation consisting of a single surface source generating body and surface waves propagating through a half -space soil model. Despite the simplicity of the model, t he outcomes obtained from this simulation offer key insights into the attenuation of a wavefield generated by a surface source and elucidate the capabilities of the NFDBFa approach. Figure 4 depicts a schematic plan view illustrating the source location and MAM array configurations employed in the half -space simulation. The wavefield was generated by a point source acting in the vertical dire ction at coordinates (0, 0, 0) in an x, y, z cartesian coordinate system . The source was a single Ricker wavelet with a center frequency of 5 Hz. This source function produces broadband energy over a frequency range of approximately 1 to 10 Hz. The wavefield emanating from the source was recorded using five circular MAM arrays, each comprising 10 sensors, with one sensor at the center and nine sensors evenly spaced around the perimeter. In this paper, the arrays are named using the convention 'C' followed by the diameter of the array, where 'C' denotes that the array is circular. Therefore, the first array, located two 18 kilometers away from the source and with a diameter of one kilometer, is denoted as C1000 at 2 km. The remaining four arrays, concentrically -centered five kilometers from the source, have diameters of 60 m (C60), 300 m (C300), 1000 m (C1000 at 5 km), and 2000 m (C2000). It is noteworthy that, although currently only the vertical component of the displacement wavefield is utilized in NFDBFa, each sensor recorded both horizontal and vertical displacement components , and plans for utilizing all components from noise recordings are ongoing . Additionally, the NFDBFa processing operated independently of any knowledge about the source location, mirroring the conditions of an ambient noise MAM survey, ensuring an unbiased analysis. The half -space constitutive soil parameters are presented in Figure 5a, where Vp, and υ are the compression wave velocity, and Poisson’s ratio, respectively. Due to the large spatial extent of the model and the substantial computational expense associated with running a simulation over such a vast domain, a 2D simulation was conducted rather than a 3D simulation . In the 2D simulations , the sensor locations were projected onto a 2D plane, as illustrated in Figure 5a . This entailed setting the y -coordinate to zero for each surface sensor location shown in Figure 4, resulting in their positions being determined exclusively by their x -axis coordinates. For example, the s ensor initially situated at coordinates (2321.4, 383, 0) in an x, y, z system (as depicted in Figure 4), transformed to (2321.4, 0) in the 2D x, z system presented in Figure 5. However, it is important to note that, during NFDBFa processing, the coordi nates assigned to each sensor were derived from those shown in Figure 4; consequently, the aforementioned sensor retained coordinates of (2321.4, 383, 0) during processing . This approach not only substantially reduced the computational cost of the simulations , but also ensured that the arrays were measuring plane waves. The simulation required 4 hours and 20 minutes of computation utilizing 256 threads on the high -performance cluster Lonestar6. Before describing the application of the NFDBFa method, some preliminary features of the amplitude decay versus distance are discussed, as they directly influence attenuation estimates. To better observe this decay pattern, the wavefield emanating from the source was recorded every ten meters along the surface. Those time histories were then filtered at discrete frequencies so the amplitude decay at each frequency could be observed. The decay of Fourier amplitudes with distance from the vertical Ricker wavelet source for frequencies 1, 2, 3, 4, and 5 Hz are shown in Figures 5b and 5c. In Figure 5b, the amplitudes for each frequency are normalized by their 19 respective maximum values at the source and plotted on a log scale , while the distances are not normalized and plotted on a linear scale . In contrast, in Figure 5c, the distances from the source are normalized by the wavelength ( λ) corresponding to each plane wave frequency and plotted on a linear scale . The figures depict a sharp amplitude decrease near the source due to nearfield effects. Following this, amplitude oscillations with diminishing power are superimposed over a linear decay trend . Note that a linear decay trend in log amplitude scale corresponds to an exponential decay in linear amplitude scale . These amplitude oscillations tend to flatten greatly after propagating approximately 10 λ away from the source . It is noteworthy that these oscillations, although verified using other software packages , such as the ElastoDynamics Toolbox (EDT; Schevenels et al., 2009), challenge conventional intuition regarding wave attenuation in a half - space . Neither the geometric spreading of Rayleigh waves nor the attenuation due to material damping should exhibit such oscillations in a half -space, as detailed by Lai (1 998). The oscillating amplitude decay pattern in a half -space model is a result of body wave amplitude decay oscillations, as shown by Tokimatsu (1995) and Holzlohner (1980) . Hence, when estimating phase attenuation using ambient noise , it is essential for the MAM arrays to be at a sufficient distance (more than approximately 10 λ) away from any potential surface sources , such that wave amplitude oscillations do not contaminate the expected trend of amplitude decay with distance . Figure 4. Plan view of the source (star symbol) and receiver (inverted triangle symbols) configurations used for synthetic wavefield simulation s. The source was a single Ricker wavelet with a center frequency of 5 Hz. The wavefield was recorded using five MAM arrays. The first array (C1000 at 2 km) has a diameter of 1 km and is positioned 2 km from the source. The remaining four arrays are concentrically -centered 5 km away from the source and have diameters of 60 m (C60), 300 m (C300), 1 km (C1000 at 5 km), and 2 km (C2000), respectively. 20 Figure 5 . Half-space wavefield simulation: Panel (a) presents a cross -section view of the configuration of the source and receivers shown in Figure 4, along with the half -space soil properties. Panel (b) shows the decay of particle vertical displacement as a function of distance from the source for five distinct frequencies, each normalized by its maximum amplitude at the source. Panel (c) presents the particle displacement decay pattern s from Panel b, with distance now normalized by the wavelength for eac h frequency. Panel (d) shows the particle ellipticit ies for each frequenc y, express ed as the horizontal particle displacement divided by the vertical particle displacement, with the dotted horizontal line indicating the theoretical ellipticity calculated based on the Poisson’s ratio of the half -space soil model. It is worth noting that in layered media, oscillating amplitude decay of Rayleigh waves due to geometric spreading has been reported and accounted for in attenuation studies, as observed in the work of Lai (1998). Thus , in layered media , wave amplitude oscillations can be more pronounced 21 and may extend beyond 10 λ from the surface source, as demonstrated by Tokimatsu (1995). This may be thought of as a type of near-field effect specific to attenuation studies, wherein the wavefield amplitude decay patterns are significantly more complicated at distances less than approximately 10 λ from source. This is distinct from , and more severe than, the typical range of near-field effects for phase velocity estimations, which generally deteriorate between 0.5 λ and 2 λ from the source , depending on th e subsurface velocity structure (Tokimatsu 1995; Rix et al., 2001) . To further demonstrate the more severe near -field effects associated with amplitude decay , Figure 5d presents the simulated wavefield ellipticity , expressed through the horizontal -to-vertical (H/V) ratio of particle displacement , measured with distance in wavelengths for the same frequencies outlined in Figure 5b. The ellipticity also display s oscillations that decrease and stabilize at normalized distances greater than about 10 λ from the source. This observation underscores that the near-field amplitude decay oscillation s stem from body waves, as Rayleigh wave ellipticity in a half -space is determined solely by Poisson’s ratio (Tokimatsu 1995) and should not oscillate. In Figure 5d, we observe that the calculated ellipticities oscillate around the theoretical value anticipated for Rayleigh wave ellipticity in a half -space with Poisson’s ratio equal to 0.33 , depicted by the dotted horizontal line in Figure 5d. The synthetic time histories recorded by the C1000 at 2 km and the C1000 at 5 km MAM arrays (refer to Figures 4 and 5) were processed using the FDBF and NFDBFa methods to estimate phase velocity and attenuation, respectively, as illustrated in Figure 6. Figure 6 aims to highlight the impact of wave amplitude decay patterns on the attenuation estimates. In terms of abilities to resolve phase velocity, both the C1000 arrays seem to perform approximately the same, whether 2 km away from the source (Figure 6a) or 5 km away from the source (Figure 6b). However, u pon inspecting Figures 6 c and 6d, it becomes evident that the array located 5 km from the source (i.e., Figure 6 d) provides more reliable attenuation estimates at lower frequencies compared to the array closer to the source . This observation can be explained by referring to Figure 5b, where the amplitude decay patterns measured by the array positioned 2 km from th e source are shaded in pink. It is apparent that in close proximity to the source, the low -frequency waves have not traveled a sufficient number of wavelengths, resulting in amplitude decay that does not conform to pure exponentials (i.e., linear decay in log scale) . However, by the time these waves reach the array 22 positioned 5 km from the source (blue shading in Figure 5b) , the oscillations in amplitude decay have diminished significantly, approaching a pure exponential decay. Therefore, it is noteworthy that in a n ambient -noise survey, even though the source location is unknown, if the noise source is close to the array in terms of wavelengths traveled by the desired frequency, it may lead to unreliable and scattered attenuation results. Nonetheless , Figures 6c and 6d clearly demonstrate the reliability of the new NFDBFa approach in retrieving phase attenuation estimates over a broad range of frequencies. Finally, the performance of the NFDBFa in the presence of incoherent noise is investigated. For this purpose, Figure 7 illustrates the influence of incoherent noise and array size on phase attenuation estimates using the same half -space simulation results. The analysis focuses on the four arrays of different sizes concentrically -centered 5 km from the source (refer to Figures 4 and 5a). Incoherent noise was introduced to the signal, with a target signal -to-noise ratio (SNR) at 20 dB, which resulted in the f requency -dependent amplitude decay patterns depicted in Figure 7a (compared to Figure 5b). Figures 7b, 7c, 7d, and 7e display the attenuation estimates obtained using the C60, C300, C1000, and C2000 MAM arrays, respectively. It becomes evident that larger arrays yield more accurate attenuation estimates in the presence of incoherent noise . Figure 7a elucidates the rationale behind this enhanced performance for larger arrays across all frequencies . The C2000 MAM array samples a significantly larger area, enabling it to discern the exponential amplitude decay even in the presence of noise. The C60 MAM array sampl es a significantly smaller area, and thus is considerably more sensitive to amplitude fluctuations caused by incoherent noise, resulting in the significant scatter observed in the attenuation estimates shown in Figure 7b. Figure 8 further illustrates the impact of array size on resolving attenuation coefficients by showcasing the f-𝛼 spectra for a frequency of 3 Hz that were calculated from the wavefield recorded by the four concentrically -centered arrays located 5 km from the source . Notably, the mainlobe (dark blue shaded area) is considerably narrower for larger arrays, resulting in more reliable estimates of phase attenuation. Two key points warrant attention here. First, the feasibility of employing larger arrays might be restricted due to limitations in access at a given site , or to help maintain approximately a one -dimensional (1D) subsurface condition beneath the array , which is an implicit assumption in the analysis technique (i.e., no lateral spatial variability). Meeting this 23 assumption becomes more challenging as the array size expands. Second, it is essential to highlight that the method used to determine the optimal MAM array size for attenuation estimates differs from the one employed in obtaining phase velocity estimates. In dispersion estimation, smaller MAM arrays are more effective at capturing high frequency phase velocities , whereas larger arrays are better suited for resolving lower frequency phase velocities (Foti et al., 2018; Vantassel and Cox, 2022) . However, according to the results depicted in Figure 7, the larger arrays demonstrated superior ability in resolving phase attenuation across the entire considered frequency range compared to the smaller arrays. Figure 6 . Half-space wavefield simulation: phase velocity (top) and phase attenuation (bottom) dispersion data estimated with FDBF and NFDBFa , respectively, from 1 km arrays positioned at two distinct distances from the ambient noise source: (left) at two kilometers (C1000 at 2 km), and (right) at five kilometers (C1000 at 5 km) . 24 Figure 7 . Half-space wavefield simulation with noise: Panel (a) shows the amplitude decay of the same five frequencies depicted in Figure 5 but now with added incoherent noise to the signal, setting the signal - to-noise ratio (SNR) at 20 dB. Panels (b) to (e) present the predicted phase attenuation data from the NFDBFa analysis for four arrays concentrically -centered at five kilometers from the source, with diameters of 60 m (C60), 300 m (C300), 1 km (C1000 at 5 km ), and 2 km (C2000), respectively. 25 Figure 8. Half-space wavefield simulation with noise: Panels (a) through (d) present the f-𝛼 spectra obtained through NFDBFa analysis for a frequency of 3 Hz. The spectra are derived from the wavefield recorded by the four arrays concentrically -centered five kilometers from the source with diameters of 60 m (C60), 300 m (C300), 1 km (C1000), and 2 km (C2000), respectively, as d epicted in Figure 7. Layer above a half -space model. The performance of the NFDBFa approach on a synthetic model consisting of a single layer above a half -space is illustrated in this subsection . The model's constitutive small -strain parameters and the source and receiver configurations are provided in Figure 9a. For this synthetic study, 150 vertical point sources with varying forcing functions and trigger times were activated. The sources were triggered five kilometers away from the center of a one -kilometer diameter circular array consisting of 10 sensors : one in the center and nine equally distributed around its perimeter (just like the C1000 at 5 km MAM array depicted in Figure 4) . The waveforms recorded by the array are depicted in Figure 9b. These waveforms were subsequently processed using FDBF and NFDBFa to derive the Rayleigh wave phase velocity dispersion data shown in Figure 9c and the phase -attenuation data shown in Figure 9d , respectively . The theoretical Rayleigh -wave phase 26 velocity dispersion and attenuation curves for the model are also presented in Figures 9c and 9d, respectively. In these figures, the fundamental theoretical mode is denoted as Mode 1, while the 1st-higher mode is denoted as Mode 2. Figure 9 . Layered model simulation: Panel (a) presents the soil properties utilized in the simulation for the soil layer and the half -space, along with the surface sources and 1 -km receiver array located 5 km from the source (C1000 at 5 km) . Panel (b) displays the waveforms collected from the C1000 array. In Panel (c), the good agreement between the theoretical Rayleigh -wave phase velocity curves (Mode 1 and Mode 2) and the experimental phase velocity data obtained through the FDBF approach on the original wavefield is demonstrated. Finally, Panel (d) showcases the good agreement between the theoretical phase attenuation curves and the experimental phase attenuation data extracted from the converted wavefield using the proposed NFDBFa approach are depicted. 27 The FDBF method is able to extract experimental phase velocity dispersion data from the synthetic wavefield that well -matches the theoretical dispersion curves and captures the transition from Mode 1 to Mode 2 at approximately 7 Hz. A strong agreement is also observed between the theoretical attenuation curve s and the experimental attenuation data extracted from the synthetic wavefield using the NFDBFa method , particularly for Mode 1 . Interestingly , the attenuation data shifts to Mode 2 at the same frequency where the phase velocity dispersion data transitions to Mode 2 . A similar observation about possible links between the frequencies where phase velocity and attenuation mode transitions occur was also reported by Aimar et al., ( 2024a ) using MASW data. While further studies are needed to validate the observations that phase velocity and phase attenuation data tend to jump modes at identical frequencies, this is a potentially important point, as patterns in attenuation modes are much more complex than phase velocity modes . The effectiveness of the proposed NFDFBa approach has been successfully demonstrated through the analyses conducted on synthetic data sets, as discussed above. Now, we shift our focus to applying this approach to real field data, offering a thorough demonstration of its effectiveness in a practical, real -world situation. Field application and validation A surface wave field -testing campaign was conducted at the Drainage Farm Site in Logan, Utah, USA (refer to Figure 10) , a property owned by Utah State University (USU). Structural geology indicates that Southern Cache Valley, encompassing the Drainage Farm Site and located in the northeastern part of the Basin and Range province, is a graben bounded by high -angle normal faults (Williams, 1962). The site is underlain by Paleozoic rocks, which are overlain by Tertiary formations such as the Wasatch and Salt Lake f ormations, composed of conglomerate, siltstone, and tuffaceous sandstone. In certain areas of Cache Valley, these formations reach thicknesses of up to 2,440 m (Evans et al., 1996). The near -surface geology of the Drainage Farm Site is characterized by sediments from ancient Lake Bonneville, which receded to form the Provo shoreline. These sediments include alluvial, lacustrine, and deltaic deposits (Williams, 1962; Evans et al., 1996). Well logs presented by Williams (1962) reveal a lternating layers of silt and clay, sand, and gravel above the Salt Lake formation. Moreover, limited deep well logs from the vicinity of the Drainage Farm Site indicate that rock can be encountered at depths ranging from 176 m to more than 350 m (Perez, 1969). 28 Figure 10 . Plan view of the MASW and MAM arrays employed for testing at the Drainage Farm Site in Logan, Utah, USA . The concentric MAM arrays featured diameters of 60 m (C60) , 300 m (C300) , and 700 m (C700) , while the MASW array comprised 24, 4.5-Hz vertical geophones, spanning 46 m. The goal of the testing was to collect a high -quality surface wave dataset that could be used for attenuation studies to validate the proposed NFDBFa technique . The field testing involved both active -source MASW testing and ambient noise MAM testing. The sensor array configurations utilized for MASW and MAM at the Drainage Farm Site are illustrated in Figure 10. MASW testing was performed using 24, 4.5 -Hz vertical geophones placed with a spacing of two meters between successive geophones, resulting in an array length of 46 m. Wavefields with strong Rayleigh wave content were actively generated by striking vertical ly on a strike -plate with a sledgehammer. The sledgehammer was used at eight distinct "shot" locations that were offset by 5, 10, 15, and 20 m relative to the first /last geophone off each end of the array . Five distinct sledgehammer blows were recorded at each location for subsequent stacking to increase the signal - to-noise ratio (Foti et al. 2018) . MAM testing utilized three concentric circular arrays that were aligned with the middle of the MASW array, as depicted in Figure 10. The three arrays were 700- m, 300 -m, and 60 -m in diameter , and will be referred to as C700, C300, and C60, respectively. Each array consisted of nine evenly distributed three -component broadband seismometers (Nanometrics Inc. Trillium Compact 120s se ismometers) along its circumference to capture 29 ambient vibrations. The three arrays did not record data simultaneously; instead, the nine sensors were used to collect noise data for each of the MAM arrays one array at a time . First, t he sensors recorded seismic noise for 13 hours and 30 minutes for the C700 array. Subsequently, the sensors were relocated to their designated locations for the C60 and C300 arrays, recording ambient noise for an hour and a half and three hours, respective ly. For Rayleigh -wave phase velocity dispersion analysis, MASW data were analyzed using the FDBF method with cylindrical -wave steering (Zywicki and Rix, 2005) , as coded in the open -source surface wave processing package swprocess ( Vantassel, 2021 ). This processing was coupled with the multiple source -offset technique for identifying near -field contamination and quantifying dispersion uncertainty (Cox and Wood, 2011; Vantassel and Cox, 2022 ). As a result, eight phase velocity estimates were obtained for each fre quency, corresponding to one phase velocity estimate from each of the eight shot locations. MASW Rayleigh wave dispersion data influenced by near - field effects or significant offline noise were trimmed before calculating phase velocity dispersion statistics. The three -component beamforming approach (Wathelet et al., 2018) coded in the open -source software package Geopsy ( Wathelet et al., 2020 ) was used to generate Rayleigh -wave phase velocity dispersion data for each of the MAM arrays. The recorded time for each array was discretized into blocks, with each block further divided into at least 30 -time windows. The window lengths were selected to contain at least 30 cycles (periods) at the lowest processing frequency that could be extracted from each MAM array (Va ntassel and Cox, 2022). For each MAM array, eight phase velocity estimates were extracted at each analyzed frequency using the three -component beamforming (Wathelet et al., 2018) approach to ensure consistency with the eight phase velocity estimates obtained from the MASW processing . Spurious dispersion data stemming from high - amplitude noise in the near -field (e.g., traffic noise close to the sensors) and incoherent noise were manually eliminated before calculating dispersion statistics. Ambient noise phase velocity dispersion data from all MAM arrays were combined with the active phase velocity dispersion data obtained from MASW processing, as shown in Figure 11a. The combined data, used to compute mean and ± one standard deviation dispersion estimates (Vantassel and Cox, 2022 ), are displayed in Figure 11b relative to the individual MASW and MAM dispersion data points for the Drainage Farm Site. 30 Figure 11 . Experimental phase velocity and attenuation data extracted from MASW and MAM testing at the Drainage Farm Site in Logan, UT , USA . Panel (a) displays the experimental phase velocity dispersion data of Rayleigh waves processed from an MASW array and three circular MAM arrays, with diameters of 60 m, 300 m, and 700 m. Panel (b) showcases the mean and ± one standard deviation of the experimental Rayleigh wave phase velocity dispersion data derived from the combined MASW and MAM datasets. Panel (c) displays the experimental phase attenuation data from MASW and three circular MAM arrays. Panel (d) illustrates the mean ± one standard deviation of the experimental phase attenuation data calculated from the combined MASW , C300 , and C700 MAM arrays. The cylindrical FDBF a (CFDBFa) approach, as proposed by Aimar et al. ( 2024a ), was employed to derive attenuation estimates from the MASW data. Mirroring the MASW phase velocity dispersion analysis, the multiple source -offset technique was utilized for quantifying attenuation uncertainty. Thus, e ight attenuation estimates were extracted from the MASW data at each analyzed frequency using CFDBFa. For the MAM attenuation estimates, the new NFDBFa approach introduced in this study was employed. The recorded time for each array was discretized into eight blocks, with each block further divided into 30 windows. Consequently, the window length employed for each MAM array can be determine d by dividing the total recording time of the array by the product of 8 blocks and 30 windows (i.e., 240) . Similar to MAM phase velocity dispersion analysis, t he window lengths were selected to contain at least 30 periods at the lowest 31 processing frequency that could be extracted from each MAM array (Vantassel and Cox, 2022) . Averaging the estimates from all windows within each block yielded a single data point per block, thus providing eight unique attenuation estimate s per frequency. This processing approach ensured that an equal number of attenuation data points were obtained at each frequency for all of the MASW and MAM arrays. The combined ambient -noise attenuation data from all MAM arrays and the active attenuation data from the MASW array are plotted together in Figure 11c. A good agreement is observed between the attenuation estimates derived from the MASW array and those obtained from the C300 and C700 arrays for frequencies ranging from 4 to 10 Hz. The MASW testing did not generate c oherent attenuation data a t frequencies less than 4 Hz, due to the limitations of the active sledgehammer source. However, the MAM testing was able to extract coherent attenuation data a t frequencies below 1 Hz. The agreement observed between the active - source and ambient noise attenuation estimates serves as compelling evidence for the efficacy of the proposed NFDBFa approach. However, it is notable that there is significant scatter in the attenuation estimates obtained using the C60 array. This variabilit y is likely attributed to the challenges previously discussed in regards to using smaller MAM arrays for attenuation studies, as the phase velocity data extracted from the C60 array was very good (refer to Figure 11a) . Hence, the attenuation data from the C60 array was removed prior to calculating attenuation statistics. The combined attenuation estimates from the MASW, C300, and C700 arrays, and the mean and ± one standard deviation attenuation estimates obtained from those three arrays, are depicted in Figure 11d. While a noticeable agreement exists among the three arrays, there is significantly greater scatter in the attenuation estimates (Figure 11d) compared to the phase velocity dispersion estimates (Figure 11c). Th is observation aligns with the findings reported by Aimar (2022). The application of the new NFDBFa approach in this field test showcases its effectiveness in estimating attenuation coefficient s from ambient noise wavefield data. Finally, the statistical experimental Rayleigh -wave phase velocity and attenuation parameters derived from both the MASW and MAM testing (refer to Figures 11b and 11d) were used to invert for Vs and Ds profiles at the Drainage Farm Site. This was achieved through the Monte Carlo -based joint inversion of phase velocity and phase attenuation data developed by Aimar et al. (2024b). Although the effectiveness of the joint inversion procedure has been proven for active surface wave data (Lai and Rix, 1998; Aimar et al., 2024b), its application to combined dispersion data from MASW and MAM testing, covering a broad frequency range, is n ovel. This is because past 32 studies on inverting MAM -based attenuation data to retrieve damping properties at large depths typically adopted an uncoupled inversion approach, based on a separate inversion of Rayleigh - wave phase velocity and α (e.g., Prieto et al., 2009; Parolai, 2014). The inversion s performed herein involved 50,000 five -layer trial Earth models with progressively increasing thicknesses, covering a comprehensive range of layer thicknesses, Vs, and Ds values. The layering was informed by a preliminary inversion study based solely on phase velocity dispersion data, which is omitted here for simplicity. Realistic values were fixed for the Poisson’s ratio and mass densities. A constant Dp/Ds ratio of 1.4 was used during the inversion , similar to the approach taken by Bergamo et al. (2023) . Forward dispersion and attenuation modeling were conducted using the Computer Programs for Seismology software (Herrmann, 2013). The fit to the experimental data was quantitatively assessed using a normalized root mean square (RMS) error that accounts for estimation uncertainty, similar to the metric proposed by Wathelet et al. (2004) . The ten best inversion results are shown in Figure 12. The theoretical phase velocity and attenuation curves are shown relative to the experimental data in Figures 12a and 12b , respectively . The Vs and Ds profiles are shown in Figures 12c and 12d, respectively, down to a depth of 400 m, which is approximately 1/ 2 of the maximum resolved phase velocity wavelength. The Vs profiles in Figure 12c collectively feature a shallow layer about 25 m thick with velocities ranging from approximately 90 to 185 m/s, including a low -velocity zone , which is consistent with known near - surface layering . Below this, there is generally a thicker layer extending down to approximately 180 m with velocities varying around 500 m/s. At depths of 150 -200 m, a stiff layer with velocities around 1500 m/s is commonly identified across the profiles. These depths, while variable, are consistent with th e location of Salt Lake Formation rock surface, as discussed above. The Ds profiles in Figure 12d indicate that damping in the top 25 m is less than approximately 1%. Below this depth, there is a noticeable variability in the estimated Ds values between the ten best profiles. Nonetheless, Ds can be observed to increase to approximately 2% to 4% in the deeper soil deposits, which consist of alternating clay, sand, and gravel layers. At the top of the Salt Lake Formation rock surface, Ds collectively decreases again to less than 2% for all of the ten best profiles . The large variability in Ds can be attributed to the complex geology of the site, the significant standard deviation in the experimental attenuation data, and the moderately low sensitivity of theoretical attenuation curves to Ds at greater depths (e.g., Badsar et al., 2012; Aimar et al., 2024b) . 33 Figure 12 . Inversion results for the experimental Rayleigh -wave phase velocity and attenuation data collected at the Drainage Farm Site in Logan, UT , USA . The figure highlights the ten best-fitting models, with Panels (a) and (b) comparing the theoretical curves for phase velocity and attenuation, respectively, against the experimental data represented by mean values with ± one standard deviation error bars. Panels (c) and (d) display the Vs and Ds profiles, respectively , for the ten best theoretical models. Despite these limitations, the joint inversion procedure successfully provided in-situ estimates of Ds at depths not reached by conventional site characterization techniques . This confirm s the advantages of combining MASW and MAM data for the combined estimation of stiffness and dissipation parameters of soil deposits. Conclusions A new methodology for estimating frequency -dependent attenuation coefficients through the analysis of ambient noise wavefield data recorded by 2D arrays of surface seismic sensors has been presented. The approach relies on the application of an attenuation -specific wavefield 34 conversion and frequency -domain beamforming (FDBF) . It has been termed the noise FDBF attenuation (NFDBFa) method. Importantly, using an FDBF approach , as opposed to a noise cross - correlation approach , enables the direction of ambient noise propagation to be determined for each noise window and frequency, and does not require an equipartitioned ambient noise wavefield . Furthermore, using a n FDBF approach enables the phase velocity and attenuation data generated from active -source testing like MASW to be combined with phase velocity and attenuation data generated from ambient noise testing like MAM in order to span a broader frequency range. This enables the joint inversion of phase velocity and attenuation to be performed as a means to extract shear wave velocity and small -strain damping ratio profiles to significantly greater depths than previously possible using only active -source data. Numerical simulations were conducted to deepen our understanding of the proposed NFDBFa method. These simulations aimed to evaluate how the proximity of the MAM array to the noise source , the presence of incoherent noise , and the size of the array affect the estimates of phase attenuation. The results demonstrated that near -field effects are more pronounced and extend over greater distances for phase attenuation estimates in comparison to those considered for phase velocity estimation. Furthermore, it was d iscovered that larger array sizes consistently provided more accurate phase attenuation estimates across all considered frequencies, contrary to the conventional MAM design criteria used for phase velocity dispersion estimation, where larger arrays are typ ically preferred for resolving lower frequencies while smaller arrays excel at resolving higher frequencies. This distinction emphasizes the need for unique design criteria when planning a MAM array for attenuation estimation. The proposed NFDBFa approach underwent validation through numerical wave propagation simulations, comparing predicted frequency -dependent phase attenuation values against theoretical phase attenuation curve s for two synthetic models . Furthermore, validation of the developed technique was reinforced using MASW and MAM field data collected at the Drainage Farm Site in Logan, Utah, USA . The phase velocity and attenuation data extracted from the MASW and MAM recordings agreed well over a common bandwidth, while the ambient noise MAM data allowed the phase velocity and attenuation estimates to be extracted at significantly lower frequencies. The joint inversion of the experimental Rayleigh -wave phase velocity and phase attenuation data obtained from both MASW and MAM testing facilitated the estimation of 35 shear wave velocity and small -strain damping ratio profiles to significant depths (400 m) at the Drainage Farm Site. As noted herein and in other studies like Aimar et al. (2024a), attenuation data are significantly more variable and more complex to understand (e.g., mod al curves that repeatedly cross one another) than phase velocity data. As such, there is a need for future studies to better understand attenuation data and how to invert them to retrieve reliable in -situ profiles of the small -strain damping ratio . Future efforts should involve additional numerical and experimental testing of diverse subsurface conditions , coupled with comparisons to damping estimates obtained from invasive tests. With the validity of this approach demonstrated on the vertical component, future research will also explore the utilization of the three components of the noise wavefield to enhance attenuation estimates beyond the current method's capabilities. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The numerical simulations were run on the Texas Advanced Computing Center’s (TACC’s) cluster Lonestar6, with an allocation provided by DesignSafe -CI (Rathje et al., 2017). This work was supported by the U.S. National Science Foundation (NSF) Grant Number CMMI -2120155. However, any opinions, findings, conclusions, or recommendations expressed in this article are those of the authors and do not necessarily reflect the views of the NSF. Research Data and Code Availability The field test data used to validate the NFDBFa approach presented in this paper are available in the dataset by Abbas et al. (2024). 36 References Abbas A, Cox BR, Dawadi N, Jackson N, and Cannon K (2024) Geotechnical site characterization at the Drainage Farm Site . 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2024-04-09

The small-strain damping ratio plays a crucial role in assessing the response of soil deposits to earthquake-induced ground motions and general dynamic loading. The damping ratio can theoretically be inverted for after extracting frequency-dependent Rayleigh wave attenuation coefficients from wavefields collected during surface wave testing. However, determining reliable estimates of in-situ attenuation coefficients is much more challenging than achieving robust phase velocity dispersion data, which are commonly measured using both active-source and ambient-wavefield surface wave methods. This paper introduces a new methodology for estimating frequency-dependent attenuation coefficients through the analysis of ambient noise wavefield data recorded by two-dimensional (2D) arrays of surface seismic sensors for the subsequent evaluation of the small-strain damping ratio. The approach relies on the application of an attenuation-specific wavefield conversion and frequency-domain beamforming. Numerical simulations are employed to verify the proposed approach and inform best practices for its application. Finally, the practical efficacy of the proposed approach is showcased through its application to field data collected at a deep, soft soil site in Logan, Utah, USA, where phase velocity and attenuation coefficients are extracted from surface wave data and then simultaneously inverted to develop deep shear wave velocity and damping ratio profiles.

A Frequency-Domain Beamforming Procedure for Extracting Rayleigh Wave Attenuation Coefficients and Small-Strain Damping Ratio from 2D Ambient Noise Array Measurements

2404.06650v1

1Spin injection characteristics of Py/graphene/Pt by gigahertz a nd terahertz magnetization dynamics driven by femtosecond laser pulse H. Idzuchi1-3*, S. Iihama4,5#, M. Shimura6, A. Kumatani1,2,6,7, S. Mizukami1,2,5, Y . P. Chen3,8,9,1,2,5 1 WPI Advanced Institute for Material s Research (AIMR), Tohoku University Sendai 980-8577, Japan 2 Center for Science and Innovation in Spintronics (CSIS), Tohoku University Sendai 980-8577, Japan 3 Purdue Quantum Science and Engineering Ins titute and Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA 4 Frontier Research Institute for Interdiscipli nary Sciences (FRIS), Tohoku University, Sendai 980-8578, Japan 5 Center for Spintronics Research Network (CSRN), Tohoku University, Sendai 980-8577, Japan 6 Graduate School of Environmental Studies, Tohoku University, Sendai 980-8579, Japan 7 WPI-International Center fo r Materials Nanoarchitectonics (MANA), National Institute for Material Science, Tsukuba 305-0044, Japan 8 School of Electrical and Computer Engi neering and Birck N anotechnology Center Purdue University, West Lafayette, Indiana 47907, USA 9Institute of Physics and Astronomy and Villum Center for Hybrid Quantum Materials and Devices, Aarhus University, 8000, Aarhus-C, Denmark *idzuchi@tohoku.ac.jp #) H. Idzuchi and S. Iihama cont ributed equally to this work. 2Abstract Spin transport characteristics of graphene has been extensively studied so far. The spin transport along c- axis is however reported by rather limited number of papers. We have studied spin transport characteristics through graphene along c-axis with permalloy(Py)/graphene(Gr)/Pt by gigahertz (GHz) and terahertz (THz) magnetization dynamics driven by femtosecond laser pulses. The relatively simple sample structure does not require electrodes on the sample. The graphene layer was pr epared by chemical vapor deposition and transferred on Pt film. The quality of graphene layer was chara cterized by Raman microscopy. Time resolved magneto-optical Kerr effect is used to characterize gi gahertz magnetization dynamics. Magnetization precession is clearly observed both for Pt/Py and Pt/Gr/Py. The Gilbert damping constant of Pt/Py was 0.015, indicates spin pumping effect from Py to Pt. T he Gilbert damping constant of Pt/Gr/Py is found to be 0.011, indicates spin injection is blocked by graph ene layer. We have also performed the measurement of THz emission for Pt/Py and Pt/Gr/Py. While the T Hz emission is clearly observed for Pt/Py, a strong reduction of THz emission is observed for Pt/Gr/Py. Wi th these two different experiments, and highly anisotropic resistivity of graphite, we conclude that th e vertical spin transport is strongly suppressed by the graphene layer. 3Recently, two-dimensional (2D) materials have attracted conside rable attention. Two-dimensional materials provide handful access on highly crystalline samples, offering new spintronics research directions. Since spin currents can flow in nonmagnetic materials, so far s uch spin transport is widely studied in in- plane direction of nonmagnetic 2D materials [1,2,3]. In three-d imensional materials such as Pt, spin transport in out-of-plane direction is often studied with spin Hall effect. The spin current is converted to charge transport with certain geometry: the spin polarization, the flow direction of spin current and the direction of detecti on voltage need to be a ll perpendicular to each other. This geometrical constraint makes it difficult to study out-of-plane spin transport through c-axis of 2D material while the spin transport in- plane has been relatively well studied. Here, we optically inve stigated spin transport characteristics in c-axis of graphene by using gigahertz ( GHz) and terahertz (THz) magnet ization dynamics excited by a femtosecond pulse laser. This makes it more easily to satisfy s uch geometrical conditions as the injector and detector are not required to be electrically connected. Figure 1 represents our sample structure as well as a brief mea surement set up. Here, we employed spin pumping and THz emission, both induced by magnetization dy namics excited by a pulsed laser as described below. Previously, vertical spin transport in multila yers of graphene have been studied by ferro magnetic resonance. Patra et al fabricated the sample on a co-p laner wave guide and used broad band frequency to characterize spin transport. They found the Gilber t damping is significantly enhanced for Py/Gr compared to Py/Pt where Py stands for permalloy (Ni 80Fe20) [4]. Later Gannett et al studied series of the samples with different thickness of Py to characterize transpor t properties, which shows no detectable enhancement for Py/Gr/Cu [5]. While the interface of graphene c an be complicated, in our experiment the sample structure is simple (just a multilayer film) and complim entary characteristics are obtained by two methods, which should help reveal t he intrinsic interface spin transport properties. In this study, spin transport was studied on Pt/graphene/Py an d Pt/Py . Pt, graphene, and Py were chosen for representative materials for spin Hall effect, 2D ma terial, and soft ferromagnet. Pt film was prepared by sputtering with the thickness of 3 nm on silicon su bstrate and glass substrate. The graphene film was transferred onto Pt in ambient condition. Graphene fil m was prepared on thin copper foil by a standard chemical vapor deposition (CVD) method and transferred onto the Pt film. Raman microscopy was used to characterize the number of layers in graphene where the laser wavelength is 532 nm. We have observed clear peaks of D, G, and 2D bands from left to right a s shown in Fig.2a. Particularly from the 2D peak, we confirmed the crystallinity of the graphene film and t he film was not folded [6,7]. The Py film and MgO capping layer was sputtere d on graphene film with the base pressure of ~ 10-7 T o r r . T h e s t a t i c magnetization process of the film was examined by magneto optic al Kerr effect. For measuring GHz magnetization dynamics induced by femtosecond laser pulse (Fig. 1a), time-resolved magneto-optical Kerr effect (TRMOKE) was employed [8]. The wavelength, pulse duratio n, and repetition rate for both the pump and probe laser pulses were 800 nm , 120 fs, and 1 kHz, respecti vely. The pump laser beam was irradiated on the sample from the film normal and the incident angle of th e probe laser beam was ~ 5 degree measured from the film normal. Kerr rotation angle of the reflected prob e beam was detected via balanced photo- 4detector. The pump laser pulse was modulated by the mechanical chopper with the frequency of 360 Hz and then the pump-laser induced change in Kerr rotation angle w as detected by a Lock-in amplifier. A magnetic field was applied with a 10 degree angle measured from the film normal. The magnetization precession can be excited by the reduction of demagnetizing fie ld due to laser heating. The damping of magnetization precession reflects the transfer of spin into adj acent normal metal layer, referred as spin- pumping effect [9, 10]. To study spin-transport induced by THz magnetization dynamics, THz time-domain spectroscopy was employed [11] (Fig. 1b), in which THz spin-cur rent can be generated by ultrafast demagnetization of Py layer and its angular momentum can be tra nsferred to Pt layer [12,13]. Then, THz electric field can be generated through spin-to-charge conversi on (inverse spin Hall effect) in Pt layer. Wavelength, pulse duration, repetition rate for the laser pulse were 800 nm, 120 fs, and 80 MHz, respectively. The femtosecond laser was irradiated from substrate side and th en THz electric field emitted from the film side was measured. The THz electric field was detected by elect ro-optic sampling method using a ZnTe (110) crystal. All measurements were conducted at room temperat ure. The spin transport of graphene in vertical direction can be accessed by spin pumping with additional layers. We compare Pt/P y bilayer with Pt/graphene/Py trilayers to characterize spin transport properties across graphene layer. Figure 2b shows typical TRMOK E signal for Pt/Gr/Py(10nm)/MgO on Si substrate under the external magnetic field of 10.7 kOe in a di rection tilted by 10 degrees from perpendicular to the substrate. We have clearly observed spin precession slow ly decaying over long period right after initial ultrafast dynamics, for the samples of both with and wi thout graphene layer. Those oscillations are fitted to the following equations 𝐴𝐵⋅e x p ሺെ𝑣𝑡 ሻ𝐶⋅e x p ሺെ𝑡 𝜏⁄ሻsinሺ2𝜋𝑓𝑡 𝜙 ሻ where, A, B, , C, f, , and 0 are signal offset, magnitude of exponential background signal due to recovery of magnetization, decay rate, oscillation amplitude, oscillatio n frequency, oscillation life-time, and initial phase, respectively. The TRMOKE signals are well fitted by the above equation shown as solid curve in Fig. 2(b). The f and 1/ values evaluated by fitting with different applied magnetic fi elds are shown in Fig. 3. f and 1/ can be calculated theoretically using Landau-Lifshitz Gilbert (LLG) equation as [8,14], 𝑓ୋ ൌఊ ଶగඥ𝐻ଵ𝐻ଶ, ( 1 ) ଵ ఛైైృൌଵଶ𝛼𝛾ሺ𝐻ଵ𝐻 ଶሻ, ( 2 ) 𝐻ଵൌ𝐻c o s ሺ𝜃െ𝜃 ுሻെ4 𝜋 𝑀 ୣcosଶ𝜃, (3) 𝐻ଶൌ𝐻c o s ሺ𝜃െ𝜃 ுሻെ4 𝜋 𝑀 ୣcos 2𝜃 , (4) where H, H (=10 degree in this study), , 4Meff, , and are external magnetic f ield, field angle, magnetization angle, effective de magnetizing field, gyromagneti c ratio and Gilbert damping constant respectively. is given by the relation, =gB/ℏ. The is determined by the energy minimum condition as, 5𝐻s i n ሺ 𝜃 ுെ𝜃 ሻ2 𝜋 𝑀 ୣsin 2𝜃 ൌ 0 , (5) The measured f is well fitted by Eq. (1) with the parameters g = 2.09 (2.06) and 4 Meff = -9.8 (-8.8) kOe for Pt / Gr / Py (Pt / Py) film. The 1/ calculated using Eq. (2) are sh own in Fig. 3(b) and 3(c). 1/ for Pt / Gr / Py / MgO sample (Fig. 3(c)) can be well explained by Eq. (2) wi th = 0.011. However, 1/ for Pt / Py / MgO cannot be explained by Eq. (2), which is due to inhomogeneo us linewidth broadening. Therefore, 1/ enhancement due to inhomogeneous linewidth broadening is consid ered as follows, ଵ ఛ౪౪ൌଵ ఛైైృଵଶቚௗሺଶగ ైైృ ሻ ௗఏಹቚΔ𝜃ு , ( 6 ) where, the first term is identi cal to Eq. (2) and the second te rm is 1/ enhancement due to distribution of H which may be related to surface roughness of the film [14]. The black solid and blue dashed curves in Fig. 3(b) are the calculated results of the first and second terms o f Eq. (6). Hext dependence of 1/ for Pt / Py / MgO filmis well explained by the summation of two contributions in Eq. (6) with the parameters, = 0.015 and H = 0.05 rad [green broken curv e in Fig. 3(b)]. The enhancement of is due to spin-pumping effect at Pt / Py interface associated with dissipation of angular mom entum. This indicates strong suppression of spin current with graphene, cons istent with Gannett et al [5]. Previously, it was shown that graphene has long spin diffusion length by means of lateral spin transport w here spin current is flowing in-plane with long spin lifetime probed by Hanle effect [1,15]. Transport alo ng c-direction can be rather different from the one in ab plane. In early studies, graphite crystal shows rather anisotr opic charge transport properties and the resistivity of c-axis is reported to larger than the one for ab plane by a factor of 102 to 103 [16]. The resistive nature of the graphene along c axis may prevent spin transport. In the THz method, by irradiating femtosecond laser pulse on th is kind of multilayers, THz electric field can be generated [12, 13]. Ultrafast spin current in nonm agnetic layer can be generated by the ultrafast demagnetization in Py layer, and spin-charge conversion via inv erse spin-Hall effect in Pt layer create terahertz charge current and electric field. In our bilayer Pt/ Py, we observed clear THz emission and its signal is inverted with reversed bias magnetic field, as shown in the top panel (a) of Fig. 4. Interestingly, on two Pt/graphene/Py samples, the THz signal was largely suppress ed [Fig. 4(b) and 4(c)]. This implies strong suppression of spin injection from Py to Pt by graphene monolay er in the terahertz frequency region. The interpretation is qualitatively c onsistent with spin pumping st udy (using ultrafast laser heating and GHz magnetization dynamics). The strong reduction of THz signal is attributed to the strong suppression of spin- transport by inserting graphene monolayer with high resistivity along the c-axis. Strong reduction of THz emission was also reported for Co/ZnO/Pt junctions[17]. With th ese two different characterization methods, we conclude graphene monolayer effectively blocks vertical spin current. For the second sample of Pt/graphene/Py (Fig.4c), a small peak appeared around 1 ps. We notice this may or may not be a delayed THz emission, whose precise mechanism (e.g., how it may be rela ted to the graphene barrier, whether it may also be generated by Py itself etc.) is not clear yet at th is stage and open for future study. In conclusion, we have investigat ed spin injection characterist ics of Py/graphene/Pt by means of 6gigahertz and terahertz magneti zation dynamics driven by a femt osecond laser pulse. Graphene layer was grown by CVD method and the Raman characteristics on Pt showed the characteristics of single layer graphene film. We have clearly observed GHz magnetization prece ssion induced by the laser pulse for the samples of both with and without graphene (Py/Pt). Graphene is observed to give an apparent suppression of the damping enhancement due to spin-pumping effect at Py / P t interface, indicating reduction of angular momentum dissipation by graphene monolayer. THz emission induce d by femto-second laser pulse was observed for Py/Pt bilayer, while the THz emission was strongly suppressed for Py/graphene/Pt, which clear indicates graphene blocks spin current in transport along c-axis. Both experiments on spin pumping and THz method can be understood by th e strongly suppressed spin tr ansport across the graphene layer. Data Availability Statements The data that support the findings of this study are available from the corresponding author upon reasonable request. Acknowledgments We acknowledge the support from AIMR common equipment unit. Thi s work was supported in part by Advanced Institute for Materials R esearch (AIMR) under World Pr emier International Research Center Initiative (WPI) of MEXT, Japan, and by AIMR fusion research pr ogram, by the Mazda Foundation, and by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), JSPS KAKENHI (G rant Number 18H 03858, 18H0447 3, 20H04623, and 20K14399). 7References [1] N. Tombros et al., Nature 448, 571 (2007). [2] Y. P. Liu et al., Appl. Phys. Lett. 102, 033105 (2013). [3] H. Idzuchi, A. Fert, Y. Otani, Phys. Rev. B 91, 241407 (2015). [4] A. K. Patra et al., Appl. Phys. Lett. 101 162407 (2012). [5] W. Gannett et al., J. Appl. Phys. 117 213907 (2015). [6] L. M. Malard et al., Phys. Rep. 473 51 (2009). [7] A. C. Ferrari, D. M. Basko, Nat. Nanotech. 8, 235 (2013). [8] S. Iihama et al., Phys. Rev. B 89, 174416 (2014). [9] Y. Tserkovnyak et al., Phys. Rev. Lett. 88, 117601 (2002). [10] S. Mizukami et al., Phys. Rev. B 66, 104413 (2002). [11] Y. Sasaki et al., Appl. Phys. Lett. 111, 102401 (2017). [12] T. Kampfrath et al., Nat. Nanotech. 8 256 (2013). [13] T. Seifert et al., Nat. Photon. 10 485 (2016). [14] S. Mizukami et al., Jpn. J. Appl. Phys. 40, 580 (2001). [15] D. Khokhriakov et al., Carbon 161, 892 (2020). [16] W. Primak and L. H. Fuchs, Phys. Rev. 95 22 (1954). [17] G. Li et al., J. Phys. D: Appl. Phys. 51, 134001 (2018). 8Figures Fig. 1. Concept and schematic image of experimental set up of this stu dy. Graphene, spin injector (Py) and detector (Pt) are depicted in black hexagon, purple box, and gray box, r espectively. (a) The set up for pump-probe measurement for magnetization dynamics. The probe beam is sligh tly (~ 5 deg) tilted from the film normal. The magnetic field is applied with a 10 degree angle measured from the film normal. (b) The set up for THz emission. The magnetic field is applied in plane. Fig.1 9 Fig. 2. (a) Raman spectroscopy of a graphene layer transferred on to a glass/Pt substrate (glass with Pt sputtered). Three main Raman peaks (D, G, and 2D) are clearly observed. Ins et shows the 2D peak, clearly different from bilayer or multilayers graphene [6]. (b) Magneto-optical Kerr effect (MOKE ) signal plotted as a function of pump-probe delay time for Pt/Gr/Py(10nm)/MgO under an external magnetic fi eld of 10.7 kOe. The field is applied with a 10degree angle measured from the film normal. The measurement s et up is schematically shown in Fig.1a. Fig.2 10 Fig. 3. Characterization of GHz magnetization dynamics for multilayers with and without graphene. The measurement set up is schemati cally shown in Fig.1a. (a) preces sion frequency as a function of the magnetic field. The field is applied with a 10 degree angle measured from the f ilm normal. Closed red squares and open green circles indicate data from the Pt/Gr/Py/MgO and Pt/Py/MgO respectively where the thickness of Py is 10 nm for both case. The solid curves are obtained by Eq. (1) with the parameters in the main text. Inverse lifetime as a function of the magnetic field for (b) Pt / Py / MgO and (c) Pt / Gr / Py / MgO films. The black solid curves shown in (b) and (c) correspond to the calculated valu e using LLG eq. (Eq. (2)). The green dotted and blue broken curves shown in (b) are the left hand side and the second term in the right hand si de in Eq. (6), respectively, with the parameters in the main text. Fig.3 11 Fig. 4. Detection of THz electric field generated by femto-second laser pulse on (a) Pt(3) / Py(2) /MgO , (b) Pt(3) / Gr / Py(2) / MgO and (c) Pt(3) / Gr / Py(5) / MgO made on glass substrates. The measurement set up is schematically shown in Fig.1b. The numbers in bracket indicate the thickness of the layers in the unit of nanometers. The magnetic field was applied in in-plane direction. Blue solid and red ope n symbols are the signal obtained with opposite magnetic field direction. Fig.4

2020-10-15

Spin transport characteristics of graphene has been extensively studied so far. The spin transport along c-axis is however reported by rather limited number of papers. We have studied spin transport characteristics through graphene along c-axis with permalloy(Py)/graphene(Gr)/Pt by gigahertz (GHz) and terahertz (THz) magnetization dynamics driven by femtosecond laser pulses. The relatively simple sample structure does not require electrodes on the sample. The graphene layer was prepared by chemical vapor deposition and transferred on Pt film. The quality of graphene layer was characterized by Raman microscopy. Time resolved magneto-optical Kerr effect is used to characterize gigahertz magnetization dynamics. Magnetization precession is clearly observed both for Pt/Py and Pt/Gr/Py. The Gilbert damping constant of Pt/Py was 0.015, indicates spin pumping effect from Py to Pt. The Gilbert damping constant of Pt/Gr/Py is found to be 0.011, indicates spin injection is blocked by graphene layer. We have also performed the measurement of THz emission for Pt/Py and Pt/Gr/Py. While the THz emission is clearly observed for Pt/Py, a strong reduction of THz emission is observed for Pt/Gr/Py. With these two different experiments, and highly anisotropic resistivity of graphite, we conclude that the vertical spin transport is strongly suppressed by the graphene layer.

Spin injection characteristics of Py/graphene/Pt by gigahertz and terahertz magnetization dynamics driven by femtosecond laser pulse

2010.07694v1

1 Impact of Fe 80B20 insertion on the properties of dual-MgO perpendicular magnetic tunnel junctions Enlong Liu1, Taeyoung Lee2 and Hyunsoo Yang1 1 Department of Electrical and computer Engineering, National University of Singapore, 117576, Singapore 2 GLOBALFOUNDRIES Singapore Pte. Ltd., Singapore 738 406, Singapore E-mail: eleyang@nus.edu.sg Abstract We explore the impact of Fe 80B20 inserted at both Co 20Fe60B20/MgO interfaces of dual-MgO free layers (FLs) in bottom-pinned magnetic tunne l junctions (MTJs). MTJ stacks are annealed for 30 min at 350 °C and 400 °C in a vacuum after film deposition. Current-in-plane tunneling measurements are carried out to characteri ze magnetotransport properties of the MTJs. Conventional magnetometry measurements and ferromagnetic resonance are conducted to estimate the saturation magnetization, the effective perpendicular anisotropy field and the Gilbert damping of dual-MgO FLs as a function of the Fe 80B20 thickness and annealing temperatures. With ultrathin Fe 80B20 (0.2 0.4 nm) inserted, perpendicular magnetic anisotropy (PMA) of FLs increases with si milar tunnel magneto-resistance (TMR) and low damping values. As Fe 80B20 layer thickness further increases (0.6 1.2 nm), both TMR and PMA degrade, and damping increases dramatically. This study demonstrates a novel approach to tune properties of MTJ stacks with dual-MgO FLs up to 400 °C annealing, which enables MTJ stacks for various applications. 1. Introduction Magnetic tunnel junctions (MTJs) with perpendi cular magnetic anisotropy (PMA) have been studied in the recent decades as the crucial el ement for next generation memory applications, such as spin-transfer-torque and spin-orb it-torque magnetic random access memory (STT/SOT-MRAM), due to their non-volatility, en ergy effectiveness, high endurance, and scalability [1–3]. Many efforts have been made in the engineering of the data-storage layer in MTJs, i.e. the free layer (FL), whose magnetic moment can be switched by the writing current. Especially, dual-MgO FLs with a structure as MgO/CoFeB/spacer/CoFeB/MgO have been under intense development [4–6]. On the one hand, dual-MgO FLs can provide high PMA and low damping to guarantee the high thermal stability and low switching current, after scaling- down of MTJ devices [7]. On the other hand, the aformentioned properties of dual-MgO FLs can be maintained after post-annealing up to 400 °C, which is required for the CMOS back-end-of-line (BEOL) process [8]. To further optimize dual-MgO FLs performanc e, previous studies focused on different topics. Among them, non-magnetic spacer engineering and element composition effects have drawn lots of interest. Researches on non-ma gnetic spacer sandwiched between two CoFeB layers in dual-MgO FLs have been widely conducte d. Materials such as Mo [9–11], Ta [8,12], and W [13–15] were explored as the spacer to identify its impact on PMA and damping before 2 and after annealing. Other works examined the effect of element (Fe or B) composition in CoFeB layers in MTJ stacks on several parameters, including tunnel magneto-resistance (TMR) [16], PMA [17–19], and annealing stability [20]. It has been demonstrated that a thickness gradient in the B content can modify the properties of CoFeB/MgO bilayer system such as damping and anisotropy [21]. PMA of th e FL was also reported to be improved with increasing the Fe composition in CoFeB, on which its TMR is almost independent [22]. However, it is still an open question how a gradient of Fe in dual-MgO FLs impacts the overall properties of MTJ stacks. In such a case, th e PMA of dual-MgO FLs would benefit from an increased Fe concentration, while the TMR of the MTJs is expected to be improved due to the formation of Fe/MgO/Fe interface after annealing [23]. Here we propose an insertion of ultrathin Fe 80B20 (hereafter FeB) at the interface of Co20Fe60B20/MgO in dual-MgO FLs to achieve tunable magnetic properties and annealing stability. By optimizing the thickness of the FeB insertion layers at both CoFeB/MgO interfaces, a large PMA can be obtained after 350 °C annealing and further improved at 400 °C, which is accompanied with a low damping constant. The result of a low saturation magnetization, large anisotropy field and low damping at the same time in the FeB-inserted dual-MgO FLs makes it promising for low switching currents in MTJ devices without reducing the thermal stability [24]. In addition, the tunability of FL performance by FeB insertion enlarges its potential for various spintronic appl ications where CoFeB-based MTJs are present, such as SOT-MRAM, spin logic devices and STT nano-oscillators. 2. Experimental Bottom-pinned perpendicular MTJs with [Co/Pt ] multilayers as a perpendicular synthetic antiferromagnet (p-SAF) were in-situ deposited at room temperature by magnetron sputtering on W/Ru/W/Ru/W bottom electrodes (BE) and ca pped by the Ta/Ru top electrode (TE) in a ULVAC Magest S200 multi-chamber machine. All samples were first annealed with a 0.5 T magnetic field perpendicular to film plane in a magnetic vacuum annealing oven at 350 °C for 30 min. The TMR and resistance-area product (RA) of the MTJ stack was measured via current- in-plane tunnelling method (CIPT) [25]. The hyst eresis loops of blanket stacks were measured by a vibrating sample magnetometer (VSM) with the magnetic field perpendicular to the sample plane. Field-modulated ferromagneti c resonance (FMR) measurements with the frequency range of 10-25 GHz were conducted to ex tract the resonance field and linewidth of the FL versus the frequency, from which the effective perpendicular anisotropy field and Gilbert damping of the FL can be estimated. All FMR measurements were conducted with samples placed film-side down on a coplanar waveguide in an electromagnet with a field range up to 0.5 T and perpendicular to the sample pl ane. The same batch of samples were then annealed at 400 °C for 30 min to study the annealing impact. 3. Results and discussion 3.1 Stack characterization without FeB insertion The detailed stack structure of MTJs used in this study is provided schematically in figure 1(a). It consists of (thickness in nm): Hard layer (HL): Pt (5)/[Co (0.25)/Pt (0.2)]/Co (0.6) Reference layer (RL): Co (0.6)/Pt (0.2)/ Co (0.3)/Pt (0.2)/Co (0.5)/W (0.3)/Co 20Fe60B20 (0.8) 3 Free layer (FL): Co 20Fe60B20 (1.2)/W (0.4)/ Co 20Fe60B20 (0.8). The FL is sandwiched between the MgO tunnel barrier and the 2nd MgO layer to form a dual- MgO structure. The magnetic hysteresis loop of the full stack in figure 1(b) indicates good PMA in each functional layer after 350 °C annealing. From the minor hysteresis loop shown in figure 1(c), PMA of the FL is still maintained after 400 °C annealing, and the coercive field also increases slightly. The reduction of the saturation magnetization per area ( 𝑀௦∙𝑡) of the FL after 400 °C annealing can be attributed to magnetic dead layer formation, which will be discussed in the following sections. After 400 °C annealing, a sloped plateau is observed around 500 mT in the red curve in figure 1(b), indicating a decreased PMA in the RL [26]. In addition, the TMR of the stack is reduced from 110% to 50%. Since the FL PMA is maintained, the TMR reduction is attributed to a PMA loss in the RL. Thus, the impact of Fe B insertion in dual-MgO FLs on TMR is studied in stacks after 350 °C annealing. 3.2 Impact of FeB insertion on TMR and RA Figure 2(a) and (b) show schematically the FeB insertion position in the dual-MgO FL. Top FeB is inserted between the 2nd MgO layer and CoFeB above W spacer, while bottom FeB is between the MgO tunnel barrier and CoFeB below W spacer. To eliminate the difference in the thickness of FL after insertion, the total thickness of FeB insertion plus remaining CoFeB is kept at 0.8 nm and 1.2 nm for layers above and below W spacer, respectively. As such, the thickness of the top FeB insertion layer is chosen as 0.2, 0.4, 0.6 and 0.8 nm. For the bottom FeB insertion layer, its thickness options are 0.2, 0.4, 0.6, 0.8, 1.0 and 1.2 nm. The TMR and RA values as a function of the FeB insertion layer at different CoFeB/MgO interfaces are summarized in figure 2(c) and (d), respectively. The trend for both TMR and RA is similar regardless of FeB insertion position. The TMR value is similar for the thickness of FeB < 0.4 nm, but the RA value reduces for thin FeB insertion < 0.6 nm and increases with thicker FeB. In addition, it is found that top FeB insertion leads to a more pronounced RA reduction. This phenomenon indicates that with top FeB insertion, the contribution to RA from the 2 nd MgO layer can be reduced. Since the MgO tunnel barrier (0.85 nm) is thicker than the 2nd MgO layer (0.7 nm), its RA change due to bottom FeB insertion is not as significant as in the top FeB case. Overall, a similar TMR value at low RA is realized when ultrathin FeB (0.2 0.4 nm) is inserted, suggesting that th e formation of Fe/MgO interface benefits magnetotransport properties of MTJ stacks. The TMR starts to drop when FeB thicker than 0.6 nm was inserted at the bottom CoFeB/MgO interface, but the RA value does not show any significant change. This TMR drop is attributed mainly to a lower spin polarization when thicker FeB replaces CoFeB in the FL [27]. 3.3 Impact of FeB insertion and annealing on FL magnetic properties In figure 1(c), an example of hysteresis loop of the FL is shown. The saturation magnetization per area of the FL can be estimated by using 𝑀௦∙𝑡ൌ𝑚 /𝐴, where 𝑚 is the magnetic moment, 𝑡 is the thickness of FL, and 𝐴 is the sample area. The effective anisotropy field ( 𝜇𝐻) is derived from FMR measurements, as shown by exemplary data in figure 3(a) from dual-MgO FLs without FeB insertion, with 0.2 nm top FeB insertion and 0.2 nm bottom FeB insertion, all after 350 C annealing. For each sample, the power absorption by FL versus the applied field 4 scan is measured at various frequencies, fro m which the ferromagne tic resonance field ( 𝜇𝐻௦) and linewidth ( 𝜇Δ𝐻) is estimated. The relation between 𝜇𝐻௦ and frequency 𝑓 is described by the Kittel equation for the out-of-plane applied field [28]: 𝑓ൌఊ ଶగ൫𝜇𝐻௦𝜇𝐻൯ (1) where 𝛾 is the gyromagnetic ratio. From figure 3(a), the x-intercept is the 𝜇𝐻. From 𝑀௦∙𝑡 and 𝜇𝐻, the effective perpendicular anisotropy energy can be calculated as 𝐾∙𝑡ൌଵ ଶ𝜇𝐻ሺ𝑀௦∙𝑡ሻ. (2) Figure 4 summarized the results calculated by the above method. First, 𝑀௦∙𝑡 of FLs can be described simply by 𝑀௦∙𝑡ൌ𝑀 ௦ி∙2.0െ𝑡ி∙ሺ𝑀௦ிെ𝑀௦ிሻ, where 𝑡ி is the inserted thickness of FeB. It is found in figure 4(a) and (b) that 𝑀௦∙𝑡 decreases with thicker inserted FeB, i.e. the slope is negative. Thus, in our stack 𝑀௦ி is smaller than 𝑀௦ி. Next, for the top FeB insertion case in figure 4(a), the amount of change in 𝑀௦∙𝑡 at the same FeB insertion thickness after 350 °C annealing is larger than that of the bottom insertion case in figure 4(b). This indicates that the top FeB inserted layer is more damaged than the bottom inserted FeB. Another major difference between top and bottom insertion cases is a larger 𝑀௦∙ 𝑡 reduction after 400 ºC annealing compared to that after 350 °C annealing, with thick bottom FeB inserted (> 0.4 nm). As the bottom-inserted FeB is less damaged, it still reduces the 𝑀௦∙𝑡 value on further annealing at 400 ºC. However, for the top inserted case, there is not much change after 400 ºC annealing, supporting that the top FeB insertion layer is damanged, mainly due to the 2nd MgO deposition. The FeB thickness dependence of 𝜇𝐻 and 𝐾∙𝑡 can be discussed together. In the top FeB insertion case after 350 ºC annealing, 𝜇𝐻 in figure 4(c) increases monotonically with FeB insertion. Due to the reduction in 𝑀௦∙𝑡, however, this increase cannot lead to a higher PMA. In figure 4(e), the PMA of FL after 350 ºC annealing changes little, and even slightly decreases with thicker FeB. However, after 400 ºC annealing, 𝜇𝐻 overall increases without any clear dependence on the FeB thickness, and the PMA of FL reaches the maximum value when 0.2 nm FeB is inserted. On the other hand, the behavior of 𝜇𝐻 and 𝐾∙𝑡 in bottom-inserted FeB cases is different. In general, the PMA can be improved more significantly with bottom-inserted FeB than top-inserted FeB cases. In 350 ºC annealing cases, both 𝜇𝐻 and 𝐾∙𝑡 increase till 1.0 nm FeB insertion as shown in figure 4(d) and figure 4(f), respectively. While in 400 ºC annealing cases, 𝐾∙𝑡 is significantly improved in thin FeB (0.2 – 0.6 nm) cases. To summarize, the impact of FeB on the PMA of FL differs according to the insertion position. Regardless of annealing temperature, FeB insertion at the bottom CoFeB/MgO interface induces a larger PMA, probably due to less damage and the formation of Fe/MgO interface. By changing the FeB insertion position and thickness, magnetic properties of dual-MgO FL can be tuned in a wide range. It can be noticed that the vertical error bar of anisotropy field is huge when the FeB thickness is 0.8 nm in the top insertion case, and 1.0 or 1.2 nm in bottom insertion cases. It reflects that the resonance field is difficult to be determined precisely in those cases, and thus the fits of Eq.(1) contains large uncertainties. It is probabl y due to a large damping constant in thick FeB insertion cases, which will be discussed in the following section. 3.4 Impact of FeB insertion layer on FL damping 5 In order to evaluate the effect of FeB insertion on the FL damping, FMR measurements are conducted. In figure 3(b), 𝜇Δ𝐻 versus the external excitation frequency for the three samples was plotted. The linewidth of the resonance is linear in frequency: 𝜇Δ𝐻 ൌ 𝜇 Δ𝐻ସగఈ ఊ𝑓 (3) where 𝜇Δ𝐻 is the inhomogeneous linewidth broadening and 𝛼 is the Gilbert damping coefficient. Figure 5 summarizes 𝛼 as a function of top or bottom FeB insertion thickness under different annealing conditions. For top FeB insertion (figure 5(a)), the influence of FeB insertion on 𝛼 shows a moderate dependence on its thickness after 350 C annealing. For bottom FeB insertion (figure 5(b)), however, 𝛼 reaches the minimum at 0.4 nm FeB insertion and increases dramatically beyond the measurement range w ith the FeB thickness in both annealing conditions. For those cases, the resonance is to o broadened to be resolved, reflecting a very large damping [21]. It also leads to a huge uncertainty in the resonance field and hence 𝜇𝐻 determination, as mentioned in the preivous section. An increase of 𝛼 is observed at 0.6 nm FeB in the 400 C annealing condition. However, a reduction in 𝛼 after 400 C annealing is obtained when ultrathin FeB (0.2 0.4 nm) is inserted at either interface. This suggests that the annealing treatment has different effects on 𝛼 of the samples with various FeB insertion thicknesses. Perhaps different amount of FeB insertion leads to changes in the Fe concentration, micr ostructures and crystallization of dual-MgO FLs after boron depletion upon annealing and thereby to different damping behaviors [29,30]. Finally, Table 1 summarizes and compares MTJ stacks after 350 C annealing with FeB insertion at top, bottom, or both CoFeB/MgO interfaces. As the FeB insertion thickness and position differ, the PMA of the dual-MgO FL can be tuned in a wide range, while the damping is almost independent. From the systematic studies on insertion thickness in the previous sections, top and bottom FeB are optimized to be 0.2 nm and 0.4 nm, respectively. As a result, the MTJ stack with such FeB insertion at both interfaces in dual-MgO FLs can be engineered with a high TMR, low RA, large 𝐾 ∙𝑡, low 𝑀௦∙𝑡, high 𝜇𝐻, and low damping constant. 4. Conclusion In this paper, we explore the impact of Fe 80B20 layer inserted at two interfaces of Co40Fe60B20/MgO in dual-MgO FLs in MTJ stacks and its annealing stability. With ultrthin FeB (0.2 0.4 nm) inserted at the top or bottom CoFeB/MgO interface, the TMR can be maintained with lower RA values, while the top-FeB insertion results in a more RA drop with a similar TMR. In both cases, the FL saturation magnetization reduces with increasing the Table1. Comparison of magnetotransport and magnetic proper ties of dual-MgO FLs with different FeB insertion after 350 C 30 min annealing. FL types TMR RA 𝑀௦∙𝑡 𝜇𝐻 𝐾∙𝑡 % m2 10-5 A mT mJ∙m2 10-3 MgO/CoFeB(1.2)/W/CoFeB(0.8)/MgO 109.3 8.5 184.0 2.7 320 1 0.294 0.004 12.5 2.6 MgO/CoFeB(1.2)/W/CoFeB(0.6)/ FeB(0.2) /MgO 114.9 6.9 177.7 3.1 340 5 0.302 0.007 15.0 4.9 MgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.8)/MgO 108.6 8.3 181.2 1.7 349 2 0.316 0.006 11.3 5.0 MgO/ FeB(0.2) /CoFeB(1.0)/W/CoFeB(0.6)/ FeB(0.2) /MgO 108.2 7.5 175.7 2.6 381 2 0.335 0.005 12.1 0.7 MgO/ FeB(0.4) /CoFeB(0.8)/W/CoFeB(0.6)/ FeB(0.2) /MgO 110.5 8.6 168.3 2.9 434 2 0.365 0.006 4.4 0.7 6 inserted FeB thickness, while the FL effective anisotropy field increases. However, the PMA of dual-MgO FLs with FeB inserted at the bo ttom interface shows a larger improvement than its top FeB insertion counterpart, even after 400 C annealing. At the same time, the FeB (0.2 0.4 nm) insertion at either interface reduces the damping constant in the FL. By optimizing the FeB insertion layer thickness, the dual-MgO FL with a low saturation magnetization, high effective anisotropy field and low damping can be achieved after 400 C annealing. However, the performance degrades if a thicker FeB is used to replace CoFeB in dual-MgO FLs. This study demonstrates a novel approach to tune dual-MgO FL properties other than typical boron composition or non-magnetic spacer engineering. By using the FeB insertion layer at CoFeB/MgO interfaces, magnetic properties of the FL and the magnetotransportation of MTJs can be engineered in a wide range, which enables MTJs to meet different performance requirements for various spintronic applications. Acknowledgements This work was supported by NRF In vestigatorship (NRFI06-2020-0015). References [1] Ikeda S, Miura K, Yamamoto H, Mizunuma K, Gan H D, Endo M, Kanai S, Hayakawa J, Matsukura F and Ohno H 2010 A perpendicular-anisotropy CoFeB-MgO magnetic tunnel junction. Nat. 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Magn. 48 3829–32 [18] Honjo H, Ikeda S, Sato H, Sato S, Watanabe T, Miura S, Nasuno T, Noguchi Y, Yasuhira M, Tanigawa T, Koike H, Muraguc hi M, Niwa M, Ito K, Ohno H and Endoh T 2016 Improvement of Thermal Tolerance of CoFeB–MgO Perpendicular-Anisotropy Magnetic Tunnel Junctions by Controlling Boron Composition IEEE Trans.Magn. 52 1–4 [19] Srivastava S, Chen A P, Dutta T, Ramaswamy R, Son J, Saifullah M S M, Yamane K, Lee K, Teo K-L, Feng Y P and Yang H 2018 Effect of (Co xFe1-x)80B20 Composition on the Magnetic Properties of the Free Layer in Double-Barrier Magnetic Tunnel Junctions Phys. Rev. Appl. 10 024031 [20] Swerts J, Liu E, Couet S, Mertens S, Rao S, Kim W, Garello K, Souriau L, Kundu S, Crotti D, Yasin F, Jossart N, Sakhare S, Devolder T, Van Beek S, O’Sullivan B, Van Elshocht S, Furnemont A and Kar G S 2017 Solving the BEOL compatibility challenge of top-pinned magnetic tunnel junction stacks 2017 IEEE International Electron Devices Meeting (IEDM) vol 2017-Decem (IEEE) pp 38.6.1-38.6.4 [21] Drobitch J L, Hsiao Y-C, Wu H, Wang K L, Lynch C S, Bussmann K, Bandyopadhyay S and Gopman D B 2020 Effect of CoFe dusting layer and annealing on the magnetic properties of sputtered Ta/W/CoF eB/CoFe/MgO layer structures J. Phys. D. Appl. Phys. 53 105001 [22] Bersweiler M, Sato H and Ohno H 2017 Magnetic and Free-Layer Properties of MgO/(Co)FeB/MgO Structures: Dependence on CoFeB Composition IEEE Magn. Lett. 8 1–3 [23] Yuasa S, Nagahama T, Fukushima A, Suzuki Y and Ando K 2004 Giant room- temperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions Nat. Mater. 3 868–71 [24] Thomas L, Jan G, Serrano-Guisan S, Liu H, Zhu J, Lee Y-J, Le S, Iwata-Harms J, Tong R-Y, Patel S, Sundar V, Shen D, Yang Y, He R, Haq J, Teng Z, Lam V, Liu P, 8 Wang Y-J, Zhong T, Fukuzawa H and Wang P 2018 STT-MRAM devices with low damping and moment optimized for LLC applications at 0x nodes 2018 IEEE International Electron Devices Meeting (IEDM) vol 2018-Decem (IEEE) pp 27.3.1- 27.3.4 [25] Worledge D C and Trouilloud P L 2003 Magnetoresistance measurement of unpatterned magnetic tunnel junction wafers by current-in-plane tunneling Appl. Phys. Lett. 83 84–6 [26] Devolder T, Couet S, Swerts J, Liu E, Lin T, Mertens S, Furnemont A and Kar G 2017 Annealing stability of magnetic tunnel junctions based on dual MgO free layers and [Co/Ni] based thin synthetic antiferromagnet fixed system J. Appl. Phys. 121 113904 [27] Zayets V 2019 Measurements of spin polarization of FeB and FeCoB nanomagnets by the Anomalous Hall effect arXiv: 1902.06451 [28] Beaujour J M, Ravelosona D, Tudosa I, Fullerton E E and Kent A D 2009 Ferromagnetic resonance linewidth in ultrathin films with perpendicular magnetic anisotropy Phys. Rev. B 80 180415(R) [29] Weber R, Han D-S, Boventer I, Jais wal S, Lebrun R, Jakob G and Kläui M 2019 Gilbert damping of CoFe-alloys J. Phys. D. Appl. Phys. 52 325001 [30] Konoto M, Imamura H, Taniguchi T, Yakushiji K, Kubota H, Fukushima A, Ando K and Yuasa S 2013 Effect of MgO Cap Layer on Gilbert Damping of FeB Electrode Layer in MgO-Based Magnetic Tunnel Junctions Appl. Phys. Express 6 073002 9 Figure1. (a) Stack layout of blanket MTJs without FeB insertion. Free layer (FL), reference layer (RL) and hard layer (HL) are indicated. Thicknesses of sublayers are shown in nm with parentheses. Blanket films were annealed first at 350 C and then at 400 C, both for 30 min. (b) Major loop and (c) minor loop of the stack in (a) measured by VSM after different annealing conditions with the magnetic field perpendicular to the sample plane. Figure2. Schematic of FeB insertion at (a) top and (b) bottom CoFeB/MgO interface in dual- MgO FL in the stack shown in figure 1(a). The total thickness of inserted FeB plus remaining CoFeB is kept at 0.8 nm and 1.2 nm for layers above and below W spacer, respectively. The impact of FeB insertion on TMR (c) a nd RA (d) of the MTJ stacks after 350 C annealing are shown. 10 Figure3. (a) The external excitation frequency as a function of ferromagnetic resonance field and (b) the linewidth versus frequency of FL in MTJ stacks without FeB insertion (black open circles), with 0.2 nm top FeB insertion (red open triangles), and with 0.2 nm bottom FeB insertion (blue open squares). Solid lines are fits. Figure4. Effect of FeB insertion and annealing conditions on 𝑀௦∙𝑡 ((a) and (b)), 𝜇𝐻 ((c) and (d)), and 𝐾∙𝑡 ((e) and (f)). (a), (c) and (e) show th e impact from FeB insertion at the top CoFeB/MgO interface, while (b), (d) and (f) show the impact from bottom interface. 11 Figure5. Gilbert damping as a function of FeB insertion thickness at (a) top interface and (b) bottom interface under two annealing conditions.

2021-04-22

We explore the impact of Fe80B20 inserted at both Co$_{20}$Fe$_{80}$B$_{20}$/MgO interfaces of dual-MgO free layers (FLs) in bottom-pinned magnetic tunnel junctions (MTJs). MTJ stacks are annealed for 30 min at 350 $^\circ$C and 400 $^\circ$C in a vacuum after film deposition. Current-in-plane tunneling measurements are carried out to characterize magnetotransport properties of the MTJs. Conventional magnetometry measurements and ferromagnetic resonance are conducted to estimate the saturation magnetization, the effective perpendicular anisotropy field and the Gilbert damping of dual-MgO FLs as a function of the Fe$_{80}$B$_{20}$ thickness and annealing temperatures. With ultrathin Fe$_{80}$B$_{20}$ (0.2 - 0.4 nm) inserted, perpendicular magnetic anisotropy (PMA) of FLs increases with similar tunnel magneto-resistance (TMR) and low damping values. As Fe$_{80}$B$_{20}$ layer thickness further increases (0.6 - 1.2 nm), both TMR and PMA degrade, and damping increases dramatically. This study demonstrates a novel approach to tune properties of MTJ stacks with dual-MgO FLs up to 400 $^\circ$C annealing, which enables MTJ stacks for various applications.

Impact of Fe$_{80}$B$_{20}$ insertion on the properties of dual-MgO perpendicular magnetic tunnel junctions

2104.10918v1

arXiv:1910.07731v1 [math.AP] 17 Oct 2019Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms Abdelhamid Mohammed Djaouti aFaculty of Exact and Computer Sciences, Hassiba Ben Bouali U niversity, Ouled Fares, 02180 Chlef, Algeria. Abstract We prove the global existence of small data solution in all space dimen sion for weakly coupled systems of semi-linear effectively damped wave, with d ifferent time-dependent coefficients in the dissipation terms. Moreover, no nlinearity termsf(t,u) andg(t,v) satisfying some properties of the parabolic equation. We study the problem in several classes of regularity. Keywords: Weakly coupled hyperbolic systems, damped wave equations, Cauchy problem, global existence, L2-decay, effective dissipation, small data solutions 2010 MSC: 35L52, 35L71 1. Introduction Let us consider the Cauchy problem for semilinear classical damped w ave equation with power nonlinearity utt−∆u+ut=f(u), u(0,x) =u0(x), ut(0,x) =u1(x), (1) wheret∈[0,∞), x∈Rnand f(0) = 0, f(u)−f(˜u)/lessorsimilar|u−˜u|(|u|−|˜u|)p−1. (2) Email address: djaouti −abdelhamid@yahoo.fr; a.mohammeddjaouti@univ-chlef.dz (Abdelhamid Mohammed Djaouti) Preprint submitted to Journal of L ATEX Templates October 18, 2019Having the estimates proved in [14] for the corresponding homoge neous prob- 5 lem, the authors in [19] proved for given compactly supported initial data (u0,u1)∈H1(Rn)×L2(Rn) and for p≤pGN(n) :=n n−2ifn≥3 the local (in time) existence of energy solutions u∈ C([0,T),H1(Rn))∩ C1([0,T),L2(Rn)). Moreover, they proved the global (in time) existence for small dat a solutions by using the technique of potential well and modified potential well. P rob- 10 lem (1) was also devoted in papers [5, 10, 11, 24, 27] where Fujita e xponent PFuj(n) := 1+2 nhas an important role as critical exponent, which means that we have global (in time) existence of small data weak solutions for p > pFuj(n), while local (in time) existence for p >1 with large data. Assuming a time-dependent coefficient in the dissipation term, we con sider first 15 the homogeneous problem utt−∆u+b(t)ut= 0, u(0,x) =u0(x), ut(0,x) =u1(x).(3) Among other classifications introduced in [25] and [26] of the dissipat ion term b(t)ut, we areinterested in this paper in the effective casewhere b=b(t) satisfies the following properties: •bis a positive and monotonic function with tb(t)→ ∞ast→ ∞, 20 •((1+t)2b(t))−1∈L1(0,∞), •b∈ C3[0,∞) and|b(k)(t)|/lessorsimilarb(t) (1+t)kfork= 1,2,3, •1 b/∈L1(0,∞) and thereexists a constant a∈[0,1) suchthat tb′(t)≤ab(t). Examples of functions belong to this class are: •b(t) =µ (1+t)rfor some µ >0 andr∈(−1,1), 25 •b(t) =µ (1+t)r(log(cr,γ+t))γfor some µ >0 andγ >0, •b(t) =µ (1+t)r(log(cr,γ+t))γfor some µ >0 andγ >0. 2Herecr,γis a sufficiently large positive constant. In[1]theauthorsderivedsuchestimatesforsolutionstothefamily ofparameter- dependent Cauchy problems30 utt−∆u+b(t)ut= 0, v(τ,x) = 0, vt(τ,x) =f(u)(τ,x). (4) Using theses estimates together with Duhamel’s Principle the author s proved in the same paper the global existence of small data solutions to the following semilinear Cauchy problem utt−∆u+b(t)ut=f(u), u(0,x) =u0(x), ut(0,x) =u1(x),(5) wheref(u) satisfied condition (2). In 2013, D’Abbicco in [2] proved the global existence of small data so lution for low space dimension and derived the decay estimates to the Cauchy p roblem utt−∆u+b(t)ut=f(t,u), u(0,x) =u0(x), ut(0,x) =u1(x), wheref(0) = 0 and f(t,v)−f(t,˜v)/lessorsimilar(1+/integraltextt 01 b(r)dr)γ|v−˜v|(|v|−|˜v|)p−1. In this paper we study in all space dimension the Cauchy problem of we akly 35 coupled system of semilinear effectively damped waves utt−∆u+b1(t)ut=f(t,v), u(0,x) =u0(x), ut(0,x) =u1(x), vtt−∆v+b2(t)vt=g(t,u), v(0,x) =v0(x), vt(0,x) =v1(x),(6) where (1+B1(t,0))1 α/lessorsimilar(1+B2(t,0))/lessorsimilar(1+B1(t,0))β, (7) f(0) = 0, f (t,v)−f(t,˜v)/lessorsimilar(1+B1(t,0))γ1|v−˜v|(|v|−|˜v|)p−1,(8) g(0) = 0, g (t,u)−g(t,˜u)/lessorsimilar(1+B2(t,0))γ2|u−˜u|(|u|−|˜u|)q−1,(9) forB1(t,τ) =/integraltextt τ1 b1(r)dr;B2(t,τ) =/integraltextt τ1 b2(r)dr;α,β∈R∗ +andγ1,γ2∈[−1,∞). Recently, K. Nishihara and Y. Wakasugi studied in [20] the particula r case of (6), where b1(t) =b2(t) = 1,f(t,v) =|v|pandg(t,u) =|u|q. Using the weighted 40 energy method they proved the global existence if the inequality max{p;q}+1 pq−1<n 2(10) 3is satisfied. In [15] and [16] the authors studied the above system with the same nonlinearities assumed in [20] by taking equivalent coefficients b1(t) and b2(t), or in other word α=β= 1. The global existence for small initial data solutions was proved assuming different classes of regularity of dat a and for all 45 space dimensions. Considering (6) in [17], the authors proved a globa l existence result for a particular case from the set of effective dissipation ter ms which is b1(t) =µ (1+t)r1andb2(t) =µ (1+t)r2withthefollowingnonlinearities f(t,v) =|v|p andf(t,u) =|u|q. 1.1. Notations50 We introduce for s >0 andm∈[1,2) the function space Am,s:= (Hs(Rn)∩Lm(Rn))×(Hs−1(Rn)∩Lm(Rn)) with the norm /ba∇⌈bl(u,v)/ba∇⌈blAm,s:=/ba∇⌈blu/ba∇⌈blHs+/ba∇⌈blu/ba∇⌈blLm+/ba∇⌈blv/ba∇⌈blHs−1+/ba∇⌈blv/ba∇⌈blLm. We denote by ˜ pand ˜qthe modified power nonlinearities of power nonlinearities appeared in (8) and (9). Then ˜p= (p−1)β+1 if β≥1, (p−m 2)β+m 2if 0< β <1,(11) and ˜q= (q−1)α+1 if α≥1, (q−m 2)α+m 2if 0< α <1.(12) Remark 1.1. Ifmax{α;β}<1, then,(1 +B1(t,0))≈(1+B2(t,0)).This 55 case was studied in previous papers. Then we will restrict ou rselves in this work to the remaining cases. 2. Main results We study the Cauchy problem (6) in several cases with respect to t he reg- ularity of the data in order to cover all space dimonsions, and the mo dified 60 4exponents of power nonlinearities ˜ p,˜qand the parameters α,β,γ 1,γ2. There- fore, we introduce the following classification of regularity: Data fr om energy spaces= 1, data from Sobolev spaces with suitable regularity s∈(1,n 2+ 1] and, finally, large regular data s >n 2+1. 2.1. Data from the energy space65 In this section we are interested in the system (6), where the data are taken from the function space Am,1. In Theorem 2.1 we treat the case where both modified exponents power nonlinearities ˜ pand ˜qare above modified Fujita ex- ponents which are pFuj,m,γ 1:= 1+2m(γ1+1) n,pFuj,m,γ 2:= 1+2m(γ2+1) n respectively. Theorem 2.1. Let the data (u0,u1),(v0,v1)are assumed to belong to Am,1× Am,1form∈[1,2). Moreover, let the modified exponents satisfy ˜p > pFuj,m,γ 1,˜q > pFuj,m,γ 2. (13) The exponents pandqof the power nonlinearities satisfy 2 m≤min{p;q} ≤max{p;q}<∞ifn≤2, 2 m≤min{p;q} ≤max{p;q} ≤pGN(n)ifn >2.(14) Then, there exists a constant ǫ0such that if /ba∇⌈bl(u0,u1)/ba∇⌈blAm,1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,1≤ǫ0, then there exists a uniquely determined global (in time) ene rgy solution to (6) in/parenleftBig C/parenleftbig [0,∞),H1(Rn)/parenrightbig ∩C1/parenleftbig [0,∞),L2(Rn)/parenrightbig/parenrightBig2 . Furthermore, the solution satisfies the following decay est imate: 70 /ba∇⌈bl∇j∂l tu(t,·)/ba∇⌈blL2(Rn)/lessorsimilarb1(t)−l/parenleftbig 1+B1(t,0)/parenrightbig−n 2(1 m−1 2)−j 2−l ×/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,1/parenrightbig , 5/ba∇⌈bl∇j∂l tv(t,·)/ba∇⌈blL2(Rn)/lessorsimilarb2(t)−l/parenleftbig 1+B2(t,0)/parenrightbig−n 2(1 m−1 2)−j 2−l ×/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,1/parenrightbig , wherej+l= 0,1. Remark 2.2. We ramark for γ1=γ2= 0, that the system (6) behave like one single equation because the modified power nonlineariti es˜pand˜qinfluenced separatly only by the modified Fujita exponent pFuj,m(n) =2m n+ 1.Then we 75 cannot feel the interplay between the powers of nonlinearit ies in the existence conditions. Remark 2.3. The final addmissible ranges for the exponents pandqof power nonlinearities can be fixed using several parameters w hich are: α,β,the powersγ1,γ2, the dimonsion of the space nand the parameter of additional 80 regularity m.As example for the dimonsion n= 1,if we take 0< β <1,then ˜p < p.We distinguish two cases: •Ifγ1≥ −1 2,thenp≥2 mis valide for ˜p > pFuj,m,γ 1which is equivant to p >1 β/parenleftbig 2m(γ1+1)−m 2+1/parenrightbig +m 2. •Ifγ1∈[−1,−1 2),then the solution existe for p >max/braceleftbigg1 β/parenleftBig 2m(γ1+1)−m 2+1/parenrightBig +m 2;2 m/bracerightbigg . The general case for the admissible ranges from below can be s ummarized as follows: Interplay parameter αNonlinearity parameter γ1Admissible range for p 0< β <1 γ1≥ −1+n 2p >1 β+2m(γ1+1) nβ−m 2β+m 2 γ1∈[−1,−1+n 2) p >max/braceleftBig 1 β+2m(γ1+1) nβ−m 2β+m 2;2 m/bracerightBig β≥1 γ1≥ −1+nβ 2p >2m(γ1+1) nβ+1 γ1∈[−1,−1+nβ 2) p >max/braceleftBig 2m(γ1+1) nβ+1;2 m/bracerightBig 6Following similar way one can get the admissible range for qwith respect to the 85 parameters αandγ2. Example 2.4. Let us choose the dimension space n= 2, the parameters γ1=−1,γ2=−1 3and the coefficients of the dissipation terms b1(t) = (1+t)−1 2 andb2(t) = (1+t)1 2which implies β=1 α= 3.Using (13) from previous theorem form= 2we get˜p >1,˜q >7 3.Theses conditions together with (14) after 90 applying (11) and (12) imply the following admissible range for the exponents of power nonlinearities p >1,q >13 9. (15) Example 2.5. If we change the second coefficient b2(t) =(1+t)1 2 (log(e+t))δ, then we consider the Cauchy problem utt−∆u+(1+t)−1 2ut= (1+B1(t,0))−1|v|p,(u,ut)(0,x) = (u0,u1)(x), vtt−∆v+(1+t)1 2 (log(e+t))δvt= (1+B2(t,0))−1 3|u|q,(v,vt)(0,x) = (v0,v1)(x), whereδ >0. From the problem we can conclude α= 1,β= 3. In this case we have to garantee ˜p >1and˜q >7 3.Finally we conclude the admissible range for the exponents of power nonlinearities95 p >1,q >7 3. (16) The case where we have only one exponent ˜ por ˜qis below modified Fujita exponent, we distinguish four cases with respect to the values of αandβas follows: 1. ˜p≤1 +2m(γ1+1) n,˜q >1 +2m(γ2+1) nwith min {α;β} ≥1 or min {α;β} ≤ 1≤max{α;β}. 100 2. ˜p >1 +2m(γ1+1) n,˜q≤1 +2m(γ2+1) nwith min {α;β} ≥1 or min {α;β} ≤ 1≤max{α;β}. Theorem 2.6. Letm∈[1,2),α≥1andβ >0. The data (u0,u1),(v0,v1) are assumed to belong to Am,1× Am,1. Moreover, let the modified exponents 7satisfy105 ˜p <2m(γ1+1) n+1, (17) ˜q >2m(γ2+1) n+1. (18) and n 2> m/parenleftbigg˜q+α+γ1˜q+γ1(α−1)+γ2 ˜p˜q−1+(α−1)(˜p−1)/parenrightbigg . (19) The exponents pandqof the power nonlinearities satisfy 2 m≤min{p;q} ≤max{p;q}<∞ifn≤2, 2 m≤min{p;q} ≤max{p;q} ≤pGN(n)ifn >2.(20) Then, there exists a constant ǫ0such that if /ba∇⌈bl(u0,u1)/ba∇⌈blAm,1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,1≤ǫ0, then there exists a uniquely determined global (in time) ene rgy solution to (6) in/parenleftBig C/parenleftbig [0,∞),H1(Rn)/parenrightbig ∩C1/parenleftbig [0,∞),L2(Rn)/parenrightbig/parenrightBig2 . Furthermore, the solution satisfies the following decay est imates: /ba∇⌈bl∇j∂l tu(t,·)/ba∇⌈blL2(Rn) /lessorsimilarb1(t)−l/parenleftbig 1+B1(t,0)/parenrightbig−n 2(1 m−1 2)−j 2−l+κ(˜p)/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,1/parenrightbig , /ba∇⌈bl∇j∂l tv(t,·)/ba∇⌈blL2(Rn) /lessorsimilarb2(t)−l/parenleftbig 1+B2(t,0)/parenrightbig−n 2(1 m−1 2)−j 2−l/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,1/parenrightbig , wherej+l= 0,1and 110 κ(˜p) =γ1−n 2m(˜p−1)+1, represent the loss of decay in comparison with the correspon ding decay estimates for the solution uof the linear Cauchy problem with vanishing right hand-side . Remark 2.7. If we would choose ˜p=pFuj,m(n)in condition (17), then we get an arbitrarily small loss of decay κ(˜p) =ε. 8We summarize the remaining results for all cases with respect to α,β,˜pand ˜q 115 as follows: •If we assume in the statement of previous theorem that α <1 andβ≥1, the we get instead of (19) the following condition n 2> m/parenleftbigg˜q+1+γ1˜q+γ2+m 2(α−1)(γ1+1) ˜p˜q−1+m 2(α−1)(˜p−1)/parenrightbigg . •If ˜p >2m(γ1+1) n+ 1,˜q≤2m(γ2+1) n+ 1, then instead of (19) we have to assume n 2> m/parenleftBig ˜p+β+γ2˜p+γ2(β−1)+γ1 ˜p˜q−1+(β−1)(˜q−1)/parenrightBig forα >0, β≥1,(21) n 2> m/parenleftBig˜p+1+γ2˜p+γ1+m 2(β−1)(γ2+1) ˜p˜q−1+m 2(β−1)(˜q−1)/parenrightBig forα≥1, β <1.(22) 2.2. Data from Sobolev spaces with suitable regularity In this section the regularity of data has strong influence on the ad missible 120 range of the modified exponents or the exponents of power nonline arities. For this reasonwe assume that the data havedifferent suitable larger r egularity, i.e., (u0,u1)∈Hs1(Rn)×Hs1−1(Rn), s1∈/parenleftBig 1,1+n 2/bracketrightBig , (v0,v1)∈Hs2(Rn)×Hs2−1(Rn), s2∈/parenleftBig 1,1+n 2/bracketrightBig , with an additional regularity Lm(Rn),m∈[1,2). In this section we shall use a generalized (fractional) Gagliardo-Nirenberg inequality used in the papers [9] and [22]. Furthermore, we shall use a fractional Leibniz rule and a fr actional 125 chain rule which are explained the Appendix. Theorem 2.8. Letn≥4,s1∈(max{1;3+ 2γ1},n 2+ 1],s2∈(max{1;3+ 2γ2},n 2+1],0< s2−s1<1and⌈s1⌉ /n⌉}ationslash=⌈s2⌉. The data (u0,u1),(v0,v1)are supposed to belong to Am,s1×Am,s2withm∈[1,2). Furthermore, we require ˜p >2m n/parenleftBigs1+1+2γ1 2/parenrightBig +1,˜q >2m n/parenleftBigs2+1+2γ2 2/parenrightBig +1.(23) The exponents pandqof the power nonlinearities satisfy the conditions 130 ⌈s1⌉< p, ⌈s2⌉< q if n≤2s1, ⌈s1⌉< p, ⌈s2⌉< q≤1+2 n−2s1if2s1< n≤2s2, ⌈s1⌉< p≤1+2 n−2s2,⌈s2⌉< q≤1+2 n−2s1if n >2s2.(24) 9Then, there exists a constant ǫ0such that if /ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2≤ǫ0, then there exists a uniquely determined globally (in time) e nergy solution to (6) in /parenleftBig C/parenleftbig [0,∞),Hs1(Rn)/parenrightbig ∩C1/parenleftbig [0,∞),Hs1−1(Rn)/parenrightbig/parenrightBig ×/parenleftBig C/parenleftbig [0,∞),Hs2(Rn)/parenrightbig ∩C1/parenleftbig [0,∞),Hs2−1(Rn)/parenrightbig/parenrightBig . Furthermore, the solution satisfies for l= 0,1the estimates /ba∇⌈bl|D|s1−l∂l tu(t,·)/ba∇⌈blL2(Rn)/lessorsimilarb1(t)−l/parenleftbig 1+B1(t,0)/parenrightbig−n 2(1 m−1 2)−l−s1−l 2 ×/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2/parenrightbig , /ba∇⌈bl|D|s2−l∂l tv(t,·)/ba∇⌈blL2(Rn)/lessorsimilarb2(t)−l/parenleftbig 1+B2(t,0)/parenrightbig−n 2(1 m−1 2)−l−s2−l 2 ×/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2/parenrightbig . Particular cases: •Ifβ≥1 ands1≥3+2γ1,then under the assumptions of Theorem 2.8, 135 the condition p >⌈s1⌉implies ˜p >2m n/parenleftBig s1+1+2γ1 2/parenrightBig +1. •Ifα≥1 ands2≥3+2γ2,then under the assumptions of Theorem 2.8, the condition p >⌈s2⌉implies ˜q >2m n/parenleftBig s2+1+2γ2 2/parenrightBig +1. 2.3. Large regular data This case has been classified to benefit from the embedding in L∞(Rn), 140 where the data are supposed to have a high regularity, this means, that (u0,u1)∈Hs1(Rn)×Hs1−1(Rn), s1>n 2+1, (v0,v1)∈Hs2(Rn)×Hs2−1(Rn), s2>n 2+1. Theorem 2.9. Letn≥4,(u0,u1),(v0,v1)∈ Am,s1× Am,s2,m∈[1,2), min{s2;s1}>n 2+1, ands1−s2∈(−1,1). Moreover, let p > s1, q > s 2 10and ˜p >2m n/parenleftBigs1+1+2γ1 2/parenrightBig +1,˜q >2m n/parenleftBigs2+1+2γ2 2/parenrightBig +1. Then, there exists a constant ǫ0such that if /ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2≤ǫ0, then there exists a uniquely determined globally (in time) e nergy solution to (6) 145 in /parenleftBig C/parenleftbig [0,∞),Hs1(Rn)/parenrightbig ∩C1/parenleftbig [0,∞),Hs1−1(Rn)/parenrightbig/parenrightBig ×/parenleftBig C/parenleftbig [0,∞),Hs2(Rn)/parenrightbig ∩C1/parenleftbig [0,t],Hs2−1(Rn)/parenrightbig/parenrightBig . Furthermore, the solution satisfies for l= 0,1the estimates: /ba∇⌈bl|D|s1−l∂l tu(t,·)/ba∇⌈blL2(Rn)/lessorsimilarb1(t)−l/parenleftbig 1+B1(t,0)/parenrightbig−n 2(1 m−1 2)−l−s1−l 2 ×/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2/parenrightbig , /ba∇⌈bl|D|s2−l∂l tv(t,·)/ba∇⌈blL2(Rn)/lessorsimilarb2(t)−l/parenleftbig 1+B2(t,0)/parenrightbig−n 2(1 m−1 2)−l−s2−l 2 ×/parenleftbig /ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2/parenrightbig . 3. Philosophy of our approach and proofs 3.1. Some tools First we recall the following result from [1]. 150 Lemma 3.1. The primitive B(t,τ)satisfies the following properties: B(t,τ)≈B(t,0)for allτ∈/bracketleftBig 0,t 2/bracketrightBig , (25) B(τ,0)≈B(t,0)for allτ∈/bracketleftBigt 2,t/bracketrightBig , (26) /integraldisplayt t 21 b(τ)/parenleftbig 1+B(t,τ)/parenrightbig−j 2−ldτ/lessorsimilar(1+B(t,0))1−j 2−llog(1+B(t,0))lforj+l= 0,1. (27) In order to use Duhamels principle we need the following results in the p roofs of our main results. 11Theorem 3.2. The Sobolev solutions to the Cauchy problem utt−∆u+b(t)ut= 0, u(0,x) =u0(x), ut(0,x) =u1(x) satisfy the following estimates: For data from the energy space (s= 1): /ba∇⌈bl∇j∂l tu(t,·)/ba∇⌈blL2/lessorsimilar(b(t))−l/parenleftbig 1+B(t,0)/parenrightbig−n 2(1 m−1 2)−j 2−l/ba∇⌈bl(u0,u1)/ba∇⌈blAm,1, wherej+l= 0,11; 155 for high regular data (s >1): /ba∇⌈blu(t,·)/ba∇⌈blL2/lessorsimilar/parenleftbig 1+B(t,0)/parenrightbig−n 2(1 m−1 2)/ba∇⌈bl(u0,u1)/ba∇⌈blAm,s, /ba∇⌈blut(t,·)/ba∇⌈blL2/lessorsimilarb(t)−1/parenleftbig 1+B(t,0)/parenrightbig−n 2(1 m−1 2)−1/ba∇⌈bl(u0,u1)/ba∇⌈blAm,s, /ba∇⌈bl|D|su(t,·)/ba∇⌈blL2/lessorsimilar/parenleftbig 1+B(t,0)/parenrightbig−n 2(1 m−1 2)−s 2/ba∇⌈bl(u0,u1)/ba∇⌈blAm,s, /ba∇⌈bl|D|s−1ut(t,·)/ba∇⌈blL2/lessorsimilarb(t)−1/parenleftbig 1+B(t,0)/parenrightbig−n 2(1 m−1 2)−s−1 2−1/ba∇⌈bl(u0,u1)/ba∇⌈blAm,s. The proof of this theorem can be concluded from [25] and [26]. Theorem 3.3. The Sobolev solutions to the parameter-dependent family of Cauchy problems vtt−∆v+b(t)vt= 0, v(τ,x) = 0, vt(τ,x) =v1(x) satisfy the following estimates: For data from the energy space (s= 1): /ba∇⌈bl∇j∂tv(t,·)/ba∇⌈blL2/lessorsimilarb(t)−1b(τ)−l/parenleftbig 1+B(t,τ)/parenrightbig−n 2(1 m−1 2)−j 2−l/ba∇⌈blv1/ba∇⌈blL2∩Lm,(28) wherej+l= 0,1; 160 for high regular data (s >1): /ba∇⌈blv(t,·)/ba∇⌈blL2/lessorsimilarb(τ)−1/parenleftbig 1+B(t,τ)/parenrightbig−n 2(1 m−1 2)/ba∇⌈blv1/ba∇⌈blHs−1∩Lm, /ba∇⌈blvt(t,·)/ba∇⌈blL2/lessorsimilarb(τ)−1b(t)−1/parenleftbig 1+B(t,τ)/parenrightbig−n 2(1 m−1 2)−1/ba∇⌈blv1/ba∇⌈blHs−1∩Lm, /ba∇⌈bl|D|sv(t,·)/ba∇⌈blL2/lessorsimilarb(τ)−1/parenleftbig 1+B(t,τ)/parenrightbig−n 2(1 m−1 2)−s 2/ba∇⌈blv1/ba∇⌈blHs−1∩Lm, /ba∇⌈bl|D|s−1vt(t,·)/ba∇⌈blL2/lessorsimilarb(τ)−1b(t)−1/parenleftbig 1+B(t,τ)/parenrightbig−n 2(1 m−1 2)−s−1 2−1/ba∇⌈blv1/ba∇⌈blHs−1∩Lm.(29) The proof of this theorem can be concluded from [1] and [18]. 123.2. Proofs We define the norm of the solution space X(t) by /ba∇⌈bl(u,v)/ba∇⌈blX(t)= sup τ∈[0,t]/braceleftbig M1(τ,u)+M2(τ,v)/bracerightbig , where we shall choose M1(τ,u) andM2(τ,v) with respect to the goals of each theorem. LetNbe the mapping on X(t) which is defined by N: (u,v)∈X(t)→N(u,v) =/parenleftbig uln+unl,vln+vnl/parenrightbig , where uln(t,x) :=E1,0(t,0,x)∗(x)u0(x)+E1,1(t,0,x)∗(x)u1(x), unl(t,x) :=/integraldisplayt 0E1,1(t,τ,x)∗(x)|v(τ,x)|pdτ, vln(t,x) :=E2,0(t,0,x)∗(x)v0(x)+E2,1(t,0,x)∗(x)v1(x), vnl(t,x) :=/integraldisplayt 0E2,1(t,τ,x)∗(x)|u(τ,x)|qdτ. We denote by E1,0=E1,0(t,0,x) andE1,1=E1,1(t,0,x) the fundamental solutions to the Cauchy problem utt−∆u+b1(t)ut= 0, u(0,x) =u0(x), u t(0,x) =u1(x), and byE2,0=E2,0(t,0,x) andE2,1=E2,1(t,0,x) the fundamental solutions to the the Cauchy problem vtt−∆v+b2(t)vt= 0, v(0,x) =v0(x), v t(0,x) =v1(x). Our aim is to prove the estimates165 /ba∇⌈blN(u,v)/ba∇⌈blX(t) /lessorsimilar/ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2+/ba∇⌈bl(u,v)/ba∇⌈blp X(t)+/ba∇⌈bl(u,v)/ba∇⌈blq X(t),(30) /ba∇⌈blN(u,v)−N(˜u,˜v)/ba∇⌈blX(t)/lessorsimilar/ba∇⌈bl(u,v)−(˜u,˜v)/ba∇⌈blX(t) ×/parenleftbig /ba∇⌈bl(u,v)/ba∇⌈blp−1 X(t)+/ba∇⌈bl(˜u,˜v)/ba∇⌈blp−1 X(t)+/ba∇⌈bl(u,v)/ba∇⌈blq−1 X(t)+/ba∇⌈bl(˜u,˜v)/ba∇⌈blq−1 X(t)/parenrightbig .(31) 13We can immediately obtain from the introduced normofthe solution sp aceX(t) the following inequality: /ba∇⌈bl(uln,vln)/ba∇⌈blX(t)/lessorsimilar/ba∇⌈bl(u0,u1)/ba∇⌈blAm,s1+/ba∇⌈bl(v0,v1)/ba∇⌈blAm,s2. We complete the proof of all results separately by showing (31) with the in- equality /ba∇⌈bl(unl,vnl)/ba∇⌈blX(t)/lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)+/ba∇⌈bl(u,v)/ba∇⌈blq X(t). (32) which leads to (30). Proof of Theorem 2.1: We choose the space of solutions X(t) =/parenleftbig C([0,t],H1)∩C1([0,t],L2)/parenrightbig2, and the following norms M1(τ,u) = (1+ B1(τ,0))n 2(1 m−1 2)/ba∇⌈blu(τ,·)/ba∇⌈blL2(Rn) +(1+B1(τ,0))n 2(1 m−1 2)+1 2/ba∇⌈bl∇u(τ,·)/ba∇⌈blL2(Rn) +b1(τ)(1+B1(τ,0))n 2(1 m−1 2)+1/ba∇⌈blut(τ,·)/ba∇⌈blL2(Rn), M2(τ,v) = (1+ B2(τ,0))n 2(1 m−1 2)/ba∇⌈blv(τ,·)/ba∇⌈blL2(Rn) +(1+B2(τ,0))n 2(1 m−1 2)+1 2/ba∇⌈bl∇v(τ,·)/ba∇⌈blL2(Rn) +b2(τ)(1+B2(τ,0))n 2(1 m−1 2)+1/ba∇⌈blvt(τ,·)/ba∇⌈blL2(Rn). To prove (32) we need to estimate all terms appearing in /ba∇⌈bl(unl,vnl)/ba∇⌈blX(t). Let 170 us begin to estimate/vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2. Using (28) with m= 2 forτ∈[t 2,t] we get /vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilar/integraldisplayt 2 0b1(t)−1b1(τ)−1(1+B1(t,τ))−n 2(1 m−1 2)−1/ba∇⌈blf(τ,v)/ba∇⌈blLm∩L2dτ +/integraldisplayt t 2b1(t)−1b1(τ)−1(1+B1(t,τ))−1/ba∇⌈blf(τ,v)/ba∇⌈blL2dτ. (33) By a fractional version of Gagliardo-Nirenberg inequality (see prop osition 4.1 ) and (8), we obtain /ba∇⌈blf(τ,v)/ba∇⌈blL2/lessorsimilar(1+B1(τ,0))γ1(1+B2(τ,0))−n 2mp+n 4/ba∇⌈bl(u,v)/ba∇⌈blp X(t),(34) /ba∇⌈blf(τ,v)/ba∇⌈blLm/lessorsimilar(1+B1(τ,0))γ1(1+B2(τ,0))−n 2mp+n 2m/ba∇⌈bl(u,v)/ba∇⌈blp X(t),(35) 14where we use condition (14). Plugging the last estimates in (33) and u sing (7), (25) and (26) we get175 /vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)/integraldisplayt 2 0b1(t)−1b1(τ)−1(1+B1(t,τ))−n 2(1 m−1 2)−1 ×(1+B1(τ,0))γ1(1+B2(τ,0))−n 2mp+n 4dτ +/ba∇⌈bl(u,v)/ba∇⌈blp X(t)/integraldisplayt t 2b1(t)−1b1(τ)−1(1+B1(t,τ))−1 ×(1+B1(τ,0))γ1(1+B2(τ,0))−n 2mp+n 2mdτ /lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)/integraldisplayt 2 0b1(t)−1b1(τ)−1(1+B1(t,τ))−n 2(1 m−1 2)−1 ×(1+B1(τ,0))(−n 2mp+n 2m)β+γ1dτ +/ba∇⌈bl(u,v)/ba∇⌈blp X(t)/integraldisplayt t 2b1(t)−1b1(τ)−1(1+B1(t,τ))−1 ×(1+B1(τ,0))(−n 2mp+n 4)β+γ1dτ /lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,0))−n 2(1 m−1 2)−1 ×/integraldisplayt 2 0b1(τ)−1(1+B1(τ,0))(−n 2mp+n 2m)β+γ1dτ +/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(τ,0))(−n 2mp+n 4)β+γ1 ×/integraldisplayt t 2b1(τ)−1(1+B1(t,τ))−1dτ. We distinguish two cases with respect to the value of β. Ifβ≥1, then we get /vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1 ×/integraldisplayt 2 0b1(τ)−1(1+B1(τ,0))−n 2m(˜p−1)+γ1dτ +/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(τ,0))−n 2m(˜p−1)−n 2(1 m−1 2)β+γ1 ×/integraldisplayt t 2b1(τ)−1(1+B1(t,τ))−1dτ /lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1, for ˜p >2m(γ1+1) n+1. If 0< β <1, then we get /vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1 15×/integraldisplayt 2 0b1(τ)−1(1+B1(τ,0))−n 2m(˜p−1)−n 2(1 m−1 2)(1−β)+γ1dτ +/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(τ,0))−n 2m˜p+n 4+γ1 ×/integraldisplayt t 2b1(τ)−1(1+B1(t,τ))−1dτ /lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1, for ˜p >2m(γ1+1) n+1.Finally, we obtain /vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1.(36) Analogously, we can prove180 /vextenddouble/vextenddouble∇unl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B1(t,0))−n 2(1 m−1 2)−1 2/ba∇⌈bl(u,v)/ba∇⌈blp X(t), (37) /vextenddouble/vextenddoubleunl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B1(t,0))−n 2(1 m−1 2)/ba∇⌈bl(u,v)/ba∇⌈blp X(t). (38) Forthesecondcomponent vnl,usingGagliardo-Nirenberginequalityfrompropo- sition 4.1 we get /ba∇⌈blg(τ,u)/ba∇⌈blL2/lessorsimilar(1+B2(τ,0))γ1(1+B1(τ,0))−n 2mq+n 4/ba∇⌈bl(u,v)/ba∇⌈blq X(t), /ba∇⌈blg(τ,u)/ba∇⌈blLm/lessorsimilar(1+B2(τ,0))γ1(1+B1(τ,0))−n 2mq+n 2m/ba∇⌈bl(u,v)/ba∇⌈blq X(t). Taking account of the last estimates, we can prove similarly to (36) t o (38) the following estimates185 /vextenddouble/vextenddoublevnl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilarb2(t)−1(1+B2(t,0))−n 2(1 m−1 2)−1/ba∇⌈bl(u,v)/ba∇⌈blq X(t),(39) /vextenddouble/vextenddouble∇vnl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B2(t,0))−n 2(1 m−1 2)−1 2/ba∇⌈bl(u,v)/ba∇⌈blq X(t), (40) /vextenddouble/vextenddoublevnl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B2(t,0))−n 2(1 m−1 2)/ba∇⌈bl(u,v)/ba∇⌈blq X(t), (41) for ˜q >2m(γ2+1) n+1.Finally, (36) to (41) implies (32). The proof of (31) is completely analogous to the proof of (30). In t his way we complete the proof of Theorem 2.1. Proof of Theorem 2.6: We choose the same space of solutions X(t) and 190 the norm M2(τ,v) used in the proof of Theorem 2.6. We modify the norm 16M1(τ,v) as follows: M1(τ,u) = (1+ B1(τ,0))n 2(1 m−1 2)−κ(˜p)/ba∇⌈blu(τ,·)/ba∇⌈blL2(Rn) +(1+B1(τ,0))n 2(1 m−1 2)+1 2−κ(˜p)/ba∇⌈bl∇u(τ,·)/ba∇⌈blL2(Rn) +b1(τ)(1+B1(τ,0))n 2(1 m−1 2)+1−κ(˜p)/ba∇⌈blut(τ,·)/ba∇⌈blL2(Rn). We begin the proof of (32) by the term/vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2. Using (28) with m= 2 for τ∈[t 2,t] together with Gagliardo-Nirenberg inequality and following the same steps of the proof of (36) we get195 /vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1 ×/integraldisplayt 2 0b1(τ)−1(1+B1(τ,0))(−n 2mp+n 2m)β+γ1dτ +/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(τ,0))(−n 2mp+n 4)β+γ1 ×/integraldisplayt t 2b1(τ)−1(1+B1(t,τ))−1dτ /lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1+κ(˜p), forβ >0.Then we have /vextenddouble/vextenddoubleunl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−1+κ(˜p).(42) In the same way one can prove /vextenddouble/vextenddouble∇unl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B1(t,0))−n 2(1 m−1 2)−1 2+κ(˜p)/ba∇⌈bl(u,v)/ba∇⌈blp X(t),(43) /vextenddouble/vextenddoubleunl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B1(t,0))−n 2(1 m−1 2)+κ(˜p)/ba∇⌈bl(u,v)/ba∇⌈blp X(t).(44) Now for vnl, we can prove using Gagliardo-Nirenberg inequality and the defini- tion of the solution space X(t) the following estimates 200 /ba∇⌈blg(τ,u)/ba∇⌈blL2/lessorsimilar(1+B2(τ,0))γ1(1+B1(τ,0))−n 2mq+n 4+κ(˜p)q/ba∇⌈bl(u,v)/ba∇⌈blq X(t), /ba∇⌈blg(τ,u)/ba∇⌈blLm/lessorsimilar(1+B2(τ,0))γ1(1+B1(τ,0))−n 2mq+n 2m+κ(˜p)q/ba∇⌈bl(u,v)/ba∇⌈blq X(t). Taking account of the last estimates we can prove analogously to (4 2) to (44) the following /vextenddouble/vextenddoublevnl t(t,·)/vextenddouble/vextenddouble L2/lessorsimilarb(t)−1(1+B(t,0))−n 2(1 m−1 2)−1/ba∇⌈bl(u,v)/ba∇⌈blq X(t),(45) 17/vextenddouble/vextenddouble∇vnl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B(t,0))−n 2(1 m−1 2)−1 2/ba∇⌈bl(u,v)/ba∇⌈blq X(t), (46) /vextenddouble/vextenddoublevnl(t,·)/vextenddouble/vextenddouble L2/lessorsimilar(1+B(t,0))−n 2(1 m−1 2)/ba∇⌈bl(u,v)/ba∇⌈blq X(t), (47) where we use the condition γ2−n 2m(˜q−1)+κ(˜p)qα+ε <−1 which is equivalent to condition (19). Consequently, (42) to (47) imp lies (32) and the proof of Theorem 2.6 is completed. Proof of Theorem 2.8: Let us choose the space of solutions X(t) =/parenleftbig C([0,t],Hs1)∩C1([0,t],Hs1−1)/parenrightbig ×/parenleftbig C([0,t],Hs2)∩C1([0,t],Hs2−1)/parenrightbig with the norm /ba∇⌈bl(u,v)/ba∇⌈blX(t)= sup τ∈[0,t]/braceleftbig M1(τ,u)+M2(τ,v)/bracerightbig , where M1(τ,u) =/parenleftbig 1+B1(τ,0)/parenrightbign 2(1 m−1 2)/ba∇⌈blu(τ,·)/ba∇⌈blL2(Rn) +b1(τ)/parenleftbig 1+B1(τ,0)/parenrightbign 2(1 m−1 2)+1/ba∇⌈blut(τ,·)/ba∇⌈blL2(Rn) +b1(τ)/parenleftbig 1+B1(τ,0)/parenrightbign 2(1 m−1 2)+s1−1 2+1/ba∇⌈bl|D|s1−1ut(τ,·)/ba∇⌈blL2(Rn) +/parenleftbig 1+B1(τ,0)/parenrightbign 2(1 m−1 2)+s1 2/ba∇⌈bl|D|s1u(τ,·)/ba∇⌈blL2(Rn), and205 M2(τ,v) =/parenleftbig 1+B2(τ,0)/parenrightbign 2(1 m−1 2)/ba∇⌈blv(τ,·)/ba∇⌈blL2(Rn) +b2(τ)/parenleftbig 1+B2(τ,0)/parenrightbign 2(1 m−1 2)+1/ba∇⌈blvt(τ,·)/ba∇⌈blL2(Rn) +b2(τ)/parenleftbig 1+B2(τ,0)/parenrightbign 2(1 m−1 2)+s2−1 2+1/ba∇⌈bl|D|s2−1vt(τ,·)/ba∇⌈blL2(Rn) +/parenleftbig 1+B2(τ,0)/parenrightbign 2(1 m−1 2)+s2 2/ba∇⌈bl|D|s2v(τ,·)/ba∇⌈blL2(Rn). To prove (32) we show how to estimate the norms /ba∇⌈bl|D|s1−1unl t(t,·)/ba∇⌈blL2(Rn)and /ba∇⌈bl|D|s2−1vnl t(t,·)/ba∇⌈blL2(Rn). From the estimate (29) it follows /ba∇⌈bl|D|s1−1unl t(t,·)/ba∇⌈blL2(Rn) 18/lessorsimilar/integraldisplayt 2 0b1(τ)−1b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−s1−1 2−1 ×/ba∇⌈blf(τ,v)/ba∇⌈blLm(Rn)∩L2(Rn)∩˙Hs1−1(Rn)dτ +/integraldisplayt t 2b1(τ)−1b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−s1−1 2−1 ×/ba∇⌈blf(τ,v)/ba∇⌈blLm(Rn)∩L2(Rn)∩˙Hs1−1(Rn)dτ. Under the assumptions of Theorem 2.8 and the choice of the above in troduced norm, for 0 ≤τ≤tthe inequalities (34) and (35) remain true. We calculate the norm210 /ba∇⌈blf(τ,v)/ba∇⌈bl˙Hs−1. Using (4.3) from the Propositions 4.3 and Proposition 4.1, we may conc lude for p >⌈s1−1⌉and 0≤τ≤tthe following estimate: /ba∇⌈blf(τ,v)/ba∇⌈bl˙Hs1−1/lessorsimilar(1+B1(τ,0))γ1/vextenddouble/vextenddoublev(τ,·)/vextenddouble/vextenddoublep−1 Lq1/vextenddouble/vextenddouble|D|s1−1(τ,·)/vextenddouble/vextenddouble Lq2 /lessorsimilar(1+B1(τ,0))γ1/vextenddouble/vextenddoublev(τ,·)/vextenddouble/vextenddouble(p−1)(1−θ1) L2 ×/vextenddouble/vextenddouble|D|s2v(τ,·)/vextenddouble/vextenddouble(p−1)θ1 L2/vextenddouble/vextenddoublev(τ,·)/vextenddouble/vextenddouble1−θ2 L2/vextenddouble/vextenddouble|D|s2v(τ,·)/vextenddouble/vextenddoubleθ2 L2 /lessorsimilar(1+B1(τ,0))γ1(1+B2(τ,0))−n 2mp+n 4−s1−1 2/ba∇⌈bl(u,v)/ba∇⌈blp X(t), where p−1 q1+1 q2=1 2, θ1=n s/parenleftBig1 2−1 q1/parenrightBig ∈[0,1], θ2=n s2/parenleftBig1 2−1 q2/parenrightBig +s1−1 s2∈/bracketleftBigs1−1 s2,1/bracketrightBig . To satisfy the last conditions for the parameters θ1andθ2we choose q2=2n n−2 andq1=n(p−1). This choice implies the condition 1+2 n≤p≤1+2 n−2s2. Consequently, we obtain215 /ba∇⌈bl|v(τ,·)|p/ba∇⌈bl˙Hs1−1/lessorsimilar(1+B2(τ,0))−n 2mp+n 4−s2−1 2/ba∇⌈bl(u,v)/ba∇⌈blp X(t).(48) Consequently, we get /ba∇⌈bl|D|s1−1unl t(t,·)/ba∇⌈blL2(Rn) /lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−s1−1 2−1 19×/integraldisplayt 2 0b1(τ)−1(1+B1(τ,0))(−n 2mp+n 2m)β+γ1dτ +/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(τ,0))(−n 2mp+n 4)β+γ1 ×/integraldisplayt t 2b1(τ)−1(1+B1(t,τ))−s1−1 2−1dτ /lessorsimilar/ba∇⌈bl(u,v)/ba∇⌈blp X(t)b1(t)−1(1+B1(t,τ))−n 2(1 m−1 2)−s1−1 2−1, where ˜p >2m n/parenleftBig s1+1+2γ1 2/parenrightBig +1. 4. Appendix Here we state some inequalities which come into play in our proofs. Proposition 4.1. Let1< p,p 0,p1<∞,σ >0ands∈[0,σ).Then the 220 following fractional Gagliardo-Nirenberg inequality hol ds for all u∈Lp0∩˙Hσ p1: /ba∇⌈blu/ba∇⌈bl˙Hsp/lessorsimilar/ba∇⌈blu/ba∇⌈bl1−θ Lp0/ba∇⌈blu/ba∇⌈blθ ˙Hσp1, (49) where θ=θs,σ:=1 p0−1 p+s n 1 p0−1 p1+σ nands σ≤θ≤1. For the proof see [9] and [4, 6, 7, 8, 12, 13]. Proposition 4.2. Let us assume s >0and1≤r≤ ∞,1< p1,p2,q1,q2≤ ∞satisfying the relation 1 r=1 p1+1 p2=1 q1+1 q2. Then the following fractional Leibniz rule holds: /ba∇⌈bl|D|s(fg)/ba∇⌈blLr/lessorsimilar/ba∇⌈bl|D|sf/ba∇⌈blLp1/ba∇⌈blg/ba∇⌈blLp2+/ba∇⌈blf/ba∇⌈blLq1/ba∇⌈bl|D|sg/ba∇⌈blLq2, for allf∈˙Hs p1∩Lq1andg∈˙Hs q2∩Lp2. For more details concerning fractional Leibniz rule see [6].225 Proposition 4.3. Let us choose s >0,p >⌈s⌉and1< r,r1,r2<∞satis- fying 1 r=p−1 r1+1 r2. 20Let us denote by F(u)one of the functions |u|p,±|u|p−1u.Then the following fractional chain rule holds: /ba∇⌈bl|D|sF(u)/ba∇⌈blLr/lessorsimilar/ba∇⌈blu/ba∇⌈blp−1 Lr1/ba∇⌈bl|D|su/ba∇⌈blLr2, (50) For the proof see [21]. Proposition 4.4. Letp >1andu∈Hs m, where s∈(n m,p). Then the following estimates hold: /ba∇⌈bl|u|p/ba∇⌈blHsm/lessorsimilar/ba∇⌈blu/ba∇⌈blHsm/ba∇⌈blu/ba∇⌈blp−1 L∞, /ba∇⌈blu|u|p−1/ba∇⌈blHs m/lessorsimilar/ba∇⌈blu/ba∇⌈blHs m/ba∇⌈blu/ba∇⌈blp−1 L∞. For the proof see [23]. We can derive from Proposition 4.4 the following corollary.230 Corollary 4.5. Under the assumptions of Proposition 4.4 it holds /ba∇⌈bl|u|p/ba∇⌈bl˙Hsm/lessorsimilar/ba∇⌈blu/ba∇⌈bl˙Hsm/ba∇⌈blu/ba∇⌈blp−1 L∞, /ba∇⌈blu|u|p−1/ba∇⌈bl˙Hsm/lessorsimilar/ba∇⌈blu/ba∇⌈bl˙Hsm/ba∇⌈blu/ba∇⌈blp−1 L∞. For the proof see [22]. Lemma 4.6. Let0<2s∗< n <2s. Then for any function f∈˙Hs∗∩˙Hs one has the estimate /ba∇⌈blf/ba∇⌈blL∞≤ /ba∇⌈blf/ba∇⌈bl˙Hs∗+/ba∇⌈blf/ba∇⌈bl˙Hs. For the proof see [3]. References [1] M. D’Abbicco, S. Lucente, M. Reissig, Semi-linear wave equations with ef- fective damping , Chin. Ann. Math. 34B (3) (2013), 345–380. 235 [2] M. D’Abbicco, Small data solutions for semilinear wave equations with ef- fective with effective damping , Discrete Contin. Dyn. Syst. (2013), 183–191. 21[3] M. D’Abbicco, The threshold of effective damping for semilinear wave equa- tions, Math. Meth. Appl. Sci. 38 (2015), 1032–1045. [4] F. Christ, M. Weinstein, Dispersion of Small-Amplitude Solutions of the 240 Generalized Korteweg-de Vries Equation , J. Funct. Anal. 100 (1991), 87–109. [5] H. Fujita, On the blowing up of solutions of the Cauchy Problem for ut= ∆u+u1+α, J. Fac.Sci. Univ. Tokyo 13 (1966), 109–124. [6] L. Grafakos, Classical and modern Fourier analysis , Prentice Hall, (2004). [7] L.Grafakos and S. Oh, The KatoPonce inequality , Comm. Partial Differen- 245 tial Equations 39 (6) (2014), 1128–1157. [8] A.GulisashviliandM.Kon, Exact smoothing properties of Schrndinger semi- groups, Amer. J. Math. 118 (1996), 1215–1248. [9] H. Hajaiej, L. Molinet, T. Ozawa, B. Wang, Necessary and sufficient con- ditions for the fractional Gagliardo-Nirenberg inequalit ies and applications to 250 Navier-Stokes and generalized boson equations , Harmonic Analysis and Non- linear Partial Differential Equations B26 (2011), 159–175. [10] R. Ikehata, Y. Mayaoka, T. Nakatake, Decay estimates of solutions for dissipative wave equations in RN with lower power nonlinear ities, J. Math. Soc. Japan, 56 (2004), 365–373. 255 [11] R. Ikehata, K. Tanizawa, Global existence of solutions for semilinear damped wave equations in Rnwith noncompactly supported initial data , Non- linear Analysis., 61 (2005), 1189–1208. [12] T. Kato and G. Ponce, Well-posedness and scattering results for the gener- alized Kortewegde-Vries equation via the contraction prin ciple, Comm. Pure 260 Appl. Math. 46 (4) (1993), 527–620. [13] C.E. Kenig, G. Ponce and L. Vega, Commutator estimates and the Euler and NavierStokes equations , Comm. Pure Appl. Math. 41 (1988), 891–907. 22[14] A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations , Publ. RIMS. 12 (1976) 169–189. 265 [15] A. Mohammed Djaouti, M. Reissig, Weakly coupled systems of semilinear damped waves with time-dependent coefficient and different po wer nonlinear- ities, submitted for publication. [16] A. Mohammed Djaouti, M. Reissig, On the benefit of different additional regularity for the weakly coupled systems of semilinear effe ctively damped 270 waves, submitted for publication. [17] A. Mohammed Djaouti, M. Reissig, Weakly coupled systems of semilinear effectively damped waves with different time-dependent coeffi cients in the dis- sipation terms and different power nonlinearities , submitted for publication. [18] A. Mohammed Djaouti, Semilinear Systems of Weakly Coupled Damped 275 Waves, PhD Thesis, TU Bergakademie Freiberg, 2018. [19] M. Nakao, K and Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations , Math. Z. 214 (1993) 325–342. [20] K. Nishihara, Y. 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2019-10-17

We prove the global existence of small data solution in all space dimension for weakly coupled systems of semi-linear effectively damped wave, with different time-dependent coefficients in the dissipation terms. Moreover, nonlinearity terms $ f(t,u) $ and $ g(t,v) $ satisfying some properties of the parabolic equation. We study the problem in several classes of regularity.

Modified different nonlinearities for weakly coupled systems of semilinear effectively damped waves with different time-dependent coefficients in the dissipation terms

1910.07731v1

arXiv:1102.5384v2 [cond-mat.mes-hall] 16 Sep 2011Dynamics of Skyrmion Crystal in Metallic Thin Films Jiadong Zang1,2,3,∗Maxim Mostovoy4, Jung Hoon Han5, and Naoto Nagaosa2,3† 1Department of Physics, Fudan University, Shanghai 200433, China 2Department of Applied Physics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 3Cross-Correlated Materials Research Group (CMRG), and Correlated Electron Research Group (CERG), RIKEN-ASI, Wako, Saitama 351-0198, Japan 4Zernike Institute for Advanced Materials, University of Gro ningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands 5Department of Physics, BK21 Physics Research Division, Sungkyunkwan University, Suwon 440-746, Korea (Dated: June 6, 2018) We study the collective dynamics of the Skyrmion crystal (Sk X) in thin films of ferromagnetic metals resulting from the nontrivial Skyrmion topology. It is shown that the current-driven motion of the crystal reduces the topological Hall effect and the Sky rmion trajectories bend away from the direction of the electric current (the Skyrmion Hall effect) . We find a new dissipation mechanism in non-collinear spin textures that can lead to a much faster spin relaxation than Gilbert damping, calculate the dispersion of phonons in the SkX, and discuss e ffects of impurity pinning of Skyrmions. PACS numbers: 73.43.Cd,72.25.-b,72.80.-r Introduction: Skyrmion is a topologically nontrivial soliton solution of the nonlinear sigma model. It was noted early on that Skyrmions in three spatial dimen- sions have physical properties of baryons and the peri- odic Skyrmion crystal (SkX) configurations were used to model nuclear matter[1, 2]. Skyrmions in two spatial dimensions play an important role in condensed matter systems, such as quantum Hall ferromagnets[3, 4]. It was suggested that SkX configurations can be stabilized by the Dzyaloshinskii-Moriya (DM) interaction in ferro- magnetswithoutinversionsymmetry[5]. Suchastatewas recently observed in a neutron scattering experiment in the A-phase of the ferromagnetic metal MnSi[6]. Recent Monte Carlo simulations indicated much greater stability of the SkX when a bulk ferromagnet is replaced by a thin film[7]. This result was corroborated by the real-space observation of SkX in Fe 0.5Co0.5Si thin film in a wide magnetic field and temperature range[8]. Lorentz force microscopy showed that Skyrmions form a triangular lattice with the magnetization vector antipar- allel to the applied magnetic field in the Skyrmion center and parallel at the periphery, as was also concluded from the neutron experiment[6]. The next important step is to explore dynamics of Skyrmion crystals and the ways to control them in anal- ogy to the actively studied current- and field-driven mo- tion of ferromagnetic domain walls[9]. Recent observa- tion ofthe rotationalmotion ofthe SkX in MnSi suggests that Skyrmionscanbe manipulated bymuchsmallercur- rents than domain walls[10]. In this Letter we study the coupled dynamics of spins and charges in the SkX, focusing on effects of the non- ∗Electronic address: jdzang@fudan.edu.cn †Electronic address: nagaosa@ap.t.u-tokyo.ac.jptrivial Skyrmion topology and effective gauge fields in- duced by the adiabatic motion of electrons in the SkX. We derive equation of motion for the collective vari- ables describing the SkX, calculate its phonon disper- sion, and discuss a new form of damping, which can be the dominant spin-relaxation mechanism in half-metals. In addition, we consider new transport phenomena, such as the topological Hall effect in a sliding SkX and the Skyrmion Hall effect. We also discuss the Skyrmion pin- ning by charged impurities and estimate the critical cur- rent above which the SkX begins to slide. Low-energy excitations in Skyrmion crystal: An iso- lated Skyrmion has two zero modes corresponding to translations along the xandydirections. Since an ap- pliedmagneticfieldopensagapinthecontinuumofspin- wave excitations, the low-energy magnetic modes in SkX are expected to be superpositions of the Skyrmion dis- placements, or the phonons. The phonon modes, as well asthecouplingofSkyrmiondisplacementstotheexternal current, can be consistently described in the framework of elasticity theory. We begin with the spin Hamiltonian HS=/integraltextd3x/bracketleftbigJ 2a(∇n)2+D a2n·[∇×n]−µ a3H·n/bracketrightbig , whereJis the exchange constant and Dis the DM coupling that stabilizes the SkX configuration n(x) in some interval of the magnetic field H=Hˆz[5, 11]. We calculate the ‘har- monic lattice energy’ by considering a deformation of the SkX,˜n(x,t) =n(x−u(x,t)), where the collective coor- dinateu(x,t) varies slowly at the scale of the SkX lattice constant. The result is: HS=dηJ/integraldisplayd2x ξ2[(∇ux)2+(∇uy)2],(1) wheredis the film thickness and ξ∼aJ Dis the characteristic length scale of SkX[11], with abeing2 the lattice spacing. The dimensionless quantity η= 1 8π/integraltext ucd2x(∂in·∂in) encodes the information about D andH, and is called shape factor in what follows. When an electroncurrentis flowingthroughthe metal- lic film, the conduction electrons interact with local mag- netic moments through the Hund’s rule coupling HH= −JHSψ†σ·nψ, whereψis the electron operator. In the case of small current density and the Skyrmion size much larger than the Fermi wavelength of conduction electrons, one can apply the adiabatic approximation in which the electron spins align perfectly with the local moment.ψis projected into the fully polarized state by ψ=χ|n/angb∇acket∇ightwithσ·n|n/angb∇acket∇ight=|n/angb∇acket∇ight. Then the electron action Sel=/integraltext dtd3x[i¯hψ†˙ψ+¯h2 2mψ†∇2ψ+JHSψ†σ·nψ] can be rewritten as Sel=/integraltext dtd3x[i¯hχ†˙χ−ea0−1 2mχ†(−i¯h∇− e ca)2χ+JHSχ†χ], whereaµ=¯hc 2e(1−cosθ)∂µϕwithθ andϕbeing the spherical angles describing the direc- tion of the local magnetization[12, 13]. The gauge po- tentialaµgives rise to internal electric and magnetic fields,eandh, acting on spin-polarized electrons pass- ing through the SkX in analogy with the electromag- netic gauge field. Crucially, the internal magnetic field b=∇×a=¯hc 2e(n·∂xn×∂yn)ˆzis intimately re- lated to the topological charge Qof Skyrmions by[14] Q=1 4π/integraltext ucd2x(n·∂xn×∂yn) =±1,where the integra- tion goes over the unit cell of the SkX. In the language of internal gauge field, this topological feature is nothing but thequantizationofinternalfluxΩ =/integraltexth·dSinunits ofhc/e. The coupling of the electric current to the inter- nal gauge field induced by the SkX, Hint=−1 c/integraltextd3xj·a, has a simple form in terms of the collective coordinates introduced above: Hint=d¯hQ e/integraldisplayd2x ξ2(uxjy−uyjx). (2) The crucial difference between the SkX and a con- ventional crystal is the form of kinetic energy. The spin dynamics originates from the Berry phase action, SBP=d γ/integraltext dtd2x(cosθ−1) ˙ϕ. Here,γ=a3 ¯h(S+x/2), where xis the filling of the conduction band, and S+x/2 is the total spin averagely per lattice site. In terms of uthe kinetic energy has the form SBP=dQ γ/integraldisplay dtd2x ξ2(ux˙uy−uy˙ux). (3) This form of the Berry phase shows that the collec- tive variables uxanduydescribing local displacements of Skyrmions form a pair of canonical conjugate vari- ables, replacing cos θandϕ. This characteristic prop- erty of SkX leads to several unusual responses to ap- plied electric currents and fields. It originates from the Skyrmion topology and distinguishes SkX from non- topological spin textures such as spirals and domain wall arrays. Using Eqs.(1), (2) and (3), we obtain equation of mo- tion foru: ˙u=−e¯hγ 2j+QγηJ e¯hˆz×∇2u. (4)Two consequences follow immediately. First, the dis- persion of phonons in the SkX obtained from Eq.(4) is quadratic, ¯hω=ηJa2 (S+x 2)k2, (5) in contrast to the linear phonon dispersion in usual crys- talsandsimilartothedispersionofmagnonsin auniform ferromagnet. Since uxanduyplay the role of the coor- dinate and momentum, the longitudinal and transverse phonon modes in the SkX merge into a single mode cor- responding to the rotational motion of Skyrmions, which leads to the quadratic dispersion. Secondly, the SkX can move as a whole driven by the charge current j, with a velocityV/bardbl=˙ u=−e¯hγ 2j. This rigid motion of SkX leads to several interesting results discussed below. Hall effect due to SkX motion: In such nontrivial spin textures, the external magnetic field (less than 0.2T for MnSi) is more than one order of magnitude smaller than the internal one, so that it would be neglected in what follows. As can be seen from Eq. (2), the collective coordinates uxanduyplay the role of electromagnetic gauge potentials Ayand−Ax, respectively. It is thus expected that the temporal variation of uinduced by the current leads to a transverse potential drop. This Hall-type effect can also be intuitively understood using theinternalmagneticfield bintroducedabove. Amoving spin texture n(x−V/bardblt) induces an internal electric field eanalogous to the electric field of a moving magnetic flux and related to the internal magnetic field by e= −1 c/bracketleftbig V/bardbl×b/bracketrightbig .For SkX with b=bzˆz, this electric field generates an electric current in the direction transverse toV/bardblresulting in the Hall conductivity: ∆σxy σxx≈ −x 2S+xe/angb∇acketleftbz/angb∇acket∇ightτ mc, (6) wheremis the electron mass and τis the relaxation time. The average internal magnetic field is /angb∇acketleftbz/angb∇acket∇ight=QΦ0 2πξ2, where Φ 0is the elementary flux and 2 πξ2is the area of the unit cell of the SkX. This Hall conductivity has the sameorderofmagnitude asthe oneresulting fromthe so- called topologicalHall effect observedin a staticSkX[15]. The latter effect is nothing but the Hall effect induced bybviae=1 c[v×b], andσTop xy/σxx≈e/angb∇acketleftbz/angb∇acket∇ightτ/mc, wherevis the electron velocity. Our new effect differs by the factor of −x 2S+xfrom the topological Hall effect. Its physical origin can be easily understood by noting the total force acting on a single conduction electron is F=−e c[(v−V/bardbl)×b], i.e. the Lorentz force on elec- trons due to the internal magnetic field of the SkX de- pendsontherelativevelocityofelectronsandSkyrmions. When the SkX begins to slide above the threshold elec- tric current jc[16], the net topological Hall voltage will be suddenly reduced by the factor2S 2S+x, which is how the effect of the spin-motive force and the collective shift of Skyrmions can be identified experimentally. New damping mechanism and Skyrmion Hall effect: Previously we have systematically discussed the novel3 effects related to the internal magnetic field. A natu- ral question thus arises as to whether there is any new phenomena associated with the intrinsic internal electric field, which is ei=−∂ia0−1 c˙ai=¯h 2e(n·∂in×˙n). Due to the time derivative in this expression, its effect is ab- sent in the static spin texture. However, in the present case, the motion of SkX makes it nonvanishing, and leads to an additional current j′byj′=σewithσthe con- ductivity of electrons. Substituting this current into the Landau-Lifshitz-Gilbert equation[12, 13] ˙n=¯hγ 2e[j·∇]n−γ/bracketleftbigg n×δHS δn/bracketrightbigg +α[˙n×n],(7) the time derivative ˙nreceives a correction given by δ˙n=¯hγσ 2e(e·∇)n=α′(n·∂in×˙n)∂in.(8) The corresponding dimensionless damping constant is α′=1 (2S+x)a3σ αfsξ2c, whereαfs≈1/137 is the fine struc- ture constant. The time derivative in the r.h.s. of Eq.(8) shows that the current induced by internal electric field leads to dissipation. In contrast to Gilbert damping this new mechanism does not require relativistic effects and only involves the Hund’s rule coupling that conserves the total spin. The relaxation of the uniform magnetization, described by Gilbert damping, is clearly impossible with- out the spin-orbit coupling, which breaks the conserva- tion of the total spin[17]. This argument, however, does not apply to inhomogeneousmagnetic textures where the breaking of the rotational symmetry by noncollinear spin orders enables the relaxation without the spin-orbit cou- pling (note that α′vanishes as ξ→ ∞). Despite the non-relativistic origin, α′depends on the DM coupling, as the latter determines the Skyrmion size. Estimates of α′made below show that in half-metals it can greatly exceedα. The effect of this new dissipation can be observed by tracing the trajectoryof Skyrmion motion. Including the newdissipationterm, themodifiedequationofmotion(4) for the rigid collective coordinates u(t) has the form ˙u=−e¯hγ 2j−Q(αη+α′η′)ˆz×˙u, (9) where the second shape factor η′is given by η′= Q 4π/integraltext ucd2x(n·∂xn×∂yn)(∂in·∂in)//integraltext ucd2x(∂in·∂in). The new dissipation term in Eq.(9) is obtained by mul- tiplying Eq.(8) with ∂jn, using ˙n=−(˙u·∇)n, and integrating over one unit cell. The whole damping term leads to a transverse motion with velocity V⊥≈Q(αη+α′η′)/bracketleftbig V/bardbl׈z/bracketrightbig . (10) This Skyrmion Hall effect can be observed by real-space images of Lorentz force microscopy. The corresponding Hall angle is θ= arctan(αη+α′η′).The estimate given below shows that main contribution to θcomes from the new dissipation mechanism.Pinning of Skyrmion crystal: Next we consider the pinning of the SkX by charged impurities. The pinning results from spatial fluctuations of the impurity density andvariationsofthespindirectionintheSkX.Variations of the density of charged impurities δnigive rise to local variations of the electron density neand since the double exchangeconstant Jisproportionaltothelatter, wehave δJ∼Jδni/ne. The energy per Skyrmion ES∼Jd/a. Denote the number of impurities in this volume by N1 with/angb∇acketleftN1/angb∇acket∇ight=ni2πξ2dand the variance δN1=√N1, we obtain the typical variation of the Skyrmion energy: V1=δJd a∼J ne2πξ2a/radicalbig N1=J neaξ/radicalbigg nid 2π.(11) The potential energy density is then V0=V1/(2πξ2). Substituting ni∼(la2)−1, wherelis the electron mean free path, and ne=x a3, we obtain V0∼J (2π)3/2x/radicalBig d la ξ3. The pinning regime of the whole SkX depends on the ra- tioofthe pinningenergy V1andthe elasticenergy ESofa single Skyrmion[18]. Let L2be the number of Skyrmions in the domain where u∼ξ. The energy gain due to the impurity pinning in the domain is ∼ −V1L, while the elastic energy cost ∼Jd ais independent of the do- main size. Minimizing the total energy per Skyrmion, Jd aL2−V1 L, we obtain L∼Jd aV1.L≫1 corresponds to the case of weak (or collective) pinning of SkX, while L∼1 corresponds to the strong pinning regime. The pinning potential gives rise to the spin transfer torque−Qγξ2 2d/bracketleftbigˆz×δV δu/bracketrightbig in the right-hand side of Eq.(4). In the steady state of moving SkX this torque has to be compensated by the interaction with the electric current. The critical current density is then jc∼e ¯hξ2 d/angbracketleftbigg∂V ∂u/angbracketrightbigg steady state∼e ¯hξV0 dL,(12) in the weak pinning regime, while in the strong pinning caseLhas to be substituted by 1. Similarly, one can estimate the gap in the spin wave spectrum due to the pinning: ¯hωpin∼¯hγξ2 d/angbracketleftbigg∂2V ∂u2/angbracketrightbigg ∼¯hγξ2 dV0L L2ξ2=a3 dSV0 L.(13) Estimates: For estimates we consider MnSi where Mn ions form a (distorted) cubic sublattice with a= 2.9˚A. The length of the reciprocal lattice vectors of the SkX ∼0.035˚A−1correspondsto ξ∼77˚AandD∼0.1J. The kinetic energy scales as ¯ h2/mξ2<JH∼1eV, so that the adiabaticapproximationisjustified. Theelectrondensity ne= 3.8·1022cm−3[15] corresponds to x=nea3∼0.9 charge carriers per lattice site, while the residual resis- tivityρ∼2µΩ·cm[15] gives an estimate of the impurity concentration xi∼5·10−3. From the magnon dispersion in the spiral state[19], J∼3 meV. Using these parameters we get α′∼0.1, which shows thatthedampingresultingfromtheelectriccurrentsgen- erated by non-collinear spin textures can be the domi- nant mechanism of spin relaxation in half-metals, where4 the Gilbert damping constant αis one-three orders of magnitude smaller[20, 21]. We note, however, that since the typical electron mean free path ξ∼500˚A is larger than the Skyrmion size ξ, the relation between the cur- rent and internal electric field is nonlocal. The topo- logical Hall angle, θH∼ehzτ mc=1 αsfneξ2σ catT= 0 and the change in θHinduced by the sliding Skyrmion crystal [see Eq.(6)] are also ∼0.1. For a 10nm thick film the parameter L∼xξ√ 2πdl a2∼103,i.e. the pin- ning of Skyrmions by charged impurities is exceedingly weak and the corresponding values of the pinning fre- quency ¯hωpin∼5·10−11meV and the critical current jc∼0.2A·cm−2. This value is much lower than that for domain walls[9] and smaller than the ultralow thresh- old current jc∼102A·cm−2observed in bulk MnSi[10], which may be attributed to other pinning mechanisms and the different dimensionality of the system. This low current density also justifies the adiabatic approximation applied in this work. Comparison with vortex dynamics: Finally, we com- pare the dynamics of SkX with that of the vortex lines (VL) in type II superconductors. Asimilarquadraticdis- persion was obtained for VL [22]. However, in supercon- ductors it results from long-ranged interactions between vortices, while the interactions between Skyrmions are short-ranged (if one ignores the relatively weak dipole- dipole interactions). The absence of long-range interac- tions in SkX ensures the stability of the quadratic disper- sion. Furthermore, the kinetic terms in VL and SkX are completely different. For VL uxanduyare two indepen- dent variables,while forSkXtheareconjugatedvariablesas in Eq.(3). Therefore VL are massive, while Skymions are not. When a supercurrent flows through the type II superconductor, the charged Cooper pairs are deflected by the VL through the Lorentz force, which in turn gives risetothetransversemotionoftheVL.TheVLdynamics is usually assumed to be overdamped[23], the kinetic en- ergy of VL is neglected, and the Lorentz force is assumed to be counterbalanced by the friction force. In contrast, the damping of Skyrmions is relatively weak. The spin torque resulting from the strong Hund’s rule coupling re- sults in a nearly longitudinal motion Skyrmion motion. The Hall motion of VL results in a longitudinal voltage drop, which is not important for SkX motion due to the small Hall angle. During the completion of this paper we became aware of a recent paper by Kim and Onoda addressed the dy- namics of Skyrmions in an itinerant double-exchange fer- romagnet using a Chern-Simons-Maxwell approach[24]. Their focus, however, seems to differ from ours. We are grateful for the insightful discussions with Prof. Yoshi- noriTokura. ThisworkissupportedbyGrant-in-Aidsfor Scientific Research (No. 17105002, 19019004, 19048008, 19048015,and21244053)fromtheMinistryofEducation, Culture, Sports, Science and Technology of Japan, and also by Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program). JZ is supported by Fudan Research Program on Postgrad- uates. MM is supported by the Stichting voor Funda- mental Onderzoekder Materie(FOM). HJH issupported by Mid-career Researcher Program through NRF grant funded by the MEST Grant No. R01- 2008-000-20586-0. [1] T. H. R. Skyrme, Proc. Roy. Soc. London A 260, 127 (1961); Nuc. Phys. 31, 556 (1962). [2] I. Klebanov, Nucl. Phys. B 262, 133 (1985). [3] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H. Rezayi, Phys. Rev. B 47, 16419 (1993). [4] C. Timm, S. M. Girvin, and H. A. Fertig, Phys. Rev. B 58, 10634 (1998). [5] A.N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP 68, 101-103 (1989); U.K. Rosler, A.N. Bogdanov, and C. Pfleiderer, Nature 4 42, 797-801 (2006). [6] S. M¨ uhlbauer, B. Binz, F. Joinetz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science 323, 915 (2009). [7] S. D. Yi, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B80, 054416 (2009); Ulrich K. Roeßler, Andrei A. Leonov, Alexei N. Bogdanov, arXiv:1009.4849. [8] X. Z. Yu et al., Nature (London) 465, 901 (2010). [9] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004). [10] F. Jonietz et al., Science 330, 1648 (2010). [11] J. H. Han, J. Zang, Z. Yang, J. H. Park, and N. Nagaosa, Phys. Rev. B 82, 094429 (2010). [12] Ya. B. Bazaliy, B. A. Jones, and S. C. Zhang, Phys. Rev. B57, R3213 (1998).[13] G. Tatara, H. Kohno, J. Shibata, Phys. Rep. 468, 213 (2008). [14] R. Rajaraman, Solitons and Instantons , North-Holland, 1987. [15] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong, Phys. Rev. Lett. 102, 186601 (2009); A. Neubauer et al., Phys. Rev. Lett. 102, 186602 (2009). [16] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204 (2005). [17] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. 75, 113706 (2006). [18] H. FukuyamaandP. A.Lee, Phys.Rev.B 17, 535(1978); P. A. Lee and T. M. Rice, Phys. Rev. B 19, 3970 (1979). [19] S. V. Grigoriev et al., Phys. Rev. B 74, 214414 (2006). [20] T. Kubota et al., Appl. Phys. Lett. 94, 122504 (2009). [21] C. Liu et al., Appl. Phys. Lett. 95, 022509 (2009). [22] P. G. de Gennes and J. Matricon, Rev. Mod. Phys. 36, 45 (1964); A. L. Fetter, and P. C. Hohenberg, Phys. Rev. 159, 330 (1967). [23] M. Tinkham, Introduction to Superconductivity , McGraw-Hill, 1996 [24] K. S. Kim and S. Onoda, arXiv:1012.0631.

2011-02-26

We study the collective dynamics of the Skyrmion crystal (SkX) in thin films of ferromagnetic metals resulting from the nontrivial Skyrmion topology. We show that the current-driven motion of the crystal reduces the topological Hall effect and the Skyrmion trajectories bend away from the direction of the electric current (the Skyrmion Hall effect). We find a new dissipation mechanism in non-collinear spin textures that can lead to a much faster spin relaxation than Gilbert damping, calculate the dispersion of phonons in the SkX, and discuss effects of impurity pinning of Skyrmions.

Dynamics of Skyrmion Crystals in Metallic Thin Films

1102.5384v2

arXiv:1609.07901v1 [cond-mat.mtrl-sci] 26 Sep 2016Relativistic theory of spin relaxation mechanisms in the La ndau-Lifshitz-Gilbert equation of spin dynamics Ritwik Mondal,∗Marco Berritta, and Peter M. Oppeneer Department of Physics and Astronomy, Uppsala University, P .O. Box 516, Uppsala, SE-75120, Sweden (Dated: October 14, 2018) Starting from the Dirac-Kohn-Sham equation we derive the re lativistic equation of motion of spin angular momentum in a magnetic solid under an external e lectromagnetic field. This equation of motion can be rewritten in the form of the well-known Landa u-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with a n electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously d erive, from fundamental principles, a general expression for the anisotropic damping tensor whic h is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chir al, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises fur thermore both electronic interband and intraband transitions. We also show that when the externall y applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torq ue exerted on the spin moment. PACS numbers: 75.78.-n, 76.20.+q, 71.15.Rf I. INTRODUCTION In their seminal 1935-paper, L. D. Landau and E. M. Lifshitz proposed the equation of motion governing the dy- namics of a continuum magnetization [1]. Eighty years after its original formulation, the Landau-Lifshitz (LL) equati on continues to play a fundamental role in the understanding of magnetization dynamics [2] and forms the cornerstone of contemporary micromagnetic simulations (see, e.g., Refs. [3, 4]). Originally, the Landau-Lifshitz equation was derived on the basis of phenomenological considerations [1]. It define s the time-evolution of a volume magnetization M(r,t)as ∂M ∂t=−γM×Heff−λM×[M×Heff],(1) whereγis the gyromagnetic ratio, Heffis the effective mag- netic field, and λis an isotropic damping parameter. The first term describes the precession of the local magnetiza- tionM(r,t)around the effective field Heff. The second term describes the magnetization relaxation such that the magnetization vector relaxes to the direction of the effecti ve field. The damping term in the LL equation was reformu- lated by Gilbert [5, 6] to give the Landau-Lifshitz-Gilbert (LLG) equation, ∂M ∂t=−γM×Heff+αM×∂M ∂t, (2) whereαis the Gilbert damping constant. Note that both damping parameters αandλare here scalars, which cor- responds to the assumption of an isotropic medium. Both LL and LLG equations preserve the length of the magneti- zation during the dynamics and are mathematically equiv- alent (see, e.g. [7]). A number of explanations have been proposed for the microscopic origin of the spin relaxation in magnetic met- als [8–18]. Already in their original work Landau and Lif- shitz attributed the damping constant to relativistic effects ∗Ritwik.Mondal@physics.uu.se[1]. More specific microscopic theories of spin relaxation in ferromagnetic metals have been developed in the last decennia. Kamberský proposed the breathing Fermi sur- face model [8] and the related torque-correlation model [14, 19]. Brataas et al. proposed a scattering theory for- mulation [15] of the Gilbert damping which is equiva- lent to a Kubo linear-response formulation. A different form of the relaxation term caused by spatial dispersion of the exchange interaction—this in contrast to the isotropic medium assumption made in the LL equation—was pro- posed by Bar’yakhtar and co-workers [10, 20, 21]. More recently the debate on what the appropriate the- ory to describe damping would be has focused on first- principles electronic structure calculations and, in how far these could provide quantitative values of the Gilbert damping [22–30]. Recent ab initio calculations of the Gilbert damping constant for transition-metal alloys pre- dicted values that correspond to the experimental values within a range of a factor of two to three [22–24, 26–28], with significant deviations however for the pure elemen- tal ferromagnets. This indicates that there is still a need to improve the fundamental understanding of the origin of spin-moment relaxation. Also, very recent publications have questioned the existing understanding of the Gilbert damping [31, 32]. Here we develop a theoretical description of spin relax- ation on the basis of the relativistic Density Functional Theory (DFT). To this end, we start from the relativis- tic Dirac-Kohn-Sham (DKS) equation that adequately de- scribes the electronic states in a magnetic solid. From thes e we derive the general equation of motion for spin angular momentum, which adopts the form of the LLG equation. Within this framework we obtain explicit expressions for the tensorial form of the Gilbert damping term, which we find to contain an isotropic Gilbert-like contribution and anisotropic Ising-like and chiral Dzyaloshinskii-Moryia -like contributions. Our derivation follows similar steps as a pr e- vious derivation by Hickey and Moodera [17], however, as discussed below, it includes previously missing terms and thus leads to different expressions for the spin relaxation.2 II. THE RELATIVISTIC DIRAC HAMILTONIAN As mentioned before, relativistic effects such as the spin- orbit interaction are at the heart of spin angular momen- tum dissipation in solids. To examine how these fundamen- tal physical interactions lead to magnetization damping we choose therefore to start from the most general relativisti c Hamiltonian, the DKS Hamiltonian. This Hamiltonian de- scribes the one-electron quantum state in an effective spin- polarized field due to other electrons and nuclei in the solid , in addition to externally applied fields. For spin-polarize d electrons in a magnetic material the DKS Hamiltonian is given as [33–35] HD=cα·(p−eA)+/parenleftbig β−1/parenrightbig mc2+V1+eΦ1 −µBβΣ·Bxc. (3) HereVis the unpolarized Kohn-Sham selfconsistent po- tential,Bxcis the spin-polarized part of the exchange- correlation potential in the material, A=A(r,t)is the vec- tor potential of an externally applied electromagnetic fiel d, eΦ(r,t)is the scalar potential of this field, p=−i/planckover2pi1∇, and µBise/planckover2pi1 2m, the Bohr magneton. 1is the4×4identity matrix andα,β, andΣare the well-known Dirac matrices in Dirac bi-spinor space, which contain the Pauli spin matrices σ and the2×2identity matrix. At this point, it is important to observe that there are two fundamentally different fields present in the DKS Hamiltonian. There are the Maxwellfields, that is, (implicitly) the external magnetic inducti on B(r,t) =∇×A(r,t)as well as the external electric field, E(r,t) =−∂A(r,t) ∂t−∇Φ. The strongest field in a magnetic material is however the exchange field, which stems from the Pauli exclusion principle. The exchange field Bxcis fundamentally different from the standard magnetic induc- tion, as it obviously acts only on the spin degree of freedom (see, e.g., [34]) and does not couple to the orbital angular momentum. Also, it doesn’t fulfill the Maxwell equations as the auxiliary electromagnetic field (e.g., ∇·B= 0) and it cannot be included as a vector potential Axcin the linear momentum, i.e. p−eAxc, but instead needs to be treated as a separate term in Eq. (3). Next, we want to investigate the relativistic spin evo- lution of spin-polarized electrons in a magnetic solid. To achieve this we need the positive energy, that is, the elec- tron solutions that are given by the large component of the Dirac bi-spinor. To arrive at an elucidating formula- tion in terms of the spin operator we employ the Foldy- Wouthuysen transformation approach [35, 36] on the DKS equation for the case where an exchange field Bxcis explic- itly present (for details, see Ref. [37]). Doing so, one ob- tains a Hamiltonian for the electron solutions only, which we expand in orders of 1/c2to select the largest relativistic contributions. This leads to a semi-relativistic, extende d Pauli Hamiltonian (see Ref. [37]), HEP=(p−eA)2 2m+V−µBσ·B−µBσ·Bxc eff+eΦ−(p−eA)4 8m3c2−1 8m2c2/parenleftbig p2V/parenrightbig −e/planckover2pi12 8m2c2∇·E +i 4m2c2σ·(pV)×(p−eA)−e/planckover2pi1 8m2c2σ·{E×(p−eA)−(p−eA)×E} +iµB 4m2c2[(p×Bxc)·(p−eA)]. (4) Except from the last term in Eq. (4), all the appearing relati vistic corrections involving the exchange interaction can be added together giving an effective exchange field [38], Bxc eff=Bxc−1 8m2c2/braceleftBig/bracketleftbig p2Bxc/bracketrightbig +2(pBxc)·(p−eA)+2(p·Bxc)(p−eA)+4[Bxc·(p−eA)](p−eA)/bracerightBig ≡Bxc+Bxc corr. (5) The Hamiltonian HEPexactly includes all spin-dependent relativistic terms (of the order of 1/c2) and all the terms involving Bxcand the external electromagnetic fields. We emphasize that for our purpose of unveiling the relativisti c mechanisms of spin dissipation it is obviously not sufficient to work with the conventional Pauli Hamiltonian, which only consists of the five first terms in the nonrelativistic limit. The correct form of all relativistic terms can solely be obtained when one starts from the DKS equation with exchange field. We remark that in a previous study Hickey and Moodera [17] used a Pauli Hamiltonian different from the above one, without exchange field and without crystal potential and thus without the intrinsic spin-orbit intera c- tion [the first term in the second line of Eq. (4)]. The meaning of the terms in Hamiltonian (4) can bereadily understood, see Ref. [37] for details. The fourth term on the right is a Zeeman-like term due the presence of the relativistically corrected exchange field, which acts a s an effective mean field. The ninth term is the one, which in a central potential V, gives rise to the conventional form of the spin-orbit coupling. The tenth term is a kind of spin-orbit interaction but due to the external fields. The very last term is a relativistic correction which depends on theBxcfield but is independent of the spin. As we will see in the following, the terms that are responsible for spin relaxation are the relativistic terms that involve a direct coupling of the spin operator with either the exchange field Bxcor one of the externally applied fields ( EorA).3 III. SPIN EQUATION OF MOTION The spin angular momentum operator is given by S= (/planckover2pi1/2)σ. To obtain an equation of motion for the spin op- erator we have to evaluate the commutator [S,HEP(t)]. It is obvious from the expression of HEPthat only the terms which are explicitly spin dependent will contribute as oth- erwise the commutator vanishes. We can thus extract from HEPthe spin Hamiltonian HS(t) =H0+Hint soc+Hext soc (6) where the Zeeman-like fields are added up to an effective magnetic induction, H0=−e mS·(B+Bxc+Bxc corr)≡ −e mS·Beff.(7) The part H0contains the main nonrelativistic contribution, all other terms in the spin Hamiltonian HSare of relativis- tic origin. The intrinsic spin-orbit coupling is given by th e Hamiltonian Hint soc=i 2/planckover2pi1m2c2S·(pV)×(p−eA). (8) The crystal potential stems from the nuclei-electron and electron-electron interactions and thus should have trans - lational symmetry. Consequently, also the intrinsic spin- orbit Hamiltonian has translational symmetry [39]. If the position of any j-th nucleus is Rj, the electron position is r, and the electron position with respect to the nucleus is represented by rj, then the crystal potential can be rep- resented by a sum of atom-centered potentials. Making now in addition the central potential approximation (no angular dependence) for each of the atom-centered poten- tials, the potential can be written as V(rj) =V(|r−Rj|). The translational symmetry is realized by the fact that rj=r−Rj. With the definition of spin-orbit interac- tion strength ξ(rj) =1 2m2c21 rdV(rj)/dr, and the Coulomb gauge,∇·A= 0, for homogeneous magnetic fields, i.e., A= (B×r)/2, this Hamiltonian can further be written as Hint soc=1 2m2c21 rdV drS·L−er 4m2c2dV drS·B +e 4m2c21 rdV dr(S·r)(r·B) =/summationdisplay jξ(rj)/bracketleftBig S·L−e 2/parenleftBig r2S·B−(S·r)(r·B)/parenrightBig/bracketrightBig .(9) We note, first, that the full spin-orbit Hamiltonian, Hint soc+ Hext soc, is gauge invariant [40], but for deriving expressions we need to make a choice. The Coulomb gauge is a suit- able choice here, yet it can be used exactly only when a slowly varying and homogeneous magnetic field is present. This gauge further implies that only the transversal parts ofEand ofAare retained, the latter being gauge invari- ant. Doing so, we have thus recovered the “usual” spin-orbit coupling term and other ultra-relativistic terms. The external spin-orbit coupling Hamiltonian is given by Hext soc=−e 4m2c2S·{E×(p−eA)−(p−eA)×E}, which has a similar form as Hint soc[Eq. (8)], but contains the external Maxwell fields instead. Making use of Maxwell’sequation ∇×E=−∂B/∂t, this Hamiltonian can be rewritten as Hext soc=−e 2m2c2S·(E×p)+ie/planckover2pi1 4m2c2S·∂B ∂t +e2 2m2c2S·(E×A). (10) The last term in the Hamiltonian Hext socdescribes the in- teraction of the photon spin angular momentum density, js=ǫ0(E×A)[41], with the electron spins [40, 42]. A related interaction energy due to a coupling of the angu- lar momentum density of the electromagnetic field with the magnetic moment was proposed recently on phenomenolog- ical grounds [43]. The relativistic light-spin interactio n in the Hamiltonian (10) adopts thus the form Hext light−spin=e2 2m2c2ǫ0S·js. (11) This term, being second order in the external fields can become important in the strong field regime. As we focus in first instance on the damping, we will not consider it in the derivation of the spin damping, but we come back to it later on. Now we have the necessary parts of the spin Hamiltonian and we are ready to calculate the spin dynamics equations. According to the definition of magnetization, this quantity is given by the expectation value of spin angular momentum [44] M=/summationdisplay jgµB VTr/braceleftbig ρSj/bracerightbig , (12) whereVis a suitably chosen volume element. The sum- mation is taken over all the electrons jand the definition of the density matrix is ρ=/summationtext ipi|ψi/an}b∇acket∇i}ht/an}b∇acketle{tψi|, where the set of wave functions |ψi/an}b∇acket∇i}htare in a mixed state and piare the occupation numbers. As is customary in spin dynamics models [12–20, 24, 26, 45] the contribution of the orbital angular momentum to the total magnetization has been neglected because it is quenched for the common transition metals (e.g., Fe, Ni, Co etc.). The equation of motion of the magnetization is obtained by taking the time deriva- tive on both sides of Eq. (12), and using that ∂ρ/∂t= 0 for quasiadiabatic processes [46], which gives ∂M ∂t=gµB V1 i/planckover2pi1/summationdisplay jTr/braceleftbig ρ[Sj,HS(t)]/bracerightbig . (13) To obtain the magnetization dynamics we substitute the spin Hamiltonian HS(t) =H0+Hint soc+Hext socin the right- hand side of Eq. (13) and work out the trace term-by-term. Before presenting the result we consider briefly the ap- proximations made in the derivation. Notably, Eq. (13) is valid for local processes and will hence provide a local damping mechanism. However, it is known that nonlo- calcontributions to the damping exist (see, e.g., [47–49]) that can be caused by spin transport from one region to another [50–52]. Such effects can be treated using the con- tinuity equation, ∂ρ/∂t+∇·J= 0, withJthe current operator, leading to an additional spin current term (see, e.g., [52, 53]). A further remark due at this point concerns4 the time dependence of the exchange field. In line with the above, we adopt the adiabatic approximation that is valid for systems not too far from the ground state [54]. Working out the commutator, we find that the first or- der dynamical equation of motion is given by the mostly nonrelativistic part in the spin Hamiltonian, H0. Using the commutation relations for spin angular momentum, [Sj,Sk] =i/planckover2pi1ǫjklSl, the first order equation of motion be- comes ∂M ∂t/vextendsingle/vextendsingle/vextendsingle0 =−γM×Beff, (14) whereγ=g|e|/2mis the gyromagnetic ratio and g≈2for spin degrees of freedom. Using B=µ0(H+M), the right- hand term can be rewritten in the conventional form as −γ0M×Heff, whereγ0=µ0γ. This equation provides the common understanding of the Larmor precessional motion of magnetization around an effective magnetic field, with a distinction that there is a relativistic correction Bxc corrto this field that has not been noted before. Next we treat the relativistic spin-orbit effects in the magnetization dynamics. As we will see, these are the ones that lead to local damping, i.e., the spin relaxation mech- anisms in a magnetic solid are of relativistic origin [1, 9]. First, we focus on the relativistic intrinsic spin-orbit co u- pling Hamiltonian Hint socin Eq. (9). Due to the quenching of the orbital angular momentum, the first term vanishes. The dynamics due to the remaining two terms in the Hamil- tonian is calculated as ∂M ∂t/vextendsingle/vextendsingle/vextendsingleint soc=e 4m2c2/angbracketleftBig rdV dr/angbracketrightBig M×B −e 4m2c2M×/angbracketleftBig r1 rdV dr(r·B)/angbracketrightbig =e 2/summationdisplay j/bracketleftBig /an}b∇acketle{tξ(rj)r2/an}b∇acket∇i}htM×B −M×/an}b∇acketle{tξ(rj)r(r·B)/an}b∇acket∇i}ht/bracketrightBig . (15) The first term in the dynamics of Eq. (15) can be seen as a further relativistic correction to the magnetization prec es- sion. The second term has a form similar to the first term, but with opposite sign. The terms can be combined, but they do not contribute to any relaxation processes as they do not contain a time variation of the magnetic induction. Next we consider the dynamics related to Hext soc. We will see below that it is mainly the relativistic extrinsic spin- orbit coupling, i.e., the first two terms of Eq. (10), which give rise to dominant local spin relaxation mechanisms in magnetic solids. In addition, we observe here that these correspond to the transverse spin relaxation. We consider here the long wavelength approximation, where the wave- length of the field is much larger than the size of the system. In other words the GHz/THz electromagnetic field inside the ferromagnetic film is assumed uniform throughout the film as long as the film thickness is sufficiently small. We can thus use the Coulomb gauge, i.e., A= (B×r)/2. This gauge allows us to obtain the explicit time dependence of the Hamiltonian. The transverse electric field in the Hamil- tonian is then written as E=1 2(r×∂B/∂t). Employing the gauge, the first two terms in Eq. (10) can be re-writtenin an explicit, time-dependent form: Hext soc=ie/planckover2pi1 4m2c2S·∂B ∂t/parenleftbigg 1−(r·p) i/planckover2pi1/parenrightbigg +e 4m2c2(S·r)/parenleftbigg∂B ∂t·p/parenrightbigg . (16) At this point it is needed to inspect the hermiticity of the Hamiltonian. It can be shown that the total spin-orbit Hamiltonian in Eq. (16) is hermitian (see Appendix A), however, for the individual terms it is different. Writing down the Hamiltonian in component form with the usual summation convention, we obtain Hext soc=ie/planckover2pi1 4m2c2Si∂Bi ∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright anti−hermitian−e 4m2c2/summationdisplay i/negationslash=jSi∂Bi ∂trjpj /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright non−hermitian +e 4m2c2/summationdisplay i/negationslash=jSiri∂Bj ∂tpj /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright hermitian. (17) Previously, Hickey and Moodera considered the effect of the spin-orbit Hamiltonian on damping, but only obtained the first two terms in Eq. (10) [17]. They proposed then only the anti-hermitian part of the Hamiltonian as an intrinsic source of Gilbert damping [17]. Anti-hermitian Hamilto- nians understandably are always dissipative [55, 56]. Con- sequently, their choice of taking the anti-hermitian term only was criticized, given that the full spin-orbit Hamilto - nian should be hermitian and that it therefore should not exhibit dissipation [55]. In our case the total spin-orbit Hamiltonian (16) is man- ifestly hermitian, yet we will show below that it does give rise to spin moment damping. The point is, that even when the full Hamiltonian is hermitian, it only has this property when one considers the dynamics of the full sys- tem. It is however customary in spin moment dynamics [12–20, 24, 26, 45] to integrate out the orbital degree of freedom and other magnetic degrees of freedom (as back- ground fluctuations of the system) thus restricting the fo- cus on the single spin moment dynamics. In the thereby restricted Hilbert space the hermiticity is lost and hence the whole Hamiltonian can contribute to the damping. Calculating now the commutation relation [S,Hext soc]and taking the summation of the trace over all electrons, the spin moment dynamics adopts the form ∂M ∂t/vextendsingle/vextendsingle/vextendsingleext soc=−ie/planckover2pi1 4m2c2M×∂B ∂t/parenleftBigg 1−/angbracketleftbig r·p/angbracketrightbig i/planckover2pi1/parenrightBigg −e 4m2c2M×/angbracketleftBig r/parenleftbigg∂B ∂t·p/parenrightbigg/angbracketrightBig . (18) A rewriting of these terms is required to elucidate further the spin relaxation. IV. THE DAMPING EQUATIONS To obtain explicit expressions for the damping terms, we employ the general relation between magnetic induction B,5 magnetization M, and magnetic field H, given as B= µ0(M+H). We take the time derivative on both sides, ∂B ∂t=µ0/bracketleftbigg∂M ∂t+∂H ∂t/bracketrightbigg . (19) This relation is generally valid, also for the stationary ca se, even though the magnetization M(t)and magnetic field H(t)are time dependent. At this point it is instructive to consider what kinds of magnetic fields H(t)can occur. The simplest case is when at some time t0only a static fieldH0 is present, then obviously only the first term in Eq. (19) contributes. If the field H(t)is explicitly time dependent, we can distinguish to cases: an ac driven, periodic magnetic field, as is commonly used in measurements, or a more gen- eral field, for example a magnetic field pulse. In the latter case, one could proceed to derive the spin dynamics by keeping explicitly the term∂H ∂t. As a result, one obtains a LLG-like equation, where however the magnetic field cou- ples into the damping term. The thus-obtained modified LLG equation is given and analyzed further below, in Sect. V. In the former case, the effect of the magnetic response becomes apparent when an ac magnetic field is applied. For ferromagnetic materials, where there is a net magneti- zation present even in the absence of the applied field, the magnetic susceptibility can be introduced by the definition : χ=∂M/∂H. Using a chain rule for the time derivative, ∂H ∂t=∂H ∂M∂M ∂t, Eq. (19) can be written as ∂B ∂t=µ0/parenleftbig 1+χ−1/parenrightbig ·∂M ∂t, (20) where 1is the3×3identity matrix. This relation has been used in the ensuing magnetization dynamics. Substituting Eq. (20) in the first term of Eq. (18), we obtain ∂M(1) ∂t/vextendsingle/vextendsingle/vextendsingleext soc=−ie/planckover2pi1µ0 4m2c2M×/bracketleftbigg (1+χ−1)·∂M ∂t/bracketrightbigg/parenleftbigg 1−/an}b∇acketle{tr·p/an}b∇acket∇i}ht i/planckover2pi1/parenrightbigg . (21) This term can already be recognized to have the form of the Gilbert damping, M×/bracketleftbig α·∂M ∂t/bracketrightbig , yet with a tensorial damping constant. For the full damping we have to combine with the second term in Eq. (18), which is rewritten as ∂M(2) ∂t/vextendsingle/vextendsingle/vextendsingleext soc=−eµ0 4m2c2M×/angbracketleftBig r/parenleftbigg/bracketleftBig (1+χ−1)·∂M ∂t/bracketrightBig ·p/parenrightbigg/angbracketrightBig . (22) To join the terms we proceed with using vector components. Equation (21) becomes ∂M(1) ∂t/vextendsingle/vextendsingle/vextendsingleext soc=−eµ0 4m2c2/summationdisplay ijklnMk/bracketleftbigg (1+χ−1)ij∂Mj ∂t/bracketrightbigg ×(i/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht)εkilˆel, (23) withεijkthe Levi-Civita tensor and ˆea unit vector. This term can be written as ∂M(1) ∂t/vextendsingle/vextendsingle/vextendsingleext soc=/summationdisplay ijklMk∂Mj ∂tΩijεkilˆel, (24)withΩij=−eµ0 4m2c2/summationtext n(i/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht)(1+χ−1)ij. The sec- ond term (22) can be written in a similar form, but with a tensor ∆ij=−eµ0 4m2c2/summationtext n/an}b∇acketle{tripn/an}b∇acket∇i}ht(1+χ−1)nj. Combining these two terms gives the total damping term, ∂M ∂t/vextendsingle/vextendsingle/vextendsingleext soc=/summationdisplay ijklMk/bracketleftBig Ωij+∆ij/bracketrightBig∂Mj ∂tεkilˆel,(25) where it is convenient to define A ij≡Ωij+∆ij, Aij=−eµ0 4m2c2/summationdisplay n/bracketleftBig i/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht+/an}b∇acketle{trnpi/an}b∇acket∇i}ht/bracketrightBig (1+χ−1)ij =−eµ0 8m2c2/summationdisplay n,k/bracketleftBig /an}b∇acketle{tripk+pkri/an}b∇acket∇i}ht−/an}b∇acketle{trnpn+pnrn/an}b∇acket∇i}htδik/bracketrightBig ×(1+χ−1)kj.(26) Note that a summation over iis not intended in the right- hand side expressions. In vector form the spin-orbit damp- ing term becomes ∂M ∂t/vextendsingle/vextendsingle/vextendsingleext soc=M×/bracketleftBig A·∂M ∂t/bracketrightBig . (27) Summarizing our result, we observe that we have obtained a damping parameter A ijof Gilbert type that is however in its general form not a scalar but a tensor. The tenso- rial character of the Gilbert damping was also concluded recently in other investigations [16, 57]. In this form it accounts for transversal spin relaxation that conserves th e length of the magnetization, i.e., ∂(M·M)/∂t= 0. Every tensor can be decomposed in a symmetric and an anti-symmetric part. Hence, the damping tensor can be decomposed into a scalar ( α) multiplied by the unit matrix, a symmetric tensor ( I), and an anti-symmetric tensor ( A, withAij=1 2(Aij−Aji)). The latter tensor can in turn be expressed as Aij=εijkDkwithDbeing a vector. Finally, the damping dynamics can then be written as ∂M ∂t/vextendsingle/vextendsingle/vextendsingleext soc=αM×∂M ∂t+M×/bracketleftBig I·∂M ∂t/bracketrightBig +M×/bracketleftBig D×∂M ∂t/bracketrightBig . (28) The first term is the conventional Gilbert damping. It orig- inates from the decomposition of the symmetric part of the tensor into an isotropic Heisenberg-like (α1)contribution as well as an anisotropic Ising-like ( I) contribution which leads to the second term. Along with that it is not surpris- ing that the last term implies a Dzyaloshinskii-Moriya-lik e contribution. The anisotropic nature of the Gilbert damp- ing has been noted before [18, 57], but not the appearance of the Dzyaloshinskii-Moriya-like damping. This type of damping could be related to the chiral damping of mag- netic domain walls that was reported recently [58]. For the case of a constant, scalar Gilbert damping param- eter it is straightforward to transform the LLG equation to obtain the LL equation with the phenomenological damp- ing term proposed by Landau and Lifshitz [1]. However, this is no longer the case for tensorial Gilbert damping, for which the transformation is much more involved. The spin-dynamics equation in the Landau-Lifshitz form now6 becomes (see Appendix B) /parenleftbig Ψ21+G/parenrightbig ·∂M ∂t= −γ0ΨM×Heff−γ0M×/bracketleftBig (α1+I)·(M×Heff)/bracketrightBig ,(29) whereΨ = 1+ M·Dand the tensor Gis defined through G=α2M21−/bracketleftBig (M·I·M)−tM2/bracketrightBig (α1+I) −/parenleftBig tM−M·I/parenrightBig M·I−M2I2+M/parenleftBig M·I2/parenrightBig ,(30) with the trace, t= Tr( I). In general the trace of such a matrix Iis non-zero, however its value will depend on how the symmetric tensor Asym ij=1 2(Aij+Aji) =Iij+αδij is decomposed. If the decomposition in Ising and Heisen- berg parts is such that the isotropic part is chosen as α=1 3Tr(Asym ij), then the trace of Iwill vanish, t= 0. Note that the term (15) due to the intrinsic spin-orbit interacti on has been left out, as it is expected to give only a small cor- rection to the effective magnetic field. The damping term thus adopts the form −γ0M×[Λ·(M×Heff)], similar to the phenomenological damping considered by Landau and Lifshitz [1], but with damping tensor Λ. A more general form of the LL damping as a tensor was already considered much earlier (see, e.g. [59]), and it is reflected also in our derivation. However, a distinction is that here the leading ∂M/∂tterm on the left-hand side in Eq. (29) is, in its gen- eral form, multiplied not with a scalar ( 1+α2M2) but with a tensor which moreover depends on the direction of M. It is worth noting that in the absence of the Dzyaloshinskii-Moriya and anisotropic relaxation contri bu- tions, i.e., setting D=I= 0we retrieve the original LL and LLG equations with scalar damping parameters. The va- lidity range of our derived equations of spin motion is thus larger than the originally proposed equations of motion. It should also be emphasized that the Dzyaloshinskii-Moriya- like contribution appears in the Gilbert damping, however, it does not appear in the damping term of the LL equation (29). Instead, it leads to the renormalization of the stan- dard dynamical terms in the LL equation as can be seen from the appearance of the quantity Ψin Eq. (29). We lastly note that the here obtained relaxation terms do not allow a variation with respect to the coordinates i.e., they do not include effects of spatial dispersion. V. DISCUSSION 1. Analysis of the damping expression Equation (26) for the Gilbert damping pertains to the relaxation of spin motion in the presence of spin-orbit in- teraction. This damping is of relativistic origin as is ex- emplified by its 1/c2dependence. The expression for the Gilbert tensor is different from that obtained previously [17], where only the constant term i/planckover2pi1in the square bracket was found. The new parts /an}b∇acketle{tripj/an}b∇acket∇i}htrelate to how the elec- tronic band energies Eνkof Bloch states |νk/an}b∇acket∇i}htdisperse with k-space direction. It can be rewritten as (see Appendix C) /an}b∇acketle{tripj/an}b∇acket∇i}ht=−i/planckover2pi1 2m/summationdisplay ν,ν′,kf(Eνk)−f(Eν′k) Eνk−Eν′kpi νν′pj ν′ν,(31)wherepνν′≡ /an}b∇acketle{tνk|p|ν′k/an}b∇acket∇i}htandf(Eνk)is the Fermi function. The sum contains interband and intraband contributions. The intraband (Fermi surface) contribution (ν=ν′)can be written as /an}b∇acketle{tripj/an}b∇acket∇i}ht=−im 2/planckover2pi1/summationdisplay νk/parenleftbigg∂f ∂E/parenrightbigg Eνk/parenleftbigg∂Eνk ∂ki/parenrightbigg/parenleftbigg∂Eνk ∂kj/parenrightbigg .(32) This expression has a similarity with other previously de- rived expressions, as e.g. the breathing Fermi surface mode l [8, 24] that has been applied to metallic ferromagnets. The expression for the /an}b∇acketle{tripj/an}b∇acket∇i}htterms has furthermore a form sim- ilar to that for the conductivity tensor in linear-response theory [60]; it is in particular well-suited for ab initio cal- culations. We note further that the influence of electron interaction with quasiparticles can be introduced by repla c- ingEνk−Eν′kbyEνk−Eν′k+iδ, where the small δgives a finite relaxation time to the electronic states. For numerical evaluation of the damping tensor the sus- ceptibility tensor χis furthermore needed, which is in gen- eral wavevector and frequency dependent, χ(q,ω). Thus, also the Gilbert damping tensor is here a frequency and q-dependent quantity, in accordance with recent measure- ments [32]. Suitable expressions for χhave been considered previously in the context of Gilbert damping [13, 16, 45]. Linear-response formulations that express χas a spin-spin correlation function include the Pauli and Van Vleck sus- ceptibility contributions [61], and expressions for the or - bital susceptibility have been derived as well [62]. These expressions are fitting for ab initio calculations of χwithin a DFT framework. The spin-orbit interaction will have an additional influence on χ, however, unlike the main Gilbert damping contribution which is proportional to the spin- orbit coupling, this will only be a higher order effect. We can consequently distinguish here two origins for the damping: the first one is related to the terms /an}b∇acketle{tripj/an}b∇acket∇i}ht, which represent dissipation contributions into the orbital degr ees of freedom. The second nature is due to the magnetic sus- ceptibilityχwhich represents losses through the magnetic structure of the material. Both effects are simultaneously present, and nonzero, for metallic ferromagnets as well as insulators. It is also important to mention that the damping ten- sor in the our derivation does not include spin-relaxation effects due to interaction of spin-polarized electrons with quasiparticles as magnon or phonons or scattering with de- fects. Longitudinal spin relaxation due to spin-flip pro- cesses caused by electron-phonon scattering have been re- cently calculated ab initio for the transition-metal ferro- magnets [63–65], and magnon spin-flip scattering has been considered as well [66]. Spin angular momentum transfer due to explicit coupling of the spins to the lattice has been treated in several models [67–69]. As mentioned above, although the spin-lattice dissipation channel is not encom - passed in our derivation, an approximate way to include its influence has been introduced before, by a suitable spec- tral broadening of the Bloch electron energies (see, e.g., [24, 70]). Lastly, we remark that in the present derivation we ob- tain only first-order time-derivatives of M(r,t). Second- order time-derivatives of M(r,t)have recently been related to moment of inertia of the magnetization [71].7 2. Exchange field and nonlocal contributions Thus far we have not explicitly discussed the exchange interaction. The influence of the exchange field can be accounted for in various levels of approximation, for ex- ample, within the Heisenberg model or evaluated within time-dependent DFT [72, 73]. In the former, a suitable simplification of the exchange interaction in a magnetic solid is to express it through the Heisenberg Hamiltonian Hxc=−/summationtext α>βJαβSα·Sβ, where the Jαβare exchange constants and Sαis the atomic spin on atom α. Using this Hamiltonian to express the exchange field leads to Landau- Lifshitz-Gilbert equations of motion for the dynamics of atomic moments (see, e.g., [74–76]). More general, the exchange field depends on the spatial position which implies that there can exist an influence of spatial nonuniformity of the exchange field on the spin re- laxation. An influence on the dynamics occurring due to magnetization inhomogeneity ( ∇2M) appearing in the ef- fective field was already suggested by Landau and Lifshitz [1]. Such a term is in fact needed to properly describe spin wave dispersions [77]. A nonlocal damping mecha- nism due to spatial dispersion of the exchange field was proposed by Bar’yakhtar on the basis of phenomenological considerations such as symmetry arguments and Onsager’s relations [10]. This leads to a modified expression for the damping term in the Landau-Lifshitz-Bar’yakhtar equation which contains the derivative of the exchange field ∇2Bxc [10, 20]. The existence of such nonlocal damping term can be related to the continuity equation connecting the spin density and spin current; it is important for obtaining the correct asymptotic behavior of spin wave damping at large wavevectors k[20] known for magnetic dielectrics, see [59]. Such nonlocal damping is important, too, for describing spin current flow in magnetic metallic heterostructures [78 ]. These nonlocal damping terms are furthermore related to the earlier proposed magnetization damping effects due to spin diffusion [52, 79–81] that have been studied recently [82]. As a consequence of the spin current flow the local length of the magnetization is not conserved. In the present work such nonlocal terms are not included since we focus on the local dissipation and have thus omitted the spin cur- rent contribution of the continuity equation. A future full treatment that takes into account both local and nonlocal spin dissipation mechanisms would permit to describe mag- netization dynamics and spin transport on an equal footing in a broader range of inhomogeneous systems. 3. General time-dependent magnetic fields When the driving magnetic field is not an ac harmonic field the dependence of M(r,t)onH(t)will induce a more complex dynamics. In this case it is possible to derive a closed expression for the spin dynamics by explicitly keep- ing the term∂H ∂tin Eq. (19). A similar derivation as pre- sented in Sect. IV for the ac driving field leads then to the following expression for the magnetization dynamics ∂M ∂t=−γ0M×Heff+M×/bracketleftBig ¯A·/parenleftBig∂M ∂t+∂H ∂t/parenrightBig/bracketrightBig ,(33)where the damping tensor ¯A is given by ¯Aij=−eµ0 8m2c2/summationdisplay n/bracketleftBig /an}b∇acketle{tripj+pjri/an}b∇acket∇i}ht−/an}b∇acketle{trnpn+pnrn/an}b∇acket∇i}htδij/bracketrightBig .(34) The time-dependent magnetic field thus leads to a new, modified spin dynamics equation which has, to our knowl- edge, not been derived before. The time-derivate of H(t) introduces here an additional torque, M×∂H ∂t. This field- derivative torque might offer new ways to achieve fast mag- netization switching. Consider for example an initially steep magnetic field pulse that thereafter relaxes slowly back to its initial value. The derivative of such field will ex - ert a large but shortly lasting torque on the magnetization, which could initiate switching. Irradiation of magnetic th in films with a picosecond THz field pulse was recently shown to trigger ultrafast magnetization dynamics [83], and suit - able shaping of the THz magnetic field pulse could hence offer a route to achieve switching on a picosecond time scale. 4. The optical spin torque The interaction of the spin moment with the optical spin angular moment jsis given by the Hamiltonian Hext light−spin. We note that such relativisitic interaction is important fo r recent attempts to manipulate the magnetization in a ma- terial using optical angular momentum, i.e., helicity of th e laser field [40, 84, 85]. This interaction leads to spin dy- namics of the form ∂M ∂t/vextendsingle/vextendsingle/vextendsingleext light−spin=−e2 2m2c2ǫ0M×js, (35) whereM×jsis the optical spin torque exerted by the optical angular moment on the spin moment. This equa- tion expresses that the spin moment in a material can be manipulated by acting on it with the optical spin angular moment of an external electromagnetic field in the strong field regime. VI. CONCLUSIONS On the basis of the relativistic Dirac-Kohn-Sham equa- tion we have derived the spin Hamiltonian to describe ad- equately the dynamics of electron spins in a solid, tak- ing into account all the possible spin-related relativisti c effects up to the order 1/c2and the exchange field and ex- ternal electromagnetic fields. From this manifestly hermi- tian spin Hamiltonian we have calculated the spin equation of motion which adopts the form of the Landau-Lifshitz- Gilbert equation for applied harmonic fields. For univer- sal time-dependent external magnetic fields we obtain a more general dynamics equation which involves the field- derivative torque. Our derivation does notably not rely on phenomenological assumptions but provides a rigorous treatment on the basis of fundamental principles, specifi- cally, Dirac theory with all relevant fields included. We have shown the existence of a relativistic correction to the precessional motion in the obtained LLG equation and have derived an expression for the spin relaxation terms of relativistic origin. One of the most prominent8 results of the presented article is the derived expression f or the tensorial Gilbert damping, which has been shown to contain an isotropic Gilbert contribution, an anisotropic Ising-like contribution, and a chiral, Dzyaloshinskii- Moriya-like contribution. Transforming the LLG equation to the Landau-Lifshitz equation of motion, we showed that the LLG equation with anisotropic tensorial Gilbert damping cannot trivially be written as a LL equation with an anisotropic LL damping term, but an additional matrix appears in front of the ∂M/∂tterm. The Dzyaloshinskii- Moriya-like contribution serves as a renormalization fact or to the common LL dynamical terms. The obtained expression for the Gilbert damping tensor in the case of a periodic driving field depends on the spin-spin suscep- tibility response function along with a term representing the electronic spin damping due to dissipation into the orbital degrees of freedom. As there exist an on-going discussion on what the fundamental origin of the Gilbert damping is and how it can accurately be evaluated from first-principles calculations [28, 30–32], we point out tha t the two components of the derived damping expression (spin-spin and current-current response functions) are suitable for future ab initio calculations within the density functional formalism. ACKNOWLEDGMENTS We thank B. A. Ivanov, P. Maldonado, A. Aperis, K. Carva, and H. Nembach for helpful discussions. We alsothank the anonymous reviewers for valuable comments. This work has been supported by the European Com- munity’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 281043, FemtoSpin, the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation (Contract No. 2015.0060), and the Swedish Na- tional Infrastructure for Computing (SNIC). Appendix A: Hermiticity of Hamiltonian Hext soc The extrinsic spin-orbit Hamiltonian Hext soc, given in Eq. (16), can indeed be shown to be hermitian, however its in- dividual terms are not all hermitian. Adapting the Einstein summation convention, this Hamiltonian can be written in component form as Hext soc=e 4m2c2/parenleftBig i/planckover2pi1Si∂tBi −Si∂tBirjpj+Siri∂tBjpj/parenrightBig ,(A1) with∂t≡∂/∂t. To demonstrate that it is hermitian, we take the Hermitian conjugate, and rewrite it in a few steps. /bracketleftBig Hext soc/bracketrightBig† =e 4m2c2/parenleftBig −i/planckover2pi1Si∂tBi−Si∂tBipjrj+Si∂tBjpjri/parenrightBig =e 4m2c2/parenleftBig −i/planckover2pi1Si∂tBi−Si∂tBirjpj+Si∂tBjripj−Si∂tBi(pjrj)+Si∂tBj(pjri)/parenrightBig =e 4m2c2/parenleftBig −i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)−(S·∂tB)(p·r)+S·{(∂tB·p)r}/parenrightBig =e 4m2c2/parenleftBig −i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)+i/planckover2pi1(S·∂tB)(∇·r)−i/planckover2pi1S·{(∂tB·∇)r}/parenrightBig =e 4m2c2/parenleftBig −i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)+3i/planckover2pi1S·∂tB−i/planckover2pi1S·∂tB/parenrightBig =e 4m2c2/parenleftBig i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)/parenrightBig =Hext soc. (A2) For the individual terms of the Hamiltonian it is straightfo rward to show their hermitian or non-hermitian character: Hext soc= =ie/planckover2pi1 4m2c2Si∂tBi /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright anti−hermitian−e 4m2c2/summationdisplay i/negationslash=jSi∂tBirjpj /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright non−hermitian+e 4m2c2/summationdisplay i/negationslash=jSiri∂tBjpj /bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright hermitian. (A3) As noted before all three terms of the hermitian Hamiltonian contribute to the spin relaxation process. Appendix B: From LLG to LL equations of motion We found that the generalized LLG equation of spin dynamics c an be written in the form [see Eq. (27)] ∂M ∂t=−γM×Beff+M×/bracketleftBig A·∂M ∂t/bracketrightBig . (B1)9 As discussed earlier, using the tensor decomposition, one c an also write ∂M ∂t=−γM×Beff+αM×∂M ∂t+M×/bracketleftBig I·∂M ∂t/bracketrightBig +M×/bracketleftBig D×∂M ∂t/bracketrightBig . (B2) The Dzyaloshinskii-Moriya-like damping terms can be expan ded, using a×(b×c) =b(a·c)−c(a·b), to give M×/bracketleftBig D×∂M ∂t/bracketrightBig =−∂M ∂t(M·D). Since the magnetization length is conserved we therefore h aveM·∂M/∂t= 0. Defining (1+M·D) = Ψ, the LLG equation of spin motion reduces to Ψ∂M ∂t=−γM×Beff+αM×∂M ∂t+M×/bracketleftBig I·∂M ∂t/bracketrightBig . (B3) Note that Ψis both a magnetization and Dzyaloshinskii-Moriya vector d ependent quantity. Next, we have to calculate the second and third terms on the right-hand side of Eq. (B3). Taking a cross product with Mon both sides of the last equation gives ΨM×∂M ∂t=−γM×(M×Beff)+αM×/parenleftBig M×∂M ∂t/parenrightBig +M×/parenleftBig M×/bracketleftBig I·∂M ∂t/bracketrightBig/parenrightBig =−γM×(M×Beff)−αM2∂M ∂t−M2/bracketleftBig I·∂M ∂t/bracketrightBig +M/parenleftBig M·/bracketleftBig I·∂M ∂t/bracketrightBig/parenrightBig . (B4) Similarly, to evaluate the last term of Eq. (B3), we take the d ot product with the symmetric part of the tensor, followed by a cross product with the magnetization, ΨM×/bracketleftBig I·∂M ∂t/bracketrightBig =−γM×/bracketleftBig I·(M×Beff)/bracketrightBig +αM×/bracketleftBig I·/parenleftBig M×∂M ∂t/parenrightBig/bracketrightBig +M×/parenleftBig I·/braceleftBig M×/bracketleftBig I·∂M ∂t/bracketrightBig/bracerightBig/parenrightBig .(B5) At this point we already observe that the first term on the righ t hand side has adopted a form of the LL damping but with a tensor. The second and third terms are treated in the fo llowing. The second term can be written in component form as αM×/bracketleftBig I·/parenleftBig M×∂M ∂t/parenrightBig/bracketrightBig =αMlImkMi∂Mj ∂tεijkεlmnˆen. (B6) We use the following relation for the product of two anti-sym metric Levi-Civita tensors εijkεlmn=δil(δjmδkn−δjnδkm)−δim(δjlδkn−δjnδkl)+δin(δjlδkm−δjmδkl), (B7) and, defining the trace of the symmetric tensor Tr(I) =t, a little bit of tensor algebra results in αM×/bracketleftBig I·/parenleftBig M×∂M ∂t/parenrightBig/bracketrightBig =αM2/parenleftBig I·∂M ∂t/parenrightBig −αtM2∂M ∂t+α/parenleftBig M·I·M/parenrightBig∂M ∂t−αM/bracketleftBig M·/parenleftBig I·∂M ∂t/parenrightBig/bracketrightBig .(B8) Now we proceed to calculate the last part of Eq. (B5); the comp onents of this term are given by M×/parenleftBig I·/braceleftBig M×/bracketleftBig I·∂M ∂t/bracketrightBig/bracerightBig/parenrightBig =MmInlMkIij∂Mj ∂tεkilεmnoˆeo. (B9) Using once again the relation in Eq. (B7) and expanding in diff erent components we find M×/parenleftBig I·/braceleftBig M×/bracketleftBig I·∂M ∂t/bracketrightBig/bracerightBig/parenrightBig =/bracketleftBig (M·I·M)−tM2/bracketrightBig/parenleftBig I·∂M ∂t/parenrightBig +/parenleftBig tM−M·I/parenrightBig/bracketleftBig M·/parenleftBig I·∂M ∂t/parenrightBig/bracketrightBig +(M·M)/bracketleftBig I·/parenleftBig I·∂M ∂t/parenrightBig/bracketrightBig −M/bracketleftBig M·/braceleftbigg I·/parenleftBig I·∂M ∂t/parenrightBig/bracerightbigg/bracketrightBig . (B10) Now we have the necessary terms to formulate the LL equation o f motion. Taking these together, the LLG dynamics of Eq. (B3) can be written as Ψ2∂M ∂t=−γΨM×Beff−γM×/bracketleftBig (α1+I)·(M×Beff)/bracketrightBig −G·∂M ∂t, (B11) with the general tensorial form of Gwhich is given by G=α2M21−/bracketleftBig (M·I·M)−tM2/bracketrightBig (α1+I)−/parenleftBig tM−M·I/parenrightBig M·I−M2I2+M/parenleftBig M·I2/parenrightBig . UsingB=µ0(H+M), the transformation from the LLG to the LL equation results i n the form /parenleftBig Ψ21+G/parenrightBig ·∂M ∂t=−γ0ΨM×Heff−γ0M×/bracketleftBig (α1+I)·(M×Heff)/bracketrightBig . (B12) As mentioned before, in general the Landau-Lifshitz dampin g cannot be described by a scalar. We find that in the damping term the effect of the anisotropic Ising-like dampin g is present, while the influence of the Dzyaloshinskii-Mori ya- like damping is accounted for through the renormalizing qua ntityΨ.10 Appendix C: Expressions for matrix elements We provide here suitable expressions for ab initio calcu- lations of the matrix elements /an}b∇acketle{tripj/an}b∇acket∇i}ht. We consider thereto the Bloch states |νk/an}b∇acket∇i}htin a crystal to calculate the expecta- tion value /an}b∇acketle{tripj/an}b∇acket∇i}ht=/summationdisplay ν,ν′,k/an}b∇acketle{tνk|ri|ν′k/an}b∇acket∇i}ht/an}b∇acketle{tν′k|pj|νk/an}b∇acket∇i}htf(Eνk),(C1) wheref(Eνk)is the Fermi-Dirac function. The momentum and position operators are connected through the Ehrenfest theorem, p=im /planckover2pi1[H,r], which we employ to obtain matrix elements of the position operator /an}b∇acketle{tν′k|r|νk/an}b∇acket∇i}ht=−i/planckover2pi1 m/an}b∇acketle{tν′k|p|νk/an}b∇acket∇i}ht (Eν′k−Eνk). (C2)Substitution in equation (C1) gives /an}b∇acketle{tripj/an}b∇acket∇i}ht=−i/planckover2pi1 m/summationdisplay ν,ν′,kf(Eνk)pi νν′pj ν′ν Eνk−Eν′k =−i/planckover2pi1 2m/summationdisplay ν,ν′,kf(Eνk)−f(Eν′k) Eνk−Eν′kpi νν′pj ν′ν.(C3) The double sum over quantum numbers can be further rewritten by separating according to interband matrix el- ements (ν/ne}ationslash=ν′) and intraband matrix elements ( ν=ν′). The latter part becomes /an}b∇acketle{tripj/an}b∇acket∇i}ht=−1 2i/planckover2pi1 m/summationdisplay ν,k/parenleftbigg∂f ∂E/parenrightbigg Eνkpi ννpj νν, (C4) which can be reformulated using pi νν=m /planckover2pi1(∂Eνk/∂k)ito give expression (32). 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2016-09-26

Starting from the Dirac-Kohn-Sham equation we derive the relativistic equation of motion of spin angular momentum in a magnetic solid under an external electromagnetic field. This equation of motion can be written in the form of the well-known Landau-Lifshitz-Gilbert equation for a harmonic external magnetic field, and leads to a more general magnetization dynamics equation for a general time-dependent magnetic field. In both cases with an electronic spin-relaxation term which stems from the spin-orbit interaction. We thus rigorously derive, from fundamental principles, a general expression for the anisotropic damping tensor which is shown to contain an isotropic Gilbert contribution as well as an anisotropic Ising-like and a chiral, Dzyaloshinskii-Moriya-like contribution. The expression for the spin relaxation tensor comprises furthermore both electronic interband and intraband transitions. We also show that when the externally applied electromagnetic field possesses spin angular momentum, this will lead to an optical spin torque exerted on the spin moment.

Relativistic theory of spin relaxation mechanisms in the Landau-Lifshitz-Gilbert equation of spin dynamics

1609.07901v1

Roles of chiral renormalization on magnetization dynamics in chiral magnets Kyoung-Whan Kim,1,Hyun-Woo Lee,2,yKyung-Jin Lee,3,4Karin Everschor-Sitte,5Olena Gomonay,5,6and Jairo Sinova5,7 1Institut für Physik, Johannes Gutenberg-Universität Mainz, Mainz 55128, Germany 2Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea 3Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 4KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea 5Institut für Physik, Johannes Gutenberg Universität Mainz, Mainz 55128, Germany 6National Technical University of Ukraine “KPI," Kyiv 03056, Ukraine 7Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 162 53 Praha 6, Czech Republic (Dated: July 28, 2021) Inmetallicferromagnets,theinteractionbetweenlocalmagneticmomentsandconductionelectronsrenormal- izesparametersoftheLandau-Lifshitz-GilbertequationsuchasthegyromagneticratioandtheGilbertdamping, and makes them dependent on the magnetic configurations. Although the effects of the renormalization for nonchiral ferromagnets are usually minor and hardly detectable, we show that the renormalization does play a crucial role for chiral magnets. Here the renormalization is chiral and as such we predict experimentally identi- fiableeffectsonthephenomenologyofmagnetizationdynamics. Inparticular,ourtheoryfortheself-consistent magnetization dynamics of chiral magnets allows for a concise interpretation of domain wall creep motion. We also argue that the conventional creep theory of the domain wall motion, which assumes Markovian dynamics, needscriticalreexaminationsincethegyromagneticratiomakesthemotionnon-Markovian. Thenon-Markovian nature of the domain wall dynamics is experimentally checkable by the chirality of the renormalization. Renormalization is a useful concept to understand interac- tion effects between a physical system and its environment. In metallic ferromagnets, magnetic moments experience such renormalization due to their coupling to conduction electrons through exchange interactions. Spin magnetohydrodynamic theory [1–3] examines the renormalization of dynamical pa- rameters in the Landau-Lifshitz-Gilbert (LLG) equation as follows. Magnetization dynamics exerts a spin motive force (SMF) [4, 5] on conduction electrons, and the resulting spin currentgeneratesspin-transfertorque(STT)[6–8]thataffects themagnetizationdynamics itself. Thisself-feedbackofmag- netizationdynamics[9]renormalizestheGilbertdampingand the gyromagnetic ratio. However, its consequences rarely go beyond quantitative corrections in nonchiral systems [10–14] and are commonly ignored. Chiralmagnetsareferromagnetsthatpreferaparticularchi- rality of magnetic texture due to spin-orbit coupling (SOC) and broken inversion symmetry. Examples include ferro- magnets in contact with heavy metals, such as Pt [15] and those with noncentrosymmetric crystal structures [16]. Mag- netization dynamics in chiral magnets are usually described by generalizing the conventional LLG equation to include the chiral counterpart of the exchange interaction called the Dzyaloshinskii-Moriya interaction (DMI) [17–19] and that of STTcalledspin-orbittorque(SOT)[20–23]. Thisdescription is incomplete, however, since it ignores the renormalization by the self-feedback of magnetization dynamics. Although the renormalization in chiral magnets has been demonstrated theoretically for a few specific models [24–27], most experi- mentalanalysesofchiralmagnetsdonottakeintoaccountthe renormalization effect. Inthiswork,wedemonstratethattherenormalizationinchi- ralmagnetsshouldbechiralregardlessofmicroscopicdetails and these effects should be nonnegligible in chiral magnets with large SOT observed in many experiments [21–23, 28–30]. Unlike in nonchiral systems, the chiral renormalization generates experimentally identifiable effects by altering the phenomenology of magnetization dynamics. This provides a useful tool to experimentally access underlying physics. We illustratethiswiththefield-drivenmagneticdomainwall(DW) motion with a controllable chirality by an external magnetic field [31, 32]. We find that not only is the steady state DW velocity chiral due to the chiral damping [25], but also the effective mass of the DW [33] is chiral due to the chiral gy- romagnetic ratio. The chiral gyromagnetic ratio also signifi- cantly affects the DW creep motion, which is one of the tech- niques to measure the strength of the DMI [32]. We argue that the chiral gyromagnetic ratio is the main mechanism for the non-energetic chiral DW creep velocity [34], contrary to the previous attribution to the chiral damping [25, 34]. We also highlight the importance of the tilting angle excitation and its delayed feedback to the DW motion. This has been ignoredinthetraditionalcreeptheory[35,36]foralongtime, since its effects merely alter the velocity prefactor which is indistinguishable from other contributions, such as the impu- rity correlation length [37]. However, in chiral magnets, it is distinguishablebymeasuringtheDWvelocityasa functionof chirality (not a single value). To get deep insight into the chiral renormalization, we adopt the self-feedback mechanism of magnetization dynam- ics through conduction electrons and develop a general, con- cise, and unified theory for chiral magnets. There are several previous reports on the anisotropic or chiral renormalization of the magnetic damping [24–26, 38] and the gyromagnetic ratio[27,38,39]intheRashbamodel[40]. Tounifyandgen- eralize the previous works, we start from the general Onsager reciprocityrelationandpredictallthecoreresultsoftheprevi- ous reports. Our theory can be generalized to situations with anyphenomenologicalspintorqueexpression,whichcaneven bedeterminedbysymmetryanalysisandexperimentswithoutarXiv:1803.00017v2 [cond-mat.mes-hall] 14 Mar 20182 Magnetization under chiral self-feedback Effective equation of motion for magnetization SOT chiral SMF LLG ( γ, α) chiral LLG ( ζγ , G )(a) (b) FIG.1. (a)Magnetizationdynamicsdescribedbytheunrenormalized LLG equation. The dynamics of magnetization and that of electrons arecoupledtoeachotherbytheexchangeinteraction. (b)Aftertracing out the electron degree of freedom, the gyromagnetic ratio (  ) and the magnetic damping ( G) are chirally renormalized [Eq. (1)]. knowing its microscopic mechanism. We provide a tabular picture(SeeTableIbelow)forphysicalunderstandingofeach contribution to the chiral renormalization. Furthermore, one can utilize the generality of the Onsager relation to include magnon excitations [26], thermal spin torques [41], and even mechanical vibrations [42] in our theory. Toexaminetheconsequencesofthechiralrenormalization, westartfromthefollowingrenormalizedLLGequation,which we derive in the later part of this paper, ( )1
@tm=mHe+ 1mG
@tm+ 1Text;(1) where mis the unit vector along magnetization, is the un- renormalized gyromagnetic ratio, Heis the effective mag- neticfield,and Textreferstospintorqueinducedbyanexternal current.andG,whicharegenerallytensorsandfunctionsof mand its gradients, address respectively the renormalization ofthegyromagneticratioandthemagneticdamping,depicted in Fig. 1. If the renormalization is neglected, Eq. (1) reduces to the conventional LLG equation with = 1andG=, whereistheunrenormalizedGilbertdamping. Otherwise  andGare dependent on the chirality of magnetic texture. At the end of this paper, we show that the chiral renormalization is completely fixed once the expressions of STT and SOT are given. We first examine implications of the chiral renormaliza- tion on a few exemplary types of field-driven DW dynamics (Fig.2). Westartfrom He=H0+Hext+Hth,where H0is the energetic contribution (without an external field), Hext= (Hx;0;Hz)is the external field, and Hthis a thermal fluctu- ation field. We use the DW profile m(x) = (sinsech[(x X)=];cossech[(xX)=];tanh[(xX)=])whereX, andaretheposition,thetiltingangle,andthewidthofthe DW, respectively. Taking Xandas the collective coordi- xyz φφ HxHz v( )FIG. 2. Chiral dynamics of a DW between domains with m=^z (red and blue respectively). The DW chirality is characterized by the DW tilting angle [the positivity (negativity) of corresponds to the left-handed (right-handed) chirality], and can be controlled by an in-plane field ( Hx). The DW motion is driven by an applied field (Hz). Measuring the DW velocity as a function of (orHx), the difference between v()andv()gives the information of the chiral renormalization. nates, Eq. (1) gives X e dX dt+1 ed dt=FX+X; (2a) 1 edX dt+ ed dt=F+; (2b) whereFX== ( =2)R (H0+Hext)
(@X=m)dxrefertothe force onXand.X== ( =2)R Hth
(@X=m)dxis the thermal force on Xand. Theeffectivedamping X= eandthegyromagneticratio e are given by X e= 2Z (@Xm
G
@Xm)dx; (3a)  e=1 2Z (@m
G
@m)dx; (3b) 1 e=1 2Z (m@m)
1
@Xm dx:(3c) Note that without the chiral renormalization, Eq. (2) reduces totheThieleequations[43]with X= e=ande= 1. We emphasizethat X= eandedependonthetiltingangle and thus on the chirality of the DW. Figure 3 shows the depen- denciesoftheseparameters. Theasymmetricdependenceson confirmtheirchiraldependences. Notethat,evenforpurely field-drivenDWmotion,thechiraldependencesoftheparam- etersaredeterminedbytheexpressionof current-induced spin torque. We first consider the steady-state dynamics of DW in the flow regime, where the effects of the pinning and the thermal forces are negligible. Then, translational symmetry along Xguarantees the absence of contribution from H0toFX, thus only the external field contribution survives in the right- hand side of Eq. (2a), FX+X Hz. In a steady state (d=dt = 0), Eq. (2a) gives the DW velocity as v ow=  X eHz; (4)3 α

ϕα


α
ϕϕα
ϕ
ζ
ϕζ















 ϕ π Righthandedchirality Left handedchirality FIG. 3. The effective dynamical parameters, X e(the red, solid curve), e(the red, dashed curve), and 1 e(the blue curve), as a function of the DW tilting angle . We take the phenomenological expression of spin torque in magnetic bilayers [21–23, 30], which is a typical example with large SOT: T= ( ~=2eMs)f(js
r)m 1m(js
r)m+kSO(^zjs)m2kSOm[(^zjs) m]g, where each term refers to the adiabatic STT [44], nonadiabatic STT[45,46],fieldlikeSOT[47,48],anddampinglikeSOT[30,49– 51], induced by the spin current js. Here,Ms= 1000 emu =cm3is thesaturationmagnetization, e>0isthe(negative)electroncharge, ^zistheinterfacenormaldirection, kSO= 1:3 (nm)1characterizes thestrengthoftheSOT.Wetake 1= 0:05,2= 5,= 8 nm,and the electrical conductivity 1 0= 6 cm. The parameters are on the order of the typical values for Pt/Co systems [28, 45, 52]. whichisinverselyproportionaltothechiraldamping X eeval- uatedatthesteady-statetiltingangle eqforwhichd=dt = 0. Aseqcanbemodulatedby Hx,themeasurementof v owasa function ofHxprovides a direct test of the chiral dependence onX e. As an experimental method to probe the chiral dependence ofe, we propose the measurement of the DW mass, called the Döring mass [33]. It can be performed by examining the response of DW under a potential trap to an oscillating fieldHz[53]. Unlike v ow,is not stationary for this case, and dynamics of it is coupled to that of X. Such coupled dynamicsof andXmakeserelevant. IntheSupplemental Material[54],weintegrateoutthecoupledequations[Eq.(2)] to obtain the effective Döring mass, mDW=1 2 e2MsS jF0 (eq)j; (5) whereSis the cross-sectional area of the DW. Here, erep- resents a measurement of its value for =eq, which can be variedbyHx.mDWprovidesanexperimentalwaytomeasure the chiral dependence of e. In the creep regime of the DW where the driving field is much weaker than the DW pinning effects, the implication of thechiralrenormalizationgobeyondmerelychiralcorrections totheDWvelocity. TherecentcontroversiesonthechiralDW creep speed vcreepmeasured from various experiments [32, 34, 55, 56] require more theoretical examinations. Typically,vcreepis believed to follow the Arrhenius-like law vcreep = v0exp(H z=kBT)[35, 36], where v0is a prefactor, is the creep exponent typically chosen to be 1/4 [57], and is a parameter proportional to the DW energy density. Based on the observation that the DMI affects , an experiment [32] attributedthechiraldependenceof vcreeptotheDMI.However later experiments [34, 55, 56] found features that cannot be explained by the DMI. In particular, Ref. [34] claimed that the chiral dependence of vcreepis an indication of the chiral damping [25], based on the observation v0/(X e)1. On the other hand, our analysis shows that the explanation of the chirality dependence may demand more fundamental change to the creep law, which assumes the dynamics of to be essentially decoupled from that of Xand thus irrelevant for vcreep. As a previous experiment on the DW creep motion in a diluted semiconductor [58] argued the coupled dynamics of andXto be important, it is not a prioriclear whether the assumptionofdecoupling Xandholdsinthecreepregime. We consider the coupling between the dynamics of Xand asfollows. Afterthedynamicsof Xexcites,thedynamics ofresults in a feedback to Xwith a delay time . Since the dynamics at a time tis affected by its velocity at past t, it is non-Markovian. The traditional creep theory takes the Markovian limit ( !0), thus=eqat any instantaneous time,decoupledfromthedynamicsof X. Toshowthecrucial roleofafinitefeedbacktime ,wecalculatetheescaperateof the DW over a barrier, which is known to be proportional to v0[37] and apply the Kramer’s theory [59] for barrier escape and its variations for non-Markovian systems [60, 61]. After somealgebraintheSupplementalMaterial[54],Eq.(2)gives v0/(X e)102 eX e e(Markovian ); e0&2 eX e e(non-Markovian ); (6) where0is called the reactive frequency [61] and is on the order of 2times the attempt frequency ( 1 GHz[37]). We emphasizethatthetworegimesshowverydifferentbehaviorin thesenseofunderlyingphysicsaswellasphenomenology. The validityoftheMarkovianassumptiondependsonthetimescale ofcomparedto 2 eX e e. Sincethedampingissmall,itis notguaranteedforoursituationtobeintheMarkovianregime. Indeed,wedemonstrateintheSupplementalMaterial[54]that thesecondregime(non-Markovian)inEq.(6)ismorerelevant withrealisticvalues,thusthechiralityof v0mainlyoriginates from the gyromagnetic ratio, not the damping [34]. One can measure the chiral dependence of X eandefrom the flow regime[Eqs.(4)and(5)]andcomparetheirchiraldependences tothecreepregimetoobservethenon-Markoviannatureofthe DWdynamics. Thisadvantageoriginatesfromthepossibility thatonecanmeasuretheDWvelocityasa functionofchirality, in contrast to nonchiral magnets where one measures the DW velocity as a single value. So far, we present the role of the chiral renormalization for given renormalized tensors Gand. To provide underlying physical insight into it, we present a analytic derivation of Eq. (1) in general situations. We start from the LLG equation4 1@tm=mHe+ 1m@tm+ 1Tandreferto the scenario illustrated in Fig. 1. Note that There includes a contribution from an internally generated SMF ( Tint) as well as that from an external current [ Textin Eq. (1)]. We write down the spin torque in a general current-linear form T= ( ~=2eMs)mP uAu(m)js;u,whereurunsoverx;y;z. Here the spin current jsis split into an internally generated SMF [4, 5] js;intand the external current js;ext. The former isproportionalto @tm,thusitrenormalizesthegyromagnetic ratio and the damping. The latter generates Textin Eq. (1). Theexpressionof js;intisgivenbytheOnsagerreciprocityof STT and SMF [62]: js;int;u=(0~=2e)Au(m)
@tm, where0is the charge conductivity [63]. Substituting this to Tint= ( ~=2eMs)mP uAu(m)js;int;ugivestheeffective LLGequation 1@tm=mHe+ 1mA
@tm+ 1Text, whereA=+P uAu(m) Au(m),= ~20=4e2Msand isthedirecttensorproduct. Asaresult, Tintis taken care of by renormalizing intoAin the LLG equation. Therenormalizeddampingandgyromagneticratioaregiven by separating different contributions of Awith different time reversal properties. A damping contribution is required to be dissipative (odd in time reversal), whereas a gyromagnetic term should be reactive (even in time reversal). Separating these gives Eq. (1) where G= (A+AT)=2and1= 1m(AAT)=2. Theparticularchoicefortheadiabatic STTandthenonadiabaticSTT Au(m) =m@um+@um reproduces the renormalized LLG equation for nonchiral sys- tems [1–3]. When one uses Au(m)for a particular chiral system, Eq. (1) produces the effective LLG equation for it, as reported by a numerical study for a one-dimensional Rashba model [27]. In chiral magnets, it is known that spin torque includes two more contributions called fieldlike SOT [47, 48] and damp- inglikeSOT[30,49–51]. Thecharacterizationoffieldlikeand dampinglikeSOTisregardlessofthechoiceofSOC,sinceitis determined by the time reversal characteristic. Since Au(m) consistsoffourcontributions,thereare16contributionsinthe feedback tensor
A=P uAu(m) Au(m)for each u. We tabulate all terms of
Ain Table I. The contributions STT:Ax(m) SMF: Ax(m)Adiabatic m@xmNonadiabatic 1@xmFLT kSOm(^ym)DLT 2kSO^ym m@xmG1G1 1@xm1G1G kSOm(^ym)G1G1 2kSO^ym1G1G TABLE I. Example characterization of contributions in Ax(m) Ax(m). Counting orders of gradients and mgives the conven- tional (white), chiral (lighter gray), or anisotropic (darker gray) con- tributions to the gyromagnetic ratio ( 1) or the damping (G). The form of the fieldlike SOT (FLT) and dampinglike SOT (DLT) are taken from magnetic bilayers [30, 47–51] for illustration, but the characterization procedure works generally.withthewhitebackgroundarezerothorderinSOCbutsecond order in gradient and are the conventional nonlocal contribu- tions [3, 65]. Those with the lighter gray background are first order in gradient and chiral [27]. Those with the darker gray color are zeroth order in gradient and anisotropic [66]. In thisway,ourtheoryprovidesaunifiedpictureontheprevious works. Whether a term contributes to 1orGis deter- mined by the order in m. After a direct product of STT and SMF, a term even (odd) in mgivesG(1), since it gives a time irreversible (reversible) contribution appearing in the LLGequationas mA
@tm. Thesameanalysiswithsimple order countings works for any Au(m). It holds even if our theory is generalized to other physics, such as magnons [26], thermal effects [41], and mechanical effects [42]. AsanexampleofapplicationsofTableI,weadoptthespin- Hall-effectdrivenSOT[21,67,68],wherealargedampinglike SOTarises. FromTableI,onecanimmediatelyfigureoutthat thecombinationofthedampinglikeSOTandtheconventional SMF(themosttoprightcell)givesachiralgyromagneticratio contribution. As another example, one notices that the com- bination of the dampinglike SOT and its Onsager counterpart (the fourth term in the SMF) gives an anisotropic damping contribution. Note that the Onsager counter part of the spin- Hall-effect driven SOT is the inverse spin Hall effect driven by spin pumping current generated by the magnetization dy- namics. In this way, Table I provides useful insight for each contribution. Table I also allows for making the general conclusion that the magnitude of the chiral gyromagnetic ratio is determined bythesizeofthedampinglikeSOT( 2)andthatofthenona- diabatic STT ( 1). This is an important observation since many experiments on magnetic bilayers and topological in- sulators [21–23, 30] shows a large dampinglike SOT. This conclusionisregardlessofthemicroscopicdetailsoftheSOT, becauseadampinglikecontributionissolelydeterminedbyits time-reversal property. To summarize, we demonstrate that the chiralities of the gyromagnetic ratio and Gilbert damping have significant im- plicationswhichgofurtherbeyondmerelythechangeinmag- netization dynamics. The chirality plays an important role in investigating underlying physics because physical quantities, which were formerly treated as constants, can now be con- trolled through their chiral dependence. An example is the non-Markovian character of the DW creep motion, which is difficult to be verified in nonchiral systems. From the non- Markovian nature of the DW creep motion, we conclude that the non-energetic origin of chiral DW creep originates from the chiral gyromagnetic ratio rather than the chiral damping. Wealsoprovideageneral,concise,andunifiedtheoryoftheir chiralities, which provide useful insight on the self-feedback of magnetization. We acknowledge M. D. Stiles, Y. Tserkovnyak, A. Thiav- ille, S.-W. Lee, V. Amin, and D.-S. Han for fruitful discus- sion. This work is supported by the Alexander von Humboldt Foundation, the ERC Synergy Grant SC2 (No. 610115), the TransregionalCollaborativeResearchCenter(SFB/TRR)1735 SPIN+X, and the German Research Foundation (DFG) (No. EV 196/2-1 and No. SI 1720/2-1). K.W.K also acknowl- edgessupportbyBasicScienceResearchProgramthroughthe National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1A6A3A03008831). H.W.L. wassupportedbyNRF(2011-0030046). 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Phys. 87, 1213 (2015).Supplementary Materials for “Roles of chiral renormalization of magnetization dynamic s in chiral magnets" Kyoung-Whan Kim,1Hyun-Woo Lee,2Kyung-Jin Lee,3, 4Karin Everschor-Sitte,1Olena Gomonay,1, 5and Jairo Sinova1, 6 1Institut für Physik, Johannes Gutenberg Universität Mainz, Mainz 5512 8, Germany 2PCTP and Department of Physics, Pohang University of Science and Te chnology, Pohang 37673, Korea 3Department of Materials Science and Engineering, Korea University, S eoul 02841, Korea 4KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea 5National Technical University of Ukraine “KPI", Kyiv 03056, Ukraine 6Institute of Physics, Academy of Sciences of the Czech Republic, Cuk rovarnická 10, 162 53 Praha 6 Czech Republic I. THE NON-MARKOVIAN NATURE OF THE DW DYNAMICS A. Integrating out φ In the linear response regime, we may take Fφ≈ −|F′ φ(φeq)|(φ−φeq)and the dynamical coefficients ζeffandαX/φ effto be evaluated at φ=φeq. Without loss of generality, we assume the initial conditio nX(0) = 0 andφ(0) = φeq. We then define the Laplace transforms L[f(t)](s) =/integraltext∞ 0e−stf(t)dt. We denote L[X] =QandL[φ−φeq] =P. Then the Laplace transform of Eq. (2) in the main text is sαX eff λQ+s ζeffP=L[FX] +L[ξX], (S1a) −s ζeffQ+sαφ effλP=−|F′ φ(φeq)|P+L[ξφ], (S1b) Eliminating Pin Eq. (S1) gives 1 ζ2 effs2 |F′ φ(φeq)|+sαφ effλQ+sαX eff λQ=−γHz s+L[Fpin] +/parenleftBigg L[ξX]−s ζeffL[ξφ] b+sαφ effλ/parenrightBigg , (S2) which is an equation of Xonly. Taking the inverse Laplace transform, we obtain the fo llowing non-Markovian equation: 1 λ/integraldisplayt 0f(t−u)X′(u)du=FX+˜ξX(t). (S3) Here f(t)is a feedback function from φ, whose explicit form is f(t) =L−1/bracketleftBigg αX eff+1 ζ2 effαφ effsτ 1 +sτ/bracketrightBigg =/parenleftBigg αX eff+1 ζ2 effαφ eff/parenrightBigg δ(t)−1 ζ2 effαφ effτe−t/τΘ(t), (S4) andτ=αφ effλ/|F′ φ(φeq)|is the relaxation time of φdegree of freedom. The correlation relation for the effectiv e thermal fluctuation field ˜ξX(t)is given by the fluctuation-dissipation theorem ∝angbracketleft˜ξX(t)˜ξX(t′)∝angbracketright ∝Tf(|t−t′|)where Tis the temperature. The noise is ‘colored’ in the sense that it is no longer a white random noise. B. Order-of-magnitude estimation of τ To estimate the order of magnitude of τ, we use the fact that the magntude of |Fφ|is determined by the DMI or the hard axis anisotropy: |F′ φ(φeq)| ≈γλ(π/2)×(2H⊥orDλ−1). We take the DMI field Dλ−1being 30 mT [1] for a rough estimation. Then, |F′ φ(φeq)|/λ≈γ×30 mT ≈5 GHz , so that τ=αφ effλ/|F′ φ(φeq)| ≈αφ eff×0.2 ns, which is small compared to the time scale of the dynamics of X.2 C. First order approximation - chiral mass correction Since τis small, compared to the times scale of the dynamics of X, we may expand f(t)byτ, in the sense of the gradient expansion in time space. Then, f(t)≈ L[αX eff+ (1/ζ2 effαφ eff)sτ] =αX effδ(t) + (τ/ζ2 effαφ eff)δ′(t). Putting this into Eq. (S3) gives τ ζ2 effαφ eff1 λd2X dt+αX eff λdX dt=FX+˜ξX(t), (S5) where the first term represents a massive term. To obtain the D W mass, we need to find the factor which makes FXhave the dimension of force. Note that the force generated by pushing the DW is calculated by Ms/integraltext Heff·∂Xmd3x= (2MsS/γ)FX. Therefore, the mass is defined by multiplying the factor 2MsS/γ, mDW=1 ζ2 eff2MsSτ γαφ effλ, (S6) which is equivalent to Eq. (5) in the main text. D. Higher order contributions - chiral creep To calculate v0, one needs to solve a barrier escape problem. For an energy ba rrierEb, Kramer [2] derived the thermal escapes rate Γ =ν 2π/radicalBigg |F′(Xm)| |F′(XM)|e−Eb/kBT, (S7) where F′(Xm)andF′(XM)are the derivatives of the force (second derivatives of the p inning energy landscape) evaluated at the potential well and the saddle point respectively. νis called the reactive frequency [3] which we calculate belo w. Then, v0 is proportional to Γ. According to the Kramer’s theory, ν∝1/αX efffor a high damping and Markovian limit, which was also confirmed by the functional renormalization group techniqu e [4]. However, we generalize this result to a non-Markovian situa tion [Eq. (S3)]. To do this, we apply the theory of escape rate for a non-Markovian equation of motion [3, 5], based on which, th e reactive frequency νcorresponding to Eq. (S3) is given by the positive root of the following algebraic equation: 1 λνL[f(t)](ν) =|F′(XM)|, (S8) whose exact solution can be calculated from Eq. (S4). As a res ult, ν=2ν0 (1−τν0) +/radicalBig (1 +τν0)2+ 4τν0/ζ2 effαX effαφ eff≈ ν0∝1 αX effν0τ≪ζ2 effαX effαφ eff, ζeff/radicalBigg ν0αX effαφ eff τ∝ζeffν0τ/greaterorsimilarζ2 effαX effαφ eff,(S9) where ν0=λ|F′(XM)|/αX effis the reactive frequency for τ= 0. In the second limit, we assume that the damping parameters are small, thus the last term in the denominator in Eq. (S9) do minates the other terms in the denominator. The two limits sh ows completely different dependences of νon the dynamical parameters. Therefore, it is important to d etermine the relevant regime. Assuming Fpinis random, |F′(XM)| ≈ |F′(Xm)|in Eq. (S7) gives ν0/2πto be the typical attempt frequency ≈1 GHz [6]. From τ≈αφ eff×0.2 ns estimated above, we obtain τν0≈αφ effwhich is an order of magnitude larger than ζ2 effαX effαφ eff, thus the second regime in Eq. (S9) is more relevant, contrary to the tr aditional creep theory just taking τ= 0. [1] S.-G. Je, D.-H. Kim, S.-C. You, B.-C. Min, K.-J. Lee, and S.- B. Choe, Phys. Rev. B 88, 214401 (2013). [2] H. A. Kramers, Physica (Amsterdam) 7, 284 (1940). [3] E. Pollak, H. Grabert, and P. Hänggi, J. Chem. Phys. 91, 4073 (1989). [4] P. Chauve, T. Giamarchi, and P. Le Doussal, Phys. Rev. B 62, 6241 (2000) [5] R. F. Grote and J. T. Hynes, J. Chem. Phys. 73, 2715 (1980). [6] K. Gorchon, S. Bustingorry, J. Ferré, V. Jeudy, A. B. Kolton, a nd T. Giamarchi, Phys. Rev. Lett. 113, 027205 (2014).

2018-02-28

In metallic ferromagnets, the interaction between local magnetic moments and conduction electrons renormalizes parameters of the Landau-Lifshitz-Gilbert equation such as the gyromagnetic ratio and the Gilbert damping, and makes them dependent on the magnetic configurations. Although the effects of the renormalization for nonchiral ferromagnets are usually minor and hardly detectable, we show that the renormalization does play a crucial role for chiral magnets. Here the renormalization is chiral and as such we predict experimentally identifiable effects on the phenomenology of magnetization dynamics. In particular, our theory for the self-consistent magnetization dynamics of chiral magnets allows for a concise interpretation of domain wall creep motion. We also argue that the conventional creep theory of the domain wall motion, which assumes Markovian dynamics, needs critical reexamination since the gyromagnetic ratio makes the motion non-Markovian. The non-Markovian nature of the domain wall dynamics is experimentally checkable by the chirality of the renormalization.

Roles of chiral renormalization on magnetization dynamics in chiral magnets

1803.00017v2

Microscopic theory of Gilbert damping in metallic ferromagnets A. T. Costa and R. B. Muniz Instituto de F sica, Universidade Federal Fluminense, 24210-346 Niter oi, RJ, Brazil. We present a microscopic theory for magnetization relaxation in metallic ferromagnets of nanoscopic dimensions that is based on the dynamic spin response matrix in the presence of spin- orbit coupling. Our approach allows the calculation of the spin excitation damping rate even for perfectly crystalline systems, where existing microscopic approaches fail. We demonstrate that the relaxation properties are notcompletely determined by the transverse susceptibility alone, and that the damping rate has a non-negligible frequency dependence in experimentally relevant situations. Our results indicate that the standard Landau-Lifshitz-Gilbert phenomenology is not always ap- propriate to describe spin dynamics of metallic nanostructure in the presence of strong spin-orbit coupling. Magnetization relaxation in metals is at the heart of spin current generation and detection processes currently under investigation, many of them candidates to play protagonist roles in innovative spintronic devices. The Landau-Lifshitz-Gilbert (LLG) equation is widely used to describe the spin dynamic properties of magnetic ma- terials [1, 2]. It includes an important system-dependent parameter, called the Gilbert damping constant, usually denoted by G, that regulates the relaxation of the mag- netization towards stability, after it is driven out of equi- librium. Recently, a lot of eort has been put into the determination of this damping rate [2{8], which charac- terizes the pumping and absorption of pure spin currents in nanostructures that are of great interest in the eld of spintronic. In most of them spin-orbit interaction is sig- nicant, and responsible for a desirable interplay between charge spin and angular momentum excitations. There is a general agreement between practitioners in the eld that a proper microscopic theory of magnetiza- tion relaxation in metals requires a good description of the electronic structure of the system including spin- orbit coupling [3{8]. The conventional approach is to combine a realistic electronic structure with some kind of adiabatic approximation to derive expressions that can be directly related to the Landau-Lifshitz-Gilbert phe- nomenology. This strategy has been employed by Kam- bersk y [3] and many others since [4{8]. This conven- tional approach has important limitations. It neglects the coupling between transverse spin, longitudinal spin and charge excitations (which is an important consequence of the spin-orbit coupling), and incorrectly pedicts the di- vergence of the damping parameter for a perfectly crys- talline system. Actually, for ferromagnets that display rotation symmetry in spin space, the Goldstone theorem ensures that any experiment which measures the total transverse magnetic moment of the sample will produce a resonant response with zero linewidth [9]. In the pres- ence of spin-orbit interaction, however, this symmetry is explicitly broken, and the resonant spectrum acquires a nite linewidth [10]. We put forward a more fundamental microscopic ap- proach to the calculation of the spin dynamics damp-ing rate that takes fully into account the eects of SOC on the spectrum of spin excitations of itinerant sys- tems. Namely, we consider the coupling of transverse spin excitations to longitudinal spin and charge excita- tions, induced by the spin-orbit interaction. We calculate the FMR spectrum at nite frequencies and arbitrary anisotropy values, without employing any adiabatic ap- proximation. We will show that those ingredients are es- sential to correctly describe the magnetization relaxation in very clean metallic ferromagnets of nanoscopic dimen- sions, and that the Landau-Lifshitz-Gilbert phenomenol- ogy fails to capture essential features of the magnetiza- tion dynamics in those systems. This letter is organized as follows: we will present brie y our formalism, discuss its main features and present numerical results for two model systems that il- lustrate common but qualitatively dierent situations. General Formalism - The spectrum of spin excitations of a ferromagnet can be obtained from the spectral den- sity associated with the transverse spin susceptibility +(l;l0; ) =Z dtei thhS+ l(t);S l0(0)ii; (1) where hhS+ l(t);S l0(0)iii(t)h[S+ l(t);S l0(0)]i; (2) and S+ l=X ay l"al#: (3) The operator ay lcreates one electron in the atomic ba- sis statelocalized at lattice site lwith spin. Although we are usually interested in +(l;l0; ) as dened above, its equation of motion involves the orbital-resolved sus- ceptibility, + 00(l;l0;t)hhay l"(t)al#(t);ay l00#(0)al00"(0)ii:(4) In the absence of spin-orbit coupling (SOC) and within the random phase approximation (RPA), the equation ofarXiv:1502.02068v1 [cond-mat.mes-hall] 6 Feb 20152 motion for + 00(l;l0;t) is closed and +(l;l0; ) can be expressed in the well-known RPA form, +( ) = [1 +U+ 0( )]1+ 0( ) (5) where+ 0( ) is the mean-eld (sometimes called non- interacting, or Hartree-Fock) susceptibility. This expres- sion is schematic and must be understood as a matrix in orbital and site indices, in real space, or a wave-vector dependent matrix in reciprocal space. The crucial point, however, is that, in the absence of spin-orbit coupling, within the RPA, the transverse spin susceptibility is un- coupled from any other susceptibility. This ceases to be true when SOC is included, as we demonstrated in ref. 10:+becomes coupled to three other susceptibil- ities, namely (2) 00(l;l0;t)hhay l"(t)al"(t);ay l00"(0)al00"(0)ii;(6) (3) 00(l;l0;t)hhay l#(t)al#(t);ay l00#(0)al00#(0)ii;(7) (4) 00(l;l0;t)hhay l#(t)al"(t);ay l00"(0)al00#(0)ii:(8) The system of equations of motion obeyed by these four susceptibilities can be cast into a form strongly re- sembling the RPA result by introducing a block-vector ~ ((1);(2);(3);(4))T, with(1)+. With an equivalent denition for the mean-eld susceptibili- ties(m) 0we write ~ ( ) =~ 0( )~ ( ); (9) where the \super-matrix"  is proportional to the eec- tive Coulomb interaction strength and involves convolu- tions of single particle Green functions. Explicit forms for its matrix elements are found in Ref. 10. The nu- merical analysis of the susceptibilities (2),(3)and(4) show that their absolute values are many orders of mag- nitude smaller than those of (1)=+. It is, thus, tempting to argue that the transverse susceptibility is ap- proximately decoupled from (2),(3)and(4)and that it can be calculated via the usual RPA expression with the single particle Green functions obtained with spin- orbit coupling taken into account. This is not a good approximation in general, since the matrix elements of  that couple (1)to the other susceptibilities are far from negligible. Our numerical calculations indicate that they are essential to determine correctly the features of the FMR mode around the resonance frequency. Thus, the behaviour of (1)in the presence of spin-orbit coupling cannot be inferred from (1) 0in the zero-frequency limit alone, as it is usually assumed in the literature on the cal- culation of the Gilbert damping parameter [3{5, 8, 11]. Numerical Results - We start the discussion by pre- senting results for the Gilbert constant Gfor unsup- ported ultrathin Co lms. Here we determine Gfrom the ratio between the FMR linewidth
and the reso- nance frequency 0. First we turn o spin-orbit coupling 0 10 20 30 40 50 η (meV)00.010.020.030.040.05αGFIG. 1: Gilbert damping constant Gas a function of the imaginary part added to the real energy, for an ultrathin lm of two atomic layers of Co where SOC has been turned o. It is clear that Gvanishes as !0. to check the consistency of our approach. Even with SOC turned o we still nd a nite linewidth for the FMR mode. It comes, as we will shortly demonstrate, from the small imaginary part that is usually added to the energy in the numerical calculations of the sin- gle particle Green functions, in order to move their poles from the real axis. We calculate Gfor various values of and extrapolate to !0+, as shown in Fig. 1. It is clear that lim !0+G= 0. Thus, our approach correctly predicts that the Gilbert damping constant vanishes in the absence of SOC, as it should. Indeed, it is easy to show [9] that the FMR mode is a stationary state of the mean-eld hamiltonian and, as such, has innite lifetime in the limit !0+. Now we discuss the dependence ofGonfor a xed, non-zero value of the spin-orbit coupling strength . We used LCAO parameters appro- priate for bulk Co to describe the electronic structure of all Co lms we investigated. The quantitative details of the ferromagnetic ground state and excitation spec- tra are sensitive to the LCAO parameters used, but their qualitative behaviour is very robust to small changes in the electronic structure. Our strategy is to use the same set of LCAO parameters for all lm thicknesses to avoid modications in Gcoming directly from changes in the LCAO parameters. This allows us to focus on geometric eects and on the -dependence. Figure 2 shows the dependence of the Gilbert damping constantGon the imaginary part for Co lms of var- ious thicknesses. Clearly Gapproaches nite values as !0. Cobalt has a small spin-orbit coupling constant. We would like to investigate the eect of increasing the strength of the SOC on the damping rate. Instead of articially increasing in Co we consider a more realis- tic setting where a double layer of Co is attached to a3 05 10 15 η (meV)00.010.020.030.04αG FIG. 2: Gilbert damping constant Gas a function of the imaginary part added to the energy, for Co ultra thin lms of various thicknesses: 1 (circles), 2 (squares), 4 (diamonds) and 6 (triangles) atomic layers. The strength of the SOC is = 85 meV. The solid lines are guides to the eye. non-magnetic substrate with high SOC parameter, such as Pt. This system has a particularly interesting fea- ture: the magnetization easy axis is perpendicular to the plane. However, we found that, for the LCAO param- eters we employed, the magnetization in-plane is also a stable conguration, with a small magnetocrystalline anisotropy. The damping rate, however, is much larger in the 2Co/2Pt system than in the unsupported Co lms. This is a nice example of how the anisotropy energy is strongly in uenced by the system's symmetry, but the damping rate is relatively insensitive to it, depending strongly on the intensity of the spin-orbit coupling. It is also an extremely convenient situation to test an assump- tion very frequently found in the literature on Gilbert damping, although sometimes not explicitly stated: that the FMR linewidth
is linearly dependent on the res- onance frequency 0and that
!0 as 0!0. This is not an unreasonable hypothesis, considering the weak static elds commonly used in FMR experiments and the smallness of the spin-orbit coupling constant, compared to other energy scales of a ferromagnet. Our calculations for the Co lms conrm that this relationship is approx- imately held. In this case, the Gilbert constant Gmay be extracted from the FMR spectrum by simply tting it to a Lorentzian and is practically eld-independent. However, our results for 2Co/2Pt indicate that the FMR linewidth is nite as 0!0, leading to a signicantly frequency-dependent G, as shown in Fig. 3. In order to illustrate how the determination of a damping parame- ter is aected by the nite value of
as 0!0 we extracted the linewidths from the calculated spectra for the 2Co/2Pt system by tting Lorentzians to our calcu- lated spectral densities. The results are shown in Fig. 3. One of its most important consequences is that, if one wishes to dene a value of Gfor the system above, it 00.5 1 Ω (meV)05e+051e+061.5e+062e+06FMR spectral density01 2 3 4 B (T) 00.20.40.60.81Ω0 (meV)(a) 01 2 3 4 B (T) 0.10.120.140.160.180.2αG0 0.2 0.4 0.6 0.8 1 Ω 0 (meV)00.050.1ΔΩ (meV)(b)FIG. 3: a) Spectral densities of the FMR mode for the 2Co/2Pt system subjected to various static magnetic elds (from -0.3 T to 4 T). The inset shows the resonance frequency as a function of the Zeeman eld B. b) The Gilbert damping parameterGas a function of applied Zeeman eld B. The inset shows the FMR line width as a function of resonance frequency 0. The strengths of the SOC are Co= 85 meV andPt= 600 meV. must be dened as a function of the Zeeman eld, as is illustrated in Fig. 3. In principle this poses a problem for the procedure usually employed to determine FMR spectra experimentally, since there the free variable is the Zeeman eld, not the frequency of the exciting eld. In Fig. 4 we illustrate this issue by plotting the FMR spectral density as a function of the Zeeman eld for two xed pumping frequencies, 24 GHz and 54 GHz. The curves have nice Lorentzian shapes, but the values for the Gilbert damping parameter Gextracted from these curves depend on the pumping frequency ( G= 0:034 for 0= 0:10 meV and G= 0:042 for 0= 0:22 meV). Also, they do not correspond to any of the values shown in Fig. 3b, although the Zeeman eld values that de- termine the linewidth in Fig. 4 lie within the range of Zeeman eld values showed in Fig. 3b. Thus, if Gis dened as
= 0, its value for a given sample depends4 -0.4-0.2 0 0.2 B (T) 05e+051e+061.5e+062e+06FMR spectral density FIG. 4: Spectral densitty of the FMR mode for the 2Co/2Pt system plotted as a function of the Zeeman eld Bat a xed pumping frequencies: p= 24 GHz (squares) and p= 54 GHz (circles). The solid curves are Lorentzian ts to the calculated points. on wether the FMR spectrum is obtained in a xed fre- quency or xed Zeeman eld set ups. Our results also imply that the existing expressions for the damping con- stantGare not valid in general, specially for very clean systems with large spin-orbit coupling materials. The conventional approaches express Gas the ratio
= 0 in the 0!0 limit. As we have just shown, this limit does not exist in general, since
approaches a nite value as 0!0. In experimental papers [12, 13] the FMR linewidth is assumed to have a zero-frequency oset, just as we de- scribed. This is usually attributed to extrinsic broad- ening mechanisms, such as two-magnon scattering [14], due to the combination between inhomogeneities in the magnetic lms and dipolar interactions. This is certainly the case in systems with small SOC, such as Fe lms de- posited on GaAs or Au [12]. However, we have shown that there can be zero-frequency oset of intrinsic ori- gin if the SOC is large. The eect of this intrinsic oset should be easily separated from that of the two-magnon scattering mechanism, since the latter is not active when the magnetization is perpendicular to the plane of the lm [14]. We would like to remark that Stoner enhancement in Pt plays a very important role in the determination of the damping rate. We had shown previously [15] that, in the absence of spin-orbit coupling, Stoner enhancement had a very mild eect on the damping rate in the Co/Pd(001) system. In the presence of SOC, however, the eect can be very large indeed. Both magnetocrystalline anisotropy and damping rate are signicantly dierent in the en- hanced and non-enhanced cases, as shown in Fig. 5. The Gilbert parameter is also very dierent in the two cases: enh G= 0:11, whereas nonenh G = 0:33. Thus, proper 01 2 3 Ω (meV)02e+054e+056e+058e+05A(Ω)FIG. 5: a) Spectral densities of the FMR mode for the 2Co/2Pt system with Stoner enhancement in Pt turned on (black line) and o (red line). treatment of Stoner enhancement in substrates like Pd an Pt is essential for the correct determination of spin relaxation features. We presented a microscopic approach to the calcu- lation of the Gilbert damping parameter Gfor ultra- thin metallic magnetic lms, illustrated by results for Co lms and Co/Pt bilayers. Our approach is based on the evaluation of the dynamic transverse susceptibility in the presence of spin-orbit coupling, taking into account realistic electronic structures and the coupling between transverse spin, longitudinal spin and charge excitations. It predicts nite values of Gin the limit of perfectly crystalline lms, a regime where methods based on the torque correlation formula nd a diverging Gilbert damp- ing parameter. We showed that the coupling between transverse, longitudinal and charge excitations, due to spin-orbit coupling, is of fundamental importance for the correct determination of FMR spectra in metallic sys- tems. We have also shown that the damping rate ex- tracted from the FMR spectrum for xed pumping fre- quency diers considerably from that extracted from the FMR spectrum for xed Zeeman eld. In this case the Gilbert damping parameter Gbecomes frequency de- pendent, in contrast to what is assumed in the standard Landau-Lifshitz-Gilbert phenomenology. Moreover, we have numerical indications that the Gilbert parameter is not well dened in the limit of vanishing resonance frequency, a fact that is very relevant to calculational schemes based on the adiabatic approximation. Inci- dentally, Stoner enhancement in materials like Pt and Pd also plays an important role in the determination of FMR frequencies and damping rates. These results may lead to important modications of the interpretation of damping \constants", either calculated or inferred from5 experimental results, for systems where spin-orbit cou- pling is strong. We believe these issues may be crucial for the correct description of relaxation in very clean sys- tems of nanoscopic dimensions, specially in the presence of relatively weak magnetocrystalline anisotropy. The authors acknowledge partial nancial support from CNPq and FAPERJ. We are grateful to Professor Caio Lewenkopf for a critical reading of the manuscript and to Dr. Mariana Odashima for enlightening discus- sions. RBM acknowledges fruitful discussions with Prof. D. M. Edwards and A.Umerski. [1] T. Gilbert, Magnetics, IEEE Transactions on 40, 3443 (2004), ISSN 0018-9464. [2] Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, Rev. Mod. Phys. 77, 1375 (2005), URL http://link. aps.org/doi/10.1103/RevModPhys.77.1375 . [3] V. Kambersk y, Phys. Rev. B 76, 134416 (2007), URL http://link.aps.org/doi/10.1103/PhysRevB.76. 134416 . [4] I. Garate and A. MacDonald, Phys. Rev. B 79, 064403 (2009), URL http://link.aps.org/doi/10. 1103/PhysRevB.79.064403 . [5] I. Garate and A. MacDonald, Phys. Rev. B 79, 064404 (2009), URL http://link.aps.org/doi/10. 1103/PhysRevB.79.064404 . [6] A. Starikov, P. Kelly, A. Brataas, Y. Tserkovnyak, and G. Bauer, Phys. Rev. Lett. 105, 236601 (2010), URLhttp://link.aps.org/doi/10.1103/PhysRevLett.105. 236601 . [7] H. Ebert, S. Mankovsky, D. K odderitzsch, and P. J. Kelly, Phys. Rev. Lett. 107, 066603 (2011), URL http://link.aps.org/doi/10.1103/PhysRevLett.107. 066603 . [8] E. Barati, M. Cinal, D. M. Edwards, and A. Umerski, Phys. Rev. B 90, 014420 (2014), URL http://link.aps. org/doi/10.1103/PhysRevB.90.014420 . [9] R. B. Muniz and D. L. Mills, Phys. Rev. B 68, 224414 (2003), URL http://link.aps.org/doi/10. 1103/PhysRevB.68.224414 . [10] A. T. Costa, R. B. Muniz, S. Lounis, A. B. Klautau, and D. L. Mills, Phys. Rev. B 82, 014428 (2010), URL http: //link.aps.org/doi/10.1103/PhysRevB.82.014428 . [11] Y. Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, and P. J. Kelly, Phys. Rev. Lett. 113, 207202 (2014), URL http://link.aps.org/doi/10. 1103/PhysRevLett.113.207202 . [12] R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204 (2001), URL http://link.aps.org/ doi/10.1103/PhysRevLett.87.217204 . [13] E. Montoya, B. Heinrich, and E. Girt, Phys. Rev. Lett. 113, 136601 (2014), URL http://link.aps.org/doi/ 10.1103/PhysRevLett.113.136601 . [14] R. Arias and D. Mills, Phys. Rev. B 60, 7395 (1999), URL http://link.aps.org/doi/10.1103/PhysRevB. 60.7395 . [15] D. L. R. Santos, P. Venezuela, R. B. Muniz, and A. T. Costa, Phys. Rev. B 88, 054423 (2013), URL http:// link.aps.org/doi/10.1103/PhysRevB.88.054423 .

2015-02-06

We present a microscopic theory for magnetization relaxation in metallic ferromagnets of nanoscopic dimensions that is based on the dynamic spin response matrix in the presence of spin-orbit coupling. Our approach allows the calculation of the spin excitation damping rate even for perfectly crystalline systems, where existing microscopic approaches fail. We demonstrate that the relaxation properties are not completely determined by the transverse susceptibility alone, and that the damping rate has a non-negligible frequency dependence in experimentally relevant situations. Our results indicate that the standard Landau-Lifshitz-Gilbert phenomenology is not always appropriate to describe spin dynamics of metallic nanostructure in the presence of strong spin-orbit coupling.

Microscopic theory of Gilbert damping in metallic ferromagnets

1502.02068v1

Approximate rogue wave solutions of the forced and damped Nonlinear Schr odinger equation for water waves Miguel Onorato and Davide Proment Dipartimento di Fisica, Universit a degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy, EU INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy, EU Abstract We consider the eect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transforma- tion, we map the forced/damped Nonlinear Schr odinger (NLS) equation into the standard NLS with constant coecients. The transformation is valid as long asjtj 1, with the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the eects of wind and dissipation on the formation of rogue waves. Keywords: rogue waves, water waves, breathers 1. Introduction Modulational instability, also known as the Benjamin-Feir instability in the water wave community, has been discovered in the late sixties indepen- dently by Benjamin and Feir [1] and Zakharov [2] (see [3] for an historical review on the subject and possible applications). It describes the exponen- tial growth of an initially sinusoidal long wave perturbation of a plane wave solution of the one dimensional water wave problem. For water waves the condition of instability in innite water depth is that 2p 2a0k0N > 1, where a0is the amplitude of the plane wave and k0is the corresponding wave number;N=k0=
Kis the number of waves under the perturbation of wavenumber
K. The modulaitonal instability is frequently studied within the Nonlinear Schr odinger (NLS) equation that describes weakly nonlinear and dispersive waves in the narrow band approximation. In this context, Preprint submitted to Physics Letters A March 22, 2012arXiv:1203.4735v1 [nlin.CD] 21 Mar 2012the nonlinear stages of the modulational instability are described by exact solutions of the NLS, known as Akhmediev breathers [4, 5]. Other exact NLS solutions which describe the focussing of an initially non-small pertur- bation have been derived in [6, 7]. Such solutions have been considered as prototypes of rogue waves [8, 9]. Within the one dimensional NLS equation, the modulational instability is well understood. What is probably less clear is the modulation of waves and the formation of rogue waves in forced (by wind) or damped (by dissipation) conditions. In this regard in the past there has been a number of experimental works, [10, 11, 12], which did not gave a clear picture on the eect of the wind on the modulaitonal instability. A careful discussion of the discrepancy of the results presented in the above papers can be found in [12]. According to their discussion the role of the wind is twofold: i) the wind changes the growth rate of the instability; ii) the natural selection of the sideband frequency is altered with respect to the no wind conditions. Concerning damping eects, it has been showed in [13] that any amount of dissipation stabilizes the modulational instability, questioning the role of the modulational instability in the formation of rogue waves, [14]. More recently, the role of dissipation and wind in the modulational instability has been considered together within the NLS equation, [15] (then conrmed by fully nonlinear simulations, [16]). The authors performed a linear stability analysis and numerical simulations and found that, in the presence of wind, young waves are more sensitive to modulational instability than old waves. The just mentioned numerical results (except the one in [16]) are all based on the following forced and damped Nonlinear Schr odinger equation: i@A @t@2A @x2jAj2A=iA: (1) Ais the wave envelope, andare two coecients that depend on the wavenumber, k0, of the carrier wave. The right-hand side is responsible for the forcing, >0, and/or dissipation, <0. The two eects are additive so that is in general the sum of forcing coecient plus a damping one. The wind forcing depends on the ratio between air and water density and the dissipation on the water viscosity, therefore the absolute value of is always a small quantity. Finding analytical solutions of equation (1) is not an obvious task. In the present paper we take advantage of the smallness of and, after a suitable transformation, we are able to nd breather solutions of 2the forced-damped NLS equation. In the following sections we rst describe the transformation and then present the rogue wave analytical solutions. 2. Reduction of the forced/damped NLS to the standard NLS We considered the NLS equation discussed in [15] i@A @t+cg@A @x 1 8!0 k2 0@2A @x21 2!0k2 0jAj2A=iA (2) with =1 2g2a w !0u c2 2k2 0 (3) hereis the Von Karman constant and uis the friction velocity, gis the gravity acceleration, aandware the air and water density, respectively; is a coecient to be determined from the solution of Rayleigh equation associated to the stability of the wind wave problem (see also [17] for a justication of the wind forcing term); cis the phase velocity, is the water kinematic viscosity. In [15] the equation is written in a nondimensional form and the coecient K= =!0is introduced). The surface elevation is related to the envelope as follows: (x;t) =1 2 A(x;t) exp[i(k0x!0t)] +c:c : (4) Note that we use a dierent denition of the surface elevation from the one in [15] where the 1/2 factor is not included (the consequence is that the coecient in the nonlinear term in equation (2) diers by a factor of 4 from the one in equation (3.1) in [15]). If is the small parameter in the derivation of the NLS, then it is assumed that the right-hand side term in (2) is of the order of2as the nonlinear and the dispersive term. We consider the following new variable: B(x;t) =A(x;t)et(5) and by selecting a coordinate system moving with the group velocity we get: i@B @t@2B @x2exp2tjBj2B= 0 (6) 3wereandare the coecients of the dispersive and nonlinear term, re- spectively. Written in the above form the eect of the forcing/damping term enters as a factor in front of nonlinear term and has the role of enhanc- ing/decreasing the nonlinearity of the system as the wave evolve in time. Recalling that is usually small, we Taylor expand the exponential and re-write the equation as follows: i@B @t@2B @x2p(t)jBj2B= 0 (7) withp(t) = 1=(12t). Let's introduce the following change of coordinates: (x;t) =p(t)x;  (t) =p(t)t (8) and scale the wave envelope function Bas follows (;) =B(x;t)p p(t)exp ip(t)x2 2 : (9) After the transformation, the equation (6) results in: i@ @@2 @2j j2 = 0 (10) i.e., the NLS equation with constant coecients. We have transformed the forced/damped NLS equation into the standard NLS equation whose solu- tions can be studied analytically. From a physical point of view the trans- formation (and consequently the validity of the solutions) is valid as long as 2jtj 1 (the transformation is singular for 2 jtj= 1). We underline that the transformation of the forced/damped NLS equation to the standard one has been possible only for 1 =p(t) equal to a linear function in t. For other functional dependences, the transformation does not seem to be possi- ble. Our result is consistent with ones reported in [18, 19] where analytical solutions of the variable coecient NLS equation are described. 3. Rogue wave solutions In the following we will present three analytical solutions corresponding to the Peregrine, the Akhmediev and the Kuznetsov-Ma breathers for the standard NLS. 4The Peregrine solution also known as rational solution, has been origi- nally proposed in [5]. It has the peculiarity of being not periodic in time and in space: it is a wave that \appears out of nowhere and disappears without trace" [20, 21]; its maximum amplitude reaches three times the amplitude of the unperturbed waves. For the above reasons it has been considered as spe- cial prototype of freak wave, [21]. The Peregrine solution has been recently reproduced experimentally in wave tank laboratories [22] and in optical bers [23]. Below we present an exact analytical solution of equation (7) which is the analogous of the Peregrine solutions but for the forced/damped case: B(x;t) =B0G(x;t)4(1i2B2 0p(t)t) +(2B2 0p(t)t)2+ 2B2 0(p(t)x)21 (11) with G(x;t) =p p(t) exp ip(t)x2 2B2 0p(t)t : (12) In gure 1 we show an example of such solution for steepness 0.1 and forcing coecientK= 0:0004 (the same value has been used in [15]). The axis are normalized by the wave period, the wavelength and the initial wave amplitude B0. The eect of the wind/dissipation is to increase/reduced the amplitude of the plane wave. As in the case of the standard NLS, the wave appears only once in time and space. The Akhmediev solution [4] describes the modulational instability in its nonlinear regime; it is periodic in space. It is characterized by an ampli- cation factor which ranges from 1 to 3 (this last value corresponds to the Peregrine solution). In the presence of a forcing or damping, the breather has the following analytical form: B(x;t) =B0G(x;t) p 2~2cosh[p(t)t]ip 2~sinh[p(t)t)]p 2 cosh[p(t)t]p 2~2cos[p(t)x]1! (13) and =k0 N;~= B0r ;~= ~p 2~2;  =B2 0~: (14) The function G(x;t) is reported (12). It should be noted that the function is periodic in space with a period that changes in time. In gure 2 we show an example of such solution for steepness 0.1, N= 5 and forcing coecient ofK= 0:0004. 5Figure 1: The Peregrine solution of the forced NLS equation. Figure 2: The Akhmediev solution of the forced NLS equation. 6Figure 3: The Kuznetsov-Ma solution of the forced NLS equation. The Kuznetsov-Ma solution [6] is periodic in time and decrease exponen- tially in space. While for the Akhmediev breather the large time (positive or negative) limit is a plane wave plus a small perturbation, the modulation for the Ma breather is never small. The solution for the forced/damped equation is here reported: B(x;t) =B0G(x;t) p 2~2cos[p(t)t] +ip 2~sin[p(t)t)]p 2 cos[p(t)t]p 2 + ~2cosh[p(t)x]1! (15) with =B0~r  ;~= ~p 2 + ~2;  =B2 0~: (16) ~is a parameter related to the amplication factor. In gure 3 we show an example of such solution for steepness 0.1, ~ =p 2 and forcing coecient of K= 0:0004. The periodicity (appearance of maxima) changes in time and increase in the presence of forcing and decrease for the damping case. 4. Conclusion In the present Letter we have considered the problem of generation of rogue waves in the presence of wind forcing or dissipation. Our work is based on the one dimensional forced/damped NLS equation. 7Under the assumption of 2 jtj 1, where is the forcing ( >0) or damping ( <0) term, we have shown how the equation can be mapped in the standard NLS equation with constant coecients. In this framework, we have found explicit analytical breather solutions. As mentioned the eect of wind/dissipation is to increase/reduce in time the coecient in front of the nonlinear term. This has an impact on the modulational instability; in particular, an initially stable (unstable) wave packet could be destabilized (stabilized) by the wind (dissipation). Similar results have been obtained for the interaction of waves and current (see [24]). The present results should be tested in wind waves tank facilities. Acknowledgments The E.U. project EXTREME SEAS (SCP8-GA- 2009-234175) is acknowledged. M.O. thanks Dr. GiuliNico for discussions and ONR (grant N000141010991) for support. We are thankful to J. Dudley for pointing us out reference [18]. References [1] T. B. Benjamin, J. E. Feir, The disintegration of wave trains on deep water. Part I. Theory, J. Fluid Mech. 27 (1967) 417{430. [2] V. Zakharov, Stability of period waves of nite amplitude on surface of a deep uid, J. Appl. Mech. Tech. Phys. 9 (1968) 190{194. [3] V. Zakharov, L. Ostrovsky, Modulation instability: the beginning, Phys- ica D: Nonlinear Phenomena 238 (5) (2009) 540{548. [4] N. Akhmediev, V. Eleonskii, N. Kulagin, Exact rst-order solutions of the nonlinear Schr odinger equation, Theoretical and Mathematical Physics 72 (2) (1987) 809{818. [5] D. Peregrine, Water waves, nonlinear schr odinger equations and their solutions, J. Austral. Math. Soc. Ser. B 25 (1) (1983) 16{43. [6] Y. Ma, The perturbed plane-wave solutions of the cubic Schr odinger equation, Studies in Applied Mathematics 60 (1979) 43{58. [7] E. Kuznetsov, Solitons in a parametrically unstable plasma, in: Akademiia Nauk SSSR Doklady, Vol. 236, 1977, pp. 575{577. [8] K. B. Dysthe, K. Trulsen, Note on breather type solutions of the nls as models for freak-waves, Physica Scripta T82 (1999) 48{52. 8[9] A. Osborne, M. Onorato, M. Serio, The nonlinear dynamics of rogue waves and holes in deep{water gravity wave train, Phys. Lett. A 275 (2000) 386{393. [10] L. Bliven, N. Huang, S. Long, Experimental study of the in uence of wind on benjamin-feir sideband instability, J. Fluid Mech 162 (1986) 237{260. [11] T. Hara, C. Mei, Frequency downshift in narrowbanded surface waves under the in uence of wind, Journal of Fluid Mechanics 230 (1) (1991) 429{477. [12] T. Waseda, M. Tulin, Experimental study of the stability of deep-water wave trains including wind eects, Journal of Fluid Mechanics 401 (1) (1999) 55{84. [13] H. Segur, D. Henderson, J. Carter, J. Hammack, C. Li, D. Phei, K. Socha, Stabilizing the benjamin-feir instability, Journal of Fluid Me- chanics 539 (2005) 229{272. [14] H. Segur, D. Henderson, J. Hammack, Can the benjamin-feir instability spawn a rogue wave?, in: Proceedings of the Aha Huliko'a Hawaiian Winter Workshop, University of Hawaii, Retrieved August, Vol. 5, 2008. [15] C. Kharif, R. Kraenkel, M. Manna, R. Thomas, The modulational in- stability in deep water under the action of wind and dissipation, Journal of Fluid Mechanics 664 (1) (2010) 138{149. [16] C. Kharif, J. Touboul, Under which conditions the benjamin-feir insta- bility may spawn an extreme wave event: A fully nonlinear approach, The European Physical Journal-Special Topics 185 (1) (2010) 159{168. [17] S. Leblanc, Amplication of nonlinear surface waves by wind, Physics of Fluids 19 (2007) 101705. [18] Q. Tian, Q. Yang, C. Dai, J. Zhang, Controllable optical rogue waves: Recurrence, annihilation and sustainment, Optics Communications 284 (2011) 2222{2225. [19] J. Zhang, C. Dai, Q. Yang, J. Zhu, Variable-coecient f-expansion method and its application to nonlinear schr odinger equation, Optics communications 252 (4-6) (2005) 408{421. 9[20] N. Akhmediev, A. Ankiewicz, M. Taki, Waves that appear from nowhere and disappear without a trace, Physics Letters A 373 (6) (2009) 675{678. [21] V. Shrira, V. Geogjaev, What makes the Peregrine soliton so special as a prototype of freak waves?, Journal of Engineering Mathematics 67 (1) (2010) 11{22. [22] A. Chabchoub, N. Homann, N. Akhmediev, Rogue wave observation in a water wave tank, Physical Review Letters 106 (20) (2011) 204502. [23] B. Kibler, J. Fatome, C. Finot, G. Millot, F. Dias, G. Genty, N. Akhme- diev, J. Dudley, The Peregrine soliton in nonlinear bre optics, Nature Physics 6 (10) (2010) 790{795. [24] M. Onorato, D. Proment, A. Tooli, Triggering rogue waves in op- posing currents, Phys. Rev. Lett. 107 (2011) 184502. doi:10.1103/ PhysRevLett.107.184502 . URL http://link.aps.org/doi/10.1103/PhysRevLett.107.184502 10

2012-03-21

We consider the effect of the wind and the dissipation on the nonlinear stages of the modulational instability. By applying a suitable transformation, we map the forced/damped Nonlinear Schr\"odinger (NLS) equation into the standard NLS with constant coefficients. The transformation is valid as long as |{\Gamma}t| \ll 1, with {\Gamma} the growth/damping rate of the waves due to the wind/dissipation. Approximate rogue wave solutions of the equation are presented and discussed. The results shed some lights on the effects of wind and dissipation on the formation of rogue waves.

Approximate rogue wave solutions of the forced and damped Nonlinear Schrödinger equation for water waves

1203.4735v1

arXiv:1309.5523v4 [math.AP] 30 Mar 2017Pattern formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations October 2, 2018 C. Melcher1& J.D.M. Rademacher2 Abstract The Landau-Lifshitz-Gilbert-Slonczewski equation describes mag netization dynamics in the presence of an applied field and a spin polarized current. In the case of axial sym- metry and with focus on one space dimension, we investigate the eme rgence of space-time patterns in the form of wavetrainsand coherent structures, wh ose local wavenumbervaries in space. A major part of this study concerns existence and stabilit y of wavetrains and of front- and domain wall-type coherent structures whose profiles a symptote to wavetrains or the constant up-/down-magnetizations. For certain polarizat ion the Slonczewski term can be removed which allows for a more complete charaterization, inc luding soliton-type solutions. Decisive for the solution structure is the polarization par ameteras well as size of anisotropy compared with the difference of field intensity and curre nt intensity normalized by the damping. 1 Introduction This paper concerns the analysis of spatio-temporal patter n formation for the axially sym- metric Landau-Lifshitz-Gilbert-Slonczewski equation fo r which the applied magnetic field and current are aligned with or orthogonal to the material aniso tropy. In one space dimension we thus consider ∂tm=m×/bracketleftBig α∂tm−∂2 xm+(µm3−h)ˆe3+β 1+ccpm3m׈e3/bracketrightBig (1) asamodelforthemagnetization dynamics m=m(x,t)∈S2(i.e.misadirectionfield)driven by an external field h=hˆe3and current j=β 1+ccpm3ˆe3with polarization parameter ccp∈ (−1,1). The parameters α >0 andµ∈Rare the Gilbert damping factor and the anisotropy constant, respectively. A brief overview of thephysical ba ckground and interpretation of terms is given below in Section 2. The constant up- or down-magnetization states m=±ˆe3are always steady states of (1) and magnetic domain walls spatially separating these state s are of major interest. While the combination of field and current excitations gives rise t o a variety of pattern formation phenomena, see e.g. [5, 14, 15, 19, 23], not much mathematica lly rigorous work is available so far, in particular for the dissipative case α >0 that we consider. The case of axial symmetry is not only particularly convenient from a technical perspe ctive. It offers at the same time valuable insight in the emergence of space-time patterns an d displays strong similarities to better studied dynamical systems such as real and complex Gi nzburg-Landau equations. In 1Lehrstuhl I f¨ ur Mathematik and JARA-FIT, RWTH Aachen Unive rsity, 52056 Aachen, Germany, melcher@rwth-aachen.de 2Fachbereich 3 – Mathematik, Universit¨ at Bremen, Postfach 33 04 40, 28359 Bremen, Germany, rademach@math.uni-bremen.de 1m1m2m3 (a) (b) Figure 1: Illustration of a wavetrain profile m(ϕ) (a) in the 2-sphere showing their constant altitude and (b) as a space-time plot of, e.g., m2. In (a) the thick arrow represents m= (m,m3), the thin line its trajectory as a function of ϕ=kx−ωt. this framework we examine the existence and stability of wav etrain solutions of (1), i.e., solutions of the form m(x,t) =ei(kx−ωt)m0, where the complex exponential acts on m0∈S2by rotation about the ˆe3-axis (cf. Figure 1 for an illustration). In the special case ccp= 0 it turns out that (1) can be transformed to the variational LLG-equation with β= 0: in a rotating frame about the m3-axis with frequency −β/αthe current dependent term vanishes and hchanges to h−β/α; see§2.1. This allows for a complete characterization of wavetrains and their L2-stability. We also investigate the existence of coherent structure sol utions which are locally in space of wavetrain form m(x,t) =eiϕ(x,t)m0(x−st) where ϕ(x,t) =φ(x−st)+Ωt (2) such that m0(ξ) = (sin θ(ξ),0,cosθ(ξ)). Samples are plotted in Figures 2, 3, 4. In the variational case ccp= 0 we completely characterize the existence of small amplit ude coherent structuresandstationary ( s= 0) ones, whichinfactcorrespondtostandingwaves intheab ove rotating frame. Throughthecoherent structureviewpoint w e recover a family of ‘homogenous’ domain wall type solutions of arbitrary velocity, having no azimuthal profile, i.e., constant φ and thus vanishing local wavenumber d ϕ/dx. For general ccpand large speeds, |s| ≫1, we prove existence of a family of more general front-type coh erent structures with nontrivial local wavenumbers, which can also form a spatial interface b etween±ˆe3and wavetrains. The analysis of these kinds of solutions is inspired by and bears similarities with that of the real and complex Ginzburg-Landau equations. More specifically, the parameter space for existence of wave trains and coherent structures is largely organized by the stability of the equilibrium sta tesm=±ˆe3. The nature of bifurcations that we find motivates the following notions to organize the parameter space of (1): We refer to parameters as being •‘supercritical’ if ±ˆe3are both unstable •‘subcritical’ if ±ˆe3have different stability •‘subsubcritical’ if ±ˆe3are both stable. 20102030405060/Minus0.50.00.5 xm2 050100150200/Minus0.8/Minus0.400.4 xm2 /Minus0.8/Minus0.4 0 0.4/Minus0.50.00.5 m1m2 /Minus0.8/Minus0.6/Minus0.4/Minus0.20.00.2/Minus0.8/Minus0.400.4 m1m2 (a) (b) Figure 2: Plots of coherent structure profiles for supercrit ical anisotropy with µ= 7,h−Ω = −1,Ω =β/α,ccp= 0, and first integral C= 1, cf. (43). (a) A quasi-periodic solution that maps to a solution with period ≈16 in the reduced equations (44). (b) The same solution type with period 200 closer to a soliton-type solution with w avetrain as its asymptotic state. From a physical viewpoint µandαare material specific, while h,βare control parameters. In our exposition we choose µas a primary parameter and speak of super-, sub- or subsubcri tical anisotropy; one may also choose βorhat the price of less convenient conditions. Our results may be summarized somewhat informally as follow s. The up- and down-magnetization equilibria ±ˆe3.(Lemma 1) Let β±:=β/(1±ccp). The constant state m=ˆe3is strictly stable if and only if µ < h−β+/αandm=−ˆe3 if and only if µ <−(h−β−/α). Instabilities are of Hopf-type for the essential spectru m with onset via spatially homogenous modes of frequency β/α. In other words, stability of ±ˆe3changes when the difference of signed anisotropy ±µand the force balance h−β± α, of magnetic field strength minus the ratio of current-polariza tion intensity and damping factor, changes sign. This corresponds to the well known instabilit y threshold in the more broadly studied ODE for solutions that are homogeneous in space. Not ably, for ccp= 0 the anisotropy is subsubcritical precisely for −µ >|h−β/α|(‘easy-axis’), and supercritical precisely for µ >|h−β/α|(‘easy-plane’). Fast and small amplitude coherent structure. (Theorems 6, 7 and corollaries) For each sufficiently large speed there exists a family of front-type c oherent structures parametrized by theazimuthal frequency. Theirprofilesconnect ±ˆe3witheach otheror, if therearewavetrains, there are fronts connecting these and/or ±ˆe3in the order of altitudes. An example is plotted in Figure 3. Small amplitude coherent structures are of fron t type and, for ccp= 0, exist only for super- and subcritical anistropy. 3−1 −0.5 0 0.5 1−101 m1m3 −1 −0.5 0 0.5 1−101 m1m2 0 100 200 30000.511.522.53 xθ, q (a) (b) Figure 3: Profile of a ‘fast’ front connecting the wavetrain a nd the unstable −ˆe3computed with the coherent structure ODE guided by the asymptotic pre diction of equation (35). Here µ= 1,h= 0.5,s= 5,Ω = 2,ccp= 0 and the asymptotic wavetrain on the left has wavenumber k= Ω/s= 0.4, and is spectrally stable. Wavetrains. In the case ccp= 0 (Theorems 2, 4) for each wavenumber k∈Rat most one wavetrain exists, and moreover: 1.Supercritical anisotropy: Wavetrains exist precisely for kwith|k|>/radicalbig µ+|h−β/α|or 0≤ |k|</radicalbig µ−|h−β/α|. There is (explicitly known) k∗∈(0,/radicalbig µ−|h−β/α|such that all wavetrains with |k|< k∗are stable and sideband unstable for |k|> k∗. 2.Subcritical anisotropy: Wavetrain exists precisely for kwith|k|>/radicalbig µ+|h−β/α|, but are all unstable. 3.Subsubcritical anisotropy: Wavetrains exist for all k, but are all unstable. The overall picture for wavetrains of (1) with ccp= 0 can be viewed as a combination of those in a supercritical and a subcritical real Ginzburg-La ndau equation; see Figure 11. For general ccp∈(−1,1) additional effects are (1) a nontrivial nonlinear dispersi onrelation ω(k) with nonzero group velocitiesd dkω(k), (2) the occurrence of ‘hyperbolic’ and ‘elliptic’ bifurcation points of wavetrains and (3) coexistence of sta ble wavetrains and stable ±ˆe3. Wavetrains for k2> µare always unstable (Theorems 2, 3), but for ccp/ne}ationslash= 0 wavetrains are potentially convectively but not absolutely unstable, tho ugh we do not investigate this here. Domain walls for ccp= 0.(Theorem 5) For any µ <0 there exists a family of fronts whose spatial profiles connect ±ˆe3withθ′=√−µsin(θ), and that are ‘homogeneous’ in the sense thatq≡0 so there is no azimuthal profile. They corresponds to well kn own domains walls of the LLG-equation3. Here we readily locate these within the coherent structure framework. Stationary coherent structures for ccp= 0.(Theorems 8, 9) 1.Supercritical anisotropy: For fixed parameters there exist various stationary coheren t structures ( s= 0) including ‘homogeneous’ ones (cf. Figure 4). An interes ting case of 3After acceptance of the present manuscript for publication , we found these were also obtained in [12]. 4−15−10−5051015−1−0.500.51 xm1 −15−10−5051015−1−0.500.51 xm1 −15−10−5051015−1−0.500.51 xm1 −15−10−5051015−1−0.500.51 xm3 −15−10−5051015−1−0.500.51 xm3 −15−10−5051015−1−0.500.51 xm3 (a) (b) (c) Figure 4: Snapshots of sample homogeneous coherent structu res with spatially periodic pro- files, having d φ/dξ=q= 0. Compare Figure 12. Here ccp= 0,α= 1 and Ω = βis arbitrary. (a) Near a pair of domains walls ( µ=−1,h= 10−4−β). (b) Near an upward ‘phase slip soliton’ with plateaus near the oscillation at m3=h−β(µ= 1,h= 0.8−β), and (c) near an upward-downward pair of such solitons ( µ= 1,h= 0.8−β). the latter is a symmetric pair of ‘phase slip’ soliton-type c oherent structures, whose spatially asymptotic states are the same spatially homogen eous oscillation ( k= 0), but the intermediate profile crosses either ˆe3or−ˆe3, so that the asymptotic states differ azimuthally by 180◦. There also exists a non-homogeneous soliton-type solutio n with asymptotic state being a wavetrain (cf. Figure 2). 2.Sub- and subsubcritical anisotropy: All stationary coherent structures have periodic profiles except a homogeneous phase slip soliton with spatia lly asymptotic state ±ˆe3for sgn(h−β/α) =±1. Higher space dimensions. The model for Nspace dimensions has the second derivative with respect to xin (1) replaced by a Laplace operator/summationtextN j=1∂2 xj. Wavetrain type solutions are then of the form m(x,t) =m∗(k·x−ωt), wherek= (k1,...,kN). Notably, for kj= 0, 2≤j≤Nthese are solutions from one space dimension extended trivially (constant) in the additional directions. Conveniently, the rotation symmetry (gauge invariance in t he Ginzburg-Landau context), means that the analyses of ±ˆe3and these wavetrains is already covered by that of the one- dimensional case: the linearization is space-independent and therefore there is no symmetry breaking due to different kj. Indeed, all relevant quantities are rotation symmetric, d epending only onk2=/summationtextN j=1k2 jorℓ2=/summationtextN j=1ℓ2 j, whereℓ= (ℓ1,...,ℓN) is the Fourier wavenumber vec- tor of the linearization. In particular, the instabilities occur simultaneously for all directions. 5Concerning coherent structures, in higher space dimension the defining equation (see (33) below) turns into an elliptic PDE in general. The analysis in this paper only covers the trivial constant extension into higher dimensions. This paper is organized as follows. In Section 2, the terms in the model equation (1) and its well-posedness are discussed. Section 3 concerns the st ability of the trivial steady states ±ˆe3and in§4 existence and stability of wavetrains are analyzed. Secti on 5 is devoted to coherent structures. Acknowledgement. JR has been supported in part by the NDNS+ cluster of the Dutch Science Fund (NWO). We thank the anonymous reviewers for sug gestions that helped improve the manuscript, and Lars Siemer as well as Ivan Ovsyannikov f or their critical reading. 2 Review of Landau-Lifshitz-Gilbert-Slonczewski equatio ns The classical equation of dissipative magnetization dynam ics, the Landau-Lifshitz-Gilbert equation [11, 22] for unit vector fields m=m(x,t)∈S2, ∂tm=m×(α∂tm−γheff). features a damped precession of maround the effective field heff=−δE(m), i.e., minus the variational derivative of the interaction energy E=E(m). The gyromagnetic ratio γ >0 is a parameter which appears as the typical precession frequen cy. By rescaling time, one can always assume γ= 1. The Gilbert damping factor α >0 is a constant that can be interpreted dynamically as the inverse of the typical relaxation time. I t is useful to take into account that there are several equivalent forms of LLG. Elementary algeb raic manipulations taking into account that −m×m×ξ=ξ−(m·ξ)myield the so-called Landau-Lifshitz form (1+α2)∂tm=−m×(αm×heff+heff), (3) introduced in the original work [22]. In case α >0, the energy E(m) is not conserved but is a Lyapunov functional, i.e., more precisely (recall heff=−δE(m)) d dtE(m(t)) =−α/bardbl∂tm(t)/bardbl2or equivalentlyd dtE(m(t)) =−α 1+α2/bardblm×heff/bardbl2. Gilbert damping enables the magnetization to approach (spi ral down to) a steady state, i.e. satisfying m×heff= 0 (Browns equation), as t→ ∞. Spin-torque interaction. The system can be driven out of equilibrium conventionally b y an external magnetic field hwhich appears as part of the effective field. In modern spin- tronic applications, magnetic systems are excited by spin p olarized currents (with direction of polarization ˆep∈S2) giving rise to a spin torque m×m×jwherej=βˆep 1+ccpm·ˆep, (4) which has been introduced in [2, 34]. Here, the parameters β >0 andccp∈(−1,1) depend on the intensity of the current and ratio of polarization [4] . Typically we have ˆep=ˆe3. Ac- cordingly, the modified Landau-Lifshitz-Gilbert equation , also called Landau-Lifshitz-Gilbert- Slonczewski equation (LLGS), reads ∂tm=m×(α∂tm−heff+m×j). (5) 6One may extend the notion of effective field to include current i nteraction by letting Heff=heff−m×j, where the second term is usually called Slonczewski term. In this framework (5) can also be written in the form (3) with heffreplaced by Heff. Observe, however, that the Slonczewski term(andhence Heff)isingeneralnon-variational andthattheenergyisnolong eraLyapunov functional. Introducing the potential Ψ( m) =β ccpln(1+ccpm·ˆep) ofj(forccp/ne}ationslash= 0) reveals theskew variational structure m×[α∂tm+δE(m)] =−m×m×[∂tm+δΨ(m)], see[6]. Inthemicromagnetic model theunderlyinginteract ion energies areintegral functionals inmcontaining in particular exchange (Dirichlet) interactio n, dipolar stray-field interaction, crystal anisotropy and Zeeman interaction with external ma gnetic field, see e.g. [16]. In this paper we shall mainly focus on the spatially one-dimensiona l situation and consider energies of the form E(m) =1 2/integraldisplay/parenleftbig |∂xm|2+µm2 3/parenrightbig dx−/integraldisplay h·mdx. (6) Here,h∈R3is a constant applied magnetic field. The parameter µ∈Rfeatures easy plane anisotropy for µ >0andeasy axis anisotropyfor µ <0, respectively, accordingtoenergetically preferred subspaces. This term comprises crystalline and s hape anisotropy effects. Shape anisotropy typically arises from stray-field interactions which prefer magnetizations tangential to the sample boundaries. Hence µ >0 corresponds to a thin-film perpendicular to the ˆe3- axis whereas µ <0 corresponds to a thin wire parallel to the ˆe3-axis. The effective field corresponding to (6) reads heff=∂2 xm−µm3ˆe3+h. (7) With the choices h=hˆe3and ˆep=ˆe3, the Landau-Lifshitz-Gilbert-Slonczewski equation (5) exhibits the aforemented rotation symmetry about the ˆe3-axis. The presence of a spin torquem×m×jexerted by a constant current may induce switching between m agnetization states or magnetization oscillation [3, 4, 7]. For the latte r effect, the energy supply due to the electric current compensates the energy dissipation du e to damping enabling a stable oscillation, called precessional states . In applications the typical frequency is in the range of GHz, so that a precessional state would basically act as a mic rowave generator. In the class of spatially homogeneous states, precessional states are p eriodic orbits with m3=const.and of constant angular velocity β/αwhenccp= 0. It is more subtle, however, to understand the occurrence and stability of spatially non-homogeneous pre cessional states. This is the theme of this paper. Extensions and related work. There is a wealth of literature studying the dynamics of related Landau-Lifschitz models with and without damping a nd axial symmetry and including effects other than spin-torque interaction and as general ref erence we mention the book [6] as well as the review article [21]. More specifically, spatiall y non-trivial states and their stabililty have been considered in [18], where the spin-torque part of t he effective field is replaced by a demagnetization term solving Maxwell’s equation. Also co upled nano-oscillators of LLGS type have been considered widely, e.g. recently in [33, 35]. Recently, for a situation without axial symmetry, Turing patterns of spin states have been num erically found in [23]. 7Non-symmetric variants of our equation (1) have been used e. g. in the description of the field driven motion of a flat domain wall connecting antipo dal steady states m3=±1 asx1→ ±∞. A prototypical situation is the field driven motion of a flat B loch wall in an uniaxial the bulk magnet governed by ∂tm=m×/parenleftbig α∂tm−∂2 xm+µ1m1ˆe1+(µ3m3−h)ˆe3/parenrightbig . (8) In this case µ1>0> µ3, whereµ1corresponds to stray-field and µ3to crystalline anisotropy. Explicit traveling wave solutions were obtained in unpubli shed work by Walker, see e.g. [16], and reveal interesting effects such as the existence of a termi nal velocity (called Walker ve- locity) and the notion of an effective wall mass. A mathematica l account on Walker’s explicit solutions and investigations on their stability, possible extensions to finite layers and curved walls can be found e.g. in [9, 25, 30]. Observe that our axiall y symmetric model is obtained in the limit µ1ց0. On the other hand, the singular limit µ3→+∞leads to trajectories confined to the {m3= 0}plane (equator map), and can be interpreted as a thin-film lim it. In suitable parameter regimes it can be shown that the limit e quation is a dissipative wave equation governing the motion of N´ eel walls [8, 24, 26]. Well-posedness of LLGS. It is well-known that Landau-Lifshitz-Gilbert equations a nd its variants have the structure of quasilinear parabolic syste ms. In the specific case of (1), one has the extended effective field Heff=heff−m×j, more precisely Heff=∂2 xm−f(m) where f(m) = (µm3−h)ˆe3+β 1+ccpm3m׈e3.(9) Hence the corresponding Landau-Lifshitz form (3) of (1) rea ds (1+α2)∂tm=−m×/bracketleftBig ∂2 xm−f(m)/bracketrightBig −αm×m×/bracketleftBig ∂2 xm−f(m)/bracketrightBig . (10) Taking into account m×∂2 xm=∂x(m×∂xm) and −m×m×∂2 xm=∂2 xm+|∂xm|2m,(11) valid for msufficiently smooth and |m|= 1, one sees that (10) has the form ∂tm=∂x(A(m)∂xm)+B(m,∂xm) (12) with analytic functions A:R3→R3×3andB:R3×R3→R3such that A(m) is uniformly elliptic for α >0, in fact ξ·A(m)ξ=α 1+α2|ξ|2for allξ∈R3. Well-posedness results for α >0 can now be deduced from techniques based on higher order energy estimates as in [27, 28] or maximal regularity a nd interpolation as in [29]. In particular, weshallrelyonresultsconcerningperturbati onsofwavetrains, travelingwaves, and steady states. Suppose m∗=m∗(x,t) is a smooth solution of (1) with bounded derivatives up to all high orders (only sufficiently many are needed) and m0:R→S2is such that m0−m∗(·,0)∈H2(R). Then there exist T >0 and a smooth solution m:R×(0,T)→S2 of (1) such that m−m∗∈C0([0,T);H2(R))∩C1([0,T);L2(R)) with lim tց0/bardblm(t)−m0/bardblH2= 0 and lim tրT/bardblm(t)−m∗(·,t)/bardblH2=∞ifT <∞. 8The solution is unique in its class and the flow map depends smo othly on initial conditions and parameters. Given the smoothness of solutions, we may compute pointwise ∂t|m|2= 2m·∂tm, so that for|m|= 1 the cross product form of the right hand side of (10) gives ∂t|m|2= 0. Hence, the set of unit vector fields, {|m|= 1}, is an invariant manifold of (12) consisting of the solution s to (1) that we are interested in. In addition to well-posedness, also stability and spectral theory for (12), see, e.g., [29], carry over to (1). In particular, the computations of L2-spectra in the following sections are justified and yield nonlinear stability for strictly stable spectrum and nonlinear instability for the unstable (essential) spectrum. Landau-Lifshitz-Gilbert-Slonczewski versus complex Ginzburg-Landau equations. Stereographic projection of (1) yields (α+i)ζt=∂2 xζ−2¯ζ(∂xζ)2 1+|ζ|2+µ(1−|ζ|2)ζ 1+|ζ|2−(h+iβ)ζwhereζ=m1+im2 1+m3, valid for magnetizations avoiding the south pole. Studying LLG-type equations via stereographic projection has a long history and has been employed in several of the aforementioned references, see, e.g., the review [21] and the references therein.mThere is also a global connection betw een LLG and CGL in the spirit of the classical Hasimoto transformation [13], which turns the (undamped) Landau-Lifshitz equation in one space dimension ( heff=∂2 x) into the focussing cubic Schr¨ odinger equation [20, 36]. The idea is to disregard the customary coordinates representation and to introduce instead a pull-back frame on the tangent bundle along m. In the case of µ=β=h= 0, i.e. heff=∂xm, this leads to (α+i)Dtu=D2 xu (13) whereu=u(x,t) is the complex coordinate of ∂xmin the moving frame representation, andDxandDtare covariant derivatives in space and time giving rise to cu bic and quintic nonlinearities, see [27, 28] for details. 2.1 Symmetry and the variational structure for ccp= 0 The aforementioned rotation symmetry of (1) about the m3-axis of all terms manifests as an equivariance of the right hand side of (12) with respect to any such rotation Rϕ: let m=Rϕ/tildewiderm, then ∂x(A(m)∂xm)+B(m,∂xm) =Rϕ(∂x(A(/tildewiderm)∂x/tildewiderm)+B(/tildewiderm,∂x/tildewiderm)). For a rotating frame m=eiΩtm, write the rotation about the ˆe3-axis asRΩtand note that time derivatives become ∂tm=RΩt(ΩR∗ ΩtR′ Ωt/tildewiderm+∂t/tildewiderm), where R∗ ΩtR′ Ωtm=m׈e3. Therefore, having ccp= 0, (1) is also an equation for /tildewidermwith the parameter βchanged to β+αΩ(from∂tmwithin the brackets) and htoh+Ω(from∂tmon the left hand side). In other words, for ccp= 0, changing spin torque current has the same effect as changin g the magnetic field. 9Choosing Ω=−β/αyieldsβ= 0 in (1), which is therefore variational with respect to the energy (6) as discussed above. This has strong structura l consequences for the coherent structures (2) and allows for a (largely) complete characte rization. In particular, it turns out that the existence of coherent structures that are stati onary (s= 0), but not necessarily time independent, requires their superimposed azimuthal f requency Ω to satisfy Ω = β/α; see §5.3. The reduction to β= 0 thus implies Ω = 0 and therefore turns the stationary coher ent structures into standing waves and hence to time-independe nt solutions. 3 Hopf instabilities of the steady states m=±ˆe3 As a starting point and to motivate the subsequent analysis o f more complex patterns, let us consider the stability of the constant magnetizations ±ˆe3. It is well-known that a Hopf bifurcation of these states occurs in the ODE associated to ( 1) in the absence of diffusion, that is, for spatially constant solutions. In the following , we account in addition for spatial dependence. Use the shorthand β±=β/(1±ccp). Substituting m=±ˆe3+δn+o(δ), where n= (n,0)∈TmS2, into (1) gives, at order δ, the linear equation ∂tn= (±µ−h)n׈e3±ˆe3×(α∂tn−∂2 xn+β±n׈e3), which may be written in complex form as ∂tn±β±n=i/parenleftbig α∂tn−∂2 xn−(h∓µ)n/parenrightbig . Its eigenvalue problem diagonalizes in Fourier space (for x) and yields the matrix eigenvalue problem/vextendsingle/vextendsingle/vextendsingle/vextendsingle±β±−λΛ −Λ±β±−λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle= 0, where Λ = ±µ−h∓αλ∓ℓ2withℓthe Fourier wave number. The determinant reads (±β±−λ)2=−Λ2⇔ ±β±−λ=σiΛ, σ∈ {±1}. Considering real and imaginary parts this leads to (1+α2)Re(λ) =±β±−α(ℓ2±h−µ) =α(µ∓(h−β±/α)−ℓ2) Im(λ) =σ(∓Re(λ)+β±/α), so that the maximal real part has ℓ= 0. At criticality, where Re( λ) = 0 the imaginary parts are ±β±/α, which (if nonzero) corresponds to a so-called Hopf-instab ility of the (purely essential) spectrum and we expect the emergence of oscillat ing solutions whose frequency at onset isβ±/α[31]. Since the critical mode has ℓ= 0, the onset of instability coincides with the aforementioned Hopf-bifurcation of the ODE associated with diffusionless (1). The formulas for real- and imaginary parts immediately give the results mentioned in §1 and Lemma 1 The constant state m=ˆe3is (strictly) L2-stable if and only if µ < µ+:=h−β+/α andm=−ˆe3if and only if µ < µ−:=−(h−β−/α). Instabilities are of Hopf-type for the essential spectrum and have frequency β/α. Forccp= 0the anisotropy is subsubcritical precisely for −µ >|h−β/α|, and supercritical precisely for µ >|h−β/α|. 104 Wavetrains To exploit the rotation symmetry about the ˆe3-axis, we change to polar coordinates in the planar components m= (m1,m2) of the magnetization m= (m,m3). With m=rexp(iϕ) equation (1) changes to /parenleftbiggα−1 1α/parenrightbigg/parenleftbiggr2∂tϕ ∂tm3/parenrightbigg =/parenleftbigg∂x(r2∂xϕ) ∂2 xm3+|∂xm|2m3/parenrightbigg +r2/parenleftbiggβ/(1+ccpm3) h−µm3/parenrightbigg ,(14) where |∂xm|2= (∂xr)2+r2(∂xϕ)2+(∂xm3)2andr2+m2 3= 1. This can be seen as follows. In view of (3), with heffreplaced by the extended effective field Heff=heff−m×jas in (9) and taking into account (11), (1) reads α∂tm+m×∂tm=∂2 xm+(h−µm3)ˆe3+/parenleftbig |∂xm|2+µm2 3−hm3/parenrightbig m−β 1+ccpm3m׈e3. The third component of the above equation is the second compo nent of (14), whereas the first component of (14) is obtained upon inner multiplication by m⊥= (m⊥,0) = (ieiϕ,0) and taking into account that m׈e3=−m⊥. Therotation symmetryhasturnedintotheshiftsymmetry ϕ/ma√sto→ϕ+const. Infullspherical coordinates m=/parenleftbigeiϕsinθ cosθ/parenrightbig , (14) further simplifies to /parenleftbiggα−1 1α/parenrightbigg/parenleftbiggsinθ∂tϕ −∂tθ/parenrightbigg =/parenleftbigg2cosθ∂xθ∂xϕ+sinθ∂2 xϕ −∂2 xθ+sinθcosθ(∂xϕ)2/parenrightbigg +sinθ/parenleftbiggβ/(1+ccpcos(θ)) h−µcosθ/parenrightbigg .(15) 4.1 Existence of wavetrains Wavetrains are solutions of the form m(x,t) =m∗(kx−ωt), where kis referred to as the wavenumber and ωas the frequency. A natural type of wavetrains are relative e quilibria with respect tothephaseshiftsymmetry forwhich ϕ=kx−ωtandm3,rareconstant. SeeFigure1 for an illustration. Theorem 1 Wavetrains with frequency ωand wavenumber kare in one-to-one correspon- dence to solutions of Γ(ω,k) :=ccpαω(ω+h)−(β+αω)(k2−µ) = 0, under the constraint |(ω+h)/(k2−µ)| ≤1. In particular, for each kthere are at most two values of ωthat yield a wavetrain, and for each ω/ne}ationslash=−β/αthere is at most one value of k2 that gives a wavetrain, unless ccpαω(ω+h) = 0forω=−β/α. Moreover, for |ccp|<1, (a)ω/ne}ationslash= 0,k2/ne}ationslash=µ,sgn(ω) =−sgn(β)andω∈[min{−β±/α},max{−β±/α}]. (b) As|k| → ∞we have ω→ −β/αandm3→0. (c) Bifurcations of k∼0fromk= 0for fixed ωare unfolded for increasing µ. 11/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0 kΩ1.55 /Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0 kΩ1.57 Figure 5: Plots of wavetrain locations in the ( k,ω)-plane when parameters pass through an elliptic bifurcation point. Labels are the values of µ. Other parameters are fixed at ccp= 0.5,h= 2,α= 1,β= 2.1sothat ω±=−β±,β+= 1.4,β−= 4.2andµ+= 0.6,µ−= 2.2. Therefore, µ= 1.55,1.57 are both subcritical with ˆe3stable and −ˆe3unstable. Shaded regions have|m3|>1. The elliptic point lies at µsn≈1.558,ωsn=−2.558. (d) Bifurcations for m/ne}ationslash=±ˆe3occur atω=ωsn:=β±√ β(β−4αh) 2αwith|m3|<1if|1 ccp−h µ|<2 and are either: ... a hyperbolic point for k/ne}ationslash= 0, ifh=β/αand then ω=−h,k2=−ccph+µ, ... a hyperbolic point for k= 0, ifsgn((ccph−µ)(ccph+µ)) = 1, ... an elliptic point for k= 0, ifsgn((ccph−µ)(ccph+µ)) =−1. In the following we discuss the existence problem and thereb y prove each statement of the theorem. The terms elliptic and hyperbolic refers to the use in [32] an d will be explained below. Note that the latter two bifurcation types do not occur for ccp= 0. That case is considered in detail in §4.2. We note that Γ depends on konly through k2−µso thatµ≥0 is the same asµ= 0 up to change in wavenumber and solutions for fixed ωcan increase |k|only through increasing µ. Substituting the wavetrain ansatz into (15) yields the alge braic equations /parenleftbiggα−1 1α/parenrightbigg/parenleftbigg−sin(θ)ω 0/parenrightbigg =/parenleftbigg0 sin(θ)cos(θ)k2/parenrightbigg +sin(θ)/parenleftbiggβ/(1+ccpcos(θ)) h−µcos(θ)/parenrightbigg . Thus either θ≡0 modπor (recall |ccp|<1) −αω=β 1+ccpm3,−ω= (k2−µ)m3+h. (16) In the first case we have r= 0, which corresponds to the constant upward or downward magnetizations, ( r,m3) = (0,±1) with unspecified kandω. In the second case, we notice aside that absence of dissipation ( α= 0) requires absence of current ( β= 0) and there is a two-parameter set of wavetrains given by the second equatio n. The case we are interested in isα >0 and then β= 0 requires ω= 0, and this falls into the special case ccp= 0. 12/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0 kΩ0.55 /Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0 kΩ0.64 /Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0 kΩ0.65 Figure 6: Analogue of Figure 5 with fixed parameters as there, whenµpasses through a hyperbolic bifurcation point at k= 0 with µsn≈0.641,ωsn=−1.641. Note that between µ= 0.55 andµ= 0.64 the upper branch enters the region |m3| ≤1 at a bifurcation of in this caseˆe3atµ=µ+= 0.6. Therefore, µ= 0.64,0.65 are supercritical with ±ˆe3both unstable. /Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5 kΩ1.99 /Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5 kΩ2 /Minus2/Minus1012/Minus3.5/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0/Minus0.5 kΩ2.001 Figure 7: Analogue of Figure 5 when parameters pass through h yperbolic bifurcation points withk/ne}ationslash= 0 so that h=β/α. Notably this cannot be unfolded by variation of µ. Labels are the values of βand other parameters are fixed at ccp= 0.5,h= 2,α= 1,µ= 1.2. This is subcritical with ˆe3unstable since µ+= 2(1−β/3),µ−= 2(β−1) and 1.2∈(µ+,µ−). In the generic case β,ccp/ne}ationslash= 0, the first equation implies that ω≈0 is not possible for |m3| ≤1 and we obtain m3=−1 ccp/parenleftbiggβ αω+1/parenrightbigg , m 3=−ω+h k2−µ(17) where for k2=µwe have ω=−hand the first equation holds. Eliminating m3and rearranging terms gives the existence condition in term s ofωandk as zeros of Γ( ω,k) as in the theorem. For ω/ne}ationslash=−β/αthis gives k2as a quadratic function ofωinverse to the nonlinear dispersion relation ω(k). The exceptional ω=−β/αoccurs precisely when h=β/αand implies m3= 0, i.e. a solution on the equator (or ccp= 0). The strict monotonicity of Γ in kaway from k= 0 also means that upon parameter change new solution branches can emerge only through local extrema of Γ atk= 0, i.e., an ‘elliptic’ point. Specifically, this occurs if at a critical point ∂2 kΓ(ω,0) = 2(β+αω) has the same sign 13as∂2 ωΓ(ω,0) = 2αccpand, e.g. Γ(0 ,0) =βµvaries. In particular, Γ( ω,k) =∂ωΓ(ω,k) = 0 occurs at αω2+β(2ω+h) = 0 (18) and is a fold point of wavetrains (in the form of homogeneous o scillations) with fixed k. For k= 0 these have at frequency and parameters (recall ccp/ne}ationslash= 0) ωsn=−ccph+µ 2ccp,4βµccp=α(µ+ccph)2. (19) Note that |m3|<1 fork= 0 is by (17) equivalent to |ωsn+h|<|µ|which yields |1 ccp−h µ|<2. At such critical point we also have ∂2 kΓ(ω,0) =−α ccp(ccph−µ)(ccph+µ), (20) so that the relative size of ccphandµdetermines whether such a bifurcation point is elliptic or hyperbolic in the language of [31]. In terms of critical pa rameters, substituting ω=ωsn+˜ω and, for instance µ=µsn+ ˜µto unfold with µand other parameters fixed we obtain ˜µ=k2−ccpα β+αωsn˜ω2(21) which gives the options of hyperbola or ellipses for level se ts. Hyperbolic points are saddle points of Γ and at such points the connectivity of existing br anches changes. For k/ne}ationslash= 0 this occurs in particular, if h=β/αwhen the two branches of Γ = 0 are the line ω=−hfor any kandω=1 ccp(k2−µ). We plot examples of these situations in Figures 5, 6, 7. More globally, since Γ is quadratic in ωthere are at most two solutions for each kand by strict monotonicity in k2, away from h=β/α, there is at most one solution for each ωor the whole line ω=−β/α. The only complication is the constraint |m3| ≤1 – dispersion curves touch the boundary m3= 1 at bifurcations of ±ˆe3, which were studied in §3. The figures illustrate the essential scenarios. 4.2 Existence in the case ccp= 0 In this case the existence conditions can be conveniently wr itten as ω=−β α(22) cos(θ) =h−β/α µ−k2,(µ/ne}ationslash=k2). (23) As expected from the variational structure in rotating coor dinates discussed in §2.1, all wave- trains oscillate with frequency given by the ratio of applie d current and dissipation. In par- ticular, in this case the natural representation of wavetra ins is that ( θ,k)-plane rather than (ω,k) as above. An involution symmetry involving parameters is (h−β/α,θ)→(β/α−h,θ+π), (24) 14/Minus4/Minus20240Π2Π kΘ /Minus4/Minus20240Π2Π kΘ /Minus4/Minus20240Π2Π kΘ (a) (b) (c) Figure 8: Plots of equilibrium locations in the ( k,m3)-plane including the trivial equilibria ±ˆe3plotted with thick line if stable (when not intersecting wav etrain parameters). Compare Figure 9. See (23). (a) supercritical (easy plane) anisotro py (µ= 1,h−β/α= 1/2), (b) subcritical anisotropy ( µ= 1,h−β/α= 2), where no homogeneous oscillations ( k= 0) exist, and (c) subsubcritical (easy axis) anisotropy µ=−1,h−β/α= 0.9. supercritical subcritical subsubcritical−|h−β/α||h−β/α| σˆe3stable±ˆe3unstable ±ˆe3stableµ k Figure 9: Sketch of existence region (shaded) in the ( k,µ)-plane, with boundary given by (25) forσ:= sgn(h−β/α)/ne}ationslash= 0. The sign of σdetermines which of ±ˆe3is stable in the subcritical range. so that the sign of h−β/αis irrelevant for the qualitative picture. Solvability of (23) requires that |µ−k2|>|h−β/α|(unlessr= 0), so that only for super- and subsubcritical anisotropy, |µ|>|h−β/α|, (25) there exist wavetrains with wavenumber in an interval aroun dk= 0. In other words, non- trivial spatially nearly homogeneous oscillations requir e sufficiently small (in absolute value) differencebetweenappliedmagneticfieldandoscillationfre quency(ratioofappliedcurrentand dissipation). Thetransitionintothisregimegoesviathe‘ Hopf’instability from §3. Combining (25) with Lemma 1 and straightforward analysis of (23) gives the following lemma. The three types of solution sets are plotted in Figure 8. Clearly, the s olution sets are symmetric with respect to the signs of kandθ, respectively. Theorem 2 There are three types of wavetrain parameter sets solving (23): 151. For supercritical anisotropy there is one connected comp onent of wavetrain parameters including k= 0, and two connected components with unbounded |k|, each with constant sign ofk. 2. For subcritical anisotropy there are two connected compo nents with unbounded |k|, each with constant sign of k. 3. For subsubcritical anisotropy there are two connected co mponents, each a graph over the k-axis. The Hopf-type instabilities of ±ˆe3noted in §3 at the transition from sub- to supercritical anisotropy is a supercritical bifurcation in the sense that solutions emerge at the loss of stability of the basic solution, here ±ˆe3, while that from sub- to subsubcritical is subcritical in th e sense that solutions emerge at the gain of stability. Proof. Let us consider the existence region of wavetrains in wavenu mber-parameter space. From (23), r= 0 atθ≡0 modπgives the boundary for nontrivial amplitude, µ=k2±(h−β/α), (26) as a pair quadratic parabolas in ( k,µ)-space. The solution set in this projection is sketched in Figure9. Remark that this set is non-empty for any parameter setα,β,h∈Rof (1). However, as in the general case not all wavenumbers are possible due to the geometric constraint. Notably, the existence region consists of two disjoint sets , one contained in {µ >0}with convex boundary and one extending into {µ <0}with concave boundary. 4.3 Stability of wavetrains In this section we discuss spectral stability of wavetrains and in summary we obtain the following result. Theorem 3 (a) Wavetrains bifurcating from ±ˆe3atk= 0are stable if µ±>max{0,ccpβ±/α}, which implies supercritical bifurcation, i.e., for increasing µ. They are unstable if µ±< max{0,ccpβ±/α}, which means subcritical bifurcation, i.e., decreasing µ. (b) Forµ >0there isk∗>0such that precisely the wavetrains with wavenumber |k|< k∗ andµ > k2+αω2ccp/βare sideband stable. These are fully spectrally stable if βccp≥0 orα≥0is sufficiently small. A wavetrain and ˆe3or−ˆe3can be simultaneously stable only for ccp/ne}ationslash= 0. (c) For each kat most one wavetrain can be stable. Wavetrains with r∼0andk/ne}ationslash= 0are unstable. All wavetrains with k2> µare unstable. (d) Near the hyperbolic or elliptic bifurcation points at k= 0the sideband stable wavetrains lie in a sector that is to leading order bounded by |ω−ωsn|=S|k|whose opening angle less than πand which includes ωwith a selected sign of ω−ωsn. 16/Minus2/Minus1012/Minus3.0/Minus2.5/Minus2.0/Minus1.5/Minus1.0 kΩ1.57 /Minus1.0/Minus0.50.00.51.0/Minus2.1/Minus2.0/Minus1.9/Minus1.8/Minus1.7/Minus1.6/Minus1.5/Minus1.4 kΩ0.64 /Minus1.0/Minus0.50.00.51.0/Minus2.4/Minus2.2/Minus2.0/Minus1.8/Minus1.6 kΩ1.99 /Minus1.0/Minus0.50.00.51.0/Minus2.10/Minus2.05/Minus2.00/Minus1.95/Minus1.90 kΩ2.001 Figure 10: Dispersion curves and stability regions. Wavetr ains on the dispersion curves in the green shaded region are unstable due to unstable eigenvalue ofA(0,0) using (29), and in the red shaded region due to unstable sideband only, using ˜λ′′ 0(0) with k2from Γ = 0. Parameters are as in (a): Figure 5; (b): Figure 6; (c),(d): Figure 7. Only the wavetrains near k= 0 are stable (except for the lower branch in (b)). Remark 1 Item (a) should be compared with the Hopf instabilities discu ssed in§3. The case ccp= 0is simplest: the constraint simply means that wavetrains bi furcating at the subsub- to subcritical transition are unstable. For general ccpthe condition also accounts for interaction with folds. Concerning item (b), notably in the easy axis case µ <0all wavetrains are unstable. The condition ccpβ≥0is not sharp. However, we do not know whether a ‘Hopf’ instabi lity can occur for otherwise stable wavetrains if ccpβ <0is sufficiently large negative. Item (d) is an analog to the results in [32]. In order to study spectral stability of wavetrains we consid er the comoving frame y= x−cphtwithcph=ω/kthe wavespeed so that the wavetrain is an equilibrium of (14) . For convenience, time is rescaled to t= (1+α2)˜t. The explicit formulation of (14) then reads ∂˜t/parenleftbiggϕ m3/parenrightbigg =/parenleftbiggα(∂y(r2∂yϕ)/r2+˜β(m3))+(∂2 ym3+|∂ym|2m3)/r2+h−µm3+cph∂yϕ α(∂2 ym3+|∂ym|2m3+r2(h−µm3))−∂y(r2∂yϕ)−r2˜β(m3)+cph∂ym3/parenrightbigg , (27) wherer2= 1−m2 3and˜β(m3) :=β/(1+ccpm3). LetF= (F1,F2)tdenote the right hand side of (27). Wavetrains have constant randm3 so that quadratic terms in their derivatives can be discarde d for the linearization LofFin a wavetrain, and from |∂ym|2onlyr2(∂yϕ)2is relevant. The components of Lare ∂ϕF1=α∂2 y+cph∂y+2km3∂y ∂m3F1=r−2∂2 y+k2−µ−2αkm3r−2∂y+α˜β′(m3) ∂ϕF2= 2αkm3r2∂y−r2∂2 y ∂m3F2=α∂2 y+αr2(k2−µ)+cph∂y−2αm3(m3k2+h−µm3)+2m3(k∂y+˜β(m3)) −r2˜β′(m3) =α∂2 y+αr2(k2−µ)+cph∂y+2m3k∂y−r2˜β′(m3), where the last equation is due to (17) and ˜β′(m3) =−ccpβ/(1+ccpm3)2. Since all coefficients are constant, the eigenvalue problem Lu=λu 17is solved by the characteristic equation arising from an exp onential ansatz u= exp(νy)u0, which yields the matrix A(ν,cph) :=/parenleftbiggαν2+(cph+2km3)ν−r−2ν(−ν+2αkm3)+k2−µ+α˜β′(m3) r2ν(−ν+2αkm3)αν2+(cph+2km3)ν+αr2(k2−µ)−r2˜β′(m3)/parenrightbigg . The characteristic equation then reads dcph(λ,ν) :=|A(ν,cph)−λ|=|A(ν,0)−(λ−νcph)|=d0(λ−cphν,ν) d0(λ,ν) =λ2−t(ν)λ+d(ν) (28) with trace and determinant of A(ν,0) t(ν) := trA(ν,0) = 2ν(αν+2km3)+αr2(k2−µ)−r2˜β′(m3) d(ν) := detA(ν,0) = (1+ α2)ν/parenleftBig ν(ν2+4k2m2 3+r2(k2−µ))−2r2˜β′(m3)km3/parenrightBig = (1+α2)ν/parenleftBigg ν/parenleftBigg ν2+(3k2+µ)(˜β(m3)/α−h)2 (k2−µ)2+k2−µ/parenrightBigg −2r2˜β′(m3)km3/parenrightBigg . In the last equation (23) was used. The characteristic equation is also referred to as the complex (linear) dispersion relation . Thespectrum of L, for instance in L2(R), consists of solutions for ν= iℓand is purely essential spectrum (in the sense that λ−Lis not a Fredholm operator with index zero). Indeed, setting ν= iℓcorresponds to Fourier transforming in ywith Fouriermode ℓ. Note that the solution d(0,0) = 0 stems from spatial translation symmetry in y. For the same reason the real part of solutions λof (28) for any given ν∈iRdoes not depend on cph, which means that spectral stability is independent of cphand is therefore completely determined by d0(λ,iℓ) = 0. As a first observation concerning stability we note that at r= 0,|m3|= 1 the solutions λ± tod0(λ,iℓ) = 0 have Re( λ±) =−ℓ(ℓ±2k) whose maximum is k2. Hence, wavetrains at (and thus near) r= 0 are unstable for k/ne}ationslash= 0. Fold stability. First note the eigenvalues A(0,0) are 0 and τ(k2) :=t(0) =r2(α(k2−µ)−˜β′(m3)) so that a ‘fold instability’ occurs precisely for r= 0 (compare §3) orα(k2−µ) =˜β′(m3). We readily compute that the latter corresponds to the critic al points in (18). Note that ˜β′(m3) =−α2ω2ccp/β. In particular, for the loops of wavetrains emerging from an elliptic point, the upper and lower ω-values have opposite signs of τ(0); specifically the lower is stable ifβccp>0, cf. Figure 5. At the bifurcation points of ±ˆe3(see§3) where r= 0, we have µ=µ±andω=−β±/α which yields Theorem 3(a) except for the super-/subcritica lity. To see this recall that ±ˆe3 destabilize always through increasing µ. Whether wavetrains with k= 0 emerge from ±ˆe3 depends on the sign of ∂µm3. From (16) we find µ=h m3−β αm3(1+ccpm3)and compute ∂m3µ(±1) =±τ(0) αr2/vextendsingle/vextendsingle/vextendsingle/vextendsingle m3=±1. 18Hence, fold-stability implies ∂µm3(µ+)<0 or∂µm3(µ−)>0, respectively, so that increasing µyields|m3|<1 and thus emergence of solutions. More generally, sign changes of τ(k2) correspond to fold points and a curve of spectrum crosses the origin. We plot some fold stability boundaries i n Figure 10. Using τ(k2) we have that wavetrains with µ < k2+ccpα βω2are unstable. In particular, for ccpβ≥0 all wavetrains withk2> µare unstable. Using the existence condition we may also writ e this condition independent of kas ω(β(2ω+h)+αω2) β(β+αω)<0, (29) which explains the changes in the fold stability indicator a tω=−β/αin the figure. Coming back to k= 0, at the bifurcation points of ±ˆe3(see§3) where r= 0, we have µ=µ±. Fold-stability is then µ > αc cpω2/βwhich holds if ±/parenleftbigg h−β± α/parenrightbigg >βccp α(1±ccp)2⇔ ±h >β α2ccp±1 (1±ccp)2, as noted in Theorem 3 item (a). We next check the other possible marginal stability configur ations case by case. Sideband instability. A sideband instability occurs when the curvature of the curv e of essential spectrum attached to the origin changes sign so th at the essential spectrum extends into positive real parts. Let ˜λ0(ℓ) denote the curve of spectrum of A(iℓ,0) attached to the origin, that is ˜λ0(0) = 0, and let′denote the differentiation with respect to ℓ. Derivatives of ˜λ0can be computed by implicit differentiation of d0(λ,iℓ) =λ2−t(iℓ)+d(iℓ) = 0. This gives ˜λ′ 0(0) = id′(0) t(0)=−i˜β′(m3)2(1+α2)km3 α(k2−µ)−˜β′(m3)therefore ˜λ′′ 0(0) =−2d′(d′−t′t)+d′′t2 t3/vextendsingle/vextendsingle/vextendsingle/vextendsingle ℓ=0=: Λ(k2)2(1+α2)r4 τ3(k2), where the ω-dependence is suppressed. Some calculations yield Λ as a cu bic polynomial in K=k2given by Λ = a3K3+a2K2+a1K+a0with a3=−α2(4−3r2), a2=α(2˜β′r2+αµ(8−5r2)), a1=−α2µ2(4−r2)−4(1−r2)α2˜β′(m3)2−(˜β′(m3)2+4α˜β′(m3)µ)r2, a0=µ(˜β′(m3)+αµ)2r2. Notably, a0=µτ(0)2and also a3<0 sincer∈(0,1). A wavetrain with wavenumber kis therefore (strictly) sideband stable precisely when Λ( k2)τ(k2)<0 andsideband unstable for Λ(k2)τ(k2)>0. As mentioned above, τ(k2)>0 for large enough k2so that all such (already unstable) wavetrains are also sideband unstable as a3<0 holds always. Sinceτ(k2)>0 is the unstable fold case, we next assume τ(k2)<0 so that sideband stability is precisely Λ( k2)>0. 19Homogeneous oscillations. Fork= 0 we obtain ˜λ′′ 0(0) = Λ(0)2(1+α2)r4 τ3(0)=−2(1+α2)µ τ(0), so thatµ >0 is required for sideband stability (given fold stability τ(0)<0). Let us study this situation near the Hopf instability of ±ˆe3, whereµ≈µ±=±(h−β±/α) withµ > µ±andω=β±/α. Hence, sideband stability of the emerging wavetrains requ ires h > β+/αorh < β−/α, that is, µ±>0. Compare Theorem 3(a). Near homogeneous. To leading order Λ( k2) = 0 isa1k2+a0=O(k4) and via (21) we have a1=a1(˜ω,k2),a0=a0(˜ω,k2), where at bifurcation a0(0) =∂ωa0(0) = 0. Upon expanding we therefore find sideband instabilities to leading order at ˜ω2=a1(0)−∂k2a0(0) ∂˜ω2a0(0)k2+O(|k|3+|˜ω|3), where∂˜ω2a0(0) =µsnr2 sn/parenleftBig 2 βα2ωsnccp/parenrightBig2 and∂k2a0(0) =αµsnr2 snwithrsn= 1−/parenleftBig ωsn+h µ/parenrightBig2 ther-coordinate of the wavetrain at the bifurcation point. Some algebra yields a1(0) = −4α(1+α2)(h+ωsn)2. In accordance with the results in [31], this means that wavet rains in a sector in the ( ω,k)- plane near the fold point are sideband stable, while wavetra ins outside this sector are sideband unstable. We expect that the opening angle of this sector can be changed while keeping the dispersion curves essentially fixed. Here we do not pursu e this further, but note that since sideband instabilities do occur and the stable region cannot include the fold points, the prefactor of k2is positive and ˜ ωhas a selected sign. The sideband boundaries for some examples are plotted in Figure 10. General wavetrains. Concerning the sign of Λ in general, Λ(0) = a0has the sign of µand thus all wavetrains for µ <0 are sideband unstable. For µ >0 we have Λ( µ) = −4α2(˜β′(m3))2m2 3µ <0, so a sign change occurs at some Ksb∈(0,µ), which implies sideband instability for k2≥µ. Moreover, µ >0 andccpβ≤0 imply a1<0 anda2>0. Since a3<0 this means both roots of Λ′, (2a2±/radicalbig 4(a2 2−3a3a1))/(6a3) lie at negative Kand so Λ is monotone decreasing for all K >0. Therefore Ksbis the unique sideband instability in this case. On the other hand, for a1>0 the roots of Λ′have opposite signs so that due to the sign change in the interval (0 ,µ) the positive one must be a local maximum so that also in this caseKsbis the unique sideband instability. Moreover, in all cases w avetrains with k2> µare unstable since Λ <0 in this range. Hopf instability. A Hopf instability occurs when the essential spectrum touch es the imag- inary axis at nonzero values. In particular, there is γ/ne}ationslash= 0 so that d0(iγ,iℓ) = 0. At k= 0 we have Im( d0(iγ,iℓ)) =γ(2αℓ2−τ(0)) so that in the fold stable case τ(0)<0 there is real ℓ forγ/ne}ationslash= 0. Recall τ(0) =−(˜β′(m3)+αµ)r2. Therefore, there is no (relevant) Hopf instability neark= 0. More generally, solving Im( d0(iγ,iℓ)) forγand substituting the result into Re( d0(iγ,iℓ)) we obtain up to a factor (1+ α2)ℓ2 ℓ2+(µ−k2)r2+4α2m2 3k2G(ℓ2), G(L) =−4L2−4L(k2−µ)r2+(˜β′(m3)2+(k2−µ)2)r4 (2αL−τ(k2))2 20/Minus4/Minus20240Π2Π kΘsupercritical subcritical subsubcriticalµ k (a) (b) Figure 11: Analogues of Figures 8(a) and 9 with stable range o f wavetrains in (a) bold line, and in (b) the dark shaded region. where all terms except possibly G(L) are positive in the interesting range µ > k2. Note that for sufficiently small α >0 there is no root besides ℓ= 0 and thus no Hopf and in fact no sideband instability. But there seems to be no satisfying ex plicit bound. (While the same seems to occur for m3∼0, such wavetrains have k2> µand are thus unstable.) However, Gis nondecreasing for ˜β′(m3)≤0, i.e.,ccpβ≥0, since then G′(L) =−4˜β′(m3)r22L+(µ−k2−α˜β′(m3))r4 (2αL−τ(k2))3≥0, in the fold stable case τ(k2)≤0. Thus, besides ℓ=γ= 0, there is at most one solution d0(iγ,iℓ) = 0 for ccpβ≥0, which rules out a Hopf instability as this requires two suc h solutions. We do not know whether or not Hopf instabilities c an occur for general αand ccpβ <0. Turing instability. A Turing instability occurs when the spectrum touches the or igin for nonzero ℓ, that is, there is ℓ/ne}ationslash= 0 so that d0(0,iℓ) = 0, which means det A(iℓ,0) = 0. Since Im(detA(iℓ,0)) =−2(1 +α2)˜β′(m3)ℓkm3r2and our previous considerations already cover zeros of this, such instabilities do not occur. This exhausts the list of possible marginal stability. 4.4 Stability of wavetrains for ccp= 0 Forccp= 0 the wavetrain frequency ωis independent of the wavenumber kin (22) so that the phase velocity ω/kof all wavetrains is −β/(αk) and the group velocity d ω/dk, which describes the motion of perturbations by localized wave pac kets, vanishes for all wavetrains. Due to Theorem 3 destabilizations of stable wavetrains can o nly occur through a unique sideband instability |k|=k∗>0. This value can be explicitly determined since d′(0) = 0 and 21d′′(0) =−2(1+α2)((3K+µ)r2−4K) so that ˜λ′′ 0(0) = 0 at k2 ∗=µr2 4−3r2. Taking into account that k2=µfor a wavetrain can occur only if h=β/αwe thus have Theorem 4 Consider ccp= 0. All wavetrains whose wavenumber ksatisfies |k|> k∗are unstable. For µ >0wavetrains with wavenumber |k|< k∗are spectrally stable, while those with|k|> k∗are unstable. In case h/ne}ationslash=β/αa sideband instability occurs at k=±k∗. There is no secondary instability for k2< µ. Nontrivial spectrally stable wavetrains exist only for supercritical anisotropy, |h−β/α|< µ. The overall picture for wavetrains of (1) with ccp= 0 is thus a combination of the scenarios from a supercritical and a subcritical real Ginzburg-Landa u equation ∂tA=∂2 xA+ ˜µA∓ A|A|2, A(x,t)∈C, which describes the dynamics near pattern forming Turing i nstabilities and possesses the gauge-symmetry A→eiϕA. The interested reader is referred to the review [1] and the references therein. 5 Coherent structures The coexistence of wavetrains and constant magnetizations raises the question how these interact. In this section we study solutions that have spati ally varying local wave number. In particular, we consider solutions that spatially connect w avetrains or ±ˆe3in a coherent way. In order to locate such solutions induced by symmetry we make the ansatz ξ=x−st ϕ=φ(ξ)+Ωt θ=θ(ξ),(30) with constant s,Ω. Solutions of (14) of this form are generalized travelling waves to (1) with speed sthat have a superimposed oscillation about ˆe3with frequency Ω. This ansatz is completely analogous to that used in the aforementioned s tudies of the real and complex Ginzburg-Landau equations [1]. Let¯β(θ) :=˜β(cos(θ)) =β/(1 +ccpcos(θ)) denote the ( β,ccp)-dependent term of (15). Substituting ansatz (30) into (15) with′= d/dξandq=φ′gives, after division by sin( θ), the ODEs /parenleftbiggα−1 1α/parenrightbigg/parenleftbiggsin(θ)(Ω−sq) sθ′/parenrightbigg =/parenleftbigg2cos(θ)θ′q+sin(θ)q′ −θ′′+sin(θ)cos(θ)q2/parenrightbigg +sin(θ)/parenleftbigg¯β(θ) h−µcos(θ)/parenrightbigg ,(31) on the cylinder ( θ,q)∈S1×R, which is the same as {(m3,r,q)∈R3:m2 3+r2= 1}. Wavetrains. Steady states with vanishing ξ-derivative of θandqhaveϕ=q(x−st)+Ωt and thus correspond to the wavetrains discussed in §4 with wavenumber k=qand frequency ω=sq−Ω. Hence, the ansatz (30) removes all wavetrains whose waven umber and frequency do not lie on the line {ω=sk−Ω}in (ω,k)-space. 22We may visualize this by drawing the line ω=sk−Ω into the wavetrain existence and stability plots in the ( ω,k)-plane such as Figure 10. Specifically, for s/ne}ationslash= 0, steady states of (31) with m3/ne}ationslash=±1 are wavetrains with wavenumber qfor which there exists θsuch that q=Ω−¯β(θ)/α s. (32) In particular, for ccp= 0 we have constant ¯β(θ) =βso that equilibria of (31) (other than ±ˆe3) have uniquely selected wavenumber q. Hence, heteroclinic solutions to (31) for ccp= 0 can only be domain walls connecting ±ˆe3or connect one of ±ˆe3to a wavetrain. Coherent structure ODEs. Writing (31) as an explicit ODE gives θ′=p p′= sin(θ)/parenleftbig h+(q2−µ)cos(θ)−(Ω−sq)/parenrightbig −αsp q′=α(Ω−sq)−β/(1+ccpcos(θ))−s+2cos(θ)q sin(θ)p,(33) whose study is the subject of the following sections. For lat er use we also note the ‘desingu- larization’ by the (singular) coordinate change p= sin(θ)˜pso that ˜p′=p′/sin(θ)−˜p2cos(θ), which gives θ′= sin(θ)˜p ˜p′=h+(q2−µ)cos(θ)−(Ω−sq)−αs˜p−cos(θ)˜p2 q′=α(Ω−sq)−β/(1+ccpcos(θ))−(s+2cos(θ)q)˜p.(34) Hence, (33) is equivalent to (34) except at zeros of sin( θ). Inparticular, for ˜ p= 0the equilibria of (34) with sin( θ)/ne}ationslash= 0 are those of (33), but θ=nπ,n∈Zare invariant subspaces which may contain equilibria with ˜ p/ne}ationslash= 0. Next, we first consider various moving heteroclinic coheren t structures with s/ne}ationslash= 0 and for ccp= 0 give a complete analysis of stationary coherent structur es (s= 0). Coherent structures also emerge near the elliptic and hyperbolic wavetrain bifu rcations that arise from fold points forccp/ne}ationslash= 0. However, the detailed analysis of this case is beyond the scope of this paper. 5.1 Homogeneous domain walls Classical domain walls connect antipodal equilibria at x=±∞. For the model equation (8) explicit (Walker) solutions are known to exist below a criti cal fieldh. These solutions exhibit a tilting of the azimuthal angle ϕ=const.in order to balance precessional forces. An analogue situation arises in our context when q=q′= 0. In this case we have ϕ= Ωtand therefore no spatially varying azimuthal profile.4 Theorem 5 Non-equilibrium coherent structure solutions with q≡0fors/ne}ationslash= 0and|ccp|<1 exist for µ <0andccp= 0orβ= 0only, and have Ω =h+αβ 1+α2,s2=−(β−αh)2 µ(1+α2)2. They are oscillating heteroclinic fronts connecting ±ˆe3that solve θ′=σ√−µsin(θ)whereσ= sgn(s(αh−β)). The family of such fronts is smooth and extends to s= 0, whereh= Ω =β/α, and fronts exist for both signs of σ. 4After acceptance of the present manuscript for publication , we found that the sufficiency of ccp= 0 for existence of such domains walls in Theorem 5 is contained in [ 12]. 23Proof.Recall¯β(θ) =β/(1+ccpcos(θ)) sod dξ¯β(θ) =ccpβsin(θ)θ′/(1+ccpcos(θ))2. Suppose a solution ( θ,p,q) to (33) has q≡0, so also q′≡0. Then the third equation of (33) fors/ne}ationslash= 0 yields, using the first equation, θ′=αΩ−¯β(θ) ssin(θ). Differentiation gives θ′′=αΩ−¯β(θ) scos(θ)θ′−ccpβsin2(θ) s(1+ccpcos(θ))2θ′ = sin(θ)αΩ−¯β(θ) s2/parenleftbigg (αΩ−¯β(θ))cos(θ)−ccpβ(1−cos2(θ)) (1+ccpcos(θ))2)/parenrightbigg . On the other hand, the second equation of (33) requires θ′′= sin(θ)(h−µcos(θ)−Ω−α(αΩ−¯β(θ))). First consider ccp= 0 orβ= 0 so that ¯β(θ) =βand these two right hand sides for θ′′simplify. Equating them and comparing the coefficients of cos( θ)j,j= 0,1, yields h= Ω+α(αΩ−β) andµ=−/parenleftBig αΩ−β s/parenrightBig2 , which means ( αΩ−β)/s=σ√−µforσ= sgn(s(αΩ−β)). Taken together, the parameter conditions can be equivalently cas t as the equations for Ω, s2andσ in the theorem statement. Hence, for ccp= 0 orβ= 0 these parameter choices and θ′=σ√−µsin(θ) are necessary conditions for q≡0. As a scalar equation, the only non-trivial and bounded sol utions are heteroclinic orbits between equilibria. Taking Ω = β/α+s˜µfor some ˜ µ/ne}ationslash= 0 gives a smooth parametrization up to s= 0 and σ= sgn(˜µ). Conversely, for ccp= 0 orβ= 0 and these choices of parameters, any ( θ,p,q)(ξ) where θ(ξ) satisfies θ′=σ√−µsin(θ),p=θ′andq≡0 is a solution to (33). It remains to show that if β/ne}ationslash= 0 then ccp= 0 is necessary for a non-trivial solution withq≡0: Subtracting the two right hand sides for θ′′from above and multiplication with (1+ccpcos(θ))3gives a polynomial in cos( θ) of degree four. A straightforward computation shows that the 4th order term to vanish requires µ=−α2Ω2/s2. Using this the coefficients ajof cos(θ)j,j= 0,1,2,3, in this polynomial can be computed as a0=−h+(1+α2)Ω−αβ+(β−αΩ)βccp/s2, a1= 3ccp((1+α2)Ω−h)+β2/s2−αβ(2ccp+(2+c2 cp)Ω/s2), a2=c2 cp(3((1+α2)Ω−h)−αβ)−3αβΩccp/s2, a3=c3 cp((1+α2)Ω−h)−αβΩc2 cp/s2. We first solve a0= 0 trivially for hand proceed with somewhat tedious, but straightforward calculations: substituting this hintoa1= 0 (which is linear in Ω) we solve for Ω, which uses |ccp|<1. Substituting the resulting h,Ω gives a2=ccpβ 2s2/parenleftbig 3β(c2 cp−1)+αccps2/parenrightbig , so that for a2= 0 either ccp= 0 (since β/ne}ationslash= 0; note that then also a3= 0) ors2= 3βc2 cp−1 αccp. In the latter case, substituting the previous h,Ω and this s2intoa3would give a3=αβc3 cp/3/ne}ationslash= 0. Hence,ccp= 0 is necessary as claimed. 24Remark 2 1. Fors= 0further coherent structure solutions exist, but not as doma in walls. See Theorem 8 below. 2. In§5.2.1 we find fast domain walls and fronts that have non-trivi alq. 3. Numerical simulations suggest that these domain walls ar e dynamically stable solutions in the subsubcritical case. They are unstable in the subcriti cal case |h−β/α|>−µ, which occurs for large |s|, since then either ˆe3or−ˆe3is unstable. 4. The profile of these domain walls depends only on the paramet erµ. In particular, the subfamily parameterized by hhas arbitrarily large speed but constant shape, though the oscillation frequency Ωdepends on h. 5.2 Moving front-type coherent structures Using the desingularized system (34), we prove existence of some non-stationary coherent structures of front-type, spatially connecting wavetrain s or±ˆe3. 5.2.1 Near the fast limit |s| ≫1 Theorem 6 For any bounded set of (α,β,µ,Ω0,Ω1)andccp∈(−1,1)there exists s0>0and neighborhood UofM0:={θ∈[0,π],˜p=q−Ω1= 0}such that for all |s| ≥s0the following holds for (34)withΩ = Ω 0+Ω1s. The heteroclinic orbits of (34)inUform a smooth family in the parameter s−1for each sign of s, which reverses their orientation. These heteroclinics and are in one-to-one correspondence with those of the ODE d dηθ=−α 1+α2sin(θ)/parenleftbigg¯β(θ) α−h+(Ω2 1−µ)cos(θ)/parenrightbigg , (35) on the spatial scale η=ξ/s, which also gives the θ-profile to leading order in s−1. Moreover, for such a heteroclinic orbit (θh,˜ph,qh)(ξ)withθσ:= lim ξ→σ∞θh(ξ)∈ {0,π}forσ= 1or σ=−1, theq-limit is/are lim ξ→σ∞qh(ξ) = Ω1−1 s/parenleftbiggh+α¯β(θσ)−σµ 1+α2−Ω0/parenrightbigg +O(s−2). (36) In particular, for Ω1/ne}ationslash= 0or(1+α2)Ω0/ne}ationslash=h+α¯β(θσ)−σµlocal wavenumbers are nontrivial: qh/ne}ationslash≡0. Before proving the theorem we note the consequences of this f or coherent structures and domains walls in in (33) and (1). Corollary 1 The heteroclinic solutions of Theorem 6 are in one-to-one corr espondence with heteroclinic solutions to (33)and thus heteroclinic coherent structures in (1)that lie in Uand connectθ= 0,πor a wavetrain with r/ne}ationslash= 0. Forθ∈(0,π)all properties carry over to (33) with the bijection given by p= sin(θ)˜p. 25Proof.Recall that (34) and (33) are equivalent for θ∈(0,π). Since the limit of the vector field of (33) along such a heteroclinic from (34) is zero by con struction in all cases. Hence, for each of the heteroclinic orbits in (34) of Theorem 6, there ex ist a heteroclinic orbit in (33) in the sense of the corollary statement. Corollary 2 For any ‘bandgap’ parameter set of (33)such that there exist no wavetrains satisfying (32)for any|s|> s1, for some s1>0, there exist fast domain wall type coherent structures spatially connecting ±ˆe3for all sufficiently large velocity |s|. Proof. Choosing Ω 1=kthere are by assumption no equilibria in (35) besides ±ˆe3, which are therefore connected by a heteroclinic orbit. Theorem 6 t hen implies the claim. Such ‘bandgaps’ occur in particular if µ >0 for Ω2 1∼µ. Remark 3 1. Concerning stability, Lemma 1 and Theorem 4 imply that for ccp= 0the domain walls connecting ±ˆe3might be stable in the subsubcritical case only since otherw ise one of the asymptotic states is unstable: the unique wavetrain with θ∈(0,π)in the subsubcritical case and ˆe3or−ˆe3in the sub- and supercritical cases. However, it may be that s ome fronts are stable in a suitable weighted sense as invasion fr onts into an unstable state. 2. For increasing speeds these solutions are decreasingly l ocalized, hence far from a sharp transition. 3. The uniqueness statement in the corollaries is limited, si nce in the (θ,p,q)-coordinates the neighborhood Ufrom the theorem is ‘pinched’ near θ= 0,π: a uniform neighborhood in(θ,˜p)has a sinus-shaped boundary in (θ,p). Remark 4 Part of the family homogeneous domains walls from Theorem 5, w hereccp= 0, is a continuation to smaller |s|of homogeneous ( q≡0) fronts in the family of Theorem 6. The latter are decreasingly localized, which requires in the fo rmer that√µ=O(s). Specifically, µ=−(β−αh)2 s2(1+α2)2andΩ0=h+αβ 1+α2,Ω1= 0in the heteroclinics of Theorem 6. Then µ→0as s2→ ∞so thatµ= 0in the leading order equation (35)and in(36)µis removed from the orders−1term. Since σs√−µ=−(β−αΩ0)andβ−αΩ =−α 1+α2/parenleftBig h−β α/parenrightBig indeed(35)equals the equation in Theorem 5. In particular, (36)is consistent with q≡0. Finally, remark that the ODE (35) is the spatial variant of th e temporal heteroclinic connection in (15): setting all space derivatives to zero, t heθ-equation of (15) reads −∂tθ=α 1+α2sin(θ)/parenleftbigg h−¯β(θ) α−µcos(θ)/parenrightbigg , which is (35) with µreplaced by Ω2 1−µand up to possible direction reversal. Since Ω 1=q on the slow manifold M0(i.e. at leading order), the reduced flow equilibria reprodu ce the wavetrain existence condition (23). This kind of relation b etween temporal dynamics and fast travelling waves holds formally (but in general not rigorou sly) for any evolution equation in one space dimension. Here the symmetry makes the temporal OD E scalar. 26Proof (Theorem 6) Let us set s=ε−1so that the limit to consider is ε→0. Since we will rescale space with εandε−1this means sign changes of sreflect the directionality of solutions. The existence proof relies on a geometric singular perturba tion argument and we shall use the terminology from this theory, see [10, 17], and also some times suppress the ε-dependence ofθ,˜p,q. Upon multiplying the ˜ p- andq-equations of (34) by εwe obtain the, for ε/ne}ationslash= 0 equivalent, ‘slow’ system θ′= sin(θ)˜p ε˜p′=−α˜p+q−Ω1+ε(h+(q2−µ)cos(θ)−Ω0−cos(θ)˜p2) εq′=−˜p−α(q−Ω1)+ε(αΩ0−¯β(θ)−2cos(θ)q˜p).(37) Settingε= 0 gives the algebraic equations A/parenleftbigg˜p q/parenrightbigg =−Ω1/parenleftbigg−1 α/parenrightbigg ,whereA=−/parenleftbiggα−1 1α/parenrightbigg . Since det A= 1+α2>0 the unique solution is ˜ p=q−Ω1= 0 and thus the ‘slow manifold’ isM0as defined in the theorem, with ‘slow flow’ given by θ′= sin(θ)˜p. Since ˜p= 0 atε= 0,M0is a manifold (a curve) of equilibria at ε= 0, so that the slow flow is in fact ‘superslow’ and will be considered explicitly below. Since the slow manifold is one-dimensional (and persists for ε >0 as shown below) it suffices to consider equilibria for ε >0. These lie on the one hand at θ=θ0∈ {0,π}, if A/parenleftbigg˜p q/parenrightbigg +Ω1/parenleftbigg−1 α/parenrightbigg +εF(˜p,q) = 0, F(˜p,q) :=/parenleftbiggh+σ(q2−µ)−Ω0−σ˜p2 αΩ0−¯β(θ0)−2σq˜p/parenrightbigg , whereσ= cos(θ0)∈ {−1,1}. Since det A=−(1 +α2)<0 the implicit function theorem provides a locally unique curve of equilibria (˜ pε,qε) for sufficiently small ε, where d dε/vextendsingle/vextendsingle/vextendsingle/vextendsingle ε=0/parenleftbigg˜pε qε/parenrightbigg =−A−1F(0,0) =−A−1/parenleftbiggh−σµ−Ω0 αΩ0−β±/parenrightbigg . This proves the claimed location of asymptotic states. On the other hand, for θ/ne}ationslash= 0 system (34) is equivalent to (33). From the previous consi d- erations of equilibria (=wavetrains) we infer that the uniq ue equilibria in an ε-neighborhood ofM0are those at θ=θ0, (˜p,q) = (˜pε,qε) together with the possible additional θ-values of wavetrains, where kis now replaced by q=ε(Ω0−¯β(θ)/α). Towards the persistence of M0as a perturbed invariant manifold for |ε|>0, let us switch to the ‘fast’ system by rescaling the time-like variable to ζ=ξ/ε. With˙θ= dθ/dζetc., this gives ˙θ=εsin(θ)˜p ˙˜p=−α˜p+q−Ω1+ε(h+(q2−µ)cos(θ)−Ω0−cos(θ)˜p2) ˙q=−˜p−α(q−Ω1)+ε(αΩ0−¯β(θ)−2cos(θ)q˜p).(38) 27Note that M0is (also) a manifold of equilibria at ε= 0 in this system and the linearization of (38) inM0for transverse directions to M0is given by A. Since the eigenvalues of A,−α±i, are away from the imaginary axis, M0is normally hyperbolic and therefore persists as an ε-close invariant one-dimensional manifold Mε, smooth in εand unique in a neighborhood of M0. See [10]. The aforementioned at least two and at most three e quilibria lie in Mε, and, Mεbeing one-dimensional, these must be connected by heterocl inic orbits. For the connectivity details it is convenient to derive an ex plicit expression of the leading order flow. We thus switch to the superslow time scale η=εξand setp= ˜p/ε,q= (q−Ω1)/ε, which changes (37) to (subdindex ηmeans d/dη) θη= sin(θ)p εpη=−αp+q+h+(Ω2 1−µ)cos(θ)−Ω0+εcos(θ)(ε(q2−p2)+2qΩ1) εqη=−p−αq+αΩ0−¯β(θ)−2εcos(θ)(εqp+Ω1p).(39) Atε= 0, solving the algebraic equations for ( p,q) gives /parenleftbiggp q/parenrightbigg =−A−1/parenleftbiggh+(Ω2 1−µ)cos(θ)−Ω0 αΩ0−¯β(θ)/parenrightbigg =1 1+α2/parenleftbiggαh+α(Ω2 1−µ)cos(θ)−¯β(θ) (1+α2)Ω0−h−(Ω2 1−µ)cos(θ)−α¯β(θ)/parenrightbigg whose first component gives pso that the leading order superslow flow on the invariant man- ifold is indeed given by (35). 5.2.2 The case of small amplitudes In this section we consider small amplitude coherent struct ures, which means qmust lie near a bifurcation point of wavetrains. Here we focus on the inter section points of the solution curves from (16) with θ=θ0= 0,π, which gives cos(θ0)/parenleftbig q2−µ/parenrightbig =¯β(θ0) α−h. (40) Forccp= 0 this is possible for super- and subcritical anisotropy on ly, compare Figure 8. We show that these intersection points are pitchfork-type b ifurcations in (34) that give rise to front-type coherent structures. As in the previous s ection, we locate such solutions in (33) from an analysis of (34). It is convenient to write (16) in terms of m3= cos(θ) so equilibria of (34) with ˜ p= 0 solve ˜Γ(m3) :=˜β(m3) α−/parenleftbig q2(m3)−µ/parenrightbig m3−h= 0, whereq(m3) is the selected qfrom (32). Recall ˜β(m3) =β 1+ccpm3. Theorem 7 Consider θ=θ0∈ {0,π}and setm0 3:= cos(θ0). Suppose that parameters of (34) are such that s/ne}ationslash= 0,˜Γ(m0 3) = 0and∂m3˜Γ(m0 3)/ne}ationslash= 0. Then the equilibrium point (θ0,0,q(m0 3)) of(34)undergoes a pitchfork bifurcation upon any perturbation of horµ. More precisely, let Sε= (αε,βε,hε,µε,Ωε,ccp(ε),sε),ε∈(−ε0,ε0)for some ε0>0, be a curve in the parameter space of (34)with|ccp(ε)|<1,αε>0,sε/ne}ationslash= 0and such that S0 satisfies ˜Γ(m0 3) = 0and˜γ:=∂m3˜Γ(m0 3)/ne}ationslash= 0. 28Then(34)has a curve of equilibria (θ0,˜pε,qε), with possibly adjusted ε0, such that ˜p0= 0,q0=q(m0 3)and two equilibria with θ/ne}ationslash=θ0bifurcate from (θ0,0,q(m0 3))for increasing ε if, with parameters Sε,m0 3˜γ∂ε˜Γ(m0 3)|ε=0>0. The bifurcating equilibria are connected to (θ0,˜pε,qε)by heteroclinic orbits which converge to (θ0,˜pε,qε)asℓξ→ ∞forℓ=−m0 3˜pε. Specifically, this occurs if hε=h0−m0 3˜γεorµε=µ0+˜γεandℓ= ˜γs, or ifβε=β0+m0 3˜γε andℓ=−(2q(m0 3)m0 3+s)˜γ, with all other parameters fixed in each case. Analogously to Corollary 1 we have Corollary 3 The heteroclinic solutions of Theorem 7 are in one-to-one corr espondence with heteroclinic solutions to (33), connecting to θ≡0orθ=πinU. Bounded solutions for θ/ne}ationslash∈ {0,π}are also in one-to-one correspondence. Proof (Theorem 7) Note that ˜Γ(m3) = 0 with |m3|<1 is equivalent to (and if |m3|= 1 sufficient for) the existence of an equilibrium of (34) at θwith cos( θ) =m3,q=q(m3) from (40) and ˜ p= 0. Assuming ˜ γ=∂m3˜Γ(m0 3)/ne}ationslash= 0 and ∂ν˜Γ(m0 3)/ne}ationslash= 0 forν=horν=µimplies existence of a locally unique curve of equilibria m3(ν) that transversely crosses m0 3. The case of parameters Sεis analogous with a curve m3(ε), where ∂εm3(0) =−∂ε˜Γ(m0 3)/∂m3˜Γ(m0 3)|ε=0 having the sign of −m0 3means bifurcation of two equilibria for ε >0. Itremainstoshowthatthecentermanifoldassociatedtothe bifurcationisone-dimensional, and to obtain the directionality of heteroclinics. For the former it suffices to show that the linearization at the bifurcation point has only a simple eigenvalue on the imaginary axis, namely at zero. Th e linearization of (34) in any point with ˜ p= 0,θ=θ0gives the 3 ×3 matrix ˜A= 00 0 0 0B , B=/parenleftbigg−αs 2qm0 3+s −(s+2qm0 3)−αs/parenrightbigg , which has a kernel with eigenvector (1 ,0,0)t. The remaining eigenvalues are those of B, which are −αs±(s+m0 32q)i. Since these lie off the imaginary axis for s/ne}ationslash= 0 there is indeed at most one simple zero eigenvalue on the imaginary axis. Thi s implies the existence of a one-dimensional center manifold which includes all equili bria and heteroclinic connections near (θ0,0,q(m0 3)) for nearby parameters. Equilibria in the symmetry plane {m3=m0 3}are solutions of (˜ p′,q′) =:G(˜p,q) = 0 with Ggiven by (34). Since DG(0,q(m0 3)) =Bis invertible we obtain a curve ( θ0,˜pε,qε) for parameters at Sεas claimed. Theuniquenessofbifurcatingequilibriaoneithersideoft hesymmetryplaneandinvariance of the one-dimensional center manifold implies existence a nd local uniqueness of heteroclinic connections for sgn( ε) = sgn( m0 3˜γ∂ε˜Γ(m0 3)|ε=0). In order to determine the directionality of these, a perturbation in the kernel gives θ′= sin(θ0+δ)˜p=m0 3δ˜p+O(δ2) so that for m0 3˜pε<0 the equilibrium at ( θ0,˜pε,qε) is stable in the center manifold, and unstable for reversed sign. Since existence of heteroclinics requires sgn( ε) = sgn(m0 3˜γ∂ε˜Γ(m0 3)|ε=0) this implies stability ifℓ:=−˜γ∂ε˜Γ(m0 3)∂ε˜p|ε=0<0 and thus convergence to ( θ0,˜pε,qε) asℓξ→ ∞. Forhε=h0−m0 3˜γεwith otherwise fixed parameters ∂ε˜Γ(m0 3) =m0 3˜γso heteroclinics exist for ε >0. On the other hand, ∂ε˜pε|ε=0is the first component −m0 3˜γαsdet(B) of −B−1∂hG(0,q(m0 3))(−m0 3˜γ), where αdet(B)>0. Hence, ℓ=s˜γas claimed. The cases ε=µ,βare determined analogously using ∂µ˜Γ(m0 3) =m0 3,∂µG(0,q(m0 3) = (−m0 3,0) and ∂β˜Γ(m0 3) = ((1+ ccpm0 3)α)−1>0,∂βG(0,q(m0 3) = (0,−((1+ccpm0 3)α)−1). 290Π 2Π/Minus2/Minus1012 ΘΘ'supercritical 0Π 2Π/Minus2/Minus1012 ΘΘ'subcritical 0Π 2Π/Minus2/Minus1012 ΘΘ'subsubcritical 0Π 2Π/Minus2/Minus1012 ΘΘ'domainwalls (a) (b) (c) (d) Figure 12: Phase plane streamplots of (42) with Mathematica . (a)-(c) have h−Ω = 1/2. (a) supercritical anisotropy (here µ= 1), (b) subcritical (here µ= 0), (c) subsubcritical (here µ=−1), (d) subsubcritical case that allows for standing domain walls,h= Ω,µ=−1. 5.3 Stationary coherent structures for ccp= 0 In this section we consider the case s= 0 (which does not imply time-independence) and ccp= 0 (which will imply integrability), so that equations (33) reduce to θ′′= sin(θ)/parenleftbig h−Ω+(q2−µ)cos(θ)/parenrightbig q′=αΩ−β−2cot(θ)θ′q.(41) In case Ω /ne}ationslash=β/αthere are no equilibria and it will be shown at the end of this s ection that there are no coherent structure-type solutions in that case. Recall from §2.1 that for ccp= 0 we may choose coordinates so that β= 0, which means Ω = 0 and thus stationary coherent structures are turned into standing waves. Howeve r, we choose not to remove the parameter βin order to emphasize the typically oscillatory nature of so lutions to (1) and for consistency in parameter relations. Nevertheless, the sym metries and integrals that we will find are consequences of this reducibility. For Ω = β/α, system (41) is invariant under the reflection q→ −qso that{q= 0}is an invariant plane which separates the three dimensional phas e space. In particular, there cannot be connections between equilibria (=wavetrains) with oppo site signs of q, that is, sign reversed spatial wavenumbers . 5.3.1 Homogeneous solutions ( q= 0) Solutions in the invariant set {q= 0}have the form m(ξ) =r(ξ)exp(itΩ) and (31) turns into a second order ODE on the circle {m2 3+r2= 1}. The ODE for θfrom (41) is given by the nonlinear pendulum equation θ′′= sin(θ)(h−Ω−µcos(θ)), (42) which is invariant under θ→ −θand is Hamiltonian with potential energy P0(θ) = cos(θ)(h−Ω−µ 2cos(θ)). The symmetry (24) applies and we therefore assume in the foll owing that Ω = β/α < h. 30We plot the qualitatively different vector fields of (42) in Fig ure 12 and some profiles in Figure 4. Coherent structure solutions are completely ch aracterized via the figure, which we formulate next explicitly for the original PDE with m= (m,m3),m=reiϕ,r= sin(θ), m3= cos(θ). Homoclinic profiles may beinterpreted as (dissipative) s olitons. Theheteroclinic connections in item 2(a) can be viewed as (dissipative) soli tons with ‘phase slip’. Theorem 8 Lets= 0andΩ =β/αand consider solutions to (14)of the form (30)withϕ constant in ξ, i.e.,q= 0. These oscillate in time pointwise about the ˆe3-axis with frequency Ω =β/α. Assume without loss of generality, due to (24), thath >Ω. 1. Subcritical anisotropy h−Ω>|µ|>0. There exist no nontrivial wavetrains with q= 0, and the coherent structure solutions with q= 0are a pair of homoclinic profiles to ˆe3, and three one-parameter families of periodic profiles, one boun ded and two semi-unbounded. The homoclinic profiles each cross once through −ˆe3in opposite θ-directions. The limit points of the bounded curve of periodic profiles are −ˆe3and the union of homoclinic profiles. Each of the homoclinics is the limit point of one of t he semi-unbounded families, each of which has unbounded θ-derivatives. The profiles from the bounded family each cross−ˆe3once during a half-period, the profiles of the unbounded fami ly cross both ±ˆe3 during one half-period. 2. Suppose super- or subsubcritical anisotropy |µ|> h−Ω. There exists a wavetrain with k= 0, which is stable in the supercritical case ( µ >0) and unstable in the subsubcritical case (µ <0). In(42)this appears in the form of two equilibria being the symmetri c pair of intersection points of the wavetrain orbit and a meridian on the sphere, phase shifted byπinϕ-direction. Details of the following can be read off Figure 12 analogous to item 1. (a) In the supercritical case ( µ > h−Ω) the coherent structure solutions with q= 0 are two pairs of heteroclinic connections between the wavet rain and its phase shift, and four curves of periodic profiles; two bounded and two semi -unbounded. (b) In the subsubcritical case ( µ < h−Ω) the coherent structure solutions with q= 0are two pairs of homoclinic connections to ±ˆe3, respectively, and five curves of periodic profiles, three bounded and two semi-unbounded. 3. The degenerate case h= Ω,µ <0is the only possibility for profiles of stationary coherent structures to connect between ±ˆe3, which then come in a pair as in the corresponding panel of Figure 12. The remaining coherent structures with q= 0are analogous to the supercritical case with ±ˆe3and the pair of wavetrain and its phase shift interchanged. Proof.As for wavetrains discussedin §4, thecondition cos( θ) = (h−Ω)/µyields theexistence criterion |h−Ω|<|µ|for an equilibrium to (42) in (0 ,π). The derivative of the right hand side of (42) at θ= 0 ish−Ω−µ, which dictates the type of all equilibria and only saddles generate heteroclinic or homoclinic solutions. It remains to study the connectivity of stable and unstable m anifolds of saddles, which is given by the difference in potential energy P0(θ). Since P0(0)−P0(π) = 2(h−Ω) the claims follow. 31/Minus4/Minus20240Π2Π kΘ0Π 2Π/Minus0.8/Minus0.400.4 Θ 0Π 2Π/Minus0.400.4 Θ (a) (b) Figure 13: (a) Figure 11(a) with solutions to (43) for θ∈(0,π) andC= 0.1 (thick solid line), andC= 0.4 (thick dashed line). (b) Upper panel: potential P(θ) for parameters as in (a) and C= 0.1. Lower panel: same with C= 0.4. 5.3.2 Non-homogeneous solutions ( q/ne}ationslash= 0) In order to study (41) for q/ne}ationslash= 0, we note the following first integral. Since Ω = β/α, the equation for qcan be written as (log|q|)′=−2(log|sin(θ)|)′, and therefore explicitly integrated. With integration con stantC= sin(θ(0))2|q(0)|this gives q=C sin(θ)2. (43) Substituting this into the equation for θyields the nonlinear pendulum θ′′= sin(θ)(h−Ω−µcos(θ))+C2cos(θ) sin(θ)3, (44) with singular potential energy P(θ) =P0(θ)+1 2C2cot(θ)2. The energy introduces barriers at multiples of πso that solutions for q/ne}ationslash= 0 cannot pass ±ˆe3, and asCincreases the energy landscape becomes qualitatively inde pendent of Ω ,h,µ. Asanimmediateconsequenceoftheenergybarriersandthefa ctthatonlystablewavetrains are saddle points we get Theorem 9 Lets= 0andΩ =β/α. Consider solutions to (14)of the form (30). Intersec- tions in(q,θ)-space of the curve Cgiven by (43)with the wavetrain existence curves Wfrom (23)(withkreplaced by q) are in one-to-one correspondence with equilibria of (41). 32Consider supercritical anisotropy 0≤h−Ω< µand assume Cis such that Ctransversely intersects the component of Wwhich intersects {q= 0}. Then the intersection point with smallerq-value corresponds to a spectrally stable wavetrain, and in (41)there is a pair of homoclinic solutions to this wavetrain. All other intersec tion points are unstable wavetrains. All other non-equilibrium solutions of the form (30)withs= 0are periodic in ξ, and this is also the case for all other parameter settings. The homoclinic orbit is a soliton-type solution to (1) with a symptotic state a wavetrain (cf. Figure 2). Notably, the tangential intersection of CandWis at the sideband instability. In Figure 13 we plot an illustration in case of supercritical anisotropy µ >Ω−h >0. In the upper panel of (b) the local maxima each generate a pair of homoclinic solutions to the stable wavetrain it represents. The values of qthat it visits lie on the bold curve in (a), whose intersections with the curve of equilibria are the local max ima and minima. The lower panel in (b) has C= 0.4 and the local maxima disappeared. All solutions are period ic and lie on the thick dashed curve in (a). 5.3.3 Absence of wavetrains ( Ω/ne}ationslash=β/α) In this case the first integral Q= log(|q|sin(θ)2) is monotone, Q′=Ω−β/α q/ne}ationslash= 0, and therefore |q|is unbounded as ξ→ ∞orξ→ −∞. In view of (44), we also infer that θ oscillates so that there are no relevant solutions of the for m (30). References [1]Aranson, I., and Kramer, L. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys. 74 (2002), 99–143. [2]Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54 , 13 (Oct. 1996), 9353–9358. [3]Berkov, D., and Miltat, J. Spin-torque driven magnetization dynamics: Micromag- netic modeling. J. Magn. Magn. Mat. 320 , 7 (April 2008), 1238–1259. [4]Bertotti, G. Spin-transfer-drivenmagnetization dynamics. In Magnetic Nanostructures in Modern Technology , B. Azzerboni, G. Asti, L. Pareti, and M. Ghidini, Eds. Sprin ger Netherlands, 2008, pp. 37–60. 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A phenomenological theory of damping in ferromagnetic mate rials.Mag- netics, IEEE Transactions on 40 , 6 (nov. 2004), 3443 – 3449. [12]Goussev, A., Robbins, J. M., and Slastikov, V. Domain-wall motion in ferromag- netic nanowires driven by arbitrary time-dependent fields: An exact result. Phys. Rev. Lett. 104 (Apr 2010), 147202. [13]Hasimoto, H. A soliton on a vortex filament. J. Fluid Mech 51 , 3 (1972), 477–485. [14]Hoefer, M., Ablowitz, M., Ilan, B., Pufall, M., and Silva, T. Theory of mag- netodynamics induced by spin torque in perpendicularly mag netized thin films. Physical review letters 95 , 26 (2005), 267206. [15]Hoefer, M., Sommacal, M., and Silva, T. Propagation and control of nanoscale magnetic-droplet solitons. Phys. Rev. B. 85 (2012), 214433. [16]Hubert, A., and Sch ¨afer, R. Magnetic Domains: The Analysis of Magnetic Mi- crostructures . Springer, August 1998. [17]Jones, C. K. Geometric singularperturbationtheory. Johnson, Russell (ed.), Dynamical systems.Lecturesgiven atthe2ndsessionoftheCentroInte rnazionaleMatematico Estivo (CIME)heldinMontecatini Terme, Italy, June13-22, 1994. B erlin: Springer-Verlag. Lect. Notes Math. 1609, 44-118 (1995)., 1995. [18]Kosaka, C., Nakamura, K., Murugesh, S., and Lakshmanan, M. Equatorial and related non-equilibrium states in magnetization dynamics of ferromagnets: generalization of Suhl’s spin-wave instabilities. Phys. D 203 , 3-4 (2005), 233–248. [19]Kravchuk, V. P., Volkov, O. M., Sheka, D. D., and Gaididei, Y. Periodic magnetization structures generated by transverse spin cur rent in magnetic nanowires. Physical Review B 87 , 22 (2013), 224402. [20]Lakshmanan, M. Continuum spin system as an exactly solvable dynamical syst em. Physics Letters A 61 , 1 (1977), 53–54. [21]Lakshmanan, M. The fascinating world of the Landau-Lifshitz-Gilbert equa tion: an overview. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. 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2013-09-21

The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization dynamics in the presence of an applied field and a spin polarized current. In the case of axial symmetry and with focus on one space dimension, we investigate the emergence of space-time patterns in the form of wavetrains and coherent structures, whose local wavenumber varies in space. A major part of this study concerns existence and stability of wavetrains and of front- and domain wall-type coherent structures whose profiles asymptote to wavetrains or the constant up-/down-magnetizations. For certain polarization the Slonczewski term can be removed which allows for a more complete charaterization, including soliton-type solutions. Decisive for the solution structure is the polarization parameter as well as size of anisotropy compared with the difference of field intensity and current intensity normalized by the damping.

Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations

1309.5523v4

arXiv:2103.03885v1 [cond-mat.str-el] 5 Mar 2021Universal spin wave damping in magnetic Weyl semimetals Predrag Nikoli´ c1,2 1Department of Physics and Astronomy, George Mason Universi ty, Fairfax, VA 22030, USA and 2Institute for Quantum Matter at Johns Hopkins University, B altimore, MD 21218, USA (Dated: March 9, 2021) We analyze the decay of spin waves into Stoner excitations in magnetic Weyl semimetals. The lifetime of a mode is found to have a universal dependence on i ts frequency and momentum, and on a few parameters that characterize the relativistic Weyl spectrum. At the same time, Gilbert damping by Weyl electrons is absent. The decay rate of spin wa ves is calculated perturbatively using the s-d model of itinerant Weyl or Dirac electrons coup led to local moments. We show that many details of the Weyl spectrum, such as the momentum-spac e locations, dispersions and sizes of the Weyl Fermi pockets, can be deduced indirectly by probi ng the spin waves of local moments using inelastic neutron scattering. I. INTRODUCTION Weyl semimetals are condensed matter realizations of massless fermions with a chiral relativistic three- dimensional spectrum1–3. Topologically protected gap- less Fermi “arc” states on the system boundaries, and unconventional transport properties such as the intrinsic anomalous Hall effect, set Weyl semimetals apart from other weakly interacting conductors. One way to ob- tain a Weyl spectrum involves breaking the time-reversal symmetry in a material that has Dirac quasiparticles. The presence of magnetization, for example, will remove the spin degeneracy of a Dirac node by splitting it into a dipole of opposite-chirality Weyl nodes in momentum space. Magnetism then becomes intimately related to the presence of Weyl electrons. Alternatively, Weyl spec- trum of itinerant electrons can be created by a broken inversion symmetry, e.g. due to the crystal structure, and then coupled to magnetism if the material possesses additional local moments or undergoes a spin density wave instability. Some of these theoretical scenarios are slowly finding their actualization in experimentally stud- ied magnetic Weyl semimetals4–13. Here we analyze an important imprint of Weyl elec- trons on the magnetic dynamics – the damping of spin waves via particle-hole (Stoner) excitations. This basic interactioneffect revealsthe definingfeatures ofthe Weyl spectrum, relativity and chirality. We will show that the lifetime of spin waves exhibits a universal dependence on the modefrequency andmomentum whichcanbe used to extract detailed properties of the underlying Weyl elec- trons. By measuring the mode lifetime throughout the first Brillouin zone, it is possible to discern the locations of the Weyl nodes in momentum space, their relative chi- ralities, slope of the energy versusmomentum dispersion, and the size of the Fermi pockets on the Weyl nodes. The spin wave lifetime is obtained from the width of the scattering intensity peaks in inelastic neutron scattering experiments, provided that a sufficient energy resolution is available and other sources of decoherence (thermal broadening, disorder, phonons) do not mask the elec- tronic source. Even though neutron scattering is a powerful Green’sfunction probe, its ability to detect fermionic quasipar- ticles is normally ruined by the incoherent continuum of excitations that can absorb an angular momentum quantum. Interestingly, this problem is reduced in Weyl semimetals14, and fortunately it is also possible to in- directly characterize the quasiparticles via collective ex- citations. The latter has been achieved in the neutron studies of samarium hexaboride (SmB 6)15,16, where the measured dispersion of a “spin exciton” has revealed a non-trivial topology of the underlying electronic quasi- particles. An energygap protects the exciton’s coherence in SmB 6, but the gaplessquasiparticlesin Weyl semimet- als will generally induce ubiquitous damping of collective modes. Such a damping can in fact reveal the existence and properties of chiral fermionic quasiparticles. The Weyl electron characterization through damping could potentially overcome various issues that plague other ap- proaches, such as correlation effects in the case of band- structure calculations, limited resolution in the case of ARPES, sensitivity to conventional bands (that coexist with Weyl nodes) in transport measurements, etc. Closely related to the physics we pursue here is the ex- tensively studied damping in metallic ferromagnets17–29. Stoner excitations provide a mechanism for the decay of spin waves, and also typically give rise to Gilbert damping30– the dissipated precession of uniform mag- netization in an external magnetic field. Many works have been devoted to the calculation of Gilbert damp- ing since it is possible to measure it by ferromagnetic resonance31,32and time-resolved magneto-optical Kerr effect33,34. A careful consideration of the relativistic electron dynamics has revealed that Gilbert damping originates in the spin-orbit coupling and depends on the electrons’ mass25. In the case of massless Weyl electrons, we show here that Gilbert damping is ab- sent. However, spin waves unavoidably decay via Stoner excitations35–39,41,42, and their damping features “non- reciprocity” – different polarization modes that carry the same momentum have different damping rates. This accompanies the non-dissipative aspects of chiral spin- momentumlocking44,45. Spinwave“non-reciprocity”has been anticipated in spiral magnets46, magnetic interfaces with a Dzyaloshinskii-Moriya interaction derived from2 the Rashba spin-orbit coupling43,47–52, and observed in several experiments53–58. In the context of magnetic Weyl semimetals, initial theoretical studies have been fo- cused on the domain wall dynamics59,60. The rest of this paper is organized as follows. Section II presentsthe approachand the main results ofthe anal- ysis, focusing on the observable physical characteristics of the spin wave damping by Weyl electrons. Section III is devoted to the technical development of the damping theory. It contains separate derivations of the dissipative termsintheeffectivespinaction(IIIA),spinwavedamp- ing (IIIB), and Gilbert damping from the semiclassical field equation (IIIC). The last section IV summarizesthe conclusions and discusses the broader applicability and limitations of the damping theory. II. SUMMARY OF THE RESULTS In this paper, we work with the s-d model of Weyl electrons coupled to local moments. We perturbatively calculate the dissipative non-Hermitian parts of the mo- ments’effectiveaction,whichdeterminetherate γofspin wavedamping. γalsodependsonthe magneticorderand the wave’s propagation direction relative to the magneti- zation, but it is always controlled by the components of the universal damping rate tensor given by γab mn(q) =a3J2 KΩ2 128πSv3fab mn/parenleftbigg|Ω| vq,|Ω| 2|µ|;sign(µ,Ω)/parenrightbigg (1) for ferromagnetic local moments of spin magnitude S. The upper indices a,b∈ {x,y,z}refer to spin projec- tions. The universal scaling functions fab mnare dimen- sionless, the factor a3is the unit-cell volume of the local moment’s lattice, JKis the Kondo or Hund coupling en- ergy scale, vandµare the Fermi velocity and Fermi energy of the Weyl electrons respectively, and Ω is the real spin wave frequency (we use the units /planckover2pi1= 1). The spin wave momentum qin this expression is measured relative to the difference ∆ Q=Qm−Qnbetween the wavevectors Qm,Qnof any two Weyl nodes in the first Brillouin zone. Coherent collective excitations that span the entire first Brillouin zone can be used to separately address many pairs of Weyl nodes – by tuning the total wavevector ∆ Q+qto the vicinity of ∆ Q. Representa- tive functions fab mnfor the Weyl nodes with finite Fermi surfaces are plotted in Figures 1 and 2 Wemakeanalyticalprogressandgainvaluablephysical insight through several idealizations: all Weyl nodes are assumed to be identical, sphericallysymmetric and living at the same node energy. Their chiralities χm=±1 and locations Qmare arbitrary (as long as the total chirality in the first Brillouin zone vanishes). Under these condi- tions, only three tensor components of γabare finite and independent, γ/bardbl/bardbl,γ⊥⊥andγ⊥⊥′. Here and throughout the paper ∝bardblindicates the spin direction parallel to the mode’s wavevector q, and⊥,⊥′are the spin directions(a) (b) FIG. 1. The plots of functions (a) f⊥⊥and (b) f/bardbl/bardblfor the damping rates of transverse and longitudinal spin waves re- spectively, contributed by the Fermi surfaces on a particul ar pair of Weyl nodes. Solid red lines are for the same-chiralit y nodes, and the dashed blue lines are for the opposite-chiral ity nodes.|Ω|= 1.4|µ|was assumed in this example. FIG. 2. The plots of selected universal functions fabfeatured in the damping rate γ∼Ω2f(vq/|Ω|;xµ). The functions are parametrized by xµ= 2|µ/Ω|, with finer dashes corre- sponding to larger Weyl Fermi pockets (solid lines refer to the Fermi level that crosses the Weyl nodes). Shown func- tionsincludetransverse( ⊥⊥)andchiral( ⊥⊥′)dampingchan- nels shaped by electron scattering between equal-chiralit y (+) andopposite-chirality ( −)Weyl nodes. Longitudinal channels (∝bardbl∝bardbl) are similar to the shown transverse channels, compare with Fig.1.3 (a) (b) FIG. 3. Examples of the damping rate map in momentum space for (a) µ∝negationslash= 0 and (b) µ= 0 (with and without a Fermi surface of Weylelectrons respectively). Brightness depic ts the rateγ(q) of spin wave damping, and the red crosshair shows the reference ∆ Qfor the local wavevector q= 0. These are qz= 0 slices through the full 3D map. Observing patterns of this kind in the full Brillouin zone scan will indicate the Weyl-electron origin of damping and reveal the complete set of ∆Q=Qm−Qnwavevectors from which the individual node wavevectors Qmcan be deduced (assuming, for exam- ple,/summationtext mQm= 0). The bright outer ring, which shrinks and closes when 2 |µ|<|Ω|, originates in the inter-band electron scattering and gains strength from the rapidly growing Weyl electron density of states. Note that various details in the se maps, such as the anisotropy and ring sizes, will generally depend on the concrete spin-wave dispersion Ω( q+∆Q), po- larization, type and orientation of magnetic order, as well as the chiralities and symmetries of the Weyl nodes. which areperpendicular to qand eachother. The full ex- pression for damping rates is presented in Section IIIB; in Weyl ferromagnets, it becomes γmn=γ⊥⊥ mn±γ⊥⊥′ mn (2) for the two polarizations of spin waves propagating along the magnetization direction. The essential utility of the universal damping comes from its qualitative features that reflect the relativistic nature of Weyl electrons. If the Fermi energy µlies away from the energy of the Weyl nodes, Fermi surfaces will form. Then, the spin wave damping rate is expected to exhibitasetofminimumsandmaximumsasafunctionof thefrequencyΩandmomentum q. Thelocationsofthese extremums depend on the parameters that characterize the Weyl nodes: Fermi velocity v, chemical potential µ and even their relative chiralities χmχn=±1. Fig.3 demonstrates how the locations Qmof Weyl nodes can be extracted from the full Brillouin zone map of the spin wave’s damping rate γ(q). Once the wavevectors Qmare known, Fig.4 illustrates how the observation of enough extremums enables indirect measurements of the Weyl electron spectra on multiple Weyl nodes. The presence of Weyl Fermi pockets also introduces spin-momentum locking into the damping rates ( γ⊥⊥′ mn∝ne}ationslash= 0), but only on the pairs of Weyl nodes with opposite chiralities. As a vqΩ 02
2  Ω=vq Ω=vq-2 FIG. 4. A density plot of the collective mode damping rate γ(q,Ω) induced by Weyl electrons. Thin solid green lines in- dicateγ= 0, and the thin dashed green line indicates the local maximum of γ. The thick dashed yellow line represents the dispersion Ω( q+ ∆Q) of a hypothetical spin-wave exci- tation (note that the origin of the plot corresponds to the momentum difference ∆ Qof two Weyl nodes in the first Bril- louin zone). The spin-wave’s damping rate will exhibit loca l minimums and maximums at the shown red points, which are characteristic for the relativistic spectrum of Weyl elect rons. Resolving two of these points is enough for the determinatio n of the Weyl Fermi velocity vand the chemical potential µof the Weyl nodes addressed via ∆ Q. Resolving three points al- lows an independent verification that Weyl nodes are indeed responsible for the damping. The two-parameter scaling of the damping rate (1) across a range of energies is the most general signature of Weyl electrons, and can be used to verif y the Weyl-electron origin of damping even if the visible spin wave dispersion does not cross any of the shown characterist ic points. consequence, the two spin wave modes that carry oppo- site spin currents at the same wavevector qhavedifferent peak widths in inelastic neutron scattering. The above qualitative features of damping disappear if the Fermi energy sits exactly at the Weyl nodes. How- ever, the damping rate then becomes a universalfunction ofa single parameter |Ω|/vq. This kind of scalingis a sig- nature of the relativistic Weyl electrons – it is caused by “inter-band” transitions in which an electron below the Weyl nodeisexcited toastate abovethe Weyl node. The plots of universal functions fab mnthat appear in Eq. 1 at µ= 0 are shown in Fig.2. The magnitude of the damping rate depends on the Kondo/Hund scale JKwhich may not be known. However, the spin wave damping caused by Weyl electrons is always related to the effec- tive strength Jof the Weyl-electron-induced Ruder- man–Kittel–Kasuya–Yosida(RKKY)interactionsamong4 qqJKJK FIG. 5. The Feynman diagram for two-spin interactions. Thick external lines represent local moment fields and thin lines represent Weyl electron propagators. The two-spin co u- plings include Heisenberg, Kitaev and Dzyaloshinskii-Mor iya interactions, but the Weyl-electron origin of spin dynamic s also creates a dissipation channel in which spin waves decay into electron-hole pairs. the local moments45: γ J∼1 (aΛ)3/parenleftBigq Λ/parenrightBig2 ×/parenleftbiggΩ vq/parenrightbigg2 , J∼vΛ/parenleftbigga3Λ2JK v/parenrightbigg2 . (3) Here, Λisthemomentumcut-offforthelinearWeylspec- trum,|q|<Λ. SinceaΛ<1 and the characteristic fea- tures of the universal damping appear near |Ω| ∼vq, the damping rates are generally comparable to the energy scaleJof the induced RKKY interactions. For example, the RKKY energy scale in the magnetic Weyl semimetal NdAlSi13can be crudely estimated as J∼1 meV. Even if the damping rate is more than an order of magnitude below this value of J, it should be detectable with high resolutionneutron instruments (a spin echospectrometer can achieve energy resolution below 10 µeV). III. DISSIPATION BY WEYL ELECTRONS Here we calculate the Gaussian dissipative part of the effective action for local moments which arises due to their coupling to itinerant Weyl electrons. The non- dissipative part of this action, computed in Ref.45, cap- tures the induced RKKY interactions among the lo- cal moments: Heisenberg, Kitaev and Dzyaloshinskii- Moriya. All Gaussian terms δnaΓabδnbof the action ob- tain from a single two-point Feynman diagram which in- volves momentum integration of a singular function; the principal part of this integral yields the interactions, and the contribution of its pole singularity amounts to dissi- pation. We will focus only on the latter, following the procedure from Ref.45. The essential dynamics of local moments ˆnicoupled to conduction electrons ψiis given by the Hamiltonian: H0=Hn+/summationdisplay kǫkψ† kψk+JK/summationdisplay iˆniψ† iσψi.(4) Both the local moments and electrons live on a lattice whose sites are labeled by i, but we will immediately take the continuum limit. The basic two-spin correla- tions∝an}bracketle{tˆna iˆnb j∝an}bracketri}htare contained in the second-order Feynman diagram shown in Fig.5: Γab mn(q) =iJ2 K 2/integraldisplayd4k (2π)4tr/bracketleftBig Gm/parenleftBig k−q 2/parenrightBig σaGn/parenleftBig k+q 2/parenrightBig σb/bracketrightBig (5)The Weyl electron Green’s functions Gn(ω,k) =/bracketleftBig ω−Hn(k)+isign(ǫn(k))0+/bracketrightBig−1 (6) are treated as spinor matrices and refer to the low-energy electronic states near any Weyl node nwhose wavevector in the first Brillouin zone is Qn; the wavevector kis a “small” displacement |k|<Λ fromQn, where Λ is the momentum cut-off for the linear Weyl dispersion. These low-energy electrons are described by the Hamiltonian Hn(k+Qn) =vχnσk−µ , (7) whereµis the chemical potential that determines the Weyl Fermi pocket character and size, vis the Fermi velocity, and χn=±1 is the Weyl node chirality. We assume for simplicity that all Weyl nodes are spherically symmetric, share the same node energy, chemical poten- tial and Fermi velocity, but have arbitrary wavevectors Qnand chiralities χn=±1 (as long as the chiralities of all nodes in the first Brillouin zone add up to zero). By this construction, the expression (5) is associated with a pairm,nof Weyl nodes, and qis a “small” wavevector measured relative to Qm−Qn. We will carry out all calculations with the formal as- sumption that no external or effective magnetic field is exerted on electrons. Realistically, however, we are inter- ested in magnetic Weyl semimetals whose local moments may carry a non-zero net magnetization ˆn0that presents itself as an effective magnetic field B=−JKˆn0to elec- trons. This is of no concern because the correction of the spectrum (7) amounts merely to a shift of the wavevector k→k−B/vχn. Hence, an effective magnetic field only alters the locations Qnof the Weyl nodes in momentum space, which are arbitrary in our formalism. The full effective action matrix Γ for local moments takes contributions from all Weyl node pairs: Γ(Q,Ω) =/summationdisplay m,nΓmn(Q−Qm+Qn,Ω).(8) In this sense, it is possible to experimentally address a particular pair of Weyl nodes, or a set of pairs, by prob- ingthe momentumspaceinthe vicinityof Q∼Qm−Qn. The dissipative part of Γ mnwill contain information about the addressed Weyl nodes. A. Calculation of the dissipative terms in the effective spin Lagrangian The calculationof(5) is lengthy, sowe will onlyoutline its key steps. The trace has been evaluated before45, and the frequency integration yields:5 Γab mn(q) =−J2 K 2/integraldisplayd3k (2π)3/bracketleftBigg Xab(Ω,q;vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle+Ω 2−µ,k) 2vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/productdisplay s=±1θ/parenleftbig µ−vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/parenrightbig Ω+vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle+i0+F(sχn,χm) −Xab(Ω,q;−vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle+Ω 2−µ,k) 2vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/productdisplay s=±1θ/parenleftbig µ+vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/parenrightbig Ω−vχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle+i0+F(sχn,−χm) +Xab(Ω,q;vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle−Ω 2−µ,k) 2vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/productdisplay s=±1θ/parenleftbig µ−vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/parenrightbig vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle−i0+F(χn,sχm) −Xab(Ω,q;−vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle−Ω 2−µ,k) 2vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/productdisplay s=±1θ/parenleftbig µ+vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/parenrightbig −vχn/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle−i0+F(−χn,sχm) Here,θ(x) is the step function, and two more functions, Xab(Ω,q;ω,k) andF(s+,s−) are introduced to simplify notation. The function Xab(Ω,q;ω,k) obtains from the numerator of the trace in (5). Introducing the Kronecker symbolδaband the Levi-Civita symbol ǫabc, we have: Xab(Ω,q;ω,k) =/bracketleftbigg (ω+µ)2−Ω2 4/bracketrightbigg δab +v2χmχn/bracketleftbigg 2/parenleftbigg kakb−qaqb 4/parenrightbigg −δab/parenleftbigg kckc−qcqc 4/parenrightbigg/bracketrightbigg +ivǫabc/bracketleftbigg χm/parenleftbigg ω+Ω 2+µ/parenrightbigg/parenleftbigg kc−qc 2/parenrightbigg −χn/parenleftbigg ω−Ω 2+µ/parenrightbigg/parenleftbigg kc+qc 2/parenrightbigg/bracketrightbigg . (9) The function F(s+,s−) withs+,s−=±1 keeps track of the infinitesimal imaginary terms in the denominators of Green’s functions: F(s+,s−) = sign/parenleftBig vs+/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle−µ/parenrightBig −sign/parenleftBig vs−/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle/parenrightBig =θ/parenleftbigg |qk|−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigµ v/parenrightBig2 −k2−q2 4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg ×/bracketleftbigg/parenleftbigg sign(µ)+s++s− 2/parenrightbigg sign(qk)+s+−s− 2/bracketrightbigg +(s+−s−)θ/parenleftbigg k2+q2 4−|qk|−/parenleftBigµ v/parenrightBig2/parenrightbigg .(10) At this point, we use the relationship 1 x±i0+=P1 x∓iπδ(x) (11) to isolate the dissipative processes that curb the x→0 resonances. Dropping all terms that involve the principal partP, we get: /tildewideΓab mn(q) =iπJ2 K 8v2/summationdisplay sm,snsmsn/integraldisplayd3k (2π)3F′(snχn,smχm) χmχn/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle ×Xab/parenleftbigg Ω,q;vsmχm/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle+Ω 2−µ,k/parenrightbigg (12) ×δ/parenleftBig Ω+vsmχm/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle/parenrightBig ×/bracketleftBig θ/parenleftBig µ−vsmχm/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle/parenrightBig −θ/parenleftBig µ−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle/parenrightBig/bracketrightBigWe introduced F′= sign(F)(1−δF,0), and the sum goes oversm,sn=±1. All chirality factors χm,χn=±1 that appear outside of Xabare clearly eliminated by the summation over sm,sn, so it will be convenient do define s−=smχm=±1 ands+=snχn=±1. The Dirac δ-function in (12) imposes: s+/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle−s−/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle=Ω v. (13) This pins the magnitude of the wavevector kto k=|Ω| 2v/radicalBigg Ω2−v2q2 Ω2−v2q2cos2θ, (14) assuming qk=qkcosθ, and further requires satisfying one of these two conditions: |Ω|>vq ∧s±=±sign(Ω) |Ω|<vq|cosθ| ∧s+=s−= sign(Ωcos θ) The wavevector k= (k,θ,φ) integration in (12) is now conveniently performed in the spherical coordinate sys- temreferencedtotheexternalwavevector q. Theintegral overk=|k|is immediately solved due to the Dirac δ- function and we merely need to replace the occurrences of|k|with (14). The integral over φaffects only the quantities (9) leading to/integraltext dφXab= 2πv2/tildewideXabwith the following non-zero components: /tildewideX/bardbl/bardbl=s+s−/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle (15) +χmχn/bracketleftbigg k2(2cos2θ−1)−q2 4/bracketrightbigg /tildewideX⊥⊥=s+s−/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle +χmχn/parenleftbigg −k2cos2θ+q2 4/parenrightbigg /tildewideX⊥⊥′=iǫ/bardbl⊥⊥′/bracketleftbigg −/parenleftBig χms+/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle+χns−/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle/parenrightBigq 2 +/parenleftBig χms+/vextendsingle/vextendsingle/vextendsinglek+q 2/vextendsingle/vextendsingle/vextendsingle−χns−/vextendsingle/vextendsingle/vextendsinglek−q 2/vextendsingle/vextendsingle/vextendsingle/parenrightBig kcosθ/bracketrightbigg . Here and onward, the upper spin indices denote direc- tions∝bardblparallel to q, and two mutually perpendicular6 directions ⊥,⊥′which are also perpendicular to q. Note thatǫ/bardbl⊥⊥′implements a chiral “right-hand-rule” rela- tionship between the three spin directions. The integral overθis finite and conveniently evaluated numerically. At the end, we arrive at: /tildewideΓab mn(q) =iJ2 KΩ2 128πv3fab mn/parenleftbigg|Ω| vq,|Ω| 2|µ|;sign(µ,Ω)/parenrightbigg (16) where the dimensionless functions f=α+βhave con- tributions from intra-band αand inter-band βelectronscattering. Note that the inter-band processes require transferringan electron between the two states whoseen- ergies have opposite signs, and thus can occur only when |Ω|>2|µ|. Defining λ=vq |Ω|, x=2|µ| |Ω|, κ=/radicalBigg 1−λ2 1−λ2ξ2(17) with|ξ|=|cosθ|, we have: α⊥⊥ mn=1/integraldisplay 0dξθ/parenleftbig 2κλξ−|x2−κ2−λ2|/parenrightbig κ2(18) ×/bracketleftBigg/parenleftBigg 1−χmχn−κ2ξ2+λ2 /radicalbig (κ2+λ2)2−(2κλξ)2/parenrightBigg θ(1−λ)+/parenleftBigg 1+χmχn−κ2ξ2+λ2 /radicalbig (κ2+λ2)2−(2κλξ)2/parenrightBigg θ(λξ−1)/bracketrightBigg α/bardbl/bardbl mn=1/integraldisplay 0dξθ/parenleftbig 2κλξ−|x2−κ2−λ2|/parenrightbig κ2 ×/bracketleftBigg/parenleftBigg 1−χmχnκ2(2ξ2−1)−λ2 /radicalbig (κ2+λ2)2−(2κλξ)2/parenrightBigg θ(1−λ)+/parenleftBigg 1+χmχnκ2(2ξ2−1)−λ2 /radicalbig (κ2+λ2)2−(2κλξ)2/parenrightBigg θ(λξ−1)/bracketrightBigg α⊥⊥′ mn=−iǫ/bardbl⊥⊥′1/integraldisplay 0dξθ/parenleftbig 2κλξ−|x2−κ2−λ2|/parenrightbig κ2 ×/summationdisplay s=±1(χm+χn)sign(µ)+s(χm−χn)sign(Ω) 2/radicalbig κ2+λ2−2sκλξ(κξ−sλ)/bracketleftBig θ(1−λ)−sθ(λξ−1)/bracketrightBig , and β⊥⊥ mn=1/integraldisplay −1dξθ/parenleftbig κ2+λ2−2κλ|ξ|−x2/parenrightbig κ2 1−χmχn−κ2ξ2+λ2 /radicalBig (κ2+λ2)2−(2κλξ)2 θ(1−λ) (19) β/bardbl/bardbl mn=1/integraldisplay −1dξθ/parenleftbig κ2+λ2−2κλ|ξ|−x2/parenrightbig κ2 1−χmχnκ2(2ξ2−1)−λ2 /radicalBig (κ2+λ2)2−(2κλξ)2 θ(1−λ) β⊥⊥′ mn=−iǫ/bardbl⊥⊥′1/integraldisplay −1dξθ/parenleftbig κ2+λ2−2κλ|ξ|−x2/parenrightbig κ2/summationdisplay s=±1s(χm−χn)sign(Ω) 2/radicalbig κ2+λ2−2sκλ|ξ|(κ|ξ|−sλ)θ(1−λ). The functions fabhave the same characteristics in all spin channels a,b∈ {⊥⊥,∝bardbl∝bardbl,⊥⊥′}. Their plots in Fig- ures 1, 2, 4 illustrate that fabvanish for |Ω|<v|q|−2|µ|, 2|µ|−v|q|>|Ω|>v|q|and|Ω|=v|q|. Thedissipationat|Ω|>max(2|µ|,v|q) is dominated by the collective mode decay into “high energy” particle-hole pairs which are excited across the Weyl node. Outside of this frequency- momentum region, the decay occurs by generating “low7 energy” particle-hole pairs across the Fermi surface on the Weyl node. This “low energy” channel is weaker, but has several features that clearly reveal the relativis- tic properties of the Weyl spectrum. Fig. 4 shows how theminimumsandmaximumsofacollectivemodedamp- ing rate can be used to characterize the Fermi surface of Weyl electrons. B. Spin wave damping The actual damping rate of collective excitations gen- erally obtains from a mixture of spin channels. Consider the spin waves with wavevectors ∆ Q+qin the vicinity of the momentum-space separation ∆ Q=Qm−Qnbe- tween two particular Weyl nodes. Let −SΩab 0(q) be the intrinsic part of the effective Lagrangian density δLeff for the local moment fluctuations δn, excluding the spin Berry phase SΩδab(Sis the spin magnitude of local moments). This can contain any exchange interactions of the localized electrons and crystal field anisotropies. The Lagrangian density terms induced by the itinerant Weyl electrons are all contained in the Γabtensor (5). The principal part of (5) yields a variety of induced RKKY interactions45, while its dissipative components /tildewideΓabare collected in (16). The presence of magnetic or- der in the ground state further affects the dynamics of spin waves because the small spin fluctuations δnof low- energy modes must be orthogonal to the local spins ˆn. This can be incorporated into the general analysis44, but we will simplify the discussion here by considering only a ferromagnetic ground state ˆn(r) =ˆn0. The spectrum of damped spin waves is extracted from the Gaussian part of the Lagrangian density in momentum space δLeff= (δna)∗/bracketleftBig SΩδab−SΩab 0(q)+a3Γab(Ω,q)/bracketrightBig δnb(20) The factor of a unit-cell volume a3converts the energy density Γabto the energy per lattice unit-cell, and the factor of1 2in the Berry phase term Ω is appropriate for the local moments with spin S=1 2. Introducing gab= Ωab 0−a3 SΓab(21) to simplify notation, the spin wave modes obtain by di- agonalizing PMP, wherePab=δab−ˆna 0ˆnb 0projects-out the high-energy amplitude fluctuations (keeps δn⊥ˆn0) and Mab= Ωδab−g⊥⊥(δab−ˆqaˆqb)−g/bardbl/bardblˆqaˆqb−g⊥⊥′ǫabcˆqc is the matrix embedded in (20). An arbitrary choice of the background magnetization ˆn0=ˆzreveals two polar- ization modes δn= (δnx,δny) atq=qˆq δn±∝/parenleftBigg g/bardbl/bardbl−g⊥⊥ 2(ˆq2 x−ˆq2 y)±δǫ (g/bardbl/bardbl−g⊥⊥)ˆqxˆqy−g⊥⊥′ˆqz/parenrightBigg (22)with energies Ω±=g⊥⊥ 0+g/bardbl/bardbl−g⊥⊥ 2(1−ˆq2 z)±δǫ(23) whereδǫ=1 2/radicalbig (g/bardbl/bardbl−g⊥⊥)2(1−ˆq2z)2−(2g⊥⊥′ˆqz)2. These polarizations are generally elliptical, but become circularδn∝(±i,1) with Ω ±=g⊥⊥∓ig⊥⊥′for the modes that propagate along the magnetization direction (q∝bardblˆn0), and linear δn+∝ˆq,δn−∝ˆn0׈qwith Ω+=g/bardbl/bardbl, Ω−=g⊥⊥respectively for the modes that propagate in the plane perpendicular to the magneti- zation (q⊥ˆn0). The character and non-degeneracy of the two polarization modes is the hallmark of the RKKY interactions induced through the spin-orbit cou- pling: Dzyaloshinskii-Moriya(DM) in the caseof circular polarizations, and Kitaev in the case of linear polariza- tions. The equation (23) has to be solved self-consistently since the components of the gabtensor on its right-hand side depend on frequency, but the revealed form of its solutions ensures all of the spin wave properties that we discuss. The two circular polarizations at the same wavevector q∝bardblˆn0carry opposite spin currents ja i=−iqiǫabc(δnb)∗δnc∝ ∓|g⊥⊥′|2qiδaz,(24) so their energy difference Ω ±=g⊥⊥∓ig⊥⊥′due to the DM interaction implies spin-momentum locking. Note that the DM interactions appears as gab DM∝ǫabc(iqc), so it does shift the spin wave energy. The dissipative com- ponents/tildewidegab∝/tildewideΓabofgabimpart an imaginary part on the pole frequency Ω, which corresponds to the damping rate. The signs of both /tildewideΓ⊥⊥,/tildewideΓ/bardbl/bardbl(f⊥⊥,f/bardbl/bardbl>0) in- deed correspond to damping and not an instability, and the chiral contributions are not large enough to overturn this at any Ω. The chiral dissipative part extracted from (16) is real,/tildewidegab DM∝ǫabcqc, and hence introduces differ- ent damping rates for the two circular spin waves. These qualitative conclusions hold for the elliptical modes as well. C. The absence of uniform precession damping The universal dependence of (16) on |Ω|/vqintroduces anon-analyticbehavioratΩ ,q→0inthedampingterms /tildewideLof the spin Lagrangian density. Therefore, one cannot strictly expand /tildewideLin powers of Ω ,qto represent the dis- sipation as a result of local processes. /tildewideLcan be approx- imated by an expansion only in special limits. Suppose the spin waves have dispersion |Ω|=uqat low ener- gies (in the vicinity of ∆ Q=Qm−Qn→0 for intra- node scattering m=n). If the spin wave velocity uis smaller than the Weyl electrons’ velocity v, then a suf- ficiently large qpushes the spin waves into the regime |Ω|< vq−2|µ|where/tildewideΓab= 0 in (16) and the damping is absent (see Fig.2). Alternatively, if u≫v, then the8 spin waves are in the regime |Ω| ≫vqand their damp- ing at energies |Ω|>2|µ|is approximately characterized by the dominant local terms /tildewideΓ/bardbl/bardbl,/tildewideΓ⊥⊥∼i(AΩ2+Bq2) and a smaller chiral term /tildewideΓ⊥⊥′∼DqΩ. Together with the non-dissipative Hermitian terms χ−1 0, the electron- induced part of the local moments’ effective Lagrangian density (20) contains Γab|Ω|≫vq− −−−− →1 2/bracketleftBig (χ−1 0)ab+i(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig (25) withAab=A⊥⊥(δab−qaqb/q2)+A/bardbl/bardblqaqb/q2andlikewise forBab. By construction (5), Γ ≡1 2χ−1is the inverse time-ordered correlation function ∝an}bracketle{tδsa(q,Ω)δsb(q′,Ω′)∝an}bracketri}ht=iχab(q,Ω)δ(q+q′)δ(Ω+Ω′) for the small fluctuations δsof the Weyl electron spins away from their equilibrium magnetization. We will con- sider only the simplest case of a collinear ferromagnet in the following analysis. The equilibrium state will be given by the uniform magnetization of local moments ˆn0 and electrons ∝an}bracketle{ts0∝an}bracketri}ht ∝bardblˆn0. A semiclassical representation of the local moment dy- namics is given by the field equation for ˆn. The pres- ence of non-Hermitian damping terms in the effective ac- tion for local moments prevents us from deriving the field equation by considering the stationary action condition. Instead, we can use linear response theory to learn about the semiclassical dynamics. The retarded electrons’ spin correlation function χR(q,Ω) =/braceleftBigg χ(q,Ω),Ω>0 χ†(q,Ω),Ω<0(26) is readily obtained from (25) (χ−1 R)ab|Ω|≫vq− −−−− →(χ−1 0)ab(27) +sign(Ω)/bracketleftBig i(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig , and then the response of electron spins to the local mo- ment field is ∝an}bracketle{tδsa(r,t)∝an}bracketri}ht=JK a3/integraldisplay dt′d3r′χab R(r−r′,t−t′)δnb(r′,t′). (28) This follows from the Kondo interaction JKin (4) be- tween the “perturbation” field nand the responding electrons spin s=ψ†σψon a lattice site (the unit-cell volumea3effectively converts the integration over coor- dinates to a summation over lattice sites). Note that χab R(q,Ω) = (χab R)∗(−q,−Ω) is established globally in momentumspace(notnecessarilyintheimmediatevicin- ity of the Weyl node wavevector ∆ Q)61, so that its in- verse Fourier transform χab R(r,t) is real. The thermody- namic potential for local moments is simply F[ˆn] =JK∝an}bracketle{ts∝an}bracketri}htˆn. (29)The local moment dynamics is driven by an effective “magnetic” field in units of energy Heff(r,t) =−δF[ˆn] δˆn(r,t)=−JK∝an}bracketle{ts(r,t)∝an}bracketri}ht(30) Taking into account the Berry phase of local moments yields the usual semiclassical field equation ∂ˆn ∂t=ˆn×Heff. (31) with Ha eff(r,t)≈ −JKˆna 0−J2 K a3/integraldisplay dδtd3δrχab R(δr,δt) (32) ×δˆnb(r+δr,t+δt) ≈ −JKˆna 0−J2 K a3/integraldisplay dδtd3δrχab R(δr,δt) ×/bracketleftbigg 1+δr∇+δt∂ ∂t+···/bracketrightbigg δˆnb(r,t) This is seen to generate Gilbert damping which dissi- pates the precession of uniform magnetization in typical ferromagnets ∂ˆn ∂t=ˆn×Heff=···+ˆn×αG∂ˆn ∂t(33) with the damping tensor αab G=−J2 K a3/integraldisplay dδtd3δrχab R(δr,δt)δt (34) =iJ2 K a3/integraldisplaydΩ 2π/integraldisplay dδte−iΩδt∂χab R(0,Ω) ∂Ω =−J2 K a3∂Imχab R ∂Ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle (q,Ω)=0. The real part of χR(q,Ω) generally does not contribute because it is an even function of Ω at q= 0 (even though it diverges for gapless spin waves when Ω →0). In the case of damping induced by Weyl electrons, the imagi- nary part of χRbecomes zero when 2 |µ|−vq>|Ω| ≥vq, following the behavior of the time-ordered χ−1≡Γab that was discussed earlier (see Fig.4). Therefore, χRis real in the limit Ω ,q→0 and the decay of spin waves into Stoner excitations of the Weyl electrons does not generate Gilbert damping. The complete equation of motion for local moments canbe extractedfrom(31)and(32), but thenon-analytic frequency dependence of the dissipative terms in (27) in- troduces (via its Fourier transform) non-local relation- ships between the fields ˆn(t) at different times t. If one were to ignore this issue, or approximate the non-local effect by couplings over small time intervals, then a local field equation would be obtained from the expansion in- dicated in (32). We will not pursue this here any further.9 IV. CONCLUSIONS AND DISCUSSION We analyzed the dynamics of local magnetic moments coupled to itinerant Weyl electrons, and focused on the dissipation of spin waves via the continuum of Stoner particle-holeexcitations. Wedescribedthisdissipationat the level of the effective Lagrangian of local moments, or equivalently the spin-spin correlation function (dynamic susceptibility). For the spin waves at wavevector∆ Q+q and frequency Ω in the vicinity of the momentum differ- ence∆Q=Qm−QnbetweentwoWeylnodes,thedamp- ing rate is proportional to Ω2and a universal function of |Ω|/v|q|wherevis the Weyl electron (Fermi) velocity. The presence of Fermi pockets with chemical potential µ introduces additional dependence of the damping rate on |Ω/µ|. If the Weyl nodes are well-separated in momen- tum space, then there is no cross-talk between them in the damping rates and the momentum-space locations of the Weyl nodes can be discerned from the wavevectors at which the spin wave dissipation is locally maximized. The Weyl-electron origin of dissipation can be experi- mentally verified by the universal relativistic properties of damping over a range of mode frequencies and mo- menta, while various parameters of the Weyl spectrum can be extracted from the momentum space locations of the characteristic damping features (e.g. local maxi- mums and points where damping vanishes). The damp- ing rates involving Weyl electrons also generally exhibit “non-reciprocity”or chirality– the modes ofdifferent po- larizationsthat propagateatthe samemomentum qhave different lifetimes. We presented a procedure to obtain the field equation for the semi-classical dynamics of the localmomentmagnetizationfield, andfoundthatthedis- sipation on Weyl electrons does not give rise to Gilbert damping. One important conclusion of this study is that the spin wave damping rate reveals the relativistic nature of Weyl electrons – both through its universal depen- dence on |Ω|/v|q|and the places in momentum space where it vanishes. We computed the damping rate asso- ciated with Stoner excitations, but similar results should hold for zero-spin particle-hole excitations as well. Then, otherkindsofcollectivemodescoupledtoWeylelectrons, e.g. the phonons of the crystal or a charge density wave, shouldexhibitsimilaruniversalityintheirdampingrates. Thiswouldbeinterestingtoexploreinthefuturesincein- elastic neutron scattering is sensitive to phonons as well. The developed theory is very general within its limi- tations. It makes no assumptions about the Weyl node locations, so it applies to Diracsemimetals aswell (where the opposite-chirality Weyl nodes coexist at the samewavevectors). It also makes no assumptions about the magnetic order, so it holds for ferromagnets, antiferro- magnets and paramagnets, with or without local spin anisotropy. In this regard, however, the damping rates of spin waves are affected by the nature of magnetic or- der; we demonstrated the calculations only in the fer- romagnetic (and implicitly also the paramagnetic) case. Analytical progress was made by simplifying the model to spherically symmetric Weyl nodes that all live at the same energy. This is the main limitation of the current theory, although many implications of realistic model ex- tensions can be readily anticipated. Energy differences between the nodes are easily included by associating dif- ferent chemical potentials to the nodes, while a small Weyl node anisotropy is expected to introduce a simi- lar anisotropy in the induced dynamics and dissipation of local moments. It is possible that type-II Weyl nodes falloutsideofthistheory’sdomain, sotheirexplorationis left for future study. We also did not consider corrections due to finite temperature and disorder. Theusefulnessofthistheoryfortheexperimentalchar- acterization of magnetic Weyl semimetals is guarantied in principle, but depends on several factors in reality. The needed level of detail is not easy to achieve in the measurements of spin wave spectra. It requires at least very clean samples, low temperatures, as well as a suffi- ciently high energy resolution and adequate statistics to resolvewithlowerrorbarsthe energy/momentumdepen- dence of the inelastic neutron scattering. These aspects of measurements can always be improved, but there are also material-related constraints: phonons, for example, must not coexist with spin waves at the same momenta and frequencies. 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2021-03-05

We analyze the decay of spin waves into Stoner excitations in magnetic Weyl semimetals. The lifetime of a mode is found to have a universal dependence on its frequency and momentum, and on a few parameters that characterize the relativistic Weyl spectrum. At the same time, Gilbert damping by Weyl electrons is absent. The decay rate of spin waves is calculated perturbatively using the s-d model of itinerant Weyl or Dirac electrons coupled to local moments. We show that many details of the Weyl spectrum, such as the momentum-space locations, dispersions and sizes of the Weyl Fermi pockets, can be deduced indirectly by probing the spin waves of local moments using inelastic neutron scattering.

Universal spin wave damping in magnetic Weyl semimetals

2103.03885v1

A unied rst-principles study of Gilbert damping, spin- ip diusion and resistivity in transition metal alloys Anton A. Starikov,1Paul J. Kelly,1Arne Brataas,2Yaroslav Tserkovnyak,3and Gerrit E. W. Bauer4 1Faculty of Science and Technology and MESA+Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands 2Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway 3Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 4Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands (Dated: October 25, 2018) Using a formulation of rst-principles scattering theory that includes disorder and spin-orbit coupling on an equal footing, we calculate the resistivity , spin ip diusion length lsfand the Gilbert damping parameter for Ni 1xFexsubstitutional alloys as a function of x. For the tech- nologically important Ni 80Fe20alloy, permalloy, we calculate values of = 3:50:15Ohm-cm, lsf= 5:50:3 nm, and = 0:00460:0001 compared to experimental low-temperature values in the range 4 :24:8Ohm-cm for , 5:06:0 nm forlsf, and 0:0040:013 forindicating that the theoretical formalism captures the most important contributions to these parameters. PACS numbers: 72.25.Rb, 71.70.Ej, 72.25.Ba, 75.40.Gb, 76.60.Es Introduction. The drive to increase the density and speed of magnetic forms of data storage has focussed at- tention on how magnetization changes in response to ex- ternal elds and currents, on shorter length- and time- scales [1]. The dynamics of a magnetization Min an ef- fective magnetic eld Heis commonly described using the phenomenological Landau-Lifshitz-Gilbert equation dM dt= MHe+M"~G(M) M2sdM dt# ;(1) whereMs=jMjis the saturation magnetization, ~G(M) is the Gilbert damping parameter (that is in general a tensor) and the gyromagnetic ratio =gB=~is ex- pressed in terms of the Bohr magneton Band the Land e gfactor, which is approximately 2 for itinerant ferromag- nets. The time decay of a magnetization precession is frequently expressed in terms of the dimensionless pa- rametergiven by the diagonal element of ~G= Msfor an isotropic medium. If a non-equilibrium magnetization is generated in a disordered metal (for example by inject- ing a current through an interface), its spatial decay is described by the diusion equation @2
 @z2=
 l2 sf(2) in terms of the spin accumulation
, the dierence be- tween the spin-dependent electrochemical potentials s for up and down spins, and the spin- ip diusion length lsf[2, 3]. In spite of the great importance of andlsf, our understanding of the factors that contribute to their numerical values is at best sketchy. For clean ferromag- netic metals [4] and ordered alloys [5] however, recent progress has been made in calculating the Gilbert damp- ing using the Torque Correlation Model (TCM) [6] and the relaxation time approximation in the framework ofthe Boltzmann equation. Estimating the relaxation time for particular materials and scattering mechanisms is in general a non-trivial task and application of the TCM to non-periodic systems entails many additional complica- tions and has not yet been demonstrated. Thus, the the- oretical study of Gilbert damping or spin- ip scattering in disordered alloys and their calculation for particular materials with intrinsic disorder remain open questions. Method. In this paper we calculate the resistivity , spin- ip diusion length lsfand Gilbert damping param- eterfor substitutional Ni 1xFexalloys within a single rst-principles framework. To do so, we have extended a scattering formalism [7] based upon the local spin den- sity approximation (LSDA) of density functional theory (DFT) so that spin-orbit coupling (SOC) and chemical disorder are included on an equal footing. Relativistic eects are included by using the Pauli Hamiltonian. For a disordered region of ferromagnetic (F) alloy sand- wiched between leads of non-magnetic (N) material, the scattering matrix Srelates incoming and outgoing states in terms of re ection ( r) and transmission matrices ( t) at the Fermi energy. To calculate the scattering ma- trix, we use a \wave-function matching" (WFM) scheme [7] implemented with a minimal basis of tight-binding linearized mun-tin orbitals (TB-LMTOs) [8]. Atomic- sphere-approximation (ASA) potentials [8] are calculated self-consistently using a surface Green's function (SGF) method also implemented [9] with TB-LMTOs. Charge and spin densities for binary alloy AandBsites are calcu- lated using the coherent potential approximation (CPA) [10] generalized to layer structures [9]. For the transmis- sion matrix calculation, the resulting spherical potentials are assigned randomly to sites in large lateral supercells (SC) subject to maintenance of the appropriate concen- tration of the alloy [7]. Solving the transport problem using lateral supercells makes it possible to go beyondarXiv:1010.1626v3 [cond-mat.mtrl-sci] 19 May 20112 eective medium approximations such as the CPA. Be- cause we are interested in the properties of bulk alloys, the leads can be chosen for convenience and we use Cu leads with a single scattering state for each value of crys- tal momentum, kk. The alloy lattice constants are de- termined using Vegard's law and the lattice constants of the leads are made to match. Though NiFe is fcc only for the concentration range 0 x0:6, we use the fcc structure for all values of x. For the self-consistent SGF calculations (without SOC), the two-dimensional (2D) Brillouin zone (BZ) cor- responding to the 1 1 interface unit cell was sampled with a 120120 grid. Transport calculations including spin-orbit coupling were performed with a 32 32 2D BZ grid for a 55 lateral supercell, which is equivalent to a 160160 grid in the 1 1 2D BZ. The thickness of the ferromagnetic layer ranged from 3 to 200 monolay- ers of fcc alloy; for the largest thicknesses, the scattering region contained more than 5000 atoms. For every thick- ness of ferromagnetic alloy, we averaged over a number of random disorder congurations; the sample to sample spread was small and typically only ve congurations were necessary. Resistivity. We calculate the electrical resistivity to illustrate our methodology. In the Landauer-B uttiker formalism, the conductance can be expressed in terms of the transmission matrix tasG= (e2=h)Tr tty [11, 12]. The resistance of the complete system consisting of ideal leads sandwiching a layer of ferromagnetic alloy of thick- nessLisR(L) = 1=G(L) = 1=GSh+ 2Rif+Rb(L) where GSh= 2e2=h
Nis the Sharvin conductance of each lead withNconductance channels per spin, Rifis the interface resistance of a single N jF interface, and Rb(L) is the bulk resistance of a ferromagnetic layer of thickness L[7, 13]. When the ferromagnetic slab is suciently thick, Ohmic behaviour is recovered whereby Rb(L)Las shown in the inset to Fig. 1 for permalloy (Py = Ni 80Fe20) and the bulk resistivity can be extracted from the slope ofR(L) [14]. For currents parallel and perpendicular to the magnetization direction, the resistivities are dierent and have to be calculated separately. The average resis- tivity is given by  = (k+ 2?)=3, and the anisotropic magnetoresistance ratio (AMR) is ( k?)=. For permalloy we nd values of  = 3:50:15Ohm- cm and AMR = 19 1%, compared to experimental low- temperature values in the range 4 :24:8Ohm-cm for and 18% for AMR [15]. The resistivity calculated as a function of xis compared to low temperature literature values [15] in Fig. 1. The plateau in the calculated values around the Py composition appears to be seen in the experiments by Smit and Jaoul et al. [15]. The overall agreement with previous calculations is good [16]. In spite of the smallness of the SOC, the resistivity of Py is underestimated by more than a factor of four when it is omitted, underlining its importance for understanding transport properties. 0 20 40 60 80 1000123456ρ [µΩ ⋅ cm] Fe concentration [%]With SOC Without SOCCadeville McGuire Jaoul Smit 0 10 20 30 123R|| [fΩ ⋅ m2]L [nm]FIG. 1. Calculated resistivity as a function of the concen- trationxfor fcc Ni 1xFexbinary alloys with (solid line) and without (dashed-dotted line) SOC. Low temperature experi- mental results are shown as symbols [15]. The composition Ni80Fe20is indicated by a vertical dashed line. Inset: resis- tance of CujNi80Fe20jCu as a function of the thickness of the alloy layer. Dots indicate the calculated values averaged over ve congurations while the solid line is a linear t. Three sources of disorder which have not been taken into account here will increase the calculated values of ; short range potential uctuations that go beyond the single site CPA, short range strain uctuations re ecting the diering volumes of Fe and Ni and spin disorder. These will be the subject of a later study. Gilbert Damping. Recently, Brataas et al. showed that the energy loss due to Gilbert damping in an N jFjN scattering conguration can be expressed in terms of the scattering matrix S[17]. Using the Landau-Lifshitz- Gilbert equation (1), the energy lost by the ferromagnetic slab is, dE dt=d dt(He
M) =He
dM dt=1 2dm dt~G(M)dm dt (3) where m=M=Msis the unit vector of the magnetization direction for the macrospin mode. By equating this en- ergy loss to the energy ow into the leads [18] associated with \spin-pumping" [19], IPump E =~ 4TrdS dtdSy dt =~ 4TrdS dmdm dtdSy dmdm dt ; (4) the elements of the tensor ~Gcan be expressed as ~Gi;j(m) = 2~ 4Re Tr@S @mi@Sy @mj : (5) Physically, energy is transferred slowly from the spin de- grees of freedom to the electronic orbital degrees of free- dom from where it is rapidly lost to the phonon degrees of freedom. Our calculations focus on the role of elastic scattering in the rate-limiting rst step. Assuming that the Gilbert damping is isotropic for cu- bic substitutional alloys and allowing for the enhance- ment of the damping due to the F jN interfaces [19{21],3 0 20 40 60 80 10002468101214α [x 10−3] Fe concentration [%]Rantschler Ingvarsson Mizukami NakamuraPatton Bailey Bonin NibargerInaba Lagae Oogane0 5 10 15 20 2500.050.10.15G/(γ ⋅ µs A) [nm] L [nm] FIG. 2. Calculated zero temperature (solid line) and exper- imental room temperature (symbols) values of the Gilbert damping parameter as a function of the concentration xfor fcc Ni 1xFexbinary alloys [21{23]. Inset: total damping of CujNi80Fe20jCu as a function of the thickness of the alloy layer. Dots indicate the calculated values averaged over ve congurations while the solid line is a linear t. the total damping in the system with a ferromagnetic slab of thickness Lcan be written ~G(L) =~Gif+~Gb(L) where we express the bulk damping in terms of the dimension- less Gilbert damping parameter ~Gb(L) = Ms(L) =  sAL, wheresis the magnetization density and Ais the cross section. The results of calculations for Ni 80Fe20 are shown in the inset to Fig. 2, where the derivatives of the scattering matrix in (5) were evaluated numerically by taking nite dierences. The intercept at L= 0, ~Gif, allows us to extract the damping enhancement [20] but here we focus on the bulk properties and leave consid- eration of the material dependence of the interface en- hancement for later study. The value of determined from the slope of ~G(L)=( sA) is 0:00460:0001 that is at the lower end of the range of values 0 :0040:013 measured at room temperature for Py [21{23]. Fig. 2 shows the Gilbert damping parameter as a func- tion ofxfor Ni 1xFexbinary alloys in the fcc structure. From a large value for clean Ni, it decreases rapidly to a minimum at x0:65 and then grows again as the limit of clean fccFe is approached. Part of the decrease in with increasing xcan be explained by the increase in the magnetic moment per atom as we progress from Ni to Fe. The large values of calculated in the dilute al- loy limits can be understood in terms of conductivity-like enhancement at low temperatures [24] that has been ex- plained in terms of intraband scattering [4, 6]. The trend exhibited by the theoretical (x) is seen to be re ected by experimental room temperature results. In spite of a large spread in measured values, these seem to be sys- tematically larger than the calculated values. Part of this discrepancy can be attributed to an increase in with temperature [22, 25]. Spin diusion. When an unpolarized current is in- jected from a normal metal into a ferromagnet, the polar- ization will return to the value characteristic of the bulk 0 5 10 15 20 25 3000.20.40.60.81 z [nm] 1+β 2 1−β 2p↑ p↓FIG. 3. Fractional spin-current densities for electrons injected atz= 0 from Cu into Ni 80Fe20alloy. Calculated values (sym- bols) and ts to Eq. (6) (solid lines). ferromagnet suciently far from the injection point, pro- vided there are processes which allow spins to ip. Fol- lowing Valet-Fert [3] and assuming there is no spin- ip scattering in the N leads, we can express the fractional spin current densities p"(#)=J"(#)=Jas a function of distancezfrom the interface as p"(#)(z) =1 2 2 1exp(z=lsf)r if( + sech) (r if+lsf Ftanh) ; (6) whereJis the total current through the device, J" andJ#are the currents of majority and minority elec- trons, respectively, lsfis the spin-diusion length,  F= (#+")=4 is the bulk resistivity and is the bulk spin asymmetry ( #")=(#+"). The interface resistance r if= (r# if+r" if)=4, the interface resistance asymmetry = (rif#r" if)=(r# if+r" if) and the interface spin-relaxation expressed through the spin- ip coecient [26] must be taken into consideration resulting in a nite polarization of current injected into the ferromagnet. The correspond- ing expressions are plotted as solid lines in Fig. 3. To calculate the spin-diusion length we inject non- polarized states from one N lead and probe the transmis- sion probability into dierent spin-channels in the other N lead for dierent thicknesses of the ferromagnet. Fig. 3 shows that the calculated values can be tted using ex- pressions (6) if we assume that J=J=G=G, yielding values of the spin- ip diusion length lsf= 5:50:3 nm and bulk asymmetry parameter = 0:6780:003 for Ni80Fe20alloy compared to experimentally estimated val- ues of 0:70:1 forand in the range 5 :06:0 nm for lsf[27]. lsfandare shown as a function of concentration x in Fig. 4. The convex behaviour of is dominated by and tracks the large minority spin resistivity #whose origin is the large mismatch of the Ni and Fe minority spin band structures that leads to a x(1x) concen- tration dependence of #(x) [16]. The majority spin band structures match well so "is much smaller and changes relatively weakly as a function of x. The increase of lsf in the clean metal limits corresponds to the increase of4 0 20 40 60 80 100510152025l sf [nm] Fe concentration [%]← lsfβ → 0 20 40 60 80 1000.50.60.70.80.9 β FIG. 4. Spin-diusion length (solid line) and polarization as a function of the concentration xfor Ni 1xFexbinary alloys. the electron momentum and spin- ip scattering times in the limit of weak disorder. In summary, we have developed a unied DFT-based scattering theoretical approach for calculating transport parameters of concentrated alloys that depend strongly on spin-orbit coupling and disorder and have illustrated it with an application to NiFe alloys. Where comparison with experiment can be made, the agreement is remark- ably good oering the prospect of gaining insight into the properties of a host of complex but technologically important magnetic materials. This work is part of the research programs of \Sticht- ing voor Fundamenteel Onderzoek der Materie" (FOM) and the use of supercomputer facilities was sponsored by the \Stichting Nationale Computer Faciliteiten" (NCF), both nancially supported by the \Nederlandse Organ- isatie voor Wetenschappelijk Onderzoek" (NWO). It was also supported by \NanoNed", a nanotechnology pro- gramme of the Dutch Ministry of Economic Aairs and by EC Contract No. IST-033749 DynaMax. [1] See the collection of articles in Ultrathin Magnetic Struc- tures I-IV , edited by J. A. C. Bland and B. Heinrich (Springer-Verlag, Berlin, 1994-2005). [2] P. C. van Son, H. van Kempen, and P. Wyder, Phys. Rev. Lett., 58, 2271 (1987); 60, 378 (1988). [3] T. Valet and A. Fert, Phys. Rev. B, 48, 7099 (1993). [4] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett., 99, 027204 (2007); J. Appl. Phys., 103, 07D303 (2008); V. Kambersk y, Phys. Rev. B, 76, 134416 (2007). [5] C. Liu, C. K. A. Mewes, M. Chshiev, T. Mewes, and W. H. Butler, Appl. Phys. Lett., 95, 022509 (2009). [6] V. Kambersk y, Czech. J. Phys., 26, 1366 (1976). [7] K. Xia, P. J. Kelly, G. E. W. Bauer, I. Turek, J. Ku- drnovsk y, and V. Drchal, Phys. Rev. B, 63, 064407 (2001); K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W. Bauer, 73, 064420 (2006). [8] O. K. Andersen, Z. Pawlowska, and O. Jepsen, Phys. Rev. B, 34, 5253 (1986); O. K. Andersen, 12, 3060(1975). [9] I. Turek, V. Drchal, J. Kudrnovsk y, M. Sob, and P. Weinberger, Electronic Structure of Disordered Al- loys, Surfaces and Interfaces (Kluwer, Boston-London- Dordrecht, 1997). [10] P. Soven, Phys. Rev., 156, 809 (1967). [11] M. B uttiker, Y. Imry, R. Landauer, and S. Pinhas, Phys. Rev. B, 31, 6207 (1985). [12] S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995). [13] K. M. Schep, J. B. A. N. van Hoof, P. J. Kelly, G. E. W. Bauer, and J. E. Ingleseld, Phys. Rev. B, 56, 10805 (1997). [14] The non-linearity of the resistance for L.20 in the inset to Fig. 1 has nothing to do with spin-orbit coupling. It already results when spin-dependent resistances of the formR(L) = 2=GSh+ 2R if+Lare added in parallel in the two-current series-resistor (2CSR) model; the total resistance only becomes linear for values of Lso large that the bulk resistances Lare much larger than the interface and Sharvin terms. [15] J. Smit, Physica, 17, 612 (1951); T. R. McGuire and R. I. Potter, IEEE Trans. Mag., 11, 1018 (1975); O. Jaoul, I. Campbell, and A. Fert, J. Magn. & Magn. Mater., 5, 23 (1977); M. C. Cadeville and B. Loegel, J. Phys. F: Met. Phys., 3, L115 (1973). [16] J. Banhart and H. Ebert, Europhys. Lett., 32, 517 (1995); J. Banhart, H. Ebert, and A. Vernes, Phys. Rev. B, 56, 10165 (1997). [17] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. Lett., 101, 037207 (2008). [18] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, Phys. Rev. Lett., 87, 236601 (2001); M. Moskalets and M. B uttiker, Phys. Rev. B, 66, 035306 (2002); 66, 205320 (2002). [19] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett., 88, 117601 (2002); Phys. Rev. B, 66, 224403 (2002). [20] M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer, Phys. Rev. B, 71, 064420 (2005). [21] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. & Magn. Mater., 226{230 , 1640 (2001); Jpn. J. Appl. Phys., 40, 580 (2001). [22] W. Bailey, P. Kabos, F. Manco, and S. Russek, IEEE Trans. Mag., 37, 1749 (2001). [23] C. E. Patton, Z. Frait, and C. H. Wilts, J. Appl. Phys., 46, 5002 (1975); S. Ingvarsson, G. Xiao, S. Parkin, and R. Koch, Appl. Phys. Lett., 85, 4995 (2004); H. Naka- mura, Y. Ando, S. Mizukami, and H. Kubota, Jpn. J. Appl. Phys., 43, L787 (2004); J. O. Rantschler, B. B. Maranville, J. J. Mallett, P. Chen, R. D. McMichael, and W. F. Egelho, IEEE Trans. Mag., 41, 3523 (2005); R. Bonin, M. L. Schneider, T. J. Silva, and J. P. Nibarger, J. Appl. Phys., 98, 123904 (2005); L. La- gae, R. Wirix-Speetjens, W. Eyckmans, S. Borghs, and J. de Boeck, J. Magn. & Magn. Mater., 286, 291 (2005); J. P. Nibarger, R. Lopusnik, Z. Celinski, and T. J. Silva, Appl. Phys. Lett., 83, 93 (2003); N. Inaba, H. Asanuma, S. Igarashi, S. Mori, F. Kirino, K. Koike, and H. Morita, IEEE Trans. Mag., 42, 2372 (2006); M. Oogane, T. Wak- itani, S. Yakata, R. Yilgin, Y. Ando, A. Sakuma, and T. Miyazaki, Jpn. J. Appl. Phys., 45, 3889 (2006). [24] S. M. Bhagat and P. Lubitz, Phys. Rev. B, 10, 179 (1974); B. Heinrich, D. J. Meredith, and J. F. Cochran, J. Appl.5 Phys., 50, 7726 (1979). [25] D. Bastian and E. Biller, Phys. Stat. Sol. A, 35, 113 (1976). [26] W. Park, D. V. Baxter, S. Steenwyk, I. Moraru, W. P.Pratt, Jr., and J. Bass, Phys. Rev. B, 62, 1178 (2000). [27] J. Bass and W. P. Pratt Jr., J. Magn. & Magn. Mater., 200, 274 (1999); J. Phys.: Condens. Matter, 19, 183201 (2007).

2010-10-08

Using a formulation of first-principles scattering theory that includes disorder and spin-orbit coupling on an equal footing, we calculate the resistivity $\rho$, spin flip diffusion length $l_{sf}$ and the Gilbert damping parameter $\alpha$ for Ni$_{1-x}$Fe$_x$ substitutional alloys as a function of $x$. For the technologically important Ni$_{80}$Fe$_{20}$ alloy, permalloy, we calculate values of $\rho = 3.5 \pm 0.15$ $\mu$Ohm-cm, $l_{sf}=5.5 \pm 0.3$ nm, and $\alpha= 0.0046 \pm 0.0001$ compared to experimental low-temperature values in the range $4.2-4.8$ $\mu$Ohm-cm for $\rho$, $5.0-6.0$ nm for $l_{sf}$, and $0.004-0.013$ for $\alpha$ indicating that the theoretical formalism captures the most important contributions to these parameters.

A unified first-principles study of Gilbert damping, spin-flip diffusion and resistivity in transition metal alloys

1010.1626v3

arXiv:1809.01042v1 [cond-mat.mtrl-sci] 4 Sep 2018Separation of the two-magnon scattering contribution to da mping for the determination of the spin mixing conductance A. Conca,1,∗S. Keller,1M. R. Schweizer,1E. Th. Papaioannou,1and B. Hillebrands1 1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany (Dated: September 5, 2018) We present angle dependent measurements of the damping prop erties of epitaxial Fe layers with MgO, Al and Pt capping layers. Based on the preferential dist ribution of lattice defects following the crystal symmetry, we make use of a model of the defect density to separate the contribution of two- magnon scattering to the damping from the isotropic contrib ution originating in the spin pumping effect, the viscous Gilbert damping and the magnetic proximi ty effect. The separation of the two- magnon contribution, which depends strongly on the defect d ensity, allows for the measurement of a value of the effective spin mixing conductance which is clos er to the value exclusively due to spin pumping. The influence of the defect density for bilayers sys tems due to the different capping layers and to the unavoidable spread in defect density from sample t o sample is thus removed. This shows the potential of studying spin pumping phenomena in fully or dered systems in which this separation is possible, contrary to polycrystalline or amorphous meta llic thin films. INTRODUCTION In bilayers systems formed by a ferromagnetic (FM) layer in contact with a metallic non-magnetic (NM) one, a pure spin current can be generated and injected in the latterwhen the ferromagneticresonanceisexcited. Typi- cally, a microwavemagnetic field is used for this purpose. The whole processis commonly referredto as spin pump- ing [1, 2]. If the non-magnetic layer is formed by a heavy metal with large spin-orbit coupling (Pt, Ta or similar), the spin current can be detected by using the inversespin Hall effect (ISHE) for conversion into a charge current. Since the spin current leaving the magnetic layer car- ries away angular momentum from the magnetization precession,it representsanadditionallosschannelforthe magnetic system and consequently causes an increase in the measured Gilbert damping parameter α[1]: ∆αsp=γ/planckover2pi1 4πMsdFMg↑↓(1) whereg↑↓is the real part of the spin mixing conductance which is controlling the magnitude of the generated spin current and γis the gyromagnetic ratio. This expression is only valid for sufficiently thick NM layers where no reflection of the spin current takes place at the film surface or interface with other materials, i.e. no spin current is flowing back into the magnetic layer. In principle, it allows the estimation of g↑↓by measur- ing the increase in damping compared to the intrinsic value. However, to perform this measurement is not straightforward. If the estimation of g↑↓for a FM/Pt system is needed, ideally one should measure the effec- tive Gilbert damping parameter α0for a single stand- ing magnetic layer acting as a reference sample with no losses due to spin pumping and repeat the same after de- positing a thick Pt layer. However, most of the commonferromagnetic materials, with exception of the magnetic insulators like YIG, will change its properties due to ox- idation processes. Therefore, a capping layer is required and one has to find an appropriate one, in the sense that its introduction must not modify the damping properties of the magnetic layer. Examples in the literature show that this is far to be a trivial task [3–5]. In addition to this, the emergence of a finite magnetic polarization in Pt in contact with a ferromagnetic layers has an impact on damping which further hinders the estimation of g↑↓ [5–12]. For the reference layers, the most convenient candi- dates as capping material are oxides like MgO, for which it has been proven that they are able to block the flow of spin current and therefore to deactivate spin pumping [13–15], or metals with weak spin-orbit interaction like Al or Ru. But even for these cases, it has been shown that an increase of damping not related to spin pumping is possible. Ruiz et al.show for instance that a MgO capping layer increases strongly the damping in permal- loy while this is not the case for Al capping layer [5]. The reason has nothing to do with the metallic char- acter of the capping layer since the increase for Ru is even larger than with MgO. The same work [5] already provides a hint for a possible reason since the increase of damping roughly scales with the value of the inter- face perpendicular anisotropy constant K⊥ S. Theoretical works [16] show that the counterplay between the de- magnetizing field responsible for the in-plane orientation of the magnetization and the perpendicular anisotropy field can induce inhomogeneous magnetization states for certain field strengths combinations which are responsi- ble for an increased damping. In this sense, this effect has been also adduced to explain the damping thickness dependence in Co 2FeAl/MgO systems [3]. Here we present angle dependent measurements of the damping properties of epitaxial Fe layers with MgO, Al2 FIG. 1. (Color online) Dependence of the FMR linewidth on the frequency for different orientations φHof the external magnetic field with respect to the [100] crystallographic axis of Fe fo r (a) Fe/Al and (b) Fe/Pt systems. The lines correspond to a li near fit to extract the effective damping parameter αeff. Forφ= 30◦a strong non-linearity due to magnetic dragging is observed . For visibility reasons, each data set is shifted vertically by 1.25 mT with respect the previous one. and Pt capping layers. Fully epitaxial systems consti- tute a perfect ordered model with almost ideal and well defined interfaces. Here, we will show that the angle de- pendence of damping allows for a measurement of the strength of the two-magnon scattering and of its contri- bution to the effective damping parameter. With the separation of this contribution we access the increase of damping caused only by spin pumping and magnetic proximity effect and to an estimation of g↑↓without the contamination of defects effects. EXPERIMENTAL DETAILS The samples were deposited by e-beam evaporation on MgO(100) substrates in a molecular beam epitaxy (MBE) chamber with a base pressure P b= 5× 10−10mbar. A set of Fe/Pt bilayers with fixed Fe thick- ness (12 nm) and varying Pt thickness were prepared. Additional reference samples, where Pt is substituted by MgO or Al, have also been prepared. The Fe and Pt films were grown with a deposition rate of 0.05 ˚A/s. The samples were deposited with a substrate temperature of 300◦C and subsequently annealed at the same tempera- ture. The characterization by X-ray diffractometry (XRD) (presented elsewhere [17]) shows that the Fe/Pt bilayers are fully epitaxial with the Fe unit cell rotated by 45◦ with respect to the MgO substrate unit cell and with Pt rotated again 45◦with respect to Fe. In the case of Fe/Al, epitaxial growth of the upper layer could not be achieved. The dynamic properties and material parameters were studied by measuring the ferromagnetic resonance using a strip-line vector network analyzer (VNA-FMR). Forthis, the samples were placed facing the strip-line and the˜S12transmission parameter was recorded. RESULTS AND DISCUSSION Figures 1 shows the dependence of the measured FMR line width ∆ Hon the frequency for the reference layer with Al capping (a) and a Fe/Pt system (b). The data is shown for different orientation of the external static magnetic field varying from φH= 0◦([100], easy axis) to φH= 45◦([110], hard axis). For visibility reasons, each data set is shifted vertically by 1.25 mT with respect to the previous one. As commented before, the choice of capping layer can have a large influence on the linewidth and effective dampingofthe magneticlayer,evenforlightmetals. The magnetic proximity effect (MPE) in the case of Pt also contributes to an increase on damping, [5, 9–12] which additionally challenges the measurement of the contri- bution from the spin pumping. Taking into account all these considerations, the effective increase on damping when comparing a reference system and a system with a heavy metal can be separated as follows: αeff=α0+αmpe+αsp+αi. (2) Hereα0is the intrinsic damping parameter which can be defined as characteristic of the material under in- vestigation (growth conditions however may influence it strongly) and it is the sum of the losses by two-magnon scattering and by energy transfer to the phonon system. αmpeis the contribution due to the dynamic coupling be- tween the ordered spins in Pt due to the MPE and the3 FIG. 2. (Color online) (a) Dependence of the FMR resonance fieldHFMRon the in-plane direction of the static magnetic field for two values of the resonant frequency. (b) Dependenc e of the in-plane angle of the magnetization vector φMon the external field direction φH. Both angles are measured relative tothe[100] axis. Thedottedline represents thecase of perf ect collinearity between magnetization and external field. magnetization in the magnetic layer. αspis the result of the losses by the spin current generated in the fer- romagnetic layer by the precession of the magnetization and that flows into the Pt layer (spin pumping). The last termαisummarizes the increase of damping due to other interfacial effects such as interface PMA as commented above, spin memory loss [18] or isotropic scattering at interface defects [19]. Several efforts have been made in order to separate some of the contributions to αeff. In a recent work with CoFeB/Pt [9] we were able to separate αmpedue to the dependence on the Pt thickness. As already reported by Caminale et al.[11], a linear Pt thickness dependence of the spin-current absorption in spin-sink layers exhibiting MPE and of αmpeis expected [12]. A detailed vector network analyzer FMR study has also been recently re- ported to separate the different contributions in NiFe/Pt systems [20]. Theterm α0isaresultoftwocontributions[22]. Oneis the pure Gilbert damping, which is of viscous nature and generates a dissipation of energy and angular momentum tothe lattice. The secondoneisthe transfertospin-wavemodes with k/negationslash= 0 from the FMR mode via two-magnon scattering. For a pure Gilbert-like viscous damping the linewidth dependence on the frequency is purely linear: µ0∆H=µ0∆H0+4παf γ. (3) Here, ∆H0is the inhomogeneous broadening and is re- lated to film quality. The lines in Figs. 1 (a) and (b) are a fit to this ex- pression. It has to be mentioned that although a viscous damping generates a linear dependence, on the contrary it is not possible to assume that the observation of a lin- earbehavior provesthat only viscous damping is present. The reason for that is that two-magnon scattering can mimic also a linear dependence [21–23]. For both sam- ples, and for the MgO capped sample not shown here, forφ= 30◦a strongly non linear behavior with a large increase in linewidth values for smaller frequencies is ob- served. For this reason, the hollow points in Fig. 1 have been excluded from the fit. The non-linearity at low fre- quencies cannot be explained by viscous damping and it is caused by magnetic dragging. The magnetic dragging effect describesthe increaseofthe linewidth of precessing magneticlayerswithlargemagneticcrystalineanisotropy due to the non-collinearity of the magnetization and the external magnetic field. In Fig. 2 (a), the dependence of the resonance field HFMRon the in-plane direction of the external magnetic field is shown for two fixed fre- quency values. As a result of the four-fold anisotropy ex- pectedfromthecubiclatticeofFeandassumingaperfect collinearity between magnetization vector and external field,HFMRcan be modeled as: [10, 24] µ0HFMR=µ0˜HFMR+2K1 Mscos(4φ),(4) whereK1is the cubic anisotropyconstant, φthe in-plane azimuthal angle and ˜HFMRis the averaged resonance field value. The fraction2K1 Msis directly the anisotropy field H B. In Fig. 2(a) a deviation from this model is ob- served for angles between the hard and easy axis and it is due to magnetic dragging, i.e., the magnetization is not aligned to the external field due to the effect of the anisotropy field. The fact that the deviation from the model in Eq.4 is smallerfor largerfrequencies(i.e. larger applied field) alsosupportsthis interpretation. The same behavior observed for φ= 30◦has been also been re- ported for ultrathin Fe films [25] or for insulating LSMO films [28] and attributed to magnetic dragging. The de- gree of non-collinearity can be estimated by solving the equilibrium condition for the angle defining the orienta- tion of the magnetization φMfor each value of φH: Hsin(φM−φH)+HB 4sin(4φM) = 0,(5)4 where the value for the cubic anisotropy field was taken from [10]. Fig. 2(b) shows the obtained value of φMfor the data shown in Fig. 2(a). The angle between mag- netization and magnetic field can be as large as 10◦for 13 GHz and it is decreased to a maximum around 4.5◦ for 18 GHz. The magnetic dragging effect is largest for φHbetween the easy and hard axis and vanishes along the main crystallographic axes. Figure 3 shows the value of the effective damping pa- rameter αeffas obtained from the fits in Fig. 1 for the three capping layers. In all of them, an eight-fold sym- metry on the in-plane angle φHis observed with maxima along the easy and hard axis of the Fe layers and min- ima in between. For the Fe/Al and Fe/MgO samples, where spin pumping has no influence, αeff=α0+αi while for the Fe/Pt sample, where both losses through spin pumping and due to the MPE are active, we obtain the situation shown in Eq. 2. It is remarkable that the different origins of the damping do not change the overall symmetry of the angular dependence. It has though an impact on the absolute values, which are larger for the Fe/Pt sample. In the literature concerning epitaxial layers, it is possi- ble to find different symmetries for the dependence of the FMRlinewidthorthedampingparameteronthein-plane field direction. For the Heusler alloy Co 2FeAl both four- and eight-fold symmetries for the linewidth have been reported. The situation differs depending on the thick- nessofthe film [23] andalsobetween different groups[30] pointing out to a role of the growth conditions. For Fe 3Si films and Fe/V multilayer systems a four-fold symmetry is reported [22, 26] and for ultrathin Fe layers, where the role of the interface is strong, a two-fold symmetry of αeffhas been measured [25]. Eight-fold symmetry has been also observed in epitaxial FeSi systems [26, 29]. In a different work on Fe layers, a decrease on the obtained αvalue along the intermediate orientation between the twomainaxisrelativetothe onemeasuredalongtheeasy and hard axis was reported [27], pointing to an angular dependence very similar to ours. Concerning insulating systems, two- and four-fold symmetries have been ob- served in LSMO films [28]. Two-magnon scattering can only occur if scattering centers in form of defects are present. If, as expected, these are present as point lattice defects or dislocation linesalongthemaincrystallographicdirections, itisclear that the scattering intensity should reflect the symmetry of the lattice. This fact would for certain explain a four- or eight-fold anisotropy in damping observed in some on the reports mentioned above and the maxima in αefffor our samples for φ= 0◦,45◦,90◦,135◦. FollowingZakeri et al. andAria et al., the contribution to damping due to two-magnonscattering can be written as [21, 26]:α2M=/summationdisplay /angbracketleftxi/angbracketrightΓ/angbracketleftxi/angbracketrightf(φH−φ/angbracketleftxi/angbracketright), (6) where Γ /angbracketleftxi/angbracketrightrepresents the strength of the two-magnon scattering contribution along the in-plane crystallo- graphic direction /angbracketleftxi/angbracketright. The function f(φH−φ/angbracketleftxi/angbracketright) al- lows for an angle dependent two-magnon contribution to damping with respect to the orientation of the external fieldHrelative to the crystallographic directions /angbracketleftxi/angbracketright. The physical interpretation of the function f(φH−φ/angbracketleftxi/angbracketright) lays in the Fourier transform of the defects in the film [26, 34]. By using the ansatz f(φH−φ/angbracketleftxi/angbracketright) = cos2(4φH− φ/angbracketleftxi/angbracketright) we can fit the damping dependence using a simpli- fied version: αeff=αiso+α2M=αiso+Γ2Mcos2(4φH−φ[100]) (7) whereαisoincludes now all the isotropic contributions to damping, i.e. αmpe,αsp, pure Gilbert damping and po- tentially isotropic interface contributions from the term αi, mainly spin memory loss and interface PMA related effects. The red lines in Fig. 3 show the fit to this model. The obtained parameters are summarized in Table. I. A very low value below 1 ×10−3is obtained for αisofor the Fe/Al sample. Since αsp,MPE= 0 is expected and due to the low value we consider that the obtained αisomust be very close to the value corresponding only to pure vis- cous Gilbert damping corresponding to high quality Fe. However, strictly speaking, the obtained value is only an upper limit since still other effects might contribute. Concerning 3d metals with no half-metallic character, a very low damping value of 0.7 ×10−3has been reported by Leeet al.for CoFe [35]. This value is comparable to theαisomeasured here for Fe/Al. The fact that the CoFe samples in which the low value was obtained are also fully epitaxial with an exceptionally high crystalline quality explains the similarity in values. The low defect density in CoFe almost suppresses two-magnon scatter- ing in the CoFe samplesand thereforeis comparablewith ourαisowhere that contribution is already separated. For the Fe/MgO sample the value for αisoincreases by a factor larger than 2 although also here αsp,MPE= 0. αiso Γ2M (10−3) (10−3) Fe/Al 0.8 ±0.3 3.6 ±0.4 Fe/Pt 3.4 ±0.3 2.4 ±0.4 Fe/MgO 1.9 ±0.1 1.3 ±0.1 TABLE I. Isotropic contribution αisoand two-magnon scat- tering contribution Γ 2Mto the total effective damping param- eterαeff.5 FIG. 3. (Color online) Angular dependence of the effective da mping parameter αeffin the in-plane direction of the static magnetic field φHfor (a) Fe/Al, (b) Fe/Pt and (c) Fe/MgO. The red lines are a fit t o Eq. 7. The main differences between Fe/Al and Fe/MgO are that the MgO is single crystalline while Al is polycrys- talline and the contrast between the metallic character of Al with the insulating oxide. The lattice mismatch between MgO and Fe is around 4% and introduces there- fore a certain degree of stress in the Fe layer which is not present when the capping is polycrystalline Al and whichcanhaveanimpactondamping. Atthesametime, since the Gilbert damping is sensitive to the density of states and this one is modified at the interface by the kind of bonds between the Fe atom and the atoms from the cappinglayer, the simple materialdifferencemay also explain the difference. In this sense it is remarkable that the low damping value by Lee et al.commented before is only observed for CoFe with a MgO capping layer and a largervalue is measuredwhen MgAl 2O4is used [35]. Our data confirms the important role of the capping layer on damping observed in other works [5]. A further increase in the value of αisois observed for the Fe/Pt sample where additional losses through spin pumping and MPE are present. Unfortunately the data presented in this paper does not allow to disentangle these two contributions. For this reason, when using Eq. 1 for the calculation of spin mixing conductance, it makes sense to refer to an effective value g↑↓ effwhich is at the same time an upper limit for the corresponding value for spin pumping alone. Using the Fe/Al sample as a reference we obtain a value for the spin mixing conduc- tance of 3 .7±0.9×1019m−2. This value is lower than the one presented in our previous report [10] and shows that the value of g↑↓ effcan be easily overestimated if the effect of two-magnon scattering on damping is not sepa- rated, with the consequent overestimation of the injected spin current and underestimation of the spin Hall angle from the ISHE voltage [17]. The advantage of using epi- taxial magnetic layers is that they allow the separation of the contribution of the two-magnon scattering due to the strongangulardependenceand welldefinedcrystallo- graphicdirections. Thesameisnotpossibleincommonly used material as CoFeB or NiFe where the amorphous or polycrystalline nature of the layers blends the scatteringdependence on the in-plane angle. The parameter Γ 2Mprovides further insight into the origin of total damping in the samples. This parameter is larger for the Fe/Al sample in comparison to the fully epitaxial bilayers being almost three times larger than for Fe/MgO. As a result, the total damping in the Fe/Al sample is dominated by the two-magnon scattering due alsoto the low αisowhile the sameis not true in the other two systems. It has to be taken into account that, since as scattering centers for magnon scattering the defects at the interfaces play a role, they can be dominant in thin films. From TEM images (presented for instance in [10]), we can prove the existence of a highly ordered interface in the fully epitaxial samples. Of course, the same is not true for the case with polycrystalline Al capping. We believe that the dominant role of the interface here is possible, also due to the overall low defect density in the bulk of the Fe layer. For completeness we want to discuss two additional effects potentially affecting the linewidth and damping. Due to the spread of internal and anisotropy field due to mosaicity in the film, there is a contribution to the line broadening which has the following form [26, 33]: ∆Hmosaic=/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂HFMR ∂φH/vextendsingle/vextendsingle/vextendsingle/vextendsingle∆φH, (8) where ∆φHis the average spread of the direction of the easy axes in the film plane. From Fig. 1(c) it is clear that this contribution should increase the linewidth in theregion φ= 15−30◦andequivalentonesbutthis isnot observed pointing to a weak impact of mosaicity. In any case, the mosaicity term is frequency independent and will be only visible in the inhomogeneous linebroadening ∆H0and will not affect the determination of αeff. The discussion followingthe introduction ofEqs.6 and 7 was focused on crystalline lattice defects as the origin of two-magnon scattering. However any kind on inhomo- geneity in the magnetic state of the sample may play the same role. The presence of magnetic dragging, visible for instance for φ= 30◦in Fig. 1 can create a slight inhomo- geneity in the magnetization state for field orientations6 close to the hard axis direction and an increase of damp- ing around the hard axis orientation. In any case, this contribution follows also the symmetry of the lattice and it is accounted in the Γ 2Mparameter. Although certain theoretical works point to an anisotropic Gilbert damping in fully epitaxial systems due to its dependence on the density of states at the Fermi energy [31, 32], experimentally this has been only seen in ultrathin Fe films [22] due to the modification of the electronic structure induced by the interfacial spin- orbit coupling. The anisotropy in αeffpresented here can be fully explained by two-magnon scattering, and there- fore an isotropic Gilbert damping can be assumed. CONCLUSIONS Making use of the well defined dependence of the two- magnon scattering mechanism on the in-plane field di- rection, we have been able to separate this contribution to damping from the isotropic contributions originating from the viscous Gilbert damping mechanism, from spin pumping and from the magnetic proximity effect in Pt. The method can be implemented thanks to the pref- erential ordering of crystalline defects with respect to the crystallographic directions in epitaxial systems and therefore cannot be extended to amorphous or polycrys- talline magnetic films. This shows the potential of the study of spin pumping related phenomena in ordered systems. Without the contribution of the two-magnon scattering, which depends strongly on the chosen cap- ping layer and defect density, a value of the effective spin mixing conductance g↑↓ effis obtained which is closer to theg↑↓associated only to spin pumping. This approach allows for a better estimation of the spin Hall angle in metals. ACKNOWLEDGEMENTS Financial support by M-era.Net through the HEUMEM project and by the Carl Zeiss Stiftung is gratefully acknowledged. ∗conca@physik.uni-kl.de [1] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, No. 4, 1375 (2005). [2] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002). [3] A. Conca, A. Niesen, G. Reiss, and B. Hillebrands, J. Phys. D: Appl. Phys. 51, 165303 (2018). [4] A. Natarajarathinam, Z. R. Tadisina, T. Mewes, S. Watts, E. Chen, and S. Gupta, J. Appl. Phys. 112, 053909 (2012)[5] A. Ruiz-Calaforra, T. Br¨ acher, V. Lauer, P. Pirro, B. Heinz, M. Geilen, A. V. Chumak, A. 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[12] A. Ghosh, S. Auffret, U. Ebels, and W. E. Bailey, Phys. Rev. Lett. 109, 127202 (2012). [13] K. Eid, R. Fonck, M. A. Darwish, W. P. Pratt, and J. Bass, J. Appl. Phys. 91, 8102 (2002). [14] L. Mihalceanu, S. Keller, J. Greser, D. Karfaridis, K. Symeonidis, G. Vourlias, T. Kehagias, A. Conca, B. Hillebrands, and E. Th. Papaioannou, Appl. Phys. Lett. 110, 252406 (2017). [15] O. Mosendz, J. E. Pearson, F. Y. Fradin, S. D. Bader, A. Hoffmann, Appl. Phys. Lett. 96, 022502 (2010). [16] N. A. Usov, and O. N. Serebryakova, J. Appl. Phys. 121, 133905 (2017). [17] S. Keller, L. Mihalceanu, M. R. Schweizer, P. Lang, B. Heinz, M. Geilen, T. Br¨ acher, P. Pirro, T. Meyer, A. Conca, D. Karfaridis, G. Vourlias, T. Kehagias, B. Hillebrands and E. Th. Papaioannou, New J. Phys. 20053002 (2018). [18] M. A. Khasawneh, C. Klose, W. P. Pratt, and N. O. Birge, Phys. Rev. B 84, 014425 (2011). [19] S. Ingvarsson, L. Ritchie, X. Y. Liu, Gang Xiao, J. C. Slonczewski, P. L. Trouilloud, and R. H. Koch, Phys. 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2018-09-04

We present angle dependent measurements of the damping properties of epitaxial Fe layers with MgO, Al and Pt capping layers. Based on the preferential distribution of lattice defects following the crystal symmetry, we make use of a model of the defect density to separate the contribution of two-magnon scattering to the damping from the isotropic contribution originating in the spin pumping effect, the viscous Gilbert damping and the magnetic proximity effect. The separation of the two-magnon contribution, which depends strongly on the defect density, allows for the measurement of a value of the effective spin mixing conductance which is closer to the value exclusively due to spin pumping. The influence of the defect density for bilayers systems due to the different capping layers and to the unavoidable spread in defect density from sample to sample is thus removed. This shows the potential of studying spin pumping phenomena in fully ordered systems in which this separation is possible, contrary to polycrystalline or amorphous metallic thin films.

Separation of the two-magnon scattering contribution to damping for the determination of the spin mixing conductance

1809.01042v1

1 Coherent Magnetization Precession in GaMnAs induced by Ultrafast Optical Excitation J. Qi, Y. Xu, N. Tolk Department of Physics and Astronomy, Vanderbilt University, Nashville, TN, 37235 X. Liu, J. K. Furdyna Department of Physics, University of Notre Dam e, Notre Dame, IN 46556 I. E. Perakis Department of Physics, University of Crete, Heraklion, Greece We use femtosecond optical pulses to induce, control and monitor magnetization precession in ferromagnetic Ga 0.965 Mn0.035 As. At temperatures below ~40 K we observe coherent oscillations of the local Mn spins, triggered by an ultrafast photoinduced reorientation of the in -plane easy axis. The amplitude saturation of the oscillations above a certain pump intensity indicates that the easy axis remains unchanged above ~T C/2. We find that the observed magnetization precession damping (Gilbert damping) is strongly dependent on pump laser intensity, but largely independent on ambient temperature. We provide a physical interpretation of the observed light -induced col lective Mn -spin relaxation and precession. The magnetic semiconductor GaMnAs has received considerable attention in recent years, largely because of its potential role in the development of spin -based devices1,2. In this itinerant ferromagnet, the collec tive magnetic order arises from the interaction between mobile valence band holes and localized Mn spins. Therefore, the magnetic properties are sensitive to external excitations that change the carrier density and distribution. Ultrafast pump -probe magnet o-optical spectroscopy is an ideal technique for controlling and characterizing the magnetization dynamics in the magnetic materials, and has been applied to the GaMnAs system by several groups3,4. Although optically induced precessional motion of magne tization has been studied in 2 other magnetic systems5, magnetization precession in ferromagnetic GaMnAs has been observed only recently4 and has yet to be adequately understood. In this paper, we report comprehensive temperature and photoexcitation intens ity dependent measurements of photoinduced magnetization precession in Ga 1-xMnxAs (x = 0.035) with no externally imposed magnetic field. By comparing and contrasting the temperature and intensity dependence of the precession frequency, damping, and amplitu de, we identify the importance of light -induced nonlinear effects and obtain new information on the relevant physica l mechanisms. Our measurements of the photoinduced magnetization show coherent oscillations, arising from the precession of collective Mn sp ins. Amplitude of the magnetization precession saturates above certain pump intensity is a strong indication that direction of the magnetic easy axis remains unchanged at temperatures above about half the Curie temperature (T C). The precession is explained by invoking an ultrafast change in the orientation of the in-plane easy axis, due to an impulsive change in the magnetic anisotropy induced by the laser pulse. We also find that the Gilbert damping coefficient, which characterizes the Mn -spin relaxation, depends only weakly on the ambient temperature but changes dramatically with pump intensity . Our results suggest a general model for photoinduced precessional motion and relaxation of magnetization in the GaMnAs system under compressive strain. Time -resolved magneto -optical Kerr effect (MOKE) measurements were performed on a 300 nm thick ferromagnetic Ga 1-xMnxAs (x = 0.035) sample, with background hole density p ≈1020 cm-3 and T C ≈70 K. The sample was grown by low temperature molecular beam epitaxy on a G aAs(001) substrate, and was therefore under compressive strain. The pump -probe experiment employed a Coherent MIRA 900 Ti:Sapphire laser, which produced ~150 -fs-wide pulses in the 720 nm (1.719 eV) to 890 nm (1.393 eV) wavelength range with a repetition ra te of 76 MHz. The pump beam was incident normal to the sample, while the probe was at an angle of about 30o to the surface normal. The polarization of the pump beam could be adjusted to be 3 linear, right -circular ( σ+), or left -circular ( σ-) polarization. Th e probe beam was linearly polarized. This configuration produced a combination of polar and longitudinal MOKE, with the former dominating6. The temporal Kerr rotation signal was detected using a balanced photodiode bridge, in combination with a lock -in amplifier. Both pump and probe beams were focused onto the sample with a spot diameter of about 100 µm, with an intensity ratio of 15:1. The pump light typically had a pulse energy of 0.065 nJ, and a fluence of 0.85 µJ/cm2. Figure 1(a) shows our time -resolve d Kerr rotation (TRKR) measurements at temperature of 20 K. The amplitude of the temporal Kerr rotation signal was found to be symmetric with respect to right or left -circularly polarized photo -excitation. In particular, we observe a superimposed oscillato ry behavior at temperatures less than ~40 K, indicating magnetization precession. It is important to note that these oscillations were observed not only with σ+ and σ- polarized but also with linearly polarized pump light . The phase difference among the os cillations for σ+ or σ- pump excitation is less than ~5o. This negligible phase difference implies that the oscillatory behavior is not due to the non -thermal circular polarization -dependent carrier spin dynamics5. After the initial few picoseconds, where equilibration between the electronic and lattice systems occurs, the oscillations can be fitted well by the following equation (see Fig. 1(b)): ) cos()/ exp( )( 0j w t q + − = t t AtK (1), where A0, w, t, and j are the amplitude, precession frequency, decay time, and initial phase of the oscillation, respectively. Some fitted parameters are shown in figure 2 as a function of pump intensity and temperature. On a sub -picosecond time scale, the photoexcited electrons/holes scatter and thermalize with the Fermi sea via electron -electron interactions. Following this initial non-thermal temporal regime, the properties of the GaMnAs system can be characterized approximately by time -dependent carrier and lattice temperatures heT/ 4 and lT. Subsequently, the carrier system transfers its energy to the lattice within a few picoseconds via the electron -phonon interaction. This leads to a quasi -equilibration of heT/and lT, which then relax back to the equilibrium tempera ture via a slow (ns) thermal diffusion process. Mn precession was also observed in Ref. 4 and was attributed to the change of uniaxial anisotropy due to the increase in hole concentration7. In our experiment, for a typical pump intensity of 0.065 nJ/pulse, we estimate that the relative increase in hole concentration is ~0.1%. The resulting transient increase in local temperature and hole concentration leads to an impulsive change in the in -plane magnetic anisotropies and in the easy axis orientation . As a r esult, the effective magnetic field experienced by the Mn spins changes, thereby triggering the observed precession. It is known that the magnetic anisotropy parameters (i.e., uniaxial anisotropy constant K1u and cubic anisotropy constant K1c), which dete rmine the direction of the easy axis in the GaMnAs system, are functions of temperature and hole concentration4,7,8. Thus when the GaMnAs system is excited by an optical pulse, a transient change in local hole concentration Δp and local temperature ΔT, reflecting variation of both the carrier temperature )(/t TheΔ and the lattice temperature ()lTtΔ, can lead to changes in the magnetic anisotropy parameters. Below the Curie temperature, the direction of the in -plane magnetic eas y axes (given by the angle f) depends on the interplay between K1u and -K1c. After the optical excitation, the new angle of the in -plane easy axes is given by 100 100((),()) ()((),())u cKTTtppt tKTTtpptff +Δ+Δ=+Δ+Δ , where T0 and p0 are the initial (ambient) temperature and hole conce ntration7,8. Therefore, the in -plane easy axes may quickly assume a new direction following photoexcitation if ΔT(t) and Δp(t) are sufficiently strong. This transient change in the magnetic easy axis, due to the change in the minimum of the magnetic free e nergy as function of Mn spin induced by the photoexcitation, triggers a precessional motion of the magnetization around the new effective magnetic field. 5 Within the mean field treatment of the p-d magnetic exchange interaction, the Mn spins precess aroun d the effective magnetic field Mn effH, which is determined by the sum of the anisotropy field Mn anisH and the hole -spin mean field JNholem. The dynamics of the hole magnetization m is determined by its precession around the mean field JNMnM due to the Mn spins , and by its rapid relaxation due to the strong spin-orbit interaction in the valence band with a rate ΓSO of the order of tens of femtoseconds9,10. Here, m (M) is the hole (Mn) magnetization , J is the exchange constant , and Nhole (NMn) is the number of holes (Mn -spins)2. For small fluctuations of the magnetization orientation around the easy axis, the magnetization dynamics can be described by the Landau -Lifshitz -Gilbert (LLG) equation2, which is appropriate to apply to our experimental data at low -pump intensities (e.g., 0.065 nJ/pulse). In the adiabatic limit, where the hole spins precess and relax much faster than the Mn spins, we can eliminate the hole spins by transforming to the rotating frame11. In this way we obt ain an effective LLG equation for the Mn magnetization M, whose precession is governed by the anisotropy field MnanisH and the effective Gilbe rt damping coefficient including the damping a0 due to e.g. spin -lattice interactions and the contribution due to the p-d exchange interaction, which depends on the hole concentration, the ratio of hole spin relaxation energy over exchange interaction energy, and the ratio m/M of the collective hole and Mn spins[9,10]. The LLG equation predict s an oscillatory behavior of the magnetization due to the precession of the local Mn moments around the magnetic anisotropy field MnanisH (T0+??(t), p0+?p(t)). The precession frequency is proportional to this anisotropy field, which is given by the gradient of magnetic free energy and is proportional to the anisotropy constants K1u and K1c.2 The magnitude of MnanisH decreases as the ambient temperature T0 or the transient temperature DT increases, primarily due to the decrease in K1c8. This leads to the decrease of the precession frequency as the ambient 6 temperature or the pump intensity increases, and is consistent with the be havior observed in Fig. 2. It should be pointed out that, in the Fourier transform of each temporal signal of the oscillations, only one oscillatory mode was observed (see also in Fig. 1(b) and Fig. 1(c)). This indicates that only a single uniform -precessi on magnon was excited in our experiment, and the scattering among uniform -precession magnons can be neglected when interpreting damping of the Mn spin precession. As can be seen in Fig. 2, the amplitude of the oscillations increas es as the ambient tempera ture T0 decreas es or as the pump intensity increases. This result is in accord with the fact that the relative change DT/T0 and Δp/p0, which determines the magnitude of f(t) and the photo -induced tilt in the easy axis, increase as T0 decreases or as the pu mp intensity increases. It is important to note that in our experiment the amplitude of the oscillations saturates as the pump intensity exceeds about 4 I0 (I0=0.065nJ/pulse) at T0=10 K. Thus, the observed saturation may indicate that the magnetic easy axi s is stabilized at pump intensity larger than 4 I0. We estimated that the increase of local hole concentration Δp/p0 is about 0.4%, and the local temperature increase ΔT /T0 is about 160% using the value of specific heat of 1mJ/g/K for GaAs12 for pump inte nsity ~4 I0. This results in the transient local temperature T0+ΔT close to TC/2. Because the magnetic easy axis is already along the [110] direction for T0+ΔT close to or higher than T C/28, our observed phenomenon is in agreement with the previous reported results. Finally we turn to the oscillation damping, which is intimately related to collective localized -spin lifetimes, and consequently to spintronic device development. Figure 3 shows the fitted Gilbert damping coefficient a obtained by using the LLG equation as a function of pump intensity and ambient temperature, respectively. It can be seen that in Figure 3(a) a changes weakly with the ambient temperature and has an average value ~0.135 for a fixed pump intensity of 0.065 nJ/pulse. However, Figure 3(b) shows that a increases nonlinearly as pump intensity increases. To interpret these results, we note that, as discussed above, the p-d kinetic -exchange coupling between 7 the local Mn moments and the itinerant carrier spins contributes significantly to Gilbert damping10. In particular, a increases with increasing ratio m/M. The ratio m/M is known to increase nonlinearly with increasing temperature13 and should therefore depend nonlinearly on the photoexcitation. The Gilbert damping coefficient due to the e xchange interaction also increases as hole density p and hole spin relaxation rate SO MnJMNΓ increase. Here, ΓSO and Δp/p (<0.01) are relatively small. Thus we can conclude that the damping coefficient due to the p-d exchange interaction should increase with increasing ambient temperature ( T0) or increasing pump intensity (ΔT and Δp). On the other hand, we also note that damping may arise from an extrinsic inhomogeneous MnanisH broadening attributed to a local temperatu re gradient due to inhomogeneities in the laser beam intensity profile and the detailed structure of the sample. This extrinsic damping effect is expected to decrease as the ambient temperature increases or the pump intensity decreases. Thus, the data in F ig. 3(a), which shows a only weakly dependent on ambient temperature, may result from the competition between these two mechanisms, both of which, however, predict the result in Fig. 3(b) that the damping coefficient increases nonlinearly with the increase of pump intensity. It is important to note that the LLG equation is valid only at low pump intensities. At high pump intensities, an alternative theoretical approach must be introduced[14]. So our new experimental results in the time domain are not access ible with static FMR experiments, and provide new information on the physical factors that contribute to the damping effect. In conclusion, we have studied the photoinduced magnetization dynamics in Ga0.965 Mn0.035 As by time -resolved MOKE with zero externa l magnetic fields. At temperatures below ~40 K, a precessional motion associated with correlated local Mn moments was observed. This precession is attributed to an ultrafast reorientation of the in -plane magnetic easy axis from an impulsive change in the m agnetic anisotropy due to photoexcitation. Our results indicate that the magnetic easy axis does not 8 change at temperatures above about T C/2. We find the Gilbert damping coefficient is independent of ambient temperature but depends nonlinearly on the pump intensity, We attribute this nonlinearity to the hole -Mn spin exchange interaction and the extrinsic anisotropy field broadening due to temperature gradients in the sample. Our results show that ultrafast optical excitation provides a way to control the am plitude, precession frequency and damping of the oscillations arising from coherent localized Mn spins in the GaMnAs system. This work was supported by ARO Grant W911NF -05-1-0436 (VU), NSF Grant DMR06 -03752 (ND), and by the EU STREP program HYSWITCH (Cret e). 1 H. Ohno, Science 281, 951 (1998) 2 J. Jungwirth, J.Sinova, J.Masek, J.Kucera, and A.H. MacDonald, Rev. Mod. Phys. 78, 809 (2006) 3 J. Wang, C. Sun, Y. Hashimoto, J. Kono, G.A. Kh odaparast, L. Cywinski, L.J. Sham, G.D. Sanders, C.J. Stanton and H. Munekata , J. Phys.:Condens. Matter 18, R501 (2006) 4 A. Oiwa, H. Takechi, and H. Munekata, Journal of Superconductivity 18, 9 (2005) 5 A.V. Kimel, A. Kirilyuk, F. Hansteen, R.V. Pisarev, and Th. Rasing , J.Phys.:Condens. Matter 19, 043201(2007) 6 V. A. Kotov and A. K. Zvezdin. Modern Magnetooptics and Magneto Optical Materials . (Institute of Physics, London, 1997) 7 T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B 63, 195205 (2001) 8 U.Wel p, V.K.Vlasko -Vlasov, X.Liu, J.K.Furdyna, and T.Wojtowicz, Phys. Rev. Lett. 90, 167206 (2003) 9 J. Chovan, E. G. Kavousanaki, and I. E. Perakis, Phys. Rev. Lett. 96, 057402 (2006) and unpublished. 10J.Sinova, T.Jungwirth, X.Liu, Y.Sasaki, J.K.Furdiyna, W.A .Atkinson, and A.H.MacDonald, Phys. Rev. B 69, 085209 (2004) 11 Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phys. Lett. 84, 5234 (2004) 9 12 J. S. Blakemore, J. Appl. Phys. 53, R123 (1982) 13 S. Das Sarma, E. H. Hwang, and A. Kaminski, Phys. Rev. B 67, 155201(2003) 14 H. Suhl, IEEE Trans.Magn. 34, 1834(1998) 10 0 200 400 600 800-0.4-0.20.00.20.4FFT (a.u.)Kerr rotation (mdeg) Linear σ- σ+T=20K Kerr rotation (mdeg) Delay time (ps)0 200 400 600 800-0.010.000.01(b) Delay time (ps) (a) 0 5 10 15 20 25 30024(c) Frequency (GHz) Figure 1. (a) Kerr rotation measurements for Ga 1-xMnxAs (x = 0.035) excited by linearly -polarized and circularly -polarized light ( σ+ and σ-) at a temperature of 20 K. The photon energy was 1 .56 eV. Oscillations due to magnetization precession are superimposed on the decay curves. (b) Oscillation data (open circles) extracted from (a). The solid line is the fitted result. (c) Fourier transformation profile for the oscillation data in (b). 10 20 30 40102030 Pump Intensity( I0) Temperature (K) ω (GHz)204060 A0 (µdeg) I=I0 50100150200 T0=10K 246810142128 Figure 2 Amplitude A0 and angular frequency w as a function of temperature T0 at constant pump intensity I=I0; and as a function of pump intensity (in units of I0) at T0 = 10 K. I0 = 0.065 nJ/pulse. 11 0.5 1.0 1.5 2.00.100.150.200.25Damping Coefficient α Pump Intensity (I0)(b) T0=10K10 20 30 400.090.120.150.18 Damping Coefficient α Temperature (K)(a) I=I0 Figure 3 (a) Gilbert damping coefficient a as a function of temperature (T 0) at a constant pump intensity I=I0. I0=0.065 nJ/pulse; (b) Gilbert damping coefficient a as a function of pump intensity in units of I0 at T0= 10 K.

2007-06-28

We use femtosecond optical pulses to induce, control and monitor magnetization precession in ferromagnetic Ga0.965Mn0.035As. At temperatures below ~40 K we observe coherent oscillations of the local Mn spins, triggered by an ultrafast photoinduced reorientation of the in-plane easy axis. The amplitude saturation of the oscillations above a certain pump intensity indicates that the easy axis remains unchanged above ~TC/2. We find that the observed magnetization precession damping (Gilbert damping) is strongly dependent on pump laser intensity, but largely independent on ambient temperature. We provide a physical interpretation of the observed light-induced collective Mn-spin relaxation and precession.

Coherent Magnetization Precession in GaMnAs induced by Ultrafast Optical Excitation

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